mirror of
https://github.com/logos-storage/plonky2.git
synced 2026-01-09 09:13:09 +00:00
694 lines
16 KiB
Rust
694 lines
16 KiB
Rust
use std::ops::{Add, Div, Mul, Neg, Sub};
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use ethereum_types::U512;
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use rand::distributions::{Distribution, Standard};
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use rand::Rng;
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pub trait FieldExt:
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Sized
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+ Copy
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+ std::ops::Add<Output = Self>
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+ std::ops::Neg<Output = Self>
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+ std::ops::Sub<Output = Self>
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+ std::ops::Mul<Output = Self>
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+ std::ops::Div<Output = Self>
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{
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const ZERO: Self;
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const UNIT: Self;
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fn inv(self) -> Self;
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}
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pub const BLS_BASE: U512 = U512([
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0xb9feffffffffaaab,
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0x1eabfffeb153ffff,
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0x6730d2a0f6b0f624,
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0x64774b84f38512bf,
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0x4b1ba7b6434bacd7,
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0x1a0111ea397fe69a,
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0x0,
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0x0,
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]);
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#[derive(Debug, Copy, Clone, PartialEq)]
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pub struct Fp381 {
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pub val: U512,
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}
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impl Fp381 {
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pub fn new(val: usize) -> Fp381 {
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Fp381 {
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val: U512::from(val),
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}
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}
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}
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impl Distribution<Fp381> for Standard {
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fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Fp381 {
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let xs = rng.gen::<[u64; 8]>();
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Fp381 {
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val: U512(xs) % BLS_BASE,
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}
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}
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}
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impl Add for Fp381 {
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type Output = Self;
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fn add(self, other: Self) -> Self {
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Fp381 {
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val: (self.val + other.val) % BLS_BASE,
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}
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}
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}
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impl Neg for Fp381 {
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type Output = Self;
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fn neg(self) -> Self::Output {
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Fp381 {
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val: (BLS_BASE - self.val) % BLS_BASE,
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}
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}
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}
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impl Sub for Fp381 {
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type Output = Self;
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fn sub(self, other: Self) -> Self {
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Fp381 {
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val: (BLS_BASE + self.val - other.val) % BLS_BASE,
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}
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}
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}
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impl Fp381 {
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fn lsh_128(self) -> Fp381 {
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let b128: U512 = U512([0, 0, 1, 0, 0, 0, 0, 0]);
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// since BLS_BASE < 2^384, multiplying by 2^128 doesn't overflow the U512
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Fp381 {
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val: self.val.saturating_mul(b128) % BLS_BASE,
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}
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}
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fn lsh_256(self) -> Fp381 {
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self.lsh_128().lsh_128()
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}
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fn lsh_512(self) -> Fp381 {
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self.lsh_256().lsh_256()
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}
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}
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#[allow(clippy::suspicious_arithmetic_impl)]
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impl Mul for Fp381 {
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type Output = Self;
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fn mul(self, other: Self) -> Self {
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// x1, y1 are at most ((q-1) // 2^256) < 2^125
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let x0 = U512(self.val.0[..4].try_into().unwrap());
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let x1 = U512(self.val.0[4..].try_into().unwrap());
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let y0 = U512(other.val.0[..4].try_into().unwrap());
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let y1 = U512(other.val.0[4..].try_into().unwrap());
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let z00 = Fp381 {
