Rename FFT to NTT

draft-irtf-cfrg-vdaf.md

@@ -233,6 +233,15 @@ informative:
     date: 2020
     target: https://web.archive.org/web/20221025174046/https://firefox-source-docs.mozilla.org/toolkit/components/telemetry/collection/origin.html
 
+  SML24:
+    title: "A Complete Beginner Guide to the Number Theoretic Transform (NTT)"
+    target: https://eprint.iacr.org/2024/585
+    author:
+      - ins: A. Satriawan
+      - ins: R. Mareta
+      - ins: H. Lee
+    date: 2024
+
 --- abstract
 
 This document describes Verifiable Distributed Aggregation Functions (VDAFs), a
@@ -1947,22 +1956,22 @@ def vec_neg(vec: list[F]) -> list[F]:
 ~~~
 {: #field-helper-functions title="Common functions for finite fields."}
 
-### FFT-Friendly Fields {#field-fft-friendly}
+### NTT-Friendly Fields {#field-ntt-friendly}
 
 Some VDAFs require fields that are suitable for efficient computation of the
-discrete Fourier transform, as this allows for fast polynomial interpolation.
-(One example is Prio3 ({{prio3}}) when instantiated with the FLP of
-{{flp-bbcggi19-construction}}.) Specifically, a field is said to be
-"FFT-friendly" if, in addition to satisfying the interface described in
+number theoretic transform (NTT) {{SML24}}, as this allows for fast polynomial
+interpolation. (One example is Prio3 ({{prio3}}) when instantiated with the FLP
+of {{flp-bbcggi19-construction}}.) Specifically, a field is said to be
+"NTT-friendly" if, in addition to satisfying the interface described in
 {{field}}, it implements the following method:
 
 * `Field.gen() -> Field` returns the generator of a large subgroup of the
-  multiplicative group. To be FFT-friendly, the order of this subgroup MUST be a
+  multiplicative group. To be NTT-friendly, the order of this subgroup MUST be a
   power of 2. In addition, 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`.
 
-FFT-friendly fields also define the following parameter:
+NTT-friendly fields also define the following parameter:
 
 * `GEN_ORDER: int` is the order of a multiplicative subgroup generated by
   `Field.gen()`.
@@ -2450,7 +2459,7 @@ of proofs.
 ## Construction {#prio3-construction}
 
 This section specifies `Prio3`, an implementation of the `Vdaf` interface
-({{vdaf}}). It has three generic parameters: an `FftField ({{fft-field}}), an
+({{vdaf}}). It has three generic parameters: an `NttField ({{ntt-field}}), an
 `Flp` ({{flp}}) and a `Xof` ({{xof}}). It also has an associated constant,
 `PROOFS`, with a value within the range of `[1, 256)`, denoting the number of
 FLPs generated by the Client ({{multiproofs}}).
@@ -3299,7 +3308,7 @@ fixed points `alpha[0], ..., alpha[M-1]`, other than to require that the points
 are distinct. In this document, the fixed points are chosen so that the gadget
 polynomial can be constructed efficiently using the Cooley-Tukey FFT ("Fast
 Fourier Transform") algorithm. Note that this requires the field to be
-"FFT-friendly" as defined in {{field-fft-friendly}}.
+"NTT-friendly" as defined in {{field-ntt-friendly}}.
 
 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
@@ -3451,10 +3460,6 @@ follows:
 
     * Let `padded_w = w + field.zeros(P_i - len(w))`.
 
-    > NOTE We pad `w` to the nearest power of 2 so that we can use FFT for
-    > interpolating the wire polynomials. Perhaps there is some clever math for
-    > picking `wire_inp` in a way that avoids having to pad.
-
     * Let `poly_wire_i[j-1]` be the lowest degree polynomial for which
       `poly_wire_i[j-1](alpha_i^k) == padded_w[k]` for all `k` in `[P_i]`.
