From 9853e63a6bae6c47ea79be79eef2cc63e7ed8847 Mon Sep 17 00:00:00 2001 From: Joey Beauvais-Feisthauer Date: Thu, 14 Dec 2023 20:32:55 -0500 Subject: [PATCH] Implement small prime-power fields --- ext/crates/fp/Cargo.toml | 2 + ext/crates/fp/src/field/fp.rs | 128 +++ ext/crates/fp/src/field/largefq.rs | 99 +++ ext/crates/fp/src/field/limb.rs | 128 +-- ext/crates/fp/src/field/mod.rs | 305 +++---- .../fp/src/field/small_conway_polys.txt | 810 ++++++++++++++++++ ext/crates/fp/src/field/smallfq.rs | 248 ++++++ ext/crates/fp/src/prime/mod.rs | 9 +- ext/crates/fp/src/vector/impl_fpvectorp.rs | 2 +- ext/crates/fp/src/vector/impl_slicemutp.rs | 2 +- 10 files changed, 1440 insertions(+), 293 deletions(-) create mode 100644 ext/crates/fp/src/field/fp.rs create mode 100644 ext/crates/fp/src/field/largefq.rs create mode 100644 ext/crates/fp/src/field/small_conway_polys.txt create mode 100644 ext/crates/fp/src/field/smallfq.rs diff --git a/ext/crates/fp/Cargo.toml b/ext/crates/fp/Cargo.toml index e8455e8e9..3aab68b3b 100644 --- a/ext/crates/fp/Cargo.toml +++ b/ext/crates/fp/Cargo.toml @@ -12,8 +12,10 @@ cfg-if = "1.0.0" itertools = { version = "0.10.0", default-features = false } serde = "1.0.0" serde_json = "1.0.0" +dashmap = "5" maybe-rayon = { path = "../maybe-rayon" } +once_cell = "1.19.0" [dev-dependencies] criterion = { version = "0.3.5", features = ["html_reports"] } diff --git a/ext/crates/fp/src/field/fp.rs b/ext/crates/fp/src/field/fp.rs new file mode 100644 index 000000000..05abd8dac --- /dev/null +++ b/ext/crates/fp/src/field/fp.rs @@ -0,0 +1,128 @@ +use crate::{constants::BITS_PER_LIMB, limb::Limb, prime::Prime}; + +use super::{limb::LimbMethods, Field, FieldElement}; + +/// A prime field. This is just a wrapper around a prime. +#[derive(Debug, Copy, Clone, PartialEq, Eq, Hash)] +pub struct Fp

(pub(crate) P); + +impl Field for Fp

{ + type Characteristic = P; + + fn characteristic(self) -> Self::Characteristic { + self.0 + } + + fn degree(self) -> u32 { + 1 + } + + fn zero(self) -> Self::Element { + 0 + } + + fn one(self) -> Self::Element { + 1 + } + + fn add(self, a: Self::Element, b: Self::Element) -> Self::Element { + self.0.sum(a, b) + } + + fn mul(self, a: Self::Element, b: Self::Element) -> Self::Element { + self.0.product(a, b) + } + + fn inv(self, a: Self::Element) -> Option { + if a == 0 { + None + } else { + // By Fermat's little theorem, `a^(p-1) = 1`. Therefore, `a^(-1) = a^(p-2)`. + Some(self.0.pow_mod(a, self.0.as_u32() - 2)) + } + } + + fn neg(self, a: Self::Element) -> Self::Element { + if a > 0 { + self.0.as_u32() - a + } else { + 0 + } + } + + fn frobenius(self, a: Self::Element) -> Self::Element { + a + } +} + +impl LimbMethods for Fp

