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Internet-DraftVDAFOctober 2024
Barnes, et al.Expires 20 April 2025[Page]
+
+
+
+
Workgroup:
+
CFRG
+
Internet-Draft:
+
draft-irtf-cfrg-vdaf-latest
+
Published:
+
+ +
+
Intended Status:
+
Informational
+
Expires:
+
+
Authors:
+
+
+
R. L. Barnes
+
Cisco
+
+
+
D. Cook
+
ISRG
+
+
+
C. Patton
+
Cloudflare
+
+
+
P. Schoppmann
+
Google
+
+
+
+
+

Verifiable Distributed Aggregation Functions

+
+

Abstract

+

This document describes Verifiable Distributed Aggregation Functions (VDAFs), a +family of multi-party protocols for computing aggregate statistics over user +measurements. These protocols are designed to ensure that, as long as at least +one aggregation server executes the protocol honestly, individual measurements +are never seen by any server in the clear. At the same time, VDAFs allow the +servers to detect if a malicious or misconfigured client submitted an invalid +measurement. Two concrete VDAFs are specified, one for general-purpose +aggregation (Prio3) and another for heavy hitters (Poplar1).

+
+
+

+Discussion Venues +

+

This note is to be removed before publishing as an RFC.

+

Discussion of this document takes place on the + Crypto Forum Research Group mailing list (cfrg@ietf.org), + which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg.

+

Source for this draft and an issue tracker can be found at + https://github.com/cfrg/draft-irtf-cfrg-vdaf.

+
+
+
+

+Status of This Memo +

+

+ This Internet-Draft is submitted in full conformance with the + provisions of BCP 78 and BCP 79.

+

+ Internet-Drafts are working documents of the Internet Engineering Task + Force (IETF). Note that other groups may also distribute working + documents as Internet-Drafts. The list of current Internet-Drafts is + at https://datatracker.ietf.org/drafts/current/.

+

+ Internet-Drafts are draft documents valid for a maximum of six months + and may be updated, replaced, or obsoleted by other documents at any + time. It is inappropriate to use Internet-Drafts as reference + material or to cite them other than as "work in progress."

+

+ This Internet-Draft will expire on 20 April 2025.

+
+
+ +
+
+ ▲

+Table of Contents +

+ +
+
+
+
+

+1. Introduction +

+

(RFC EDITOR: remove this paragraph.) The source for this draft and the +reference implementation can be found at +https://github.com/cfrg/draft-irtf-cfrg-vdaf.

+

The ubiquity of the Internet makes it an ideal platform for measurement of +large-scale phenomena, whether public health trends or the behavior of computer +systems at scale. There is substantial overlap, however, between information +that is valuable to measure and information that users consider private.

+

For example, consider an application that provides health information to users. +The operator of an application might want to know which parts of their +application are used most often, as a way to guide future development of the +application. Specific users' patterns of usage, though, could reveal sensitive +things about them, such as which users are researching a given health +condition.

+

In many situations, the measurement collector is only interested in aggregate +statistics, e.g., which portions of an application are most used or what +fraction of people have experienced a given disease. Thus systems that provide +aggregate statistics while protecting individual measurements can deliver the +value of the measurements while protecting users' privacy.

+

This problem is often formulated in terms of differential privacy (DP) +[Dwo06]. Roughly speaking, a data aggregation system that is differentially +private ensures that the degree to which any individual measurement influences +the value of the aggregate result can be precisely controlled. For example, in +systems like RAPPOR [EPK14], each user samples noise from a well-known +distribution and adds it to their measurement before submitting to the +aggregation server. The aggregation server then adds up the noisy measurements, +and because it knows the distribution from which the noise was sampled, it can +estimate the true sum with reasonable accuracy.

+

Even when noise is added to the measurements, collecting them in the clear +still reveals a significant amount of information to the collector. On the one +hand, depending on the "amount" of noise a client adds to its measurement, it +may be possible for a curious collector to make a reasonable guess of the +measurement's true value. On the other hand, the more noise the clients add, +the less reliable will be the server's estimate of the aggregate. Thus systems +relying solely on a DP mechanism must strike a delicate balance between privacy +and utility.

+

Another way of constructing a privacy-preserving measurement system is to use +multi-party computation (MPC). The goal of such a system is that no participant +in the protocol should learn anything about an individual measurement beyond +what it can deduce from the aggregate. MPC achieves this goal by distributing +the computation of the aggregate across multiple aggregation servers, one of +which is presumed to be honest, i.e., not under control of the attacker. +Moreover, MPC can be composed with various DP mechanisms to ensure the +aggregate itself does not leak too much information about any one of the +measurements [MPRV09].

+

This document describes two classes of MPC protocols, each aiming for a +different set of goals.

+

In a Distributed Aggregation Function (DAF, Section 4), each client splits its +measurement into multiple secret shares, one for each aggregation +server. DAFs require two properties of the secret sharing scheme. First, we can +reconstruct the underlying measurement by simply adding up all of the shares. +(Typically the shares are vectors over some finite field.) Second, given all +but one of the shares, it is impossible to learn anything about the underlying +measurement. These properties give rise to a simple strategy for privately +aggregating the measurements: each aggregation server adds up its measurement +shares locally before revealing their sum to the data collector; then all +the data collector has to do is add up these sums to get the aggregate result.

+

This strategy is compatible with any aggregation function that can be +represented as the sum of some encoding of the measurements. Examples include: +summary statistics such as sum, mean, and standard deviation; estimation of +quantiles, e.g., median; histograms; linear regression; or counting data +structures, e.g., Bloom filters. However, not all functions fit into this +rubric, as it is constrained to linear computations over the encoded +measurements.

+

In fact, our framework admits DAFs with slightly more +functionality, computing aggregation functions of the form

+
+
+F(agg_param, meas_1, ..., meas_M) =
+    G(agg_param, meas_1) + ... + G(agg_param, meas_M)
+
+
+

where meas_1, ..., meas_M are the measurements, G is a possibly +non-linear function, and agg_param is a parameter of that function chosen by +the data collector. This paradigm, known as function secret sharing [BGI15], +allows for more sophisticated data analysis tasks, such as grouping metrics by +private client attributes [MPDST25] or computing heavy hitters [BBCGGI21]. +(More on the latter task below.)

+

The second class of protocols defined in this document are called Verifiable +Distributed Aggregation Functions (VDAFs, Section 5). In addition to being +private, VDAFs are verifiable in the following sense. By design, a secret +sharing of a valid measurement, e.g., a number between 1 and 10, is +indistinguishable from a secret sharing of an invalid measurement, e.g., a +number larger than 10. This means that DAFs are vulnerable to attacks from +malicious clients attempting to disrupt the computation by submitting invalid +measurements. VDAFs are designed to allow the servers to detect and remove +these measurements prior to aggregation. We refer to this property as +robustness.

+

Achieving robustness without sacrificing privacy requires the servers to +interact with one another over a number of rounds of communication. DAFs on the +other hand are non-interactive, making them easier to deploy; but they do not +provide robustness on their own. This may be tolerable in some applications. +For instance, if the client's software is executed in a trusted execution +environment, it may be reasonable to assume that no client is malicious.

+

The DAF and VDAF abstractions encompass a variety of MPC techniques in the +literature. These protocols vary in their operational and security +requirements, sometimes in subtle but consequential ways. This document +therefore has two important goals:

+
    +
  1. +

    Provide higher-level protocols, like [DAP], with +a simple, uniform interface for accessing privacy-preserving measurement +schemes, document relevant operational and security requirements, and +specify constraints for safe usage:

    +
      +
    1. +

      General patterns of communications among the various actors involved in +the system (clients, aggregation servers, and the collector of the +aggregate result);

      +
      2. +
    2. +

      Capabilities of a malicious coalition of parties attempting to divulge +information about client measurements; and

      +
      3. +
    3. +

      Conditions that are necessary to ensure that malicious clients cannot +corrupt the computation.

      +
      4. +
    +
    2. +
  2. +

    Provide cryptographers with design criteria that provide a clear deployment +roadmap for new constructions of privacy-preserving measurement systems.

    +
    3. +
+

This document also specifies two concrete VDAF schemes, each based on a protocol +from the literature.

+ +

The remainder of this document is organized as follows: Section 2 lists +definitions and conventions used in the remainder of the document; Section 3 +gives a brief overview of DAFs and VDAFs, the parties involved in the +computation, and the requirements for non-collusion; Section 4 defines the syntax +for DAFs; Section 5 defines the syntax for VDAFs; Section 6 defines various +functionalities that are common to our constructions; Section 7 specifies +Prio3; Section 8 specifies Poplar1; and Section 9 enumerates security +considerations for DAFs and VDAFs in general and our constructions in +particular.

+
+
+

+1.1. Change Log +

+

(RFC EDITOR: remove this section.)

+

(*) Indicates a change that breaks wire compatibility with the previous draft.

+

12:

+
    +
  • +

    (V)DAF: Add an application context string parameter to sharding and +preparation. The motivation for this change is to harden Prio3 against +offline attacks. More generally, however, it allows designing schemes for +which correct execution requires agreement on the application context. +Accordingly, both Prio3 and Poplar1 have been modified to include the context +in the domain separation tag of each XOF invocation. (*)

    +
    • +
  • +

    Prio3: Improve soundness of the base proof system and the circuits of some +variants. Generally speaking, wherever we evaluate a univariate polynomial at +a random point, we can instead evaluate a multivariate polynomial of lower +degree. (*)

    +
    • +
  • +

    Prio3: Replace the helper's measurement and proof share seeds with a single +seed. (*)

    +
    • +
  • +

    Prio3Sum: Update the circuit to support a more general range check and avoid +using joint randomness. (*)

    +
    • +
  • +

    Prio3Histogram, Prio3MultihotCountVec: Move the final reduction of the +intermediate outputs out of the circuit. (*)

    +
    • +
  • +

    IDPF: Add the application context string to key generation and evaluation and +bind it to the fixed AES key. (*)

    +
    • +
  • +

    IDPF: Use XofTurboShake128 for deriving the leaf nodes in order to ensure the +construction is extractable. (*)

    +
    • +
  • +

    IDPF: Simplify the public share encoding. (*)

    +
    • +
  • +

    XofTurboShake128: Change SEED_SIZE from 16 bytes to 32 to mitigate offline +attacks on Prio3 robustness. In addition, allow seeds of different lengths so +that we can continue to use XofTurboShake128 with IDPF. (*)

    +
    • +
  • +

    XofTurboShake128, XofFixedKeyAes128: Increase the length prefix for the +domain separation tag from one by to two bytes. This is to accommodate the +application context. (*)

    +
    • +
  • +

    Reassign codepoints for all Prio3 variants and Poplar1. (*)

    +
    • +
  • +

    Security considerations: Add a section on defense-in-depth measures taken by +Prio3 and Poplar1 and more discussion about choosing FLP parameters.

    +
    • +
+

11:

+
    +
  • +

    Define message formats for the Poplar1 aggregation parameter and IDPF public +share.

    +
    • +
  • +

    IDPF: Require the IDPF binder must be a random nonce.

    +
    • +
  • +

    VDAF: Replace the pseudocode description of the ping-ping topology with +Python and sketch the star topology.

    +
    • +
  • +

    DAF: Align aggregation parameter validation with VDAF.

    +
    • +
  • +

    Replace Union[A, B] type with A | B.

    +
    • +
  • +

    Rename FFT ("Fast Fourier Transform") with NTT ("Number Theoretic +Transform").

    +
    • +
+

10:

+
    +
  • +

    Define Prio3MultihotCountVec, a variant of Prio3 for aggregating bit vectors +with bounded weight.

    +
    • +
  • +

    FLP: Allow the output of the circuit to be a vector. This makes it possible +to skip joint randomness derivation in more cases.

    +
    • +
  • +

    Poplar1: On the first round of preparation, handle None as an error. +Previously this message was interpreted as a length-3 vector of zeros.

    +
    • +
  • +

    Prio3: Move specification of the field from the FLP validity circuit to the +VDAF itself.

    +
    • +
  • +

    Clarify the extent to which the attacker controls the network in our threat +models for privacy and robustness.

    +
    • +
  • +

    Clean up various aspects of the code, including: Follow existing +object-oriented programming patterns for Python more closely; make the type +hints enforceable; and avoid shadowing variables.

    +
    • +
  • +

    Poplar1: Align terminology with [BBCGGI23].

    +
    • +
  • +

    IDPF: Add guidance for encoding byte strings as indices.

    +
    • +
+

09:

+
    +
  • +

    Poplar1: Make prefix tree traversal stricter by requiring each node to be a +child of a node that was already visited. This change is intended to make it +harder for a malicious Aggregator to steer traversal towards +non-heavy-hitting measurements.

    +
    • +
  • +

    Prio3: Add more explicit guidance for choosing the field size.

    +
    • +
  • +

    IDPF: Define extractability and clarify (un)safe usage of intermediate prefix +counts. Accordingly, add text ensuring public share consistency to security +considerations.

    +
    • +
+

08:

+
    +
  • +

    Poplar1: Bind the report nonce to the authenticator vector programmed into +the IDPF. (*)

    +
    • +
  • +

    IdpfPoplar: Modify extend() by stealing each control bit from its +corresponding seed. This improves performance by reducing the number of AES +calls per level from 3 to 2. The cost is a slight reduction in the concrete +privacy bound. (*)

    +
    • +
  • +

    Prio3: Add support for generating and verifying mutliple proofs per +measurement. This enables a trade-off between communication cost and runtime: +if more proofs are used, then a smaller field can be used without impacting +robustness. (*)

    +
    • +
  • +

    Replace SHAKE128 with TurboSHAKE128. (*)

    +
    • +
+

07:

+
    +
  • +

    Rename PRG to XOF ("eXtendable Output Function"). Accordingly, rename PrgSha3 +to XofShake128 and PrgFixedKeyAes128 to XofFixedKeyAes128. "PRG" is a misnomer +since we don't actually treat this object as a pseudorandom generator in +existing security analysis.

    +
    • +
  • +

    Replace cSHAKE128 with SHAKE128, re-implementing domain separation for the +customization string using a simpler scheme. This change addresses the +reality that implementations of cSHAKE128 are less common. (*)

    +
    • +
  • +

    Define a new VDAF, called Prio3SumVec, that generalizes Prio3Sum to a vector +of summands.

    +
    • +
  • +

    Prio3Histogram: Update the codepoint and use the parallel sum optimization +introduced by Prio3SumVec to reduce the proof size. (*)

    +
    • +
  • +

    Daf, Vdaf: Rename interface methods to match verbiage in the draft.

    +
    • +
  • +

    Daf: Align with Vdaf by adding a nonce to shard() and prep().

    +
    • +
  • +

    Vdaf: Have prep_init() compute the first prep share. This change is +intended to simplify the interface by making the input to prep_next() not +optional.

    +
    • +
  • +

    Prio3: Split sharding into two auxiliary functions, one for sharding with +joint randomness and another without. This change is intended to improve +readability.

    +
    • +
  • +

    Fix bugs in the ping-pong interface discovered after implementing it.

    +
    • +
+

06:

+
    +
  • +

    Vdaf: Define a wrapper interface for preparation that is suitable for the +"ping-pong" topology in which two Aggregators exchange messages over a +request/response protocol, like HTTP, and take turns executing the +computation until input from the peer is required.

    +
    • +
  • +

    Prio3Histogram: Generalize the measurement type so that the histogram can be +used more easily with discrete domains. (*)

    +
    • +
  • +

    Daf, Vdaf: Change the aggregation parameter validation algorithm to take the +set of previous parameters rather than a list. (The order of the parameters +is irrelevant.)

    +
    • +
  • +

    Daf, Vdaf, Idpf: Add parameter RAND_SIZE that specifies the number of +random bytes consumed by the randomized algorithm (shard() for Daf and Vdaf +and gen() for Idpf).

    +
    • +
+

05:

+
    +
  • +

    IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixed-key mode for +AES-128 based on a construction from [GKWWY20]. This change is intended to +improve performance of IDPF evaluation. Note that the new PRG is not suitable +for all applications. (*)

    +
    • +
  • +

    Idpf: Add a binder string to the key-generation and evaluation algorithms. +This is used to plumb the nonce generated by the Client to the PRG.

    +
    • +
  • +

    Plumb random coins through the interface of randomized algorithms. +Specifically, add a random input to (V)DAF sharding algorithm and IDPF +key-generation algorithm and require implementations to specify the length of +the random input. Accordingly, update Prio3, Poplar1, and IdpfPoplar to match +the new interface. This change is intended to improve coverage of test +vectors.

    +
    • +
  • +

    Use little-endian byte-order for field element encoding. (*)

    +
    • +
  • +

    Poplar1: Move the last step of sketch evaluation from prep_next() to +prep_shares_to_prep().

    +
    • +
+

04:

+
    +
  • +

    Align security considerations with the security analysis of [DPRS23].

    +
    • +
  • +

    Vdaf: Pass the nonce to the sharding algorithm.

    +
    • +
  • +

    Vdaf: Rather than allow the application to choose the nonce length, have each +implementation of the Vdaf interface specify the expected nonce length. (*)

    +
    • +
  • +

    Prg: Split "info string" into two components: the "customization string", +intended for domain separation; and the "binder string", used to bind the +output to ephemeral values, like the nonce, associated with execution of a +(V)DAF.

    +
    • +
  • +

    Replace PrgAes128 with PrgSha3, an implementation of the Prg interface based +on SHA-3, and use the new scheme as the default. Accordingly, replace +Prio3Aes128Count with Prio3Count, Poplar1Aes128 with Poplar1, and so on. SHA-3 +is a safer choice for instantiating a random oracle, which is used in the +analysis of Prio3 of [DPRS23]. (*)

    +
    • +
  • +

    Prio3, Poplar1: Ensure each invocation of the Prg uses a distinct +customization string, as suggested by [DPRS23]. This is intended to make +domain separation clearer, thereby simplifying security analysis. (*)

    +
    • +
  • +

    Prio3: Replace "joint randomness hints" sent in each input share with "joint +randomness parts" sent in the public share. This reduces communication +overhead when the number of shares exceeds two. (*)

    +
    • +
  • +

    Prio3: Bind nonce to joint randomness parts. This is intended to address +birthday attacks on robustness pointed out by [DPRS23]. (*)

    +
    • +
  • +

    Poplar1: Use different Prg invocations for producing the correlated randomness +for inner and leaf nodes of the IDPF tree. This is intended to simplify +implementations. (*)

    +
    • +
  • +

    Poplar1: Don't bind the candidate prefixes to the verifier randomness. This is +intended to improve performance, while not impacting security. According to +the analysis of [DPRS23], it is necessary to restrict Poplar1 usage such +that no report is aggregated more than once at a given level of the IDPF tree; +otherwise, attacks on privacy may be possible. In light of this restriction, +there is no added benefit of binding to the prefixes themselves. (*)

    +
    • +
  • +

    Poplar1: During preparation, assert that all candidate prefixes are unique +and appear in order. Uniqueness is required to avoid erroneously rejecting a +valid report; the ordering constraint ensures the uniqueness check can be +performed efficiently. (*)

    +
    • +
  • +

    Poplar1: Increase the maximum candidate prefix count in the encoding of the +aggregation parameter. (*)

    +
    • +
  • +

    Poplar1: Bind the nonce to the correlated randomness derivation. This is +intended to provide defense-in-depth by ensuring the Aggregators reject the +report if the nonce does not match what the Client used for sharding. (*)

    +
    • +
  • +

    Poplar1: Clarify that the aggregation parameter encoding is OPTIONAL. +Accordingly, update implementation considerations around cross-aggregation +state.

    +
    • +
  • +

    IdpfPoplar: Add implementation considerations around branching on the values +of control bits.

    +
    • +
  • +

    IdpfPoplar: When decoding the control bits in the public share, assert +that the trailing bits of the final byte are all zero. (*)

    +
    • +
+

03:

+
    +
  • +

    Define codepoints for (V)DAFs and use them for domain separation in Prio3 and +Poplar1. (*)

    +
    • +
  • +

    Prio3: Align joint randomness computation with revised paper [BBCGGI19]. +This change mitigates an attack on robustness. (*)

    +
    • +
  • +

    Prio3: Remove an intermediate PRG evaluation from query randomness generation. +(*)

    +
    • +
  • +

    Add additional guidance for choosing FFT-friendly fields.

    +
    • +
+

02:

+
    +
  • +

    Complete the initial specification of Poplar1.

    +
    • +
  • +

    Extend (V)DAF syntax to include a "public share" output by the Client and +distributed to all of the Aggregators. This is to accommodate "extractable" +IDPFs as required for Poplar1. (See [BBCGGI21], Section 4.3 for details.)

    +
    • +
  • +

    Extend (V)DAF syntax to allow the unsharding step to take into account the +number of measurements aggregated.

    +
    • +
  • +

    Extend FLP syntax by adding a method for decoding the aggregate result from a +vector of field elements. The new method takes into account the number of +measurements.

    +
    • +
  • +

    Prio3: Align aggregate result computation with updated FLP syntax.

    +
    • +
  • +

    Prg: Add a method for statefully generating a vector of field elements.

    +
    • +
  • +

    Field: Require that field elements are fully reduced before decoding. (*)

    +
    • +
  • +

    Define new field Field255.

    +
    • +
+

01:

+
    +
  • +

    Require that VDAFs specify serialization of aggregate shares.

    +
    • +
  • +

    Define Distributed Aggregation Functions (DAFs).

    +
    • +
  • +

    Prio3: Move proof verifier check from prep_next() to +prep_shares_to_prep(). (*)

    +
    • +
  • +

    Remove public parameter and replace verification parameter with a +"verification key" and "Aggregator ID".

    +
    • +
+
+
+
+
+
+
+

+2. Conventions +

+

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", +"MAY", and "OPTIONAL" in this document are to be interpreted as +described in BCP 14 [RFC2119] [RFC8174] when, and only when, they +appear in all capitals, as shown here.

+

Algorithms in this document are written in Python (compatible with Python 3.12 +or later). A fatal error in a program (e.g., failure to parse one of the +function parameters) is usually handled by raising an exception.

+

In Python, array indexing starts with 0, e.g., x[0] is the first element +and x[len(x)-1] is the last of x. We can also index from the end of the +list, e.g., x[-1] is the last element of x.

+

Exponentiation is denoted by x ** y in Python. We also sometimes use more +conventional notation for exponentiation, namely x^y.

+

Division of x by y is denoted by x / y. Python includes a similar +operator, x // y, which is short for floor(x / y).

+

Type hints are used to define input and output types:

+ +

This document defines several byte-string constants. When comprised of printable +ASCII characters, they are written as Python 3 byte-string literals (e.g., +b'some constant string').

+

A global constant VERSION of type int is defined, which algorithms are free +to use as desired. Its value SHALL be 12.

+

This document describes algorithms for multi-party computations in which the +parties typically communicate over a network. Wherever a quantity is defined +that must be transmitted from one party to another, this document prescribes +a particular encoding of that quantity as a byte string.

+

Some common functionalities:

+ +
+
+
+
+

+3. Overview +

+
+
+
+
+                 +--------------+
+           +---->| Aggregator 0 |----+
+     input |     +--------------+    | aggregate
+    shares |             ^           | shares
+           |             |           |
+           |             V           |
+           |     +--------------+    |
+           | +-->| Aggregator 1 |--+ |
+           | |   +--------------+  | |
++--------+-+ |           ^         | +->+-----------+
+| Client |---+           |         +--->| Collector |--> aggregate
++--------+-+                         +->+-----------+
+           |            ...          |
+           |                         |
+           |             |           |
+           |             V           |
+           |    +----------------+   |
+           +--->| Aggregator N-1 |---+
+                +----------------+
+
+
+
Figure 1: +Overall data flow of a (V)DAF. +
+
+

In a DAF- or VDAF-based private measurement system, we distinguish between +three types of actors: Clients, Aggregators, and the Collector. The overall +flow of the measurement process is as follows:

+ +

Aggregators are a new class of actor relative to traditional measurement systems +where Clients submit measurements to a single server. They are critical for +both the privacy properties of the system and, in the case of VDAFs, the +validity of the measurements obtained. The privacy properties of the system +are assured by non-collusion among Aggregators, and Aggregators are the +entities that perform validation of Client measurements. Thus Clients trust +Aggregators not to collude (typically it is required that at least one +Aggregator is honest; see Section 9.8), and Collectors trust +Aggregators to correctly run the protocol.

+

Within the bounds of the non-collusion requirements of a given (V)DAF instance, +it is possible for the same entity to play more than one role. For example, the +Collector could also act as an Aggregator, effectively using the other +Aggregator(s) to augment a basic client-server protocol.

+

In this document, we describe the computations performed by the actors in this +system. It is up to the higher-level protocol making use of the (V)DAF to +arrange for the required information to be delivered to the proper actors in the +proper sequence. In general, we assume that all communications are confidential +and mutually authenticated, with the exception that Clients submitting +measurements may be anonymous.

+
+
+
+
+

+4. Definition of DAFs +

+

By way of a gentle introduction to VDAFs, this section describes a simpler class +of schemes called Distributed Aggregation Functions (DAFs). Unlike VDAFs, DAFs +do not provide verifiability of the computation. Clients must therefore be +trusted to compute their input shares correctly. Because of this fact, the use +of a DAF is NOT RECOMMENDED for most applications. See Section 9 for +additional discussion.

+

A DAF scheme is used to compute a particular "aggregation function" over a set +of measurements generated by Clients. Depending on the aggregation function, the +Collector might select an "aggregation parameter" and disseminate it to the +Aggregators. The semantics of this parameter is specific to the aggregation +function, but in general it is used to represent the set of "queries" that can +be made by the Collector on the batch of measurements. For example, the +aggregation parameter is used to represent the candidate prefixes in the +Poplar1 VDAF Section 8.

+

Execution of a DAF has four distinct stages:

+ +

Sharding and preparation are done once per measurement. Aggregation and +unsharding are done over a batch of measurements (more precisely, over the +recovered output shares).

+

A concrete DAF specifies the algorithm for the computation needed in each of +these stages. The interface, denoted Daf, is defined in the remainder of this +section. In addition, a concrete DAF defines the associated constants and types +enumerated in the following table.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 1: +Constants and types defined by each concrete DAF. +
ParameterDescription
+ ID: int +Algorithm identifier for this DAF, in [0, 2^32).
+ SHARES: int +Number of input shares into which each measurement is sharded.
+ NONCE_SIZE: int +Size of the nonce associated with the report.
+ RAND_SIZE: int +Size of the random byte string consumed by the sharding algorithm.
+ Measurement +Type of each measurement.
+ PublicShare +Type of each public share.
+ InputShare +Type of each input share.
+ AggParam +Type of the aggregation parameter.
+ OutShare +Type of each output share.
+ AggShare +Type of the aggregate share.
+ AggResult +Type of the aggregate result.
+
+

The types in this table define the inputs and outputs of DAF methods at various +stages of the computation. Some of these values need to be written to the +network in order to carry forward the computation. In particular, it is +RECOMMENDED that concrete instantiations of the Daf interface specify a +standard encoding the PublicShare, InputShare, AggParam, and AggShare.

+
+
+

+4.1. Sharding +

+
+
+
+
+    Client
+    ======
+
+    measurement
+      |
+      V
+    +----------------------------------------------+
+    | shard                                        |
+    +----------------------------------------------+
+      |              |              |     |
+      |              |         ...  |    public_share
+      |              |              |     |
+      |    +---------|-----+--------|-----+
+      |    |         |     |        |     |
+      V    |         V     |        V     |
+     input_share_0  input_share_1  input_share_[SHARES-1]
+      |    |         |     |   ...  |     |
+      V    V         V     V        V     V
+    Aggregator 0   Aggregator 1    Aggregator SHARES-1
+
+
+
Figure 2: +Illustration of the sharding algorithm. +
+
+

The sharding algorithm run by each Client is defined as follows:

+
    +
  • +

    daf.shard(ctx: bytes, measurement: Measurement, nonce: bytes, rand: bytes) +-> tuple[PublicShare, list[InputShare]] consumes the "application context" +(defined below), a measurement, and a nonce and produces the public share, +distributed to each of the Aggregators, and the input shares, one for each +Aggregator.

    +

    +Pre-conditions:

    +
      +
    • +

      nonce MUST have length equal to daf.NONCE_SIZE and MUST be generated +using a cryptographically secure random number generator (CSPRNG).

      +
      • +
    • +

      rand consists of the random bytes consumed by the algorithm. It MUST have +length equal to daf.RAND_SIZE and MUST be generated using a CSPRNG.

      +
      • +
    +

    +Post-conditions:

    +
      +
    • +

      The number of input shares MUST equal daf.SHARES.

      +
      • +
    +
    • +
+

Sharding is bound to a specific "application context". The application context +is a string intended to uniquely identify an instance of the higher level +protocol that uses the DAF. The goal of binding the application to DAF +execution is to ensure that aggregation succeeds only if the Clients and +Aggregators agree on the application context. (Preparation binds the +application context, too; see Section 4.2.) Note that, unlike VDAFs +(Section 5), there is no explicit signal of disagreement; it may only manifest +as a garbled aggregate result.

+

The nonce is a public random value associated with the report. It is referred +to as a nonce because normally it will also be used as a unique identifier for +that report in the context of some application. The randomness requirement is +especially important for VDAFs, where it may be used by the Aggregators to +derive per-report randomness for verification of the computation. See +Section 9.2 for details.

+
+
+
+
+

+4.2. Preparation +

+
+
+
+
+    Aggregator 0   Aggregator 1           Aggregator SHARES-1
+    ============   ============           ===================
+
+    input_share_0  input_share_1          input_share_[SHARES-1]
+      |              |                 ...  |
+      V              V                      V
+    +-----------+  +-----------+          +-----------+
+    | prep      |  | prep      |          | prep      |
+    +-----------+  +-----------+          +-----------+
+      |              |                 ...  |
+      V              V                      V
+    out_share_0    out_share_1         out_share_[SHARES-1]
+
+
+
Figure 3: +Illustration of preparation. +
+
+

Once an Aggregator has received the public share and its input share, the next +step is to prepare the input share for aggregation. This is accomplished using +the preparation algorithm:

+
    +
  • +

    daf.prep(ctx: bytes, agg_id: int, agg_param: AggParam, nonce: bytes, +public_share: PublicShare, input_share: InputShare) -> OutShare consumes the +public share and one of the input shares generated by the Client, the +application context, the Aggregator's unique identifier, the aggregation +parameter selected by the Collector, and the report nonce and returns an +output share.

    +

    +Pre-conditions:

    +
      +
    • +

      agg_id MUST be in the range [0, daf.SHARES) and match the index of +input_share in the sequence of input shares produced by the Client.

      +
      • +
    • +

      nonce MUST have length daf.NONCE_SIZE.

      +
      • +
    +
    • +
+

The Aggregators MUST agree on the value of the aggregation parameter. +Otherwise, the aggregate result may be computed incorrectly by the Collector.

+
+
+
+
+

+4.3. Validity of Aggregation Parameters +

+

In general, it is permissible to aggregate a batch of reports multiple times. +However, to prevent privacy violations, DAFs may impose certain restrictions on +the aggregation parameters selected by the Collector. Restrictions are +expressed by the aggregation-parameter validity function:

+
    +
  • +

    daf.is_valid(agg_param: AggParam, previous_agg_params: list[AggParam]) -> +bool returns True if agg_param is allowed given the sequence +previous_agg_params of previously accepted aggregation parameters.

    +
    • +
+

Prior to accepting an aggregation parameter from the Collector and beginning +preparation, each Aggregator MUST validate it using this function.

+
+
+
+
+

+4.4. Aggregation +

+
+
+
+
+    Aggregator j
+    ============
+
+    +------------+    +------------+
+    | agg_init   |--->| agg_update |<--- out_share_0
+    +------------+    +------------+
+                         |
+                         V
+                      +------------+
+                      | agg_update |<--- out_share_1,
+                      +------------+
+                        |
+                        V
+                       ...
+                        |
+                        V
+                      +------------+
+                      | agg_update |<--- out_share_M
+                      +------------+
+                        |
+                        V
+                      agg_share_j
+
+
+
Figure 4: +Illustration of aggregation. The number of measurements in the batch is denoted by M. +
+
+

Once an Aggregator holds an output share, it adds it into its aggregate share +for the batch. This streaming aggregation process is implemented by the +following pair of algorithms:

+
    +
  • +

    daf.agg_init(agg_param: AggParam) -> AggShare returns an empty aggregate +share. It is called to initialize aggregation of a batch of measurements.

    +
    • +
  • +

    daf.agg_update(agg_param: AggParam, agg_share: AggShare, out_share: +OutShare) -> AggShare accumulates an output share into an aggregate share +and returns the updated aggregate share.

    +
    • +
+

In many situations it is desirable to split an aggregate share across multiple +storage elements, then merge the aggregate shares together just before +releasing the completed aggregate share to the Collector. DAFs facilitate this +with the following method:

+
    +
  • +

    daf.merge(agg_param: AggParam, agg_shares: list[AggShare]) -> AggShare +merges a sequence of aggregate shares into a single aggregate share.

    +
    • +
+
+
+

+4.4.1. Aggregation Order +

+

For most DAFs and VDAFs, the outcome of aggregation is not sensitive to the +order in which output shares are aggregated. This means that aggregate shares +can be updated or merged with other aggregate shares in any order. For +instance, for both Prio3 (Section 7) and Poplar1 (Section 8), the aggregate +shares and output shares both have the same type, a vector over some finite +field (Section 6.1); and aggregation involves simply adding vectors together.

+

In theory, however, there may be a DAF or VDAF for which correct execution +requires each Aggregator to aggregate output shares in the same order.

+
+
+
+
+
+
+

+4.5. Unsharding +

+
+
+
+
+    Aggregator 0    Aggregator 1        Aggregator SHARES-1
+    ============    ============        ===================
+
+    agg_share_0     agg_share_1         agg_share_[SHARES-1]
+      |               |                   |
+      V               V                   V
+    +-----------------------------------------------+
+    | unshard                                       |
+    +-----------------------------------------------+
+      |
+      V
+    agg_result
+
+    Collector
+    =========
+
+
+
Figure 5: +Illustration of unsharding. +
+
+

After the Aggregators have aggregated all measurements in the batch, each sends +its aggregate share to the Collector, who runs the unsharding algorithm to +recover the aggregate result:

+
    +
  • +

    daf.unshard(agg_param: AggParam, agg_shares: list[AggShare], +num_measurements: int) -> AggResult consumes the aggregate shares +and produces the aggregate result.

    +

    +Pre-conditions:

    +
      +
    • +

      The length of agg_shares MUST be SHARES.

      +
      • +
    • +

      num_measurements MUST equal the number of measurements in the batch.

      +
      • +
    +
    • +
+
+
+
+
+

+4.6. Execution of a DAF +

+

Secure execution of a DAF involves simulating the following procedure over an +insecure network.

+
+
+def run_daf(
+        daf: Daf[
+            Measurement,
+            AggParam,
+            PublicShare,
+            InputShare,
+            OutShare,
+            AggShare,
+            AggResult,
+        ],
+        ctx: bytes,
+        agg_param: AggParam,
+        measurements: list[Measurement]) -> AggResult:
+    agg_shares: list[AggShare]
+    agg_shares = [daf.agg_init(agg_param)
+                  for _ in range(daf.SHARES)]
+    for measurement in measurements:
+        # Sharding
+        nonce = gen_rand(daf.NONCE_SIZE)
+        rand = gen_rand(daf.RAND_SIZE)
+        (public_share, input_shares) = \
+            daf.shard(ctx, measurement, nonce, rand)
+
+        # Preparation, aggregation
+        for j in range(daf.SHARES):
+            out_share = daf.prep(ctx, j, agg_param, nonce,
+                                 public_share, input_shares[j])
+            agg_shares[j] = daf.agg_update(agg_param,
+                                           agg_shares[j],
+                                           out_share)
+
+    # Unsharding
+    num_measurements = len(measurements)
+    agg_result = daf.unshard(agg_param, agg_shares,
+                             num_measurements)
+    return agg_result
+
+
+

The inputs to this procedure include the parameters of the aggregation function +computed by the DAF: an aggregation parameter and a sequence of measurements. +It also includes the application context. The procedure prescribes how a DAF is +executed in a "benign" environment in which there is no adversary and the +messages are passed among the protocol participants over secure point-to-point +channels. In reality, these channels need to be instantiated by some "wrapper +protocol", such as [DAP], that realizes these channels using suitable +cryptographic mechanisms. Moreover, some fraction of the Aggregators (or +Clients) may be malicious and diverge from their prescribed behaviors. +Section 9 describes the execution of the DAF in various adversarial +environments and what properties the wrapper protocol needs to provide in each.

+
+
+
+
+
+
+

+5. Definition of VDAFs +

+

VDAFs are identical to DAFs except that preparation is an interactive process +carried out by the Aggregators. If successful, this process results in each +Aggregator computing an output share. The process will fail if, for example, +the underlying measurement is invalid.

+

Failure manifests as an exception raised by one of the algorithms defined in +this section. If an exception is raised during preparation, the Aggregators +MUST remove the report from the batch and not attempt to aggregate it. +Otherwise, a malicious Client can cause the Collector to compute a malformed +aggregate result.

+

The remainder of this section defines the VDAF interface, which we denote by +Vdaf in the remainder. The attributes are listed in Table 2 are +defined by each concrete VDAF.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 2: +Constants and types defined by each concrete VDAF. +
ParameterDescription
+ ID: int +Algorithm identifier for this VDAF, in [0, 2^32).
+ SHARES: int +Number of input shares into which each measurement is sharded.
+ ROUNDS: int +Number of rounds of communication during preparation.
+ NONCE_SIZE: int +Size of the report nonce.
+ RAND_SIZE: int +Size of the random byte string consumed during sharding.
+ VERIFY_KEY_SIZE: int +Size of the verification key used during preparation.
+ Measurement +Type of each measurement.
+ PublicShare +Type of each public share.
+ InputShare +Type of each input share.
+ AggParam +Type of the aggregation parameter.
+ OutShare +Type of each output share.
+ AggShare +Type of the aggregate share.
+ AggResult +Type of the aggregate result.
+ PrepState +Type of the prep state.
+ PrepShare +Type of each prep share.
+ PrepMessage +Type of each prep message.
+
+

Some of type types in the table above need to be written to the network in +order to carry out the computation. It is RECOMMENDED that concrete +instantiations of the Vdaf interface specify a method of encoding the +PublicShare, InputShare, AggParam, AggShare, PrepShare, and +PrepMessage.

+

Each VDAF is identified by a unique, 32-bit integer, denoted ID. Identifiers +for each VDAF specified in this document are defined in Table 18. The +following method is used by both Prio3 and Poplar1:

+
+
+def domain_separation_tag(self, usage: int, ctx: bytes) -> bytes:
+    """
+    Format domain separation tag for this VDAF with the given
+    application context and usage.
+
+    Pre-conditions:
+
+        - `usage` in `[0, 2^16)`
+    """
+    return format_dst(0, self.ID, usage) + ctx
+
+
+

The output, called the "domain separation tag", is used in our constructions +for domain separation. Function format_dst() is defined in Section 6.2.3.

+
+
+

+5.1. Sharding +

+

Sharding is as described for DAFs in Section 4.1. The public share and +input shares encode additional information used during preparation to validate +the output shares before they are aggregated (e.g., the "proof shares" in +Section 7).

+

Like DAFs, sharding is bound to the application context via the application +context string. Again, this is intended to ensure that aggregation succeeds +only if the Clients and Aggregators agree on the application context. Unlike +DAFs, however, disagreement on the context should manifest as a preparation +failure, causing the report to be rejected without garbling the aggregate +result. The application context also provides some defense-in-depth against +cross protocol attacks; see Section 9.9.

+
+
+
+
+

+5.2. Preparation +

+
+
+
+
+    Aggregator 0   Aggregator 1           Aggregator SHARES-1
+    ============   ============           ===================
+
+    input_share_0  input_share_1          input_share_[SHARES-1]
+      |              |                 ...  |
+      V              V                      V
+    +-----------+  +-----------+          +-----------+
+    | prep_init |  | prep_init |          | prep_init |
+    +-----------+  +-----------+          +-----------+
+      |       |      |       |         ...  |       |
+      V       |      V       |              V       |
+    +---------|--------------|----------------------|-+   \
+    |         |              | prep_shares_to_prep  | |   |
+    +---------|--------------|----------------------|-+   |
+      |       |      |       |         ...  |       |     |
+      V       V      V       V              V       |     | x ROUNDS
+    +-----------+  +-----------+          +-----------+   |
+    | prep_next |  | prep_next |          | prep_next |   |
+    +-----------+  +-----------+          +-----------+   |
+      |       |      |                 ...  |       |     |
+      V       V      V                      V       V     /
+     ...            ...                    ...
+      |              |                 ...  |
+      V              V                      V
+    out_share_0    out_share_1         out_share_[SHARES-1]
+
+
+
Figure 6: +Illustration of interactive VDAF preparation. +
+
+

Preparation is organized into a number of rounds. The number of rounds depends +on the VDAF: Prio3 (Section 7) has one round and Poplar1 (Section 8) has two.

+

Aggregators retain some local state between successive rounds of preparation. +This is referred to as "preparation state" or "prep state" for short.

+

At the start of each round, each Aggregator broadcasts a message called a +"preparation share", or "prep share" for short. The prep shares are then +combined into a single message called the "preparation message", or "prep +message". The prep message MAY be computed by any one of the Aggregators.

+

The prep message is disseminated to each of the Aggregators to begin the next +round, or compute the output shares in case of the last round. An Aggregator +begins the first round with its input share and it begins each subsequent round +with the current prep state and the previous prep message. Its output in the +last round is its output share and its output in each of the preceding rounds +is a prep share.

+

Just as for DAFs (Section 4.2), preparation involves an aggregation +parameter. The aggregation parameter is consumed by each Aggregator before the +first round of communication.

+

Unlike DAFs, VDAF preparation involves a secret "verification key" held by each +of the Aggregators. This key is used to verify validity of the output shares +they compute. It is up to the high level protocol in which the VDAF is used to +arrange for the distribution of the verification key prior to generating and +processing reports. See Section 9 for details.

+

Preparation is implemented by the following set of algorithms:

+
    +
  • +

    vdaf.prep_init(verify_key: bytes, ctx: bytes, agg_id: int, agg_param: +AggParam, nonce: bytes, public_share: PublicShare, input_share: InputShare) +-> tuple[PrepState, PrepShare] is the deterministic preparation-state +initialization algorithm run by each Aggregator. It consumes the shared +verification key, the application context, the Aggregator's unique +identifier, the aggregation parameter chosen by the Collector, the report +nonce, the public share, and one of the input shares generated by the Client. +It produces the Aggregator's initial prep state and prep share.

    +

    +Protocols MUST ensure that public share consumed by each of the Aggregators is +identical. This is security critical for VDAFs such as Poplar1.

    +

    +Pre-conditions:

    +
      +
    • +

      verify_key MUST have length vdaf.VERIFY_KEY_SIZE.

      +
      • +
    • +

      agg_id MUST be the integer in [0, vdaf.SHARES) that matches the +index of input_share in the sequence of input shares output by the +Client.

      +
      • +
    • +

      nonce MUST have length vdaf.NONCE_SIZE.

      +
      • +
    +
    • +
  • +

    vdaf.prep_shares_to_prep(ctx: bytes, agg_param: AggParam, prep_shares: +list[PrepShare]) -> PrepMessage is the deterministic preparation-message +pre-processing algorithm. It combines the prep shares produced by the +Aggregators in the previous round into the prep message consumed by each +Aggregator to start the next round.

    +
    • +
  • +

    vdaf.prep_next(ctx: bytes, prep_state: PrepState, prep_msg: PrepMessage) -> +tuple[PrepState, PrepShare] | OutShare is the deterministic +preparation-state update algorithm run by each Aggregator. It updates the +Aggregator's prep state (prep_state) and returns either its next prep state +and prep share for the next round or, if this is the last round, its output +share.

    +
    • +
+

An exception may be raised by one of these algorithms, in which case the report +MUST be deemed invalid and not processed any further.

+

Implementation note: The preparation process accomplishes two tasks: recovery +of output shares from the input shares and ensuring that the recovered output +shares are valid. The abstraction boundary is drawn so that an Aggregator only +recovers an output share if the underlying data is deemed valid (at least, +based on the Aggregator's view of the protocol). Another way to draw this +boundary would be to have the Aggregators recover output shares first, then +verify that they are valid. However, this would allow the possibility of +misusing the API by, say, aggregating an invalid output share. Moreover, in +protocols like Prio+ [AGJOP21] based on oblivious transfer, it is necessary +for the Aggregators to interact in order to recover aggregatable output shares +at all.

+
+
+
+
+

+5.3. Validity of Aggregation Parameters +

+

Aggregation parameter validation is as described for DAFs in +Section 4.3. Again, each Aggregator MUST validate each +aggregation parameter received by the Collector before beginning preparation +with that parameter.

+
+
+
+
+

+5.4. Aggregation +

+

Aggregation is identical to DAF aggregation as described in +Section 4.4. Like DAFs, computation of the VDAF aggregate is not +usually sensitive to the order in which output shares are aggregated. See +Section 4.4.1.

+
+
+
+
+

+5.5. Unsharding +

+

Unsharding is identical to DAF unsharding as described in Section 4.5.

+
+
+
+
+

+5.6. Execution of a VDAF +

+

Secure execution of a VDAF involves simulating the following procedure over an +insecure network.

+
+
+def run_vdaf(
+        vdaf: Vdaf[
+            Measurement,
+            AggParam,
+            PublicShare,
+            InputShare,
+            list[F],  # OutShare
+            AggShare,
+            AggResult,
+            PrepState,
+            PrepShare,
+            PrepMessage,
+        ],
+        verify_key: bytes,
+        agg_param: AggParam,
+        ctx: bytes,
+        measurements: list[Measurement]) -> AggResult:
+    """
+    Pre-conditions:
+
+        - `len(verify_key) == vdaf.VERIFY_KEY_SIZE`
+    """
+    agg_shares = [vdaf.agg_init(agg_param)
+                  for _ in range(vdaf.SHARES)]
+    for measurement in measurements:
+        # Sharding
+        nonce = gen_rand(vdaf.NONCE_SIZE)
+        rand = gen_rand(vdaf.RAND_SIZE)
+        (public_share, input_shares) = \
+            vdaf.shard(ctx, measurement, nonce, rand)
+
+        # Initialize preparation
+        prep_states = []
+        outbound_prep_shares = []
+        for j in range(vdaf.SHARES):
+            (state, share) = vdaf.prep_init(verify_key, ctx, j,
+                                            agg_param,
+                                            nonce,
+                                            public_share,
+                                            input_shares[j])
+            prep_states.append(state)
+            outbound_prep_shares.append(share)
+
+        # Complete preparation
+        for i in range(vdaf.ROUNDS - 1):
+            prep_msg = vdaf.prep_shares_to_prep(ctx,
+                                                agg_param,
+                                                outbound_prep_shares)
+
+            outbound_prep_shares = []
+            for j in range(vdaf.SHARES):
+                out = vdaf.prep_next(ctx, prep_states[j], prep_msg)
+                assert isinstance(out, tuple)
+                (prep_states[j], prep_share) = out
+                outbound_prep_shares.append(prep_share)
+
+        prep_msg = vdaf.prep_shares_to_prep(ctx,
+                                            agg_param,
+                                            outbound_prep_shares)
+
+        # Aggregation
+        for j in range(vdaf.SHARES):
+            out_share = vdaf.prep_next(ctx, prep_states[j], prep_msg)
+            assert not isinstance(out_share, tuple)
+            agg_shares[j] = vdaf.agg_update(agg_param,
+                                            agg_shares[j],
+                                            out_share)
+
+    # Unsharding
+    num_measurements = len(measurements)
+    agg_result = vdaf.unshard(agg_param, agg_shares,
+                              num_measurements)
+    return agg_result
+
+
+

The inputs to this algorithm are the verification key, application context, +aggregation parameter, and a list of measurements. As explained in +Section 4.6, the secure execution of a VDAF requires the application to +instantiate secure channels between each of the protocol participants.

+
+
+
+
+

+5.7. Communication Patterns for Preparation +

+

The only stage of VDAF execution that requires interaction is preparation +(Section 5.2). There are a number of ways to coordinate this +interaction; the best strategy depends largely on the number of Aggregators +(i.e., vdaf.SHARES). This section describes two strategies, one specialized +for two Aggregators and another that is suitable for any number of Aggregators.

+

In each round of preparation, each Aggregator writes a prep share to some +broadcast channel, which is then processed into the prep message using the +public prep_shares_to_prep() algorithm and broadcast to the Aggregators to +start the next round. Our goal in this section is to realize this broadcast +channel.

+

The state machine of each Aggregator is shown below.

+
+
+
+
+                  +----------------+
+                  |                |
+                  v                |
+Start ----> Continued(prep_state, prep_round) --> Finished(out_share)
+ |                |
+ |                |
+ +--> Rejected <--+
+
+
+
Figure 7: +State machine of VDAF preparation. +
+
+

State transitions are made when the state is acted upon by the Aggregator's +local inputs and/or messages sent by its co-Aggregators. The initial state is +Start. The terminal states are: Rejected, indicating that the report cannot +be processed any further; and Finished(out_share), indicating that the +Aggregator has recovered an output share out_share. For completeness, we +define these states in Appendix B.

+

The methods described in this section are defined in terms of opaque byte +strings. A compatible Vdaf MUST specify methods for encoding public shares, +input shares, prep shares, prep messages, and aggregation parameters.

+

Implementations of Prio3 and Poplar1 MUST use the encoding scheme specified in +Section 7.2.7 and Section 8.2.6 respectively.

+
+
+

+5.7.1. The Ping-Pong Topology (Only Two Aggregators) +

+

For VDAFs with precisely two Aggregators (i.e., vdaf.SHARES == 2), the +following "ping pong" communication pattern can be used. It is compatible with +any request/response transport protocol, such as HTTP.

+

Let us call the initiating Aggregator the "Leader" and the responding +Aggregator the "Helper". The high-level idea is that the Leader and Helper will +take turns running the computation locally until input from their peer is +required:

+
    +
  • +

    For a 1-round VDAF (e.g., Prio3 in Section 7), the Leader sends its prep share +to the Helper, who computes the prep message locally, computes its output +share, then sends the prep message to the Leader. Preparation requires just +one round trip between the Leader and the Helper.

    +
    • +
  • +

    For a 2-round VDAF (e.g., Poplar1 in Section 8), the Leader sends its +first-round prep share to the Helper, who replies with the first-round prep +message and its second-round prep share. In the next request, the Leader +computes its second-round prep share locally, computes its output share, and +sends the second-round prep message to the Helper. Finally, the Helper +computes its own output share.

    +
    • +
  • +

    In general, each request includes the Leader's prep share for the previous +round and/or the prep message for the current round; correspondingly, each +response consists of the prep message for the current round and the Helper's +prep share for the next round.

    +
    • +
+

The Aggregators proceed in this ping-ponging fashion until a step of the +computation fails (indicating the report is invalid and should be rejected) or +preparation is completed. All told there are ceil((vdaf.ROUNDS+1)/2) +requests sent.

+

We specify protocol messages in the presentation language of TLS; see Section 3 of [RFC8446]. Each message is structured as follows:

+
+
+enum {
+  initialize(0),
+  continue(1),
+  finish(2),
+  (255)
+} MessageType;
+
+struct {
+  MessageType type;
+  select (Message.type) {
+    case initialize:
+      opaque prep_share<0..2^32-1>;
+    case continue:
+      opaque prep_msg<0..2^32-1>;
+      opaque prep_share<0..2^32-1>;
+    case finish:
+      opaque prep_msg<0..2^32-1>;
+  };
+} Message;
+
+
+

These messages trigger all transitions in the state machine in +Figure 7, except for the Leader's initial transition. The +Leader's state is initialized using its local inputs with the following method +on class Vdaf:

+
+
+def ping_pong_leader_init(
+        self,
+        vdaf_verify_key: bytes,
+        ctx: bytes,
+        agg_param: bytes,
+        nonce: bytes,
+        public_share: bytes,
+        input_share: bytes) -> tuple[State, Optional[bytes]]:
+    """Called by the Leader to initialize ping-ponging."""
+    try:
+        (prep_state, prep_share) = self.prep_init(
+            vdaf_verify_key,
+            ctx,
+            0,
+            self.decode_agg_param(agg_param),
+            nonce,
+            self.decode_public_share(public_share),
+            self.decode_input_share(0, input_share),
+        )
+        encoded_prep_share = self.encode_prep_share(prep_share)
+        return (
+            Continued(prep_state, 0),
+            encode(0, encoded_prep_share),  # initialize
+        )
+    except:
+        return (Rejected(), None)
+
+
+

The output is the State to which the Leader has transitioned and an encoded +Message. If the Leader's state is Rejected, then processing halts. +Otherwise, if the state is Continued, then processing continues. Function +encode is used to encode the outbound message, here the initialize variant +(hence 0).

+

To continue processing the report, the Leader sends the outbound message to the +Helper. The Helper's initial transition is computed using the following +procedure:

+
+
+def ping_pong_helper_init(
+        self,
+        vdaf_verify_key: bytes,
+        ctx: bytes,
+        agg_param: bytes,
+        nonce: bytes,
+        public_share: bytes,
+        input_share: bytes,
+        inbound: bytes) -> tuple[State, Optional[bytes]]:
+    """
+    Called by the Helper in response to the Leader's initial
+    message.
+    """
+    try:
+        (prep_state, prep_share) = self.prep_init(
+            vdaf_verify_key,
+            ctx,
+            1,
+            self.decode_agg_param(agg_param),
+            nonce,
+            self.decode_public_share(public_share),
+            self.decode_input_share(1, input_share),
+        )
+
+        (inbound_type, inbound_items) = decode(inbound)
+        if inbound_type != 0:  # initialize
+            return (Rejected(), None)
+
+        encoded_prep_share = inbound_items[0]
+        prep_shares = [
+            self.decode_prep_share(prep_state, encoded_prep_share),
+            prep_share,
+        ]
+        return self.ping_pong_transition(
+            ctx,
+            self.decode_agg_param(agg_param),
+            prep_shares,
+            prep_state,
+            0,
+        )
+    except:
+        return (Rejected(), None)
+
+
+

Procedure decode() decodes the inbound message and returns the MessageType +variant (initialize, continue, or finish) and the sequence of fields. +Procedure ping_pong_transition() takes in the prep shares, combines them into +the prep message, and computes the next prep state of the caller:

+
+
+def ping_pong_transition(
+        self,
+        ctx: bytes,
+        agg_param: AggParam,
+        prep_shares: list[PrepShare],
+        prep_state: PrepState,
+        prep_round: int) -> tuple[State, bytes]:
+    prep_msg = self.prep_shares_to_prep(ctx,
+                                        agg_param,
+                                        prep_shares)
+    encoded_prep_msg = self.encode_prep_msg(prep_msg)
+    out = self.prep_next(ctx, prep_state, prep_msg)
+    if prep_round+1 == self.ROUNDS:
+        return (
+            Finished(out),
+            encode(2, encoded_prep_msg),  # finalize
+        )
+    (prep_state, prep_share) = cast(
+        tuple[PrepState, PrepShare], out)
+    encoded_prep_share = self.encode_prep_share(prep_share)
+    return (
+        Continued(prep_state, prep_round+1),
+        encode(1, encoded_prep_msg, encoded_prep_share)  # continue
+    )
+
+
+

The output is the State to which the Helper has transitioned and an encoded +Message. If the Helper's state is Finished or Rejected, then processing +halts. Otherwise, if the state is Continued, then processing continues.