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val: x0.saturating_mul(y0) % BLS_BASE,
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};
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let z01 = Fp381 {
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val: x0.saturating_mul(y1),
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};
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let z10 = Fp381 {
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val: x1.saturating_mul(y0),
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};
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let z11 = Fp381 {
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val: x1.saturating_mul(y1),
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};
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z00 + (z01 + z10).lsh_256() + z11.lsh_512()
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}
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}
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impl FieldExt for Fp381 {
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const ZERO: Self = Fp381 { val: U512::zero() };
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const UNIT: Self = Fp381 { val: U512::one() };
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fn inv(self) -> Fp381 {
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exp_fp(self, BLS_BASE - 2)
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}
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}
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#[allow(clippy::suspicious_arithmetic_impl)]
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impl Div for Fp381 {
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type Output = Self;
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fn div(self, rhs: Self) -> Self::Output {
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self * rhs.inv()
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}
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}
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fn exp_fp(x: Fp381, e: U512) -> Fp381 {
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let mut current = x;
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let mut product = Fp381 { val: U512::one() };
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for j in 0..512 {
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if e.bit(j) {
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product = product * current;
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}
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current = current * current;
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}
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product
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}
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/// The degree 2 field extension Fp2 is given by adjoining i, the square root of -1, to Fp
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/// The arithmetic in this extension is standard complex arithmetic
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#[derive(Debug, Copy, Clone, PartialEq)]
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pub struct Fp2<T>
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where
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T: FieldExt,
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{
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pub re: T,
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pub im: T,
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}
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impl<T> Distribution<Fp2<T>> for Standard
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where
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T: Distribution<T> + FieldExt,
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Standard: Distribution<T>,
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{
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fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Fp2<T> {
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let (re, im) = rng.gen::<(T, T)>();
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Fp2 { re, im }
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}
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}
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impl<T: FieldExt> Add for Fp2<T> {
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type Output = Self;
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fn add(self, other: Self) -> Self {
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Fp2 {
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re: self.re + other.re,
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im: self.im + other.im,
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}
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}
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}
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impl<T: FieldExt> Neg for Fp2<T> {
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type Output = Self;
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fn neg(self) -> Self::Output {
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Fp2 {
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re: -self.re,
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im: -self.im,
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}
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}
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}
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impl<T: FieldExt> Sub for Fp2<T> {
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type Output = Self;
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fn sub(self, other: Self) -> Self {
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Fp2 {
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re: self.re - other.re,
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im: self.im - other.im,
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}
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}
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}
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impl<T: FieldExt> Mul for Fp2<T> {
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type Output = Self;
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fn mul(self, other: Self) -> Self {
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Fp2 {
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re: self.re * other.re - self.im * other.im,
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im: self.re * other.im + self.im * other.re,
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}
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}
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}
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impl<T: FieldExt> Fp2<T> {
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/// This function scalar multiplies an Fp2 by an Fp