{ + type Element = u32; + + fn encode(self, element: Self::Element) -> Limb { + element as Limb + } + + fn decode(self, element: Limb) -> Self::Element { + (element % self.0.as_u32() as Limb) as u32 + } + + fn bit_length(self) -> usize { + let p = self.characteristic().as_u32() as u64; + match p { + 2 => 1, + _ => (BITS_PER_LIMB as u32 - (p * (p - 1)).leading_zeros()) as usize, + } + } + + fn fma_limb(self, limb_a: Limb, limb_b: Limb, coeff: Self::Element) -> Limb { + if self.characteristic() == 2 { + limb_a ^ (coeff as Limb * limb_b) + } else { + limb_a + (coeff as Limb) * limb_b + } + } + + /// Contributed by Robert Burklund. + fn reduce(self, limb: Limb) -> Limb { + match self.characteristic().as_u32() { + 2 => limb, + 3 => { + // Set top bit to 1 in every limb + const TOP_BIT: Limb = (!0 / 7) << (2 - BITS_PER_LIMB % 3); + let mut limb_2 = ((limb & TOP_BIT) >> 2) + (limb & (!TOP_BIT)); + let mut limb_3s = limb_2 & (limb_2 >> 1); + limb_3s |= limb_3s << 1; + limb_2 ^= limb_3s; + limb_2 + } + 5 => { + // Set bottom bit to 1 in every limb + const BOTTOM_BIT: Limb = (!0 / 31) >> (BITS_PER_LIMB % 5); + const BOTTOM_TWO_BITS: Limb = BOTTOM_BIT | (BOTTOM_BIT << 1); + const BOTTOM_THREE_BITS: Limb = BOTTOM_BIT | (BOTTOM_TWO_BITS << 1); + let a = (limb >> 2) & BOTTOM_THREE_BITS; + let b = limb & BOTTOM_TWO_BITS; + let m = (BOTTOM_BIT << 3) - a + b; + let mut c = (m >> 3) & BOTTOM_BIT; + c |= c << 1; + let d = m & BOTTOM_THREE_BITS; + d + c - BOTTOM_TWO_BITS + } + _ => self.pack(self.unpack(limb)), + } + } +} + +impl

std::ops::Deref for Fp

{ + type Target = P; + + fn deref(&self) -> &Self::Target { + &self.0 + } +} + +impl FieldElement for u32 { + fn is_zero(&self) -> bool { + *self == 0 + } +} diff --git a/ext/crates/fp/src/field/largefq.rs b/ext/crates/fp/src/field/largefq.rs new file mode 100644 index 000000000..c56644619 --- /dev/null +++ b/ext/crates/fp/src/field/largefq.rs @@ -0,0 +1,99 @@ +use dashmap::DashMap as HashMap; +use once_cell::sync::Lazy; + +use crate::{ + limb::Limb, + matrix::Matrix, + prime::{Prime, ValidPrime}, + vector::inner::FqVectorP, +}; + +use super::{limb::LimbMethods, Field, FieldElement, Fp}; + +static MULT_MATRICES: Lazy> = Lazy::new(HashMap::new); +static FROB_MATRICES: Lazy> = Lazy::new(HashMap::new); + +/// A field of order `q = p^d`, where `q >= 2^16` and `d > 1`. Fields of that size are too large to +/// cache their Zech logarithms, so we represent their elements as polynomials over the prime field. +#[derive(Debug, Copy, Clone, PartialEq, Eq, Hash)] +pub struct LargeFq

{ + p: P, + d: u32, +} + +impl Field for LargeFq

{ + type Characteristic = P; + + fn characteristic(self) -> Self::Characteristic { + self.p + } + + fn degree(self) -> u32 { + self.d + } + + fn zero(self) -> Self::Element { + LargeFqElement(FqVectorP::new(Fp(self.p), self.d as usize)) + } + + fn one(self) -> Self::Element { + todo!() + } + + fn add(self, a: Self::Element, b: Self::Element) -> Self::Element { + todo!() + } + + fn mul(self, a: Self::Element, b: Self::Element) -> Self::Element { + todo!() + } + + fn neg(self, a: Self::Element) -> Self::Element { + todo!() + } + + fn inv(self, a: Self::Element) -> Option { + todo!() + } + + fn frobenius(self, a: Self::Element) -> Self::Element { + todo!() + } +} + +impl LimbMethods for LargeFq

{ + type Element = LargeFqElement

; + + fn encode(self, element: Self::Element) -> Limb { + todo!() + } + + fn decode(self, element: Limb) -> Self::Element { + todo!() + } + + fn bit_length(self) -> usize { + todo!() + } + + fn fma_limb(self, limb_a: Limb, limb_b: Limb, coeff: Self::Element) -> Limb { + todo!() + } + + fn reduce(self, limb: Limb) -> Limb { + limb + } +} + +/// A field element as a polynomial over the prime field. This is used when the order of the +/// field is large, since otherwise caching Zech logarithms uses too much memory. +/// +/// This is backed by an `FpVectorP` consisting of the coefficients of the polynomial. +#[derive(Debug, Clone, PartialEq, Eq, Hash)] +pub struct LargeFqElement(pub(super) FqVectorP>); + +impl FieldElement for LargeFqElement