+

To continue processing, the Helper sends the outbound message to the Leader. +The Leader computes its next state transition using the following method on +class Vdaf:

+
+
+def ping_pong_leader_continued(
+    self,
+    ctx: bytes,
+    agg_param: bytes,
+    state: State,
+    inbound: bytes,
+) -> tuple[State, Optional[bytes]]:
+    """
+    Called by the Leader to start the next step of ping-ponging.
+    """
+    return self.ping_pong_continued(
+        True, ctx, agg_param, state, inbound)
+
+def ping_pong_continued(
+    self,
+    is_leader: bool,
+    ctx: bytes,
+    agg_param: bytes,
+    state: State,
+    inbound: bytes,
+) -> tuple[State, Optional[bytes]]:
+    try:
+        if not isinstance(state, Continued):
+            return (Rejected(), None)
+        prep_round = state.prep_round
+
+        (inbound_type, inbound_items) = decode(inbound)
+        if inbound_type == 0:  # initialize
+            return (Rejected(), None)
+
+        encoded_prep_msg = inbound_items[0]
+        prep_msg = self.decode_prep_msg(
+            state.prep_state,
+            encoded_prep_msg,
+        )
+        out = self.prep_next(ctx, state.prep_state, prep_msg)
+        if prep_round+1 < self.ROUNDS and \
+                inbound_type == 1:  # continue
+            (prep_state, prep_share) = cast(
+                tuple[PrepState, PrepShare], out)
+            encoded_prep_share = inbound_items[1]
+            prep_shares = [
+                self.decode_prep_share(
+                    prep_state,
+                    encoded_prep_share,
+                ),
+                prep_share,
+            ]
+            if is_leader:
+                prep_shares.reverse()
+            return self.ping_pong_transition(
+                ctx,
+                self.decode_agg_param(agg_param),
+                prep_shares,
+                prep_state,
+                prep_round+1,
+            )
+        elif prep_round+1 == self.ROUNDS and \
+                inbound_type == 2:  # finish
+            return (Finished(out), None)
+        else:
+            return (Rejected(), None)
+    except:
+        return (Rejected(), None)
+
+
+

If the Leader's state is Finished or Rejected, then processing halts. +Otherwise, the Leader sends the outbound message to the Helper. The Helper +computes its next state transition using the following method on class Vdaf:

+
+
+def ping_pong_helper_continued(
+    self,
+    ctx: bytes,
+    agg_param: bytes,
+    state: State,
+    inbound: bytes,
+) -> tuple[State, Optional[bytes]]:
+    """Called by the Helper to continue ping-ponging."""
+    return self.ping_pong_continued(
+        False, ctx, agg_param, state, inbound)
+
+
+

They continue in this way until processing halts. Note that, depending on the +number of rounds of preparation that are required, there may be one more +message to send before the peer can also finish processing (i.e., outbound != +None).

+
+
+
+
+

+5.7.2. The Star Topology (Any Number of Aggregators) +

+

The ping-pong topology of the previous section is only suitable for VDAFs +involving exactly two Aggregators. If the VDAF supports more than two +Aggregators, then the star topology described in this section can +be used instead.

+

We again designate an Aggregator to initiate the computation. We refer to this +Aggregator as the Leader and to all other Aggregators as Helpers.

+

At the start of each round, the Leader requests from each Helper its prep +share. After gathering each of the prep shares, the Leader computes the next +prep message (via vdaf.prep_shares_to_prep()) and broadcasts it to the +Helpers. At this point, each Aggregator runs vdaf.prep_next() locally to +either recover an output share or, if more rounds of preparation are required, +compute its updated state and prep share. If another round is required, then +the Helper responds to the broadcast message with its next prep share.

+

The Aggregators proceed in this way until each recovers an output share or some +step of the computation fails.

+
+
+
+
+
+
+
+
+

+6. Preliminaries +

+

This section describes the primitives that are common to the VDAFs specified in +this document.

+
+
+

+6.1. Finite Fields +

+

Both Prio3 and Poplar1 use finite fields of prime order. Finite field +elements are represented by a class Field with the following associated +parameters:

+
    +
  • +

    MODULUS: int is the prime modulus that defines the field.

    +
    • +
  • +

    ENCODED_SIZE: int is the number of bytes used to encode a field element +as a byte string.

    +
    • +
+

Concrete fields, i.e., subclasses of Field, implement the following class +methods:

+
    +
  • +

    Field.zeros(length: int) -> list[Self] returns a vector of zeros of the +requested length.

    +

    +Pre-conditions:

    +
      +
    • +

      length MUST be greater than or equal 0.

      +
      • +
    +

    +Post-conditions:

    +
      +
    • +

      The length of the output MUST be length.

      +
      • +
    +
    • +
  • +

    Field.rand_vec(length: int) -> list[Self] returns a vector of random field +elements and has the same pre- and post-conditions as for Field.zeros(). +Note that this function is not used normatively in the specification of +either Prio3 or Poplar1.

    +
    • +
+

A field element is an instance of a concrete Field. Addition, +subtraction, multiplication, division, negation, and inversion are denoted, +respectively, x + y, x - y, x * y, x / y, -x, and x.inv().

+

We sometimes need to convert a field element to an int, which we denote by +x.int(). Likewise, each concrete Field implements a constructor for +converting an unsigned integer into a field element:

+
    +
  • +

    Field(integer: int) returns integer represented as a field element. The +value of integer MUST be in the range (-Field.MODULUS, Field.MODULUS); +negative values are treated as negations.

    +
    • +
+
+
+

+6.1.1. Auxiliary Functions +

+

The following class methods on Field are used to encode and decode vectors of +field elements as byte strings:

+
+
+def encode_vec(cls, vec: list[Self]) -> bytes:
+    """
+    Encode a vector of field elements `vec` as a byte string.
+    """
+    encoded = bytes()
+    for x in vec:
+        encoded += to_le_bytes(x.int(), cls.ENCODED_SIZE)
+    return encoded
+
+def decode_vec(cls, encoded: bytes) -> list[Self]:
+    """
+    Parse a vector of field elements from `encoded`.
+    """
+    if len(encoded) % cls.ENCODED_SIZE != 0:
+        raise ValueError(
+            'input length must be a multiple of the size of an '
+            'encoded field element')
+
+    vec = []
+    while len(encoded) > 0:
+        (encoded_x, encoded) = front(cls.ENCODED_SIZE, encoded)
+        x = from_le_bytes(encoded_x)
+        if x >= cls.MODULUS:
+            raise ValueError('modulus overflow')
+        vec.append(cls(x))
+    return vec
+
+
+

Field provides the following class methods for representing a value as a +sequence of field elements, each of which represents a bit of the input. These +are used to encode measurements in some variants of Prio3 +(Section 7.4).

+
+
+def encode_into_bit_vec(
+        cls,
+        val: int,
+        bits: int) -> list[Self]:
+    """
+    Encode the bit representation of `val` with at most `bits` number
+    of bits, as a vector of field elements.
+
+    Pre-conditions:
+
+        - `val >= 0`
+        - `bits >= 0`
+    """
+    if val >= 2 ** bits:
+        # Sanity check we are able to represent `val` with `bits`
+        # number of bits.
+        raise ValueError("Number of bits is not enough to represent "
+                         "the input integer.")
+    encoded = []
+    for l in range(bits):
+        encoded.append(cls((val >> l) & 1))
+    return encoded
+
+def decode_from_bit_vec(cls, vec: list[Self]) -> Self:
+    """
+    Decode the field element from the bit representation, expressed
+    as a vector of field elements `vec`.
+    """
+    bits = len(vec)
+    if cls.MODULUS >> bits == 0:
+        raise ValueError("Number of bits is too large to be "
+                         "represented by field modulus.")
+    decoded = cls(0)
+    for (l, bit) in enumerate(vec):
+        decoded += cls(1 << l) * bit
+    return decoded
+
+
+

Finally, the following functions define arithmetic on vectors over a finite +field. Note that an exception is raised by each function if the operands are +not the same length.

+
+
+def vec_sub(left: list[F], right: list[F]) -> list[F]:
+    """
+    Subtract the right operand from the left and return the result.
+    """
+    if len(left) != len(right):
+        raise ValueError("mismatched vector sizes")
+    return list(map(lambda x: x[0] - x[1], zip(left, right)))
+
+def vec_add(left: list[F], right: list[F]) -> list[F]:
+    """Add the right operand to the left and return the result."""
+    if len(left) != len(right):
+        raise ValueError("mismatched vector sizes")
+    return list(map(lambda x: x[0] + x[1], zip(left, right)))
+
+def vec_neg(vec: list[F]) -> list[F]:
+    """Negate the input vector."""
+    return list(map(lambda x: -x, vec))
+
+
+
+
+
+
+

+6.1.2. NTT-Friendly Fields +

+

Some VDAFs, including Prio3, require fields that are suitable for efficient +computation of the number theoretic transform (NTT) [SML24], as this allows +for fast polynomial interpolation. Specifically, a field is said to be +"NTT-friendly" if, in addition to the interface described in Section 6.1, it +provides the following interface:

+
    +
  • +

    Field.gen() -> Self is a class method that returns the generator of a large +subgroup of the multiplicative group. To be NTT-friendly, the order of this +subgroup MUST be a power of 2.

    +
    • +
  • +

    GEN_ORDER: int is the order of a multiplicative subgroup generated by +Field.gen(). This is the smallest positive integer for which +Field.gen()**Field.GEN_ORDER == Field(1).

    +
    • +
+

The size of the subgroup dictates how large interpolated polynomials can be. It +is RECOMMENDED that a generator is chosen with order at least 2^20.

+
+
+
+
+

+6.1.3. Parameters +

+

The tables below define finite fields used in the remainder of this document.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 3: +Parameters for the finite fields used in this document. +
ParameterField64Field128Field255
MODULUS2^32 * 4294967295 + 12^66 * 4611686018427387897 + 12^255 - 19
ENCODED_SIZE81632
Generator7^42949672957^4611686018427387897n/a
GEN_ORDER2^322^66n/a
+
+
+
+
+
+
+
+

+6.2. Extendable Output Functions (XOFs) +

+

VDAFs in this specification use eXtendable Output Functions (XOFs) for two +purposes:

+
    +
  1. +

    Extracting short, pseudorandom strings we call "seeds" from high entropy +inputs

    +
    2. +
  2. +

    Expanding seeds into long, pseudorandom outputs

    +
    3. +
+

Concrete XOFs implement a class Xof providing the following interface:

+
    +
  • +

    SEED_SIZE: int is the size (in bytes) of a seed.

    +
    • +
  • +

    Xof(seed: bytes, dst: bytes, binder: bytes) constructs an instance of the +XOF from the given seed and a domain separation tag and binder string as +defined in Section 6.2.3. The length of the seed will typically be +SEED_SIZE, but some XOFs may support multiple seed sizes. The seed MUST be +generated securely, i.e., it is either the output of a CSPRNG or a +previous invocation of the XOF.

    +
    • +
  • +

    xof.next(length: int) returns the next chunk of the output of the +initialized XOF. The length of the chunk MUST be length.

    +
    • +
+

The following methods are provided for all concrete XOFs. The first is a class +method used to derive a fresh seed from an existing one. The second is an +instance method used to compute a sequence of field elements. The third is a +class method that provides a one-shot interface for expanding a seed into a +field vector.

+
+
+def derive_seed(cls,
+                seed: bytes,
+                dst: bytes,
+                binder: bytes) -> bytes:
+    """
+    Derive a new seed.
+
+    Pre-conditions:
+
+        - `len(seed) == cls.SEED_SIZE`
+    """
+    xof = cls(seed, dst, binder)
+    return xof.next(cls.SEED_SIZE)
+
+def next_vec(self, field: type[F], length: int) -> list[F]:
+    """
+    Output the next `length` field elements.
+
+    Pre-conditions:
+
+        - `field` is sub-class of `Field`
+        - `length > 0`
+    """
+    m = next_power_of_2(field.MODULUS) - 1
+    vec: list[F] = []
+    while len(vec) < length:
+        x = from_le_bytes(self.next(field.ENCODED_SIZE))
+        x &= m
+        if x < field.MODULUS:
+            vec.append(field(x))
+    return vec
+
+def expand_into_vec(cls,
+                    field: type[F],
+                    seed: bytes,
+                    dst: bytes,
+                    binder: bytes,
+                    length: int) -> list[F]:
+    """
+    Expand the input `seed` into vector of `length` field elements.
+
+    Pre-conditions:
+
+        - `field` is sub-class of `Field`
+        - `len(seed) == cls.SEED_SIZE`
+        - `length > 0`
+    """
+    xof = cls(seed, dst, binder)
+    return xof.next_vec(field, length)
+
+
+
+
+

+6.2.1. XofTurboShake128 +

+

This section describes XofTurboShake128, an XOF based on the TurboSHAKE128 +function specified in [TurboSHAKE]. This +XOF is RECOMMENDED for all use cases for DAFs and VDAFs.

+
    +
  • +

    TODO Update the [TurboSHAKE] reference to point to the RFC instead of the +draft.

    +
    • +
+

Pre-conditions:

+
    +
  • +

    The default seed length is 32. The seed MAY have a different length, but MUST not +exceed 255. Otherwise initialization will raise an exception.

    +
    • +
  • +

    The length of the domain separation string dst passed to XofTurboShake128 +MUST NOT exceed 65535 bytes. Otherwise initialization will raise an +exception.

    +
    • +
+
+
+class XofTurboShake128(Xof):
+    """XOF wrapper for TurboSHAKE128."""
+
+    # Associated parameters
+    SEED_SIZE = 32
+
+    def __init__(self, seed: bytes, dst: bytes, binder: bytes):
+        self.l = 0
+        self.m = \
+            to_le_bytes(len(dst), 2) + dst \
+            to_le_bytes(len(seed), 1) + seed + \
+            binder
+
+    def next(self, length: int) -> bytes:
+        self.l += length
+
+        # Function `TurboSHAKE128(M, D, L)` is as defined in
+        # Section 2.2 of [TurboSHAKE].
+        #
+        # Implementation note: rather than re-generate the output
+        # stream each time `next()` is invoked, most implementations
+        # of TurboSHAKE128 will expose an "absorb-then-squeeze" API
+        # that allows stateful handling of the stream.
+        stream = TurboSHAKE128(self.m, 1, self.l)
+        return stream[-length:]
+
+
+
+
+
+
+

+6.2.2. XofFixedKeyAes128 +

+

The XOF in the previous section can be used safely wherever a XOF is needed in +this document. However, there are some situations where TurboSHAKE128 creates a +performance bottleneck and a more efficient XOF can be used safely instead.

+

This section describes XofFixedKeyAes128, which is used to implement the IDPF +of Poplar1 (Section 8.3). It is NOT RECOMMENDED to use this XOF for any +other purpose. See Section 9 for a more detailed discussion.

+

XofFixedKeyAes128 uses the AES128 blockcipher [AES] for most of the +computation, thereby taking advantage of the hardware implementations of this +blockcipher that are widely available. AES128 is used in a fixed-key mode of +operation; the key is derived during initialization using TurboSHAKE128.

+

Pre-conditions:

+
    +
  • +

    The length of the seed MUST be 16.

    +
    • +
  • +

    The length of the domain separation string dst passed to XofTurboShake128 +MUST NOT exceed 65535 bytes. Otherwise initialization will raise an +exception.

    +
    • +
+
+
+class XofFixedKeyAes128(Xof):
+    """
+    XOF based on a circular collision-resistant hash function from
+    fixed-key AES.
+    """
+
+    # Associated parameters
+    SEED_SIZE = 16
+
+    def __init__(self, seed: bytes, dst: bytes, binder: bytes):
+        if len(seed) != self.SEED_SIZE:
+            raise ValueError("incorrect seed size")
+
+        self.length_consumed = 0
+
+        # Use TurboSHAKE128 to derive a key from the binder string
+        # and domain separation tag. Note that the AES key does not
+        # need to be kept secret from any party. However, when used
+        # with an IDPF, we require the binder to be a random nonce.
+        #
+        # Implementation note: this step can be cached across XOF
+        # evaluations with many different seeds.
+        dst_length = to_le_bytes(len(dst), 2)
+        self.fixed_key = TurboSHAKE128(
+            dst_length + dst + binder,
+            2,
+            16,
+        )
+        self.seed = seed
+
+    def next(self, length: int) -> bytes:
+        offset = self.length_consumed % 16
+        new_length = self.length_consumed + length
+        block_range = range(
+            self.length_consumed // 16,
+            new_length // 16 + 1
+        )
+        self.length_consumed = new_length
+
+        hashed_blocks = [
+            self.hash_block(xor(self.seed, to_le_bytes(i, 16)))
+            for i in block_range
+        ]
+        return concat(hashed_blocks)[offset:offset+length]
+
+    def hash_block(self, block: bytes) -> bytes:
+        """
+        The multi-instance tweakable circular correlation-robust hash
+        function of [GKWWY20] (Section 4.2). The tweak here is the
+        key that stays constant for all XOF evaluations of the same
+        Client, but differs between Clients.
+
+        Function `AES128(key, block)` is the AES-128 blockcipher.
+        """
+        lo, hi = block[:8], block[8:]
+        sigma_block = concat([hi, xor(hi, lo)])
+        return xor(AES128(self.fixed_key, sigma_block), sigma_block)
+
+
+
+
+
+
+

+6.2.3. The Domain Separation Tag and Binder String +

+

XOFs are used to map a seed to a finite domain, e.g., a fresh seed or a vector +of field elements. To ensure domain separation, derivation is bound to some +distinguished domain separation tag. The domain separation tag encodes the +following values:

+
    +
  1. +

    The document version (i.e.,VERSION);

    +
    2. +
  2. +

    The "class" of the algorithm using the output (e.g., DAF, VDAF, or IDPF as +defined in Section 8.1);

    +
    3. +
  3. +

    A unique identifier for the algorithm (e.g., VDAF.ID); and

    +
    4. +
  4. +

    Some indication of how the output is used (e.g., for deriving the measurement +shares in Prio3 Section 7).

    +
    5. +
+

The following algorithm is used in the remainder of this document in order to +format the domain separation tag:

+
+
+def format_dst(algo_class: int,
+               algo: int,
+               usage: int) -> bytes:
+    """
+    Format XOF domain separation tag.
+
+    Pre-conditions:
+
+        - `algo_class` in `[0, 2^8)`
+        - `algo` in `[0, 2^32)`
+        - `usage` in `[0, 2^16)`
+    """
+    return concat([
+        to_be_bytes(VERSION, 1),
+        to_be_bytes(algo_class, 1),
+        to_be_bytes(algo, 4),
+        to_be_bytes(usage, 2),
+    ])
+
+
+

It is also sometimes necessary to bind the output to some ephemeral value that +multiple parties need to agree on. We call this input the "binder string".

+
+
+
+
+
+
+
+
+

+7. Prio3 +

+

This section describes Prio3, a VDAF for general-purpose aggregation. Prio3 is +suitable for a wide variety of aggregation functions, including (but not +limited to) sum, mean, standard deviation, histograms, and linear regression. +It is compatible with any aggregation function that has the following +structure:

+ +

Clients protect the privacy of their measurements by secret sharing them and +distributing the shares among the Aggregators. To ensure each measurement is +valid, the Aggregators run a multi-party computation on their shares, the +result of which is the output of the arithmetic circuit. This involves +verification of a "Fully Linear Proof (FLP)" (Section 7.1) generated by the Client. +FLPs are the core component of Prio3, as they specify the types of +measurements, how they are encoded, and how they are aggregated. In fact Prio3 +can be thought of as a transformation of an FLP into a VDAF.

+

Prio3 does not have an aggregation parameter. Instead, each output share is +derived from each input share by applying a fixed map. See Section 8 for an +example of a VDAF that makes meaningful use of the aggregation parameter.

+

The remainder of this section is structured as follows. The syntax for FLPs is +described in Section 7.1. The generic transformation of an FLP into Prio3 is +specified in Section 7.2. Next, a concrete FLP suitable for any +validity circuit is specified in Section 7.3. Finally, variants of Prio3 +for various types of aggregation tasks are specified in +Section 7.4. Test vectors for each variant can be found in +Appendix C.

+
+
+

+7.1. Fully Linear Proofs (FLPs) +

+

Conceptually, an FLP is a two-party protocol executed by a prover and a +verifier. In actual use, however, the prover's computation is carried out by +the Client, and the verifier's computation is distributed among the +Aggregators. The Client generates a "proof" of its measurement's validity and +distributes shares of the proof to the Aggregators. During preparation, each +Aggregator performs some computation on its measurement share and proof share +locally, then broadcasts the result in its prep share. The validity decision is +then made by the prep_shares_to_prep() algorithm (Section 5.2).

+

As usual, we describe the interface implemented by a concrete FLP in terms of +an abstract base class, denoted Flp, that specifies the set of methods and +parameters a concrete FLP must provide.

+

The parameters provided by a concrete FLP are listed in Table 4. A +concrete FLP specifies the following algorithms for generating and verifying +proofs of validity (encoding is described below in Section 7.1.1):

+
    +
  • +

    flp.prove(meas: list[F], prove_rand: list[F], joint_rand: list[F]) -> +list[F] is the proof-generation algorithm run by the prover. Its inputs are +the encoded measurement, the "prover randomness" prove_rand, and the "joint +randomness" joint_rand. The prover randomness is used only by the prover, +but the joint randomness is shared by both the prover and verifier.

    +
    • +
  • +

    flp.query(meas: list[F], proof: list[F], query_rand: list[F], joint_rand: +list[F], num_shares: int) -> list[F] is the query algorithm run by the +verifier on the encoded measurement and proof. The result of the query (i.e., +the output of this function) is called the "verifier message". In addition to +the measurement and proof, this algorithm takes as input the query randomness +query_rand and the joint randomness joint_rand. The former is used only +by the verifier. num_shares specifies the number of shares (more on this +below).

    +
    • +
  • +

    flp.decide(verifier: list[F]) -> bool is the deterministic decision +algorithm run by the verifier. It takes as input the verifier message and +outputs a boolean indicating if the measurement from which it was generated +is valid.

    +
    • +
+

Our application requires that the FLP is "fully linear" in the sense defined in +[BBCGGI19]. As a practical matter, what this property implies is that, when +run on a share of the measurement and proof, the query algorithm outputs a +share of the verifier message (hereafter the "verifier share"). Furthermore, +the privacy property of the FLP system ensures that the verifier message +reveals nothing about the measurement other than the fact that it is valid. +Therefore, to decide if a measurement is valid, the Aggregators will run the +query algorithm locally, exchange verifier shares, combine them to recover the +verifier message, and run the decision algorithm.

+

The query algorithm includes a parameter num_shares that specifies the number +of shares of the measurement and proof that were generated. If these data are +not secret shared, then num_shares == 1. This parameter is useful for +normalizing constants in arithmetic circuits so that each Aggregator properly +computes a secret share of the circuit's output. See Section 7.3 for +details.

+

An FLP is executed by the prover and verifier as follows:

+
+
+def run_flp(
+        flp: Flp[Measurement, AggResult, F],
+        meas: list[F],
+        num_shares: int) -> bool:
+    """Run the FLP on an encoded measurement."""
+
+    joint_rand = flp.field.rand_vec(flp.JOINT_RAND_LEN)
+    prove_rand = flp.field.rand_vec(flp.PROVE_RAND_LEN)
+    query_rand = flp.field.rand_vec(flp.QUERY_RAND_LEN)
+
+    # Prover generates the proof.
+    proof = flp.prove(meas, prove_rand, joint_rand)
+
+    # Shard the measurement and the proof.
+    meas_shares = additive_secret_share(
+        meas,
+        num_shares,
+        flp.field,
+    )
+    proof_shares = additive_secret_share(
+        proof,
+        num_shares,
+        flp.field,
+    )
+
+    # Verifier queries the meas shares and proof shares.
+    verifier_shares = [
+        flp.query(
+            meas_share,
+            proof_share,
+            query_rand,
+            joint_rand,
+            num_shares,
+        )
+        for meas_share, proof_share in zip(meas_shares, proof_shares)
+    ]
+
+    # Combine the verifier shares into the verifier.
+    verifier = flp.field.zeros(len(verifier_shares[0]))
+    for verifier_share in verifier_shares:
+        verifier = vec_add(verifier, verifier_share)
+
+    # Verifier decides if the measurement is valid.
+    return flp.decide(verifier)
+
+
+

The proof system is designed so that, if meas is valid, then run_flp(flp, +meas, num_shares) always returns True. On the other hand, if meas is +invalid, then as long as joint_rand and query_rand are generated uniform +randomly, the output is False with high probability. False positives are +possible: there is a small probability that a verifier accepts an invalid input +as valid. An FLP is said to be "sound" if this probability is sufficiently +small. The soundness of the FLP depends on a variety of parameters, like the +length of the input and the size of the field. See Section 7.3 for +details.

+

Note that soundness of an FLP system is not the same as robustness for the VDAF +that uses it. In particular, soundness of the FLP is necessary, but +insufficient for robustness of Prio3 (Section 7). See Section 9.7 +for details.

+

We remark that [BBCGGI19] defines a larger class of fully linear proof +systems than we consider here. In particular, what is called an "FLP" here is +called a 1.5-round, public-coin, interactive oracle proof system in their +paper.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 4: +FLP parameters. +
ParameterDescription
+ PROVE_RAND_LEN: int +Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof.
+ QUERY_RAND_LEN: int +Length of the query randomness, the number of random field elements consumed by the verifier.
+ JOINT_RAND_LEN: int +Length of the joint randomness, the number of random field elements shared by the prover and verifier.
+ MEAS_LEN: int +Length of the encoded measurement (Section 7.1.1).
+ OUTPUT_LEN: int +Length of the aggregatable output (Section 7.1.1).
+ PROOF_LEN: int +Length of the proof.
+ VERIFIER_LEN: int +Length of the verifier message generated by querying the measurement and proof.
+ Measurement +Type of the measurement.
+ AggResult +Type of the aggregate result.
+ field: type[F] +Class object for the field (Section 6.1).
+
+
+
+

+7.1.1. Encoding the Input +

+

The type of measurement being aggregated is defined by the FLP. Hence, the FLP +also specifies a method of encoding raw measurements as a vector of field +elements:

+
    +
  • +

    flp.encode(measurement: Measurement) -> list[F] encodes a raw measurement +as a vector of field elements.

    +

    +Post-conditions:

    +
      +
    • +

      The encoded measurement MUST have length flp.MEAS_LEN.

      +
      • +
    +
    • +
+

For some FLPs, the encoded measurement also includes redundant field elements +that are useful for checking the proof, but which are not needed after the +proof has been checked. An example is the Sum type defined in Section 7.4.2 +for which each measurement is an integer in range [0, max_measurement]. The +range check requires encoding the measurement with several field elements, +though just one is needed for aggregation. Thus the FLP defines an algorithm +for truncating the encoded measurement to the length of the aggregated output:

+
    +
  • +

    flp.truncate(meas: list[F]) -> list[F] maps an encoded measurement (e.g., +the bit-encoding of the measurement) to an aggregatable output (e.g., the +singleton vector containing the measurement).

    +

    +Pre-conditions:

    +
      +
    • +

      The length of the input MUST be flp.MEAS_LEN

      +
      • +
    +

    +Post-conditions:

    +
      +
    • +

      The length of the output MUST be flp.OUTPUT_LEN.

      +
      • +
    +
    • +
+

Once the aggregate shares have been transmitted to the Collector, their sum can +be converted into the aggregate result. This could be a projection from the +FLP's field to the integers, or it could include additional post-processing. +Either way, this functionality is implemented by the following method:

+
    +
  • +

    flp.decode(output: list[F], num_measurements: int) -> AggResult maps a sum +of aggregate shares to an aggregate result.

    +

    +Pre-conditions:

    +
      +
    • +

      The length of the output MUST be OUTPUT_LEN.

      +
      • +
    • +

      num_measurements MUST equal the number of measurements that were +aggregated.

      +
      • +
    +
    • +
+

We remark that, taken together, these three functionalities correspond to the +notion of "Affine-aggregatable encodings (AFEs)" from [CGB17].

+
+
+
+
+

+7.1.2. Multiple Proofs +

+

It is sometimes desirable to generate and verify multiple independent proofs +for the same input. First, this improves the soundness of the proof system +without having to change any of its parameters. Second, it allows a smaller +field to be used (e.g., replace Field128 with Field64, see Section 7.3) +without sacrificing soundness. This is useful because it reduces the overall +communication of the protocol. (This is a trade-off, of course, since +generating and verifying more proofs requires more time.) Given these benefits, +this feature is implemented by Prio3 (Section 7).

+

To generate these proofs for a specific measurement, the prover calls +flp.prove() multiple times, each time using fresh prover and joint +randomness. The verifier checks each proof independently, each time with fresh +query randomness. It accepts the measurement only if the decision algorithm +accepts on each proof.

+

See Section 9.7 for guidance on choosing the field size and number +of proofs.

+
+
+
+
+
+
+

+7.2. Specification +

+

This section specifies Prio3, an implementation of the Vdaf interface +defined in Section 5. The parameters and types required by the Vdaf interface +are defined in Table 5. The methods required for sharding, preparation, +aggregation, and unsharding are described in the remaining subsections. These +methods refer to constants enumerated in Table 6.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 5: +Parameters for Prio3. +
ParameterValue
+ flp +An instance of Flp (Section 7.1).
+ xof + + XofTurboShake128 (Section 6.2.1)
+ PROOFS +Any int in the range [1, 256).
+ VERIFY_KEY_SIZE + + xof.SEED_SIZE +
+ RAND_SIZE + + xof.SEED_SIZE * SHARES if flp.JOINT_RAND_LEN == 0 else 2 * xof.SEED_SIZE * SHARES +
+ NONCE_SIZE + + 16 +
+ ROUNDS + + 1 +
+ SHARES +Any int in the range [2, 256).
+ Measurement +As defined by flp.
+ AggParam + + None +
+ PublicShare + + Optional[list[bytes]] +
+ InputShare + + tuple[list[F], list[F], Optional[bytes]] | tuple[bytes, Optional[bytes]] +
+ OutShare + + list[F] +
+ AggShare + + list[F] +
+ AggResult +As defined by flp.
+ PrepState + + tuple[list[F], Optional[bytes]] +
+ PrepShare + + tuple[list[F], Optional[bytes]] +
+ PrepMessage + + Optional[bytes] +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 6: +Constants used by Prio3. +
VariableValue
+ USAGE_MEAS_SHARE: int +1
+ USAGE_PROOF_SHARE: int +2
+ USAGE_JOINT_RANDOMNESS: int +3
+ USAGE_PROVE_RANDOMNESS: int +4
+ USAGE_QUERY_RANDOMNESS: int +5
+ USAGE_JOINT_RAND_SEED: int +6
+ USAGE_JOINT_RAND_PART: int +7
+
+
+
+

+7.2.1. Sharding +

+

Recall from Section 7.1 that the FLP syntax calls for "joint randomness" shared by +the prover (i.e., the Client) and the verifier (i.e., the Aggregators). VDAFs +have no such notion. Instead, the Client derives the joint randomness from its +measurement in a way that allows the Aggregators to reconstruct it from their +shares. (This idea is based on the Fiat-Shamir heuristic and is described in +Section 6.2.3 of [BBCGGI19].)

+

The sharding algorithm involves the following steps:

+
    +
  1. +

    Encode the Client's measurement as specified by the FLP

    +
    2. +
  2. +

    Shard the measurement into a sequence of measurement shares

    +
    3. +
  3. +

    Derive the joint randomness from the measurement shares and nonce

    +
    4. +
  4. +

    Generate the proof using the derived joint randomness

    +
    5. +
  5. +

    Shard the proof into a sequence of proof shares

    +
    6. +
+

As described in Section 7.1.2, robustness of Prio3 can be amplified by +generating and verifying multiple proofs. To support this:

+
    +
  • +

    In step 3, derive as much joint randomness as required by PROOFS proofs

    +
    • +
  • +

    Repeat step 4 PROOFS times, each time with a unique joint randomness

    +
    • +
+

Depending on the FLP, joint randomness may not be required. In particular, when +flp.JOINT_RAND_LEN == 0, the Client does not derive the joint randomness +(Step 3).

+

The sharding algorithm is specified below:

+
+
+def shard(
+        self,
+        ctx: bytes,
+        measurement: Measurement,
+        nonce: bytes,
+        rand: bytes) -> tuple[
+            Optional[list[bytes]],
+            list[Prio3InputShare]]:
+    if len(nonce) != self.NONCE_SIZE:
+        raise ValueError("incorrect nonce size")
+    if len(rand) != self.RAND_SIZE:
+        raise ValueError("incorrect size of random bytes argument")
+
+    l = self.xof.SEED_SIZE
+    seeds = [rand[i:i + l] for i in range(0, self.RAND_SIZE, l)]
+
+    meas = self.flp.encode(measurement)
+    if self.flp.JOINT_RAND_LEN > 0:
+        return self.shard_with_joint_rand(ctx, meas, nonce, seeds)
+    else:
+        return self.shard_without_joint_rand(ctx, meas, seeds)
+
+
+

It starts by splitting the randomness into seeds. It then encodes the +measurement as prescribed by the FLP and calls one of two methods, depending on +whether joint randomness is required by the FLP. The methods are defined in the +subsections below.

+
+
+
+7.2.1.1. FLPs Without Joint Randomness +
+

The following method is used for FLPs that do not require joint randomness, +i.e., when flp.JOINT_RAND_LEN == 0. It consists of the following steps:

+
    +
  1. +

    Shard the encoded measurement into shares

    +
    2. +
  2. +

    Generate proofs and shard each into shares

    +
    3. +
  3. +

    Encode each measurement share and shares of each proof into an input share

    +
    4. +
+

Only one pair of measurement and proof(s) share (called the "Leader" shares +above) are vectors of field elements. The other shares (called the "Helper" +shares) are represented instead by an XOF seed, which is expanded into vectors +of field elements. The methods on Prio3 for deriving the prover randomness, +measurement shares, and proof shares are defined in Section 7.2.6.

+
+
+def shard_without_joint_rand(
+        self,
+        ctx: bytes,
+        meas: list[F],
+        seeds: list[bytes]) -> tuple[
+            Optional[list[bytes]],
+            list[Prio3InputShare[F]]]:
+    helper_shares, seeds = front(self.SHARES - 1, seeds)
+    (prove_seed,), seeds = front(1, seeds)
+
+    # Shard the encoded measurement into shares.
+    leader_meas_share = meas
+    for j in range(self.SHARES - 1):
+        leader_meas_share = vec_sub(
+            leader_meas_share,
+            self.helper_meas_share(ctx, j + 1, helper_shares[j]),
+        )
+
+    # Generate and shard each proof into shares.
+    prove_rands = self.prove_rands(ctx, prove_seed)
+    leader_proofs_share = []
+    for _ in range(self.PROOFS):
+        prove_rand, prove_rands = front(
+            self.flp.PROVE_RAND_LEN, prove_rands)
+        leader_proofs_share += self.flp.prove(meas, prove_rand, [])
+    for j in range(self.SHARES - 1):
+        leader_proofs_share = vec_sub(
+            leader_proofs_share,
+            self.helper_proofs_share(
+                ctx,
+                j + 1,
+                helper_shares[j],
+            ),
+        )
+
+    # Each Aggregator's input share contains its measurement share
+    # and its share of the proof(s).
+    input_shares: list[Prio3InputShare[F]] = []
+    input_shares.append((
+        leader_meas_share,
+        leader_proofs_share,
+        None,
+    ))
+    for j in range(self.SHARES - 1):
+        input_shares.append((
+            helper_shares[j],
+            None,
+        ))
+    return (None, input_shares)
+
+
+
+
+
+
+
+7.2.1.2. FLPs With Joint Randomness +
+

The following method is used for FLPs that require joint randomness, i.e., for +which flp.JOINT_RAND_LEN > 0. Joint randomness derivation involves an +additional XOF seed for each Aggregator called the "blind". The computation +involves the following steps:

+
    +
  1. +

    Compute a "joint randomness part" from each measurement share and blind

    +
    2. +
  2. +

    Compute a "joint randomness seed" from the joint randomness parts

    +
    3. +
  3. +

    Compute the joint randomness for each proof evaluation from the joint +randomness seed

    +
    4. +
+

This three-step process is designed to ensure that the joint randomness does +not leak the measurement to the Aggregators while preventing a malicious Client +from tampering with the joint randomness in a way that allows it to break +robustness. To bootstrap the required check, the Client encodes the joint +randomness parts in the public share. (See Section 7.2.2 for details.)

+

All functions used in the following listing are defined in Section 7.2.6:

+
+
+def shard_with_joint_rand(
+        self,
+        ctx: bytes,
+        meas: list[F],
+        nonce: bytes,
+        seeds: list[bytes]) -> tuple[
+            Optional[list[bytes]],
+            list[Prio3InputShare[F]]]:
+    helper_seeds, seeds = front((self.SHARES - 1) * 2, seeds)
+    helper_shares = [
+        helper_seeds[i]
+        for i in range(0, (self.SHARES - 1) * 2, 2)
+    ]
+    helper_blinds = [
+        helper_seeds[i]
+        for i in range(1, (self.SHARES - 1) * 2, 2)
+    ]
+    (leader_blind, prove_seed), seeds = front(2, seeds)
+
+    # Shard the encoded measurement into shares and compute the
+    # joint randomness parts.
+    leader_meas_share = meas
+    joint_rand_parts = []
+    for j in range(self.SHARES - 1):
+        helper_meas_share = self.helper_meas_share(
+            ctx, j + 1, helper_shares[j])
+        leader_meas_share = vec_sub(leader_meas_share,
+                                    helper_meas_share)
+        joint_rand_parts.append(self.joint_rand_part(
+            ctx, j + 1, helper_blinds[j],
+            helper_meas_share, nonce))
+    joint_rand_parts.insert(0, self.joint_rand_part(
+        ctx, 0, leader_blind, leader_meas_share, nonce))
+
+    # Generate each proof and shard it into proof shares.
+    prove_rands = self.prove_rands(ctx, prove_seed)
+    joint_rands = self.joint_rands(
+        ctx, self.joint_rand_seed(ctx, joint_rand_parts))
+    leader_proofs_share = []
+    for _ in range(self.PROOFS):
+        prove_rand, prove_rands = front(
+            self.flp.PROVE_RAND_LEN, prove_rands)
+        joint_rand, joint_rands = front(
+            self.flp.JOINT_RAND_LEN, joint_rands)
+        leader_proofs_share += self.flp.prove(
+            meas,
+            prove_rand,
+            joint_rand,
+        )
+    for j in range(self.SHARES - 1):
+        leader_proofs_share = vec_sub(
+            leader_proofs_share,
+            self.helper_proofs_share(
+                ctx,
+                j + 1,
+                helper_shares[j],
+            ),
+        )
+
+    # Each Aggregator's input share contains its measurement share,
+    # share of proof(s), and blind. The public share contains the
+    # Aggregators' joint randomness parts.
+    input_shares: list[Prio3InputShare[F]] = []
+    input_shares.append((
+        leader_meas_share,
+        leader_proofs_share,
+        leader_blind,
+    ))
+    for j in range(self.SHARES - 1):
+        input_shares.append((
+            helper_shares[j],
+            helper_blinds[j],
+        ))
+    return (joint_rand_parts, input_shares)
+
+
+
+
+
+
+
+
+

+7.2.2. Preparation +

+

This section describes the process of recovering output shares from the input +shares. The high-level idea is that each Aggregator first queries its +measurement share and proof(s) share(s) locally, then broadcasts its verifier +share(s) in its prep share. The shares of verifier(s) are then combined into +the verifier message(s) used to decide whether to accept.

+

In addition, the Aggregators must recompute the same joint randomness used by +the Client to generate the proof(s). In order to avoid an extra round of +communication, the Client includes the joint randomness parts in the public +share. This leaves open the possibility that the Client cheated by, say, +forcing the Aggregators to use joint randomness that biases the proof check +procedure some way in its favor. To mitigate this, the Aggregators also check +that they have all computed the same joint randomness seed before accepting +their output shares. To do so, they exchange their parts of the joint +randomness along with their shares of verifier(s).

+

Implementation note: the prep state for Prio3 includes the output share that +will be released once preparation is complete. In some situations, it may be +necessary for the Aggregator to encode this state as bytes and store it for +retrieval later on. For all but the first Aggregator, it is possible to save +storage by storing the measurement share rather than output share itself. It is +relatively inexpensive to expand this seed into the measurement share, then +truncate the measurement share to get the output share.

+

All functions used in the following listing are defined in Section 7.2.6:

+
+
+def prep_init(
+        self,
+        verify_key: bytes,
+        ctx: bytes,
+        agg_id: int,
+        _agg_param: None,
+        nonce: bytes,
+        public_share: Optional[list[bytes]],
+        input_share: Prio3InputShare[F]) -> tuple[
+            Prio3PrepState[F],
+            Prio3PrepShare[F]]:
+    joint_rand_parts = public_share
+    (meas_share, proofs_share, blind) = \
+        self.expand_input_share(ctx, agg_id, input_share)
+    out_share = self.flp.truncate(meas_share)
+
+    # Compute the joint randomness.
+    joint_rand: list[F] = []
+    corrected_joint_rand_seed, joint_rand_part = None, None
+    if self.flp.JOINT_RAND_LEN > 0:
+        assert blind is not None
+        assert joint_rand_parts is not None
+        joint_rand_part = self.joint_rand_part(
+            ctx, agg_id, blind, meas_share, nonce)
+        joint_rand_parts[agg_id] = joint_rand_part
+        corrected_joint_rand_seed = self.joint_rand_seed(
+            ctx, joint_rand_parts)
+        joint_rands = self.joint_rands(
+            ctx, corrected_joint_rand_seed)
+
+    # Query the measurement and proof share.
+    query_rands = self.query_rands(verify_key, ctx, nonce)
+    verifiers_share = []
+    for _ in range(self.PROOFS):
+        proof_share, proofs_share = front(
+            self.flp.PROOF_LEN, proofs_share)
+        query_rand, query_rands = front(
+            self.flp.QUERY_RAND_LEN, query_rands)
+        if self.flp.JOINT_RAND_LEN > 0:
+            joint_rand, joint_rands = front(
+                self.flp.JOINT_RAND_LEN, joint_rands)
+        verifiers_share += self.flp.query(
+            meas_share,
+            proof_share,
+            query_rand,
+            joint_rand,
+            self.SHARES,
+        )
+
+    prep_state = (out_share, corrected_joint_rand_seed)
+    prep_share = (verifiers_share, joint_rand_part)
+    return (prep_state, prep_share)
+
+def prep_shares_to_prep(
+        self,
+        ctx: bytes,
+        _agg_param: None,
+        prep_shares: list[Prio3PrepShare[F]]) -> Optional[bytes]:
+    # Unshard each set of verifier shares into each verifier message.
+    verifiers = self.flp.field.zeros(
+        self.flp.VERIFIER_LEN * self.PROOFS)
+    joint_rand_parts = []
+    for (verifiers_share, joint_rand_part) in prep_shares:
+        verifiers = vec_add(verifiers, verifiers_share)
+        if self.flp.JOINT_RAND_LEN > 0:
+            assert joint_rand_part is not None
+            joint_rand_parts.append(joint_rand_part)
+
+    # Verify that each proof is well-formed and input is valid.
+    for _ in range(self.PROOFS):
+        verifier, verifiers = front(self.flp.VERIFIER_LEN, verifiers)
+        if not self.flp.decide(verifier):
+            raise ValueError('proof verifier check failed')
+
+    # Combine the joint randomness parts computed by the
+    # Aggregators into the true joint randomness seed. This is
+    # used in the last step.
+    joint_rand_seed = None
+    if self.flp.JOINT_RAND_LEN > 0:
+        joint_rand_seed = self.joint_rand_seed(ctx, joint_rand_parts)
+    return joint_rand_seed
+
+def prep_next(
+    self,
+    _ctx: bytes,
+    prep_state: Prio3PrepState[F],
+    prep_msg: Optional[bytes]
+) -> tuple[Prio3PrepState[F], Prio3PrepShare[F]] | list[F]:
+    joint_rand_seed = prep_msg
+    (out_share, corrected_joint_rand_seed) = prep_state
+
+    # If joint randomness was used, check that the value computed by
+    # the Aggregators matches the value indicated by the Client.
+    if joint_rand_seed != corrected_joint_rand_seed:
+        raise ValueError('joint randomness check failed')
+
+    return out_share
+
+
+
+
+
+
+

+7.2.3. Validity of Aggregation Parameters +

+

Prio3 only permits a report to be aggregated once.

+
+
+def is_valid(
+        self,
+        _agg_param: None,
+        previous_agg_params: list[None]) -> bool:
+    return len(previous_agg_params) == 0
+
+
+
+
+
+
+

+7.2.4. Aggregation +

+

Aggregating a set of output shares is simply a matter of adding up the vectors +element-wise.

+
+
+def agg_init(self, _agg_param: None) -> list[F]:
+    return self.flp.field.zeros(self.flp.OUTPUT_LEN)
+
+def agg_update(self,
+               _agg_param: None,
+               agg_share: list[F],
+               out_share: list[F]) -> list[F]:
+    return vec_add(agg_share, out_share)
+
+def merge(self,
+          _agg_param: None,
+          agg_shares: list[list[F]]) -> list[F]:
+    agg = self.agg_init(None)
+    for agg_share in agg_shares:
+        agg = vec_add(agg, agg_share)
+    return agg
+
+
+
+
+
+
+

+7.2.5. Unsharding +

+

To unshard a set of aggregate shares, the Collector first adds up the vectors +element-wise, then decodes the aggregate result from the sum according to the +FLP (Section 7.1.1).

+
+
+def unshard(
+        self,
+        _agg_param: None,
+        agg_shares: list[list[F]],
+        num_measurements: int) -> AggResult:
+    agg = self.merge(None, agg_shares)
+    return self.flp.decode(agg, num_measurements)
+
+
+
+
+
+
+

+7.2.6. Auxiliary Functions +

+

This section defines a number of auxiliary functions referenced by the main +algorithms for Prio3 in the preceding sections.

+
+
+def helper_meas_share(
+        self,
+        ctx: bytes,
+        agg_id: int,
+        share: bytes) -> list[F]:
+    return self.xof.expand_into_vec(
+        self.flp.field,
+        share,
+        self.domain_separation_tag(USAGE_MEAS_SHARE, ctx),
+        byte(agg_id),
+        self.flp.MEAS_LEN,
+    )
+
+def helper_proofs_share(
+        self,
+        ctx: bytes,
+        agg_id: int,
+        share: bytes) -> list[F]:
+    return self.xof.expand_into_vec(
+        self.flp.field,
+        share,
+        self.domain_separation_tag(USAGE_PROOF_SHARE, ctx),
+        byte(self.PROOFS) + byte(agg_id),
+        self.flp.PROOF_LEN * self.PROOFS,
+    )
+
+def expand_input_share(
+        self,
+        ctx: bytes,
+        agg_id: int,
+        input_share: Prio3InputShare[F]) -> tuple[
+            list[F],
+            list[F],
+            Optional[bytes]]:
+    if agg_id > 0:
+        assert len(input_share) == 2
+        (share, blind) = input_share
+        meas_share = self.helper_meas_share(ctx, agg_id, share)
+        proofs_share = self.helper_proofs_share(ctx, agg_id, share)
+    else:
+        assert len(input_share) == 3
+        (meas_share, proofs_share, blind) = input_share
+    return (meas_share, proofs_share, blind)
+
+def prove_rands(self, ctx: bytes, prove_seed: bytes) -> list[F]:
+    return self.xof.expand_into_vec(
+        self.flp.field,
+        prove_seed,
+        self.domain_separation_tag(USAGE_PROVE_RANDOMNESS, ctx),
+        byte(self.PROOFS),
+        self.flp.PROVE_RAND_LEN * self.PROOFS,
+    )
+
+def query_rands(
+        self,
+        verify_key: bytes,
+        ctx: bytes,
+        nonce: bytes) -> list[F]:
+    return self.xof.expand_into_vec(
+        self.flp.field,
+        verify_key,
+        self.domain_separation_tag(USAGE_QUERY_RANDOMNESS, ctx),
+        byte(self.PROOFS) + nonce,
+        self.flp.QUERY_RAND_LEN * self.PROOFS,
+    )
+
+def joint_rand_part(
+        self,
+        ctx: bytes,
+        agg_id: int,
+        blind: bytes,
+        meas_share: list[F],
+        nonce: bytes) -> bytes:
+    return self.xof.derive_seed(
+        blind,
+        self.domain_separation_tag(USAGE_JOINT_RAND_PART, ctx),
+        byte(agg_id) + nonce + self.flp.field.encode_vec(meas_share),
+    )
+
+def joint_rand_seed(self,
+                    ctx: bytes,
+                    joint_rand_parts: list[bytes]) -> bytes:
+    """Derive the joint randomness seed from its parts."""
+    return self.xof.derive_seed(
+        zeros(self.xof.SEED_SIZE),
+        self.domain_separation_tag(USAGE_JOINT_RAND_SEED, ctx),
+        concat(joint_rand_parts),
+    )
+
+def joint_rands(self,
+                ctx: bytes,
+                joint_rand_seed: bytes) -> list[F]:
+    """Derive the joint randomness from its seed."""
+    return self.xof.expand_into_vec(
+        self.flp.field,
+        joint_rand_seed,
+        self.domain_separation_tag(USAGE_JOINT_RANDOMNESS, ctx),
+        byte(self.PROOFS),
+        self.flp.JOINT_RAND_LEN * self.PROOFS,
+    )
+
+
+
+
+
+
+

+7.2.7. Message Serialization +

+

This section defines serialization formats for messages exchanged over the +network while executing Prio3. Messages are defined in the presentation +language of TLS as defined in Section 3 of [RFC8446].

+

Let prio3 denote an instance of Prio3. In the remainder we use S as an +alias for prio3.xof.SEED_SIZE and F as an alias for +prio3.field.ENCODED_SIZE. XOF seeds are represented as follows:

+
+
+opaque Prio3Seed[S];
+
+
+

Field elements are encoded in little-endian byte order (as defined in +Section 6.1) and represented as follows:

+
+
+opaque Prio3Field[F];
+
+
+
+
+
+7.2.7.1. Public Share +
+

The content of the public share depends on whether joint randomness is +required for the underlying FLP (i.e., prio3.flp.JOINT_RAND_LEN > 0). If +joint randomness is not used, then the public share is the empty string. +Otherwise, if joint randomness is used, then the public share encodes the joint +randomness parts as follows:

+
+
+struct {
+    Prio3Seed joint_rand_parts[S * prio3.SHARES];
+} Prio3PublicShareWithJointRand;
+
+
+
+
+
+
+
+7.2.7.2. Input Share +
+

Just as for the public share, the content of the input shares depends on +whether joint randomness is used. If so, then each input share includes the +Aggregator's blind for generating its joint randomness part.

+

In addition, the encoding of the input shares depends on which aggregator is +receiving the message. If the aggregator ID is 0, then the input share +includes the full measurement share and proofs(s) share(s). Otherwise, if the +aggregator ID is greater than 0, then the measurement and shares of proof(s) +are represented by an XOF seed. We shall call the former the "Leader" and the +latter the "Helpers".

+

In total there are four variants of the input share. When joint randomness is +not used, the Leader's share is structured as follows:

+
+
+struct {
+    Prio3Field meas_share[F * prio3.flp.MEAS_LEN];
+    Prio3Field proofs_share[F * prio3.flp.PROOF_LEN * prio3.PROOFS];
+} Prio3LeaderShare;
+
+
+

When joint randomness is not used, the Helpers' shares are structured +as follows:

+
+
+struct {
+    Prio3Seed share;
+} Prio3HelperShare;
+
+
+

When joint randomness is used, the Leader's input share is structured as +follows:

+
+
+struct {
+    Prio3LeaderShare inner;
+    Prio3Seed blind;
+} Prio3LeaderShareWithJointRand;
+
+
+

Finally, when joint randomness is used, the Helpers' shares are structured as +follows:

+
+
+struct {
+    Prio3HelperShare inner;
+    Prio3Seed blind;
+} Prio3HelperShareWithJointRand;
+
+
+
+
+
+
+
+7.2.7.3. Prep Share +
+

When joint randomness is not used, the prep share is structured as follows:

+
+
+struct {
+    Prio3Field verifiers_share[F * V];
+} Prio3PrepShare;
+
+
+

where V = prio3.flp.VERIFIER_LEN * prio3.PROOFS. When joint randomness is +used, the prep share includes the Aggregator's joint randomness part and is +structured as follows:

+
+
+struct {
+    Prio3Field verifiers_share[F * V];
+    Prio3Seed joint_rand_part;
+} Prio3PrepShareWithJointRand;
+
+
+
+
+
+
+
+7.2.7.4. Prep Message +
+

When joint randomness is not used, the prep message is the empty string. +Otherwise the prep message consists of the joint randomness seed computed by +the Aggregators:

+
+
+struct {
+    Prio3Seed joint_rand;
+} Prio3PrepMessageWithJointRand;
+
+
+
+
+
+
+
+7.2.7.5. Aggregation +
+

Aggregate shares are structured as follows:

+
+
+struct {
+    Prio3Field agg_share[F * prio3.flp.OUTPUT_LEN];
+} Prio3AggShare;
+
+
+
+
+
+
+
+
+
+
+

+7.3. FLP Specification +

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 7: +FLP parameters for a validity circuit. +
ParameterValue
+ valid +An instance of Valid (Section 7.3.2).
+ field + + valid.field +
+ PROVE_RAND_LEN + + valid.prove_rand_len() +
+ QUERY_RAND_LEN + + valid.query_rand_len() +
+ JOINT_RAND_LEN + + valid.JOINT_RAND_LEN +
+ MEAS_LEN + + valid.MEAS_LEN +
+ OUTPUT_LEN + + valid.OUTPUT_LEN +
+ PROOF_LEN + + valid.proof_len() +
+ VERIFIER_LEN + + valid.verifier_len() +
+ Measurement +As defined by valid.
+ AggResult +As defined by valid.
+
+

This section specifies an implementation of the Flp interface (Section 7.1) based +on the construction from [BBCGGI19], Section 4.2. The types and parameters +required by this interface are listed in the table above.

+

We begin in Section 7.3.1 with an overview of the proof system and +some extensions to it. Section 7.3.2 defines validity circuits, the +core component of the proof system that determines measurement validity and how +measurements are aggregated. The proof-generation algorithm, query algorithm, +and decision algorithm are defined in Section 7.3.3, +Section 7.3.4, and Section 7.3.5 +respectively.

+
+
+

+7.3.1. Overview +

+

An FLP is a type of "zero-knowledge proof". A conventional zero-knowledge proof +system involves two parties:

+
    +
  • +

    The prover, who holds a measurement and generates a proof of the +measurement's validity

    +
    • +
  • +

    The verifier who holds an encryption of, or commitment to, the measurement +and checks the proof

    +
    • +
+

Our proof system is much the same, except the verifier is split across multiple +Aggregators, each of which has a secret share of the measurement rather than an +encryption of it.

+

Validity is defined in terms of an arithmetic circuit evaluated over the +measurement. The inputs to this circuit are elements of a finite field that +comprise the encoded measurement; the gates of the circuit are multiplication, +addition, and subtraction operations; and the output of the circuit is a single +field element. If the value is zero, then the measurement is deemed valid; +otherwise, if the output is non-zero, then the measurement is deemed invalid.

+

For example, the simplest circuit specified in this document is the following +(Section 7.4.1):

+
+
+C(x) = x * (x-1)
+
+
+

This circuit contains one subtraction gate (x-1) and one multiplication gate +(x * (x-1)). Observe that C(x) = 0 if and only if x is in [0, 2).

+

The goal of the proof system is to allow each Aggregator to privately and +correctly compute a share of C(x) from its share of x. Then all they need +to determine validity is to broadcast their shares of C(x).

+

Suppose for a moment that C is an affine arithmetic circuit, meaning its only +operations are addition, subtraction, and multiplication-by-constant. (The +circuit above is non-affine because it contains a multiplication gate with two +non-constant inputs.) Then each Aggregator can compute its share locally, since

+
+
+C(x_shares[0] + ... + x_shares[SHARES-1]) =
+    C(x_shares[0]) + ... + C(x_shares[SHARES-1])
+
+
+

(Note that, for this equality to hold, it is necessary to scale any addition of +a constant in the circuit by 1/SHARES.) However, this is not the case if C +contains multiplication gates with two non-constant inputs. Thus our goal is to +transform these multiplication gates into computations on secret shared data +that each Aggregator can perform locally.

+

The key idea is to have the prover construct a polynomial p such that p(j) +is equal to the output of the j-th multiplication gate. Polynomial evaluation +is fully linear, which means the coefficients of the polynomial can be secret +shared in a way that allows each Aggregator to compute a share of p(j) for +any j. These intermediate results can then be combined with the affine +arithmetic operations of the validity circuit to produce the final output.

+

Applying this idea to the example circuit C above:

+
    +
  1. +

    The Client, given its measurement x, constructs the lowest degree +polynomial p for which p(0) = s and p(1) = x * (x-1), where s is a +random blinding value generated by the Client. (The blinding value is to +protect the privacy of the measurement.) It then sends shares of x and +shares of the coefficients of p to each of the Aggregators.

    +
    2. +
  2. +

    Each Aggregator locally computes and broadcasts its share of p(1), which +is equal to its share of C(x).