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pub fn scale(self, x: T) -> Self {
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Fp2 {
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re: x * self.re,
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im: x * self.im,
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}
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}
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/// Return the complex conjugate z' of z: Fp2
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/// This also happens to be the frobenius map
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/// z -> z^p
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/// since p == 3 mod 4 and hence
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/// i^p = i^(4k) * i^3 = 1*(-i) = -i
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fn conj(self) -> Self {
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Fp2 {
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re: self.re,
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im: -self.im,
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}
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}
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// Return the magnitude squared of a complex number
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fn norm_sq(self) -> T {
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self.re * self.re + self.im * self.im
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}
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}
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impl<T: FieldExt> FieldExt for Fp2<T> {
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const ZERO: Fp2<T> = Fp2 {
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re: T::ZERO,
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im: T::ZERO,
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};
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const UNIT: Fp2<T> = Fp2 {
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re: T::UNIT,
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im: T::ZERO,
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};
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/// The inverse of z is given by z'/||z||^2 since ||z||^2 = zz'
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fn inv(self) -> Fp2<T> {
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let norm_sq = self.norm_sq();
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self.conj().scale(norm_sq.inv())
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}
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}
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#[allow(clippy::suspicious_arithmetic_impl)]
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impl<T: FieldExt> Div for Fp2<T> {
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type Output = Self;
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fn div(self, rhs: Self) -> Self::Output {
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self * rhs.inv()
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}
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}
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pub trait Adj {
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fn mul_adj(self) -> Self;
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}
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/// Helper function which multiplies by the Fp2 element
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/// whose cube root we will adjoin in the next extension
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impl Adj for Fp2<Fp381> {
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fn mul_adj(self) -> Self {
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Fp2 {
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re: self.re - self.im,
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im: self.re + self.im,
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}
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}
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}
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/// The degree 3 field extension Fp6 over Fp2 is given by adjoining t, where t^3 = 1 + i
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/// Fp6 has basis 1, t, t^2 over Fp2
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#[derive(Debug, Copy, Clone, PartialEq)]
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pub struct Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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pub t0: Fp2<T>,
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pub t1: Fp2<T>,
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pub t2: Fp2<T>,
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}
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impl<T> Distribution<Fp6<T>> for Standard
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where
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T: Distribution<T> + FieldExt,
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Fp2<T>: Adj,
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Standard: Distribution<T>,
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{
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fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Fp6<T> {
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let (t0, t1, t2) = rng.gen::<(Fp2<T>, Fp2<T>, Fp2<T>)>();
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Fp6 { t0, t1, t2 }
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}
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}
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impl<T> Add for Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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type Output = Self;
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fn add(self, other: Self) -> Self {
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Fp6 {
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t0: self.t0 + other.t0,
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t1: self.t1 + other.t1,
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t2: self.t2 + other.t2,
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}
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}
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}
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impl<T> Neg for Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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type Output = Self;
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fn neg(self) -> Self::Output {
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Fp6 {
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t0: -self.t0,
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t1: -self.t1,
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t2: -self.t2,
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}
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}
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}
