From 26a9fb8aeedd06a17862c569517157171cd776ee Mon Sep 17 00:00:00 2001 From: Nicolas Mohnblatt <49298616+nmohnblatt@users.noreply.github.com> Date: Fri, 1 Nov 2024 16:52:15 +0000 Subject: [PATCH] Add definitions for other advanced crypto: FHE, MPC, WE (#33) * setup new entries * fhe * add acronyms * mpc blurb * witness encryption * nits * add summary table * disambiguate variables * intro to mpc * nits * witness encryption examples --- src/SUMMARY.md | 3 +++ src/definitions/fhe.md | 21 +++++++++++++++ src/definitions/mpc.md | 30 +++++++++++++++++++++ src/definitions/witness_encryption.md | 36 ++++++++++++++++++++++++++ src/images/fhe.png | Bin 0 -> 89364 bytes 5 files changed, 90 insertions(+) create mode 100644 src/definitions/fhe.md create mode 100644 src/definitions/mpc.md create mode 100644 src/definitions/witness_encryption.md create mode 100644 src/images/fhe.png diff --git a/src/SUMMARY.md b/src/SUMMARY.md index 155548c..07268f0 100644 --- a/src/SUMMARY.md +++ b/src/SUMMARY.md @@ -17,9 +17,11 @@ - [Evaluation Form](./definitions/evaluation_form.md) - [Fast Fourier Transform](./definitions/fast_fourier_transform.md) - [Fiat-Shamir Transform](./definitions/fiat_shamir.md) +- [Fully Homomorphic Encryption (FHE)](./definitions/fhe.md) - [Instance](./definitions/instance.md) - [Lagrange Interpolation](./definitions/lagrange_interpolation.md) - [Language](./definitions/language.md) +- [Multi-party Computation (MPC)](./definitions/mpc.md) - [Nullifier](./definitions/nullifier.md) - [Oracle](./definitions/oracle.md) - [PLONKish Arithmetization](./definitions/plonkish_arithmetization.md) @@ -45,5 +47,6 @@ - [Trusted Setup](./definitions/trusted_setup.md) - [Vanishing Polynomial](./definitions/vanishing_polynomial.md) - [Witness](./definitions/witness.md) +- [Witness Encryption](./definitions/witness_encryption.md) - [Zero-Knowledge Proof](./definitions/zero_knowledge_proof.md) - [zkVM](./definitions/zkvm.md) \ No newline at end of file diff --git a/src/definitions/fhe.md b/src/definitions/fhe.md new file mode 100644 index 0000000..45c3422 --- /dev/null +++ b/src/definitions/fhe.md @@ -0,0 +1,21 @@ +# Fully Homomorphic Encryption (FHE) + +> An encryption scheme is said to be *fully homomorphic* if it allows to compute additions and multiplications on ciphertexts; decrypting the modified ciphertext reveals the result of applying those additions and multiplications to the original message. + +--- + +Regular encryption allows one party (the sender) to hide a message such that there is only one party (the receiver) that can unhide it. + +Fully homomorphic encryption (FHE) adds the possibility of computing an arbitrary function $F$ on the ciphertext before it gets decrypted. +The decryption will yield $F(\texttt{message})$. + +[![](../images/fhe.png)](../images/fhe.png) + +This process allows to outsource computation to an untrusted party without revealing the input data. +Note however that FHE alone gives no guarantees as to what function $F$ was run. + +# Practicality + +FHE is regarded as the holy grail of encryption and for a long time was thought to be impossible. +Today we are starting to see practical FHE schemes. +Note however that they are orders of magnitude slower than symmetric encryption, and non-FHE public key schemes. \ No newline at end of file diff --git a/src/definitions/mpc.md b/src/definitions/mpc.md new file mode 100644 index 0000000..d6017d8 --- /dev/null +++ b/src/definitions/mpc.md @@ -0,0 +1,30 @@ +# Multi-party Computation (MPC) + +> A protocol that allows mutually distrusting parties, each holding a secret, to jointly compute a function over their secrets without revealing any information other than the result. + +--- + +As indicated in the name, MPC is a protocol between multiple parties. +These parties each hold some data that they want to keep secret. +They also do no trust each other but, for some reason, want to jointly compute a program together. + +An MPC protocol allows these parties to evaluate the function of interest on their joint private inputs without revealing anything other than the result. + +> **The canonical example: Yao's millionaire problem.