    +
    3. +
+

In fact, our FLP is slightly more general than this. We can replace the +multiplication gate with any non-affine sub-circuit and apply the same idea. +For example, in Section 7.4.2, the validity circuit uses the following +sub-circuit multiple times:

+
+
+Range2(x) = x * (x-1) = x^2 - x
+
+
+

(This is the same functionality computed by the example circuit C above.) +Here again we can interpolate the lowest degree polynomial p for which p(j) +is the value of the j-th call to Range2 in the validity circuit. Each +validity circuit defines a sub-circuit that encapsulates its non-affine +arithmetic operations. We refer to this sub-circuit as the "gadget".

+

Finally, the proof system has one more important component. It is possible for +a malicious Client to produce a gadget polynomial p that would result in +C(x) being computed incorrectly, potentially resulting in an invalid +measurement being accepted. To prevent this, the Aggregators perform a +probabilistic test to check that the gadget polynomial was constructed +properly. This "gadget test", and the procedure for constructing the +polynomial, are described in detail in Section 7.3.3.

+
+
+
+7.3.1.1. Extensions +
+

The FLP described in Section 7.3 extends the proof system of +[BBCGGI19], Section 4.2 in a few ways.

+

First, the validity circuit in our construction includes an additional, random +input (this is the "joint randomness" derived from the measurement shares in +Prio3; see Section 7.2). This allows for circuit optimizations that +trade a small soundness error for a shorter proof. For example, consider a +circuit that recognizes the set of length-N vectors for which each element is +either one or zero. A deterministic circuit could be constructed for this +language, but it would involve a large number of multiplications that would +result in a large proof. (See the discussion in [BBCGGI19], Section 5.2 for +details). A much shorter proof can be constructed for the following randomized +circuit:

+
+
+C(x, r) = r * Range2(x[0]) + ... + r^N * Range2(x[N-1])
+
+
+

(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here x is +the length-N input and r is a random field element. The gadget circuit +Range2 is the "range-check" polynomial described above, i.e., Range2(x) = +x^2 - x. The idea is that, if x is valid, i.e., each x[j] is in +[0, 2), then the circuit will evaluate to zero regardless of the value of +r; but if some x[j] is not in [0, 2), then the output will be non-zero +with high probability.

+

The second extension implemented by our FLP allows the validity circuit to +contain multiple gadget types. (This generalization was suggested in +[BBCGGI19], Remark 4.5.) This provides additional flexibility for designing +circuits by allowing multiple, non-affine sub-circuits. For example, the +following circuit is allowed:

+
+
+C(x, r) = r * Range2(x[0]) + ... + r^L * Range2(x[L-1]) + \
+            r^(L+1) * Range3(x[L]) + ... + r^N * Range3(x[N-1])
+
+
+

where Range3(x) = x^3 - 3x^2 + 2x. This circuit checks that the first L +inputs are in [0, 2) and the last N-L inputs are in [0, 3). The same +circuit can be expressed using a simpler gadget, namely multiplication, but the +resulting proof would be longer.

+

Third, rather than interpolate the gadget polynomial at inputs 1, 2, ..., +j, ..., where j is the j-th invocation of the gadget, we use roots of +unity for the field. This allows us to construct each gadget polynomial via the +number theoretic transform [SML24], which is far more efficient than generic +formulas. Note that the roots of unity are powers of the generator for the +NTT-friendly field (see Section 6.1.2).

+

Finally, the validity circuit in our FLP may have any number of outputs (at +least one). The input is said to be valid if each of the outputs is zero. To +save bandwidth, we take a random linear combination of the outputs. If each of +the outputs is zero, then the reduced output will be zero; but if one of the +outputs is non-zero, then the reduced output will be non-zero with high +probability.

+
+
+
+
+
+
+

+7.3.2. Validity Circuits +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 8: +Validity circuit parameters. +
ParameterDescription
+ GADGETS: list[Gadget] +A list of gadgets.
+ GADGET_CALLS: list[int] +Number of times each gadget is called.
+ MEAS_LEN: int +Length of the measurement.
+ JOINT_RAND_LEN: int +Length of the joint randomness.
+ EVAL_OUTPUT_LEN: int +Length of the circuit output.
+ OUTPUT_LEN: int +Length of the aggregatable output.
+ Measurement +Type of the measurement.
+ AggResult +Type of the aggregate result.
+ field: type[F] +Class object for the field (Section 6.1.2)
+

An instance of the proof system is defined in terms of a validity circuit that +implements the Valid interface specified in this section. The parameters are +listed in the table above.

+

Each circuit has a list of gadgets, denoted GADGETS. The circuit evaluated by +the gadget MAY be non-affine, but MUST be arithmetic, i.e., composed only of +multiplication, addition, and subtraction gates. An instance of class Gadget +has the following interface:

+
    +
  • +

    ARITY: int is the number of input wires. For example, the multiplication +gadget Mul(x,y) = x*y has arity of 2.

    +
    • +
  • +

    DEGREE: int is the arithmetic degree of the gadget circuit. This is defined +to be the degree of the polynomial that computes it. This exists +because the circuit is arithmetic. For example, Mul has degree 2.

    +
    • +
  • +

    gadget.eval(field: type[F], inp: list[F]) -> F evaluates the gadget over +the given inputs and field.

    +
    • +
  • +

    gadget.eval_poly(field: type[F], inp_poly: list[list[F]]) -> list[F] is the +same as eval() except it evaluates the circuit over the polynomial ring of +the field. This is well defined because the circuit is arithmetic.

    +
    • +
+

In addition to the list of gadgets, the validity circuit specifies how many +times each gadget is called (GADGET_CALLS). It also specifies the length of +the circuit's input (MEAS_LEN), the length of the joint randomness +(JOINT_RAND_LEN), and the length of the circuit's output (EVAL_OUTPUT_LEN).

+

A validity circuit also specifies parameters and methods needed for Prio3 +aggregation. These are used to implement the interface in Section 7.1.1:

+
    +
  • +

    valid.encode(measurement: Measurement) -> list[F] returns a vector of +length MEAS_LEN representing a measurement of type Measurement.

    +
    • +
  • +

    valid.truncate(meas: list[F]) -> list[F] returns a vector of length +OUTPUT_LEN representing (a share of) an aggregatable output.

    +
    • +
  • +

    valid.decode(agg: list[F], num_measurements: int) -> AggResult returns an +aggregate result of type AggResult. This computation may depend on the +number of outputs aggregated.

    +
    • +
+

Finally, the following are helper methods used to instantiate parameters of the +Flp interface (Section 7.1):

+
+
+def prove_rand_len(self) -> int:
+    """Length of the prover randomness."""
+    return sum(g.ARITY for g in self.GADGETS)
+
+def query_rand_len(self) -> int:
+    """Length of the query randomness."""
+    query_rand_len = len(self.GADGETS)
+    if self.EVAL_OUTPUT_LEN > 1:
+        query_rand_len += self.EVAL_OUTPUT_LEN
+    return query_rand_len
+
+def proof_len(self) -> int:
+    """Length of the proof."""
+    length = 0
+    for (g, g_calls) in zip(self.GADGETS, self.GADGET_CALLS):
+        p = next_power_of_2(1 + g_calls)
+        length += g.ARITY + g.DEGREE * (p - 1) + 1
+    return length
+
+def verifier_len(self) -> int:
+    """Length of the verifier message."""
+    length = 1
+    for g in self.GADGETS:
+        length += g.ARITY + 1
+    return length
+
+
+
+
+
+
+

+7.3.3. Generating the Proof +

+
+
+
++------------------+
+| Prove            |
+| +-------------+  |
+| | Valid       |<---- meas
+| | +--------+  |<---- joint rand
+| | | Gadget |  |  |<- prove rand
+| | +--------+  |  |
+| +-------------+  |
++------------------+
+ |
+ V proof
+
+
+
Figure 8: +Components of the proof generation algorithm. +
+

The proof generation algorithm invokes the validity circuit on the encoded +measurement and joint randomness. The validity circuit in turn invokes the +gadgets defined by the circuit. The prove randomness is used to construct the +gadget polynomials that the verifier will use to compute the outputs of each +gadget.

+

To generate the gadget polynomials, the prover evaluates the validity circuit, +setting g.wires[j][k] to the value of the j-th wire for the k-th +invocation of gadget g. This is accomplished by "wrapping" each gadget in a +class ProveGadget that records the wire inputs. We list this class in +Appendix A.4.

+

Next, we compute each of the "wire polynomials" for each gadget. The j-th +wire polynomial is the lowest degree polynomial that evaluates to +g.wire[j][k] at a sequence of fixed points. We obtain the gadget polynomial by +evaluating the gadget on the wire polynomials.

+
+
+def prove(self,
+          meas: list[F],
+          prove_rand: list[F],
+          joint_rand: list[F]) -> list[F]:
+    # Evaluate the validity circuit, recording the value of each
+    # input wire for each evaluation of each gadget.
+    valid = ProveGadget.wrap(self.valid, prove_rand)
+    valid.eval(meas, joint_rand, 1)
+
+    # Construct the proof, which consists of the wire seeds and
+    # gadget polynomial for each gadget.
+    proof = []
+    for g in cast(list[ProveGadget[F]], valid.GADGETS):
+        p = len(g.wires[0])
+
+        # Compute the wire polynomials for this gadget. For each `j`,
+        # find the lowest degree polynomial `wire_poly` for which
+        # `wire_poly(alpha^k) = g.wires[j][k]` for all `k`. Note that
+        # each `g.wires[j][0]` is set to the seed of wire `j`, which
+        # is included in the prove randomness.
+        #
+        # Implementation note: `alpha` is a root of unity, which
+        # means `poly_interp()` can be evaluated using the NTT. Note
+        # that `g.wires[j]` is padded with 0s to a power of 2.
+        alpha = self.field.gen() ** (self.field.GEN_ORDER // p)
+        wire_inp = [alpha ** k for k in range(p)]
+        wire_polys = []
+        for j in range(g.ARITY):
+            wire_poly = poly_interp(self.field, wire_inp, g.wires[j])
+            wire_polys.append(wire_poly)
+
+        # Compute the gadget polynomial by evaluating the gadget on
+        # the wire polynomials. By construction we have that
+        # `gadget_poly(alpha^k)` is the `k`-th output.
+        gadget_poly = g.eval_poly(self.field, wire_polys)
+
+        for j in range(g.ARITY):
+            proof.append(g.wires[j][0])
+        proof += gadget_poly
+
+    return proof
+
+
+
+
+
+
+

+7.3.4. Querying the Proof +

+
+
+
+      proof (share)
++---------|--------+
+| Query   |        |
+| +-------|-----+  |
+| | Valid |     |<---- meas (share)
+| | +-----V--+  |<---- joint rand
+| | | Gadget |  |  |<- query rand
+| | +--------+  |  |
+| +-------------+  |
++------------------+
+ |
+ V verifier (share)
+
+
+
Figure 9: +Components of the query algorithm. +
+

The query algorithm invokes the validity circuit on the encoded measurement and +joint randomness. It evaluates the gadget polynomials encoded by the proof +(share) to produce (a share of) each gadget output. The verifier (share) +consists of (a share of) the validity circuit's output and (a share of) each +gadget test. The gadget tests consume the query randomness.

+

The goal of each gadget test is to ensure the inputs used by the prover to +generate the gadget polynomial matches the inputs we used to evaluate it. We do +this by partially reconstructing the gadget polynomial and evaluating it at a random +point: when we evaluate the gadget polynomial at the same point, we expect to +get the same result.

+

To start a gadget test, we first construct the (shares of the) wire polynomials +just as the prover did. First, we record g.wires[j][k] as the input (share) +of the j-th wire of the k-th invocation of the gadget. Again, this is +accomplished by a wrapper gadget, QueryGadget, listed in Appendix A.4. +This gadget also evaluates the gadget polynomial for each gadget invocation in +order to produce the gadget's output. Then we compute the wire polynomials from +the recorded values.

+

Next, we choose a random point t (parsed from the query randomness), evaluate +each wire polynomial at t, and evaluate the gadget polynomial at t. The +results are recorded in the verifier message passed to the decision algorithm, +where we finish the test.

+

The random point t MUST NOT be one of the fixed evaluation points used to +interpolate the wire polynomials. Otherwise, the verifier message may partially +leak the encoded measurement.

+
+
+def query(self,
+          meas: list[F],
+          proof: list[F],
+          query_rand: list[F],
+          joint_rand: list[F],
+          num_shares: int) -> list[F]:
+    # Evaluate the validity circuit, recording the value of each
+    # input wire for each evaluation of each gadget. Use the gadget
+    # polynomials encoded by `proof` to compute the gadget outputs.
+    valid = QueryGadget.wrap(self.valid, proof)
+    out = valid.eval(meas, joint_rand, num_shares)
+
+    # Reduce the output.
+    if self.valid.EVAL_OUTPUT_LEN > 1:
+        (rand, query_rand) = front(
+            self.valid.EVAL_OUTPUT_LEN,
+            query_rand,
+        )
+        v = self.field(0)
+        for (r, out_elem) in zip(rand, out):
+            v += r * out_elem
+    else:
+        [v] = out
+
+    # Construct the verifier message, which consists of the reduced
+    # circuit output and each gadget test.
+    verifier = [v]
+    for (g, t) in zip(cast(list[QueryGadget[F]], valid.GADGETS),
+                      query_rand):
+        p = len(g.wires[0])
+
+        # Abort if `t` is one of the inputs used to compute the wire
+        # polynomials so that the verifier message doesn't leak the
+        # gadget output. It suffices to check if `t` is a root of
+        # unity, which implies it is a power of `alpha`.
+        if t ** p == self.field(1):
+            raise ValueError('test point is a root of unity')
+
+        # To test the gadget, we re-compute the wire polynomials and
+        # check for consistency with the gadget polynomial provided
+        # by the prover. To start, evaluate the gadget polynomial and
+        # each of the wire polynomials at the random point `t`.
+        wire_checks = []
+        wire_inp = [g.alpha ** k for k in range(p)]
+        for j in range(g.ARITY):
+            wire_poly = poly_interp(self.field, wire_inp, g.wires[j])
+            wire_checks.append(poly_eval(self.field, wire_poly, t))
+
+        gadget_check = poly_eval(self.field, g.poly, t)
+
+        verifier += wire_checks
+        verifier.append(gadget_check)
+
+    return verifier
+
+
+
+
+
+
+

+7.3.5. Deciding Validity +

+
+
+
+ | verifier
+ V
++------------------+
+| Decide           |
++------------------+
+ |
+ V is_valid
+
+
+
Figure 10: +Components of the decision algorithm. +
+

The decision algorithm consumes the verifier message. (Each of the Aggregators +computes an additive share of the verifier message after the previous step.) The +verifier message consists of the reduced circuit output and the gadget tests.

+

To finish each gadget test, we evaluate the gadget on the wire checks: if the +encoded measurement and joint randomness used to generate the proof are the +same as the measurement (share) and joint randomness used to verify the proof, +then the output of the gadget will be equal to the gadget check; otherwise, the +output will not equal the gadget check with high probability.

+
+
+def decide(self, verifier: list[F]) -> bool:
+    # Check the output of the validity circuit.
+    ([v], verifier) = front(1, verifier)
+    if v != self.field(0):
+        return False
+
+    # Complete each gadget test.
+    for g in self.valid.GADGETS:
+        (wire_checks, verifier) = front(g.ARITY, verifier)
+        ([gadget_check], verifier) = front(1, verifier)
+        if g.eval(self.field, wire_checks) != gadget_check:
+            return False
+
+    return True
+
+
+
+
+
+
+
+
+

+7.4. Variants +

+

This section specifies instantiations of Prio3 for various aggregation tasks. +Each variant is determined by a field (Section 6.1), a validity circuit +(Section 7.3.2),and the number of proofs to generate and verify. All +gadgets are listed in Appendix A. Test vectors for each can be found in +Appendix C.

+
+
+

+7.4.1. Prio3Count +

+ + + + + + + + + + + + + + + + + + + + + + +
+Table 9: +Parameters for Prio3Count. +
ParameterValue
+ field + + Field64 (Table 3)
+ Valid + + Count(field) (this section)
+ PROOFS + + 1 +
+

Our first variant of Prio3 is for a simple counter: each measurement is either +one or zero and the aggregate result is the sum of the measurements. Its +validity circuit uses the multiplication gadget Mul specified in +Appendix A.1, which takes two inputs and multiplies them. The circuit is +specified below:

+
+
+class Count(Valid[int, int, F]):
+    GADGETS: list[Gadget[F]] = [Mul()]
+    GADGET_CALLS = [1]
+    MEAS_LEN = 1
+    JOINT_RAND_LEN = 0
+    OUTPUT_LEN = 1
+    EVAL_OUTPUT_LEN = 1
+
+    # Class object for the field.
+    field: type[F]
+
+    def __init__(self, field: type[F]):
+        self.field = field
+
+    def encode(self, measurement: int) -> list[F]:
+        return [self.field(measurement)]
+
+    def eval(
+            self,
+            meas: list[F],
+            joint_rand: list[F],
+            _num_shares: int) -> list[F]:
+        squared = self.GADGETS[0].eval(self.field,
+                                       [meas[0], meas[0]])
+        return [squared - meas[0]]
+
+    def truncate(self, meas: list[F]) -> list[F]:
+        return meas
+
+    def decode(self, output: list[F], _num_measurements: int) -> int:
+        return output[0].int()
+
+
+
+
+
+
+

+7.4.2. Prio3Sum +

+ + + + + + + + + + + + + + + + + + + + + + +
+Table 10: +Parameters for Prio3Sum. +
ParameterValue
+ field + + Field64 (Table 3)
+ Valid + + Sum(field, max_measurement) (this section)
+ PROOFS + + 1 +
+

The next variant of Prio3 supports summing of integers in a pre-determined +range. Each measurement is an integer in the range [0, max_measurement], +where max_measurement defines the largest valid measurement.

+

The range check is accomplished by encoding the measurement as a bit vector, +encoding the measurement plus an offset as a bit vector, then checking that the +two encoded integers are consistent. Let

+
    +
  • +

    bits = max_measurement.bit_length(), the number of bits needed to encode +the largest valid measurement

    +
    • +
  • +

    offset = 2^bits - 1 - max_measurement

    +
    • +
+

The first bit-encoded integer is the measurement itself. Note that only +measurements between 0 and 2^bits - 1 can be encoded this way with as many +bits. The second bit-encoded integer is the sum of the measurement and +offset. Observe that this sum can only be encoded this way if it is between +0 and 2^bits - 1, which implies that the measurement is between -offset +and max_measurement.

+

The circuit first checks that each entry of both bit vectors is a one or a +zero. It then decodes both the measurement and the offset measurement, and +subtracts the offset from the latter. It then checks if these two values are +equal. Since both the measurement and the measurement plus offset are in the +same range of [0, 2^bits), this means that the measurement itself is between +0 and max_measurement.

+

The circuit uses the polynomial-evaluation gadget PolyEval specified in +Appendix A.2. The polynomial is p(x) = x^2 - x, which is equal to 0 +if and only if x is in [0, 2). The complete circuit is specified below:

+
+
+class Sum(Valid[int, int, F]):
+    GADGETS: list[Gadget[F]] = [PolyEval([0, -1, 1])]
+    JOINT_RAND_LEN = 0
+    OUTPUT_LEN = 1
+    field: type[F]
+
+    def __init__(self, field: type[F], max_measurement: int):
+        self.field = field
+        self.bits = max_measurement.bit_length()
+        self.offset = self.field(2**self.bits - 1 - max_measurement)
+        self.max_measurement = max_measurement
+        self.GADGET_CALLS = [2 * self.bits]
+        self.MEAS_LEN = 2 * self.bits
+        self.EVAL_OUTPUT_LEN = 2 * self.bits + 1
+
+    def encode(self, measurement: int) -> list[F]:
+        encoded = []
+        encoded += self.field.encode_into_bit_vec(
+            measurement,
+            self.bits
+        )
+        encoded += self.field.encode_into_bit_vec(
+            measurement + self.offset.int(),
+            self.bits
+        )
+        return encoded
+
+    def eval(
+            self,
+            meas: list[F],
+            joint_rand: list[F],
+            num_shares: int) -> list[F]:
+        shares_inv = self.field(num_shares).inv()
+
+        out = []
+        for b in meas:
+            out.append(self.GADGETS[0].eval(self.field, [b]))
+
+        range_check = self.offset * shares_inv + \
+            self.field.decode_from_bit_vec(meas[:self.bits]) - \
+            self.field.decode_from_bit_vec(meas[self.bits:])
+        out.append(range_check)
+        return out
+
+    def truncate(self, meas: list[F]) -> list[F]:
+        return [self.field.decode_from_bit_vec(meas[:self.bits])]
+
+    def decode(self, output: list[F], _num_measurements: int) -> int:
+        return output[0].int()
+
+
+
+
+
+
+

+7.4.3. Prio3SumVec +

+ + + + + + + + + + + + + + + + + + + + + + +
+Table 11: +Parameters for Prio3SumVec. +
ParameterValue
+ field + + Field128 (Table 3)
+ Valid + + SumVec(field, length, bits, chunk_lengh) (this section)
+ PROOFS + + 1 +
+

This instance of Prio3 supports summing vectors of integers. It has three +parameters: length, bits, and chunk_length. Each measurement is a vector +of positive integers with length equal to the length parameter. Each element +of the measurement is an integer in the range [0, 2^bits). It is RECOMMENDED +to set chunk_length to an integer near the square root of length * bits +(see Section 7.4.3.1).

+

The circuit is denoted SumVec. Each measurements is encoded as a vector of +field elements with a length of length * bits. The field elements in the +encoded vector represent all the bits of the measurement vector's elements, +consecutively, in LSB to MSB order.

+

The validity circuit uses the ParallelSum gadget in Appendix A.3. +This gadget applies an arithmetic subcircuit to multiple inputs in parallel, +then returns the sum of the results. Along with the subcircuit, the +parallel-sum gadget is parameterized by an integer, denoted count, specifying +how many times to call the subcircuit. It takes in a list of inputs and passes +them through to instances of the subcircuit in the same order. It returns the +sum of the subcircuit outputs.

+

Note that only the ParallelSum gadget itself, and not its subcircuit, +participates in the FLP's wire recording during evaluation, gadget consistency +proofs, and proof validation, even though the subcircuit is provided to +ParallelSum gadget as an implementation of the Gadget interface.

+

The SumVec validity circuit checks that the encoded measurement consists of +ones and zeros. Rather than use the PolyEval gadget on each element, as in +the Sum validity circuit, it instead uses Mul subcircuits (Appendix A.1) +and "free" constant multiplication and addition gates to simultaneously +evaluate the same range check polynomial on each element, and multiply by a +constant. One of the two Mul subcircuit inputs is equal to a measurement +element multiplied by a power of one of the elements of the joint randomness +vector, and the other is equal to the same measurement element minus one. These +Mul subcircuits are evaluated by a ParallelSum gadget, and the results are +added up both within the ParallelSum gadget and after it.

+

The complete circuit is specified below:

+
+
+class SumVec(Valid[list[int], list[int], F]):
+    EVAL_OUTPUT_LEN = 1
+    length: int
+    bits: int
+    chunk_length: int
+    field: type[F]
+
+    def __init__(self,
+                 field: type[F],
+                 length: int,
+                 bits: int,
+                 chunk_length: int):
+        """
+        Instantiate the `SumVec` circuit for measurements with
+        `length` elements, each in the range `[0, 2^bits)`.
+        """
+        self.field = field
+        self.length = length
+        self.bits = bits
+        self.chunk_length = chunk_length
+        self.GADGETS = [ParallelSum(Mul(), chunk_length)]
+        self.GADGET_CALLS = [
+            (length * bits + chunk_length - 1) // chunk_length
+        ]
+        self.MEAS_LEN = length * bits
+        self.OUTPUT_LEN = length
+        self.JOINT_RAND_LEN = self.GADGET_CALLS[0]
+
+    def encode(self, measurement: list[int]) -> list[F]:
+        encoded = []
+        for val in measurement:
+            encoded += self.field.encode_into_bit_vec(
+                val, self.bits)
+        return encoded
+
+    def eval(
+            self,
+            meas: list[F],
+            joint_rand: list[F],
+            num_shares: int) -> list[F]:
+        out = self.field(0)
+        shares_inv = self.field(num_shares).inv()
+        for i in range(self.GADGET_CALLS[0]):
+            r = joint_rand[i]
+            r_power = r
+            inputs: list[Optional[F]]
+            inputs = [None] * (2 * self.chunk_length)
+            for j in range(self.chunk_length):
+                index = i * self.chunk_length + j
+                if index < len(meas):
+                    meas_elem = meas[index]
+                else:
+                    meas_elem = self.field(0)
+
+                inputs[j * 2] = r_power * meas_elem
+                inputs[j * 2 + 1] = meas_elem - shares_inv
+
+                r_power *= r
+
+            out += self.GADGETS[0].eval(
+                self.field,
+                cast(list[F], inputs),
+            )
+
+        return [out]
+
+    def truncate(self, meas: list[F]) -> list[F]:
+        truncated = []
+        for i in range(self.length):
+            truncated.append(self.field.decode_from_bit_vec(
+                meas[i * self.bits: (i + 1) * self.bits]
+            ))
+        return truncated
+
+    def decode(
+            self,
+            output: list[F],
+            _num_measurements: int) -> list[int]:
+        return [x.int() for x in output]
+
+
+
+
+
+7.4.3.1. Selection of ParallelSum Chunk Length +
+

The chunk_length parameter provides a trade-off between the arity of the +ParallelSum gadget (Appendix A.3) and the number of times the +gadget is called. The proof length is asymptotically minimized when the chunk +length is near the square root of the length of the measurement. However, the +relationship between VDAF parameters and proof length is complicated, involving +two forms of rounding:

+
    +
  • +

    The circuit pads the inputs to its last ParallelSum gadget call, up to +the chunk length.

    +
    • +
  • +

    The proof system rounds the degree of wire polynomials, determined by the +number of times a gadget is called, up to the next power of two.

    +
    • +
+

Therefore, the optimal choice of chunk_length for a concrete measurement size +will vary, and must be found through trial and error. Setting chunk_length +equal to the square root of the appropriate measurement length will result in +proofs up to 50% larger than the optimal proof size.

+
+
+
+
+
+
+

+7.4.4. Prio3Histogram +

+ + + + + + + + + + + + + + + + + + + + + + +
+Table 12: +Parameters for Prio3Histogram. +
ParameterValue
+ field + + Field128 (Table 3)
+ Valid + + Histogram(field, length, chunk_lengh) (this section)
+ PROOFS + + 1 +
+

This variant of Prio3 allows for estimating the distribution of some quantity +by computing a simple histogram. Each measurement increments one histogram +bucket, out of a set of fixed buckets. (Bucket indexing begins at 0.) For +example, the buckets might quantize the real numbers, and each measurement +would report the bucket that the corresponding client's real-numbered value +falls into. The aggregate result counts the number of measurements in each +bucket.

+

The validity circuit is denoted Histogram. It has two parameters: length, +the number of histogram buckets; and chunk_length, which is used by a +circuit optimization described below. It is RECOMMENDED to set chunk_length +to an integer near the square root of length (see +Section 7.4.3.1).

+

The measurement is encoded as a one-hot vector representing the bucket into +which the measurement falls. The circuit checks for one-hotness in two steps, +by checking that the encoded measurement consists of ones and zeros, and by +checking that the sum of all elements in the encoded measurement is equal to +one. The individual checks constitute the output of the circuit.

+

As in the SumVec validity circuit (Section 7.4.3), the first part of the +validity circuit uses the ParallelSum (Appendix A.3) gadget to +perform range checks while achieving a smaller proof size. The ParallelSum +gadget uses Mul subcircuits (Appendix A.1) to evaluate a range check +polynomial on each element, and includes an additional constant multiplication. +One of the two Mul subcircuit inputs is equal to a measurement element +multiplied by a power of an element of the joint randomness vector, and the +other is equal to the same measurement element minus one. The results are added +up both within the ParallelSum gadget and after it.

+
+
+class Histogram(Valid[int, list[int], F]):
+    EVAL_OUTPUT_LEN = 2
+    field: type[F]
+    length: int
+    chunk_length: int
+
+    def __init__(self,
+                 field: type[F],
+                 length: int,
+                 chunk_length: int):
+        """
+        Instantiate an instance of the `Histogram` circuit with the
+        given `length` and `chunk_length`.
+        """
+        self.field = field
+        self.length = length
+        self.chunk_length = chunk_length
+        self.GADGETS = [ParallelSum(Mul(), chunk_length)]
+        self.GADGET_CALLS = [
+            (length + chunk_length - 1) // chunk_length]
+        self.MEAS_LEN = self.length
+        self.OUTPUT_LEN = self.length
+        self.JOINT_RAND_LEN = self.GADGET_CALLS[0]
+
+    def encode(self, measurement: int) -> list[F]:
+        encoded = [self.field(0)] * self.length
+        encoded[measurement] = self.field(1)
+        return encoded
+
+    def eval(
+            self,
+            meas: list[F],
+            joint_rand: list[F],
+            num_shares: int) -> list[F]:
+        # Check that each bucket is one or zero.
+        range_check = self.field(0)
+        shares_inv = self.field(num_shares).inv()
+        for i in range(self.GADGET_CALLS[0]):
+            r = joint_rand[i]
+            r_power = r
+            inputs: list[Optional[F]]
+            inputs = [None] * (2 * self.chunk_length)
+            for j in range(self.chunk_length):
+                index = i * self.chunk_length + j
+                if index < len(meas):
+                    meas_elem = meas[index]
+                else:
+                    meas_elem = self.field(0)
+
+                inputs[j * 2] = r_power * meas_elem
+                inputs[j * 2 + 1] = meas_elem - shares_inv
+
+                r_power *= r
+
+            range_check += self.GADGETS[0].eval(
+                self.field,
+                cast(list[F], inputs),
+            )
+
+        # Check that the buckets sum to 1.
+        sum_check = -shares_inv
+        for b in meas:
+            sum_check += b
+
+        return [range_check, sum_check]
+
+    def truncate(self, meas: list[F]) -> list[F]:
+        return meas
+
+    def decode(
+            self,
+            output: list[F],
+            _num_measurements: int) -> list[int]:
+        return [bucket_count.int()
+                for bucket_count in output]
+
+
+
+
+
+
+

+7.4.5. Prio3MultihotCountVec +

+ + + + + + + + + + + + + + + + + + + + + + +
+Table 13: +Parameters for Prio3MultihotCountVec. +
ParameterValue
+ field + + Field128 (Table 3)
+ Valid + + MultihotCountVec(field, length, max_weight, chunk_lengh) (this section)
+ PROOFS + + 1 +
+

For this instance of Prio3, each measurement is a vector of ones and zeros, +where the number of ones is bounded. This provides a functionality similar to +Prio3Histogram except that more than one entry (or none at all) may be +non-zero. This allows Prio3MultihotCountVec to be composed with a randomized +response mechanism, like [EPK14], for providing differential privacy. (For +example, each Client would set each entry to one with some small probability.)

+

The validity circuit is denoted MultihotCountVec and has three parameters: +length, the number of entries in the count vector; max_weight, the +maximum number of non-zero entries (i.e., the weight must be at most +max_weight); and chunk_length, used the same way as in Section 7.4.3 and +Section 7.4.4.

+

Validation works as follows. Let

+
    +
  • +

    bits_for_weight = max_weight.bit_length()

    +
    • +
  • +

    offset = 2^bits_for_weight - 1 - max_weight

    +
    • +
+

The Client reports the weight of the count vector by adding offset to it and +bit-encoding the result. Observe that only a weight of at most max_weight can +be encoded with bits_for_weight bits.

+

The verifier checks that each entry of the encoded measurement is a bit (i.e., +either one or zero). It then decodes the reported weight and subtracts it from +offset + sum(count_vec), where count_vec is the count vector. The result is +zero if and only if the reported weight is equal to the true weight. The +complete circuit is defined below.

+
+
+class MultihotCountVec(Valid[list[int], list[int], F]):
+    EVAL_OUTPUT_LEN = 2
+    field: type[F]
+
+    def __init__(self,
+                 field: type[F],
+                 length: int,
+                 max_weight: int,
+                 chunk_length: int):
+        """
+        Instantiate an instance of the this circuit with the given
+        `length`, `max_weight`, and `chunk_length`.
+
+        Pre-conditions:
+
+            - `length > 0`
+            - `0 < max_weight` and `max_weight <= length`
+            - `chunk_length > 0`
+        """
+        self.field = field
+
+        # Compute the number of bits to represent `max_weight`.
+        self.bits_for_weight = max_weight.bit_length()
+        self.offset = self.field(
+            2**self.bits_for_weight - 1 - max_weight)
+
+        # Make sure `offset + length` doesn't overflow the field
+        # modulus. Otherwise we may not correctly compute the sum
+        # measurement vector entries during circuit evaluation.
+        if self.field.MODULUS - self.offset.int() <= length:
+            raise ValueError('length and max_weight are too large '
+                             'for the current field size')
+
+        self.length = length
+        self.max_weight = max_weight
+        self.chunk_length = chunk_length
+        self.GADGETS: list[Gadget[F]] = [
+            ParallelSum(Mul(), chunk_length),
+        ]
+        self.GADGET_CALLS = [
+            (length + self.bits_for_weight + chunk_length - 1)
+            // chunk_length
+        ]
+        self.MEAS_LEN = self.length + self.bits_for_weight
+        self.OUTPUT_LEN = self.length
+        self.JOINT_RAND_LEN = self.GADGET_CALLS[0]
+
+    def encode(self, measurement: list[int]) -> list[F]:
+        if len(measurement) != self.length:
+            raise ValueError('invalid Client measurement length')
+
+        # The first part is the vector of counters.
+        count_vec = list(map(self.field, measurement))
+
+        # The second part is the reported weight.
+        weight_reported = sum(count_vec, self.field(0))
+
+        encoded = []
+        encoded += count_vec
+        encoded += self.field.encode_into_bit_vec(
+            (self.offset + weight_reported).int(),
+            self.bits_for_weight)
+        return encoded
+
+    def eval(
+            self,
+            meas: list[F],
+            joint_rand: list[F],
+            num_shares: int) -> list[F]:
+        # Check that each entry in the input vector is one or zero.
+        range_check = self.field(0)
+        shares_inv = self.field(num_shares).inv()
+        for i in range(self.GADGET_CALLS[0]):
+            r = joint_rand[i]
+            r_power = r
+            inputs: list[Optional[F]]
+            inputs = [None] * (2 * self.chunk_length)
+            for j in range(self.chunk_length):
+                index = i * self.chunk_length + j
+                if index < len(meas):
+                    meas_elem = meas[index]
+                else:
+                    meas_elem = self.field(0)
+
+                inputs[j * 2] = r_power * meas_elem
+                inputs[j * 2 + 1] = meas_elem - shares_inv
+
+                r_power *= r
+
+            range_check += self.GADGETS[0].eval(
+                self.field,
+                cast(list[F], inputs),
+            )
+
+        # Check that the weight `offset` plus the sum of the counters
+        # is equal to the value claimed by the Client.
+        count_vec = meas[:self.length]
+        weight = sum(count_vec, self.field(0))
+        weight_reported = \
+            self.field.decode_from_bit_vec(meas[self.length:])
+        weight_check = self.offset*shares_inv + weight - \
+            weight_reported
+
+        return [range_check, weight_check]
+
+    def truncate(self, meas: list[F]) -> list[F]:
+        return meas[:self.length]
+
+    def decode(
+            self,
+            output: list[F],
+            _num_measurements: int) -> list[int]:
+        return [bucket_count.int() for
+                bucket_count in output]
+
+
+
+
+
+
+
+
+
+
+

+8. Poplar1 +

+

This section specifies Poplar1, a VDAF for the following task. Each Client +holds a bit-string of length BITS and the Collector chooses a sequence of +L-bit strings, where L <= BITS. We will refer to the latter as the of +"candidate prefixes". The goal is to count how many of the Clients' inputs +begin with each candidate prefix.

+

This functionality is the core component of the heavy hitters protocol of +[BBCGGI21]. The goal of this protocol is to compute the subset of inputs held +by at least T Clients for some threshold T. It invokes Poplar1 as follows:

+
    +
  1. +

    Each Client shards its string into secret shares and uploads one share to +each of the Aggregators.

    +
    2. +
  2. +

    The Collector picks an initial set of candidate prefixes, say 0 and 1, +and sends them to the Aggregators.

    +
    3. +
  3. +

    The Aggregators run VDAF preparation and aggregation on each of the reports +and send their aggregate shares to the Collector.

    +
    4. +
  4. +

    The Collector unshards the aggregate result, which consists of the hit count +for each candidate prefix. For each prefix p with hit count at least T, +the Collector adds p || 0 and p || 1 to the next generation of +candidate prefixes and repeats Step 2.

    +
    5. +
+

Poplar1 is constructed from an "Incremental Distributed Point Function (IDPF)", +a primitive described by [BBCGGI21] that generalizes the notion of a +Distributed Point Function (DPF) [GI14]. Briefly, a DPF is used to distribute +the computation of a "point function", a function that evaluates to zero on +every input except at a programmable "point". The computation is distributed in +such a way that no one party knows either the point or what it evaluates to.

+

An IDPF generalizes this "point" to a path on a full binary tree from the root +to one of the leaves. It is evaluated on an "index" representing a unique node +of the tree. If the node is on the programmed path, then the function evaluates +to a non-zero value; otherwise it evaluates to zero. This structure allows an +IDPF to provide the functionality required for the above protocol: to compute +the hit count for an index, just evaluate each set of IDPF shares at that index +and add up the results.

+

Consider the sub-tree constructed from a set of input strings and a target +threshold T by including all indices with hit count at least T. We shall +refer to this structure as the "prefix tree" for the batch of measurements and +target threshold. To compute the T-heavy-hitters for the batch, the +Aggregators and Collector first compute the prefix tree, then extract the heavy +hitters from the leaves of this tree. Note that the prefix tree leaks more +information about the set than the heavy hitters themselves; see +Section 9.4 for more discussion.

+

Poplar1 composes an IDPF with the arithmetic sketch of [BBCGGI21], Section +4.2. (The paper calls this a "secure sketch", but the underlying technique was +later generalized in [BBCGGI23], where it is called "arithmetic sketching".) +The sketch ensures that evaluating a set of input shares on a unique set of +candidate prefixes results in shares of a "one-hot" vector, i.e., a vector that +is zero everywhere except for in at most one position. Moreover, the value at +that position should be one.

+

The remainder of this section is structured as follows. The syntax of IDPFs is +defined in Section 8.1; a specification of the IDPF of [BBCGGI21] is given in +Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in +terms of a generic IDPF. Test vectors for Poplar1 can be found in +Appendix C.

+
+
+

+8.1. Incremental Distributed Point Functions (IDPFs) +

+

An IDPF is defined over a domain of size 2^BITS, where BITS is a constant. +Indices into the IDPF tree are bit strings. (In Poplar1, each Client's bit +string is an index; see Section 8.1.1 for details.) The Client +specifies an index alpha and a vector of values beta, one for each "level" +L in [0, BITS). The key generation algorithm generates one IDPF "key" for +each Aggregator. When evaluated at level L and index prefix, each IDPF key +returns an additive share of beta[L] if prefix is the L-bit prefix of +alpha and shares of zero otherwise.

+

Each of the programmed points beta is a vector of elements of some finite +field. We distinguish two types of fields: one for inner nodes (denoted +FieldInner), and one for leaf nodes (FieldLeaf). (Our instantiation of +Poplar1 (Section 8.2) will use a much larger field for leaf nodes +than for inner nodes. This is to ensure the IDPF is "extractable" as defined in +[BBCGGI21], Definition 1. See Section 9.5 for details.)

+

A concrete IDPF defines the types and parameters enumerated in Table 14. +In the remainder we write Output as shorthand for the type +list[list[FieldInner]] | list[list[FieldLeaf]]. (This type denotes either a +vector of inner node field elements or leaf node field elements.) The scheme is +comprised of the following algorithms:

+
    +
  • +

    idpf.gen(alpha: tuple[bool, ...], beta_inner: list[list[FieldInner]], beta_leaf: +list[FieldLeaf], ctx: bytes, nonce: bytes, rand: bytes) -> tuple[PublicShare, +list[bytes]] is the IDPF-key generation algorithm. Its inputs are the index +alpha, the values beta, the application context, and the report nonce.

    +

    +The output is a public part (of type PublicShare) that is sent to each +Aggregator and a vector of private IDPF keys, one for each Aggregator. The +nonce is used to derive the fixed AES key for XofFixedKeyAes128 +(Section 6.2.2). Looking ahead, this key is used for extending the +seed into the seeds for the child nodes at each level of the tree; see +Section 8.3.

    +

    +Pre-conditions:

    +
      +
    • +

      alpha MUST have length BITS.

      +
      • +
    • +

      beta_inner MUST have length BITS - 1.

      +
      • +
    • +

      beta_inner[level] MUST have length VALUE_LEN for each level in + [0, BITS - 1).

      +
      • +
    • +

      beta_leaf MUST have length VALUE_LEN.

      +
      • +
    • +

      rand MUST be generated by a CSPRNG and have length RAND_SIZE.

      +
      • +
    • +

      nonce MUST be generated by a CSPRNG (see Section 9.2 for +details) and have length idpf.NONCE_SIZE.

      +
      • +
    +

    +Post-conditions:

    +
      +
    • +

      The number IDPF keys MUST be idpf.SHARES.

      +
      • +
    +
    • +
  • +

    idpf.eval(agg_id: int, public_share: PublicShare, key: bytes, level: int, +prefixes: Sequence[tuple[bool, ...]], ctx: bytes, nonce: bytes) -> Output is +the IDPF-key evaluation algorithm run by each Aggregator. Its inputs are the +Aggregator's unique identifier, the public share distributed to all of the +Aggregators, the Aggregator's IDPF key, the "level" at which to evaluate the +IDPF, the sequence of candidate prefixes, the application context, and the +report nonce. It returns the share of the value corresponding to each +candidate prefix.

    +

    +Pre-conditions:

    +
      +
    • +

      agg_id MUST be in [0, idpf.SHARES) and match the index of key in +the sequence of IDPF keys output by the Client.

      +
      • +
    • +

      level MUST be in [0, BITS).

      +
      • +
    • +

      Each prefix MUST be distinct and have length level + 1.

      +
      • +
    • +

      The length of the nonce MUST be idpf.NONCE_SIZE.

      +
      • +
    +

    +Post-conditions:

    +
      +
    • +

      The length of the output MUST be len(prefixes)

      +
      • +
    • +

      The length of each element of the output MUST be idpf.VALUE_LEN

      +
      • +
    • +

      If level == idpf.BITS - 1, then the output field MUST be FieldLeaf and + FieldInner otherwise

      +
      • +
    +
    • +
+

In addition, the IDPF provides the following method:

+
+
+def current_field(
+        self,
+        level: int) -> type[FieldInner] | type[FieldLeaf]:
+    if level < self.BITS - 1:
+        return self.field_inner
+    return self.field_leaf
+
+
+

Finally, an implementation note. The interface for IDPFs specified here is +stateless, in the sense that there is no state carried between IDPF evaluations. +This is to align the IDPF syntax with the VDAF abstraction boundary, which does +not include shared state across VDAF evaluations. In practice, of course, it +will often be beneficial to expose a stateful API for IDPFs and carry the state +across evaluations. See Section 8.3 for details.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 14: +Constants and types defined by a concrete IDPF. +
ParameterDescription
+ SHARES: int +Number of IDPF keys output by IDPF-key generator.
+ BITS: int +Length in bits of each input string.
+ VALUE_LEN: int +Number of field elements of each output value.
+ RAND_SIZE: int +Size of the random string consumed by the IDPF-key generator.
+ NONCE_SIZE: int +Size of the random nonce generated by the Client.
+ KEY_SIZE: int +Size in bytes of each IDPF key.
+ FieldInner +Implementation of Field (Section 6.1) used for values of inner nodes.
+ FieldLeaf +Implementation of Field used for values of leaf nodes.
+ PublicShare +Type of public share for this IDPF.
+ Output +Alias of list[list[FieldInner]] | list[list[FieldLeaf]].
+ FieldVec +Alias of list[FieldInner] | list[FieldLeaf]. +
+
+
+
+

+8.1.1. Encoding Inputs as Indices +

+

How data are represented as IDPF indices is up to the application. When the +inputs are fixed-length byte strings, the most natural choice of representation +is as a bit string formed from all the bits of the byte string, first ordered by +byte position, then ordered from most significant bit to least significant bit +within each byte. This ensures that, when a byte string is a prefix of another, +so too is its corresponding index. (Index prefixes are defined in Section 8.1.) For +example,

+
+
+Byte string: 01 02
+Bit string: 00000001 00000010
+
+
+

is a prefix of

+
+
+Byte string: 01 02 03
+Bit string: 00000001 00000010 00000011
+
+
+

Additionally, lexicographic ordering is preserved by this mapping from a byte +string to a bit string.

+

When the inputs are variable length, it is necessary to pad each input to some +fixed length. Further, the padding scheme must be non-ambiguous. For example, +each input could be padded with b"\x01" followed by as many b"\x00" bytes +as needed.

+
+
+
+
+
+
+

+8.2. Specification +

+

This section specifies Poplar1, an implementation of the Vdaf interface +(Section 5). It is defined in terms of the Idpf implementation of +Section 8.3 with SHARES == 2 and VALUE_LEN == 2 and +XofTurboShake128 as specified in Section 6.2.1. The associated +constants and types required by the Vdaf interface are defined in +Table 15. The methods required for sharding, preparation, aggregation, +and unsharding are described in the remaining subsections. These methods make +use of constants defined in Table 16.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 15: +VDAF parameters for Poplar1. +
ParameterValue
+ idpf +As specified in Section 8.3.
+ xof + + XofTurboShake128 (Section 6.2.1)
+ VERIFY_KEY_SIZE + + xof.SEED_SIZE +
+ RAND_SIZE + + xof.SEED_SIZE * 3 + idpf.RAND_SIZE +
+ NONCE_SIZE + + 16 +
+ ROUNDS + + 2 +
+ SHARES + + 2 +
+ Measurement + + tuple[bool, ...] +
+ AggParam + + tuple[int, Sequence[tuple[bool, ...]]] +
+ PublicShare +As defined by idpf.
+ InputShare + + tuple[bytes, bytes, list[FieldInner], list[FieldLeaf]] +
+ OutShare + + FieldVec +
+ AggShare + + FieldVec +
+ AggResult + + list[int] +
+ PrepState + + tuple[bytes, int, FieldVec] +
+ PrepShare + + FieldVec +
+ PrepMessage + + Optional[FieldVec] +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 16: +Constants used by Poplar1. +
VariableValue
USAGE_SHARD_RAND: int1
USAGE_CORR_INNER: int2
USAGE_CORR_LEAF: int3
USAGE_VERIFY_RAND: int4
+
+
+
+

+8.2.1. Sharding +

+

The Client's measurement is an IDPF index, denoted alpha, whose type is a +sequence of bits tuple[bool, ...] (See Section 8.1.1 for +guidelines on index encoding.)

+

The programmed IDPF values are pairs of field elements (1, k) where each k is +chosen at random. This random value is used as part of the arithmetic sketching +protocol of [BBCGGI21], Appendix C.4. After evaluating their IDPF key shares +on a given sequence of candidate prefixes, the sketching protocol is used by +the Aggregators to verify that they hold shares of a one-hot vector at a given +level of the IDPF tree.

+

In addition to programming k into the IDPF output, for each level of the +tree, the Client generates random elements a, b, and c and computes

+
+
+    A = -2*a + k
+    B = a^2 + b - k*a + c
+
+
+

and sends additive shares of a, b, c, A and B to each of the +Aggregators. These help the Aggregators evaluate the sketch during preparation.

+

Putting everything together, the sharding algorithm is defined as +follows.

+
+
+def shard(
+    self,
+    ctx: bytes,
+    measurement: tuple[bool, ...],
+    nonce: bytes,
+    rand: bytes,
+) -> tuple[Poplar1PublicShare, list[Poplar1InputShare]]:
+    if len(nonce) != self.NONCE_SIZE:
+        raise ValueError("incorrect nonce size")
+    if len(rand) != self.RAND_SIZE:
+        raise ValueError("incorrect size of random bytes argument")
+
+    l = self.xof.SEED_SIZE
+
+    # Split the random input into the random input for IDPF key
+    # generation, correlated randomness, and sharding.
+    if len(rand) != self.RAND_SIZE:
+        raise ValueError('incorrect rand size')
+    idpf_rand, rand = front(self.idpf.RAND_SIZE, rand)
+    seeds = [rand[i:i + l] for i in range(0, 3 * l, l)]
+    corr_seed, seeds = front(2, seeds)
+    (shard_seed,), seeds = front(1, seeds)
+
+    xof = self.xof(
+        shard_seed,
+        self.domain_separation_tag(USAGE_SHARD_RAND, ctx),
+        nonce,
+    )
+
+    # Construct the IDPF values for each level of the IDPF tree.
+    # Each "data" value is 1; in addition, the Client generates
+    # a random "authenticator" value used by the Aggregators to
+    # evaluate the sketch during preparation. This sketch is used
+    # to verify the one-hotness of their output shares.
+    beta_inner = [
+        [self.idpf.field_inner(1), k]
+        for k in xof.next_vec(self.idpf.field_inner,
+                              self.idpf.BITS - 1)
+    ]
+    beta_leaf = [self.idpf.field_leaf(1)] + \
+        xof.next_vec(self.idpf.field_leaf, 1)
+
+    # Generate the IDPF keys.
+    (public_share, keys) = self.idpf.gen(
+        measurement,
+        beta_inner,
+        beta_leaf,
+        ctx,
+        nonce,
+        idpf_rand,
+    )
+
+    # Generate correlated randomness used by the Aggregators to
+    # evaluate the sketch over their output shares. Seeds are used
+    # to encode shares of the `(a, b, c)` triples. (See [BBCGGI21,
+    # Appendix C.4].)
+    corr_offsets: list[Field] = vec_add(
+        self.xof.expand_into_vec(
+            self.idpf.field_inner,
+            corr_seed[0],
+            self.domain_separation_tag(USAGE_CORR_INNER, ctx),
+            byte(0) + nonce,
+            3 * (self.idpf.BITS - 1),
+        ),
+        self.xof.expand_into_vec(
+            self.idpf.field_inner,
+            corr_seed[1],
+            self.domain_separation_tag(USAGE_CORR_INNER, ctx),
+            byte(1) + nonce,
+            3 * (self.idpf.BITS - 1),
+        ),
+    )
+    corr_offsets += vec_add(
+        self.xof.expand_into_vec(
+            self.idpf.field_leaf,
+            corr_seed[0],
+            self.domain_separation_tag(USAGE_CORR_LEAF, ctx),
+            byte(0) + nonce,
+            3,
+        ),
+        self.xof.expand_into_vec(
+            self.idpf.field_leaf,
+            corr_seed[1],
+            self.domain_separation_tag(USAGE_CORR_LEAF, ctx),
+            byte(1) + nonce,
+            3,
+        ),
+    )
+
+    # For each level of the IDPF tree, shares of the `(A, B)`
+    # pairs are computed from the corresponding `(a, b, c)`
+    # triple and authenticator value `k`.
+    corr_inner: list[list[Field64]] = [[], []]
+    for level in range(self.idpf.BITS):
+        field = cast(type[Field], self.idpf.current_field(level))
+        k = beta_inner[level][1] if level < self.idpf.BITS - 1 \
+            else beta_leaf[1]
+        (a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:]
+        A = -field(2) * a + k
+        B = a ** 2 + b - a * k + c
+        corr1 = xof.next_vec(field, 2)
+        corr0 = vec_sub([A, B], corr1)
+        if level < self.idpf.BITS - 1:
+            corr_inner[0] += cast(list[Field64], corr0)
+            corr_inner[1] += cast(list[Field64], corr1)
+        else:
+            corr_leaf = [
+                cast(list[Field255], corr0),
+                cast(list[Field255], corr1),
+            ]
+
+    # Each input share consists of the Aggregator's IDPF key
+    # and a share of the correlated randomness.
+    input_shares = list(zip(keys, corr_seed, corr_inner, corr_leaf))
+    return (public_share, input_shares)
+
+
+
+
+
+
+

+8.2.2. Preparation +

+

The aggregation parameter encodes a sequence of candidate prefixes. When an +Aggregator receives an input share from the Client, it begins by evaluating its +IDPF share on each candidate prefix, recovering a data_share and auth_share +for each. The Aggregators use these and the correlation shares provided by the +Client to verify that the sequence of data_share values are additive shares of +a one-hot vector.

+

Aggregators MUST ensure the candidate prefixes are all unique and appear in +lexicographic order. (This is enforced in the definition of prep_init() +below.) Uniqueness is necessary to ensure the refined measurement (i.e., the sum +of the output shares) is in fact a one-hot vector. Otherwise, sketch +verification might fail, causing the Aggregators to erroneously reject a report +that is actually valid. Note that enforcing the order is not strictly necessary, +but this does allow uniqueness to be determined more efficiently.

+
+
+def prep_init(
+        self,
+        verify_key: bytes,
+        ctx: bytes,
+        agg_id: int,
+        agg_param: Poplar1AggParam,
+        nonce: bytes,
+        public_share: Poplar1PublicShare,
+        input_share: Poplar1InputShare) -> tuple[
+            Poplar1PrepState,
+            FieldVec]:
+    (level, prefixes) = agg_param
+    (key, corr_seed, corr_inner, corr_leaf) = input_share
+    field = self.idpf.current_field(level)
+
+    # Ensure that candidate prefixes are all unique and appear in
+    # lexicographic order.
+    for i in range(1, len(prefixes)):
+        if prefixes[i - 1] >= prefixes[i]:
+            raise ValueError('out of order prefix')
+
+    # Evaluate the IDPF key at the given set of prefixes.
+    value = self.idpf.eval(
+        agg_id, public_share, key, level, prefixes, ctx, nonce)
+
+    # Get shares of the correlated randomness for evaluating the
+    # Aggregator's share of the sketch.
+    if level < self.idpf.BITS - 1:
+        corr_xof = self.xof(
+            corr_seed,
+            self.domain_separation_tag(USAGE_CORR_INNER, ctx),
+            byte(agg_id) + nonce,
+        )
+        # Fast-forward the XOF state to the current level.
+        corr_xof.next_vec(field, 3 * level)
+    else:
+        corr_xof = self.xof(
+            corr_seed,
+            self.domain_separation_tag(USAGE_CORR_LEAF, ctx),
+            byte(agg_id) + nonce,
+        )
+    (a_share, b_share, c_share) = corr_xof.next_vec(field, 3)
+    if level < self.idpf.BITS - 1:
+        (A_share, B_share) = cast(
+            list[Field],
+            corr_inner[2 * level:2 * (level + 1)],
+        )
+    else:
+        (A_share, B_share) = cast(list[Field], corr_leaf)
+
+    # Evaluate the Aggregator's share of the sketch. These are
+    # called the "masked input values" [BBCGGI21, Appendix C.4].
+    verify_rand_xof = self.xof(
+        verify_key,
+        self.domain_separation_tag(USAGE_VERIFY_RAND, ctx),
+        nonce + to_be_bytes(level, 2),
+    )
+    verify_rand = cast(
+        list[Field],
+        verify_rand_xof.next_vec(field, len(prefixes)),
+    )
+    sketch_share = [a_share, b_share, c_share]
+    out_share = []
+    for (i, r) in enumerate(verify_rand):
+        data_share = cast(Field, value[i][0])
+        auth_share = cast(Field, value[i][1])
+        sketch_share[0] += data_share * r
+        sketch_share[1] += data_share * r ** 2
+        sketch_share[2] += auth_share * r
+        out_share.append(data_share)
+
+    prep_mem = [A_share, B_share, field(agg_id)] + out_share
+    return (
+        (
+            b'evaluate sketch',
+            level,
+            cast(FieldVec, prep_mem),
+        ),
+        cast(FieldVec, sketch_share),
+    )
+
+def prep_next(
+    self,
+    _ctx: bytes,
+    prep_state: Poplar1PrepState,
+    prep_msg: Optional[FieldVec]
+) -> tuple[Poplar1PrepState, FieldVec] | FieldVec:
+    prev_sketch = cast(list[Field], prep_msg)
+    (step, level, prep_mem) = prep_state
+
+    if step == b'evaluate sketch':
+        if prev_sketch is None:
+            raise ValueError('expected value, got none')
+        elif len(prev_sketch) != 3:
+            raise ValueError('incorrect sketch length')
+        A_share = cast(Field, prep_mem[0])
+        B_share = cast(Field, prep_mem[1])
+        agg_id = cast(Field, prep_mem[2])
+        prep_mem = prep_mem[3:]
+        sketch_share = [
+            agg_id * (prev_sketch[0] ** 2
+                      - prev_sketch[1]
+                      - prev_sketch[2])
+            + A_share * prev_sketch[0]
+            + B_share
+        ]
+        return cast(
+            tuple[Poplar1PrepState, FieldVec],
+            (
+                (
+                    b'reveal sketch',
+                    level,
+                    prep_mem,
+                ),
+                sketch_share,
+            )
+        )
+
+    elif step == b'reveal sketch':
+        if prev_sketch is None:
+            return prep_mem  # Output shares
+        else:
+            raise ValueError('invalid prep message')
+
+    raise ValueError('invalid prep state')
+
+def prep_shares_to_prep(
+        self,
+        _ctx: bytes,
+        agg_param: Poplar1AggParam,
+        prep_shares: list[FieldVec]) -> Optional[FieldVec]:
+    if len(prep_shares) != 2:
+        raise ValueError('incorrect number of prep shares')
+    (level, _) = agg_param
+    field = self.idpf.current_field(level)
+    sketch = vec_add(
+        cast(list[Field], prep_shares[0]),
+        cast(list[Field], prep_shares[1]),
+    )
+    if len(sketch) == 3:
+        return cast(FieldVec, sketch)
+    elif len(sketch) == 1:
+        if sketch == field.zeros(1):
+            # In order to reduce communication overhead, let `None`
+            # denote a successful sketch verification.
+            return None
+        else:
+            raise ValueError('sketch verification failed')
+    else:
+        raise ValueError('incorrect sketch length')
+
+
+
+
+
+
+

+8.2.3. Validity of Aggregation Parameters +

+

Aggregation parameters are valid for a given input share if no aggregation +parameter with the same level has been used with the same input share before. +The whole preparation phase MUST NOT be run more than once for a given +combination of input share and level. This function checks that levels are +increasing between calls, and also enforces that the prefixes at each level are +suffixes of the previous level's prefixes.