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impl<T> Sub for Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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type Output = Self;
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fn sub(self, other: Self) -> Self {
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Fp6 {
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t0: self.t0 - other.t0,
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t1: self.t1 - other.t1,
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t2: self.t2 - other.t2,
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}
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}
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}
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impl<T> Mul for Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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type Output = Self;
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fn mul(self, other: Self) -> Self {
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Fp6 {
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t0: self.t0 * other.t0 + (self.t1 * other.t2 + self.t2 * other.t1).mul_adj(),
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t1: self.t0 * other.t1 + self.t1 * other.t0 + (self.t2 * other.t2).mul_adj(),
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t2: self.t0 * other.t2 + self.t1 * other.t1 + self.t2 * other.t0,
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}
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}
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}
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impl<T> Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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// This function scalar multiplies an Fp6 by an Fp2
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fn scale(self, x: Fp2<T>) -> Fp6<T> {
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Fp6 {
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t0: x * self.t0,
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t1: x * self.t1,
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t2: x * self.t2,
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}
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}
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}
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impl<T> Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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/// This function multiplies an Fp6 element by t, and hence shifts the bases,
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/// where the t^2 coefficient picks up a factor of 1+i as the 1 coefficient of the output
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fn sh(self) -> Fp6<T> {
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Fp6 {
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t0: self.t2.mul_adj(),
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t1: self.t0,
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t2: self.t1,
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}
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}
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}
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impl<T> Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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const FROB_T: [[Fp2<T>; 6]; 2] = [[Fp2::<T>::ZERO; 6]; 2];
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const FROB_Z: [Fp2<T>; 12] = [Fp2::<T>::ZERO; 12];
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}
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impl<T> Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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/// The nth frobenius endomorphism of a p^q field is given by mapping
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/// x to x^(p^n)
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/// which sends a + bt + ct^2: Fp6 to
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/// a^(p^n) + b^(p^n) * t^(p^n) + c^(p^n) * t^(2p^n)
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/// The Fp2 coefficients are determined by the comment in the conj method,
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/// while the values of
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/// t^(p^n) and t^(2p^n)
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/// are precomputed in the constant arrays FROB_T1 and FROB_T2
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pub fn frob(self, n: usize) -> Fp6<T> {
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let n = n % 6;
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let frob_t1 = Fp6::<T>::FROB_T[0][n];
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let frob_t2 = Fp6::<T>::FROB_T[1][n];
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if n % 2 != 0 {
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Fp6 {
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t0: self.t0.conj(),
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t1: frob_t1 * self.t1.conj(),
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t2: frob_t2 * self.t2.conj(),
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}
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} else {
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Fp6 {
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t0: self.t0,
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t1: frob_t1 * self.t1,
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t2: frob_t2 * self.t2,
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}
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}
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}
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}
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impl<T> FieldExt for Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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const ZERO: Fp6<T> = Fp6 {
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t0: Fp2::<T>::ZERO,
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t1: Fp2::<T>::ZERO,
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t2: Fp2::<T>::ZERO,
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};
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const UNIT: Fp6<T> = Fp6 {
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t0: Fp2::<T>::UNIT,
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t1: Fp2::<T>::ZERO,
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t2: Fp2::<T>::ZERO,
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};
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/// Let x_n = x^(p^n) and note that
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/// x_0 = x^(p^0) = x^1 = x
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/// (x_n)_m = (x^(p^n))^(p^m) = x^(p^n * p^m) = x^(p^(n+m)) = x_{n+m}
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/// By Galois Theory, given x: Fp6, the product