 

poc/tests/test_field.py

@@ -1,7 +1,7 @@
 import unittest
 
-from vdaf_poc.field import (FftField, Field, Field2, Field64, Field96,
-                            Field128, Field255, poly_eval, poly_interp)
+from vdaf_poc.field import (Field, Field2, Field64, Field96, Field128,
+                            Field255, NttField, poly_eval, poly_interp)
 
 
 class TestFields(unittest.TestCase):
@@ -41,20 +41,20 @@ def run_field_test(self, cls: type[Field]) -> None:
             self.assertTrue(cls.decode_from_bit_vector(
                 encoded).as_unsigned() == val)
 
-    def run_fft_field_test(self, cls: type[FftField]) -> None:
+    def run_ntt_field_test(self, cls: type[NttField]) -> None:
         self.run_field_test(cls)
 
         # Test generator.
         self.assertTrue(cls.gen()**cls.GEN_ORDER == cls(1))
 
     def test_field64(self) -> None:
-        self.run_fft_field_test(Field64)
+        self.run_ntt_field_test(Field64)
 
     def test_field96(self) -> None:
-        self.run_fft_field_test(Field96)
+        self.run_ntt_field_test(Field96)
 
     def test_field128(self) -> None:
-        self.run_fft_field_test(Field128)
+        self.run_ntt_field_test(Field128)
 
     def test_field255(self) -> None:
         self.run_field_test(Field255)

poc/tests/test_flp_bbcggi19.py

@@ -1,14 +1,14 @@
 from typing import TypeVar
 
-from vdaf_poc.field import FftField, Field64, Field96, Field128
+from vdaf_poc.field import Field64, Field96, Field128, NttField
 from vdaf_poc.flp_bbcggi19 import (Count, FlpBBCGGI19, Histogram, Mul,
                                    MultihotCountVec, PolyEval, Range2, Sum,
                                    SumOfRangeCheckedInputs, SumVec, Valid)
 from vdaf_poc.test_utils import TestFlpBBCGGI19
 
 Measurement = TypeVar("Measurement")
 AggResult = TypeVar("AggResult")
-F = TypeVar("F", bound=FftField)
+F = TypeVar("F", bound=NttField)
 
 
 class MultiGadget(Valid[int, int, Field64]):
@@ -140,22 +140,22 @@ def test_small(self) -> None:
 
 
 class TestSumVec(TestFlpBBCGGI19):
-    def run_encode_truncate_decode_with_fft_fields_test(
+    def run_encode_truncate_decode_with_ntt_fields_test(
             self,
             measurements: list[list[int]],
             length: int,
             bits: int,
             chunk_length: int) -> None:
         for field in [Field64, Field96, Field128]:
-            sumvec = SumVec[FftField](field, length, bits, chunk_length)
+            sumvec = SumVec[NttField](field, length, bits, chunk_length)
             self.assertEqual(sumvec.field, field)
             self.assertTrue(isinstance(sumvec, SumVec))
             self.run_encode_truncate_decode_test(
                 FlpBBCGGI19(sumvec), measurements)
 
     def test(self) -> None:
         # SumVec with length 2, bits 4, chunk len 1.
-        self.run_encode_truncate_decode_with_fft_fields_test(
+        self.run_encode_truncate_decode_with_ntt_fields_test(
             [[1, 2], [3, 4], [5, 6], [7, 8]],
             2,
             4,

poc/tests/test_vdaf_prio3.py

@@ -2,15 +2,15 @@
 
 from tests.test_flp import FlpTest
 from tests.test_flp_bbcggi19 import TestAverage
-from vdaf_poc.field import FftField, Field64, Field128
+from vdaf_poc.field import Field64, Field128, NttField
 from vdaf_poc.flp_bbcggi19 import FlpBBCGGI19
 from vdaf_poc.test_utils import TestVdaf
 from vdaf_poc.vdaf_prio3 import (Prio3, Prio3Count, Prio3Histogram,
                                  Prio3MultihotCountVec, Prio3Sum, Prio3SumVec,
                                  Prio3SumVecWithMultiproof)
 from vdaf_poc.xof import XofTurboShake128
 
-F = TypeVar("F", bound=FftField)
+F = TypeVar("F", bound=NttField)
 
 
 class Prio3Average(Prio3):

poc/vdaf_poc/field.py

@@ -152,7 +152,13 @@ def as_unsigned(self) -> int:
         return int(self.gf(self.val))
 
 
-class FftField(Field):
+class NttField(Field):
+    """
+    A field that is suitable for use with the NTT ("number theoretic
+    transform") algorithm for efficient polynomial interpolation. Such a field
+    defines a large multiplicative subgroup whose order is a power of 2.
+    """
+
     # Order of the multiplicative group generated by `Field.gen()`.
     GEN_ORDER: int
 
@@ -188,7 +194,7 @@ def conditional_select(self, inp: bytes) -> bytes:
         return bytes(map(lambda x: m & x, inp))
 
 
-class Field64(FftField):
+class Field64(NttField):
     """The finite field GF(2^32 * 4294967295 + 1)."""