+
+
+def is_valid(
+        self,
+        agg_param: Poplar1AggParam,
+        previous_agg_params: list[Poplar1AggParam]) -> bool:
+    """
+    Checks that levels are increasing between calls, and also
+    enforces that the prefixes at each level are suffixes of the
+    previous level's prefixes.
+    """
+    if len(previous_agg_params) < 1:
+        return True
+
+    (level, prefixes) = agg_param
+    (last_level, last_prefixes) = previous_agg_params[-1]
+    last_prefixes_set = set(last_prefixes)
+
+    # Check that level increased.
+    if level <= last_level:
+        return False
+
+    # Check that prefixes are suffixes of the last level's prefixes.
+    for prefix in prefixes:
+        last_prefix = get_ancestor(prefix, last_level)
+        if last_prefix not in last_prefixes_set:
+            # Current prefix not a suffix of last level's prefixes.
+            return False
+    return True
+
+def get_ancestor(
+        index: tuple[bool, ...],
+        level: int) -> tuple[bool, ...]:
+    """
+    Helper function to determine the prefix of `index` at
+    `level`.
+    """
+    return index[:level + 1]
+
+
+
+
+
+
+

+8.2.4. Aggregation +

+

Aggregation involves simply adding up the output shares.

+
+
+def agg_init(self, agg_param: Poplar1AggParam) -> FieldVec:
+    (level, prefixes) = agg_param
+    field = self.idpf.current_field(level)
+    return field.zeros(len(prefixes))
+
+def agg_update(self,
+               agg_param: Poplar1AggParam,
+               agg_share: FieldVec,
+               out_share: FieldVec) -> FieldVec:
+    a = cast(list[Field], agg_share)
+    o = cast(list[Field], out_share)
+    return cast(FieldVec, vec_add(a, o))
+
+def merge(self,
+          agg_param: Poplar1AggParam,
+          agg_shares: list[FieldVec]) -> FieldVec:
+    (level, prefixes) = agg_param
+    field = self.idpf.current_field(level)
+    agg = cast(list[Field], field.zeros(len(prefixes)))
+    for agg_share in agg_shares:
+        agg = vec_add(agg, cast(list[Field], agg_share))
+    return cast(FieldVec, agg)
+
+
+
+
+
+
+

+8.2.5. Unsharding +

+

Finally, the Collector unshards the aggregate result by adding up the aggregate +shares.

+
+
+def unshard(
+        self,
+        agg_param: Poplar1AggParam,
+        agg_shares: list[FieldVec],
+        _num_measurements: int) -> list[int]:
+    agg = self.merge(agg_param, agg_shares)
+    return [x.int() for x in agg]
+
+
+
+
+
+
+

+8.2.6. Message Serialization +

+

This section defines serialization formats for messages exchanged over the +network while executing Poplar1. Messages are defined in the presentation +language of TLS as defined in Section 3 of [RFC8446].

+

Let poplar1 be an instance of Poplar1. In the remainder we use Fi as an +alias for poplar1.idpf.field_inner.ENCODED_SIZE, Fl as an alias for +poplar1.idpf.field_leaf.ENCODED_SIZE, and B as an alias for +poplar1.idpf.BITS.

+

Elements of the inner field are encoded in little-endian byte order (as defined +in Section 6.1) and are represented as follows:

+
+
+opaque Poplar1FieldInner[Fi];
+
+
+

Likewise, elements of the leaf field are encoded in little-endian byte order +(as defined in Section 6.1) and are represented as follows:

+
+
+opaque Poplar1FieldLeaf[Fl];
+
+
+
+
+
+8.2.6.1. Public Share +
+

The public share of the IDPF scheme in Section 8.3 consists of a sequence +of "correction words". A correction word has three components:

+
    +
  1. +

    the XOF seed of type bytes;

    +
    2. +
  2. +

    the control bits of type tuple[bool, bool]; and

    +
    3. +
  3. +

    the payload of type list[Field64] for the first BITS-1 words and +list[Field255] for the last word.

    +
    4. +
+

The encoding is a straightforward structure of arrays, except that the control +bits are packed as tightly as possible. The encoded public share is structured +as follows:

+
+
+struct {
+    opaque packed_control_bits[packed_len];
+    opaque seed[poplar1.idpf.KEY_SIZE*B];
+    Poplar1FieldInner payload_inner[Fi*poplar1.idpf.VALUE_LEN*(B-1)];
+    Poplar1FieldLeaf payload_leaf[Fl*poplar1.idpf.VALUE_LEN];
+} Poplar1PublicShare;
+
+
+

Here packed_len = (2*B + 7) // 8 is the length of the packed control bits. +Field packed_control_bits is encoded with the following function:

+
+
+packed_control_buf = [int(0)] * packed_len
+for i, bit in enumerate(control_bits):
+    packed_control_buf[i // 8] |= bit.int() << (i % 8)
+packed_control_bits = bytes(packed_control_buf)
+
+
+

It encodes each group of eight bits into a byte, in LSB to MSB order, padding the most +significant bits of the last byte with zeros as necessary, and returns the byte +array. Decoding performs the reverse operation: it takes in a byte array +and a number of bits, and returns a list of bits, extracting eight bits from +each byte in turn, in LSB to MSB order, and stopping after the requested number +of bits. If the byte array has an incorrect length, or if unused bits in the +last bytes are not zero, it throws an error:

+
+
+control_bits = []
+for i in range(length):
+    control_bits.append(bool(
+        (packed_control_bits[i // 8] >> (i % 8)) & 1
+    ))
+leftover_bits = packed_control_bits[-1] >> (
+    (length + 7) % 8 + 1
+)
+if (length + 7) // 8 != len(packed_control_bits) or \
+        leftover_bits != 0:
+    raise ValueError('trailing bits')
+
+
+
+
+
+
+
+8.2.6.2. Input Share +
+

Each input share is structured as follows:

+
+
+struct {
+    opaque idpf_key[poplar1.idpf.KEY_SIZE];
+    opaque corr_seed[poplar1.xof.SEED_SIZE];
+    Poplar1FieldInner corr_inner[Fi * 2 * (B- 1)];
+    Poplar1FieldLeaf corr_leaf[Fl * 2];
+} Poplar1InputShare;
+
+
+
+
+
+
+
+8.2.6.3. Prep Share +
+

Encoding of the prep share depends on the round of sketching: if the first +round, then each sketch share has three field elements; if the second round, +then each sketch share has one field element. The field that is used depends on +the level of the IDPF tree specified by the aggregation parameter, either the +inner field or the leaf field.

+

For the first round and inner field:

+
+
+struct {
+    Poplar1FieldInner sketch_share[Fi * 3];
+} Poplar1PrepShareRoundOneInner;
+
+
+

For the first round and leaf field:

+
+
+struct {
+    Poplar1FieldLeaf sketch_share[Fl * 3];
+} Poplar1PrepShareRoundOneLeaf;
+
+
+

For the second round and inner field:

+
+
+struct {
+    Poplar1FieldInner sketch_share;
+} Poplar1PrepShareRoundTwoInner;
+
+
+

For the second round and leaf field:

+
+
+struct {
+    Poplar1FieldLeaf sketch_share;
+} Poplar1PrepShareRoundTwoLeaf;
+
+
+
+
+
+
+
+8.2.6.4. Prep Message +
+

Likewise, the structure of the prep message for Poplar1 depends on the +sketching round and field. For the first round and inner field:

+
+
+struct {
+    Poplar1FieldInner[Fi * 3];
+} Poplar1PrepMessageRoundOneInner;
+
+
+

For the first round and leaf field:

+
+
+struct {
+    Poplar1FieldLeaf sketch[Fl * 3];
+} Poplar1PrepMessageRoundOneLeaf;
+
+
+

Note that these messages have the same structures as the prep shares for the +first round.

+

The second-round prep message is the empty string. This is because the sketch +shares are expected to sum to a particular value if the output shares are +valid; we represent a successful preparation with the empty string and +otherwise return an error.

+
+
+
+
+
+8.2.6.5. Aggregate Share +
+

The encoding of the aggregate share depends on whether the inner or leaf field +is used, and the number of candidate prefixes. Both of these are determined by + the aggregation parameter.

+

Let prefix_count denote the number of candidate prefixes. For the inner field:

+
+
+struct {
+    Poplar1FieldInner agg_share[Fi * prefix_count];
+} Poplar1AggShareInner;
+
+
+

For the leaf field:

+
+
+struct {
+    Poplar1FieldLeaf agg_share[Fl * prefix_count];
+} Poplar1AggShareLeaf;
+
+
+
+
+
+
+
+8.2.6.6. Aggregation Parameter +
+

The aggregation parameter is encoded as follows:

+
+
+struct {
+    uint16_t level;
+    uint32_t num_prefixes;
+    opaque encoded_prefixes[prefixes_len];
+} Poplar1AggParam;
+
+
+

The fields in this struct are: level, the level of the IDPF tree of each +prefixes; num_prefixes, the number of prefixes to evaluate; and +encoded_prefixes, the sequence of prefixes encoded into a byte string of +length prefixes_len. The prefixes are encoded with the following procedure:

+
+
+prefixes_len = ((level + 1) + 7) // 8 * len(prefixes)
+encoded_prefixes = bytearray()
+for prefix in prefixes:
+    for chunk in itertools.batched(prefix, 8):
+        byte_out = 0
+        for (bit_position, bit) in enumerate(chunk):
+            byte_out |= bit << (7 - bit_position)
+        encoded_prefixes.append(byte_out)
+
+
+

Decoding involves the following procedure:

+
+
+prefixes = []
+bytes_per_prefix = ((level + 1) + 7) // 8
+for chunk in itertools.batched(encoded_prefixes, bytes_per_prefix):
+    prefix = []
+    for i in range(level + 1):
+        byte_index = i // 8
+        bit_offset = 7 - (i % 8)
+        bit = (chunk[byte_index] >> bit_offset) & 1 != 0
+        prefix.append(bit)
+    prefixes.append(tuple(prefix))
+
+
+

Implementation note: the aggregation parameter includes the level of the IDPF +tree and the sequence of indices to evaluate. For implementations that perform +per-report caching across executions of the VDAF, this may be more information +than is strictly needed. In particular, it may be sufficient to convey which +indices from the previous execution will have their children included in the +next. This would help reduce communication overhead.

+
+
+
+
+
+
+
+
+

+8.3. IDPF Specification +

+

In this section we specify a concrete IDPF suitable for instantiating Poplar1. +The constant and type definitions required by the Idpf interface are given in +Table 17.

+

Our IDPF requires an XOF for deriving the output shares, as well as a variety +of other artifacts used internally. For performance reasons, we instantiate +this object using XofFixedKeyAes128 (Section 6.2.2) wherever +possible. See Section 9.6 for security considerations.

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 17: +Constants and type definitions for our concrete IDPF. +
ParameterValue
+ xof + + XofFixedKeyAes128 (Section 6.2.2)
+ SHARES + + 2 +
+ BITS +Any positive integer.
+ VALUE_LEN +Any positive integer.
+ KEY_SIZE + + xof.SEED_SIZE +
+ FieldInner + + Field64 (Table 3)
+ FieldLeaf + + Field255 (Table 3)
+
+
+
+

+8.3.1. Overview +

+

At a high level, the IDPF maps a key generator's input (alpha, beta_1, ..., +beta_BITS) onto a binary tree with BITS+1 levels, where each edge going from +a parent node to a left child is labeled 0, and each right edge is labeled +1. Then each leaf node corresponds to a bit string of length BITS, where the +labels on the path from the root to x contain the individual bits. Finally, +all nodes in the tree have an assigned value, with the nodes on the path +from the root to alpha having values beta_1, ..., beta_BITS, and all other +nodes having value 0.

+

The IDPF construction now boils down to secret-sharing the values at each node +of that tree in an efficient way. Note that explicitly representing the tree +requires O(2^BITS) space, so the generator cannot just compute additive shares +of it and send them to the two evaluators. Instead, the evaluators will +re-generate shares of the tree using a XOF (Section 6.2).

+

The basic observation is that if both evaluators have the same seed s of +length KEY_SIZE, then expanding s using a XOF will also result in the same +expansion. If we set the length of the XOF expansion to 2*KEY_SIZE, it can +then be split again into two seeds s_l, s_r, that can again serve as XOF +seeds. Now if we view seeds as XOR-shares of integers, and if evaluators have +the same seed at the root of the tree, then their expanded trees will form a +secret-shared tree of zeros. The actual construction will additionally use a +convert() function before each expansion, which maps seeds into the +appropriate output domain (see Section 8.3.4), generating +a new seed for the next level in the process.

+

The open task now is to ensure that evaluators have different seeds at nodes +that lie on the path to alpha, while having the same seeds on all other nodes. +This is done using so-called "correction words" included in the public share. +The correction words are conditionally added to the XOF output by both +evaluators. The condition here is a secret-shared bit, called a "control bit", +which indicates whether the current node is on the path to alpha or not. On +the path, the control bits add up to 1, meaning only one evaluator will add +the correction word to its XOF output. Off the path, either none or both +evaluators add the correction word, and so the seeds at the next level stay the +same.

+

What remains is to turn the (now pseudorandom) values on the path to alpha +into the desired beta values. This is done by including "value correction +words" in the public share, which are chosen such that when added with the +pseudorandom shares at the ith node on the path to alpha, they add up to +shares of beta_i.

+

In the following two sections we describe the algorithms for key generation in +full detail.

+
+
+
+
+

+8.3.2. Key Generation +

+

The description of the IDPF-key generation algorithm makes use of auxiliary +functions extend() and convert() defined in +Section 8.3.4.

+
+
+def gen(
+        self,
+        alpha: tuple[bool, ...],
+        beta_inner: list[list[Field64]],
+        beta_leaf: list[Field255],
+        ctx: bytes,
+        nonce: bytes,
+        rand: bytes) -> tuple[list[CorrectionWord], list[bytes]]:
+    if len(alpha) != self.BITS:
+        raise ValueError("incorrect alpha length")
+    if len(beta_inner) != self.BITS - 1:
+        raise ValueError("incorrect beta_inner length")
+    if len(rand) != self.RAND_SIZE:
+        raise ValueError("incorrect rand size")
+    if len(nonce) != self.NONCE_SIZE:
+        raise ValueError("incorrect nonce size")
+
+    key = [
+        rand[:XofFixedKeyAes128.SEED_SIZE],
+        rand[XofFixedKeyAes128.SEED_SIZE:],
+    ]
+
+    seed = key.copy()
+    ctrl = [False, True]
+    public_share = []
+    for level in range(self.BITS):
+        bit = alpha[level]
+        keep = int(bit)
+        lose = 1 - keep
+
+        (s0, t0) = self.extend(level, seed[0], ctx, nonce)
+        (s1, t1) = self.extend(level, seed[1], ctx, nonce)
+        seed_cw = xor(s0[lose], s1[lose])
+        ctrl_cw = (
+            t0[0] ^ t1[0] ^ (not bit),
+            t0[1] ^ t1[1] ^ bit,
+        )
+
+        # Implementation note: these conditional XORs and
+        # input-dependent array indices should be replaced with
+        # constant-time selects in practice in order to reduce
+        # leakage via timing side channels.
+        if ctrl[0]:
+            x0 = xor(s0[keep], seed_cw)
+            ctrl[0] = t0[keep] ^ ctrl_cw[keep]
+        else:
+            x0 = s0[keep]
+            ctrl[0] = t0[keep]
+        if ctrl[1]:
+            x1 = xor(s1[keep], seed_cw)
+            ctrl[1] = t1[keep] ^ ctrl_cw[keep]
+        else:
+            x1 = s1[keep]
+            ctrl[1] = t1[keep]
+        (seed[0], w0) = self.convert(level, x0, ctx, nonce)
+        (seed[1], w1) = self.convert(level, x1, ctx, nonce)
+
+        if level < self.BITS - 1:
+            b = cast(list[Field], beta_inner[level])
+        else:
+            b = cast(list[Field], beta_leaf)
+        if len(b) != self.VALUE_LEN:
+            raise ValueError(
+                "length of beta must match the value length"
+            )
+
+        w_cw = vec_add(vec_sub(b, w0), w1)
+        # Implementation note: this conditional negation should be
+        # replaced with a constant time select or a constant time
+        # multiplication in practice in order to reduce leakage via
+        # timing side channels.
+        if ctrl[1]:
+            for i in range(len(w_cw)):
+                w_cw[i] = -w_cw[i]
+
+        public_share.append((seed_cw, ctrl_cw, w_cw))
+    return (public_share, key)
+
+
+
+
+
+
+

+8.3.3. Key Evaluation +

+

The description of the IDPF-evaluation algorithm makes use of auxiliary +functions extend() and convert() defined in +Section 8.3.4.

+
+
+def eval(
+        self,
+        agg_id: int,
+        public_share: list[CorrectionWord],
+        key: bytes,
+        level: int,
+        prefixes: Sequence[tuple[bool, ...]],
+        ctx: bytes,
+        nonce: bytes) -> list[list[Field64]] | list[list[Field255]]:
+    if agg_id not in range(self.SHARES):
+        raise ValueError('aggregator id out of range')
+    if level not in range(self.BITS):
+        raise ValueError('level out of range')
+    if len(set(prefixes)) != len(prefixes):
+        raise ValueError('prefixes must be unique')
+
+    out_share = []
+    for prefix in prefixes:
+        if len(prefix) != level + 1:
+            raise ValueError('incorrect prefix length')
+
+        # The Aggregator's output share is the value of a node of
+        # the IDPF tree at the given `level`. The node's value is
+        # computed by traversing the path defined by the candidate
+        # `prefix`. Each node in the tree is represented by a seed
+        # (`seed`) and a control bit (`ctrl`).
+        seed = key
+        ctrl = bool(agg_id)
+        y: FieldVec
+        for current_level in range(level + 1):
+            bit = int(prefix[current_level])
+
+            # Implementation note: typically the current round of
+            # candidate prefixes would have been derived from
+            # aggregate results computed during previous rounds.
+            # For example, when using the IDPF to compute heavy
+            # hitters, a string whose hit count exceeded the
+            # given threshold in the last round would be the
+            # prefix of each `prefix` in the current round. (See
+            # [BBCGGI21, Section 5.1].) In this case, part of the
+            # path would have already been traversed.
+            #
+            # Re-computing nodes along previously traversed paths is
+            # wasteful. Implementations can eliminate this added
+            # complexity by caching nodes (i.e., `(seed, ctrl)`
+            # pairs) output by previous calls to `eval_next()`.
+            (seed, ctrl, y) = self.eval_next(
+                seed,
+                ctrl,
+                public_share[current_level],
+                current_level,
+                bit,
+                ctx,
+                nonce,
+            )
+        if agg_id == 0:
+            out_share.append(cast(list[Field], y))
+        else:
+            out_share.append(vec_neg(cast(list[Field], y)))
+    return cast(
+        list[list[Field64]] | list[list[Field255]],
+        out_share,
+    )
+
+def eval_next(
+        self,
+        prev_seed: bytes,
+        prev_ctrl: bool,
+        correction_word: CorrectionWord,
+        level: int,
+        bit: int,
+        ctx: bytes,
+        nonce: bytes) -> tuple[bytes, bool, FieldVec]:
+    """
+    Compute the next node in the IDPF tree along the path determined
+    by a candidate prefix. The next node is determined by `bit`, the
+    bit of the prefix corresponding to the next level of the tree.
+    """
+
+    seed_cw = correction_word[0]
+    ctrl_cw = correction_word[1]
+    w_cw = cast(list[Field], correction_word[2])
+    (s, t) = self.extend(level, prev_seed, ctx, nonce)
+
+    # Implementation note: these conditional operations and
+    # input-dependent array indices should be replaced with
+    # constant-time selects in practice in order to reduce leakage
+    # via timing side channels.
+    if prev_ctrl:
+        s[0] = xor(s[0], seed_cw)
+        s[1] = xor(s[1], seed_cw)
+        t[0] ^= ctrl_cw[0]
+        t[1] ^= ctrl_cw[1]
+
+    next_ctrl = t[bit]
+    convert_output = self.convert(level, s[bit], ctx, nonce)
+    next_seed = convert_output[0]
+    y = cast(list[Field], convert_output[1])
+    # Implementation note: this conditional addition should be
+    # replaced with a constant-time select in practice in order to
+    # reduce leakage via timing side channels.
+    if next_ctrl:
+        for i in range(len(y)):
+            y[i] += w_cw[i]
+
+    return (next_seed, next_ctrl, cast(FieldVec, y))
+
+
+
+
+
+
+

+8.3.4. Auxiliary Functions +

+
+
+def extend(
+        self,
+        level: int,
+        seed: bytes,
+        ctx: bytes,
+        nonce: bytes) -> tuple[list[bytes], list[bool]]:
+    xof = self.current_xof(
+        level,
+        seed,
+        format_dst(1, 0, 0) + ctx,
+        nonce,
+    )
+    s = [
+        bytearray(xof.next(self.KEY_SIZE)),
+        bytearray(xof.next(self.KEY_SIZE)),
+    ]
+    # Use the least significant bits as the control bit correction,
+    # and then zero it out. This gives effectively 127 bits of
+    # security, but reduces the number of AES calls needed by 1/3.
+    t = [bool(s[0][0] & 1), bool(s[1][0] & 1)]
+    s[0][0] &= 0xFE
+    s[1][0] &= 0xFE
+    return ([bytes(s[0]), bytes(s[1])], t)
+
+def convert(
+        self,
+        level: int,
+        seed: bytes,
+        ctx: bytes,
+        nonce: bytes) -> tuple[bytes, FieldVec]:
+    xof = self.current_xof(
+        level,
+        seed,
+        format_dst(1, 0, 1) + ctx,
+        nonce,
+    )
+    next_seed = xof.next(self.KEY_SIZE)
+    field = self.current_field(level)
+    w = xof.next_vec(field, self.VALUE_LEN)
+    return (next_seed, cast(FieldVec, w))
+
+def current_xof(self,
+                level: int,
+                seed: bytes,
+                dst: bytes,
+                nonce: bytes) -> Xof:
+    if level < self.BITS-1:
+        return XofFixedKeyAes128(seed, dst, nonce)
+    return XofTurboShake128(seed, dst, nonce)
+
+
+
+
+
+
+
+
+
+
+

+9. Security Considerations +

+

VDAFs (Section 5) have two essential security goals:

+
    +
  1. +

    Privacy: an attacker that controls the Collector and a subset of Clients and +Aggregators learns nothing about the measurements of honest Clients beyond +what it can deduce from the aggregate result. We assume the attacker +controls the entire network except for channels between honest Clients and +honest Aggregators. In particular, it cannot forge or prevent transmission +of messages on these channels.

    +
    2. +
  2. +

    Robustness: an attacker that controls a subset of Clients cannot cause the +Collector to compute anything other than the aggregate of the measurements +of honest Clients. We assume the attacker eavesdrops on the network but does +not control transmission of messages between honest parties.

    +
    3. +
+

Formal definitions of privacy and robustness can be found in [DPRS23]. A VDAF +is the core cryptographic primitive of a protocol that achieves the above +privacy and robustness goals. It is not sufficient on its own, however. The +application will need to assure a few security properties, for example:

+ +

In such an environment, a VDAF provides the high-level privacy property +described above: the Collector learns only the aggregate result, and nothing +about individual measurements aside from what can be inferred from the +aggregate result. The Aggregators learn neither individual measurements nor +the aggregate result. The Collector is assured that the aggregate statistic +accurately reflects the inputs as long as the Aggregators correctly executed +their role in the VDAF.

+

On their own, VDAFs do not provide:

+
    +
  1. +

    Mitigation of Sybil attacks [Dou02]. In this attack, the adversary +observes a subset of input shares transmitted by a Client it is interested +in. It allows the input shares to be processed, but corrupts and picks bogus +measurements for the remaining Clients. Applications can guard against +these risks by adding additional controls on report submission, such as +Client authentication and rate limits.

    +
    2. +
  2. +

    Differential privacy [Dwo06]. Depending on the distribution of the +measurements, the aggregate result itself can still leak a significant +amount of information about an individual measurement or the person that +generated it.

    +
    3. +
  3. +

    Robustness in the presence of a malicious Aggregator. An Aggregator can, +without detection, manipulate the aggregate result by modifying its own +aggregate share.

    +
    4. +
  4. +

    Guaranteed output delivery [GSZ20]. An attacker that controls transmission +of messages between honest parties can prevent computation of the aggregate +result by dropping messages.

    +
    5. +
+
+
+

+9.1. The Verification Key +

+

The Aggregators are responsible for exchanging the verification key in advance +of executing the VDAF. Any procedure is acceptable as long as the following +conditions are met:

+
    +
  1. +

    To ensure robustness of the computation, the Aggregators MUST NOT reveal the +verification key to the Clients. Otherwise, a malicious Client might be able +to exploit knowledge of this key to craft an invalid report that would be +accepted by the Aggregators.

    +
    2. +
  2. +

    To ensure privacy of the measurements, the Aggregators MUST commit to the +verification key prior to processing reports generated by Clients. +Otherwise, the attacker may be able to craft a verification key that, for a +given report, causes an honest Aggregator to leak information about the +measurement during preparation.

    +
    3. +
+

Meeting these requirements is relatively straightforward. For example, the +Aggregators may designate one of their peers to generate the verification key +and distribute it to the others. To assure Clients of key commitment, the +Clients and (honest) Aggregators SHOULD bind the verification key to the +application context. For instance, the "task ID" of DAP [DAP] could be set to +the hash of the verification key; then as long as honest Aggregators only +consume reports for the task indicated by the Client, forging a new key after +the fact would reduce to finding collisions in the underlying hash function. +(Keeping the key secret from the Clients would require the hash function to be +one-way.) However, since rotating the key implies rotating the task ID, this +scheme would not allow key rotation over the lifetime of a task.

+
+
+
+
+

+9.2. The Nonce +

+

The sharding and preparation steps of VDAF execution depend on a nonce +associated with the Client's report. To ensure privacy of the underlying +measurement, the Client MUST generate this nonce using a CSPRNG. This is +required in order to leverage security analysis for the privacy definition of +[DPRS23], which assumes the nonce is chosen at random prior to generating the +report. Uniqueness of the nonce is not sufficient because the verification key +is controlled by the attacker.

+

Other security considerations may require the nonce to be non-repeating in a +given context. For example, to achieve differential privacy it is necessary to +avoid "over exposing" a report by including it too many times in a single batch +or across multiple batches. It is RECOMMENDED that the nonce generated by the +Client be used by the Aggregators for replay protection.

+
+
+
+
+

+9.3. The Public Share +

+

The Aggregators MUST ensure they have both received the same public share from +the Client. It is sufficient, for example, to exchange a hash of the public +share over a secure channel.

+
+
+
+
+

+9.4. The Aggregation Parameter +

+

As described in Section 4.3 and Section 5.3 +respectively, DAFs and VDAFs may impose restrictions on the re-use of reports. +For Prio3, reports should only be aggregated once; for Poplar1, reports may be +aggregated multiple times, but never twice at the same level of the tree. +Otherwise, we risk re-using correlated randomness, which might compromise +confidentiality of the Client's measurement.

+

Higher level applications that use DAFs or VDAFs MUST enforce aggregation +parameter validity. In particular, prior to beginning preparation with an +aggregation parameter provided by the Collector, they MUST invoke is_valid() +to decide if the parameter is valid given the sequence of previously accepted +parameters.

+

Note that aggregating a batch of reports multiple times, even with a valid +sequence of aggregation parameters, can result in information leakage beyond +what is used by the application.

+

For example, when Poplar1 is used for heavy hitters, the Aggregators learn not +only the heavy hitters themselves, but also the prefix tree (as defined in +Section 8) computed along the way. Indeed, this leakage is inherent to any +construction that uses an IDPF (Section 8.1) in the same way. Depending on the +distribution of the measurements, the prefix tree can leak a significant amount +of information about unpopular inputs. For instance, it is possible (though +perhaps unlikely) for a large set of non-heavy-hitter values to share a common +prefix, which would be leaked by a prefix tree with a sufficiently small +threshold.

+

A malicious adversary controlling the Collector and one of the Aggregators can +further turn arbitrary non-heavy prefixes into heavy ones by tampering with the +IDPF output at any position. While our construction ensures that the nodes +evaluated at one level are children of the nodes evaluated at the previous +level, this still may allow an adversary to discover individual non-heavy +strings. We call this a "steering attack".

+

The only practical, general-purpose defense against steering attacks is to +compose Poplar1 with some mechanism for differential privacy. It is therefore +RECOMMENDED to use differential privacy for any heavy-hitter type application.

+
+
+
+
+

+9.5. Safe Usage of IDPF Outputs +

+

The arithmetic sketch described in Section 8 is used by the Aggregators to check +that the shares of the vector obtained by evaluating a Client's IDPF at a +sequence of candidate prefixes has at most one non-zero value, and that the +non-zero value is 1. Depending on how the values are used, the arithmetic sketch +on its own may not be sufficient for robustness of the application. In +particular, a malicious Client may attempt to influence the computation by +choosing an IDPF that evaluates to 1 at more than one node at a given +level of the tree.

+

This issue can be mitigated by using an IDPF that is extractable as defined +in Appendix D of [BBCGGI21]. Extractability ensures that, for a particular +level of the tree, it is infeasible for an attacker to control values of the +IDPF such that it takes on chosen non-zero values at more than one node. (It +can practically only achieve the zero function, a point function, or a +pseudorandom function.)

+

The IDPF specified in Section 8.1 only guarantees extractability at the last level +of the tree. (This is by virtue of using a larger field for the leaves than for +inner nodes and using an XOF to derive leaves that is safe to model as a random +oracle. See Section 9.6.) For intermediate levels, it is feasible for a +client to produce IDPF shares with two controlled non-zero nodes.

+

This is not an issue for running heavy hitters, since (1) each node in the +prefix tree is a child of a previously traversed node, (2) the arithmetic sketch +would detect double voting at every level of the prefix tree, and (3) the IDPF +is extractable at the last level of the tree. However, the lack of +extractability at intermediate levels may result in attacks on the robustness +of certain applications.

+

Thus applications SHOULD NOT use prefix counts for intermediate levels for any +purpose beyond computing the prefix tree for heavy hitters.

+
+
+
+
+

+9.6. Safe Usage of XOFs +

+

In the security analyses of our protocols, XOFs (Section 6.2) are usually modeled +as random oracles. XofTurboShake128 is designed to be indifferentiable from a +random oracle [MRH04], making it a suitable choice for most situations.

+

The one exception is the IDPF of Section 8.3. Here, a random oracle is not +needed to prove privacy, since the analysis of [BBCGGI21], Proposition 1, only +requires a Pseudorandom Generator (PRG). As observed in [GKWY20], a PRG can be +instantiated from a correlation-robust hash function H. Informally, +correlation robustness requires that for a random r, H(xor(r, x)) is +computationally indistinguishable from a random function of x. A PRG can +therefore be constructed as

+
+
+PRG(r) = H(xor(r, 1)) || H(xor(r, 2)) || ...
+
+
+

since each individual hash function evaluation is indistinguishable from a random +function.

+

XofFixedKeyAes128 in Section 6.2.2 implements a correlation-robust +hash function using fixed-key AES. For security, it assumes that AES with a +fixed key can be modeled as a random permutation [GKWY20]. Additionally, we +use a different AES key for every client, which in the ideal cipher model leads +to better concrete security [GKWWY20].

+

We note that for robustness, the analysis of [BBCGGI21] still assumes a +random oracle to make the IDPF extractable. We therefore use XofTurboShake128 +instead for the last level of the tree. It is important that XofTurboShake128 +supports many seed lengths, in particular 16 bytes, as this is the seed size +for the inner levels.

+

While XofFixedKeyAes128 has been shown to be differentiable from a random +oracle [GKWWY20], there are no known attacks exploiting this difference. We +also stress that even if the IDPF is not extractable, Poplar1 guarantees that +every client can contribute to at most one prefix among the ones being +evaluated by the helpers.

+
+
+
+
+

+9.7. Choosing FLP Parameters +

+

Prio3 and other systems built from the FLP of Section 7.3 may benefit from +choosing a field size that is as small as possible. Generally speaking, a +smaller field results in lower communication and storage costs. Care must be +taken, however, since a smaller field also results in degraded (or even +vacuous) robustness.

+

Different variants of Prio3 (Section 7) use different field sizes: Prio3Count +and Prio3Sum use Field64; but Prio3SumVec, Prio3Histogram, and +Prio3MultihotCountVec all use Field128, a field that is twice as large as +Field64. This is due to the use of joint randomness (Section 7.1) in the latter +variants. Joint randomness allows for more flexible circuit design (see +Section 7.3.1.1), but opens up Prio3 to offline attacks in +which the attacker searches for input shares for an invalid measurement that +derive joint randomness that causes the circuit to accept. Choosing a large +enough field ensures this computation is too expensive to be feasible. (See +[DPRS23], Theorem 1.) Note that privacy is not susceptible to such attacks.

+

Another way to mitigate this issue (or improve robustness in general) is to +generate and verify multiple, independent proofs. (See Section 7.1.2.) For +Prio3, the PROOFS parameter controls the number of proofs (at least one) that +are generated and verified. In general the soundness error of the FLP is given +by the following formula:

+
+
+(circuit_soundness + flp_soundness)^PROOFS
+
+
+

where:

+
    +
  • +

    circuit_soundness is the soundness of the validity circuit +(Section 7.3.2)

    +
    • +
  • +

    flp_soundness is the base soundness of the proof system ([BBCGGI19], +Theorem 4.3)

    +
    • +
+

For circuits involving joint randomness, we aim for the soundness error to be +close to 2^-128 in order to mitigate offline attacks. Such circuits MUST use +Field128 with at least one proof or Field64 with at least three proofs. +Depending on the circuit, Field64 with two proofs might have significantly +lower soundness than Field128 with one proof.

+

We stress that weak parameters (too small a field, too few proofs, or both) can +be exploited to attack any aggregation task using those parameters. To +mitigate offline attacks, it is necessary to disable all tasks that use the +weak parameters.

+
+
+
+
+

+9.8. Choosing the Number of Aggregators +

+

Two Aggregators are required for privacy in our threat model, but some (V)DAFs, +including Prio3 (Section 7), allow for any number of Aggregators, only one of +which needs to be trusted in order for the computation to be private. To hedge +against corruptions that happen during the course of the attack, deployments +may consider involving more than two Aggregators as described for example in +Section 5.7.2. Note however that some schemes are not compatible with this mode +of operation, such as Poplar1.

+
+
+
+
+

+9.9. Defense-in-Depth Measures +

+

Prio3 and Poplar1 are designed to resist some attacks that fall outside the +main threat model for VDAFs.

+

Broadly speaking, domain separation is used to prevent cross protocol attacks, +in which data from evaluation of one VDAF translates to an attack against another. +For example:

+
    +
  1. +

    Weak entropy sources: the VDAF algorithm ID is bound to each XOF invocation, +thereby ensuring the outputs are different between VDAF invocations, even if +the underlying randomness is the same. For example, two different instances +of Prio3 would compute different measurement shares.

    +
    2. +
  2. +

    Weak parameters: Prio3 variants that require joint randomness are subject to +offline attacks against robustness. These attacks are feasible if the field +size or number of proofs is sufficiently small. (See +Section 9.7.) The joint randomness derivation is bound to both +the field (via the algorithm ID) and the number of proofs, thereby ensuring +that joint randomness derived for weak parameters is not reused for stronger +parameters. In addition, the joint randomness is bound to the application +context, meaning any work the attacker does to attack some application is +not useful for other applications that use the same parameters.

    +
    3. +
+

There are also some important limitations to be aware of. For example, Prio3 +provides domain separation between families of circuits, but does not provide +domain separation between instances of a circuit. Concretely, it is possible +for Aggregators to accept a report for Prio3SumVec from a Client who disagrees +with them on the value of bits and length. This is because there is no +binding of the circuit parameters to the computation.

+
+
+
+
+
+
+

+10. IANA Considerations +

+

IANA is requested to make one new registry:

+ +

This registry should be created under the heading "Verifiable Distributed +Aggregation Functions (VDAF)", and administered under the Specification Required +policy [RFC8126].

+

The "VDAF Identifiers" registry lists identifiers for Distributed Aggregation +Functions (DAFs) and Verifiable Distributed Aggregation Functions (VDAFs). These +identifiers are four-byte values, so the minimum possible value is 0x00000000 +and the maximum possible value is 0xffffffff.

+

Template:

+ +

The initial contents of the registry are as follows:

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 18: +Verifiable Distributed Aggregation Function Identifiers Registry +
ValueSchemeTypeReference
+ 0x00000000 +Reservedn/aRFC XXXX
+ 0x00000001 +Prio3CountVDAF + Section 7.4.1 of RFC XXXX
+ 0x00000002 +Prio3SumVDAF + Section 7.4.2 of RFC XXXX
+ 0x00000003 +Prio3SumVecVDAF + Section 7.4.3 of RFC XXXX
+ 0x00000004 +Prio3HistogramVDAF + Section 7.4.4 of RFC XXXX
+ 0x00000005 +Prio3MultihotCountVecVDAF + Section 7.4.5 of RFC XXXX
+ 0x00000006 +Poplar1VDAF + Section 8.2 of RFC XXXX
+ 0xFFFF0000 to 0xFFFFFFFF +Reserved for Private Usen/an/a
+
+

(RFC EDITOR: Please replace "RFC XXXX" above with the RFC number assigned to +this document.)

+

VDAF identifiers are used for domain separation, as described in +Section 6.2.3. Domain separation guards against cross protocol attacks and +certain failures of entropy sources. See Section 9.9.

+

The benefits of domain separation are undermined if different VDAFs are used +with the same VDAF Identifier. The "Reserved for Private Use" code points +should thus be used judiciously, because they provide no defense against such +collisions. Applications SHOULD prefer the use of registered code points.

+
+
+
+
+

+11. References +

+
+
+

+11.1. Normative References +

+
+
[AES]
+
+Dworkin, M. J., Barker, E., Nechvatal, J. R., Foti, J., Bassham, L. E., Roback, E., and J. Dray Jr, "Advanced Encryption Standard (AES)", , <https://www.nist.gov/publications/advanced-encryption-standard-aes>.
+
+
[RFC2119]
+
+Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/rfc/rfc2119>.
+
+
[RFC8126]
+
+Cotton, M., Leiba, B., and T. Narten, "Guidelines for Writing an IANA Considerations Section in RFCs", BCP 26, RFC 8126, DOI 10.17487/RFC8126, , <https://www.rfc-editor.org/rfc/rfc8126>.
+
+
[RFC8174]
+
+Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/rfc/rfc8174>.
+
+
[RFC8446]
+
+Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, , <https://www.rfc-editor.org/rfc/rfc8446>.
+
+
[TurboSHAKE]
+
+Viguier, B., Wong, D., Van Assche, G., Dang, Q., and J. Daemen, "KangarooTwelve and TurboSHAKE", Work in Progress, Internet-Draft, draft-irtf-cfrg-kangarootwelve-14, , <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-kangarootwelve-14>.
+
+
+
+
+
+
+

+11.2. Informative References +

+
+
[AGJOP21]
+
+Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A. Polychroniadou, "Prio+: Privacy Preserving Aggregate Statistics via Boolean Shares", Security and Cryptography for Networks (SCN), , <https://ia.cr/2021/576>.
+
+
[BBCGGI19]
+
+Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and Y. Ishai, "Zero-Knowledge Proofs on Secret-Shared Data via Fully Linear PCPs", Crypto, , <https://ia.cr/2019/188>.
+
+
[BBCGGI21]
+
+Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and Y. Ishai, "Lightweight Techniques for Private Heavy Hitters", IEEE Security & Privacy (S&P), , <https://ia.cr/2021/017>.
+
+
[BBCGGI23]
+
+Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and Y. Ishai, "Arithmetic Sketching", Crypto, , <https://ia.cr/2023/1012>.
+
+
[BGI15]
+
+Boyle, E., Gilboa, N., and Y. Ishai, "Function Secret Sharing", Eurocrypt, , <https://www.iacr.org/archive/eurocrypt2015/90560300/90560300.pdf>.
+
+
[CGB17]
+
+Boneh, D. and H. Corrigan-Gibbs, "Prio: Private, Robust, and Scalable Computation of Aggregate Statistics", USENIX Symposium on Networked Systems Design and Implementation (NSDI), , <https://dl.acm.org/doi/10.5555/3154630.3154652>.
+
+
[DAP]
+
+Geoghegan, T., Patton, C., Pitman, B., Rescorla, E., and C. A. Wood, "Distributed Aggregation Protocol for Privacy Preserving Measurement", Work in Progress, Internet-Draft, draft-ietf-ppm-dap-12, , <https://datatracker.ietf.org/doc/html/draft-ietf-ppm-dap-12>.
+
+
[Dou02]
+
+Douceur, J. R., "The Sybil Attack", International Workshop on Peer-to-Peer Systems (IPTPS), , <https://doi.org/10.1007/3-540-45748-8_24>.
+
+
[DPRS23]
+
+Hannah Davis, Christopher Patton, Mike Rosulek, and Phillipp Schoppmann, "Verifiable Distributed Aggregation Functions", Privacy Enhancing Technologies Symposium (PETS), , <https://ia.cr/2023/130>.
+
+
[Dwo06]
+
+Cynthia Dwork, "Differential Privacy", International Colloquium on Automata, Languages, and Programming (ICALP), , <https://link.springer.com/chapter/10.1007/11787006_1>.
+
+
[ENPA]
+
+"Exposure Notification Privacy-preserving Analytics (ENPA) White Paper", , <https://covid19-static.cdn-apple.com/applications/covid19/current/static/contact-tracing/pdf/ENPA_White_Paper.pdf>.
+
+
[EPK14]
+
+Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response", ACM Conference on Computer and Communications Security (CCS), , <https://dl.acm.org/doi/10.1145/2660267.2660348>.
+
+
[GI14]
+
+Gilboa, N. and Y. Ishai, "Distributed Point Functions and Their Applications", Eurocrypt, , <https://link.springer.com/chapter/10.1007/978-3-642-55220-5_35>.
+
+
[GKWWY20]
+
+Guo, C., Katz, J., Wang, X., Weng, C., and and Y. Yu, "Better concrete security for half-gates garbling (in the multi-instance setting)", Crypto, , <https://link.springer.com/chapter/10.1007/978-3-030-56880-1_28>.
+
+
[GKWY20]
+
+Guo, C., Katz, J., Wang, X., and Y. Yu, "Efficient and Secure Multiparty Computation from Fixed-Key Block Ciphers", IEEE Security & Privacy (S&P), , <https://eprint.iacr.org/2019/074>.
+
+
[GSZ20]
+
+Goyal, V., Song, Y., and C. Zhu, "Guaranteed Output Delivery Comes Free in Honest Majority MPC", Crypto, , <https://link.springer.com/chapter/10.1007/978-3-030-56880-1_22>.
+
+
[MPDST25]
+
+Mouris, D., Patton, C., Davis, H., Sarkar, P., and N. G. Tsoutsos, "Mastic: Private Weighted Heavy-Hitters and Attribute-Based Metrics", Privacy Enhancing Technologies Symposium (PETS), , <https://eprint.iacr.org/2024/221>.
+
+
[MPRV09]
+
+Mironov, I., Pandey, O., Reingold, O., and S. Vadhan, "Computational Differential Privacy", Crypto, , <https://link.springer.com/chapter/10.1007/978-3-642-03356-8_8>.
+
+
[MRH04]
+
+Maurer, U., Renner, R., and C. Holenstein, "Indifferentiability, impossibility results on reductions, and applications to the random oracle methodology", Theory of Cryptography (TCC), , <https://doi.org/10.1007/978-3-540-24638-1_2>.
+
+
[OriginTelemetry]
+
+"Origin Telemetry", , <https://web.archive.org/web/20221025174046/https://firefox-source-docs.mozilla.org/toolkit/components/telemetry/collection/origin.html>.
+
+
[SML24]
+
+Satriawan, A., Mareta, R., and H. Lee, "A Complete Beginner Guide to the Number Theoretic Transform (NTT)", IEEE Access, vol. 11, , <https://eprint.iacr.org/2024/585>.
+
+
[TestVectors]
+
+"Test vectors for Prio3 and Poplar1", commit hash 1891650, , <https://github.com/cfrg/draft-irtf-cfrg-vdaf>.
+
+
+
+
+
+
+
+
+

+Appendix A. FLP Gadgets +

+

This section defines gadgets used to specify variants of Prio3 defined in +Section 7.4 and to construct FLPs as defined in Section 7.3.

+
+
+

+A.1. Multiplication +

+

The multiplication gadget takes in a pair of inputs and multiplies them +together. The degree of this circuit is 2.

+
+
+class Mul(Gadget[F]):
+    ARITY = 2
+    DEGREE = 2
+
+    def eval(self, _field: type[F], inp: list[F]) -> F:
+        return inp[0] * inp[1]
+
+    def eval_poly(self,
+                  field: type[F],
+                  inp_poly: list[list[F]]) -> list[F]:
+        return poly_mul(field, inp_poly[0], inp_poly[1])
+
+
+
+
+
+
+

+A.2. Polynomial Evaluation +

+

The polynomial-evaluation gadget takes in a single input x and returns +p(x), where p is a polynomial specified by the user. Its degree is the same +as p.

+
+
+class PolyEval(Gadget[F]):
+    ARITY = 1
+    p: list[int]  # polynomial coefficients
+
+    def __init__(self, p: list[int]):
+        """
+        Instantiate this gadget with the given polynomial.
+        """
+        # Strip leading zeros.
+        for i in reversed(range(len(p))):
+            if p[i] != 0:
+                p = p[:i+1]
+                break
+        if len(p) < 1:
+            raise ValueError('invalid polynomial: zero length')
+
+        self.p = p
+        self.DEGREE = len(p) - 1
+
+    def eval(self, field: type[F], inp: list[F]) -> F:
+        p = [field(coeff) for coeff in self.p]
+        return poly_eval(field, p, inp[0])
+
+    def eval_poly(self,
+                  field: type[F],
+                  inp_poly: list[list[F]]) -> list[F]:
+        p = [field(coeff) for coeff in self.p]
+        out = [field(0)] * (self.DEGREE * len(inp_poly[0]))
+        out[0] = p[0]
+        x = inp_poly[0]
+        for i in range(1, len(p)):
+            for j in range(len(x)):
+                out[j] += p[i] * x[j]
+            x = poly_mul(field, x, inp_poly[0])
+        return poly_strip(field, out)
+
+
+
+
+
+
+

+A.3. Parallel Sum +

+

The parallel-sum gadget applies a subcircuit to multiple inputs in parallel, +then returns the sum of the results. The arity of the gadget is equal to count +times the arity of the subcircuit, where count denotes the number of times the +subcircuit is called. The degree of the gadget is equal to the degree of the +subcircuit.

+
+
+class ParallelSum(Gadget[F]):
+    subcircuit: Gadget[F]
+    count: int
+
+    def __init__(self, subcircuit: Gadget[F], count: int):
+        self.subcircuit = subcircuit
+        self.count = count
+        self.ARITY = subcircuit.ARITY * count
+        self.DEGREE = subcircuit.DEGREE
+
+    def eval(self, field: type[F], inp: list[F]) -> F:
+        out = field(0)
+        for i in range(self.count):
+            start_index = i * self.subcircuit.ARITY
+            end_index = (i + 1) * self.subcircuit.ARITY
+            out += self.subcircuit.eval(
+                field,
+                inp[start_index:end_index],
+            )
+        return out
+
+    def eval_poly(self,
+                  field: type[F],
+                  inp_poly: list[list[F]]) -> list[F]:
+        output_poly_length = self.DEGREE * (len(inp_poly[0]) - 1) + 1
+        out_sum = [field(0) for _ in range(output_poly_length)]
+        for i in range(self.count):
+            start_index = i * self.subcircuit.ARITY
+            end_index = (i + 1) * self.subcircuit.ARITY
+            out_current = self.subcircuit.eval_poly(
+                field,
+                inp_poly[start_index:end_index]
+            )
+            for j in range(output_poly_length):
+                out_sum[j] += out_current[j]
+        return poly_strip(field, out_sum)
+
+
+
+
+
+
+

+A.4. Shims for Generating and Querying Proofs +

+

This section specifies two "shim" gadgets, one for generating an FLP as +specified in Section 7.3.3 and another for querying an FLP +as specified in Section 7.3.4.

+
+
+class ProveGadget(Gadget[F]):
+    """
+    Gadget wrapper that records the input wires for each evaluation.
+    """
+
+    def __init__(self,
+                 field: type[F],
+                 wire_seeds: list[F],
+                 g: Gadget[F],
+                 g_calls: int):
+        p = next_power_of_2(1 + g_calls)
+        self.inner = g
+        self.ARITY = g.ARITY
+        self.DEGREE = g.DEGREE
+        self.wires = []
+        self.k = 0  # evaluation counter
+        for s in wire_seeds:
+            wire = field.zeros(p)
+            wire[0] = s  # set the wire seed
+            self.wires.append(wire)
+
+    def eval(self, field: type[F], inp: list[F]) -> F:
+        self.k += 1
+        for j in range(len(inp)):
+            self.wires[j][self.k] = inp[j]
+        return self.inner.eval(field, inp)
+
+    def eval_poly(self,
+                  field: type[F],
+                  inp_poly: list[list[F]]) -> list[F]:
+        return self.inner.eval_poly(field, inp_poly)
+
+    @classmethod
+    def wrap(cls,
+             valid: Valid[Measurement, AggResult, F],
+             prove_rand: list[F],
+             ) -> Valid[Measurement, AggResult, F]:
+        """
+        Make a copy of `valid` with each gadget wrapped for recording
+        the wire inputs. `prove_rand` is used to produce the wire
+        seeds for each gadget.
+        """
+        wrapped_gadgets: list[Gadget[F]] = []
+        for (g, g_calls) in zip(valid.GADGETS, valid.GADGET_CALLS):
+            (wire_seeds, prove_rand) = front(g.ARITY, prove_rand)
+            wrapped = cls(valid.field, wire_seeds, g, g_calls)
+            wrapped_gadgets.append(wrapped)
+        wrapped_valid = deepcopy(valid)
+        wrapped_valid.GADGETS = wrapped_gadgets
+        return wrapped_valid
+
+class QueryGadget(Gadget[F]):
+    """
+    Gadget wrapper that records the input wires for each evaluation.
+    Rather than evaluate the circuit, use the provided gadget
+    polynomial to produce the output.
+    """
+
+    def __init__(
+            self,
+            field: type[F],
+            wire_seeds: list[F],
+            gadget_poly: list[F],
+            g: Gadget[F],
+            g_calls: int):
+        p = next_power_of_2(1 + g_calls)
+        self.alpha = field.gen() ** (field.GEN_ORDER // p)
+        self.poly = gadget_poly
+        self.ARITY = g.ARITY
+        self.DEGREE = g.DEGREE
+        self.wires = []
+        self.k = 0
+        for s in wire_seeds:
+            wire = field.zeros(p)
+            wire[0] = s  # set the wire seed
+            self.wires.append(wire)
+
+    def eval(self, field: type[F], inp: list[F]) -> F:
+        self.k += 1
+        for j in range(len(inp)):
+            self.wires[j][self.k] = inp[j]
+        return poly_eval(field, self.poly, self.alpha ** self.k)
+
+    @classmethod
+    def wrap(cls,
+             valid: Valid[Measurement, AggResult, F],
+             proof: list[F]) -> Valid[Measurement, AggResult, F]:
+        wrapped_gadgets: list[Gadget[F]] = []
+        for (g, g_calls) in zip(valid.GADGETS, valid.GADGET_CALLS):
+            p = next_power_of_2(1 + g_calls)
+            gadget_poly_len = g.DEGREE * (p - 1) + 1
+            (wire_seeds, proof) = front(g.ARITY, proof)
+            (gadget_poly, proof) = front(gadget_poly_len, proof)
+            wrapped = cls(valid.field,
+                          wire_seeds,
+                          gadget_poly,
+                          g, g_calls)
+            wrapped_gadgets.append(wrapped)
+        wrapped_valid = deepcopy(valid)
+        wrapped_valid.GADGETS = wrapped_gadgets
+        return wrapped_valid
+
+
+
+
+
+
+
+
+

+Appendix B. VDAF Preparation State +

+

This section lists the classes used to define each Aggregator's state during +VDAF preparation (Section 5.7).

+
+
+class State:
+    pass
+
+class Start(State):
+    pass
+
+class Continued(State, Generic[PrepState]):
+    def __init__(self, prep_state: PrepState, prep_round: int):
+        self.prep_state = prep_state
+        self.prep_round = prep_round
+
+    def __eq__(self, other: object) -> bool:
+        return isinstance(other, Continued) and \
+            self.prep_state == other.prep_state and \
+            self.prep_round == other.prep_round
+
+class Finished(State, Generic[OutShare]):
+    def __init__(self, out_share: OutShare):
+        self.out_share = out_share
+
+    def __eq__(self, other: object) -> bool:
+        return isinstance(other, Finished) and \
+            self.out_share == other.out_share
+
+class Rejected(State):
+    pass
+
+
+
+
+
+
+

+Appendix C. Test Vectors +

+

Test vectors for Prio3 (Section 7) and Poplar1 (Section 8) are available at +[TestVectors]. The test vector directory, test_vec/vdaf, contains a set of +JSON files. Each file contains a test vector for an instance of class Vdaf as +defined in Section 5. A test vector covers sharding, preparation, aggregation, +and unsharding of a batch of several measurements. The test vector schema is +defined below.

+
+
+

+C.1. Schema +

+
+
+ctx:
+
+

The application context string encoded in hexadecimal.

+
+
+
+verify_key:
+
+

The verification key encoded in hexadecimal.

+
+
+
+agg_param:
+
+

The aggregation parameter encoded in hexadecimal.

+
+
+
+prep:
+
+

A list of objects with the following schema:

+
+
+measurement:
+
+

The measurement of type Measurement defined by the VDAF.

+
+
+
+nonce:
+
+

The nonce encoded in hexadecimal.

+
+
+
+rand:
+
+

The sharding randomness encoded in hexadecimal.

+
+
+
+public_share:
+
+

The expected public share encoded in hexadecimal.

+
+
+
+input_shares:
+
+

The expected list of input shares, each encoded in hexadecimal.

+
+
+
+prep_shares:
+
+

The expected list of prep shares generated by each Aggregator at each +round of preparation, encoded in hexadecimal.

+
+
+
+prep_messages:
+
+

The expected list of prep messages for each round of preparation, encoded +in hexadecimal.

+
+
+
+out_shares:
+
+

The expected list of output shares, encoded in hexadecimal.

+
+
+
+
+
+
+agg_shares:
+
+

The expected aggregate shares encoded in hexadecimal.

+
+
+
+agg_result:
+
+

The expected aggregate result of type AggResult defined by the VDAF.