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/// phi = x_0 * x_1 * x_2 * x_3 * x_4 * x_5
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/// lands in Fp, and hence the inverse of x is given by
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/// (x_1 * x_2 * x_3 * x_4 * x_5) / phi
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/// We can save compute by rearranging the numerator:
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/// (x_1 * x_3) * x_5 * (x_1 * x_3)_1
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/// By Galois theory, the following are in Fp2 and are complex conjugates
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/// x_1 * x_3 * x_5, x_0 * x_2 * x_4
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/// and therefore
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/// phi = ||x_1 * x_3 * x_5||^2
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/// and hence the inverse is given by
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/// ([x_1 * x_3] * x_5) * [x_1 * x_3]_1 / ||[x_1 * x_3] * x_5||^2
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fn inv(self) -> Fp6<T> {
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let prod_13 = self.frob(1) * self.frob(3);
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let prod_135 = (prod_13 * self.frob(5)).t0;
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let phi = prod_135.norm_sq();
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let prod_odds_over_phi = prod_135.scale(phi.inv());
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let prod_24 = prod_13.frob(1);
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prod_24.scale(prod_odds_over_phi)
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}
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}
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#[allow(clippy::suspicious_arithmetic_impl)]
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impl<T> Div for Fp6<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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type Output = Self;
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fn div(self, rhs: Self) -> Self::Output {
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self * rhs.inv()
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}
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}
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/// The degree 2 field extension Fp12 over Fp6 is given by adjoining z, where z^2 = t.
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/// It thus has basis 1, z over Fp6
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#[derive(Debug, Copy, Clone, PartialEq)]
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pub struct Fp12<T>
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where
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T: FieldExt,
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Fp2<T>: Adj,
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{
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pub z0: Fp6<T>,
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pub z1: Fp6<T>,
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}
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impl<T> FieldExt for Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
const ZERO: Fp12<T> = Fp12 {
|
|
z0: Fp6::<T>::ZERO,
|
|
z1: Fp6::<T>::ZERO,
|
|
};
|
|
|
|
const UNIT: Fp12<T> = Fp12 {
|
|
z0: Fp6::<T>::UNIT,
|
|
z1: Fp6::<T>::ZERO,
|
|
};
|
|
|
|
/// By Galois Theory, given x: Fp12, the product
|
|
/// phi = Prod_{i=0}^11 x_i
|
|
/// lands in Fp, and hence the inverse of x is given by
|
|
/// (Prod_{i=1}^11 x_i) / phi
|
|
/// The 6th Frob map is nontrivial but leaves Fp6 fixed and hence must be the conjugate:
|
|
/// x_6 = (a + bz)_6 = a - bz = x.conj()
|
|
/// Letting prod_17 = x_1 * x_7, the remaining factors in the numerator can be expresed as:
|
|
/// [(prod_17) * (prod_17)_2] * (prod_17)_4 * [(prod_17) * (prod_17)_2]_1
|
|
/// By Galois theory, both the following are in Fp2 and are complex conjugates
|
|
/// prod_odds, prod_evens
|
|
/// Thus phi = ||prod_odds||^2, and hence the inverse is given by
|
|
/// prod_odds * prod_evens_except_six * x.conj() / ||prod_odds||^2
|
|
fn inv(self) -> Fp12<T> {
|
|
let prod_17 = (self.frob(1) * self.frob(7)).z0;
|
|
let prod_1379 = prod_17 * prod_17.frob(2);
|
|
let prod_odds = (prod_1379 * prod_17.frob(4)).t0;
|
|
let phi = prod_odds.norm_sq();
|
|
let prod_odds_over_phi = prod_odds.scale(phi.inv());
|
|
let prod_evens_except_six = prod_1379.frob(1);
|
|
let prod_except_six = prod_evens_except_six.scale(prod_odds_over_phi);
|
|
self.conj().scale(prod_except_six)
|
|
}
|
|
}
|
|
|
|
impl<T> Distribution<Fp12<T>> for Standard
|
|
where
|
|
T: Distribution<T> + FieldExt,
|
|
Fp2<T>: Adj,
|
|
Standard: Distribution<T>,
|
|
{
|
|
fn sample<R: Rng + ?Sized>(&self, rng: &mut R) -> Fp12<T> {
|
|
let (z0, z1) = rng.gen::<(Fp6<T>, Fp6<T>)>();
|
|
Fp12 { z0, z1 }
|
|
}
|
|
}
|
|
|
|
impl<T> Add for Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
type Output = Self;
|
|
|
|
fn add(self, other: Self) -> Self {
|
|
Fp12 {
|
|
z0: self.z0 + other.z0,
|
|
z1: self.z1 + other.z1,
|
|
}
|
|
}
|
|
}
|
|
|
|
impl<T> Neg for Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
type Output = Self;
|
|
|
|
fn neg(self) -> Self::Output {
|
|
Fp12 {
|
|
z0: -self.z0,
|
|
z1: -self.z1,
|
|
}
|
|
}
|
|
}
|
|
|
|
impl<T> Sub for Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
type Output = Self;
|
|
|
|
fn sub(self, other: Self) -> Self {
|
|
Fp12 {
|
|
z0: self.z0 - other.z0,
|
|
z1: self.z1 - other.z1,
|
|
}
|
|
}
|
|
}
|
|
|
|
impl<T> Mul for Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
type Output = Self;
|
|
|
|
fn mul(self, other: Self) -> Self {
|
|
let h0 = self.z0 * other.z0;
|
|
let h1 = self.z1 * other.z1;
|
|
let h01 = (self.z0 + self.z1) * (other.z0 + other.z1);
|
|
Fp12 {
|
|
z0: h0 + h1.sh(),
|
|
z1: h01 - (h0 + h1),
|
|
}
|
|
}
|
|
}
|
|
|
|
impl<T> Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
// This function scalar multiplies an Fp12 by an Fp6
|
|
fn scale(self, x: Fp6<T>) -> Fp12<T> {
|
|
Fp12 {
|
|
z0: x * self.z0,
|
|
z1: x * self.z1,
|
|
}
|
|
}
|
|
|
|
fn conj(self) -> Fp12<T> {
|
|
Fp12 {
|
|
z0: self.z0,
|
|
z1: -self.z1,
|
|
}
|
|
}
|
|
}
|
|
|
|
impl<T> Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
/// The nth frobenius endomorphism of a p^q field is given by mapping
|
|
/// x to x^(p^n)
|
|
/// which sends a + bz: Fp12 to
|
|
/// a^(p^n) + b^(p^n) * z^(p^n)
|
|
/// where the values of z^(p^n) are precomputed in the constant array FROB_Z
|
|
pub fn frob(self, n: usize) -> Fp12<T> {
|
|
let n = n % 12;
|
|
Fp12 {
|
|
z0: self.z0.frob(n),
|
|
z1: self.z1.frob(n).scale(Fp6::<T>::FROB_Z[n]),
|
|
}
|
|
}
|
|
}
|
|
|
|
#[allow(clippy::suspicious_arithmetic_impl)]
|
|
impl<T> Div for Fp12<T>
|
|
where
|
|
T: FieldExt,
|
|
Fp2<T>: Adj,
|
|
{
|
|
type Output = Self;
|
|
|
|
fn div(self, rhs: Self) -> Self::Output {
|
|
self * rhs.inv()
|
|
}
|
|
}
|
|
|
|
// trait Stack {
|
|
// fn on_stack(self) -> Vec<U256>;
|
|
// }
|