{ + fn is_zero(&self) -> bool { + self.0.is_zero() + } +} diff --git a/ext/crates/fp/src/field/limb.rs b/ext/crates/fp/src/field/limb.rs index ccc872889..9d914e2c1 100644 --- a/ext/crates/fp/src/field/limb.rs +++ b/ext/crates/fp/src/field/limb.rs @@ -1,15 +1,21 @@ +// According to +// https://doc.rust-lang.org/stable/rustc/lints/listing/warn-by-default.html#private-interfaces: +// +// "Having something private in primary interface guarantees that the item will be unusable from +// outer modules due to type privacy." +// +// In our case, this is a feature. We want to be able to use the `LimbMethods` trait in this crate +// and we also want it to be inaccessible from outside the crate. +#![allow(private_interfaces)] + use std::ops::Range; use crate::{ constants::BITS_PER_LIMB, limb::{Limb, LimbBitIndexPair}, - prime::Prime, }; -use super::{ - element::{FieldElement, MultiplicativeFieldElement, PolynomialFieldElement}, - Field, Fp, LargeFq, SmallFq, -}; +use super::FieldElement; /// Methods that lets us interact with the underlying `Limb` type. /// @@ -123,7 +129,7 @@ pub trait LimbMethods: Clone + Copy + Sized { } } -struct LimbIterator { +pub(crate) struct LimbIterator { fq: F, limb: Limb, bit_length: usize, @@ -142,113 +148,3 @@ impl Iterator for LimbIterator { Some(self.fq.decode(result)) } } - -impl LimbMethods for Fp