+
+
+
+

The schema also includes whatever parameters are required to instantiate the +VDAF. These are listed in the subsections below.

+
+
+

+C.1.1. Prio3Count +

+
+
+shares:
+
+

The number of shares, an integer.

+
+
+
+
+
+
+
+

+C.1.2. Prio3Sum +

+
+
+shares:
+
+

The number of shares, an integer.

+
+
+
+max_measurement:
+
+

The largest measurement, an integer. Each element is in the range + [0, max_measurement].

+
+
+
+
+
+
+
+

+C.1.3. Prio3SumVec +

+
+
+shares:
+
+

The number of shares, an integer.

+
+
+
+length:
+
+

The length of the vector, an integer.

+
+
+
+chunk_length:
+
+

the length of each vector chunk, an integer.

+
+
+
+bits:
+
+

the bit length of each element of the vector, an integer. Each element is in +[0, 2^bits).

+
+
+
+
+
+
+
+

+C.1.4. Prio3Histogram +

+
+
+shares:
+
+

The number of shares, an integer.

+
+
+
+length:
+
+

The length of the vector, an integer.

+
+
+
+chunk_length:
+
+

the length of each vector chunk, an integer.

+
+
+
+
+
+
+
+

+C.1.5. Prio3MultihotCountVec +

+
+
+shares:
+
+

The number of shares, an integer.

+
+
+
+length:
+
+

The length of the vector, an integer.

+
+
+
+chunk_length:
+
+

the length of each vector chunk, an integer.

+
+
+
+max_weight:
+
+

The largest vector weight, an integer. The sum of the elements must be in +[0, max_weight].

+
+
+
+
+
+
+
+

+C.1.6. Poplar1 +

+
+
+bits:
+
+

The length of each input in bits, an integer.

+
+
+
+
+
+
+
+
+
+
+
+

+Acknowledgments +

+

The impetus of this work is the success of recent deployments of predecessors +of Prio3. These include the Mozilla Origin Telemetry project +[OriginTelemetry] and the Exposure Notification Private Analytics system +[ENPA] developed jointly by ISRG, Google, Apple, and others. Together these +systems have aggregated data from hundreds of millions of users.

+

As the name implies, Prio3 is a descendant of the original Prio construction +[CGB17]. A second iteration was deployed in the [ENPA] system, and like the +VDAF described here, the ENPA system was built from techniques introduced in +[BBCGGI19] that significantly improve communication cost. That system was +specialized for a particular aggregation function; the goal of Prio3 is to +provide the same level of generality as the original construction.

+

The security considerations in Section 9 are based largely on the security +analysis of [DPRS23]. Thanks to Hannah Davis and Mike Rosulek, who lent their +time to developing definitions and security proofs.

+

Thanks to Junye Chen, Henry Corrigan-Gibbs, Armando Faz-Hernández, Simon +Friedberger, Tim Geoghegan, Albert Liu, Brandon Pitman, Mariana Raykova, +Michael Rosenberg, Jacob Rothstein, Shan Wang, Xiao Wang, Bas Westerbaan, and +Christopher Wood for useful feedback on and contributions to the spec.