** +> The example we usually use to illustrate this setting is known as Yao's millionaire problem, named after its inventor Andrew Yao. +> Alice and Bob are two millionaires and want to know who is more rich. +> However they don't want to reveal how much they own to each other, nor to any third party. +> +> The problem can then be extended to allow for more than 2 parties, and any arbitrary program rather than just a comparison of integers. + +## Generic vs Task-specific Protocols +MPC protocols can be divided into two groups: those that are task-specific, and those that are generic. +A generic protocol will allow the set of parties to compute any program they like. +However, making the protocol generic prevents it from being optimized for the specific task at hand. +On the other hand, task-specific protocols can take advantage fo this specialization and can be overall faster or cheaper to run. + +## Constructing MPC +Generic 2-party MPC (2-PC) protocols can be built following the *garbled circuit* approach laid out by Andrew Yao [[Yao86]]((https://ieeexplore.ieee.org/document/4568207)). +For further reading on garbled circuits, we recommend [*A Gentle Introduction to Yao’s Garbled Circuits*](https://web.mit.edu/sonka89/www/papers/2017ygc.pdf) by Sophia Yakoubov. + +Protocols involving more parties will often use techniques such as [zero-knowledge proof](./zero_knowledge_proof.md) and [fully homomorphic encryption](./fhe.md). \ No newline at end of file diff --git a/src/definitions/witness_encryption.md b/src/definitions/witness_encryption.md new file mode 100644 index 0000000..d4f5bee --- /dev/null +++ b/src/definitions/witness_encryption.md @@ -0,0 +1,36 @@ +# Witness Encryption + +> An encryption scheme where public keys are instances (public inputs of a circuits) and private keys are witnesses (the private inputs and intermediate computation steps). + +--- + +Witness encryption is a generalization of public key encryption. + +Recall that for public key encryption, we pick one specific hard problem to generate private-public key pairs. +For example, if we choose the discrete logarithm in [elliptic curves](./elliptic_curve.md) (EC), the private key is a scalar $s$ and the public key is a point $P = sG$. +We assume that finding $s$ given $P$ is hard and use this fact to construct encryption schemes. + +In witness encryption, we allow to encrypt to any $\mathsf{NP}$-problem (see [Decision Problems](../intro_to_zk/what_is_proving.md#decision-problems)). +For example, we could **encrypt to an [arithmetic circuit](./circuit.md)** --- the same ones we use for SNARKs (!). +The **public key would be the public inputs**, while the **private key is the private inputs and all intermediate computation steps**. + +More formally, the public key is any [instance](./instance.md) of an $\mathsf{NP}$ relation, and the private key is a corresponding [witness](./witness.md). +Note that depending on the relation, a single public key (instance) could have multiple private keys (witnesses). +Depending on your use case this could be a nasty bug or a great feature! + +Here is a table summarizing the above discussion: +| Scheme | Hard Problem | Public Key | Private Key | +| ------------------------------ | ------------------------- | -------------- | ------------------------------------- | +| EC public key encryption | EC discrete logarithm | Point $P = sG$ | Scalar $s$ | +| Circuit-based encryption | Circuit satisfiability | public inputs | private inputs and intermediate wires | +| Generic **witness encryption** | any $\mathsf{NP}$-problem | instance $x$ | witness $w$ | + +## Witness Encryption in Practice +**Witness encryption schemes are not yet practical**. +Cryptographers are hard at work to make these schemes usable. + +However, we do have variants that are already available: +- **commitment-based witness encryption (CWE).** In CWE, the public key is an instance $x$ and a commitment $\mathsf{com}$. The corresponding private key is a value $w$ such that $\mathsf{com}$ was a commitment to $w$ and $w$ was a valid witness for $x$. + > **CWE example.** Following our circuit example: the public key would be the public inputs *and* a commitment to the private inputs + intermediate steps. +- **signature-based witness encryption (SWE).** In SWE, the public key is a signature key $\mathsf{pk}$ and a string `s`. The corresponding private key is a signature on `s` by the secret key $\mathsf{sk}$ that corresponds to $\mathsf{pk}$. + > **SWE example.** SWE has been used to build *timelock encryption*, otherwise known as "encrypting to the future" [[DHMW22]](https://eprint.iacr.org/2022/433) [[GMR23]](https://eprint.iacr.org/2023/189). \ No newline at end of file diff --git a/src/images/fhe.png b/src/images/fhe.png new file mode 100644 index 0000000000000000000000000000000000000000..45f7c996932c71131fd578b93209c877939948be GIT binary patch literal 89364 zcmeFZRa}(o8#W39h_r}+Af%bPWcG(%m54-K{7c($Xl>-3{OK z4%Yg9d#!{2KG_HRz!8|4ci!iY>%Q(9Z&_(kJZy4o6ciLZaj}PwQBZL8QBY8^&z}Q- zS$X;d5B!H}^H}r&N^U2`Jox3Ifr_}HloSdh`1w2vs+S20I`k6oj~x6%K|xDEML`4q zqe6d6fc<(FS3d#m*UzZXr=g5f)OA*{&RH2Q5S9`MWr;qX`SM(7~ z?I3(*u4)SXhg*1^N2lly5;~CJA^dmut$$g+@#&FjC6}h#aM)EQdxJ$5Sg~Nu7>dKb zOsxJA!hyg!5&J(aKcNt^ZKbz%|8Cn~AJWig$&y$~guhnr(O(b4ZE0n)n^&C}=FdA)-%^I68FDQPDdGa=5pC$5<13P^!XxH^f#q9 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