{ - type Element = u32; - - fn encode(self, element: Self::Element) -> Limb { - element as Limb - } - - fn decode(self, element: Limb) -> Self::Element { - (element % self.0.as_u32() as Limb) as u32 - } - - fn bit_length(self) -> usize { - let p = self.characteristic().as_u32() as u64; - match p { - 2 => 1, - _ => (BITS_PER_LIMB as u32 - (p * (p - 1)).leading_zeros()) as usize, - } - } - - fn fma_limb(self, limb_a: Limb, limb_b: Limb, coeff: Self::Element) -> Limb { - if self.characteristic() == 2 { - limb_a ^ (coeff as Limb * limb_b) - } else { - limb_a + (coeff as Limb) * limb_b - } - } - - /// Contributed by Robert Burklund. - fn reduce(self, limb: Limb) -> Limb { - match self.characteristic().as_u32() { - 2 => limb, - 3 => { - // Set top bit to 1 in every limb - const TOP_BIT: Limb = (!0 / 7) << (2 - BITS_PER_LIMB % 3); - let mut limb_2 = ((limb & TOP_BIT) >> 2) + (limb & (!TOP_BIT)); - let mut limb_3s = limb_2 & (limb_2 >> 1); - limb_3s |= limb_3s << 1; - limb_2 ^= limb_3s; - limb_2 - } - 5 => { - // Set bottom bit to 1 in every limb - const BOTTOM_BIT: Limb = (!0 / 31) >> (BITS_PER_LIMB % 5); - const BOTTOM_TWO_BITS: Limb = BOTTOM_BIT | (BOTTOM_BIT << 1); - const BOTTOM_THREE_BITS: Limb = BOTTOM_BIT | (BOTTOM_TWO_BITS << 1); - let a = (limb >> 2) & BOTTOM_THREE_BITS; - let b = limb & BOTTOM_TWO_BITS; - let m = (BOTTOM_BIT << 3) - a + b; - let mut c = (m >> 3) & BOTTOM_BIT; - c |= c << 1; - let d = m & BOTTOM_THREE_BITS; - d + c - BOTTOM_TWO_BITS - } - _ => self.pack(self.unpack(limb)), - } - } -} - -impl LimbMethods for SmallFq { - type Element = MultiplicativeFieldElement; - - fn encode(self, element: Self::Element) -> Limb { - element.0.map(|x| (x as Limb) << 1 | 1).unwrap_or(0) - } - - fn decode(self, element: Limb) -> Self::Element { - if element & 1 == 0 { - MultiplicativeFieldElement(None) - } else { - MultiplicativeFieldElement(Some((element >> 1) as u32)) - } - } - - fn bit_length(self) -> usize { - todo!() - } - - fn fma_limb(self, limb_a: Limb, limb_b: Limb, coeff: Self::Element) -> Limb { - todo!() - } - - fn reduce(self, limb: Limb) -> Limb { - todo!() - } -} - -impl LimbMethods for LargeFq { - type Element = PolynomialFieldElement; - - fn encode(self, element: Self::Element) -> Limb { - todo!() - } - - fn decode(self, element: Limb) -> Self::Element { - todo!() - } - - fn bit_length(self) -> usize { - todo!() - } - - fn fma_limb(self, limb_a: Limb, limb_b: Limb, coeff: Self::Element) -> Limb { - todo!() - } - - fn reduce(self, limb: Limb) -> Limb { - todo!() - } -} diff --git a/ext/crates/fp/src/field/mod.rs b/ext/crates/fp/src/field/mod.rs index 95d98ab5a..d789d2e70 100644 --- a/ext/crates/fp/src/field/mod.rs +++ b/ext/crates/fp/src/field/mod.rs @@ -1,199 +1,156 @@ -use crate::prime::{primes::*, Prime}; +use crate::prime::Prime; -use self::{element::MultiplicativeFieldElement, limb::LimbMethods}; +use self::limb::LimbMethods; -pub mod element; pub(crate) mod limb; -pub trait Field: Clone + Copy + LimbMethods + Sized { +pub mod fp; +// pub mod largefq; +pub mod smallfq; + +pub use fp::Fp; +// pub use largefq::LargeFq; +pub use smallfq::SmallFq; + +pub trait FieldElement: Clone { + fn is_zero(&self) -> bool; +} + +pub trait Field: LimbMethods + Sized { type Characteristic: Prime; fn characteristic(self) -> Self::Characteristic; fn degree(self) -> u32; + fn q(self) -> u32 { + self.characteristic().pow(self.degree()) + } + fn zero(self) -> Self::Element; fn one(self) -> Self::Element; fn add(self, a: Self::Element, b: Self::Element) -> Self::Element; - fn sub(self, a: Self::Element, b: Self::Element) -> Self::Element; fn mul(self, a: Self::Element, b: Self::Element) -> Self::Element; - fn div(self, a: Self::Element, b: Self::Element) -> Self::Element; - - fn inv(self, a: Self::Element) -> Self::Element; - fn neg(self, a: Self::Element) -> Self::Element; - - fn frobenius(self, a: Self::Element) -> Self::Element; -} - -/// A prime field. This is just a wrapper around a prime. -#[derive(Debug, Copy, Clone, PartialEq, Eq, Hash)] -pub struct Fp

(pub P); - -/// A field of order `q = p^d`, where `q < 2^16` and `d > 1`. Fields of that size are small enough -/// that we can cache their Zech logarithms. -#[derive(Debug, Copy, Clone, PartialEq, Eq, Hash)] -pub struct SmallFq { - p: ValidPrime, - d: u32, -} - -/// A field of order `q = prime^degree`, where `q >= 2^16` and `d > 1`. Fields of that size are too -/// large to cache their Zech logarithms, so we represent their elements as polynomials over the -/// prime field. -#[derive(Debug, Copy, Clone, PartialEq, Eq, Hash)] -pub struct LargeFq { - p: ValidPrime, - d: u32, -} - -impl SmallFq { - pub fn new(p: ValidPrime, degree: u32) -> Self { - assert!(degree > 0); - assert!(degree <= 16); - Self { p, d: degree } - } -} - -impl Field for Fp