+
+
+
+
+

+Authors' Addresses +

+
+
Richard L. Barnes
+
Cisco
+ +
+
+
David Cook
+
ISRG
+ +
+
+
Christopher Patton
+
Cloudflare
+ +
+
+
Phillipp Schoppmann
+
Google
+ +
+
+
+ + + diff --git a/david/reorder-circuit-methods/draft-irtf-cfrg-vdaf.txt b/david/reorder-circuit-methods/draft-irtf-cfrg-vdaf.txt new file mode 100644 index 00000000..578a788c --- /dev/null +++ b/david/reorder-circuit-methods/draft-irtf-cfrg-vdaf.txt @@ -0,0 +1,6645 @@ + + + + +CFRG R. L. Barnes +Internet-Draft Cisco +Intended status: Informational D. Cook +Expires: 20 April 2025 ISRG + C. Patton + Cloudflare + P. Schoppmann + Google + 17 October 2024 + + + Verifiable Distributed Aggregation Functions + draft-irtf-cfrg-vdaf-latest + +Abstract + + This document describes Verifiable Distributed Aggregation Functions + (VDAFs), a family of multi-party protocols for computing aggregate + statistics over user measurements. These protocols are designed to + ensure that, as long as at least one aggregation server executes the + protocol honestly, individual measurements are never seen by any + server in the clear. At the same time, VDAFs allow the servers to + detect if a malicious or misconfigured client submitted an invalid + measurement. Two concrete VDAFs are specified, one for general- + purpose aggregation (Prio3) and another for heavy hitters (Poplar1). + +Discussion Venues + + This note is to be removed before publishing as an RFC. + + Discussion of this document takes place on the Crypto Forum Research + Group mailing list (cfrg@ietf.org), which is archived at + https://mailarchive.ietf.org/arch/search/?email_list=cfrg. + + Source for this draft and an issue tracker can be found at + https://github.com/cfrg/draft-irtf-cfrg-vdaf. + +Status of This Memo + + This Internet-Draft is submitted in full conformance with the + provisions of BCP 78 and BCP 79. + + Internet-Drafts are working documents of the Internet Engineering + Task Force (IETF). Note that other groups may also distribute + working documents as Internet-Drafts. The list of current Internet- + Drafts is at https://datatracker.ietf.org/drafts/current/. + + Internet-Drafts are draft documents valid for a maximum of six months + and may be updated, replaced, or obsoleted by other documents at any + time. It is inappropriate to use Internet-Drafts as reference + material or to cite them other than as "work in progress." + + This Internet-Draft will expire on 20 April 2025. + +Copyright Notice + + Copyright (c) 2024 IETF Trust and the persons identified as the + document authors. All rights reserved. + + This document is subject to BCP 78 and the IETF Trust's Legal + Provisions Relating to IETF Documents (https://trustee.ietf.org/ + license-info) in effect on the date of publication of this document. + Please review these documents carefully, as they describe your rights + and restrictions with respect to this document. + +Table of Contents + + 1. Introduction + 1.1. Change Log + 2. Conventions + 3. Overview + 4. Definition of DAFs + 4.1. Sharding + 4.2. Preparation + 4.3. Validity of Aggregation Parameters + 4.4. Aggregation + 4.4.1. Aggregation Order + 4.5. Unsharding + 4.6. Execution of a DAF + 5. Definition of VDAFs + 5.1. Sharding + 5.2. Preparation + 5.3. Validity of Aggregation Parameters + 5.4. Aggregation + 5.5. Unsharding + 5.6. Execution of a VDAF + 5.7. Communication Patterns for Preparation + 5.7.1. The Ping-Pong Topology (Only Two Aggregators) + 5.7.2. The Star Topology (Any Number of Aggregators) + 6. Preliminaries + 6.1. Finite Fields + 6.1.1. Auxiliary Functions + 6.1.2. NTT-Friendly Fields + 6.1.3. Parameters + 6.2. Extendable Output Functions (XOFs) + 6.2.1. XofTurboShake128 + 6.2.2. XofFixedKeyAes128 + 6.2.3. The Domain Separation Tag and Binder String + 7. Prio3 + 7.1. Fully Linear Proofs (FLPs) + 7.1.1. Encoding the Input + 7.1.2. Multiple Proofs + 7.2. Specification + 7.2.1. Sharding + 7.2.2. Preparation + 7.2.3. Validity of Aggregation Parameters + 7.2.4. Aggregation + 7.2.5. Unsharding + 7.2.6. Auxiliary Functions + 7.2.7. Message Serialization + 7.3. FLP Specification + 7.3.1. Overview + 7.3.2. Validity Circuits + 7.3.3. Generating the Proof + 7.3.4. Querying the Proof + 7.3.5. Deciding Validity + 7.4. Variants + 7.4.1. Prio3Count + 7.4.2. Prio3Sum + 7.4.3. Prio3SumVec + 7.4.4. Prio3Histogram + 7.4.5. Prio3MultihotCountVec + 8. Poplar1 + 8.1. Incremental Distributed Point Functions (IDPFs) + 8.1.1. Encoding Inputs as Indices + 8.2. Specification + 8.2.1. Sharding + 8.2.2. Preparation + 8.2.3. Validity of Aggregation Parameters + 8.2.4. Aggregation + 8.2.5. Unsharding + 8.2.6. Message Serialization + 8.3. IDPF Specification + 8.3.1. Overview + 8.3.2. Key Generation + 8.3.3. Key Evaluation + 8.3.4. Auxiliary Functions + 9. Security Considerations + 9.1. The Verification Key + 9.2. The Nonce + 9.3. The Public Share + 9.4. The Aggregation Parameter + 9.5. Safe Usage of IDPF Outputs + 9.6. Safe Usage of XOFs + 9.7. Choosing FLP Parameters + 9.8. Choosing the Number of Aggregators + 9.9. Defense-in-Depth Measures + 10. IANA Considerations + 11. References + 11.1. Normative References + 11.2. Informative References + Appendix A. FLP Gadgets + A.1. Multiplication + A.2. Polynomial Evaluation + A.3. Parallel Sum + A.4. Shims for Generating and Querying Proofs + Appendix B. VDAF Preparation State + Appendix C. Test Vectors + C.1. Schema + C.1.1. Prio3Count + C.1.2. Prio3Sum + C.1.3. Prio3SumVec + C.1.4. Prio3Histogram + C.1.5. Prio3MultihotCountVec + C.1.6. Poplar1 + Acknowledgments + Authors' Addresses + +1. Introduction + + (RFC EDITOR: remove this paragraph.) The source for this draft and + the reference implementation can be found at https://github.com/cfrg/ + draft-irtf-cfrg-vdaf. + + The ubiquity of the Internet makes it an ideal platform for + measurement of large-scale phenomena, whether public health trends or + the behavior of computer systems at scale. There is substantial + overlap, however, between information that is valuable to measure and + information that users consider private. + + For example, consider an application that provides health information + to users. The operator of an application might want to know which + parts of their application are used most often, as a way to guide + future development of the application. Specific users' patterns of + usage, though, could reveal sensitive things about them, such as + which users are researching a given health condition. + + In many situations, the measurement collector is only interested in + aggregate statistics, e.g., which portions of an application are most + used or what fraction of people have experienced a given disease. + Thus systems that provide aggregate statistics while protecting + individual measurements can deliver the value of the measurements + while protecting users' privacy. + + This problem is often formulated in terms of differential privacy + (DP) [Dwo06]. Roughly speaking, a data aggregation system that is + differentially private ensures that the degree to which any + individual measurement influences the value of the aggregate result + can be precisely controlled. For example, in systems like RAPPOR + [EPK14], each user samples noise from a well-known distribution and + adds it to their measurement before submitting to the aggregation + server. The aggregation server then adds up the noisy measurements, + and because it knows the distribution from which the noise was + sampled, it can estimate the true sum with reasonable accuracy. + + Even when noise is added to the measurements, collecting them in the + clear still reveals a significant amount of information to the + collector. On the one hand, depending on the "amount" of noise a + client adds to its measurement, it may be possible for a curious + collector to make a reasonable guess of the measurement's true value. + On the other hand, the more noise the clients add, the less reliable + will be the server's estimate of the aggregate. Thus systems relying + solely on a DP mechanism must strike a delicate balance between + privacy and utility. + + Another way of constructing a privacy-preserving measurement system + is to use multi-party computation (MPC). The goal of such a system + is that no participant in the protocol should learn anything about an + individual measurement beyond what it can deduce from the aggregate. + MPC achieves this goal by distributing the computation of the + aggregate across multiple aggregation servers, one of which is + presumed to be honest, i.e., not under control of the attacker. + Moreover, MPC can be composed with various DP mechanisms to ensure + the aggregate itself does not leak too much information about any one + of the measurements [MPRV09]. + + This document describes two classes of MPC protocols, each aiming for + a different set of goals. + + In a Distributed Aggregation Function (DAF, Section 4), each client + splits its measurement into multiple secret shares, one for each + aggregation server. DAFs require two properties of the secret + sharing scheme. First, we can reconstruct the underlying measurement + by simply adding up all of the shares. (Typically the shares are + vectors over some finite field.) Second, given all but one of the + shares, it is impossible to learn anything about the underlying + measurement. These properties give rise to a simple strategy for + privately aggregating the measurements: each aggregation server adds + up its measurement shares locally before revealing their sum to the + data collector; then all the data collector has to do is add up these + sums to get the aggregate result. + + This strategy is compatible with any aggregation function that can be + represented as the sum of some encoding of the measurements. + Examples include: summary statistics such as sum, mean, and standard + deviation; estimation of quantiles, e.g., median; histograms; linear + regression; or counting data structures, e.g., Bloom filters. + However, not all functions fit into this rubric, as it is constrained + to linear computations over the encoded measurements. + + In fact, our framework admits DAFs with slightly more functionality, + computing aggregation functions of the form + + F(agg_param, meas_1, ..., meas_M) = + G(agg_param, meas_1) + ... + G(agg_param, meas_M) + + where meas_1, ..., meas_M are the measurements, G is a possibly non- + linear function, and agg_param is a parameter of that function chosen + by the data collector. This paradigm, known as function secret + sharing [BGI15], allows for more sophisticated data analysis tasks, + such as grouping metrics by private client attributes [MPDST25] or + computing heavy hitters [BBCGGI21]. (More on the latter task below.) + + The second class of protocols defined in this document are called + Verifiable Distributed Aggregation Functions (VDAFs, Section 5). In + addition to being private, VDAFs are verifiable in the following + sense. By design, a secret sharing of a valid measurement, e.g., a + number between 1 and 10, is indistinguishable from a secret sharing + of an invalid measurement, e.g., a number larger than 10. This means + that DAFs are vulnerable to attacks from malicious clients attempting + to disrupt the computation by submitting invalid measurements. VDAFs + are designed to allow the servers to detect and remove these + measurements prior to aggregation. We refer to this property as + robustness. + + Achieving robustness without sacrificing privacy requires the servers + to interact with one another over a number of rounds of + communication. DAFs on the other hand are non-interactive, making + them easier to deploy; but they do not provide robustness on their + own. This may be tolerable in some applications. For instance, if + the client's software is executed in a trusted execution environment, + it may be reasonable to assume that no client is malicious. + + The DAF and VDAF abstractions encompass a variety of MPC techniques + in the literature. These protocols vary in their operational and + security requirements, sometimes in subtle but consequential ways. + This document therefore has two important goals: + + 1. Provide higher-level protocols, like [DAP], with a simple, + uniform interface for accessing privacy-preserving measurement + schemes, document relevant operational and security requirements, + and specify constraints for safe usage: + + 1. General patterns of communications among the various actors + involved in the system (clients, aggregation servers, and the + collector of the aggregate result); + + 2. Capabilities of a malicious coalition of parties attempting + to divulge information about client measurements; and + + 3. Conditions that are necessary to ensure that malicious + clients cannot corrupt the computation. + + 2. Provide cryptographers with design criteria that provide a clear + deployment roadmap for new constructions of privacy-preserving + measurement systems. + + This document also specifies two concrete VDAF schemes, each based on + a protocol from the literature. + + * The Prio system [CGB17] allows for the privacy-preserving + computation of a variety of aggregate statistics, combining + additive secret sharing as described above with a mechanism for + checking the validity of each measurement. In Section 7 we + specify Prio3, a VDAF that follows the same overall framework as + the original Prio protocol, but incorporates techniques introduced + in [BBCGGI19] that result in significant performance gains. + + * The Poplar protocol [BBCGGI21] solves a problem known as private + heavy-hitters. In this problem, each client holds a bit-string, + and the goal of the aggregation servers is to compute the set of + strings that occur at least T times for some threshold T. The + core primitive in their protocol is a secret sharing of a point + function [GI14] (denoted G above) that allows the servers to + privately count how many of the clients' strings begin with a + given prefix (agg_param in the notation above). In Section 8 we + specify a VDAF called Poplar1 that implements this functionality. + + The remainder of this document is organized as follows: Section 2 + lists definitions and conventions used in the remainder of the + document; Section 3 gives a brief overview of DAFs and VDAFs, the + parties involved in the computation, and the requirements for non- + collusion; Section 4 defines the syntax for DAFs; Section 5 defines + the syntax for VDAFs; Section 6 defines various functionalities that + are common to our constructions; Section 7 specifies Prio3; Section 8 + specifies Poplar1; and Section 9 enumerates security considerations + for DAFs and VDAFs in general and our constructions in particular. + +1.1. Change Log + + (RFC EDITOR: remove this section.) + + (*) Indicates a change that breaks wire compatibility with the + previous draft. + + 12: + + * (V)DAF: Add an application context string parameter to sharding + and preparation. The motivation for this change is to harden + Prio3 against offline attacks. More generally, however, it allows + designing schemes for which correct execution requires agreement + on the application context. Accordingly, both Prio3 and Poplar1 + have been modified to include the context in the domain separation + tag of each XOF invocation. (*) + + * Prio3: Improve soundness of the base proof system and the circuits + of some variants. Generally speaking, wherever we evaluate a + univariate polynomial at a random point, we can instead evaluate a + multivariate polynomial of lower degree. (*) + + * Prio3: Replace the helper's measurement and proof share seeds with + a single seed. (*) + + * Prio3Sum: Update the circuit to support a more general range check + and avoid using joint randomness. (*) + + * Prio3Histogram, Prio3MultihotCountVec: Move the final reduction of + the intermediate outputs out of the circuit. (*) + + * IDPF: Add the application context string to key generation and + evaluation and bind it to the fixed AES key. (*) + + * IDPF: Use XofTurboShake128 for deriving the leaf nodes in order to + ensure the construction is extractable. (*) + + * IDPF: Simplify the public share encoding. (*) + + * XofTurboShake128: Change SEED_SIZE from 16 bytes to 32 to mitigate + offline attacks on Prio3 robustness. In addition, allow seeds of + different lengths so that we can continue to use XofTurboShake128 + with IDPF. (*) + + * XofTurboShake128, XofFixedKeyAes128: Increase the length prefix + for the domain separation tag from one by to two bytes. This is + to accommodate the application context. (*) + + * Reassign codepoints for all Prio3 variants and Poplar1. (*) + + * Security considerations: Add a section on defense-in-depth + measures taken by Prio3 and Poplar1 and more discussion about + choosing FLP parameters. + + 11: + + * Define message formats for the Poplar1 aggregation parameter and + IDPF public share. + + * IDPF: Require the IDPF binder must be a random nonce. + + * VDAF: Replace the pseudocode description of the ping-ping topology + with Python and sketch the star topology. + + * DAF: Align aggregation parameter validation with VDAF. + + * Replace Union[A, B] type with A | B. + + * Rename FFT ("Fast Fourier Transform") with NTT ("Number Theoretic + Transform"). + + 10: + + * Define Prio3MultihotCountVec, a variant of Prio3 for aggregating + bit vectors with bounded weight. + + * FLP: Allow the output of the circuit to be a vector. This makes + it possible to skip joint randomness derivation in more cases. + + * Poplar1: On the first round of preparation, handle None as an + error. Previously this message was interpreted as a length-3 + vector of zeros. + + * Prio3: Move specification of the field from the FLP validity + circuit to the VDAF itself. + + * Clarify the extent to which the attacker controls the network in + our threat models for privacy and robustness. + + * Clean up various aspects of the code, including: Follow existing + object-oriented programming patterns for Python more closely; make + the type hints enforceable; and avoid shadowing variables. + + * Poplar1: Align terminology with [BBCGGI23]. + + * IDPF: Add guidance for encoding byte strings as indices. + + 09: + + * Poplar1: Make prefix tree traversal stricter by requiring each + node to be a child of a node that was already visited. This + change is intended to make it harder for a malicious Aggregator to + steer traversal towards non-heavy-hitting measurements. + + * Prio3: Add more explicit guidance for choosing the field size. + + * IDPF: Define extractability and clarify (un)safe usage of + intermediate prefix counts. Accordingly, add text ensuring public + share consistency to security considerations. + + 08: + + * Poplar1: Bind the report nonce to the authenticator vector + programmed into the IDPF. (*) + + * IdpfPoplar: Modify extend() by stealing each control bit from its + corresponding seed. This improves performance by reducing the + number of AES calls per level from 3 to 2. The cost is a slight + reduction in the concrete privacy bound. (*) + + * Prio3: Add support for generating and verifying mutliple proofs + per measurement. This enables a trade-off between communication + cost and runtime: if more proofs are used, then a smaller field + can be used without impacting robustness. (*) + + * Replace SHAKE128 with TurboSHAKE128. (*) + + 07: + + * Rename PRG to XOF ("eXtendable Output Function"). Accordingly, + rename PrgSha3 to XofShake128 and PrgFixedKeyAes128 to + XofFixedKeyAes128. "PRG" is a misnomer since we don't actually + treat this object as a pseudorandom generator in existing security + analysis. + + * Replace cSHAKE128 with SHAKE128, re-implementing domain separation + for the customization string using a simpler scheme. This change + addresses the reality that implementations of cSHAKE128 are less + common. (*) + + * Define a new VDAF, called Prio3SumVec, that generalizes Prio3Sum + to a vector of summands. + + * Prio3Histogram: Update the codepoint and use the parallel sum + optimization introduced by Prio3SumVec to reduce the proof size. + (*) + + * Daf, Vdaf: Rename interface methods to match verbiage in the + draft. + + * Daf: Align with Vdaf by adding a nonce to shard() and prep(). + + * Vdaf: Have prep_init() compute the first prep share. This change + is intended to simplify the interface by making the input to + prep_next() not optional. + + * Prio3: Split sharding into two auxiliary functions, one for + sharding with joint randomness and another without. This change + is intended to improve readability. + + * Fix bugs in the ping-pong interface discovered after implementing + it. + + 06: + + * Vdaf: Define a wrapper interface for preparation that is suitable + for the "ping-pong" topology in which two Aggregators exchange + messages over a request/response protocol, like HTTP, and take + turns executing the computation until input from the peer is + required. + + * Prio3Histogram: Generalize the measurement type so that the + histogram can be used more easily with discrete domains. (*) + + * Daf, Vdaf: Change the aggregation parameter validation algorithm + to take the set of previous parameters rather than a list. (The + order of the parameters is irrelevant.) + + * Daf, Vdaf, Idpf: Add parameter RAND_SIZE that specifies the number + of random bytes consumed by the randomized algorithm (shard() for + Daf and Vdaf and gen() for Idpf). + + 05: + + * IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixed-key + mode for AES-128 based on a construction from [GKWWY20]. This + change is intended to improve performance of IDPF evaluation. + Note that the new PRG is not suitable for all applications. (*) + + * Idpf: Add a binder string to the key-generation and evaluation + algorithms. This is used to plumb the nonce generated by the + Client to the PRG. + + * Plumb random coins through the interface of randomized algorithms. + Specifically, add a random input to (V)DAF sharding algorithm and + IDPF key-generation algorithm and require implementations to + specify the length of the random input. Accordingly, update + Prio3, Poplar1, and IdpfPoplar to match the new interface. This + change is intended to improve coverage of test vectors. + + * Use little-endian byte-order for field element encoding. (*) + + * Poplar1: Move the last step of sketch evaluation from prep_next() + to prep_shares_to_prep(). + + 04: + + * Align security considerations with the security analysis of + [DPRS23]. + + * Vdaf: Pass the nonce to the sharding algorithm. + + * Vdaf: Rather than allow the application to choose the nonce + length, have each implementation of the Vdaf interface specify the + expected nonce length. (*) + + * Prg: Split "info string" into two components: the "customization + string", intended for domain separation; and the "binder string", + used to bind the output to ephemeral values, like the nonce, + associated with execution of a (V)DAF. + + * Replace PrgAes128 with PrgSha3, an implementation of the Prg + interface based on SHA-3, and use the new scheme as the default. + Accordingly, replace Prio3Aes128Count with Prio3Count, + Poplar1Aes128 with Poplar1, and so on. SHA-3 is a safer choice + for instantiating a random oracle, which is used in the analysis + of Prio3 of [DPRS23]. (*) + + * Prio3, Poplar1: Ensure each invocation of the Prg uses a distinct + customization string, as suggested by [DPRS23]. This is intended + to make domain separation clearer, thereby simplifying security + analysis. (*) + + * Prio3: Replace "joint randomness hints" sent in each input share + with "joint randomness parts" sent in the public share. This + reduces communication overhead when the number of shares exceeds + two. (*) + + * Prio3: Bind nonce to joint randomness parts. This is intended to + address birthday attacks on robustness pointed out by [DPRS23]. + (*) + + * Poplar1: Use different Prg invocations for producing the + correlated randomness for inner and leaf nodes of the IDPF tree. + This is intended to simplify implementations. (*) + + * Poplar1: Don't bind the candidate prefixes to the verifier + randomness. This is intended to improve performance, while not + impacting security. According to the analysis of [DPRS23], it is + necessary to restrict Poplar1 usage such that no report is + aggregated more than once at a given level of the IDPF tree; + otherwise, attacks on privacy may be possible. In light of this + restriction, there is no added benefit of binding to the prefixes + themselves. (*) + + * Poplar1: During preparation, assert that all candidate prefixes + are unique and appear in order. Uniqueness is required to avoid + erroneously rejecting a valid report; the ordering constraint + ensures the uniqueness check can be performed efficiently. (*) + + * Poplar1: Increase the maximum candidate prefix count in the + encoding of the aggregation parameter. (*) + + * Poplar1: Bind the nonce to the correlated randomness derivation. + This is intended to provide defense-in-depth by ensuring the + Aggregators reject the report if the nonce does not match what the + Client used for sharding. (*) + + * Poplar1: Clarify that the aggregation parameter encoding is + OPTIONAL. Accordingly, update implementation considerations + around cross-aggregation state. + + * IdpfPoplar: Add implementation considerations around branching on + the values of control bits. + + * IdpfPoplar: When decoding the control bits in the public share, + assert that the trailing bits of the final byte are all zero. (*) + + 03: + + * Define codepoints for (V)DAFs and use them for domain separation + in Prio3 and Poplar1. (*) + + * Prio3: Align joint randomness computation with revised paper + [BBCGGI19]. This change mitigates an attack on robustness. (*) + + * Prio3: Remove an intermediate PRG evaluation from query randomness + generation. (*) + + * Add additional guidance for choosing FFT-friendly fields. + + 02: + + * Complete the initial specification of Poplar1. + + * Extend (V)DAF syntax to include a "public share" output by the + Client and distributed to all of the Aggregators. This is to + accommodate "extractable" IDPFs as required for Poplar1. (See + [BBCGGI21], Section 4.3 for details.) + + * Extend (V)DAF syntax to allow the unsharding step to take into + account the number of measurements aggregated. + + * Extend FLP syntax by adding a method for decoding the aggregate + result from a vector of field elements. The new method takes into + account the number of measurements. + + * Prio3: Align aggregate result computation with updated FLP syntax. + + * Prg: Add a method for statefully generating a vector of field + elements. + + * Field: Require that field elements are fully reduced before + decoding. (*) + + * Define new field Field255. + + 01: + + * Require that VDAFs specify serialization of aggregate shares. + + * Define Distributed Aggregation Functions (DAFs). + + * Prio3: Move proof verifier check from prep_next() to + prep_shares_to_prep(). (*) + + * Remove public parameter and replace verification parameter with a + "verification key" and "Aggregator ID". + +2. Conventions + + The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", + "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and + "OPTIONAL" in this document are to be interpreted as described in + BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all + capitals, as shown here. + + Algorithms in this document are written in Python (compatible with + Python 3.12 or later). A fatal error in a program (e.g., failure to + parse one of the function parameters) is usually handled by raising + an exception. + + In Python, array indexing starts with 0, e.g., x[0] is the first + element and x[len(x)-1] is the last of x. We can also index from the + end of the list, e.g., x[-1] is the last element of x. + + Exponentiation is denoted by x ** y in Python. We also sometimes use + more conventional notation for exponentiation, namely x^y. + + Division of x by y is denoted by x / y. Python includes a similar + operator, x // y, which is short for floor(x / y). + + Type hints are used to define input and output types: + + * The type variable F is used in signatures to signify any type that + is a subclass of Field (Section 6.1). + + * bytes is a byte string. + + * int is an integer. + + * Generic is used in class definitions to explicitly declare type + variables of generic classes. + + * Any is the universal supertype, which admits values of any type. + + * Optional[T] is shorthand for T | None. Its value may be None or + have type T. + + * Self represents the containing class of the method definition in + which it appears. + + * Sequence[T] is either a list or tuple of values of type T. + + This document defines several byte-string constants. When comprised + of printable ASCII characters, they are written as Python 3 byte- + string literals (e.g., b'some constant string'). + + A global constant VERSION of type int is defined, which algorithms + are free to use as desired. Its value SHALL be 12. + + This document describes algorithms for multi-party computations in + which the parties typically communicate over a network. Wherever a + quantity is defined that must be transmitted from one party to + another, this document prescribes a particular encoding of that + quantity as a byte string. + + Some common functionalities: + + * additive_secret_share(x: list[F], num_shares: int, field: type[F]) + -> list[list[F]] takes a vector x of field elements and returns + num_shares vectors of length len(x) such that they all add up to + the input vector. Note that this function is not used normatively + in this document. + + * byte(x: int) -> bytes returns the representation of the integer x + in range [0, 256) as a single-byte byte string. + + * cast(typ: type, x: object) -> object returns the input value + unchanged. This is only present to assist with static analysis of + the Python code. Type checkers will ignore the inferred type of + the input value, and assume the output value has the given type. + + * concat(parts: list[bytes]) -> bytes returns the concatenation of + the input byte strings, i.e., parts[0] + ... + parts[len(parts)- + 1]. + + * from_be_bytes(encoded: bytes) -> int decodes a big-endian byte + string, i.e., returns the integer x for which to_be_bytes(x, + len(encoded)) == encoded. + + * from_le_bytes(encoded: bytes) -> int decodes a little-endian byte + string, i.e., returns the integer x for which to_le_bytes(x, + len(encoded)) == encoded. + + * front(len: int, x: list[Any]) -> tuple[list[Any], list[Any]] + splits x into two vectors, where the first vector is made up of + the first len elements of x and the second is made up of the + remaining elements. This function is equivalent to (x[:len], + x[len:]). + + * gen_rand(len: int) -> bytes returns a byte array of the requested + length (len) generated by a cryptographically secure pseudorandom + number generator (CSPRNG). + + * next_power_of_2(x: int) -> int returns the smallest integer + greater than or equal to x that is also a power of two. + + * range(stop) or range(start, stop[, step]) is the range function + from the Python standard library. The one-argument form returns + the integers from zero (inclusive) to stop, exclusive. The two- + and three-argument forms allow overriding the start of the range + and overriding the step between successive output values. + + * to_be_bytes(x: int, len: int) -> bytes converts an integer x whose + value is in the range [0, 2^(8*len)) to a big-endian, len-byte + string. + + * to_le_bytes(x: int, len: int) -> bytes converts an integer x whose + value is in the range [0, 2^(8*len)) to a little-endian, len-byte + string. + + * poly_eval(field: type[F], p: list[F], x: F) -> F returns the + result of evaluating the polynomial, p(x). The coefficients of + polynomials are stored in lists in ascending order of degree, + starting with the constant coefficient. The field parameter is + the class object for F and is used by the implementation to + construct field elements. (See Section 6.1.) + + * poly_interp(field: type[F], inputs: list[F], outputs: list[F]) -> + list[F] returns the coefficients of the lowest degree polynomial p + for which p(inputs[k]) == outputs[k] for all k. Normally this + will be computed using the Number Theoretic Transform (NTT) + [SML24]. + + * poly_mul(field: type[F], p: list[F], q: list[F]) -> list[F] + returns the product of two polynomials. + + * poly_strip(field: type[F], p: list[F]) -> list[F] strips the zeros + from the end of the input polynomial. That is, it returns p[:i] + where i is the index of the last non-zero coefficient of p. + + * xor(left: bytes, right: bytes) -> bytes returns the bitwise XOR of + left and right. An exception is raised if the inputs are not the + same length. + + * zeros(len: int) -> bytes returns an array of bytes of the + requested length (len). Each element of the array is set to zero. + +3. Overview + + +--------------+ + +---->| Aggregator 0 |----+ + input | +--------------+ | aggregate + shares | ^ | shares + | | | + | V | + | +--------------+ | + | +-->| Aggregator 1 |--+ | + | | +--------------+ | | + +--------+-+ | ^ | +->+-----------+ + | Client |---+ | +--->| Collector |--> aggregate + +--------+-+ +->+-----------+ + | ... | + | | + | | | + | V | + | +----------------+ | + +--->| Aggregator N-1 |---+ + +----------------+ + + Figure 1: Overall data flow of a (V)DAF. + + In a DAF- or VDAF-based private measurement system, we distinguish + between three types of actors: Clients, Aggregators, and the + Collector. The overall flow of the measurement process is as + follows: + + * To submit an individual measurement, the Client shards the + measurement into "input shares" and sends one input share to each + Aggregator. We sometimes refer to this sequence of input shares + collectively as the Client's "report". + + * The Aggregators refine their input shares into "output shares": + + - Output shares are in one-to-one correspondence with the input + shares. + + - Just as each Aggregator receives one input share of each + measurement, if this process succeeds, then each aggregator + holds one output share. + + - In VDAFs, Aggregators will need to exchange information among + themselves as part of the validation process. + + * As each Aggregator recovers output shares, it accumulates them + into an "aggregate share" for the batch of measurements. + + * The Aggregators submit their aggregate shares to the Collector, + who combines them to obtain the aggregate result over the batch. + + Aggregators are a new class of actor relative to traditional + measurement systems where Clients submit measurements to a single + server. They are critical for both the privacy properties of the + system and, in the case of VDAFs, the validity of the measurements + obtained. The privacy properties of the system are assured by non- + collusion among Aggregators, and Aggregators are the entities that + perform validation of Client measurements. Thus Clients trust + Aggregators not to collude (typically it is required that at least + one Aggregator is honest; see Section 9.8), and Collectors trust + Aggregators to correctly run the protocol. + + Within the bounds of the non-collusion requirements of a given (V)DAF + instance, it is possible for the same entity to play more than one + role. For example, the Collector could also act as an Aggregator, + effectively using the other Aggregator(s) to augment a basic client- + server protocol. + + In this document, we describe the computations performed by the + actors in this system. It is up to the higher-level protocol making + use of the (V)DAF to arrange for the required information to be + delivered to the proper actors in the proper sequence. In general, + we assume that all communications are confidential and mutually + authenticated, with the exception that Clients submitting + measurements may be anonymous. + +4. Definition of DAFs + + By way of a gentle introduction to VDAFs, this section describes a + simpler class of schemes called Distributed Aggregation Functions + (DAFs). Unlike VDAFs, DAFs do not provide verifiability of the + computation. Clients must therefore be trusted to compute their + input shares correctly. Because of this fact, the use of a DAF is + NOT RECOMMENDED for most applications. See Section 9 for additional + discussion. + + A DAF scheme is used to compute a particular "aggregation function" + over a set of measurements generated by Clients. Depending on the + aggregation function, the Collector might select an "aggregation + parameter" and disseminate it to the Aggregators. The semantics of + this parameter is specific to the aggregation function, but in + general it is used to represent the set of "queries" that can be made + by the Collector on the batch of measurements. For example, the + aggregation parameter is used to represent the candidate prefixes in + the Poplar1 VDAF Section 8. + + Execution of a DAF has four distinct stages: + + * Sharding: Each Client generates input shares from its measurement + and distributes them among the Aggregators. In addition to the + input shares, the client generates a "public share" during this + step that is disseminated to all of the Aggregators. + + * Preparation: Each Aggregator converts each input share into an + output share compatible with the aggregation function. This + computation involves the aggregation parameter. In general, each + aggregation parameter may result in a different output share. + + * Aggregation: Each Aggregator combines a sequence of output shares + into its aggregate share and sends the aggregate share to the + Collector. + + * Unsharding: The Collector combines the aggregate shares into the + aggregate result. + + Sharding and preparation are done once per measurement. Aggregation + and unsharding are done over a batch of measurements (more precisely, + over the recovered output shares). + + A concrete DAF specifies the algorithm for the computation needed in + each of these stages. The interface, denoted Daf, is defined in the + remainder of this section. In addition, a concrete DAF defines the + associated constants and types enumerated in the following table. + + +=============+==================================================+ + | Parameter | Description | + +=============+==================================================+ + | ID: int | Algorithm identifier for this DAF, in [0, 2^32). | + +-------------+--------------------------------------------------+ + | SHARES: int | Number of input shares into which each | + | | measurement is sharded. | + +-------------+--------------------------------------------------+ + | NONCE_SIZE: | Size of the nonce associated with the report. | + | int | | + +-------------+--------------------------------------------------+ + | RAND_SIZE: | Size of the random byte string consumed by the | + | int | sharding algorithm. | + +-------------+--------------------------------------------------+ + | Measurement | Type of each measurement. | + +-------------+--------------------------------------------------+ + | PublicShare | Type of each public share. | + +-------------+--------------------------------------------------+ + | InputShare | Type of each input share. | + +-------------+--------------------------------------------------+ + | AggParam | Type of the aggregation parameter. | + +-------------+--------------------------------------------------+ + | OutShare | Type of each output share. | + +-------------+--------------------------------------------------+ + | AggShare | Type of the aggregate share. | + +-------------+--------------------------------------------------+ + | AggResult | Type of the aggregate result. | + +-------------+--------------------------------------------------+ + + Table 1: Constants and types defined by each concrete DAF. + + The types in this table define the inputs and outputs of DAF methods + at various stages of the computation. Some of these values need to + be written to the network in order to carry forward the computation. + In particular, it is RECOMMENDED that concrete instantiations of the + Daf interface specify a standard encoding the PublicShare, + InputShare, AggParam, and AggShare. + +4.1. Sharding + + Client + ====== + + measurement + | + V + +----------------------------------------------+ + | shard | + +----------------------------------------------+ + | | | | + | | ... | public_share + | | | | + | +---------|-----+--------|-----+ + | | | | | | + V | V | V | + input_share_0 input_share_1 input_share_[SHARES-1] + | | | | ... | | + V V V V V V + Aggregator 0 Aggregator 1 Aggregator SHARES-1 + + Figure 2: Illustration of the sharding algorithm. + + The sharding algorithm run by each Client is defined as follows: + + * daf.shard(ctx: bytes, measurement: Measurement, nonce: bytes, + rand: bytes) -> tuple[PublicShare, list[InputShare]] consumes the + "application context" (defined below), a measurement, and a nonce + and produces the public share, distributed to each of the + Aggregators, and the input shares, one for each Aggregator. + + Pre-conditions: + + - nonce MUST have length equal to daf.NONCE_SIZE and MUST be + generated using a cryptographically secure random number + generator (CSPRNG). + + - rand consists of the random bytes consumed by the algorithm. + It MUST have length equal to daf.RAND_SIZE and MUST be + generated using a CSPRNG. + + Post-conditions: + + - The number of input shares MUST equal daf.SHARES. + + Sharding is bound to a specific "application context". The + application context is a string intended to uniquely identify an + instance of the higher level protocol that uses the DAF. The goal of + binding the application to DAF execution is to ensure that + aggregation succeeds only if the Clients and Aggregators agree on the + application context. (Preparation binds the application context, + too; see Section 4.2.) Note that, unlike VDAFs (Section 5), there is + no explicit signal of disagreement; it may only manifest as a garbled + aggregate result. + + The nonce is a public random value associated with the report. It is + referred to as a nonce because normally it will also be used as a + unique identifier for that report in the context of some application. + The randomness requirement is especially important for VDAFs, where + it may be used by the Aggregators to derive per-report randomness for + verification of the computation. See Section 9.2 for details. + +4.2. Preparation + + Aggregator 0 Aggregator 1 Aggregator SHARES-1 + ============ ============ =================== + + input_share_0 input_share_1 input_share_[SHARES-1] + | | ... | + V V V + +-----------+ +-----------+ +-----------+ + | prep | | prep | | prep | + +-----------+ +-----------+ +-----------+ + | | ... | + V V V + out_share_0 out_share_1 out_share_[SHARES-1] + + Figure 3: Illustration of preparation. + + Once an Aggregator has received the public share and its input share, + the next step is to prepare the input share for aggregation. This is + accomplished using the preparation algorithm: + + * daf.prep(ctx: bytes, agg_id: int, agg_param: AggParam, nonce: + bytes, public_share: PublicShare, input_share: InputShare) -> + OutShare consumes the public share and one of the input shares + generated by the Client, the application context, the Aggregator's + unique identifier, the aggregation parameter selected by the + Collector, and the report nonce and returns an output share. + + Pre-conditions: + + - agg_id MUST be in the range [0, daf.SHARES) and match the index + of input_share in the sequence of input shares produced by the + Client. + + - nonce MUST have length daf.NONCE_SIZE. + + The Aggregators MUST agree on the value of the aggregation parameter. + Otherwise, the aggregate result may be computed incorrectly by the + Collector. + +4.3. Validity of Aggregation Parameters + + In general, it is permissible to aggregate a batch of reports + multiple times. However, to prevent privacy violations, DAFs may + impose certain restrictions on the aggregation parameters selected by + the Collector. Restrictions are expressed by the aggregation- + parameter validity function: + + * daf.is_valid(agg_param: AggParam, previous_agg_params: + list[AggParam]) -> bool returns True if agg_param is allowed given + the sequence previous_agg_params of previously accepted + aggregation parameters. + + Prior to accepting an aggregation parameter from the Collector and + beginning preparation, each Aggregator MUST validate it using this + function. + +4.4. Aggregation + + Aggregator j + ============ + + +------------+ +------------+ + | agg_init |--->| agg_update |<--- out_share_0 + +------------+ +------------+ + | + V + +------------+ + | agg_update |<--- out_share_1, + +------------+ + | + V + ... + | + V + +------------+ + | agg_update |<--- out_share_M + +------------+ + | + V + agg_share_j + + Figure 4: Illustration of aggregation. The number of + measurements in the batch is denoted by M. + + Once an Aggregator holds an output share, it adds it into its + aggregate share for the batch. This streaming aggregation process is + implemented by the following pair of algorithms: + + * daf.agg_init(agg_param: AggParam) -> AggShare returns an empty + aggregate share. It is called to initialize aggregation of a + batch of measurements. + + * daf.agg_update(agg_param: AggParam, agg_share: AggShare, + out_share: OutShare) -> AggShare accumulates an output share into + an aggregate share and returns the updated aggregate share. + + In many situations it is desirable to split an aggregate share across + multiple storage elements, then merge the aggregate shares together + just before releasing the completed aggregate share to the Collector. + DAFs facilitate this with the following method: + + * daf.merge(agg_param: AggParam, agg_shares: list[AggShare]) -> + AggShare merges a sequence of aggregate shares into a single + aggregate share. + +4.4.1. Aggregation Order + + For most DAFs and VDAFs, the outcome of aggregation is not sensitive + to the order in which output shares are aggregated. This means that + aggregate shares can be updated or merged with other aggregate shares + in any order. For instance, for both Prio3 (Section 7) and Poplar1 + (Section 8), the aggregate shares and output shares both have the + same type, a vector over some finite field (Section 6.1); and + aggregation involves simply adding vectors together. + + In theory, however, there may be a DAF or VDAF for which correct + execution requires each Aggregator to aggregate output shares in the + same order. + +4.5. Unsharding + + Aggregator 0 Aggregator 1 Aggregator SHARES-1 + ============ ============ =================== + + agg_share_0 agg_share_1 agg_share_[SHARES-1] + | | | + V V V + +-----------------------------------------------+ + | unshard | + +-----------------------------------------------+ + | + V + agg_result + + Collector + ========= + + Figure 5: Illustration of unsharding. + + After the Aggregators have aggregated all measurements in the batch, + each sends its aggregate share to the Collector, who runs the + unsharding algorithm to recover the aggregate result: + + * daf.unshard(agg_param: AggParam, agg_shares: list[AggShare], + num_measurements: int) -> AggResult consumes the aggregate shares + and produces the aggregate result. + + Pre-conditions: + + - The length of agg_shares MUST be SHARES. + + - num_measurements MUST equal the number of measurements in the + batch. + +4.6. Execution of a DAF + + Secure execution of a DAF involves simulating the following procedure + over an insecure network. + + def run_daf( + daf: Daf[ + Measurement, + AggParam, + PublicShare, + InputShare, + OutShare, + AggShare, + AggResult, + ], + ctx: bytes, + agg_param: AggParam, + measurements: list[Measurement]) -> AggResult: + agg_shares: list[AggShare] + agg_shares = [daf.agg_init(agg_param) + for _ in range(daf.SHARES)] + for measurement in measurements: + # Sharding + nonce = gen_rand(daf.NONCE_SIZE) + rand = gen_rand(daf.RAND_SIZE) + (public_share, input_shares) = \ + daf.shard(ctx, measurement, nonce, rand) + + # Preparation, aggregation + for j in range(daf.SHARES): + out_share = daf.prep(ctx, j, agg_param, nonce, + public_share, input_shares[j]) + agg_shares[j] = daf.agg_update(agg_param, + agg_shares[j], + out_share) + + # Unsharding + num_measurements = len(measurements) + agg_result = daf.unshard(agg_param, agg_shares, + num_measurements) + return agg_result + + The inputs to this procedure include the parameters of the + aggregation function computed by the DAF: an aggregation parameter + and a sequence of measurements. It also includes the application + context. The procedure prescribes how a DAF is executed in a + "benign" environment in which there is no adversary and the messages + are passed among the protocol participants over secure point-to-point + channels. In reality, these channels need to be instantiated by some + "wrapper protocol", such as [DAP], that realizes these channels using + suitable cryptographic mechanisms. Moreover, some fraction of the + Aggregators (or Clients) may be malicious and diverge from their + prescribed behaviors. Section 9 describes the execution of the DAF + in various adversarial environments and what properties the wrapper + protocol needs to provide in each. + +5. Definition of VDAFs + + VDAFs are identical to DAFs except that preparation is an interactive + process carried out by the Aggregators. If successful, this process + results in each Aggregator computing an output share. The process + will fail if, for example, the underlying measurement is invalid. + + Failure manifests as an exception raised by one of the algorithms + defined in this section. If an exception is raised during + preparation, the Aggregators MUST remove the report from the batch + and not attempt to aggregate it. Otherwise, a malicious Client can + cause the Collector to compute a malformed aggregate result. + + The remainder of this section defines the VDAF interface, which we + denote by Vdaf in the remainder. The attributes are listed in + Table 2 are defined by each concrete VDAF. + + +==================+===================================+ + | Parameter | Description | + +==================+===================================+ + | ID: int | Algorithm identifier for this | + | | VDAF, in [0, 2^32). | + +------------------+-----------------------------------+ + | SHARES: int | Number of input shares into which | + | | each measurement is sharded. | + +------------------+-----------------------------------+ + | ROUNDS: int | Number of rounds of communication | + | | during preparation. | + +------------------+-----------------------------------+ + | NONCE_SIZE: int | Size of the report nonce. | + +------------------+-----------------------------------+ + | RAND_SIZE: int | Size of the random byte string | + | | consumed during sharding. | + +------------------+-----------------------------------+ + | VERIFY_KEY_SIZE: | Size of the verification key used | + | int | during preparation. | + +------------------+-----------------------------------+ + | Measurement | Type of each measurement. | + +------------------+-----------------------------------+ + | PublicShare | Type of each public share. | + +------------------+-----------------------------------+ + | InputShare | Type of each input share. | + +------------------+-----------------------------------+ + | AggParam | Type of the aggregation | + | | parameter. | + +------------------+-----------------------------------+ + | OutShare | Type of each output share. | + +------------------+-----------------------------------+ + | AggShare | Type of the aggregate share. | + +------------------+-----------------------------------+ + | AggResult | Type of the aggregate result. | + +------------------+-----------------------------------+ + | PrepState | Type of the prep state. | + +------------------+-----------------------------------+ + | PrepShare | Type of each prep share. | + +------------------+-----------------------------------+ + | PrepMessage | Type of each prep message. | + +------------------+-----------------------------------+ + + Table 2: Constants and types defined by each + concrete VDAF. + + Some of type types in the table above need to be written to the + network in order to carry out the computation. It is RECOMMENDED + that concrete instantiations of the Vdaf interface specify a method + of encoding the PublicShare, InputShare, AggParam, AggShare, + PrepShare, and PrepMessage. + + Each VDAF is identified by a unique, 32-bit integer, denoted ID. + Identifiers for each VDAF specified in this document are defined in + Table 18. The following method is used by both Prio3 and Poplar1: + + def domain_separation_tag(self, usage: int, ctx: bytes) -> bytes: + """ + Format domain separation tag for this VDAF with the given + application context and usage. + + Pre-conditions: + + - `usage` in `[0, 2^16)` + """ + return format_dst(0, self.ID, usage) + ctx + + The output, called the "domain separation tag", is used in our + constructions for domain separation. Function format_dst() is + defined in Section 6.2.3. + +5.1. Sharding + + Sharding is as described for DAFs in Section 4.1. The public share + and input shares encode additional information used during + preparation to validate the output shares before they are aggregated + (e.g., the "proof shares" in Section 7). + + Like DAFs, sharding is bound to the application context via the + application context string. Again, this is intended to ensure that + aggregation succeeds only if the Clients and Aggregators agree on the + application context. Unlike DAFs, however, disagreement on the + context should manifest as a preparation failure, causing the report + to be rejected without garbling the aggregate result. The + application context also provides some defense-in-depth against cross + protocol attacks; see Section 9.9. + +5.2. Preparation + + Aggregator 0 Aggregator 1 Aggregator SHARES-1 + ============ ============ =================== + + input_share_0 input_share_1 input_share_[SHARES-1] + | | ... | + V V V + +-----------+ +-----------+ +-----------+ + | prep_init | | prep_init | | prep_init | + +-----------+ +-----------+ +-----------+ + | | | | ... | | + V | V | V | + +---------|--------------|----------------------|-+ \ + | | | prep_shares_to_prep | | | + +---------|--------------|----------------------|-+ | + | | | | ... | | | + V V V V V | | x ROUNDS + +-----------+ +-----------+ +-----------+ | + | prep_next | | prep_next | | prep_next | | + +-----------+ +-----------+ +-----------+ | + | | | ... | | | + V V V V V / + ... ... ... + | | ... | + V V V + out_share_0 out_share_1 out_share_[SHARES-1] + + Figure 6: Illustration of interactive VDAF preparation. + + Preparation is organized into a number of rounds. The number of + rounds depends on the VDAF: Prio3 (Section 7) has one round and + Poplar1 (Section 8) has two. + + Aggregators retain some local state between successive rounds of + preparation. This is referred to as "preparation state" or "prep + state" for short. + + At the start of each round, each Aggregator broadcasts a message + called a "preparation share", or "prep share" for short. The prep + shares are then combined into a single message called the + "preparation message", or "prep message". The prep message MAY be + computed by any one of the Aggregators. + + The prep message is disseminated to each of the Aggregators to begin + the next round, or compute the output shares in case of the last + round. An Aggregator begins the first round with its input share and + it begins each subsequent round with the current prep state and the + previous prep message. Its output in the last round is its output + share and its output in each of the preceding rounds is a prep share. + + Just as for DAFs (Section 4.2), preparation involves an aggregation + parameter. The aggregation parameter is consumed by each Aggregator + before the first round of communication. + + Unlike DAFs, VDAF preparation involves a secret "verification key" + held by each of the Aggregators. This key is used to verify validity + of the output shares they compute. It is up to the high level + protocol in which the VDAF is used to arrange for the distribution of + the verification key prior to generating and processing reports. See + Section 9 for details. + + Preparation is implemented by the following set of algorithms: + + * vdaf.prep_init(verify_key: bytes, ctx: bytes, agg_id: int, + agg_param: AggParam, nonce: bytes, public_share: PublicShare, + input_share: InputShare) -> tuple[PrepState, PrepShare] is the + deterministic preparation-state initialization algorithm run by + each Aggregator. It consumes the shared verification key, the + application context, the Aggregator's unique identifier, the + aggregation parameter chosen by the Collector, the report nonce, + the public share, and one of the input shares generated by the + Client. It produces the Aggregator's initial prep state and prep + share. + + Protocols MUST ensure that public share consumed by each of the + Aggregators is identical. This is security critical for VDAFs + such as Poplar1. + + Pre-conditions: + + - verify_key MUST have length vdaf.VERIFY_KEY_SIZE. + + - agg_id MUST be the integer in [0, vdaf.SHARES) that matches the + index of input_share in the sequence of input shares output by + the Client. + + - nonce MUST have length vdaf.NONCE_SIZE. + + * vdaf.prep_shares_to_prep(ctx: bytes, agg_param: AggParam, + prep_shares: list[PrepShare]) -> PrepMessage is the deterministic + preparation-message pre-processing algorithm. It combines the + prep shares produced by the Aggregators in the previous round into + the prep message consumed by each Aggregator to start the next + round. + + * vdaf.prep_next(ctx: bytes, prep_state: PrepState, prep_msg: + PrepMessage) -> tuple[PrepState, PrepShare] | OutShare is the + deterministic preparation-state update algorithm run by each + Aggregator. It updates the Aggregator's prep state (prep_state) + and returns either its next prep state and prep share for the next + round or, if this is the last round, its output share. + + An exception may be raised by one of these algorithms, in which case + the report MUST be deemed invalid and not processed any further. + + Implementation note: The preparation process accomplishes two tasks: + recovery of output shares from the input shares and ensuring that the + recovered output shares are valid. The abstraction boundary is drawn + so that an Aggregator only recovers an output share if the underlying + data is deemed valid (at least, based on the Aggregator's view of the + protocol). Another way to draw this boundary would be to have the + Aggregators recover output shares first, then verify that they are + valid. However, this would allow the possibility of misusing the API + by, say, aggregating an invalid output share. Moreover, in protocols + like Prio+ [AGJOP21] based on oblivious transfer, it is necessary for + the Aggregators to interact in order to recover aggregatable output + shares at all. + +5.3. Validity of Aggregation Parameters + + Aggregation parameter validation is as described for DAFs in + Section 4.3. Again, each Aggregator MUST validate each aggregation + parameter received by the Collector before beginning preparation with + that parameter. + +5.4. Aggregation + + Aggregation is identical to DAF aggregation as described in + Section 4.4. Like DAFs, computation of the VDAF aggregate is not + usually sensitive to the order in which output shares are aggregated. + See Section 4.4.1. + +5.5. Unsharding + + Unsharding is identical to DAF unsharding as described in + Section 4.5. + +5.6. Execution of a VDAF + + Secure execution of a VDAF involves simulating the following + procedure over an insecure network. + + def run_vdaf( + vdaf: Vdaf[ + Measurement, + AggParam, + PublicShare, + InputShare, + list[F], # OutShare + AggShare, + AggResult, + PrepState, + PrepShare, + PrepMessage, + ], + verify_key: bytes, + agg_param: AggParam, + ctx: bytes, + measurements: list[Measurement]) -> AggResult: + """ + Pre-conditions: + + - `len(verify_key) == vdaf.VERIFY_KEY_SIZE` + """ + agg_shares = [vdaf.agg_init(agg_param) + for _ in range(vdaf.SHARES)] + for measurement in measurements: + # Sharding + nonce = gen_rand(vdaf.NONCE_SIZE) + rand = gen_rand(vdaf.RAND_SIZE) + (public_share, input_shares) = \ + vdaf.shard(ctx, measurement, nonce, rand) + + # Initialize preparation + prep_states = [] + outbound_prep_shares = [] + for j in range(vdaf.SHARES): + (state, share) = vdaf.prep_init(verify_key, ctx, j, + agg_param, + nonce, + public_share, + input_shares[j]) + prep_states.append(state) + outbound_prep_shares.append(share) + + # Complete preparation + for i in range(vdaf.ROUNDS - 1): + prep_msg = vdaf.prep_shares_to_prep(ctx, + agg_param, + outbound_prep_shares) + + outbound_prep_shares = [] + for j in range(vdaf.SHARES): + out = vdaf.prep_next(ctx, prep_states[j], prep_msg) + assert isinstance(out, tuple) + (prep_states[j], prep_share) = out + outbound_prep_shares.append(prep_share) + + prep_msg = vdaf.prep_shares_to_prep(ctx, + agg_param, + outbound_prep_shares) + + # Aggregation + for j in range(vdaf.SHARES): + out_share = vdaf.prep_next(ctx, prep_states[j], prep_msg) + assert not isinstance(out_share, tuple) + agg_shares[j] = vdaf.agg_update(agg_param, + agg_shares[j], + out_share) + + # Unsharding + num_measurements = len(measurements) + agg_result = vdaf.unshard(agg_param, agg_shares, + num_measurements) + return agg_result + + The inputs to this algorithm are the verification key, application + context, aggregation parameter, and a list of measurements. As + explained in Section 4.6, the secure execution of a VDAF requires the + application to instantiate secure channels between each of the + protocol participants. + +5.7. Communication Patterns for Preparation + + The only stage of VDAF execution that requires interaction is + preparation (Section 5.2). There are a number of ways to coordinate + this interaction; the best strategy depends largely on the number of + Aggregators (i.e., vdaf.SHARES). This section describes two + strategies, one specialized for two Aggregators and another that is + suitable for any number of Aggregators. + + In each round of preparation, each Aggregator writes a prep share to + some broadcast channel, which is then processed into the prep message + using the public prep_shares_to_prep() algorithm and broadcast to the + Aggregators to start the next round. Our goal in this section is to + realize this broadcast channel. + + The state machine of each Aggregator is shown below. + + +----------------+ + | | + v | + Start ----> Continued(prep_state, prep_round) --> Finished(out_share) + | | + | | + +--> Rejected <--+ + + Figure 7: State machine of VDAF preparation. + + State transitions are made when the state is acted upon by the + Aggregator's local inputs and/or messages sent by its co-Aggregators. + The initial state is Start. The terminal states are: Rejected, + indicating that the report cannot be processed any further; and + Finished(out_share), indicating that the Aggregator has recovered an + output share out_share. For completeness, we define these states in + Appendix B. + + The methods described in this section are defined in terms of opaque + byte strings. A compatible Vdaf MUST specify methods for encoding + public shares, input shares, prep shares, prep messages, and + aggregation parameters. + + Implementations of Prio3 and Poplar1 MUST use the encoding scheme + specified in Section 7.2.7 and Section 8.2.6 respectively. + +5.7.1. The Ping-Pong Topology (Only Two Aggregators) + + For VDAFs with precisely two Aggregators (i.e., vdaf.SHARES == 2), + the following "ping pong" communication pattern can be used. It is + compatible with any request/response transport protocol, such as + HTTP. + + Let us call the initiating Aggregator the "Leader" and the responding + Aggregator the "Helper". The high-level idea is that the Leader and + Helper will take turns running the computation locally until input + from their peer is required: + + * For a 1-round VDAF (e.g., Prio3 in Section 7), the Leader sends + its prep share to the Helper, who computes the prep message + locally, computes its output share, then sends the prep message to + the Leader. Preparation requires just one round trip between the + Leader and the Helper. + + * For a 2-round VDAF (e.g., Poplar1 in Section 8), the Leader sends + its first-round prep share to the Helper, who replies with the + first-round prep message and its second-round prep share. In the + next request, the Leader computes its second-round prep share + locally, computes its output share, and sends the second-round + prep message to the Helper. Finally, the Helper computes its own + output share. + + * In general, each request includes the Leader's prep share for the + previous round and/or the prep message for the current round; + correspondingly, each response consists of the prep message for + the current round and the Helper's prep share for the next round. + + The Aggregators proceed in this ping-ponging fashion until a step of + the computation fails (indicating the report is invalid and should be + rejected) or preparation is completed. All told there are + ceil((vdaf.ROUNDS+1)/2) requests sent. + + We specify protocol messages in the presentation language of TLS; see + Section 3 of [RFC8446]. Each message is structured as follows: + + enum { + initialize(0), + continue(1), + finish(2), + (255) + } MessageType; + + struct { + MessageType type; + select (Message.type) { + case initialize: + opaque prep_share<0..2^32-1>; + case continue: + opaque prep_msg<0..2^32-1>; + opaque prep_share<0..2^32-1>; + case finish: + opaque prep_msg<0..2^32-1>; + }; + } Message; + + These messages trigger all transitions in the state machine in + Figure 7, except for the Leader's initial transition. The Leader's + state is initialized using its local inputs with the following method + on class Vdaf: + + def ping_pong_leader_init( + self, + vdaf_verify_key: bytes, + ctx: bytes, + agg_param: bytes, + nonce: bytes, + public_share: bytes, + input_share: bytes) -> tuple[State, Optional[bytes]]: + """Called by the Leader to initialize ping-ponging.""" + try: + (prep_state, prep_share) = self.prep_init( + vdaf_verify_key, + ctx, + 0, + self.decode_agg_param(agg_param), + nonce, + self.decode_public_share(public_share), + self.decode_input_share(0, input_share), + ) + encoded_prep_share = self.encode_prep_share(prep_share) + return ( + Continued(prep_state, 0), + encode(0, encoded_prep_share), # initialize + ) + except: + return (Rejected(), None) + + The output is the State to which the Leader has transitioned and an + encoded Message. If the Leader's state is Rejected, then processing + halts. Otherwise, if the state is Continued, then processing + continues. Function encode is used to encode the outbound message, + here the initialize variant (hence 0). + + To continue processing the report, the Leader sends the outbound + message to the Helper. The Helper's initial transition is computed + using the following procedure: + + def ping_pong_helper_init( + self, + vdaf_verify_key: bytes, + ctx: bytes, + agg_param: bytes, + nonce: bytes, + public_share: bytes, + input_share: bytes, + inbound: bytes) -> tuple[State, Optional[bytes]]: + """ + Called by the Helper in response to the Leader's initial + message. + """ + try: + (prep_state, prep_share) = self.prep_init( + vdaf_verify_key, + ctx, + 1, + self.decode_agg_param(agg_param), + nonce, + self.decode_public_share(public_share), + self.decode_input_share(1, input_share), + ) + + (inbound_type, inbound_items) = decode(inbound) + if inbound_type != 0: # initialize + return (Rejected(), None) + + encoded_prep_share = inbound_items[0] + prep_shares = [ + self.decode_prep_share(prep_state, encoded_prep_share), + prep_share, + ] + return self.ping_pong_transition( + ctx, + self.decode_agg_param(agg_param), + prep_shares, + prep_state, + 0, + ) + except: + return (Rejected(), None) + + Procedure decode() decodes the inbound message and returns the + MessageType variant (initialize, continue, or finish) and the + sequence of fields. Procedure ping_pong_transition() takes in the + prep shares, combines them into the prep message, and computes the + next prep state of the caller: + + def ping_pong_transition( + self, + ctx: bytes, + agg_param: AggParam, + prep_shares: list[PrepShare], + prep_state: PrepState, + prep_round: int) -> tuple[State, bytes]: + prep_msg = self.prep_shares_to_prep(ctx, + agg_param, + prep_shares) + encoded_prep_msg = self.encode_prep_msg(prep_msg) + out = self.prep_next(ctx, prep_state, prep_msg) + if prep_round+1 == self.ROUNDS: + return ( + Finished(out), + encode(2, encoded_prep_msg), # finalize + ) + (prep_state, prep_share) = cast( + tuple[PrepState, PrepShare], out) + encoded_prep_share = self.encode_prep_share(prep_share) + return ( + Continued(prep_state, prep_round+1), + encode(1, encoded_prep_msg, encoded_prep_share) # continue + ) + + The output is the State to which the Helper has transitioned and an + encoded Message. If the Helper's state is Finished or Rejected, then + processing halts. Otherwise, if the state is Continued, then + processing continues. + + To continue processing, the Helper sends the outbound message to the + Leader. The Leader computes its next state transition using the + following method on class Vdaf: + + def ping_pong_leader_continued( + self, + ctx: bytes, + agg_param: bytes, + state: State, + inbound: bytes, + ) -> tuple[State, Optional[bytes]]: + """ + Called by the Leader to start the next step of ping-ponging. + """ + return self.ping_pong_continued( + True, ctx, agg_param, state, inbound) + + def ping_pong_continued( + self, + is_leader: bool, + ctx: bytes, + agg_param: bytes, + state: State, + inbound: bytes, + ) -> tuple[State, Optional[bytes]]: + try: + if not isinstance(state, Continued): + return (Rejected(), None) + prep_round = state.prep_round + + (inbound_type, inbound_items) = decode(inbound) + if inbound_type == 0: # initialize + return (Rejected(), None) + + encoded_prep_msg = inbound_items[0] + prep_msg = self.decode_prep_msg( + state.prep_state, + encoded_prep_msg, + ) + out = self.prep_next(ctx, state.prep_state, prep_msg) + if prep_round+1 < self.ROUNDS and \ + inbound_type == 1: # continue + (prep_state, prep_share) = cast( + tuple[PrepState, PrepShare], out) + encoded_prep_share = inbound_items[1] + prep_shares = [ + self.decode_prep_share( + prep_state, + encoded_prep_share, + ), + prep_share, + ] + if is_leader: + prep_shares.reverse() + return self.ping_pong_transition( + ctx, + self.decode_agg_param(agg_param), + prep_shares, + prep_state, + prep_round+1, + ) + elif prep_round+1 == self.ROUNDS and \ + inbound_type == 2: # finish + return (Finished(out), None) + else: + return (Rejected(), None) + except: + return (Rejected(), None) + + If the Leader's state is Finished or Rejected, then processing halts. + Otherwise, the Leader sends the outbound message to the Helper. The + Helper computes its next state transition using the following method + on class Vdaf: + + def ping_pong_helper_continued( + self, + ctx: bytes, + agg_param: bytes, + state: State, + inbound: bytes, + ) -> tuple[State, Optional[bytes]]: + """Called by the Helper to continue ping-ponging.""" + return self.ping_pong_continued( + False, ctx, agg_param, state, inbound) + + They continue in this way until processing halts. Note that, + depending on the number of rounds of preparation that are required, + there may be one more message to send before the peer can also finish + processing (i.e., outbound != None). + +5.7.2. The Star Topology (Any Number of Aggregators) + + The ping-pong topology of the previous section is only suitable for + VDAFs involving exactly two Aggregators. If the VDAF supports more + than two Aggregators, then the star topology described in this + section can be used instead. + + We again designate an Aggregator to initiate the computation. We + refer to this Aggregator as the Leader and to all other Aggregators + as Helpers. + + At the start of each round, the Leader requests from each Helper its + prep share. After gathering each of the prep shares, the Leader + computes the next prep message (via vdaf.prep_shares_to_prep()) and + broadcasts it to the Helpers. At this point, each Aggregator runs + vdaf.prep_next() locally to either recover an output share or, if + more rounds of preparation are required, compute its updated state + and prep share. If another round is required, then the Helper + responds to the broadcast message with its next prep share. + + The Aggregators proceed in this way until each recovers an output + share or some step of the computation fails. + +6. Preliminaries + + This section describes the primitives that are common to the VDAFs + specified in this document. + +6.1. Finite Fields + + Both Prio3 and Poplar1 use finite fields of prime order. Finite + field elements are represented by a class Field with the following + associated parameters: + + * MODULUS: int is the prime modulus that defines the field. + + * ENCODED_SIZE: int is the number of bytes used to encode a field + element as a byte string. + + Concrete fields, i.e., subclasses of Field, implement the following + class methods: + + * Field.zeros(length: int) -> list[Self] returns a vector of zeros + of the requested length. + + Pre-conditions: + + - length MUST be greater than or equal 0. + + Post-conditions: + + - The length of the output MUST be length. + + * Field.rand_vec(length: int) -> list[Self] returns a vector of + random field elements and has the same pre- and post-conditions as + for Field.zeros(). Note that this function is not used + normatively in the specification of either Prio3 or Poplar1. + + A field element is an instance of a concrete Field. Addition, + subtraction, multiplication, division, negation, and inversion are + denoted, respectively, x + y, x - y, x * y, x / y, -x, and x.inv(). + + We sometimes need to convert a field element to an int, which we + denote by x.int(). Likewise, each concrete Field implements a + constructor for converting an unsigned integer into a field element: + + * Field(integer: int) returns integer represented as a field + element. The value of integer MUST be in the range + (-Field.MODULUS, Field.MODULUS); negative values are treated as + negations. + +6.1.1. Auxiliary Functions + + The following class methods on Field are used to encode and decode + vectors of field elements as byte strings: + + def encode_vec(cls, vec: list[Self]) -> bytes: + """ + Encode a vector of field elements `vec` as a byte string. + """ + encoded = bytes() + for x in vec: + encoded += to_le_bytes(x.int(), cls.ENCODED_SIZE) + return encoded + + def decode_vec(cls, encoded: bytes) -> list[Self]: + """ + Parse a vector of field elements from `encoded`. + """ + if len(encoded) % cls.ENCODED_SIZE != 0: + raise ValueError( + 'input length must be a multiple of the size of an ' + 'encoded field element') + + vec = [] + while len(encoded) > 0: + (encoded_x, encoded) = front(cls.ENCODED_SIZE, encoded) + x = from_le_bytes(encoded_x) + if x >= cls.MODULUS: + raise ValueError('modulus overflow') + vec.append(cls(x)) + return vec + + Field provides the following class methods for representing a value + as a sequence of field elements, each of which represents a bit of + the input. These are used to encode measurements in some variants of + Prio3 (Section 7.4). + + def encode_into_bit_vec( + cls, + val: int, + bits: int) -> list[Self]: + """ + Encode the bit representation of `val` with at most `bits` number + of bits, as a vector of field elements. + + Pre-conditions: + + - `val >= 0` + - `bits >= 0` + """ + if val >= 2 ** bits: + # Sanity check we are able to represent `val` with `bits` + # number of bits. + raise ValueError("Number of bits is not enough to represent " + "the input integer.") + encoded = [] + for l in range(bits): + encoded.append(cls((val >> l) & 1)) + return encoded + + def decode_from_bit_vec(cls, vec: list[Self]) -> Self: + """ + Decode the field element from the bit representation, expressed + as a vector of field elements `vec`. + """ + bits = len(vec) + if cls.MODULUS >> bits == 0: + raise ValueError("Number of bits is too large to be " + "represented by field modulus.") + decoded = cls(0) + for (l, bit) in enumerate(vec): + decoded += cls(1 << l) * bit + return decoded + + Finally, the following functions define arithmetic on vectors over a + finite field. Note that an exception is raised by each function if + the operands are not the same length. + + def vec_sub(left: list[F], right: list[F]) -> list[F]: + """ + Subtract the right operand from the left and return the result. + """ + if len(left) != len(right): + raise ValueError("mismatched vector sizes") + return list(map(lambda x: x[0] - x[1], zip(left, right))) + + def vec_add(left: list[F], right: list[F]) -> list[F]: + """Add the right operand to the left and return the result.""" + if len(left) != len(right): + raise ValueError("mismatched vector sizes") + return list(map(lambda x: x[0] + x[1], zip(left, right))) + + def vec_neg(vec: list[F]) -> list[F]: + """Negate the input vector.""" + return list(map(lambda x: -x, vec)) + +6.1.2. NTT-Friendly Fields + + Some VDAFs, including Prio3, require fields that are suitable for + efficient computation of the number theoretic transform (NTT) + [SML24], as this allows for fast polynomial interpolation. + Specifically, a field is said to be "NTT-friendly" if, in addition to + the interface described in Section 6.1, it provides the following + interface: + + * Field.gen() -> Self is a class method that returns the generator + of a large subgroup of the multiplicative group. To be NTT- + friendly, the order of this subgroup MUST be a power of 2. + + * GEN_ORDER: int is the order of a multiplicative subgroup generated + by Field.gen(). This is the smallest positive integer for which + Field.gen()**Field.GEN_ORDER == Field(1). + + The size of the subgroup dictates how large interpolated polynomials + can be. It is RECOMMENDED that a generator is chosen with order at + least 2^20. + +6.1.3. Parameters + + The tables below define finite fields used in the remainder of this + document. + + +==============+================+=======================+==========+ + | Parameter | Field64 | Field128 | Field255 | + +==============+================+=======================+==========+ + | MODULUS | 2^32 * | 2^66 * | 2^255 - | + | | 4294967295 + 1 | 4611686018427387897 + | 19 | + | | | 1 | | + +--------------+----------------+-----------------------+----------+ + | ENCODED_SIZE | 8 | 16 | 32 | + +--------------+----------------+-----------------------+----------+ + | Generator | 7^4294967295 | 7^4611686018427387897 | n/a | + +--------------+----------------+-----------------------+----------+ + | GEN_ORDER | 2^32 | 2^66 | n/a | + +--------------+----------------+-----------------------+----------+ + + Table 3: Parameters for the finite fields used in this document. + +6.2. Extendable Output Functions (XOFs) + + VDAFs in this specification use eXtendable Output Functions (XOFs) + for two purposes: + + 1. Extracting short, pseudorandom strings we call "seeds" from high + entropy inputs + + 2. Expanding seeds into long, pseudorandom outputs + + Concrete XOFs implement a class Xof providing the following + interface: + + * SEED_SIZE: int is the size (in bytes) of a seed. + + * Xof(seed: bytes, dst: bytes, binder: bytes) constructs an instance + of the XOF from the given seed and a domain separation tag and + binder string as defined in Section 6.2.3. The length of the seed + will typically be SEED_SIZE, but some XOFs may support multiple + seed sizes. The seed MUST be generated securely, i.e., it is + either the output of a CSPRNG or a previous invocation of the XOF. + + * xof.next(length: int) returns the next chunk of the output of the + initialized XOF. The length of the chunk MUST be length. + + The following methods are provided for all concrete XOFs. The first + is a class method used to derive a fresh seed from an existing one. + The second is an instance method used to compute a sequence of field + elements. The third is a class method that provides a one-shot + interface for expanding a seed into a field vector. + + def derive_seed(cls, + seed: bytes, + dst: bytes, + binder: bytes) -> bytes: + """ + Derive a new seed. + + Pre-conditions: + + - `len(seed) == cls.SEED_SIZE` + """ + xof = cls(seed, dst, binder) + return xof.next(cls.SEED_SIZE) + + def next_vec(self, field: type[F], length: int) -> list[F]: + """ + Output the next `length` field elements. + + Pre-conditions: + + - `field` is sub-class of `Field` + - `length > 0` + """ + m = next_power_of_2(field.MODULUS) - 