 
     MODULUS = 2**32 * 4294967295 + 1
@@ -203,7 +209,7 @@ def gen(cls) -> Self:
         return cls(7)**4294967295
 
 
-class Field96(FftField):
+class Field96(NttField):
     """The finite field GF(2^64 * 4294966555 + 1)."""
 
     MODULUS = 2**64 * 4294966555 + 1
@@ -218,7 +224,7 @@ def gen(cls) -> Self:
         return cls(3)**4294966555
 
 
-class Field128(FftField):
+class Field128(NttField):
     """The finite field GF(2^66 * 4611686018427387897 + 1)."""
 
     MODULUS = 2**66 * 4611686018427387897 + 1

poc/vdaf_poc/flp_bbcggi19.py

@@ -5,13 +5,13 @@
 from typing import Any, Generic, Optional, TypeVar, cast
 
 from vdaf_poc.common import front, next_power_of_2
-from vdaf_poc.field import (FftField, poly_eval, poly_interp, poly_mul,
+from vdaf_poc.field import (NttField, poly_eval, poly_interp, poly_mul,
                             poly_strip)
 from vdaf_poc.flp import Flp
 
 Measurement = TypeVar("Measurement")
 AggResult = TypeVar("AggResult")
-F = TypeVar("F", bound=FftField)
+F = TypeVar("F", bound=NttField)
 
 
 class Gadget(Generic[F], metaclass=ABCMeta):
@@ -52,10 +52,10 @@ class Valid(Generic[Measurement, AggResult, F], metaclass=ABCMeta):
     Generic type parameters:
     Measurement -- the measurement type
     AggResult -- the aggregate result type
-    Field -- An FFT-friendly field
+    Field -- An NTT-friendly field
 
     Attributes:
-    field -- Class object for the FFT-friendly field.
+    field -- Class object for the NTT-friendly field.
     MEAS_LEN -- Length of the encoded measurement input to the validity
         circuit.
     JOINT_RAND_LEN -- Length of the random input of the validity circuit.
@@ -320,9 +320,7 @@ def prove(self, meas: list[F], prove_rand: list[F], joint_rand: list[F]) -> list
             # Compute the wire polynomials for this gadget.
             #
             # NOTE We pad the wire inputs to the nearest power of 2 so that we
-            # can use FFT for interpolating the wire polynomials. Perhaps there
-            # is some clever math for picking `wire_inp` in a way that avoids
-            # having to pad.
+            # can use NTT for interpolating the wire polynomials.
             assert self.field.GEN_ORDER % P == 0
             alpha = self.field.gen() ** (self.field.GEN_ORDER // P)
             wire_inp = [alpha ** k for k in range(P)]

poc/vdaf_poc/test_utils.py

@@ -5,7 +5,7 @@
 
 from vdaf_poc.common import (gen_rand, next_power_of_2, print_wrapped_line,
                              to_le_bytes)
-from vdaf_poc.field import FftField, poly_eval
+from vdaf_poc.field import NttField, poly_eval
 from vdaf_poc.flp import Flp, run_flp
 from vdaf_poc.flp_bbcggi19 import FlpBBCGGI19, Gadget
 from vdaf_poc.vdaf import Vdaf, run_vdaf
@@ -20,7 +20,7 @@
 PrepState = TypeVar("PrepState")
 PrepShare = TypeVar("PrepShare")
 PrepMessage = TypeVar("PrepMessage")
-F = TypeVar("F", bound=FftField)
+F = TypeVar("F", bound=NttField)
 
 
 def test_vec_gen_rand(length: int) -> bytes:

poc/vdaf_poc/vdaf_prio3.py

@@ -5,7 +5,7 @@
 
 from vdaf_poc import flp_bbcggi19
 from vdaf_poc.common import byte, concat, front, vec_add, vec_sub, zeros
-from vdaf_poc.field import FftField, Field64, Field128
+from vdaf_poc.field import Field64, Field128, NttField
 from vdaf_poc.flp import Flp
 from vdaf_poc.vdaf import Vdaf
 from vdaf_poc.xof import Xof, XofTurboShake128
@@ -20,7 +20,7 @@
 
 Measurement = TypeVar("Measurement")
 AggResult = TypeVar("AggResult")
-F = TypeVar("F", bound=FftField)
+F = TypeVar("F", bound=NttField)
 
 Prio3InputShare: TypeAlias = \
     tuple[  # leader input share