{ - type Characteristic = P; - - fn characteristic(self) -> Self::Characteristic { - self.0 - } - - fn degree(self) -> u32 { - 1 - } - - fn zero(self) -> Self::Element { - 0 - } - - fn one(self) -> Self::Element { - 1 - } - - fn add(self, a: Self::Element, b: Self::Element) -> Self::Element { - ((a as u64 + b as u64) % self.0.as_u32() as u64) as u32 - } fn sub(self, a: Self::Element, b: Self::Element) -> Self::Element { - ((a as u64 - b as u64) % self.0.as_u32() as u64) as u32 + self.add(a, self.neg(b)) } - fn mul(self, a: Self::Element, b: Self::Element) -> Self::Element { - ((a as u64 * b as u64) % self.0.as_u32() as u64) as u32 + fn div(self, a: Self::Element, b: Self::Element) -> Option { + Some(self.mul(a, self.inv(b)?)) } - fn div(self, a: Self::Element, b: Self::Element) -> Self::Element { - self.mul(a, self.inv(b)) - } - - fn inv(self, a: Self::Element) -> Self::Element { - self.0.pow_mod(a, self.0.as_u32() - 2) - } - - fn neg(self, a: Self::Element) -> Self::Element { - self.0.as_u32() - a - } - - fn frobenius(self, a: Self::Element) -> Self::Element { - a - } -} - -impl Field for SmallFq { - type Characteristic = ValidPrime; - - fn characteristic(self) -> Self::Characteristic { - self.p - } - - fn degree(self) -> u32 { - self.d - } - - fn zero(self) -> Self::Element { - MultiplicativeFieldElement(None) - } - - fn one(self) -> Self::Element { - todo!() - } - - fn add(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn sub(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn mul(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn div(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn inv(self, a: Self::Element) -> Self::Element { - todo!() - } - - fn neg(self, a: Self::Element) -> Self::Element { - todo!() - } + fn neg(self, a: Self::Element) -> Self::Element; + fn inv(self, a: Self::Element) -> Option; - fn frobenius(self, a: Self::Element) -> Self::Element { - todo!() - } + fn frobenius(self, a: Self::Element) -> Self::Element; } -impl Field for LargeFq { - type Characteristic = ValidPrime; - - fn characteristic(self) -> Self::Characteristic { - self.p - } - - fn degree(self) -> u32 { - self.d - } - - fn zero(self) -> Self::Element { - todo!() - } - - fn one(self) -> Self::Element { - todo!() - } - - fn add(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn sub(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn mul(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn div(self, a: Self::Element, b: Self::Element) -> Self::Element { - todo!() - } - - fn inv(self, a: Self::Element) -> Self::Element { - todo!() - } - - fn neg(self, a: Self::Element) -> Self::Element { - todo!() - } - - fn frobenius(self, a: Self::Element) -> Self::Element { - todo!() +#[cfg(test)] +mod test { + use crate::prime::{P2, P3}; + + use super::{Field, SmallFq}; + + #[test] + fn test_f_4() { + // Multiplication table generated by Sage. + let f4 = SmallFq::new(P2, 2); + + let mut a = vec![f4.one()]; + for _ in 1..f4.q() { + let prev = a.last().unwrap(); + a.push(f4.mul(*prev, f4.a())); + } + + let expansions = vec![a[0], a[1], f4.add(a[1], a[0]), a[0]]; + + assert_eq!(a, expansions); + } + + #[test] + fn test_f_8() { + // Multiplication table generated by Sage. + let f8 = SmallFq::new(P2, 3); + + let mut a = vec![f8.one()]; + for _ in 1..f8.q() { + let prev = a.last().unwrap(); + a.push(f8.mul(*prev, f8.a())); + } + + let expansions = vec![ + a[0], + a[1], + a[2], + f8.add(a[1], a[0]), + f8.add(a[2], a[1]), + f8.add(a[2], f8.add(a[1], a[0])), + f8.add(a[2], a[0]), + a[0], + ]; + + assert_eq!(a, expansions); + } + + #[test] + fn test_f_16() { + // Multiplication table generated by Sage. + let f16 = SmallFq::new(P2, 4); + + let mut a = vec![f16.one()]; + for _ in 1..f16.q() { + let prev = a.last().unwrap(); + a.push(f16.mul(*prev, f16.a())); + } + + let expansions = vec![ + a[0], + a[1], + a[2], + a[3], + f16.add(a[1], a[0]), + f16.add(a[2], a[1]), + f16.add(a[3], a[2]), + f16.add(a[3], f16.add(a[1], a[0])), + f16.add(a[2], a[0]), + f16.add(a[3], a[1]), + f16.add(a[2], f16.add(a[1], a[0])), + f16.add(a[3], f16.add(a[2], a[1])), + f16.add(a[3], f16.add(a[2], f16.add(a[1], a[0]))), + f16.add(a[3], f16.add(a[2], a[0])), + f16.add(a[3], a[0]), + a[0], + ]; + + assert_eq!(a, expansions); + } + + #[test] + fn test_f_9() { + // Multiplication table generated by Sage. + let f9 = SmallFq::new(P3, 2); + + let two = f9.add(f9.one(), f9.one()); + + let mut a = vec![f9.one()]; + for _ in 1..f9.q() { + let prev = a.last().unwrap(); + a.push(f9.mul(*prev, f9.a())); + } + + let expansions = vec![ + a[0], + a[1], + f9.add(a[1], a[0]), + f9.add(f9.mul(two, a[1]), a[0]), + two, + f9.mul(two, a[1]), + f9.add(f9.mul(two, a[1]), two), + f9.add(a[1], two), + a[0], + ]; + + assert_eq!(a, expansions); } } diff --git a/ext/crates/fp/src/field/small_conway_polys.txt b/ext/crates/fp/src/field/small_conway_polys.txt new file mode 100644 index 000000000..50d343e49 --- /dev/null +++ b/ext/crates/fp/src/field/small_conway_polys.txt @@ -0,0 +1,810 @@ +[[[1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], + [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], + [1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0], + [1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], + [1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0], + [1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], + [1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]], + [[2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [2, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [2, 2, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [2, 2, 2, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], + [1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], + [2, 1, 0, 0, 2, 2, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], + [[2, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [3, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], + [[6, 242, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]] \ No newline at end of file diff --git a/ext/crates/fp/src/field/smallfq.rs b/ext/crates/fp/src/field/smallfq.rs new file mode 100644 index 000000000..b04118bd6 --- /dev/null +++ b/ext/crates/fp/src/field/smallfq.rs @@ -0,0 +1,248 @@ +use std::sync::Arc; + +use dashmap::DashMap as HashMap; +use once_cell::sync::Lazy; + +use crate::{ + constants::BITS_PER_LIMB, + limb::Limb, + prime::{log2, Prime, ValidPrime}, + vector::FpVector, + PRIME_TO_INDEX_MAP, +}; + +use super::{limb::LimbMethods, Field, FieldElement, Fp}; + +static SMALL_CONWAY_POLYS: [[[u32; 17]; 15]; 54] = include!("small_conway_polys.txt"); + +type ZechTable = HashMap; + +/// A table of lazily initialized [Zech logarithms][zech_logs]. +/// +/// Key is the field, value is a fully initialized table of Zech logarithms. +/// +/// [zech_logs]: https://en.wikipedia.org/wiki/Zech%27s_logarithm +static ZECH_LOGS: Lazy>> = Lazy::new(HashMap::new); + +/// Return the Zech logarithm table for the given field. If it does not exist yet, initialize it. +/// The initialization might be fairly expensive (several ms). +fn zech_logs(fq: SmallFq