1 + vec: list[F] = [] + while len(vec) < length: + x = from_le_bytes(self.next(field.ENCODED_SIZE)) + x &= m + if x < field.MODULUS: + vec.append(field(x)) + return vec + + def expand_into_vec(cls, + field: type[F], + seed: bytes, + dst: bytes, + binder: bytes, + length: int) -> list[F]: + """ + Expand the input `seed` into vector of `length` field elements. + + Pre-conditions: + + - `field` is sub-class of `Field` + - `len(seed) == cls.SEED_SIZE` + - `length > 0` + """ + xof = cls(seed, dst, binder) + return xof.next_vec(field, length) + +6.2.1. XofTurboShake128 + + This section describes XofTurboShake128, an XOF based on the + TurboSHAKE128 function specified in [TurboSHAKE]. This XOF is + RECOMMENDED for all use cases for DAFs and VDAFs. + + TODO Update the [TurboSHAKE] reference to point to the RFC instead + of the draft. + + Pre-conditions: + + * The default seed length is 32. The seed MAY have a different + length, but MUST not exceed 255. Otherwise initialization will + raise an exception. + + * The length of the domain separation string dst passed to + XofTurboShake128 MUST NOT exceed 65535 bytes. Otherwise + initialization will raise an exception. + + class XofTurboShake128(Xof): + """XOF wrapper for TurboSHAKE128.""" + + # Associated parameters + SEED_SIZE = 32 + + def __init__(self, seed: bytes, dst: bytes, binder: bytes): + self.l = 0 + self.m = \ + to_le_bytes(len(dst), 2) + dst \ + to_le_bytes(len(seed), 1) + seed + \ + binder + + def next(self, length: int) -> bytes: + self.l += length + + # Function `TurboSHAKE128(M, D, L)` is as defined in + # Section 2.2 of [TurboSHAKE]. + # + # Implementation note: rather than re-generate the output + # stream each time `next()` is invoked, most implementations + # of TurboSHAKE128 will expose an "absorb-then-squeeze" API + # that allows stateful handling of the stream. + stream = TurboSHAKE128(self.m, 1, self.l) + return stream[-length:] + +6.2.2. XofFixedKeyAes128 + + The XOF in the previous section can be used safely wherever a XOF is + needed in this document. However, there are some situations where + TurboSHAKE128 creates a performance bottleneck and a more efficient + XOF can be used safely instead. + + This section describes XofFixedKeyAes128, which is used to implement + the IDPF of Poplar1 (Section 8.3). It is NOT RECOMMENDED to use this + XOF for any other purpose. See Section 9 for a more detailed + discussion. + + XofFixedKeyAes128 uses the AES128 blockcipher [AES] for most of the + computation, thereby taking advantage of the hardware implementations + of this blockcipher that are widely available. AES128 is used in a + fixed-key mode of operation; the key is derived during initialization + using TurboSHAKE128. + + Pre-conditions: + + * The length of the seed MUST be 16. + + * The length of the domain separation string dst passed to + XofTurboShake128 MUST NOT exceed 65535 bytes. Otherwise + initialization will raise an exception. + + class XofFixedKeyAes128(Xof): + """ + XOF based on a circular collision-resistant hash function from + fixed-key AES. + """ + + # Associated parameters + SEED_SIZE = 16 + + def __init__(self, seed: bytes, dst: bytes, binder: bytes): + if len(seed) != self.SEED_SIZE: + raise ValueError("incorrect seed size") + + self.length_consumed = 0 + + # Use TurboSHAKE128 to derive a key from the binder string + # and domain separation tag. Note that the AES key does not + # need to be kept secret from any party. However, when used + # with an IDPF, we require the binder to be a random nonce. + # + # Implementation note: this step can be cached across XOF + # evaluations with many different seeds. + dst_length = to_le_bytes(len(dst), 2) + self.fixed_key = TurboSHAKE128( + dst_length + dst + binder, + 2, + 16, + ) + self.seed = seed + + def next(self, length: int) -> bytes: + offset = self.length_consumed % 16 + new_length = self.length_consumed + length + block_range = range( + self.length_consumed // 16, + new_length // 16 + 1 + ) + self.length_consumed = new_length + + hashed_blocks = [ + self.hash_block(xor(self.seed, to_le_bytes(i, 16))) + for i in block_range + ] + return concat(hashed_blocks)[offset:offset+length] + + def hash_block(self, block: bytes) -> bytes: + """ + The multi-instance tweakable circular correlation-robust hash + function of [GKWWY20] (Section 4.2). The tweak here is the + key that stays constant for all XOF evaluations of the same + Client, but differs between Clients. + + Function `AES128(key, block)` is the AES-128 blockcipher. + """ + lo, hi = block[:8], block[8:] + sigma_block = concat([hi, xor(hi, lo)]) + return xor(AES128(self.fixed_key, sigma_block), sigma_block) + +6.2.3. The Domain Separation Tag and Binder String + + XOFs are used to map a seed to a finite domain, e.g., a fresh seed or + a vector of field elements. To ensure domain separation, derivation + is bound to some distinguished domain separation tag. The domain + separation tag encodes the following values: + + 1. The document version (i.e.,VERSION); + + 2. The "class" of the algorithm using the output (e.g., DAF, VDAF, + or IDPF as defined in Section 8.1); + + 3. A unique identifier for the algorithm (e.g., VDAF.ID); and + + 4. Some indication of how the output is used (e.g., for deriving the + measurement shares in Prio3 Section 7). + + The following algorithm is used in the remainder of this document in + order to format the domain separation tag: + + def format_dst(algo_class: int, + algo: int, + usage: int) -> bytes: + """ + Format XOF domain separation tag. + + Pre-conditions: + + - `algo_class` in `[0, 2^8)` + - `algo` in `[0, 2^32)` + - `usage` in `[0, 2^16)` + """ + return concat([ + to_be_bytes(VERSION, 1), + to_be_bytes(algo_class, 1), + to_be_bytes(algo, 4), + to_be_bytes(usage, 2), + ]) + + It is also sometimes necessary to bind the output to some ephemeral + value that multiple parties need to agree on. We call this input the + "binder string". + +7. Prio3 + + This section describes Prio3, a VDAF for general-purpose aggregation. + Prio3 is suitable for a wide variety of aggregation functions, + including (but not limited to) sum, mean, standard deviation, + histograms, and linear regression. It is compatible with any + aggregation function that has the following structure: + + * Each measurement is encoded as a vector over some finite field. + + * Measurement validity is determined by an "arithmetic circuit" + evaluated over the encoded measurement. An arithmetic circuit is + a function comprised of arithmetic operations in the field. (We + specify these in full detail in Section 7.3.2.) + + * The aggregate result is obtained by summing up the encoded + measurements and computing some function of the sum. + + Clients protect the privacy of their measurements by secret sharing + them and distributing the shares among the Aggregators. To ensure + each measurement is valid, the Aggregators run a multi-party + computation on their shares, the result of which is the output of the + arithmetic circuit. This involves verification of a "Fully Linear + Proof (FLP)" (Section 7.1) generated by the Client. FLPs are the + core component of Prio3, as they specify the types of measurements, + how they are encoded, and how they are aggregated. In fact Prio3 can + be thought of as a transformation of an FLP into a VDAF. + + Prio3 does not have an aggregation parameter. Instead, each output + share is derived from each input share by applying a fixed map. See + Section 8 for an example of a VDAF that makes meaningful use of the + aggregation parameter. + + The remainder of this section is structured as follows. The syntax + for FLPs is described in Section 7.1. The generic transformation of + an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP + suitable for any validity circuit is specified in Section 7.3. + Finally, variants of Prio3 for various types of aggregation tasks are + specified in Section 7.4. Test vectors for each variant can be found + in Appendix C. + +7.1. Fully Linear Proofs (FLPs) + + Conceptually, an FLP is a two-party protocol executed by a prover and + a verifier. In actual use, however, the prover's computation is + carried out by the Client, and the verifier's computation is + distributed among the Aggregators. The Client generates a "proof" of + its measurement's validity and distributes shares of the proof to the + Aggregators. During preparation, each Aggregator performs some + computation on its measurement share and proof share locally, then + broadcasts the result in its prep share. The validity decision is + then made by the prep_shares_to_prep() algorithm (Section 5.2). + + As usual, we describe the interface implemented by a concrete FLP in + terms of an abstract base class, denoted Flp, that specifies the set + of methods and parameters a concrete FLP must provide. + + The parameters provided by a concrete FLP are listed in Table 4. A + concrete FLP specifies the following algorithms for generating and + verifying proofs of validity (encoding is described below in + Section 7.1.1): + + * flp.prove(meas: list[F], prove_rand: list[F], joint_rand: list[F]) + -> list[F] is the proof-generation algorithm run by the prover. + Its inputs are the encoded measurement, the "prover randomness" + prove_rand, and the "joint randomness" joint_rand. The prover + randomness is used only by the prover, but the joint randomness is + shared by both the prover and verifier. + + * flp.query(meas: list[F], proof: list[F], query_rand: list[F], + joint_rand: list[F], num_shares: int) -> list[F] is the query + algorithm run by the verifier on the encoded measurement and + proof. The result of the query (i.e., the output of this + function) is called the "verifier message". In addition to the + measurement and proof, this algorithm takes as input the query + randomness query_rand and the joint randomness joint_rand. The + former is used only by the verifier. num_shares specifies the + number of shares (more on this below). + + * flp.decide(verifier: list[F]) -> bool is the deterministic + decision algorithm run by the verifier. It takes as input the + verifier message and outputs a boolean indicating if the + measurement from which it was generated is valid. + + Our application requires that the FLP is "fully linear" in the sense + defined in [BBCGGI19]. As a practical matter, what this property + implies is that, when run on a share of the measurement and proof, + the query algorithm outputs a share of the verifier message + (hereafter the "verifier share"). Furthermore, the privacy property + of the FLP system ensures that the verifier message reveals nothing + about the measurement other than the fact that it is valid. + Therefore, to decide if a measurement is valid, the Aggregators will + run the query algorithm locally, exchange verifier shares, combine + them to recover the verifier message, and run the decision algorithm. + + The query algorithm includes a parameter num_shares that specifies + the number of shares of the measurement and proof that were + generated. If these data are not secret shared, then num_shares == + 1. This parameter is useful for normalizing constants in arithmetic + circuits so that each Aggregator properly computes a secret share of + the circuit's output. See Section 7.3 for details. + + An FLP is executed by the prover and verifier as follows: + + def run_flp( + flp: Flp[Measurement, AggResult, F], + meas: list[F], + num_shares: int) -> bool: + """Run the FLP on an encoded measurement.""" + + joint_rand = flp.field.rand_vec(flp.JOINT_RAND_LEN) + prove_rand = flp.field.rand_vec(flp.PROVE_RAND_LEN) + query_rand = flp.field.rand_vec(flp.QUERY_RAND_LEN) + + # Prover generates the proof. + proof = flp.prove(meas, prove_rand, joint_rand) + + # Shard the measurement and the proof. + meas_shares = additive_secret_share( + meas, + num_shares, + flp.field, + ) + proof_shares = additive_secret_share( + proof, + num_shares, + flp.field, + ) + + # Verifier queries the meas shares and proof shares. + verifier_shares = [ + flp.query( + meas_share, + proof_share, + query_rand, + joint_rand, + num_shares, + ) + for meas_share, proof_share in zip(meas_shares, proof_shares) + ] + + # Combine the verifier shares into the verifier. + verifier = flp.field.zeros(len(verifier_shares[0])) + for verifier_share in verifier_shares: + verifier = vec_add(verifier, verifier_share) + + # Verifier decides if the measurement is valid. + return flp.decide(verifier) + + The proof system is designed so that, if meas is valid, then + run_flp(flp, meas, num_shares) always returns True. On the other + hand, if meas is invalid, then as long as joint_rand and query_rand + are generated uniform randomly, the output is False with high + probability. False positives are possible: there is a small + probability that a verifier accepts an invalid input as valid. An + FLP is said to be "sound" if this probability is sufficiently small. + The soundness of the FLP depends on a variety of parameters, like the + length of the input and the size of the field. See Section 7.3 for + details. + + Note that soundness of an FLP system is not the same as robustness + for the VDAF that uses it. In particular, soundness of the FLP is + necessary, but insufficient for robustness of Prio3 (Section 7). See + Section 9.7 for details. + + We remark that [BBCGGI19] defines a larger class of fully linear + proof systems than we consider here. In particular, what is called + an "FLP" here is called a 1.5-round, public-coin, interactive oracle + proof system in their paper. + + +=================+==========================================+ + | Parameter | Description | + +=================+==========================================+ + | PROVE_RAND_LEN: | Length of the prover randomness, the | + | int | number of random field elements consumed | + | | by the prover when generating a proof. | + +-----------------+------------------------------------------+ + | QUERY_RAND_LEN: | Length of the query randomness, the | + | int | number of random field elements consumed | + | | by the verifier. | + +-----------------+------------------------------------------+ + | JOINT_RAND_LEN: | Length of the joint randomness, the | + | int | number of random field elements shared | + | | by the prover and verifier. | + +-----------------+------------------------------------------+ + | MEAS_LEN: int | Length of the encoded measurement | + | | (Section 7.1.1). | + +-----------------+------------------------------------------+ + | OUTPUT_LEN: int | Length of the aggregatable output | + | | (Section 7.1.1). | + +-----------------+------------------------------------------+ + | PROOF_LEN: int | Length of the proof. | + +-----------------+------------------------------------------+ + | VERIFIER_LEN: | Length of the verifier message generated | + | int | by querying the measurement and proof. | + +-----------------+------------------------------------------+ + | Measurement | Type of the measurement. | + +-----------------+------------------------------------------+ + | AggResult | Type of the aggregate result. | + +-----------------+------------------------------------------+ + | field: type[F] | Class object for the field | + | | (Section 6.1). | + +-----------------+------------------------------------------+ + + Table 4: FLP parameters. + +7.1.1. Encoding the Input + + The type of measurement being aggregated is defined by the FLP. + Hence, the FLP also specifies a method of encoding raw measurements + as a vector of field elements: + + * flp.encode(measurement: Measurement) -> list[F] encodes a raw + measurement as a vector of field elements. + + Post-conditions: + + - The encoded measurement MUST have length flp.MEAS_LEN. + + For some FLPs, the encoded measurement also includes redundant field + elements that are useful for checking the proof, but which are not + needed after the proof has been checked. An example is the Sum type + defined in Section 7.4.2 for which each measurement is an integer in + range [0, max_measurement]. The range check requires encoding the + measurement with several field elements, though just one is needed + for aggregation. Thus the FLP defines an algorithm for truncating + the encoded measurement to the length of the aggregated output: + + * flp.truncate(meas: list[F]) -> list[F] maps an encoded measurement + (e.g., the bit-encoding of the measurement) to an aggregatable + output (e.g., the singleton vector containing the measurement). + + Pre-conditions: + + - The length of the input MUST be flp.MEAS_LEN + + Post-conditions: + + - The length of the output MUST be flp.OUTPUT_LEN. + + Once the aggregate shares have been transmitted to the Collector, + their sum can be converted into the aggregate result. This could be + a projection from the FLP's field to the integers, or it could + include additional post-processing. Either way, this functionality + is implemented by the following method: + + * flp.decode(output: list[F], num_measurements: int) -> AggResult + maps a sum of aggregate shares to an aggregate result. + + Pre-conditions: + + - The length of the output MUST be OUTPUT_LEN. + + - num_measurements MUST equal the number of measurements that + were aggregated. + + We remark that, taken together, these three functionalities + correspond to the notion of "Affine-aggregatable encodings (AFEs)" + from [CGB17]. + +7.1.2. Multiple Proofs + + It is sometimes desirable to generate and verify multiple independent + proofs for the same input. First, this improves the soundness of the + proof system without having to change any of its parameters. Second, + it allows a smaller field to be used (e.g., replace Field128 with + Field64, see Section 7.3) without sacrificing soundness. This is + useful because it reduces the overall communication of the protocol. + (This is a trade-off, of course, since generating and verifying more + proofs requires more time.) Given these benefits, this feature is + implemented by Prio3 (Section 7). + + To generate these proofs for a specific measurement, the prover calls + flp.prove() multiple times, each time using fresh prover and joint + randomness. The verifier checks each proof independently, each time + with fresh query randomness. It accepts the measurement only if the + decision algorithm accepts on each proof. + + See Section 9.7 for guidance on choosing the field size and number of + proofs. + +7.2. Specification + + This section specifies Prio3, an implementation of the Vdaf interface + defined in Section 5. The parameters and types required by the Vdaf + interface are defined in Table 5. The methods required for sharding, + preparation, aggregation, and unsharding are described in the + remaining subsections. These methods refer to constants enumerated + in Table 6. + + +=================+==============================================+ + | Parameter | Value | + +=================+==============================================+ + | flp | An instance of Flp (Section 7.1). | + +-----------------+----------------------------------------------+ + | xof | XofTurboShake128 (Section 6.2.1) | + +-----------------+----------------------------------------------+ + | PROOFS | Any int in the range [1, 256). | + +-----------------+----------------------------------------------+ + | VERIFY_KEY_SIZE | xof.SEED_SIZE | + +-----------------+----------------------------------------------+ + | RAND_SIZE | xof.SEED_SIZE * SHARES if flp.JOINT_RAND_LEN | + | | == 0 else 2 * xof.SEED_SIZE * SHARES | + +-----------------+----------------------------------------------+ + | NONCE_SIZE | 16 | + +-----------------+----------------------------------------------+ + | ROUNDS | 1 | + +-----------------+----------------------------------------------+ + | SHARES | Any int in the range [2, 256). | + +-----------------+----------------------------------------------+ + | Measurement | As defined by flp. | + +-----------------+----------------------------------------------+ + | AggParam | None | + +-----------------+----------------------------------------------+ + | PublicShare | Optional[list[bytes]] | + +-----------------+----------------------------------------------+ + | InputShare | tuple[list[F], list[F], Optional[bytes]] | | + | | tuple[bytes, Optional[bytes]] | + +-----------------+----------------------------------------------+ + | OutShare | list[F] | + +-----------------+----------------------------------------------+ + | AggShare | list[F] | + +-----------------+----------------------------------------------+ + | AggResult | As defined by flp. | + +-----------------+----------------------------------------------+ + | PrepState | tuple[list[F], Optional[bytes]] | + +-----------------+----------------------------------------------+ + | PrepShare | tuple[list[F], Optional[bytes]] | + +-----------------+----------------------------------------------+ + | PrepMessage | Optional[bytes] | + +-----------------+----------------------------------------------+ + + Table 5: Parameters for Prio3. + + +=============================+=======+ + | Variable | Value | + +=============================+=======+ + | USAGE_MEAS_SHARE: int | 1 | + +-----------------------------+-------+ + | USAGE_PROOF_SHARE: int | 2 | + +-----------------------------+-------+ + | USAGE_JOINT_RANDOMNESS: int | 3 | + +-----------------------------+-------+ + | USAGE_PROVE_RANDOMNESS: int | 4 | + +-----------------------------+-------+ + | USAGE_QUERY_RANDOMNESS: int | 5 | + +-----------------------------+-------+ + | USAGE_JOINT_RAND_SEED: int | 6 | + +-----------------------------+-------+ + | USAGE_JOINT_RAND_PART: int | 7 | + +-----------------------------+-------+ + + Table 6: Constants used by Prio3. + +7.2.1. Sharding + + Recall from Section 7.1 that the FLP syntax calls for "joint + randomness" shared by the prover (i.e., the Client) and the verifier + (i.e., the Aggregators). VDAFs have no such notion. Instead, the + Client derives the joint randomness from its measurement in a way + that allows the Aggregators to reconstruct it from their shares. + (This idea is based on the Fiat-Shamir heuristic and is described in + Section 6.2.3 of [BBCGGI19].) + + The sharding algorithm involves the following steps: + + 1. Encode the Client's measurement as specified by the FLP + + 2. Shard the measurement into a sequence of measurement shares + + 3. Derive the joint randomness from the measurement shares and nonce + + 4. Generate the proof using the derived joint randomness + + 5. Shard the proof into a sequence of proof shares + + As described in Section 7.1.2, robustness of Prio3 can be amplified + by generating and verifying multiple proofs. To support this: + + * In step 3, derive as much joint randomness as required by PROOFS + proofs + + * Repeat step 4 PROOFS times, each time with a unique joint + randomness + + Depending on the FLP, joint randomness may not be required. In + particular, when flp.JOINT_RAND_LEN == 0, the Client does not derive + the joint randomness (Step 3). + + The sharding algorithm is specified below: + + def shard( + self, + ctx: bytes, + measurement: Measurement, + nonce: bytes, + rand: bytes) -> tuple[ + Optional[list[bytes]], + list[Prio3InputShare]]: + if len(nonce) != self.NONCE_SIZE: + raise ValueError("incorrect nonce size") + if len(rand) != self.RAND_SIZE: + raise ValueError("incorrect size of random bytes argument") + + l = self.xof.SEED_SIZE + seeds = [rand[i:i + l] for i in range(0, self.RAND_SIZE, l)] + + meas = self.flp.encode(measurement) + if self.flp.JOINT_RAND_LEN > 0: + return self.shard_with_joint_rand(ctx, meas, nonce, seeds) + else: + return self.shard_without_joint_rand(ctx, meas, seeds) + + It starts by splitting the randomness into seeds. It then encodes + the measurement as prescribed by the FLP and calls one of two + methods, depending on whether joint randomness is required by the + FLP. The methods are defined in the subsections below. + +7.2.1.1. FLPs Without Joint Randomness + + The following method is used for FLPs that do not require joint + randomness, i.e., when flp.JOINT_RAND_LEN == 0. It consists of the + following steps: + + 1. Shard the encoded measurement into shares + + 2. Generate proofs and shard each into shares + + 3. Encode each measurement share and shares of each proof into an + input share + + Only one pair of measurement and proof(s) share (called the "Leader" + shares above) are vectors of field elements. The other shares + (called the "Helper" shares) are represented instead by an XOF seed, + which is expanded into vectors of field elements. The methods on + Prio3 for deriving the prover randomness, measurement shares, and + proof shares are defined in Section 7.2.6. + + def shard_without_joint_rand( + self, + ctx: bytes, + meas: list[F], + seeds: list[bytes]) -> tuple[ + Optional[list[bytes]], + list[Prio3InputShare[F]]]: + helper_shares, seeds = front(self.SHARES - 1, seeds) + (prove_seed,), seeds = front(1, seeds) + + # Shard the encoded measurement into shares. + leader_meas_share = meas + for j in range(self.SHARES - 1): + leader_meas_share = vec_sub( + leader_meas_share, + self.helper_meas_share(ctx, j + 1, helper_shares[j]), + ) + + # Generate and shard each proof into shares. + prove_rands = self.prove_rands(ctx, prove_seed) + leader_proofs_share = [] + for _ in range(self.PROOFS): + prove_rand, prove_rands = front( + self.flp.PROVE_RAND_LEN, prove_rands) + leader_proofs_share += self.flp.prove(meas, prove_rand, []) + for j in range(self.SHARES - 1): + leader_proofs_share = vec_sub( + leader_proofs_share, + self.helper_proofs_share( + ctx, + j + 1, + helper_shares[j], + ), + ) + + # Each Aggregator's input share contains its measurement share + # and its share of the proof(s). + input_shares: list[Prio3InputShare[F]] = [] + input_shares.append(( + leader_meas_share, + leader_proofs_share, + None, + )) + for j in range(self.SHARES - 1): + input_shares.append(( + helper_shares[j], + None, + )) + return (None, input_shares) + +7.2.1.2. FLPs With Joint Randomness + + The following method is used for FLPs that require joint randomness, + i.e., for which flp.JOINT_RAND_LEN > 0. Joint randomness derivation + involves an additional XOF seed for each Aggregator called the + "blind". The computation involves the following steps: + + 1. Compute a "joint randomness part" from each measurement share and + blind + + 2. Compute a "joint randomness seed" from the joint randomness parts + + 3. Compute the joint randomness for each proof evaluation from the + joint randomness seed + + This three-step process is designed to ensure that the joint + randomness does not leak the measurement to the Aggregators while + preventing a malicious Client from tampering with the joint + randomness in a way that allows it to break robustness. To bootstrap + the required check, the Client encodes the joint randomness parts in + the public share. (See Section 7.2.2 for details.) + + All functions used in the following listing are defined in + Section 7.2.6: + + def shard_with_joint_rand( + self, + ctx: bytes, + meas: list[F], + nonce: bytes, + seeds: list[bytes]) -> tuple[ + Optional[list[bytes]], + list[Prio3InputShare[F]]]: + helper_seeds, seeds = front((self.SHARES - 1) * 2, seeds) + helper_shares = [ + helper_seeds[i] + for i in range(0, (self.SHARES - 1) * 2, 2) + ] + helper_blinds = [ + helper_seeds[i] + for i in range(1, (self.SHARES - 1) * 2, 2) + ] + (leader_blind, prove_seed), seeds = front(2, seeds) + + # Shard the encoded measurement into shares and compute the + # joint randomness parts. + leader_meas_share = meas + joint_rand_parts = [] + for j in range(self.SHARES - 1): + helper_meas_share = self.helper_meas_share( + ctx, j + 1, helper_shares[j]) + leader_meas_share = vec_sub(leader_meas_share, + helper_meas_share) + joint_rand_parts.append(self.joint_rand_part( + ctx, j + 1, helper_blinds[j], + helper_meas_share, nonce)) + joint_rand_parts.insert(0, self.joint_rand_part( + ctx, 0, leader_blind, leader_meas_share, nonce)) + + # Generate each proof and shard it into proof shares. + prove_rands = self.prove_rands(ctx, prove_seed) + joint_rands = self.joint_rands( + ctx, self.joint_rand_seed(ctx, joint_rand_parts)) + leader_proofs_share = [] + for _ in range(self.PROOFS): + prove_rand, prove_rands = front( + self.flp.PROVE_RAND_LEN, prove_rands) + joint_rand, joint_rands = front( + self.flp.JOINT_RAND_LEN, joint_rands) + leader_proofs_share += self.flp.prove( + meas, + prove_rand, + joint_rand, + ) + for j in range(self.SHARES - 1): + leader_proofs_share = vec_sub( + leader_proofs_share, + self.helper_proofs_share( + ctx, + j + 1, + helper_shares[j], + ), + ) + + # Each Aggregator's input share contains its measurement share, + # share of proof(s), and blind. The public share contains the + # Aggregators' joint randomness parts. + input_shares: list[Prio3InputShare[F]] = [] + input_shares.append(( + leader_meas_share, + leader_proofs_share, + leader_blind, + )) + for j in range(self.SHARES - 1): + input_shares.append(( + helper_shares[j], + helper_blinds[j], + )) + return (joint_rand_parts, input_shares) + +7.2.2. Preparation + + This section describes the process of recovering output shares from + the input shares. The high-level idea is that each Aggregator first + queries its measurement share and proof(s) share(s) locally, then + broadcasts its verifier share(s) in its prep share. The shares of + verifier(s) are then combined into the verifier message(s) used to + decide whether to accept. + + In addition, the Aggregators must recompute the same joint randomness + used by the Client to generate the proof(s). In order to avoid an + extra round of communication, the Client includes the joint + randomness parts in the public share. This leaves open the + possibility that the Client cheated by, say, forcing the Aggregators + to use joint randomness that biases the proof check procedure some + way in its favor. To mitigate this, the Aggregators also check that + they have all computed the same joint randomness seed before + accepting their output shares. To do so, they exchange their parts + of the joint randomness along with their shares of verifier(s). + + Implementation note: the prep state for Prio3 includes the output + share that will be released once preparation is complete. In some + situations, it may be necessary for the Aggregator to encode this + state as bytes and store it for retrieval later on. For all but the + first Aggregator, it is possible to save storage by storing the + measurement share rather than output share itself. It is relatively + inexpensive to expand this seed into the measurement share, then + truncate the measurement share to get the output share. + + All functions used in the following listing are defined in + Section 7.2.6: + + def prep_init( + self, + verify_key: bytes, + ctx: bytes, + agg_id: int, + _agg_param: None, + nonce: bytes, + public_share: Optional[list[bytes]], + input_share: Prio3InputShare[F]) -> tuple[ + Prio3PrepState[F], + Prio3PrepShare[F]]: + joint_rand_parts = public_share + (meas_share, proofs_share, blind) = \ + self.expand_input_share(ctx, agg_id, input_share) + out_share = self.flp.truncate(meas_share) + + # Compute the joint randomness. + joint_rand: list[F] = [] + corrected_joint_rand_seed, joint_rand_part = None, None + if self.flp.JOINT_RAND_LEN > 0: + assert blind is not None + assert joint_rand_parts is not None + joint_rand_part = self.joint_rand_part( + ctx, agg_id, blind, meas_share, nonce) + joint_rand_parts[agg_id] = joint_rand_part + corrected_joint_rand_seed = self.joint_rand_seed( + ctx, joint_rand_parts) + joint_rands = self.joint_rands( + ctx, corrected_joint_rand_seed) + + # Query the measurement and proof share. + query_rands = self.query_rands(verify_key, ctx, nonce) + verifiers_share = [] + for _ in range(self.PROOFS): + proof_share, proofs_share = front( + self.flp.PROOF_LEN, proofs_share) + query_rand, query_rands = front( + self.flp.QUERY_RAND_LEN, query_rands) + if self.flp.JOINT_RAND_LEN > 0: + joint_rand, joint_rands = front( + self.flp.JOINT_RAND_LEN, joint_rands) + verifiers_share += self.flp.query( + meas_share, + proof_share, + query_rand, + joint_rand, + self.SHARES, + ) + + prep_state = (out_share, corrected_joint_rand_seed) + prep_share = (verifiers_share, joint_rand_part) + return (prep_state, prep_share) + + def prep_shares_to_prep( + self, + ctx: bytes, + _agg_param: None, + prep_shares: list[Prio3PrepShare[F]]) -> Optional[bytes]: + # Unshard each set of verifier shares into each verifier message. + verifiers = self.flp.field.zeros( + self.flp.VERIFIER_LEN * self.PROOFS) + joint_rand_parts = [] + for (verifiers_share, joint_rand_part) in prep_shares: + verifiers = vec_add(verifiers, verifiers_share) + if self.flp.JOINT_RAND_LEN > 0: + assert joint_rand_part is not None + joint_rand_parts.append(joint_rand_part) + + # Verify that each proof is well-formed and input is valid. + for _ in range(self.PROOFS): + verifier, verifiers = front(self.flp.VERIFIER_LEN, verifiers) + if not self.flp.decide(verifier): + raise ValueError('proof verifier check failed') + + # Combine the joint randomness parts computed by the + # Aggregators into the true joint randomness seed. This is + # used in the last step. + joint_rand_seed = None + if self.flp.JOINT_RAND_LEN > 0: + joint_rand_seed = self.joint_rand_seed(ctx, joint_rand_parts) + return joint_rand_seed + + def prep_next( + self, + _ctx: bytes, + prep_state: Prio3PrepState[F], + prep_msg: Optional[bytes] + ) -> tuple[Prio3PrepState[F], Prio3PrepShare[F]] | list[F]: + joint_rand_seed = prep_msg + (out_share, corrected_joint_rand_seed) = prep_state + + # If joint randomness was used, check that the value computed by + # the Aggregators matches the value indicated by the Client. + if joint_rand_seed != corrected_joint_rand_seed: + raise ValueError('joint randomness check failed') + + return out_share + +7.2.3. Validity of Aggregation Parameters + + Prio3 only permits a report to be aggregated once. + + def is_valid( + self, + _agg_param: None, + previous_agg_params: list[None]) -> bool: + return len(previous_agg_params) == 0 + +7.2.4. Aggregation + + Aggregating a set of output shares is simply a matter of adding up + the vectors element-wise. + + def agg_init(self, _agg_param: None) -> list[F]: + return self.flp.field.zeros(self.flp.OUTPUT_LEN) + + def agg_update(self, + _agg_param: None, + agg_share: list[F], + out_share: list[F]) -> list[F]: + return vec_add(agg_share, out_share) + + def merge(self, + _agg_param: None, + agg_shares: list[list[F]]) -> list[F]: + agg = self.agg_init(None) + for agg_share in agg_shares: + agg = vec_add(agg, agg_share) + return agg + +7.2.5. Unsharding + + To unshard a set of aggregate shares, the Collector first adds up the + vectors element-wise, then decodes the aggregate result from the sum + according to the FLP (Section 7.1.1). + + def unshard( + self, + _agg_param: None, + agg_shares: list[list[F]], + num_measurements: int) -> AggResult: + agg = self.merge(None, agg_shares) + return self.flp.decode(agg, num_measurements) + +7.2.6. Auxiliary Functions + + This section defines a number of auxiliary functions referenced by + the main algorithms for Prio3 in the preceding sections. + + def helper_meas_share( + self, + ctx: bytes, + agg_id: int, + share: bytes) -> list[F]: + return self.xof.expand_into_vec( + self.flp.field, + share, + self.domain_separation_tag(USAGE_MEAS_SHARE, ctx), + byte(agg_id), + self.flp.MEAS_LEN, + ) + + def helper_proofs_share( + self, + ctx: bytes, + agg_id: int, + share: bytes) -> list[F]: + return self.xof.expand_into_vec( + self.flp.field, + share, + self.domain_separation_tag(USAGE_PROOF_SHARE, ctx), + byte(self.PROOFS) + byte(agg_id), + self.flp.PROOF_LEN * self.PROOFS, + ) + + def expand_input_share( + self, + ctx: bytes, + agg_id: int, + input_share: Prio3InputShare[F]) -> tuple[ + list[F], + list[F], + Optional[bytes]]: + if agg_id > 0: + assert len(input_share) == 2 + (share, blind) = input_share + meas_share = self.helper_meas_share(ctx, agg_id, share) + proofs_share = self.helper_proofs_share(ctx, agg_id, share) + else: + assert len(input_share) == 3 + (meas_share, proofs_share, blind) = input_share + return (meas_share, proofs_share, blind) + + def prove_rands(self, ctx: bytes, prove_seed: bytes) -> list[F]: + return self.xof.expand_into_vec( + self.flp.field, + prove_seed, + self.domain_separation_tag(USAGE_PROVE_RANDOMNESS, ctx), + byte(self.PROOFS), + self.flp.PROVE_RAND_LEN * self.PROOFS, + ) + + def query_rands( + self, + verify_key: bytes, + ctx: bytes, + nonce: bytes) -> list[F]: + return self.xof.expand_into_vec( + self.flp.field, + verify_key, + self.domain_separation_tag(USAGE_QUERY_RANDOMNESS, ctx), + byte(self.PROOFS) + nonce, + self.flp.QUERY_RAND_LEN * self.PROOFS, + ) + + def joint_rand_part( + self, + ctx: bytes, + agg_id: int, + blind: bytes, + meas_share: list[F], + nonce: bytes) -> bytes: + return self.xof.derive_seed( + blind, + self.domain_separation_tag(USAGE_JOINT_RAND_PART, ctx), + byte(agg_id) + nonce + self.flp.field.encode_vec(meas_share), + ) + + def joint_rand_seed(self, + ctx: bytes, + joint_rand_parts: list[bytes]) -> bytes: + """Derive the joint randomness seed from its parts.""" + return self.xof.derive_seed( + zeros(self.xof.SEED_SIZE), + self.domain_separation_tag(USAGE_JOINT_RAND_SEED, ctx), + concat(joint_rand_parts), + ) + + def joint_rands(self, + ctx: bytes, + joint_rand_seed: bytes) -> list[F]: + """Derive the joint randomness from its seed.""" + return self.xof.expand_into_vec( + self.flp.field, + joint_rand_seed, + self.domain_separation_tag(USAGE_JOINT_RANDOMNESS, ctx), + byte(self.PROOFS), + self.flp.JOINT_RAND_LEN * self.PROOFS, + ) + +7.2.7. Message Serialization + + This section defines serialization formats for messages exchanged + over the network while executing Prio3. Messages are defined in the + presentation language of TLS as defined in Section 3 of [RFC8446]. + + Let prio3 denote an instance of Prio3. In the remainder we use S as + an alias for prio3.xof.SEED_SIZE and F as an alias for + prio3.field.ENCODED_SIZE. XOF seeds are represented as follows: + + opaque Prio3Seed[S]; + + Field elements are encoded in little-endian byte order (as defined in + Section 6.1) and represented as follows: + + opaque Prio3Field[F]; + +7.2.7.1. Public Share + + The content of the public share depends on whether joint randomness + is required for the underlying FLP (i.e., prio3.flp.JOINT_RAND_LEN > + 0). If joint randomness is not used, then the public share is the + empty string. Otherwise, if joint randomness is used, then the + public share encodes the joint randomness parts as follows: + + struct { + Prio3Seed joint_rand_parts[S * prio3.SHARES]; + } Prio3PublicShareWithJointRand; + +7.2.7.2. Input Share + + Just as for the public share, the content of the input shares depends + on whether joint randomness is used. If so, then each input share + includes the Aggregator's blind for generating its joint randomness + part. + + In addition, the encoding of the input shares depends on which + aggregator is receiving the message. If the aggregator ID is 0, then + the input share includes the full measurement share and proofs(s) + share(s). Otherwise, if the aggregator ID is greater than 0, then + the measurement and shares of proof(s) are represented by an XOF + seed. We shall call the former the "Leader" and the latter the + "Helpers". + + In total there are four variants of the input share. When joint + randomness is not used, the Leader's share is structured as follows: + + struct { + Prio3Field meas_share[F * prio3.flp.MEAS_LEN]; + Prio3Field proofs_share[F * prio3.flp.PROOF_LEN * prio3.PROOFS]; + } Prio3LeaderShare; + + When joint randomness is not used, the Helpers' shares are structured + as follows: + + struct { + Prio3Seed share; + } Prio3HelperShare; + + When joint randomness is used, the Leader's input share is structured + as follows: + + struct { + Prio3LeaderShare inner; + Prio3Seed blind; + } Prio3LeaderShareWithJointRand; + + Finally, when joint randomness is used, the Helpers' shares are + structured as follows: + + struct { + Prio3HelperShare inner; + Prio3Seed blind; + } Prio3HelperShareWithJointRand; + +7.2.7.3. Prep Share + + When joint randomness is not used, the prep share is structured as + follows: + + struct { + Prio3Field verifiers_share[F * V]; + } Prio3PrepShare; + + where V = prio3.flp.VERIFIER_LEN * prio3.PROOFS. When joint + randomness is used, the prep share includes the Aggregator's joint + randomness part and is structured as follows: + + struct { + Prio3Field verifiers_share[F * V]; + Prio3Seed joint_rand_part; + } Prio3PrepShareWithJointRand; + +7.2.7.4. Prep Message + + When joint randomness is not used, the prep message is the empty + string. Otherwise the prep message consists of the joint randomness + seed computed by the Aggregators: + + struct { + Prio3Seed joint_rand; + } Prio3PrepMessageWithJointRand; + +7.2.7.5. Aggregation + + Aggregate shares are structured as follows: + + struct { + Prio3Field agg_share[F * prio3.flp.OUTPUT_LEN]; + } Prio3AggShare; + +7.3. FLP Specification + + +================+=======================================+ + | Parameter | Value | + +================+=======================================+ + | valid | An instance of Valid (Section 7.3.2). | + +----------------+---------------------------------------+ + | field | valid.field | + +----------------+---------------------------------------+ + | PROVE_RAND_LEN | valid.prove_rand_len() | + +----------------+---------------------------------------+ + | QUERY_RAND_LEN | valid.query_rand_len() | + +----------------+---------------------------------------+ + | JOINT_RAND_LEN | valid.JOINT_RAND_LEN | + +----------------+---------------------------------------+ + | MEAS_LEN | valid.MEAS_LEN | + +----------------+---------------------------------------+ + | OUTPUT_LEN | valid.OUTPUT_LEN | + +----------------+---------------------------------------+ + | PROOF_LEN | valid.proof_len() | + +----------------+---------------------------------------+ + | VERIFIER_LEN | valid.verifier_len() | + +----------------+---------------------------------------+ + | Measurement | As defined by valid. | + +----------------+---------------------------------------+ + | AggResult | As defined by valid. | + +----------------+---------------------------------------+ + + Table 7: FLP parameters for a validity circuit. + + This section specifies an implementation of the Flp interface + (Section 7.1) based on the construction from [BBCGGI19], Section 4.2. + The types and parameters required by this interface are listed in the + table above. + + We begin in Section 7.3.1 with an overview of the proof system and + some extensions to it. Section 7.3.2 defines validity circuits, the + core component of the proof system that determines measurement + validity and how measurements are aggregated. The proof-generation + algorithm, query algorithm, and decision algorithm are defined in + Section 7.3.3, Section 7.3.4, and Section 7.3.5 respectively. + +7.3.1. Overview + + An FLP is a type of "zero-knowledge proof". A conventional zero- + knowledge proof system involves two parties: + + * The prover, who holds a measurement and generates a proof of the + measurement's validity + + * The verifier who holds an encryption of, or commitment to, the + measurement and checks the proof + + Our proof system is much the same, except the verifier is split + across multiple Aggregators, each of which has a secret share of the + measurement rather than an encryption of it. + + Validity is defined in terms of an arithmetic circuit evaluated over + the measurement. The inputs to this circuit are elements of a finite + field that comprise the encoded measurement; the gates of the circuit + are multiplication, addition, and subtraction operations; and the + output of the circuit is a single field element. If the value is + zero, then the measurement is deemed valid; otherwise, if the output + is non-zero, then the measurement is deemed invalid. + + For example, the simplest circuit specified in this document is the + following (Section 7.4.1): + + C(x) = x * (x-1) + + This circuit contains one subtraction gate (x-1) and one + multiplication gate (x * (x-1)). Observe that C(x) = 0 if and only + if x is in [0, 2). + + The goal of the proof system is to allow each Aggregator to privately + and correctly compute a share of C(x) from its share of x. Then all + they need to determine validity is to broadcast their shares of C(x). + + Suppose for a moment that C is an affine arithmetic circuit, meaning + its only operations are addition, subtraction, and multiplication-by- + constant. (The circuit above is non-affine because it contains a + multiplication gate with two non-constant inputs.) Then each + Aggregator can compute its share locally, since + + C(x_shares[0] + ... + x_shares[SHARES-1]) = + C(x_shares[0]) + ... + C(x_shares[SHARES-1]) + + (Note that, for this equality to hold, it is necessary to scale any + addition of a constant in the circuit by 1/SHARES.) However, this is + not the case if C contains multiplication gates with two non-constant + inputs. Thus our goal is to transform these multiplication gates + into computations on secret shared data that each Aggregator can + perform locally. + + The key idea is to have the prover construct a polynomial p such that + p(j) is equal to the output of the j-th multiplication gate. + Polynomial evaluation is fully linear, which means the coefficients + of the polynomial can be secret shared in a way that allows each + Aggregator to compute a share of p(j) for any j. These intermediate + results can then be combined with the affine arithmetic operations of + the validity circuit to produce the final output. + + Applying this idea to the example circuit C above: + + 1. The Client, given its measurement x, constructs the lowest degree + polynomial p for which p(0) = s and p(1) = x * (x-1), where s is + a random blinding value generated by the Client. (The blinding + value is to protect the privacy of the measurement.) It then + sends shares of x and shares of the coefficients of p to each of + the Aggregators. + + 2. Each Aggregator locally computes and broadcasts its share of + p(1), which is equal to its share of C(x). + + In fact, our FLP is slightly more general than this. We can replace + the multiplication gate with any non-affine sub-circuit and apply the + same idea. For example, in Section 7.4.2, the validity circuit uses + the following sub-circuit multiple times: + + Range2(x) = x * (x-1) = x^2 - x + + (This is the same functionality computed by the example circuit C + above.) Here again we can interpolate the lowest degree polynomial p + for which p(j) is the value of the j-th call to Range2 in the + validity circuit. Each validity circuit defines a sub-circuit that + encapsulates its non-affine arithmetic operations. We refer to this + sub-circuit as the "gadget". + + Finally, the proof system has one more important component. It is + possible for a malicious Client to produce a gadget polynomial p that + would result in C(x) being computed incorrectly, potentially + resulting in an invalid measurement being accepted. To prevent this, + the Aggregators perform a probabilistic test to check that the gadget + polynomial was constructed properly. This "gadget test", and the + procedure for constructing the polynomial, are described in detail in + Section 7.3.3. + +7.3.1.1. Extensions + + The FLP described in Section 7.3 extends the proof system of + [BBCGGI19], Section 4.2 in a few ways. + + First, the validity circuit in our construction includes an + additional, random input (this is the "joint randomness" derived from + the measurement shares in Prio3; see Section 7.2). This allows for + circuit optimizations that trade a small soundness error for a + shorter proof. For example, consider a circuit that recognizes the + set of length-N vectors for which each element is either one or zero. + A deterministic circuit could be constructed for this language, but + it would involve a large number of multiplications that would result + in a large proof. (See the discussion in [BBCGGI19], Section 5.2 for + details). A much shorter proof can be constructed for the following + randomized circuit: + + C(x, r) = r * Range2(x[0]) + ... + r^N * Range2(x[N-1]) + + (Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here + x is the length-N input and r is a random field element. The gadget + circuit Range2 is the "range-check" polynomial described above, i.e., + Range2(x) = x^2 - x. The idea is that, if x is valid, i.e., each + x[j] is in [0, 2), then the circuit will evaluate to zero regardless + of the value of r; but if some x[j] is not in [0, 2), then the output + will be non-zero with high probability. + + The second extension implemented by our FLP allows the validity + circuit to contain multiple gadget types. (This generalization was + suggested in [BBCGGI19], Remark 4.5.) This provides additional + flexibility for designing circuits by allowing multiple, non-affine + sub-circuits. For example, the following circuit is allowed: + + C(x, r) = r * Range2(x[0]) + ... + r^L * Range2(x[L-1]) + \ + r^(L+1) * Range3(x[L]) + ... + r^N * Range3(x[N-1]) + + where Range3(x) = x^3 - 3x^2 + 2x. This circuit checks that the + first L inputs are in [0, 2) and the last N-L inputs are in [0, 3). + The same circuit can be expressed using a simpler gadget, namely + multiplication, but the resulting proof would be longer. + + Third, rather than interpolate the gadget polynomial at inputs 1, 2, + ..., j, ..., where j is the j-th invocation of the gadget, we use + roots of unity for the field. This allows us to construct each + gadget polynomial via the number theoretic transform [SML24], which + is far more efficient than generic formulas. Note that the roots of + unity are powers of the generator for the NTT-friendly field (see + Section 6.1.2). + + Finally, the validity circuit in our FLP may have any number of + outputs (at least one). The input is said to be valid if each of the + outputs is zero. To save bandwidth, we take a random linear + combination of the outputs. If each of the outputs is zero, then the + reduced output will be zero; but if one of the outputs is non-zero, + then the reduced output will be non-zero with high probability. + +7.3.2. Validity Circuits + + +=========================+========================================+ + | Parameter | Description | + +=========================+========================================+ + | GADGETS: list[Gadget] | A list of gadgets. | + +-------------------------+----------------------------------------+ + | GADGET_CALLS: list[int] | Number of times each gadget is called. | + +-------------------------+----------------------------------------+ + | MEAS_LEN: int | Length of the measurement. | + +-------------------------+----------------------------------------+ + | JOINT_RAND_LEN: int | Length of the joint randomness. | + +-------------------------+----------------------------------------+ + | EVAL_OUTPUT_LEN: int | Length of the circuit output. | + +-------------------------+----------------------------------------+ + | OUTPUT_LEN: int | Length of the aggregatable output. | + +-------------------------+----------------------------------------+ + | Measurement | Type of the measurement. | + +-------------------------+----------------------------------------+ + | AggResult | Type of the aggregate result. | + +-------------------------+----------------------------------------+ + | field: type[F] | Class object for the field | + | | (Section 6.1.2) | + +-------------------------+----------------------------------------+ + + Table 8: Validity circuit parameters. + + An instance of the proof system is defined in terms of a validity + circuit that implements the Valid interface specified in this + section. The parameters are listed in the table above. + + Each circuit has a list of gadgets, denoted GADGETS. The circuit + evaluated by the gadget MAY be non-affine, but MUST be arithmetic, + i.e., composed only of multiplication, addition, and subtraction + gates. An instance of class Gadget has the following interface: + + * ARITY: int is the number of input wires. For example, the + multiplication gadget Mul(x,y) = x*y has arity of 2. + + * DEGREE: int is the arithmetic degree of the gadget circuit. This + is defined to be the degree of the polynomial that computes it. + This exists because the circuit is arithmetic. For example, Mul + has degree 2. + + * gadget.eval(field: type[F], inp: list[F]) -> F evaluates the + gadget over the given inputs and field. + + * gadget.eval_poly(field: type[F], inp_poly: list[list[F]]) -> + list[F] is the same as eval() except it evaluates the circuit over + the polynomial ring of the field. This is well defined because + the circuit is arithmetic. + + In addition to the list of gadgets, the validity circuit specifies + how many times each gadget is called (GADGET_CALLS). It also + specifies the length of the circuit's input (MEAS_LEN), the length of + the joint randomness (JOINT_RAND_LEN), and the length of the + circuit's output (EVAL_OUTPUT_LEN). + + A validity circuit also specifies parameters and methods needed for + Prio3 aggregation. These are used to implement the interface in + Section 7.1.1: + + * valid.encode(measurement: Measurement) -> list[F] returns a vector + of length MEAS_LEN representing a measurement of type Measurement. + + * valid.truncate(meas: list[F]) -> list[F] returns a vector of + length OUTPUT_LEN representing (a share of) an aggregatable + output. + + * valid.decode(agg: list[F], num_measurements: int) -> AggResult + returns an aggregate result of type AggResult. This computation + may depend on the number of outputs aggregated. + + Finally, the following are helper methods used to instantiate + parameters of the Flp interface (Section 7.1): + + def prove_rand_len(self) -> int: + """Length of the prover randomness.""" + return sum(g.ARITY for g in self.GADGETS) + + def query_rand_len(self) -> int: + """Length of the query randomness.""" + query_rand_len = len(self.GADGETS) + if self.EVAL_OUTPUT_LEN > 1: + query_rand_len += self.EVAL_OUTPUT_LEN + return query_rand_len + + def proof_len(self) -> int: + """Length of the proof.""" + length = 0 + for (g, g_calls) in zip(self.GADGETS, self.GADGET_CALLS): + p = next_power_of_2(1 + g_calls) + length += g.ARITY + g.DEGREE * (p - 1) + 1 + return length + + def verifier_len(self) -> int: + """Length of the verifier message.""" + length = 1 + for g in self.GADGETS: + length += g.ARITY + 1 + return length + +7.3.3. Generating the Proof + + +------------------+ + | Prove | + | +-------------+ | + | | Valid |<---- meas + | | +--------+ |<---- joint rand + | | | Gadget | | |<- prove rand + | | +--------+ | | + | +-------------+ | + +------------------+ + | + V proof + + Figure 8: Components of the proof generation algorithm. + + The proof generation algorithm invokes the validity circuit on the + encoded measurement and joint randomness. The validity circuit in + turn invokes the gadgets defined by the circuit. The prove + randomness is used to construct the gadget polynomials that the + verifier will use to compute the outputs of each gadget. + + To generate the gadget polynomials, the prover evaluates the validity + circuit, setting g.wires[j][k] to the value of the j-th wire for the + k-th invocation of gadget g. This is accomplished by "wrapping" each + gadget in a class ProveGadget that records the wire inputs. We list + this class in Appendix A.4. + + Next, we compute each of the "wire polynomials" for each gadget. The + j-th wire polynomial is the lowest degree polynomial that evaluates + to g.wire[j][k] at a sequence of fixed points. We obtain the gadget + polynomial by evaluating the gadget on the wire polynomials. + + def prove(self, + meas: list[F], + prove_rand: list[F], + joint_rand: list[F]) -> list[F]: + # Evaluate the validity circuit, recording the value of each + # input wire for each evaluation of each gadget. + valid = ProveGadget.wrap(self.valid, prove_rand) + valid.eval(meas, joint_rand, 1) + + # Construct the proof, which consists of the wire seeds and + # gadget polynomial for each gadget. + proof = [] + for g in cast(list[ProveGadget[F]], valid.GADGETS): + p = len(g.wires[0]) + + # Compute the wire polynomials for this gadget. For each `j`, + # find the lowest degree polynomial `wire_poly` for which + # `wire_poly(alpha^k) = g.wires[j][k]` for all `k`. Note that + # each `g.wires[j][0]` is set to the seed of wire `j`, which + # is included in the prove randomness. + # + # Implementation note: `alpha` is a root of unity, which + # means `poly_interp()` can be evaluated using the NTT. Note + # that `g.wires[j]` is padded with 0s to a power of 2. + alpha = self.field.gen() ** (self.field.GEN_ORDER // p) + wire_inp = [alpha ** k for k in range(p)] + wire_polys = [] + for j in range(g.ARITY): + wire_poly = poly_interp(self.field, wire_inp, g.wires[j]) + wire_polys.append(wire_poly) + + # Compute the gadget polynomial by evaluating the gadget on + # the wire polynomials. By construction we have that + # `gadget_poly(alpha^k)` is the `k`-th output. + gadget_poly = g.eval_poly(self.field, wire_polys) + + for j in range(g.ARITY): + proof.append(g.wires[j][0]) + proof += gadget_poly + + return proof + +7.3.4. Querying the Proof + + proof (share) + +---------|--------+ + | Query | | + | +-------|-----+ | + | | Valid | |<---- meas (share) + | | +-----V--+ |<---- joint rand + | | | Gadget | | |<- query rand + | | +--------+ | | + | +-------------+ | + +------------------+ + | + V verifier (share) + + Figure 9: Components of the query algorithm. + + The query algorithm invokes the validity circuit on the encoded + measurement and joint randomness. It evaluates the gadget + polynomials encoded by the proof (share) to produce (a share of) each + gadget output. The verifier (share) consists of (a share of) the + validity circuit's output and (a share of) each gadget test. The + gadget tests consume the query randomness. + + The goal of each gadget test is to ensure the inputs used by the + prover to generate the gadget polynomial matches the inputs we used + to evaluate it. We do this by partially reconstructing the gadget + polynomial and evaluating it at a random point: when we evaluate the + gadget polynomial at the same point, we expect to get the same + result. + + To start a gadget test, we first construct the (shares of the) wire + polynomials just as the prover did. First, we record g.wires[j][k] + as the input (share) of the j-th wire of the k-th invocation of the + gadget. Again, this is accomplished by a wrapper gadget, + QueryGadget, listed in Appendix A.4. This gadget also evaluates the + gadget polynomial for each gadget invocation in order to produce the + gadget's output. Then we compute the wire polynomials from the + recorded values. + + Next, we choose a random point t (parsed from the query randomness), + evaluate each wire polynomial at t, and evaluate the gadget + polynomial at t. The results are recorded in the verifier message + passed to the decision algorithm, where we finish the test. + + The random point t MUST NOT be one of the fixed evaluation points + used to interpolate the wire polynomials. Otherwise, the verifier + message may partially leak the encoded measurement. + + def query(self, + meas: list[F], + proof: list[F], + query_rand: list[F], + joint_rand: list[F], + num_shares: int) -> list[F]: + # Evaluate the validity circuit, recording the value of each + # input wire for each evaluation of each gadget. Use the gadget + # polynomials encoded by `proof` to compute the gadget outputs. + valid = QueryGadget.wrap(self.valid, proof) + out = valid.eval(meas, joint_rand, num_shares) + + # Reduce the output. + if self.valid.EVAL_OUTPUT_LEN > 1: + (rand, query_rand) = front( + self.valid.EVAL_OUTPUT_LEN, + query_rand, + ) + v = self.field(0) + for (r, out_elem) in zip(rand, out): + v += r * out_elem + else: + [v] = out + + # Construct the verifier message, which consists of the reduced + # circuit output and each gadget test. + verifier = [v] + for (g, t) in zip(cast(list[QueryGadget[F]], valid.GADGETS), + query_rand): + p = len(g.wires[0]) + + # Abort if `t` is one of the inputs used to compute the wire + # polynomials so that the verifier message doesn't leak the + # gadget output. It suffices to check if `t` is a root of + # unity, which implies it is a power of `alpha`. + if t ** p == self.field(1): + raise ValueError('test point is a root of unity') + + # To test the gadget, we re-compute the wire polynomials and + # check for consistency with the gadget polynomial provided + # by the prover. To start, evaluate the gadget polynomial and + # each of the wire polynomials at the random point `t`. + wire_checks = [] + wire_inp = [g.alpha ** k for k in range(p)] + for j in range(g.ARITY): + wire_poly = poly_interp(self.field, wire_inp, g.wires[j]) + wire_checks.append(poly_eval(self.field, wire_poly, t)) + + gadget_check = poly_eval(self.field, g.poly, t) + + verifier += wire_checks + verifier.append(gadget_check) + + return verifier + +7.3.5. Deciding Validity + + | verifier + V + +------------------+ + | Decide | + +------------------+ + | + V is_valid + + Figure 10: Components of the decision algorithm. + + The decision algorithm consumes the verifier message. (Each of the + Aggregators computes an additive share of the verifier message after + the previous step.) The verifier message consists of the reduced + circuit output and the gadget tests. + + To finish each gadget test, we evaluate the gadget on the wire + checks: if the encoded measurement and joint randomness used to + generate the proof are the same as the measurement (share) and joint + randomness used to verify the proof, then the output of the gadget + will be equal to the gadget check; otherwise, the output will not + equal the gadget check with high probability. + + def decide(self, verifier: list[F]) -> bool: + # Check the output of the validity circuit. + ([v], verifier) = front(1, verifier) + if v != self.field(0): + return False + + # Complete each gadget test. + for g in self.valid.GADGETS: + (wire_checks, verifier) = front(g.ARITY, verifier) + ([gadget_check], verifier) = front(1, verifier) + if g.eval(self.field, wire_checks) != gadget_check: + return False + + return True + +7.4. Variants + + This section specifies instantiations of Prio3 for various + aggregation tasks. Each variant is determined by a field + (Section 6.1), a validity circuit (Section 7.3.2),and the number of + proofs to generate and verify. All gadgets are listed in Appendix A. + Test vectors for each can be found in Appendix C. + +7.4.1. Prio3Count + + +===========+=============================+ + | Parameter | Value | + +===========+=============================+ + | field | Field64 (Table 3) | + +-----------+-----------------------------+ + | Valid | Count(field) (this section) | + +-----------+-----------------------------+ + | PROOFS | 1 | + +-----------+-----------------------------+ + + Table 9: Parameters for Prio3Count. + + Our first variant of Prio3 is for a simple counter: each measurement + is either one or zero and the aggregate result is the sum of the + measurements. Its validity circuit uses the multiplication gadget + Mul specified in Appendix A.1, which takes two inputs and multiplies + them. The circuit is specified below: + + class Count(Valid[int, int, F]): + GADGETS: list[Gadget[F]] = [Mul()] + GADGET_CALLS = [1] + MEAS_LEN = 1 + JOINT_RAND_LEN = 0 + OUTPUT_LEN = 1 + EVAL_OUTPUT_LEN = 1 + + # Class object for the field. + field: type[F] + + def __init__(self, field: type[F]): + self.field = field + + def encode(self, measurement: int) -> list[F]: + return [self.field(measurement)] + + def eval( + self, + meas: list[F], + joint_rand: list[F], + _num_shares: int) -> list[F]: + squared = self.GADGETS[0].eval(self.field, + [meas[0], meas[0]]) + return [squared - meas[0]] + + def truncate(self, meas: list[F]) -> list[F]: + return meas + + def decode(self, output: list[F], _num_measurements: int) -> int: + return output[0].int() + +7.4.2. Prio3Sum + + +===========+============================================+ + | Parameter | Value | + +===========+============================================+ + | field | Field64 (Table 3) | + +-----------+--------------------------------------------+ + | Valid | Sum(field, max_measurement) (this section) | + +-----------+--------------------------------------------+ + | PROOFS | 1 | + +-----------+--------------------------------------------+ + + Table 10: Parameters for Prio3Sum. + + The next variant of Prio3 supports summing of integers in a pre- + determined range. Each measurement is an integer in the range [0, + max_measurement], where max_measurement defines the largest valid + measurement. + + The range check is accomplished by encoding the measurement as a bit + vector, encoding the measurement plus an offset as a bit vector, then + checking that the two encoded integers are consistent. Let + + * bits = max_measurement.bit_length(), the number of bits needed to + encode the largest valid measurement + + * offset = 2^bits - 1 - max_measurement + + The first bit-encoded integer is the measurement itself. Note that + only measurements between 0 and 2^bits - 1 can be encoded this way + with as many bits. The second bit-encoded integer is the sum of the + measurement and offset. Observe that this sum can only be encoded + this way if it is between 0 and 2^bits - 1, which implies that the + measurement is between -offset and max_measurement. + + The circuit first checks that each entry of both bit vectors is a one + or a zero. It then decodes both the measurement and the offset + measurement, and subtracts the offset from the latter. It then + checks if these two values are equal. Since both the measurement and + the measurement plus offset are in the same range of [0, 2^bits), + this means that the measurement itself is between 0 and + max_measurement. + + The circuit uses the polynomial-evaluation gadget PolyEval specified + in Appendix A.2. The polynomial is p(x) = x^2 - x, which is equal to + 0 if and only if x is in [0, 2). The complete circuit is specified + below: + + class Sum(Valid[int, int, F]): + GADGETS: list[Gadget[F]] = [PolyEval([0, -1, 1])] + JOINT_RAND_LEN = 0 + OUTPUT_LEN = 1 + field: type[F] + + def __init__(self, field: type[F], max_measurement: int): + self.field = field + self.bits = max_measurement.bit_length() + self.offset = self.field(2**self.bits - 1 - max_measurement) + self.max_measurement = max_measurement + self.GADGET_CALLS = [2 * self.bits] + self.MEAS_LEN = 2 * self.bits + self.EVAL_OUTPUT_LEN = 2 * self.bits + 1 + + def encode(self, measurement: int) -> list[F]: + encoded = [] + encoded += self.field.encode_into_bit_vec( + measurement, + self.bits + ) + encoded += self.field.encode_into_bit_vec( + measurement + self.offset.int(), + self.bits + ) + return encoded + + def eval( + self, + meas: list[F], + joint_rand: list[F], + num_shares: int) -> list[F]: + shares_inv = self.field(num_shares).inv() + + out = [] + for b in meas: + out.append(self.GADGETS[0].eval(self.field, [b])) + + range_check = self.offset * shares_inv + \ + self.field.decode_from_bit_vec(meas[:self.bits]) - \ + self.field.decode_from_bit_vec(meas[self.bits:]) + out.append(range_check) + return out + + def truncate(self, meas: list[F]) -> list[F]: + return [self.field.decode_from_bit_vec(meas[:self.bits])] + + def decode(self, output: list[F], _num_measurements: int) -> int: + return output[0].int() + +7.4.3. Prio3SumVec + + +===========+=============================+ + | Parameter | Value | + +===========+=============================+ + | field | Field128 (Table 3) | + +-----------+-----------------------------+ + | Valid | SumVec(field, length, bits, | + | | chunk_lengh) (this section) | + +-----------+-----------------------------+ + | PROOFS | 1 | + +-----------+-----------------------------+ + + Table 11: Parameters for Prio3SumVec. + + This instance of Prio3 supports summing vectors of integers. It has + three parameters: length, bits, and chunk_length. Each measurement + is a vector of positive integers with length equal to the length + parameter. Each element of the measurement is an integer in the + range [0, 2^bits). It is RECOMMENDED to set chunk_length to an + integer near the square root of length * bits (see Section 7.4.3.1). + + The circuit is denoted SumVec. Each measurements is encoded as a + vector of field elements with a length of length * bits. The field + elements in the encoded vector represent all the bits of the + measurement vector's elements, consecutively, in LSB to MSB order. + + The validity circuit uses the ParallelSum gadget in Appendix A.3. + This gadget applies an arithmetic subcircuit to multiple inputs in + parallel, then returns the sum of the results. Along with the + subcircuit, the parallel-sum gadget is parameterized by an integer, + denoted count, specifying how many times to call the subcircuit. It + takes in a list of inputs and passes them through to instances of the + subcircuit in the same order. It returns the sum of the subcircuit + outputs. + + Note that only the ParallelSum gadget itself, and not its subcircuit, + participates in the FLP's wire recording during evaluation, gadget + consistency proofs, and proof validation, even though the subcircuit + is provided to ParallelSum gadget as an implementation of the Gadget + interface. + + The SumVec validity circuit checks that the encoded measurement + consists of ones and zeros. Rather than use the PolyEval gadget on + each element, as in the Sum validity circuit, it instead uses Mul + subcircuits (Appendix A.1) and "free" constant multiplication and + addition gates to simultaneously evaluate the same range check + polynomial on each element, and multiply by a constant. One of the + two Mul subcircuit inputs is equal to a measurement element + multiplied by a power of one of the elements of the joint randomness + vector, and the other is equal to the same measurement element minus + one. These Mul subcircuits are evaluated by a ParallelSum gadget, + and the results are added up both within the ParallelSum gadget and + after it. + + The complete circuit is specified below: + + class SumVec(Valid[list[int], list[int], F]): + EVAL_OUTPUT_LEN = 1 + length: int + bits: int + chunk_length: int + field: type[F] + + def __init__(self, + field: type[F], + length: int, + bits: int, + chunk_length: int): + """ + Instantiate the `SumVec` circuit for measurements with + `length` elements, each in the range `[0, 2^bits)`. + """ + self.field = field + self.length = length + self.bits = bits + self.chunk_length = chunk_length + self.GADGETS = [ParallelSum(Mul(), chunk_length)] + self.GADGET_CALLS = [ + (length * bits + chunk_length - 1) // chunk_length + ] + self.MEAS_LEN = length * bits + self.OUTPUT_LEN = length + self.JOINT_RAND_LEN = self.GADGET_CALLS[0] + + def encode(self, measurement: list[int]) -> list[F]: + encoded = [] + for val in measurement: + encoded += self.field.encode_into_bit_vec( + val, self.bits) + return encoded + + def eval( + self, + meas: list[F], + joint_rand: list[F], + num_shares: int) -> list[F]: + out = self.field(0) + shares_inv = self.field(num_shares).inv() + for i in range(self.GADGET_CALLS[0]): + r = joint_rand[i] + r_power = r + inputs: list[Optional[F]] + inputs = [None] * (2 * self.chunk_length) + for j in range(self.chunk_length): + index = i * self.chunk_length + j + if index < len(meas): + meas_elem = meas[index] + else: + meas_elem = self.field(0) + + inputs[j * 2] = r_power * meas_elem + inputs[j * 2 + 1] = meas_elem - shares_inv + + r_power *= r + + out += self.GADGETS[0].eval( + self.field, + cast(list[F], inputs), + ) + + return [out] + + def truncate(self, meas: list[F]) -> list[F]: + truncated = [] + for i in range(self.length): + truncated.append(self.field.decode_from_bit_vec( + meas[i * self.bits: (i + 1) * self.bits] + )) + return truncated + + def decode( + self, + output: list[F], + _num_measurements: int) -> list[int]: + return [x.int() for x in output] + +7.4.3.1. Selection of ParallelSum Chunk Length + + The chunk_length parameter provides a trade-off between the arity of + the ParallelSum gadget (Appendix A.3) and the number of times the + gadget is called. The proof length is asymptotically minimized when + the chunk length is near the square root of the length of the + measurement. However, the relationship between VDAF parameters and + proof length is complicated, involving two forms of rounding: + + * The circuit pads the inputs to its last ParallelSum gadget call, + up to the chunk length. + + * The proof system rounds the degree of wire polynomials, determined + by the number of times a gadget is called, up to the next power of + two. + + Therefore, the optimal choice of chunk_length for a concrete + measurement size will vary, and must be found through trial and + error. Setting chunk_length equal to the square root of the + appropriate measurement length will result in proofs up to 50% larger + than the optimal proof size. + +7.4.4. Prio3Histogram + + +===========+======================================================+ + | Parameter | Value | + +===========+======================================================+ + | field | Field128 (Table 3) | + +-----------+------------------------------------------------------+ + | Valid | Histogram(field, length, chunk_lengh) (this section) | + +-----------+------------------------------------------------------+ + | PROOFS | 1 | + +-----------+------------------------------------------------------+ + + Table 12: Parameters for Prio3Histogram. + + This variant of Prio3 allows for estimating the distribution of some + quantity by computing a simple histogram. Each measurement + increments one histogram bucket, out of a set of fixed buckets. + (Bucket indexing begins at 0.) For example, the buckets might + quantize the real numbers, and each measurement would report the + bucket that the corresponding client's real-numbered value falls + into. The aggregate result counts the number of measurements in each + bucket. + + The validity circuit is denoted Histogram. It has two parameters: + length, the number of histogram buckets; and chunk_length, which is + used by a circuit optimization described below. It is RECOMMENDED to + set chunk_length to an integer near the square root of length (see + Section 7.4.3.1). + + The measurement is encoded as a one-hot vector representing the + bucket into which the measurement falls. The circuit checks for one- + hotness in two steps, by checking that the encoded measurement + consists of ones and zeros, and by checking that the sum of all + elements in the encoded measurement is equal to one. The individual + checks constitute the output of the circuit. + + As in the SumVec validity circuit (Section 7.4.3), the first part of + the validity circuit uses the ParallelSum (Appendix A.3) gadget to + perform range checks while achieving a smaller proof size. The + ParallelSum gadget uses Mul subcircuits (Appendix A.1) to evaluate a + range check polynomial on each element, and includes an additional + constant multiplication. One of the two Mul subcircuit inputs is + equal to a measurement element multiplied by a power of an element of + the joint randomness vector, and the other is equal to the same + measurement element minus one. The results are added up both within + the ParallelSum gadget and after it. + + class Histogram(Valid[int, list[int], F]): + EVAL_OUTPUT_LEN = 2 + field: type[F] + length: int + chunk_length: int + + def __init__(self, + field: type[F], + length: int, + chunk_length: int): + """ + Instantiate an instance of the `Histogram` circuit with the + given `length` and `chunk_length`. + """ + self.field = field + self.length = length + self.chunk_length = chunk_length + self.GADGETS = [ParallelSum(Mul(), chunk_length)] + self.GADGET_CALLS = [ + (length + chunk_length - 1) // chunk_length] + self.MEAS_LEN = self.length + self.OUTPUT_LEN = self.length + self.JOINT_RAND_LEN = self.GADGET_CALLS[0] + + def encode(self, measurement: int) -> list[F]: + encoded = [self.field(0)] * self.length + encoded[measurement] = self.field(1) + return encoded + + def eval( + self, + meas: list[F], + joint_rand: list[F], + num_shares: int) -> list[F]: + # Check that each bucket is one or zero. + range_check = self.field(0) + shares_inv = self.field(num_shares).inv() + for i in range(self.GADGET_CALLS[0]): + r = joint_rand[i] + r_power = r + inputs: list[Optional[F]] + inputs = [None] * (2 * self.chunk_length) + for j in range(self.chunk_length): + index = i * self.chunk_length + j + if index < len(meas): + meas_elem = meas[index] + else: + meas_elem = self.field(0) + + inputs[j * 2] = r_power * meas_elem + inputs[j * 2 + 1] = meas_elem - shares_inv + + r_power *= r + + range_check += self.GADGETS[0].eval( + self.field, + cast(list[F], inputs), + ) + + # Check that the buckets sum to 1. + sum_check = -shares_inv + for b in meas: + sum_check += b + + return [range_check, sum_check] + + def truncate(self, meas: list[F]) -> list[F]: + return meas + + def decode( + self, + output: list[F], + _num_measurements: int) -> list[int]: + return [bucket_count.int() + for bucket_count in output] + +7.4.5. Prio3MultihotCountVec + + +===========+=========================================+ + | Parameter | Value | + +===========+=========================================+ + | field | Field128 (Table 3) | + +-----------+-----------------------------------------+ + | Valid | MultihotCountVec(field, length, | + | | max_weight, chunk_lengh) (this section) | + +-----------+-----------------------------------------+ + | PROOFS | 1 | + +-----------+-----------------------------------------+ + + Table 13: Parameters for Prio3MultihotCountVec. + + For this instance of Prio3, each measurement is a vector of ones and + zeros, where the number of ones is bounded. This provides a + functionality similar to Prio3Histogram except that more than one + entry (or none at all) may be non-zero. This allows + Prio3MultihotCountVec to be composed with a randomized response + mechanism, like [EPK14], for providing differential privacy. (For + example, each Client would set each entry to one with some small + probability.) + + The validity circuit is denoted MultihotCountVec and has three + parameters: length, the number of entries in the count vector; + max_weight, the maximum number of non-zero entries (i.e., the weight + must be at most max_weight); and chunk_length, used the same way as + in Section 7.4.3 and Section 7.4.4. + + Validation works as follows. Let + + * bits_for_weight = max_weight.bit_length() + + * offset = 2^bits_for_weight - 1 - max_weight + + The Client reports the weight of the count vector by adding offset to + it and bit-encoding the result. Observe that only a weight of at + most max_weight can be encoded with bits_for_weight bits. + + The verifier checks that each entry of the encoded measurement is a + bit (i.e., either one or zero). It then decodes the reported weight + and subtracts it from offset + sum(count_vec), where count_vec is the + count vector. The result is zero if and only if the reported weight + is equal to the true weight. The complete circuit is defined below. + + class MultihotCountVec(Valid[list[int], list[int], F]): + EVAL_OUTPUT_LEN = 2 + field: type[F] + + def __init__(self, + field: type[F], + length: int, + max_weight: int, + chunk_length: int): + """ + Instantiate an instance of the this circuit with the given + `length`, `max_weight`, and `chunk_length`. + + Pre-conditions: + + - `length > 0` + - `0 < max_weight` and `max_weight <= length` + - `chunk_length > 0` + """ + self.field = field + + # Compute the number of bits to represent `max_weight`. + self.bits_for_weight = max_weight.bit_length() + self.offset = self.field( + 2**self.bits_for_weight - 1 - max_weight) + + # Make sure `offset + length` doesn't overflow the field + # modulus. Otherwise we may not correctly compute the sum + # measurement vector entries during circuit evaluation. + if self.field.MODULUS - self.offset.int() <= length: + raise ValueError('length and max_weight are too large ' + 'for the current field size') + + self.length = length + self.max_weight = max_weight + self.chunk_length = chunk_length + self.GADGETS: list[Gadget[F]] = [ + ParallelSum(Mul(), chunk_length), + ] + self.GADGET_CALLS = [ + (length + self.bits_for_weight + chunk_length - 1) + // chunk_length + ] + self.MEAS_LEN = self.length + self.bits_for_weight + self.OUTPUT_LEN = self.length + self.JOINT_RAND_LEN = self.GADGET_CALLS[0] + + def encode(self, measurement: list[int]) -> list[F]: + if len(measurement) != self.length: + raise ValueError('invalid Client measurement length') + + # The first part is the vector of counters. + count_vec = list(map(self.field, measurement)) + + # The second part is the reported weight. + weight_reported = sum(count_vec, self.field(0)) + + encoded = [] + encoded += count_vec + encoded += self.field.encode_into_bit_vec( + (self.offset + weight_reported).int(), + self.bits_for_weight) + return encoded + + def eval( + self, + meas: list[F], + joint_rand: list[F], + num_shares: int) -> list[F]: + # Check that each entry in the input vector is one or zero. + range_check = self.field(0) + shares_inv = self.field(num_shares).inv() + for i in range(self.GADGET_CALLS[0]): + r = joint_rand[i] + r_power = r + inputs: list[Optional[F]] + inputs = [None] * (2 * self.chunk_length) + for j in range(self.chunk_length): + index = i * self.chunk_length + j + if index < len(meas): + meas_elem = meas[index] + else: + meas_elem = self.field(0) + + inputs[j * 2] = r_power * meas_elem + inputs[j * 2 + 1] = meas_elem - shares_inv + + r_power *= r + + range_check += self.GADGETS[0].eval( + self.field, + cast(list[F], inputs), + ) + + # Check that the weight `offset` plus the sum of the counters + # is equal to the value claimed by the Client. + count_vec = meas[:self.length] + weight = sum(count_vec, self.field(0)) + weight_reported = \ + self.field.decode_from_bit_vec(meas[self.length:]) + weight_check = self.offset*shares_inv + weight - \ + weight_reported + + return [range_check, weight_check] + + def truncate(self, meas: list[F]) -> list[F]: + return meas[:self.length] + + def decode( + self, + output: list[F], + _num_measurements: int) -> list[int]: + return [bucket_count.int() for + bucket_count in output] + +8. Poplar1 + + This section specifies Poplar1, a VDAF for the following task. Each + Client holds a bit-string of length BITS and the Collector chooses a + sequence of L-bit strings, where L <= BITS. We will refer to the + latter as the of "candidate prefixes". The goal is to count how many + of the Clients' inputs begin with each candidate prefix. + + This functionality is the core component of the heavy hitters + protocol of [BBCGGI21]. The goal of this protocol is to compute the + subset of inputs held by at least T Clients for some threshold T. It + invokes Poplar1 as follows: + + 1. Each Client shards its string into secret shares and uploads one + share to each of the Aggregators. + + 2. The Collector picks an initial set of candidate prefixes, say 0 + and 1, and sends them to the Aggregators. + + 3. The Aggregators run VDAF preparation and aggregation on each of + the reports and send their aggregate shares to the Collector. + + 4. The Collector unshards the aggregate result, which consists of + the hit count for each candidate prefix. For each prefix p with + hit count at least T, the Collector adds p || 0 and p || 1 to the + next generation of candidate prefixes and repeats Step 2. + + Poplar1 is constructed from an "Incremental Distributed Point + Function (IDPF)", a primitive described by [BBCGGI21] that + generalizes the notion of a Distributed Point Function (DPF) [GI14]. + Briefly, a DPF is used to distribute the computation of a "point + function", a function that evaluates to zero on every input except at + a programmable "point". The computation is distributed in such a way + that no one party knows either the point or what it evaluates to. + + An IDPF generalizes this "point" to a path on a full binary tree from + the root to one of the leaves. It is evaluated on an "index" + representing a unique node of the tree. If the node is on the + programmed path, then the function evaluates to a non-zero value; + otherwise it evaluates to zero. This structure allows an IDPF to + provide the functionality required for the above protocol: to compute + the hit count for an index, just evaluate each set of IDPF shares at + that index and add up the results. + + Consider the sub-tree constructed from a set of input strings and a + target threshold T by including all indices with hit count at least + T. We shall refer to this structure as the "prefix tree" for the + batch of measurements and target threshold. To compute the T-heavy- + hitters for the batch, the Aggregators and Collector first compute + the prefix tree, then extract the heavy hitters from the leaves of + this tree. Note that the prefix tree leaks more information about + the set than the heavy hitters themselves; see Section 9.4 for more + discussion. + + Poplar1 composes an IDPF with the arithmetic sketch of [BBCGGI21], + Section 4.2. (The paper calls this a "secure sketch", but the + underlying technique was later generalized in [BBCGGI23], where it is + called "arithmetic sketching".) The sketch ensures that evaluating a + set of input shares on a unique set of candidate prefixes results in + shares of a "one-hot" vector, i.e., a vector that is zero everywhere + except for in at most one position. Moreover, the value at that + position should be one. + + The remainder of this section is structured as follows. The syntax + of IDPFs is defined in Section 8.1; a specification of the IDPF of + [BBCGGI21] is given in Section 8.3. The Poplar1 VDAF is defined in + Section 8.2 in terms of a generic IDPF. Test vectors for Poplar1 can + be found in Appendix C. + +8.1. Incremental Distributed Point Functions (IDPFs) + + An IDPF is defined over a domain of size 2^BITS, where BITS is a + constant. Indices into the IDPF tree are bit strings. (In Poplar1, + each Client's bit string is an index; see Section 8.1.1 for details.) + The Client specifies an index alpha and a vector of values beta, one + for each "level" L in [0, BITS). The key generation algorithm + generates one IDPF "key" for each Aggregator. When evaluated at + level L and index prefix, each IDPF key returns an additive share of + beta[L] if prefix is the L-bit prefix of alpha and shares of zero + otherwise. + + Each of the programmed points beta is a vector of elements of some + finite field. We distinguish two types of fields: one for inner + nodes (denoted FieldInner), and one for leaf nodes (FieldLeaf). (Our + instantiation of Poplar1 (Section 8.2) will use a much larger field + for leaf nodes than for inner nodes. This is to ensure the IDPF is + "extractable" as defined in [BBCGGI21], Definition 1. See + Section 9.5 for details.) + + A concrete IDPF defines the types and parameters enumerated in + Table 14. In the remainder we write Output as shorthand for the type + list[list[FieldInner]] | list[list[FieldLeaf]]. (This type denotes + either a vector of inner node field elements or leaf node field + elements.) The scheme is comprised of the following algorithms: + + * idpf.gen(alpha: tuple[bool, ...], beta_inner: + list[list[FieldInner]], beta_leaf: list[FieldLeaf], ctx: bytes, + nonce: bytes, rand: bytes) -> tuple[PublicShare, list[bytes]] is + the IDPF-key generation algorithm. Its inputs are the index + alpha, the values beta, the application context, and the report + nonce. + + The output is a public part (of type PublicShare) that is sent to + each Aggregator and a vector of private IDPF keys, one for each + Aggregator. The nonce is used to derive the fixed AES key for + XofFixedKeyAes128 (Section 6.2.2). Looking ahead, this key is + used for extending the seed into the seeds for the child nodes at + each level of the tree; see Section 8.3. + + Pre-conditions: + + - alpha MUST have length BITS. + + - beta_inner MUST have length BITS - 1. + + - beta_inner[level] MUST have length VALUE_LEN for each level in + [0, BITS - 1). + + - beta_leaf MUST have length VALUE_LEN. + + - rand MUST be generated by a CSPRNG and have length RAND_SIZE. + + - nonce MUST be generated by a CSPRNG (see Section 9.2 for + details) and have length idpf.NONCE_SIZE. + + Post-conditions: + + - The number IDPF keys MUST be idpf.SHARES. + + * idpf.eval(agg_id: int, public_share: PublicShare, key: bytes, + level: int, prefixes: Sequence[tuple[bool, ...]], ctx: bytes, + nonce: bytes) -> Output is the IDPF-key evaluation algorithm run + by each Aggregator. Its inputs are the Aggregator's unique + identifier, the public share distributed to all of the + Aggregators, the Aggregator's IDPF key, the "level" at which to + evaluate the IDPF, the sequence of candidate prefixes, the + application context, and the report nonce. It returns the share + of the value corresponding to each candidate prefix. + + Pre-conditions: + + - agg_id MUST be in [0, idpf.SHARES) and match the index of key + in the sequence of IDPF keys output by the Client. + + - level MUST be in [0, BITS). + + - Each prefix MUST be distinct and have length level + 1. + + - The length of the nonce MUST be idpf.NONCE_SIZE. + + Post-conditions: + + - The length of the output MUST be len(prefixes) + + - The length of each element of the output MUST be idpf.VALUE_LEN + + - If level == idpf.BITS - 1, then the output field MUST be + FieldLeaf and FieldInner otherwise + + In addition, the IDPF provides the following method: + + def current_field( + self, + level: int) -> type[FieldInner] | type[FieldLeaf]: + if level < self.BITS - 1: + return self.field_inner + return self.field_leaf + + Finally, an implementation note. The interface for IDPFs specified + here is stateless, in the sense that there is no state carried + between IDPF evaluations. This is to align the IDPF syntax with the + VDAF abstraction boundary, which does not include shared state across + VDAF evaluations. In practice, of course, it will often be + beneficial to expose a stateful API for IDPFs and carry the state + across evaluations. See Section 8.3 for details. + + +=============+===================================================+ + | Parameter | Description | + +=============+===================================================+ + | SHARES: int | Number of IDPF keys output by IDPF-key generator. | + +-------------+---------------------------------------------------+ + | BITS: int | Length in bits of each input string. | + +-------------+---------------------------------------------------+ + | VALUE_LEN: | Number of field elements of each output value. | + | int | | + +-------------+---------------------------------------------------+ + | RAND_SIZE: | Size of the random string consumed by the IDPF- | + | int | key generator. | + +-------------+---------------------------------------------------+ + | NONCE_SIZE: | Size of the random nonce generated by the Client. | + | int | | + +-------------+---------------------------------------------------+ + | KEY_SIZE: | Size in bytes of each IDPF key. | + | int | | + +-------------+---------------------------------------------------+ + | FieldInner | Implementation of Field (Section 6.1) used for | + | | values of inner nodes. | + +-------------+---------------------------------------------------+ + | FieldLeaf | Implementation of Field used for values of leaf | + | | nodes. | + +-------------+---------------------------------------------------+ + | PublicShare | Type of public share for this IDPF. | + +-------------+---------------------------------------------------+ + | Output | Alias of list[list[FieldInner]] | | + | | list[list[FieldLeaf]]. | + +-------------+---------------------------------------------------+ + | FieldVec | Alias of list[FieldInner] | list[FieldLeaf]. | + +-------------+---------------------------------------------------+ + + Table 14: Constants and types defined by a concrete IDPF. + +8.1.1. Encoding Inputs as Indices + + How data are represented as IDPF indices is up to the application. + When the inputs are fixed-length byte strings, the most natural + choice of representation is as a bit string formed from all the bits + of the byte string, first ordered by byte position, then ordered from + most significant bit to least significant bit within each byte. This + ensures that, when a byte string is a prefix of another, so too is + its corresponding index. (Index prefixes are defined in + Section 8.1.) For example, + + Byte string: 01 02 + Bit string: 00000001 00000010 + + is a prefix of + + Byte string: 01 02 03 + Bit string: 00000001 00000010 00000011 + + Additionally, lexicographic ordering is preserved by this mapping + from a byte string to a bit string. + + When the inputs are variable length, it is necessary to pad each + input to some fixed length. Further, the padding scheme must be non- + ambiguous. For example, each input could be padded with b"\x01" + followed by as many b"\x00" bytes as needed. + +8.2. Specification + + This section specifies Poplar1, an implementation of the Vdaf + interface (Section 5). It is defined in terms of the Idpf + implementation of Section 8.3 with SHARES == 2 and VALUE_LEN == 2 and + XofTurboShake128 as specified in Section 6.2.1. The associated + constants and types required by the Vdaf interface are defined in + Table 15. The methods required for sharding, preparation, + aggregation, and unsharding are described in the remaining + subsections. These methods make use of constants defined in + Table 16. + + +=================+========================================+ + | Parameter | Value | + +=================+========================================+ + | idpf | As specified in Section 8.3. | + +-----------------+----------------------------------------+ + | xof | XofTurboShake128 (Section 6.2.1) | + +-----------------+----------------------------------------+ + | VERIFY_KEY_SIZE | xof.SEED_SIZE | + +-----------------+----------------------------------------+ + | RAND_SIZE | xof.SEED_SIZE * 3 + idpf.RAND_SIZE | + +-----------------+----------------------------------------+ + | NONCE_SIZE | 16 | + +-----------------+----------------------------------------+ + | ROUNDS | 2 | + +-----------------+----------------------------------------+ + | SHARES | 2 | + +-----------------+----------------------------------------+ + | Measurement | tuple[bool, ...] | + +-----------------+----------------------------------------+ + | AggParam | tuple[int, Sequence[tuple[bool, ...]]] | + +-----------------+----------------------------------------+ + | PublicShare | As defined by idpf. | + +-----------------+----------------------------------------+ + | InputShare | tuple[bytes, bytes, list[FieldInner], | + | | list[FieldLeaf]] | + +-----------------+----------------------------------------+ + | OutShare | FieldVec | + +-----------------+----------------------------------------+ + | AggShare | FieldVec | + +-----------------+----------------------------------------+ + | AggResult | list[int] | + +-----------------+----------------------------------------+ + | PrepState | tuple[bytes, int, FieldVec] | + +-----------------+----------------------------------------+ + | PrepShare | FieldVec | + +-----------------+----------------------------------------+ + | PrepMessage | Optional[FieldVec] | + +-----------------+----------------------------------------+ + + Table 15: VDAF parameters for Poplar1. + + +========================+=======+ + | Variable | Value | + +========================+=======+ + | USAGE_SHARD_RAND: int | 1 | + +------------------------+-------+ + | USAGE_CORR_INNER: int | 2 | + +------------------------+-------+ + | USAGE_CORR_LEAF: int | 3 | + +------------------------+-------+ + | USAGE_VERIFY_RAND: int | 4 | + +------------------------+-------+ + + Table 16: Constants used by + Poplar1. + +8.2.1. Sharding + + The Client's measurement is an IDPF index, denoted alpha, whose type + is a sequence of bits tuple[bool, ...] (See Section 8.1.1 for + guidelines on index encoding.) + + The programmed IDPF values are pairs of field elements (1, k) where + each k is chosen at random. This random value is used as part of the + arithmetic sketching protocol of [BBCGGI21], Appendix C.4. After + evaluating their IDPF key shares on a given sequence of candidate + prefixes, the sketching protocol is used by the Aggregators to verify + that they hold shares of a one-hot vector at a given level of the + IDPF tree. + + In addition to programming k into the IDPF output, for each level of + the tree, the Client generates random elements a, b, and c and + computes + + A = -2*a + k + B = a^2 + b - k*a + c + + and sends additive shares of a, b, c, A and B to each of the + Aggregators. These help the Aggregators evaluate the sketch during + preparation. + + Putting everything together, the sharding algorithm is defined as + follows. + + def shard( + self, + ctx: bytes, + measurement: tuple[bool, ...], + nonce: bytes, + rand: bytes, + ) -> tuple[Poplar1PublicShare, list[Poplar1InputShare]]: + if len(nonce) != self.NONCE_SIZE: + raise ValueError("incorrect nonce size") + if len(rand) != self.RAND_SIZE: + raise ValueError("incorrect size of random bytes argument") + + l = self.xof.SEED_SIZE + + # Split the random input into the random input for IDPF key + # generation, correlated randomness, and sharding. + if len(rand) != self.RAND_SIZE: + raise ValueError('incorrect rand size') + idpf_rand, rand = front(self.idpf.RAND_SIZE, rand) + seeds = [rand[i:i + l] for i in range(0, 3 * l, l)] + corr_seed, seeds = front(2, seeds) + (shard_seed,), seeds = front(1, seeds) + + xof = self.xof( + shard_seed, + self.domain_separation_tag(USAGE_SHARD_RAND, ctx), + nonce, + ) + + # Construct the IDPF values for each level of the IDPF tree. + # Each "data" value is 1; in addition, the Client generates + # a random "authenticator" value used by the Aggregators to + # evaluate the sketch during preparation. This sketch is used + # to verify the one-hotness of their output shares. + beta_inner = [ + [self.idpf.field_inner(1), k] + for k in xof.next_vec(self.idpf.field_inner, + self.idpf.BITS - 1) + ] + beta_leaf = [self.idpf.field_leaf(1)] + \ + xof.next_vec(self.idpf.field_leaf, 1) + + # Generate the IDPF keys. + (public_share, keys) = self.idpf.gen( + measurement, + beta_inner, + beta_leaf, + ctx, + nonce, + idpf_rand, + ) + + # Generate correlated randomness used by the Aggregators to + # evaluate the sketch over their output shares. Seeds are used + # to encode shares of the `(a, b, c)` triples. (See [BBCGGI21, + # Appendix C.4].) + corr_offsets: list[Field] = vec_add( + self.xof.expand_into_vec( + self.idpf.field_inner, + corr_seed[0], + self.domain_separation_tag(USAGE_CORR_INNER, ctx), + byte(0) + nonce, + 3 * (self.idpf.BITS - 1), + ), + self.xof.expand_into_vec( + self.idpf.field_inner, + corr_seed[1], + self.domain_separation_tag(USAGE_CORR_INNER, ctx), + byte(1) + nonce, + 3 * (self.idpf.BITS - 1), + ), + ) + corr_offsets += vec_add( + self.xof.expand_into_vec( + self.idpf.field_leaf, + corr_seed[0], + self.domain_separation_tag(USAGE_CORR_LEAF, ctx), + byte(0) + nonce, + 3, + ), + self.xof.expand_into_vec( + self.idpf.field_leaf, + corr_seed[1], + self.domain_separation_tag(USAGE_CORR_LEAF, ctx), + byte(1) + nonce, + 3, + ), + ) + + # For each level of the IDPF tree, shares of the `(A, B)` + # pairs are computed from the corresponding `(a, b, c)` + # triple and authenticator value `k`. + corr_inner: list[list[Field64]] = [[], []] + for level in range(self.idpf.BITS): + field = cast(type[Field], self.idpf.current_field(level)) + k = beta_inner[level][1] if level < self.idpf.BITS - 1 \ + else beta_leaf[1] + (a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:] + A = -field(2) * a + k + B = a ** 2 + b - a * k + c + corr1 = xof.next_vec(field, 2) + corr0 = vec_sub([A, B], corr1) + if level < self.idpf.BITS - 1: + corr_inner[0] += cast(list[Field64], corr0) + corr_inner[1] += cast(list[Field64], corr1) + else: + corr_leaf = [ + cast(list[Field255], corr0), + cast(list[Field255], corr1), + ] + + # Each input share consists of the Aggregator's IDPF key + # and a share of the correlated randomness. + input_shares = list(zip(keys, corr_seed, corr_inner, corr_leaf)) + return (public_share, input_shares) + +8.2.2. Preparation + + The aggregation parameter encodes a sequence of candidate prefixes. + When an Aggregator receives an input share from the Client, it begins + by evaluating its IDPF share on each candidate prefix, recovering a + data_share and auth_share for each. The Aggregators use these and + the correlation shares provided by the Client to verify that the + sequence of data_share values are additive shares of a one-hot + vector. + + Aggregators MUST ensure the candidate prefixes are all unique and + appear in lexicographic order. (This is enforced in the definition + of prep_init() below.) Uniqueness is necessary to ensure the refined + measurement (i.e., the sum of the output shares) is in fact a one-hot + vector. Otherwise, sketch verification might fail, causing the + Aggregators to erroneously reject a report that is actually valid. + Note that enforcing the order is not strictly necessary, but this + does allow uniqueness to be determined more efficiently. + + def prep_init( + self, + verify_key: bytes, + ctx: bytes, + agg_id: int, + agg_param: Poplar1AggParam, + nonce: bytes, + public_share: Poplar1PublicShare, + input_share: Poplar1InputShare) -> tuple[ + Poplar1PrepState, + FieldVec]: + (level, prefixes) = agg_param + (key, corr_seed, corr_inner, corr_leaf) = input_share + field = self.idpf.current_field(level) + + # Ensure that candidate prefixes are all unique and appear in + # lexicographic order. + for i in range(1, len(prefixes)): + if prefixes[i - 1] >= prefixes[i]: + raise ValueError('out of order prefix') + + # Evaluate the IDPF key at the given set of prefixes. + value = self.idpf.eval( + agg_id, public_share, key, level, prefixes, ctx, nonce) + + # Get shares of the correlated randomness for evaluating the + # Aggregator's share of the sketch. + if level < self.idpf.BITS - 1: + corr_xof = self.xof( + corr_seed, + self.domain_separation_tag(USAGE_CORR_INNER, ctx), + byte(agg_id) + nonce, + ) + # Fast-forward the XOF state to the current level. + corr_xof.next_vec(field, 3 * level) + else: + corr_xof = self.xof( + corr_seed, + self.domain_separation_tag(USAGE_CORR_LEAF, ctx), + byte(agg_id) + nonce, + ) + (a_share, b_share, c_share) = corr_xof.next_vec(field, 3) + if level < self.idpf.BITS - 1: + (A_share, B_share) = cast( + list[Field], + corr_inner[2 * level:2 * (level + 1)], + ) + else: + (A_share, B_share) = cast(list[Field], corr_leaf) + + # Evaluate the Aggregator's share of the sketch. These are + # called the "masked input values" [BBCGGI21, Appendix C.4]. + verify_rand_xof = self.xof( + verify_key, + self.domain_separation_tag(USAGE_VERIFY_RAND, ctx), + nonce + to_be_bytes(level, 2), + ) + verify_rand = cast( + list[Field], + verify_rand_xof.next_vec(field, len(prefixes)), + ) + sketch_share = [a_share, b_share, c_share] + out_share = [] + for (i, r) in enumerate(verify_rand): + data_share = cast(Field, value[i][0]) + auth_share = cast(Field, value[i][1]) + sketch_share[0] += data_share * r + sketch_share[1] += data_share * r ** 2 + sketch_share[2] += auth_share * r + out_share.append(data_share) + + prep_mem = [A_share, B_share, field(agg_id)] + out_share + return ( + ( + b'evaluate sketch', + level, + cast(FieldVec, prep_mem), + ), + cast(FieldVec, sketch_share), + ) + + def prep_next( + self, + _ctx: bytes, + prep_state: Poplar1PrepState, + prep_msg: Optional[FieldVec] + ) -> tuple[Poplar1PrepState, FieldVec] | FieldVec: + prev_sketch = cast(list[Field], prep_msg) + (step, level, prep_mem) = prep_state + + if step == b'evaluate sketch': + if prev_sketch is None: + raise ValueError('expected value, got none') + elif len(prev_sketch) != 3: + raise ValueError('incorrect sketch length') + A_share = cast(Field, prep_mem[0]) + B_share = cast(Field, prep_mem[1]) + agg_id = cast(Field, prep_mem[2]) + prep_mem = prep_mem[3:] + sketch_share = [ + agg_id * (prev_sketch[0] ** 2 + - prev_sketch[1] + - prev_sketch[2]) + + A_share * prev_sketch[0] + + B_share + ] + return cast( + tuple[Poplar1PrepState, FieldVec], + ( + ( + b'reveal sketch', + level, + prep_mem, + ), + sketch_share, + ) + ) + + elif step == b'reveal sketch': + if prev_sketch is None: + return prep_mem # Output shares + else: + raise ValueError('invalid prep message') + + raise ValueError('invalid prep state') + + def prep_shares_to_prep( + self, + _ctx: bytes, + agg_param: Poplar1AggParam, + prep_shares: list[FieldVec]) -> Optional[FieldVec]: + if len(prep_shares) != 2: + raise ValueError('incorrect number of prep shares') + (level, _) = agg_param + field = self.idpf.current_field(level) + sketch = vec_add( + cast(list[Field], prep_shares[0]), + cast(list[Field], prep_shares[1]), + ) + if len(sketch) == 3: + return cast(FieldVec, sketch) + elif len(sketch) == 1: + if sketch == field.zeros(1): + # In order to reduce communication overhead, let `None` + # denote a successful sketch verification. + return None + else: + raise ValueError('sketch verification failed') + else: + raise ValueError('incorrect sketch length') + +8.2.3. Validity of Aggregation Parameters + + Aggregation parameters are valid for a given input share if no + aggregation parameter with the same level has been used with the same + input share before. The whole preparation phase MUST NOT be run more + than once for a given combination of input share and level. This + function checks that levels are increasing between calls, and also + enforces that the prefixes at each level are suffixes of the previous + level's prefixes. + + def is_valid( + self, + agg_param: Poplar1AggParam, + previous_agg_params: list[Poplar1AggParam]) -> bool: + """ + Checks that levels are increasing between calls, and also + enforces that the prefixes at each level are suffixes of the + previous level's prefixes. + """ + if len(previous_agg_params) < 1: + return True + + (level, prefixes) = agg_param + (last_level, last_prefixes) = previous_agg_params[-1] + last_prefixes_set = set(last_prefixes) + + # Check that level increased. + if level <= last_level: + return False + + # Check that prefixes are suffixes of the last level's prefixes. + for prefix in prefixes: + last_prefix = get_ancestor(prefix, last_level) + if last_prefix not in last_prefixes_set: + # Current prefix not a suffix of last level's prefixes. + return False + return True + + def get_ancestor( + index: tuple[bool, ...], + level: int) -> tuple[bool, ...]: + """ + Helper function to determine the prefix of `index` at + `level`. + """ + return index[:level + 1] + +8.2.4. Aggregation + + Aggregation involves simply adding up the output shares. + + def agg_init(self, agg_param: Poplar1AggParam) -> FieldVec: + (level, prefixes) = agg_param + field = self.idpf.current_field(level) + return field.zeros(len(prefixes)) + + def agg_update(self, + agg_param: Poplar1AggParam, + agg_share: FieldVec, + out_share: FieldVec) -> FieldVec: + a = cast(list[Field], agg_share) + o = cast(list[Field], out_share) + return cast(FieldVec, vec_add(a, o)) + + def merge(self, + agg_param: Poplar1AggParam, + agg_shares: list[FieldVec]) -> FieldVec: + (level, prefixes) = agg_param + field = self.idpf.current_field(level) + agg = cast(list[Field], field.zeros(len(prefixes))) + for agg_share in agg_shares: + agg = vec_add(agg, cast(list[Field], agg_share)) + return cast(FieldVec, agg) + +8.2.5. Unsharding + + Finally, the Collector unshards the aggregate result by adding up the + aggregate shares. + + def unshard( + self, + agg_param: Poplar1AggParam, + agg_shares: list[FieldVec], + _num_measurements: int) -> list[int]: + agg = self.merge(agg_param, agg_shares) + return [x.int() for x in agg] + +8.2.6. Message Serialization + + This section defines serialization formats for messages exchanged + over the network while executing Poplar1. Messages are defined in + the presentation language of TLS as defined in Section 3 of + [RFC8446]. + + Let poplar1 be an instance of Poplar1. In the remainder we use Fi as + an alias for poplar1.idpf.field_inner.ENCODED_SIZE, Fl as an alias + for poplar1.idpf.field_leaf.ENCODED_SIZE, and B as an alias for + poplar1.idpf.BITS. + + Elements of the inner field are encoded in little-endian byte order + (as defined in Section 6.1) and are represented as follows: + + opaque Poplar1FieldInner[Fi]; + + Likewise, elements of the leaf field are encoded in little-endian + byte order (as defined in Section 6.1) and are represented as + follows: + + opaque Poplar1FieldLeaf[Fl]; + +8.2.6.1. Public Share + + The public share of the IDPF scheme in Section 8.3 consists of a + sequence of "correction words". A correction word has three + components: + + 1. the XOF seed of type bytes; + + 2. the control bits of type tuple[bool, bool]; and + + 3. the payload of type list[Field64] for the first BITS-1 words and + list[Field255] for the last word. + + The encoding is a straightforward structure of arrays, except that + the control bits are packed as tightly as possible. The encoded + public share is structured as follows: + + struct { + opaque packed_control_bits[packed_len]; + opaque seed[poplar1.idpf.KEY_SIZE*B]; + Poplar1FieldInner payload_inner[Fi*poplar1.idpf.VALUE_LEN*(B-1)]; + Poplar1FieldLeaf payload_leaf[Fl*poplar1.idpf.VALUE_LEN]; + } Poplar1PublicShare; + + Here packed_len = (2*B + 7) // 8 is the length of the packed control + bits. Field packed_control_bits is encoded with the following + function: + + packed_control_buf = [int(0)] * packed_len + for i, bit in enumerate(control_bits): + packed_control_buf[i // 8] |= bit.int() << (i % 8) + packed_control_bits = bytes(packed_control_buf) + + It encodes each group of eight bits into a byte, in LSB to MSB order, + padding the most significant bits of the last byte with zeros as + necessary, and returns the byte array. Decoding performs the reverse + operation: it takes in a byte array and a number of bits, and returns + a list of bits, extracting eight bits from each byte in turn, in LSB + to MSB order, and stopping after the requested number of bits. If + the byte array has an incorrect length, or if unused bits in the last + bytes are not zero, it throws an error: + + control_bits = [] + for i in range(length): + control_bits.append(bool( + (packed_control_bits[i // 8] >> (i % 8)) & 1 + )) + leftover_bits = packed_control_bits[-1] >> ( + (length + 7) % 8 + 1 + ) + if (length + 7) // 8 != len(packed_control_bits) or \ + leftover_bits != 0: + raise ValueError('trailing bits') + +8.2.6.2. Input Share + + Each input share is structured as follows: + + struct { + opaque idpf_key[poplar1.idpf.KEY_SIZE]; + opaque corr_seed[poplar1.xof.SEED_SIZE]; + Poplar1FieldInner corr_inner[Fi * 2 * (B- 1)]; + Poplar1FieldLeaf corr_leaf[Fl * 2]; + } Poplar1InputShare; + +8.2.6.3. Prep Share + + Encoding of the prep share depends on the round of sketching: if the + first round, then each sketch share has three field elements; if the + second round, then each sketch share has one field element. The + field that is used depends on the level of the IDPF tree specified by + the aggregation parameter, either the inner field or the leaf field. + + For the first round and inner field: + + struct { + Poplar1FieldInner sketch_share[Fi * 3]; + } Poplar1PrepShareRoundOneInner; + + For the first round and leaf field: + + struct { + Poplar1FieldLeaf sketch_share[Fl * 3]; + } Poplar1PrepShareRoundOneLeaf; + + For the second round and inner field: + + struct { + Poplar1FieldInner sketch_share; + } Poplar1PrepShareRoundTwoInner; + + For the second round and leaf field: + + struct { + Poplar1FieldLeaf sketch_share; + } Poplar1PrepShareRoundTwoLeaf; + +8.2.6.4. Prep Message + + Likewise, the structure of the prep message for Poplar1 depends on + the sketching round and field. For the first round and inner field: + + struct { + Poplar1FieldInner[Fi * 3]; + } Poplar1PrepMessageRoundOneInner; + + For the first round and leaf field: + + struct { + Poplar1FieldLeaf sketch[Fl * 3]; + } Poplar1PrepMessageRoundOneLeaf; + + Note that these messages have the same structures as the prep shares + for the first round. + + The second-round prep message is the empty string. This is because + the sketch shares are expected to sum to a particular value if the + output shares are valid; we represent a successful preparation with + the empty string and otherwise return an error. + +8.2.6.5. Aggregate Share + + The encoding of the aggregate share depends on whether the inner or + leaf field is used, and the number of candidate prefixes. Both of + these are determined by the aggregation parameter. + + Let prefix_count denote the number of candidate prefixes. For the + inner field: + + struct { + Poplar1FieldInner agg_share[Fi * prefix_count]; + } Poplar1AggShareInner; + + For the leaf field: + + struct { + Poplar1FieldLeaf agg_share[Fl * prefix_count]; + } Poplar1AggShareLeaf; + +8.2.6.6. Aggregation Parameter + + The aggregation parameter is encoded as follows: + + struct { + uint16_t level; + uint32_t num_prefixes; + opaque encoded_prefixes[prefixes_len]; + } Poplar1AggParam; + + The fields in this struct are: level, the level of the IDPF tree of + each prefixes; num_prefixes, the number of prefixes to evaluate; and + encoded_prefixes, the sequence of prefixes encoded into a byte string + of length prefixes_len. The prefixes are encoded with the following + procedure: + + prefixes_len = ((level + 1) + 7) // 8 * len(prefixes) + encoded_prefixes = bytearray() + for prefix in prefixes: + for chunk in itertools.batched(prefix, 8): + byte_out = 0 + for (bit_position, bit) in enumerate(chunk): + byte_out |= bit << (7 - bit_position) + encoded_prefixes.append(byte_out) + + Decoding involves the following procedure: + + prefixes = [] + bytes_per_prefix = ((level + 1) + 7) // 8 + for chunk in itertools.batched(encoded_prefixes, bytes_per_prefix): + prefix = [] + for i in range(level + 1): + byte_index = i // 8 + bit_offset = 7 - (i % 8) + bit = (chunk[byte_index] >> bit_offset) & 1 != 0 + prefix.append(bit) + prefixes.append(tuple(prefix)) + + Implementation note: the aggregation parameter includes the level of + the IDPF tree and the sequence of indices to evaluate. For + implementations that perform per-report caching across executions of + the VDAF, this may be more information than is strictly needed. In + particular, it may be sufficient to convey which indices from the + previous execution will have their children included in the next. + This would help reduce communication overhead. + +8.3. IDPF Specification + + In this section we specify a concrete IDPF suitable for instantiating + Poplar1. The constant and type definitions required by the Idpf + interface are given in Table 17. + + Our IDPF requires an XOF for deriving the output shares, as well as a + variety of other artifacts used internally. For performance reasons, + we instantiate this object using XofFixedKeyAes128 (Section 6.2.2) + wherever possible. See Section 9.6 for security considerations. + + +============+===================================+ + | Parameter | Value | + +============+===================================+ + | xof | XofFixedKeyAes128 (Section 6.2.2) | + +------------+-----------------------------------+ + | SHARES | 2 | + +------------+-----------------------------------+ + | BITS | Any positive integer. | + +------------+-----------------------------------+ + | VALUE_LEN | Any positive integer. | + +------------+-----------------------------------+ + | KEY_SIZE | xof.SEED_SIZE | + +------------+-----------------------------------+ + | FieldInner | Field64 (Table 3) | + +------------+-----------------------------------+ + | FieldLeaf | Field255 (Table 3) | + +------------+-----------------------------------+ + + Table 17: Constants and type definitions for + our concrete IDPF. + +8.3.1. Overview + + At a high level, the IDPF maps a key generator's input (alpha, + beta_1, ..., beta_BITS) onto a binary tree with BITS+1 levels, where + each edge going from a parent node to a left child is labeled 0, and + each right edge is labeled 1. Then each leaf node corresponds to a + bit string of length BITS, where the labels on the path from the root + to x contain the individual bits. Finally, all nodes in the tree + have an assigned value, with the nodes on the path from the root to + alpha having values beta_1, ..., beta_BITS, and all other nodes + having value 0. + + The IDPF construction now boils down to secret-sharing the values at + each node of that tree in an efficient way. Note that explicitly + representing the tree requires O(2^BITS) space, so the generator + cannot just compute additive shares of it and send them to the two + evaluators. Instead, the evaluators will re-generate shares of the + tree using a XOF (Section 6.2). + + The basic observation is that if both evaluators have the same seed s + of length KEY_SIZE, then expanding s using a XOF will also result in + the same expansion. If we set the length of the XOF expansion to + 2*KEY_SIZE, it can then be split again into two seeds s_l, s_r, that + can again serve as XOF seeds. Now if we view seeds as XOR-shares of + integers, and if evaluators have the same seed at the root of the + tree, then their expanded trees will form a secret-shared tree of + zeros. The actual construction will additionally use a convert() + function before each expansion, which maps seeds into the appropriate + output domain (see Section 8.3.4), generating a new seed for the next + level in the process. + + The open task now is to ensure that evaluators have different seeds + at nodes that lie on the path to alpha, while having the same seeds + on all other nodes. This is done using so-called "correction words" + included in the public share. The correction words are conditionally + added to the XOF output by both evaluators. The condition here is a + secret-shared bit, called a "control bit", which indicates whether + the current node is on the path to alpha or not. On the path, the + control bits add up to 1, meaning only one evaluator will add the + correction word to its XOF output. Off the path, either none or both + evaluators add the correction word, and so the seeds at the next + level stay the same. + + What remains is to turn the (now pseudorandom) values on the path to + alpha into the desired beta values. This is done by including "value + correction words" in the public share, which are chosen such that + when added with the pseudorandom shares at the ith node on the path + to alpha, they add up to shares of beta_i. + + In the following two sections we describe the algorithms for key + generation in full detail. + +8.3.2. Key Generation + + The description of the IDPF-key generation algorithm makes use of + auxiliary functions extend() and convert() defined in Section 8.3.4. + + def gen( + self, + alpha: tuple[bool, ...], + beta_inner: list[list[Field64]], + beta_leaf: list[Field255], + ctx: bytes, + nonce: bytes, + rand: bytes) -> tuple[list[CorrectionWord], list[bytes]]: + if len(alpha) != self.BITS: + raise ValueError("incorrect alpha length") + if len(beta_inner) != self.BITS - 1: + raise ValueError("incorrect beta_inner length") + if len(rand) != self.RAND_SIZE: + raise ValueError("incorrect rand size") + if len(nonce) != self.NONCE_SIZE: + raise ValueError("incorrect nonce size") + + key = [ + rand[:XofFixedKeyAes128.SEED_SIZE], + rand[XofFixedKeyAes128.SEED_SIZE:], + ] + + seed = key.copy() + ctrl = [False, True] + public_share = [] + for level in range(self.BITS): + bit = alpha[level] + keep = int(bit) + lose = 1 - keep + + (s0, t0) = self.extend(level, seed[0], ctx, nonce) + (s1, t1) = self.extend(level, seed[1], ctx, nonce) + seed_cw = xor(s0[lose], s1[lose]) + ctrl_cw = ( + t0[0] ^ t1[0] ^ (not bit), + t0[1] ^ t1[1] ^ bit, + ) + + # Implementation note: these conditional XORs and + # input-dependent array indices should be replaced with + # constant-time selects in practice in order to reduce + # leakage via timing side channels. + if ctrl[0]: + x0 = xor(s0[keep], seed_cw) + ctrl[0] = t0[keep] ^ ctrl_cw[keep] + else: + x0 = s0[keep] + ctrl[0] = t0[keep] + if ctrl[1]: + x1 = xor(s1[keep], seed_cw) + ctrl[1] = t1[keep] ^ ctrl_cw[keep] + else: + x1 = s1[keep] + ctrl[1] = t1[keep] + (seed[0], w0) = self.convert(level, x0, ctx, nonce) + (seed[1], w1) = self.convert(level, x1, ctx, nonce) + + if level < self.BITS - 1: + b = cast(list[Field], beta_inner[level]) + else: + b = cast(list[Field], beta_leaf) + if len(b) != self.VALUE_LEN: + raise ValueError( + "length of beta must match the value length" + ) + + w_cw = vec_add(vec_sub(b, w0), w1) + # Implementation note: this conditional negation should be + # replaced with a constant time select or a constant time + # multiplication in practice in order to reduce leakage via + # timing side channels. + if ctrl[1]: + for i in range(len(w_cw)): + w_cw[i] = -w_cw[i] + + public_share.append((seed_cw, ctrl_cw, w_cw)) + return (public_share, key) + +8.3.3. Key Evaluation + + The description of the IDPF-evaluation algorithm makes use of + auxiliary functions extend() and convert() defined in Section 8.3.4. + + def eval( + self, + agg_id: int, + public_share: list[CorrectionWord], + key: bytes, + level: int, + prefixes: Sequence[tuple[bool, ...]], + ctx: bytes, + nonce: bytes) -> list[list[Field64]] | list[list[Field255]]: + if agg_id not in range(self.SHARES): + raise ValueError('aggregator id out of range') + if level not in range(self.BITS): + raise ValueError('level out of range') + if len(set(prefixes)) != len(prefixes): + raise ValueError('prefixes must be unique') + + out_share = [] + for prefix in prefixes: + if len(prefix) != level + 1: + raise ValueError('incorrect prefix length') + + # The Aggregator's output share is the value of a node of + # the IDPF tree at the given `level`. The node's value is + # computed by traversing the path defined by the candidate + # `prefix`. Each node in the tree is represented by a seed + # (`seed`) and a control bit (`ctrl`). + seed = key + ctrl = bool(agg_id) + y: FieldVec + for current_level in range(level + 1): + bit = int(prefix[current_level]) + + # Implementation note: typically the current round of + # candidate prefixes would have been derived from + # aggregate results computed during previous rounds. + # For example, when using the IDPF to compute heavy + # hitters, a string whose hit count exceeded the + # given threshold in the last round would be the + # prefix of each `prefix` in the current round. (See + # [BBCGGI21, Section 5.1].) In this case, part of the + # path would have already been traversed. + # + # Re-computing nodes along previously traversed paths is + # wasteful. Implementations can eliminate this added + # complexity by caching nodes (i.e., `(seed, ctrl)` + # pairs) output by previous calls to `eval_next()`. + (seed, ctrl, y) = self.eval_next( + seed, + ctrl, + public_share[current_level], + current_level, + bit, + ctx, + nonce, + ) + if agg_id == 0: + out_share.append(cast(list[Field], y)) + else: + out_share.append(vec_neg(cast(list[Field], y))) + return cast( + list[list[Field64]] | list[list[Field255]], + out_share, + ) + + def eval_next( + self, + prev_seed: bytes, + prev_ctrl: bool, + correction_word: CorrectionWord, + level: int, + bit: int, + ctx: bytes, + nonce: bytes) -> tuple[bytes, bool, FieldVec]: + """ + Compute the next node in the IDPF tree along the path determined + by a candidate prefix. The next node is determined by `bit`, the + bit of the prefix corresponding to the next level of the tree. + """ + + seed_cw = correction_word[0] + ctrl_cw = correction_word[1] + w_cw = cast(list[Field], correction_word[2]) + (s, t) = self.extend(level, prev_seed, ctx, nonce) + + # Implementation note: these conditional operations and + # input-dependent array indices should be replaced with + # constant-time selects in practice in order to reduce leakage + # via timing side channels. + if prev_ctrl: + s[0] = xor(s[0], seed_cw) + s[1] = xor(s[1], seed_cw) + t[0] ^= ctrl_cw[0] + t[1] ^= ctrl_cw[1] + + next_ctrl = t[bit] + convert_output = self.convert(level, s[bit], ctx, nonce) + next_seed = convert_output[0] + y = cast(list[Field], convert_output[1]) + # Implementation note: this conditional addition should be + # replaced with a constant-time select in practice in order to + # reduce leakage via timing side channels. + if next_ctrl: + for i in range(len(y)): + y[i] += w_cw[i] + + return (next_seed, next_ctrl, cast(FieldVec, y)) + +8.3.4. Auxiliary Functions + + def extend( + self, + level: int, + seed: bytes, + ctx: bytes, + nonce: bytes) -> tuple[list[bytes], list[bool]]: + xof = self.current_xof( + level, + seed, + format_dst(1, 0, 0) + ctx, + nonce, + ) + s = [ + bytearray(xof.next(self.KEY_SIZE)), + bytearray(xof.next(self.KEY_SIZE)), + ] + # Use the least significant bits as the control bit correction, + # and then zero it out. This gives effectively 127 bits of + # security, but reduces the number of AES calls needed by 1/3. + t = [bool(s[0][0] & 1), bool(s[1][0] & 1)] + s[0][0] &= 0xFE + s[1][0] &= 0xFE + return ([bytes(s[0]), bytes(s[1])], t) + + def convert( + self, + level: int, + seed: bytes, + ctx: bytes, + nonce: bytes) -> tuple[bytes, FieldVec]: + xof = self.current_xof( + level, + seed, + format_dst(1, 0, 1) + ctx, + nonce, + ) + next_seed = xof.next(self.KEY_SIZE) + field = self.current_field(level) + w = xof.next_vec(field, self.VALUE_LEN) + return (next_seed, cast(FieldVec, w)) + + def current_xof(self, + level: int, + seed: bytes, + dst: bytes, + nonce: bytes) -> Xof: + if level < self.BITS-1: + return XofFixedKeyAes128(seed, dst, nonce) + return XofTurboShake128(seed, dst, nonce) + +9. Security Considerations + + VDAFs (Section 5) have two essential security goals: + + 1. Privacy: an attacker that controls the Collector and a subset of + Clients and Aggregators learns nothing about the measurements of + honest Clients beyond what it can deduce from the aggregate + result. We assume the attacker controls the entire network + except for channels between honest Clients and honest + Aggregators. In particular, it cannot forge or prevent + transmission of messages on these channels. + + 2. Robustness: an attacker that controls a subset of Clients cannot + cause the Collector to compute anything other than the aggregate + of the measurements of honest Clients. We assume the attacker + eavesdrops on the network but does not control transmission of + messages between honest parties. + + Formal definitions of privacy and robustness can be found in + [DPRS23]. A VDAF is the core cryptographic primitive of a protocol + that achieves the above privacy and robustness goals. It is not + sufficient on its own, however. The application will need to assure + a few security properties, for example: + + * Securely distributing the long-lived parameters, in particular the + verification key. + + * Establishing secure channels: + + - Confidential and authentic channels among Aggregators, and + between the Aggregators and the Collector; and + + - Confidential and Aggregator-authenticated channels between + Clients and Aggregators. + + * Enforcing the non-collusion properties required of the specific + VDAF in use. + + In such an environment, a VDAF provides the high-level privacy + property described above: the Collector learns only the aggregate + result, and nothing about individual measurements aside from what can + be inferred from the aggregate result. The Aggregators learn neither + individual measurements nor the aggregate result. The Collector is + assured that the aggregate statistic accurately reflects the inputs + as long as the Aggregators correctly executed their role in the VDAF. + + On their own, VDAFs do not provide: + + 1. Mitigation of Sybil attacks [Dou02]. In this attack, the + adversary observes a subset of input shares transmitted by a + Client it is interested in. It allows the input shares to be + processed, but corrupts and picks bogus measurements for the + remaining Clients. Applications can guard against these risks by + adding additional controls on report submission, such as Client + authentication and rate limits. + + 2. Differential privacy [Dwo06]. Depending on the distribution of + the measurements, the aggregate result itself can still leak a + significant amount of information about an individual measurement + or the person that generated it. + + 3. Robustness in the presence of a malicious Aggregator. An + Aggregator can, without detection, manipulate the aggregate + result by modifying its own aggregate share. + + 4. Guaranteed output delivery [GSZ20]. An attacker that controls + transmission of messages between honest parties can prevent + computation of the aggregate result by dropping messages. + +9.1. The Verification Key + + The Aggregators are responsible for exchanging the verification key + in advance of executing the VDAF. Any procedure is acceptable as + long as the following conditions are met: + + 1. To ensure robustness of the computation, the Aggregators MUST NOT + reveal the verification key to the Clients. Otherwise, a + malicious Client might be able to exploit knowledge of this key + to craft an invalid report that would be accepted by the + Aggregators. + + 2. To ensure privacy of the measurements, the Aggregators MUST + commit to the verification key prior to processing reports + generated by Clients. Otherwise, the attacker may be able to + craft a verification key that, for a given report, causes an + honest Aggregator to leak information about the measurement + during preparation. + + Meeting these requirements is relatively straightforward. For + example, the Aggregators may designate one of their peers to generate + the verification key and distribute it to the others. To assure + Clients of key commitment, the Clients and (honest) Aggregators + SHOULD bind the verification key to the application context. For + instance, the "task ID" of DAP [DAP] could be set to the hash of the + verification key; then as long as honest Aggregators only consume + reports for the task indicated by the Client, forging a new key after + the fact would reduce to finding collisions in the underlying hash + function. (Keeping the key secret from the Clients would require the + hash function to be one-way.) However, since rotating the key + implies rotating the task ID, this scheme would not allow key + rotation over the lifetime of a task. + +9.2. The Nonce + + The sharding and preparation steps of VDAF execution depend on a + nonce associated with the Client's report. To ensure privacy of the + underlying measurement, the Client MUST generate this nonce using a + CSPRNG. This is required in order to leverage security analysis for + the privacy definition of [DPRS23], which assumes the nonce is chosen + at random prior to generating the report. Uniqueness of the nonce is + not sufficient because the verification key is controlled by the + attacker. + + Other security considerations may require the nonce to be non- + repeating in a given context. For example, to achieve differential + privacy it is necessary to avoid "over exposing" a report by + including it too many times in a single batch or across multiple + batches. It is RECOMMENDED that the nonce generated by the Client be + used by the Aggregators for replay protection. + +9.3. The Public Share + + The Aggregators MUST ensure they have both received the same public + share from the Client. It is sufficient, for example, to exchange a + hash of the public share over a secure channel. + +9.4. The Aggregation Parameter + + As described in Section 4.3 and Section 5.3 respectively, DAFs and + VDAFs may impose restrictions on the re-use of reports. For Prio3, + reports should only be aggregated once; for Poplar1, reports may be + aggregated multiple times, but never twice at the same level of the + tree. Otherwise, we risk re-using correlated randomness, which might + compromise confidentiality of the Client's measurement. + + Higher level applications that use DAFs or VDAFs MUST enforce + aggregation parameter validity. In particular, prior to beginning + preparation with an aggregation parameter provided by the Collector, + they MUST invoke is_valid() to decide if the parameter is valid given + the sequence of previously accepted parameters. + + Note that aggregating a batch of reports multiple times, even with a + valid sequence of aggregation parameters, can result in information + leakage beyond what is used by the application. + + For example, when Poplar1 is used for heavy hitters, the Aggregators + learn not only the heavy hitters themselves, but also the prefix tree + (as defined in Section 8) computed along the way. Indeed, this + leakage is inherent to any construction that uses an IDPF + (Section 8.1) in the same way. Depending on the distribution of the + measurements, the prefix tree can leak a significant amount of + information about unpopular inputs. For instance, it is possible + (though perhaps unlikely) for a large set of non-heavy-hitter values + to share a common prefix, which would be leaked by a prefix tree with + a sufficiently small threshold. + + A malicious adversary controlling the Collector and one of the + Aggregators can further turn arbitrary non-heavy prefixes into heavy + ones by tampering with the IDPF output at any position. While our + construction ensures that the nodes evaluated at one level are + children of the nodes evaluated at the previous level, this still may + allow an adversary to discover individual non-heavy strings. We call + this a "steering attack". + + The only practical, general-purpose defense against steering attacks + is to compose Poplar1 with some mechanism for differential privacy. + It is therefore RECOMMENDED to use differential privacy for any + heavy-hitter type application. + +9.5. Safe Usage of IDPF Outputs + + The arithmetic sketch described in Section 8 is used by the + Aggregators to check that the shares of the vector obtained by + evaluating a Client's IDPF at a sequence of candidate prefixes has at + most one non-zero value, and that the non-zero value is 1. Depending + on how the values are used, the arithmetic sketch on its own may not + be sufficient for robustness of the application. In particular, a + malicious Client may attempt to influence the computation by choosing + an IDPF that evaluates to 1 at more than one node at a given level of + the tree. + + This issue can be mitigated by using an IDPF that is extractable as + defined in Appendix D of [BBCGGI21]. Extractability ensures that, + for a particular level of the tree, it is infeasible for an attacker + to control values of the IDPF such that it takes on chosen non-zero + values at more than one node. (It can practically only achieve the + zero function, a point function, or a pseudorandom function.) + + The IDPF specified in Section 8.1 only guarantees extractability at + the last level of the tree. (This is by virtue of using a larger + field for the leaves than for inner nodes and using an XOF to derive + leaves that is safe to model as a random oracle. See Section 9.6.) + For intermediate levels, it is feasible for a client to produce IDPF + shares with two controlled non-zero nodes. + + This is not an issue for running heavy hitters, since (1) each node + in the prefix tree is a child of a previously traversed node, (2) the + arithmetic sketch would detect double voting at every level of the + prefix tree, and (3) the IDPF is extractable at the last level of the + tree. However, the lack of extractability at intermediate levels may + result in attacks on the robustness of certain applications. + + Thus applications SHOULD NOT use prefix counts for intermediate + levels for any purpose beyond computing the prefix tree for heavy + hitters. + +9.6. Safe Usage of XOFs + + In the security analyses of our protocols, XOFs (Section 6.2) are + usually modeled as random oracles. XofTurboShake128 is designed to + be indifferentiable from a random oracle [MRH04], making it a + suitable choice for most situations. + + The one exception is the IDPF of Section 8.3. Here, a random oracle + is not needed to prove privacy, since the analysis of [BBCGGI21], + Proposition 1, only requires a Pseudorandom Generator (PRG). As + observed in [GKWY20], a PRG can be instantiated from a correlation- + robust hash function H. Informally, correlation robustness requires + that for a random r, H(xor(r, x)) is computationally + indistinguishable from a random function of x. A PRG can therefore + be constructed as + + PRG(r) = H(xor(r, 1)) || H(xor(r, 2)) || ... + + since each individual hash function evaluation is indistinguishable + from a random function. + + XofFixedKeyAes128 in Section 6.2.2 implements a correlation-robust + hash function using fixed-key AES. For security, it assumes that AES + with a fixed key can be modeled as a random permutation [GKWY20]. + Additionally, we use a different AES key for every client, which in + the ideal cipher model leads to better concrete security [GKWWY20]. + + We note that for robustness, the analysis of [BBCGGI21] still assumes + a random oracle to make the IDPF extractable. We therefore use + XofTurboShake128 instead for the last level of the tree. It is + important that XofTurboShake128 supports many seed lengths, in + particular 16 bytes, as this is the seed size for the inner levels. + + While XofFixedKeyAes128 has been shown to be differentiable from a + random oracle [GKWWY20], there are no known attacks exploiting this + difference. We also stress that even if the IDPF is not extractable, + Poplar1 guarantees that every client can contribute to at most one + prefix among the ones being evaluated by the helpers. + +9.7. Choosing FLP Parameters + + Prio3 and other systems built from the FLP of Section 7.3 may benefit + from choosing a field size that is as small as possible. Generally + speaking, a smaller field results in lower communication and storage + costs. Care must be taken, however, since a smaller field also + results in degraded (or even vacuous) robustness. + + Different variants of Prio3 (Section 7) use different field sizes: + Prio3Count and Prio3Sum use Field64; but Prio3SumVec, Prio3Histogram, + and Prio3MultihotCountVec all use Field128, a field that is twice as + large as Field64. This is due to the use of joint randomness + (Section 7.1) in the latter variants. Joint randomness allows for + more flexible circuit design (see Section 7.3.1.1), but opens up + Prio3 to offline attacks in which the attacker searches for input + shares for an invalid measurement that derive joint randomness that + causes the circuit to accept. Choosing a large enough field ensures + this computation is too expensive to be feasible. (See [DPRS23], + Theorem 1.) Note that privacy is not susceptible to such attacks. + + Another way to mitigate this issue (or improve robustness in general) + is to generate and verify multiple, independent proofs. (See + Section 7.1.2.) For Prio3, the PROOFS parameter controls the number + of proofs (at least one) that are generated and verified. In general + the soundness error of the FLP is given by the following formula: + + (circuit_soundness + flp_soundness)^PROOFS + + where: + + * circuit_soundness is the soundness of the validity circuit + (Section 7.3.2) + + * flp_soundness is the base soundness of the proof system + ([BBCGGI19], Theorem 4.3) + + For circuits involving joint randomness, we aim for the soundness + error to be close to 2^-128 in order to mitigate offline attacks. + Such circuits MUST use Field128 with at least one proof or Field64 + with at least three proofs. Depending on the circuit, Field64 with + two proofs might have significantly lower soundness than Field128 + with one proof. + + We stress that weak parameters (too small a field, too few proofs, or + both) can be exploited to attack any aggregation task using those + parameters. To mitigate offline attacks, it is necessary to disable + all tasks that use the weak parameters. + +9.8. Choosing the Number of Aggregators + + Two Aggregators are required for privacy in our threat model, but + some (V)DAFs, including Prio3 (Section 7), allow for any number of + Aggregators, only one of which needs to be trusted in order for the + computation to be private. To hedge against corruptions that happen + during the course of the attack, deployments may consider involving + more than two Aggregators as described for example in Section 5.7.2. + Note however that some schemes are not compatible with this mode of + operation, such as Poplar1. + +9.9. Defense-in-Depth Measures + + Prio3 and Poplar1 are designed to resist some attacks that fall + outside the main threat model for VDAFs. + + Broadly speaking, domain separation is used to prevent cross protocol + attacks, in which data from evaluation of one VDAF translates to an + attack against another. For example: + + 1. Weak entropy sources: the VDAF algorithm ID is bound to each XOF + invocation, thereby ensuring the outputs are different between + VDAF invocations, even if the underlying randomness is the same. + For example, two different instances of Prio3 would compute + different measurement shares. + + 2. Weak parameters: Prio3 variants that require joint randomness are + subject to offline attacks against robustness. These attacks are + feasible if the field size or number of proofs is sufficiently + small. (See Section 9.7.) The joint randomness derivation is + bound to both the field (via the algorithm ID) and the number of + proofs, thereby ensuring that joint randomness derived for weak + parameters is not reused for stronger parameters. In addition, + the joint randomness is bound to the application context, meaning + any work the attacker does to attack some application is not + useful for other applications that use the same parameters. + + There are also some important limitations to be aware of. For + example, Prio3 provides domain separation between families of + circuits, but does not provide domain separation between instances of + a circuit. Concretely, it is possible for Aggregators to accept a + report for Prio3SumVec from a Client who disagrees with them on the + value of bits and length. This is because there is no binding of the + circuit parameters to the computation. + +10. IANA Considerations + + IANA is requested to make one new registry: + + * DAF and VDAF Identifiers + + This registry should be created under the heading "Verifiable + Distributed Aggregation Functions (VDAF)", and administered under the + Specification Required policy [RFC8126]. + + The "VDAF Identifiers" registry lists identifiers for Distributed + Aggregation Functions (DAFs) and Verifiable Distributed Aggregation + Functions (VDAFs). These identifiers are four-byte values, so the + minimum possible value is 0x00000000 and the maximum possible value + is 0xffffffff. + + Template: + + * Value: The four-byte identifier for the DAF or VDAF + + * Scheme: The name of the DAF or VDAF + + * Type: Either "DAF" for a Distributed Aggregation Function or + "VDAF" for a Verifiable Distributed Aggregation Function + + * Reference: Where the algorithm is defined + + The initial contents of the registry are as follows: + + +===============+=======================+======+===============+ + | Value | Scheme | Type | Reference | + +===============+=======================+======+===============+ + | 0x00000000 | Reserved | n/a | RFC XXXX | + +---------------+-----------------------+------+---------------+ + | 0x00000001 | Prio3Count | VDAF | Section 7.4.1 | + | | | | of RFC XXXX | + +---------------+-----------------------+------+---------------+ + | 0x00000002 | Prio3Sum | VDAF | Section 7.4.2 | + | | | | of RFC XXXX | + +---------------+-----------------------+------+---------------+ + | 0x00000003 | Prio3SumVec | VDAF | Section 7.4.3 | + | | | | of RFC XXXX | + +---------------+-----------------------+------+---------------+ + | 0x00000004 | Prio3Histogram | VDAF | Section 7.4.4 | + | | | | of RFC XXXX | + +---------------+-----------------------+------+---------------+ + | 0x00000005 | Prio3MultihotCountVec | VDAF | Section 7.4.5 | + | | | | of RFC XXXX | + +---------------+-----------------------+------+---------------+ + | 0x00000006 | Poplar1 | VDAF | Section 8.2 | + | | | | of RFC XXXX | + +---------------+-----------------------+------+---------------+ + | 0xFFFF0000 to | Reserved for Private | n/a | n/a | + | 0xFFFFFFFF | Use | | | + +---------------+-----------------------+------+---------------+ + + Table 18: Verifiable Distributed Aggregation Function + Identifiers Registry + + (RFC EDITOR: Please replace "RFC XXXX" above with the RFC number + assigned to this document.) + + VDAF identifiers are used for domain separation, as described in + Section 6.2.3. Domain separation guards against cross protocol + attacks and certain failures of entropy sources. See Section 9.9. + + The benefits of domain separation are undermined if different VDAFs + are used with the same VDAF Identifier. The "Reserved for Private + Use" code points should thus be used judiciously, because they + provide no defense against such collisions. Applications SHOULD + prefer the use of registered code points. + +11. References + +11.1. Normative References + + [AES] Dworkin, M. J., Barker, E., Nechvatal, J. R., Foti, J., + Bassham, L. E., Roback, E., and J. Dray Jr, "Advanced + Encryption Standard (AES)", 2001, + . + + [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate + Requirement Levels", BCP 14, RFC 2119, + DOI 10.17487/RFC2119, March 1997, + . + + [RFC8126] Cotton, M., Leiba, B., and T. Narten, "Guidelines for + Writing an IANA Considerations Section in RFCs", BCP 26, + RFC 8126, DOI 10.17487/RFC8126, June 2017, + . + + [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC + 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, + May 2017, . + + [RFC8446] Rescorla, E., "The Transport Layer Security (TLS) Protocol + Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018, + . + + [TurboSHAKE] + Viguier, B., Wong, D., Van Assche, G., Dang, Q., and J. + Daemen, "KangarooTwelve and TurboSHAKE", Work in Progress, + Internet-Draft, draft-irtf-cfrg-kangarootwelve-14, 9 May + 2024, . + +11.2. Informative References + + [AGJOP21] Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A. + Polychroniadou, "Prio+: Privacy Preserving Aggregate + Statistics via Boolean Shares", Security and Cryptography + for Networks (SCN), 2022, . + + [BBCGGI19] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and + Y. Ishai, "Zero-Knowledge Proofs on Secret-Shared Data via + Fully Linear PCPs", Crypto, 2019, + . + + [BBCGGI21] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and + Y. Ishai, "Lightweight Techniques for Private Heavy + Hitters", IEEE Security & Privacy (S&P), 2021, + . + + [BBCGGI23] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and + Y. Ishai, "Arithmetic Sketching", Crypto, 2023, + . + + [BGI15] Boyle, E., Gilboa, N., and Y. Ishai, "Function Secret + Sharing", Eurocrypt, 2015, . + + [CGB17] Boneh, D. and H. Corrigan-Gibbs, "Prio: Private, Robust, + and Scalable Computation of Aggregate Statistics", USENIX + Symposium on Networked Systems Design and Implementation + (NSDI), 2017, + . + + [DAP] Geoghegan, T., Patton, C., Pitman, B., Rescorla, E., and + C. A. Wood, "Distributed Aggregation Protocol for Privacy + Preserving Measurement", Work in Progress, Internet-Draft, + draft-ietf-ppm-dap-12, 10 October 2024, + . + + [Dou02] Douceur, J. R., "The Sybil Attack", International Workshop + on Peer-to-Peer Systems (IPTPS), 2002, + . + + [DPRS23] Hannah Davis, Christopher Patton, Mike Rosulek, and + Phillipp Schoppmann, "Verifiable Distributed Aggregation + Functions", Privacy Enhancing Technologies Symposium + (PETS), 2023, . + + [Dwo06] Cynthia Dwork, "Differential Privacy", International + Colloquium on Automata, Languages, and Programming + (ICALP), 2006, + . + + [ENPA] "Exposure Notification Privacy-preserving Analytics (ENPA) + White Paper", 2021, . + + [EPK14] Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR: + Randomized Aggregatable Privacy-Preserving Ordinal + Response", ACM Conference on Computer and Communications + Security (CCS), 2014, + . + + [GI14] Gilboa, N. and Y. Ishai, "Distributed Point Functions and + Their Applications", Eurocrypt, 2014, + . + + [GKWWY20] Guo, C., Katz, J., Wang, X., Weng, C., and and Y. Yu, + "Better concrete security for half-gates garbling (in the + multi-instance setting)", Crypto, 2020, + . + + [GKWY20] Guo, C., Katz, J., Wang, X., and Y. Yu, "Efficient and + Secure Multiparty Computation from Fixed-Key Block + Ciphers", IEEE Security & Privacy (S&P), 2020, + . + + [GSZ20] Goyal, V., Song, Y., and C. Zhu, "Guaranteed Output + Delivery Comes Free in Honest Majority MPC", Crypto, 2020, + . + + [MPDST25] Mouris, D., Patton, C., Davis, H., Sarkar, P., and N. G. + Tsoutsos, "Mastic: Private Weighted Heavy-Hitters and + Attribute-Based Metrics", Privacy Enhancing Technologies + Symposium (PETS), 2025, + . + + [MPRV09] Mironov, I., Pandey, O., Reingold, O., and S. Vadhan, + "Computational Differential Privacy", Crypto, 2009, + . + + [MRH04] Maurer, U., Renner, R., and C. Holenstein, + "Indifferentiability, impossibility results on reductions, + and applications to the random oracle methodology", Theory + of Cryptography (TCC), 2004, + . + + [OriginTelemetry] + "Origin Telemetry", 2020, + . + + [SML24] Satriawan, A., Mareta, R., and H. Lee, "A Complete + Beginner Guide to the Number Theoretic Transform (NTT)", + IEEE Access, vol. 11, 2024, + . + + [TestVectors] + "Test vectors for Prio3 and Poplar1", commit hash 1891650, + October 2024, + . + +Appendix A. FLP Gadgets + + This section defines gadgets used to specify variants of Prio3 + defined in Section 7.4 and to construct FLPs as defined in + Section 7.3. + +A.1. Multiplication + + The multiplication gadget takes in a pair of inputs and multiplies + them together. The degree of this circuit is 2. + + class Mul(Gadget[F]): + ARITY = 2 + DEGREE = 2 + + def eval(self, _field: type[F], inp: list[F]) -> F: + return inp[0] * inp[1] + + def eval_poly(self, + field: type[F], + inp_poly: list[list[F]]) -> list[F]: + return poly_mul(field, inp_poly[0], inp_poly[1]) + +A.2. Polynomial Evaluation + + The polynomial-evaluation gadget takes in a single input x and + returns p(x), where p is a polynomial specified by the user. Its + degree is the same as p. + + class PolyEval(Gadget[F]): + ARITY = 1 + p: list[int] # polynomial coefficients + + def __init__(self, p: list[int]): + """ + Instantiate this gadget with the given polynomial. + """ + # Strip leading zeros. + for i in reversed(range(len(p))): + if p[i] != 0: + p = p[:i+1] + break + if len(p) < 1: + raise ValueError('invalid polynomial: zero length') + + self.p = p + self.DEGREE = len(p) - 1 + + def eval(self, field: type[F], inp: list[F]) -> F: + p = [field(coeff) for coeff in self.p] + return poly_eval(field, p, inp[0]) + + def eval_poly(self, + field: type[F], + inp_poly: list[list[F]]) -> list[F]: + p = [field(coeff) for coeff in self.p] + out = [field(0)] * (self.DEGREE * len(inp_poly[0])) + out[0] = p[0] + x = inp_poly[0] + for i in range(1, len(p)): + for j in range(len(x)): + out[j] += p[i] * x[j] + x = poly_mul(field, x, inp_poly[0]) + return poly_strip(field, out) + +A.3. Parallel Sum + + The parallel-sum gadget applies a subcircuit to multiple inputs in + parallel, then returns the sum of the results. The arity of the + gadget is equal to count times the arity of the subcircuit, where + count denotes the number of times the subcircuit is called. The + degree of the gadget is equal to the degree of the subcircuit. + + class ParallelSum(Gadget[F]): + subcircuit: Gadget[F] + count: int + + def __init__(self, subcircuit: Gadget[F], count: int): + self.subcircuit = subcircuit + self.count = count + self.ARITY = subcircuit.ARITY * count + self.DEGREE = subcircuit.DEGREE + + def eval(self, field: type[F], inp: list[F]) -> F: + out = field(0) + for i in range(self.count): + start_index = i * self.subcircuit.ARITY + end_index = (i + 1) * self.subcircuit.ARITY + out += self.subcircuit.eval( + field, + inp[start_index:end_index], + ) + return out + + def eval_poly(self, + field: type[F], + inp_poly: list[list[F]]) -> list[F]: + output_poly_length = self.DEGREE * (len(inp_poly[0]) - 1) + 1 + out_sum = [field(0) for _ in range(output_poly_length)] + for i in range(self.count): + start_index = i * self.subcircuit.ARITY + end_index = (i + 1) * self.subcircuit.ARITY + out_current = self.subcircuit.eval_poly( + field, + inp_poly[start_index:end_index] + ) + for j in range(output_poly_length): + out_sum[j] += out_current[j] + return poly_strip(field, out_sum) + +A.4. Shims for Generating and Querying Proofs + + This section specifies two "shim" gadgets, one for generating an FLP + as specified in Section 7.3.3 and another for querying an FLP as + specified in Section 7.3.4. + + class ProveGadget(Gadget[F]): + """ + Gadget wrapper that records the input wires for each evaluation. + """ + + def __init__(self, + field: type[F], + wire_seeds: list[F], + g: Gadget[F], + g_calls: int): + p = next_power_of_2(1 + g_calls) + self.inner = g + self.ARITY = g.ARITY + self.DEGREE = g.DEGREE + self.wires = [] + self.k = 0 # evaluation counter + for s in wire_seeds: + wire = field.zeros(p) + wire[0] = s # set the wire seed + self.wires.append(wire) + + def eval(self, field: type[F], inp: list[F]) -> F: + self.k += 1 + for j in range(len(inp)): + self.wires[j][self.k] = inp[j] + return self.inner.eval(field, inp) + + def eval_poly(self, + field: type[F], + inp_poly: list[list[F]]) -> list[F]: + return self.inner.eval_poly(field, inp_poly) + + @classmethod + def wrap(cls, + valid: Valid[Measurement, AggResult, F], + prove_rand: list[F], + ) -> Valid[Measurement, AggResult, F]: + """ + Make a copy of `valid` with each gadget wrapped for recording + the wire inputs. `prove_rand` is used to produce the wire + seeds for each gadget. + """ + wrapped_gadgets: list[Gadget[F]] = [] + for (g, g_calls) in zip(valid.GADGETS, valid.GADGET_CALLS): + (wire_seeds, prove_rand) = front(g.ARITY, prove_rand) + wrapped = cls(valid.field, wire_seeds, g, g_calls) + wrapped_gadgets.append(wrapped) + wrapped_valid = deepcopy(valid) + wrapped_valid.GADGETS = wrapped_gadgets + return wrapped_valid + + class QueryGadget(Gadget[F]): + """ + Gadget wrapper that records the input wires for each evaluation. + Rather than evaluate the circuit, use the provided gadget + polynomial to produce the output. + """ + + def __init__( + self, + field: type[F], + wire_seeds: list[F], + gadget_poly: list[F], + g: Gadget[F], + g_calls: int): + p = next_power_of_2(1 + g_calls) + self.alpha = field.gen() ** (field.GEN_ORDER // p) + self.poly = gadget_poly + self.ARITY = g.ARITY + self.DEGREE = g.DEGREE + self.wires = [] + self.k = 0 + for s in wire_seeds: + wire = field.zeros(p) + wire[0] = s # set the wire seed + self.wires.append(wire) + + def eval(self, field: type[F], inp: list[F]) -> F: + self.k += 1 + for j in range(len(inp)): + self.wires[j][self.k] = inp[j] + return poly_eval(field, self.poly, self.alpha ** self.k) + + @classmethod + def wrap(cls, + valid: Valid[Measurement, AggResult, F], + proof: list[F]) -> Valid[Measurement, AggResult, F]: + wrapped_gadgets: list[Gadget[F]] = [] + for (g, g_calls) in zip(valid.GADGETS, valid.GADGET_CALLS): + p = next_power_of_2(1 + g_calls) + gadget_poly_len = g.DEGREE * (p - 1) + 1 + (wire_seeds, proof) = front(g.ARITY, proof) + (gadget_poly, proof) = front(gadget_poly_len, proof) + wrapped = cls(valid.field, + wire_seeds, + gadget_poly, + g, g_calls) + wrapped_gadgets.append(wrapped) + wrapped_valid = deepcopy(valid) + wrapped_valid.GADGETS = wrapped_gadgets + return wrapped_valid + +Appendix B. VDAF Preparation State + + This section lists the classes used to define each Aggregator's state + during VDAF preparation (Section 5.7). + + class State: + pass + + class Start(State): + pass + + class Continued(State, Generic[PrepState]): + def __init__(self, prep_state: PrepState, prep_round: int): + self.prep_state = prep_state + self.prep_round = prep_round + + def __eq__(self, other: object) -> bool: + return isinstance(other, Continued) and \ + self.prep_state == other.prep_state and \ + self.prep_round == other.prep_round + + class Finished(State, Generic[OutShare]): + def __init__(self, out_share: OutShare): + self.out_share = out_share + + def __eq__(self, other: object) -> bool: + return isinstance(other, Finished) and \ + self.out_share == other.out_share + + class Rejected(State): + pass + +Appendix C. Test Vectors + + Test vectors for Prio3 (Section 7) and Poplar1 (Section 8) are + available at [TestVectors]. The test vector directory, test_vec/ + vdaf, contains a set of JSON files. Each file contains a test vector + for an instance of class Vdaf as defined in Section 5. A test vector + covers sharding, preparation, aggregation, and unsharding of a batch + of several measurements. The test vector schema is defined below. + +C.1. Schema + + ctx: The application context string encoded in hexadecimal. + + verify_key: The verification key encoded in hexadecimal. + + agg_param: The aggregation parameter encoded in hexadecimal. + + prep: A list of objects with the following schema: + + measurement: The measurement of type Measurement defined by the + VDAF. + + nonce: The nonce encoded in hexadecimal. + + rand: The sharding randomness encoded in hexadecimal. + + public_share: The expected public share encoded in hexadecimal. + + input_shares: The expected list of input shares, each encoded in + hexadecimal. + + prep_shares: The expected list of prep shares generated by each + Aggregator at each round of preparation, encoded in + hexadecimal. + + prep_messages: The expected list of prep messages for each round + of preparation, encoded in hexadecimal. + + out_shares: The expected list of output shares, encoded in + hexadecimal. + + agg_shares: The expected aggregate shares encoded in hexadecimal. + + agg_result: The expected aggregate result of type AggResult defined + by the VDAF. + + The schema also includes whatever parameters are required to + instantiate the VDAF. These are listed in the subsections below. + +C.1.1. Prio3Count + + shares: The number of shares, an integer. + +C.1.2. Prio3Sum + + shares: The number of shares, an integer. + + max_measurement: The largest measurement, an integer. Each element + is in the range [0, max_measurement]. + +C.1.3. Prio3SumVec + + shares: The number of shares, an integer. + + length: The length of the vector, an integer. + + chunk_length: the length of each vector chunk, an integer. + + bits: the bit length of each element of the vector, an integer. + Each element is in [0, 2^bits). + +C.1.4. Prio3Histogram + + shares: The number of shares, an integer. + + length: The length of the vector, an integer. + + chunk_length: the length of each vector chunk, an integer. + +C.1.5. Prio3MultihotCountVec + + shares: The number of shares, an integer. + + length: The length of the vector, an integer. + + chunk_length: the length of each vector chunk, an integer. + + max_weight: The largest vector weight, an integer. The sum of the + elements must be in [0, max_weight]. + +C.1.6. Poplar1 + + bits: The length of each input in bits, an integer. + +Acknowledgments + + The impetus of this work is the success of recent deployments of + predecessors of Prio3. These include the Mozilla Origin Telemetry + project [OriginTelemetry] and the Exposure Notification Private + Analytics system [ENPA] developed jointly by ISRG, Google, Apple, and + others. Together these systems have aggregated data from hundreds of + millions of users. + + As the name implies, Prio3 is a descendant of the original Prio + construction [CGB17]. A second iteration was deployed in the [ENPA] + system, and like the VDAF described here, the ENPA system was built + from techniques introduced in [BBCGGI19] that significantly improve + communication cost. That system was specialized for a particular + aggregation function; the goal of Prio3 is to provide the same level + of generality as the original construction. + + The security considerations in Section 9 are based largely on the + security analysis of [DPRS23]. Thanks to Hannah Davis and Mike + Rosulek, who lent their time to developing definitions and security + proofs. + + Thanks to Junye Chen, Henry Corrigan-Gibbs, Armando Faz-Hernández, + Simon Friedberger, Tim Geoghegan, Albert Liu, Brandon Pitman, Mariana + Raykova, Michael Rosenberg, Jacob Rothstein, Shan Wang, Xiao Wang, + Bas Westerbaan, and Christopher Wood for useful feedback on and + contributions to the spec. + +Authors' Addresses + + Richard L. Barnes + Cisco + Email: rlb@ipv.sx + + + David Cook + ISRG + Email: divergentdave@gmail.com + + + Christopher Patton + Cloudflare + Email: chrispatton+ietf@gmail.com + + + Phillipp Schoppmann + Google + Email: schoppmann@google.com diff --git a/david/reorder-circuit-methods/index.html b/david/reorder-circuit-methods/index.html new file mode 100644 index 00000000..54cdca11 --- /dev/null +++ b/david/reorder-circuit-methods/index.html @@ -0,0 +1,45 @@ + + + + cfrg/draft-irtf-cfrg-vdaf david/reorder-circuit-methods preview + + + + +

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