) -> Arc { + let table = ZECH_LOGS.entry((fq.p.to_dyn(), fq.d)).or_insert_with(|| { + let conway_poly = { + let v = SMALL_CONWAY_POLYS[PRIME_TO_INDEX_MAP[fq.p.as_usize()]][fq.d as usize - 2] + .iter() + .copied() + .take(fq.d as usize + 1) + .collect::>(); + FpVector::from_slice(fq.characteristic(), &v) + }; + let mul_by_a = |cur: FpVector| { + // Shift all entries to the right by one. We're assuming that cur is a polynomial + // representing an element of the field, so the leading coefficient is zero, and there + // is no overflow. + let mut next = FpVector::from_slice( + cur.prime(), + &std::iter::once(0) + .chain(cur.iter()) + .take(cur.len()) + .collect::>(), + ); + let leading_coeff = next.entry(next.len() - 1); + next.add(&conway_poly, Fp(cur.prime()).neg(leading_coeff)); + next + }; + + // Generate a lookup table. For every element represented as a polynomial, we store the + // power of `a` that corresponds to it. + let poly_to_power: HashMap = HashMap::new(); + let mut cur = FpVector::new(fq.characteristic(), conway_poly.len()); + cur.set_entry(0, 1); + poly_to_power.insert(cur.clone(), 0); + + for i in 1..fq.q() - 1 { + cur = mul_by_a(cur); + poly_to_power.insert(cur.clone(), i); + } + + // Loop over all elements again, but now recording logarithms. + let table = HashMap::new(); + table.insert(fq.zero(), fq.one()); + + let mut cur = FpVector::new(fq.characteristic(), conway_poly.len()); + cur.set_entry(0, 1); + for i in 0..fq.q() - 1 { + let cur_plus_1 = { + let mut cur_plus_1 = cur.clone(); + cur_plus_1.add_basis_element(0, 1); + cur_plus_1 + }; + cur = mul_by_a(cur); + + table.insert( + SmallFqElement(Some(i)), + SmallFqElement(poly_to_power.get(&cur_plus_1).as_deref().cloned()), + ); + } + for x in table.iter() { + let (a, b) = x.pair(); + println!("{a} + 1 = {b}"); + } + + Arc::new(table) + }); + Arc::clone(&table) +} + +/// A field of order `q = p^d`, where `q < 2^16` and `d > 1`. Fields of that size are small enough +/// that we can cache their Zech logarithms. +#[derive(Debug, Copy, Clone, PartialEq, Eq, Hash)] +pub struct SmallFq

{ + p: P, + d: u32, +} + +impl SmallFq

{ + pub fn new(p: P, d: u32) -> Self { + assert!(d > 1); + assert!(log2(p.pow(d) as usize) < 16); + Self { p, d } + } + + /// Return the element `-1`. If `p = 2`, this is `a^0 = 1`. Otherwise, it is `a^((q - 1) / 2)`. + pub fn negative_one(self) -> SmallFqElement { + let e = if self.p == 2 { 0 } else { (self.q() - 1) / 2 }; + SmallFqElement(Some(e)) + } + + /// The distinguished primitive element that generates the multiplicative group of the field. + pub fn a(self) -> SmallFqElement { + SmallFqElement(Some(1)) + } +} + +impl std::fmt::Display for SmallFq

{ + fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { + write!(f, "F_({}^{})", self.p, self.d) + } +} + +impl Field for SmallFq

{ + type Characteristic = P; + + fn characteristic(self) -> Self::Characteristic { + self.p + } + + fn degree(self) -> u32 { + self.d + } + + fn zero(self) -> Self::Element { + SmallFqElement(None) + } + + fn one(self) -> Self::Element { + SmallFqElement(Some(0)) + } + + fn add(self, a: Self::Element, b: Self::Element) -> Self::Element { + match (a, b) { + (SmallFqElement(None), b) => b, + (a, SmallFqElement(None)) => a, + (SmallFqElement(Some(a)), SmallFqElement(Some(b))) => { + // a^m + a^n = a^m (1 + a^(n - m)) = a^(m + Zech(n - m)) + let table = zech_logs(self); + let (a, b) = if a >= b { (a, b) } else { (b, a) }; + let zech = table.get(&SmallFqElement(Some(a - b))).unwrap(); + if let Some(zech) = zech.0 { + SmallFqElement(Some((b + zech) % (self.q() - 1))) + } else { + SmallFqElement(None) + } + } + } + } + + fn mul(self, a: Self::Element, b: Self::Element) -> Self::Element { + if let (Some(a), Some(b)) = (a.0, b.0) { + // Both `a` and `b` are less than 2^16, so there is no overflow. + SmallFqElement(Some((a + b) % (self.q() - 1))) + } else { + SmallFqElement(None) + } + } + + fn neg(self, a: Self::Element) -> Self::Element { + self.mul(a, self.negative_one()) + } + + fn inv(self, a: Self::Element) -> Option { + Some(SmallFqElement(Some((self.characteristic() - 1) - a.0?))) + } + + fn frobenius(self, a: Self::Element) -> Self::Element { + SmallFqElement(a.0.map(|x| (x * self.characteristic().as_u32()) % (self.q() - 1))) + } +} + +impl LimbMethods for SmallFq

{ + type Element = SmallFqElement; + + /// This is 2n + 1 if `element` is a^n, and 0 otherwise. + fn encode(self, element: Self::Element) -> Limb { + element.0.map(|x| (x as Limb) << 1 | 1).unwrap_or(0) + } + + fn decode(self, element: Limb) -> Self::Element { + if element & 1 == 0 { + // This only checks that the element is even, but by the definition of `encode`, this + // only happens if the element is zero. + SmallFqElement(None) + } else { + SmallFqElement(Some((element >> 1) as u32)) + } + } + + fn bit_length(self) -> usize { + // A field has q - 1 units, so SmallFqElement is either Some(a) where a is in [0, q - 2], or + // None. We add 1 bit to account for encoding the None case. + BITS_PER_LIMB - (self.q() - 1).leading_zeros() as usize + 1 + } + + fn fma_limb(self, limb_a: Limb, limb_b: Limb, coeff: Self::Element) -> Limb { + let bit_length = self.bit_length(); + let mut result: Limb = 0; + let mut shift = 0; + for (a, b) in self.unpack(limb_a).zip(self.unpack(limb_b)) { + result += self.encode(self.add(a, self.mul(coeff, b))) << shift; + shift += bit_length; + } + result + } + + fn reduce(self, limb: Limb) -> Limb { + limb + } +} + +/// A field element, stored as the exponent of a distinguished generator of the group of units. +/// `None` if the element is zero. +#[derive(Debug, Clone, Copy, PartialEq, Eq, Hash)] +pub struct SmallFqElement(pub(super) Option); + +impl FieldElement for SmallFqElement { + fn is_zero(&self) -> bool { + self.0.is_none() + } +} + +impl std::fmt::Display for SmallFqElement { + fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { + match self.0 { + None => write!(f, "0"), + Some(0) => write!(f, "1"), + Some(1) => write!(f, "a"), + Some(x) => write!(f, "a^{}", x), + } + } +} diff --git a/ext/crates/fp/src/prime/mod.rs b/ext/crates/fp/src/prime/mod.rs index 58623952f..ce2255b45 100644 --- a/ext/crates/fp/src/prime/mod.rs +++ b/ext/crates/fp/src/prime/mod.rs @@ -1,5 +1,5 @@ use std::{ - fmt::Debug, + fmt::{Debug, Display}, ops::{ Add, AddAssign, Div, DivAssign, Mul, MulAssign, Rem, RemAssign, Shl, ShlAssign, Shr, ShrAssign, Sub, SubAssign, @@ -33,7 +33,9 @@ pub const TWO: ValidPrime = ValidPrime::new(2); /// chosen at runtime. pub trait Prime: Debug + + Clone + Copy + + Display + PartialEq + Eq + PartialEq @@ -59,6 +61,11 @@ pub trait Prime: self.as_u32() as usize } + /// Computes the sum mod p. This takes care of overflow. + fn sum(self, n1: u32, n2: u32) -> u32 { + ((n1 as u64) + (n2 as u64) % (self.as_u32() as u64)) as u32 + } + /// Computes the product mod p. This takes care of overflow. fn product(self, n1: u32, n2: u32) -> u32 { ((n1 as u64) * (n2 as u64) % (self.as_u32() as u64)) as u32 diff --git a/ext/crates/fp/src/vector/impl_fpvectorp.rs b/ext/crates/fp/src/vector/impl_fpvectorp.rs index 31b45932a..f004c9e61 100644 --- a/ext/crates/fp/src/vector/impl_fpvectorp.rs +++ b/ext/crates/fp/src/vector/impl_fpvectorp.rs @@ -1,7 +1,7 @@ use itertools::Itertools; use crate::{ - field::{element::FieldElement, Field}, + field::{Field, FieldElement}, limb::{self, Limb}, prime::{Prime, ValidPrime}, simd, diff --git a/ext/crates/fp/src/vector/impl_slicemutp.rs b/ext/crates/fp/src/vector/impl_slicemutp.rs index 334b11e70..28424949a 100644 --- a/ext/crates/fp/src/vector/impl_slicemutp.rs +++ b/ext/crates/fp/src/vector/impl_slicemutp.rs @@ -4,7 +4,7 @@ use itertools::Itertools; use crate::{ constants, - field::{element::FieldElement, Field}, + field::{Field, FieldElement}, limb::Limb, prime::{Prime, ValidPrime}, };