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Dmitry Vagner 2023-01-19 00:25:40 +07:00
parent 23698b7474
commit d6167a630d
1 changed files with 13 additions and 5 deletions
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18
evm/src/bn254_arithmetic.rs
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@ -1,4 +1,5 @@
use std::{ops::{Add, Div, Mul, Neg, Sub}, mem::transmute};
use std::mem::transmute;
use std::ops::{Add, Div, Mul, Neg, Sub};
use ethereum_types::U256;
use itertools::Itertools;
@ -247,7 +248,10 @@ impl Mul for Fp6 {
/// (x_1 * x_2 * x_3 * x_4 * x_5) / phi
/// Since (x_n)_m = x_{n+m}, we save compute by rearranging the numerator:
/// (x_1 * x_3) * x_5 * (x_1 * x_3)_1
/// By Galois theory, both the following are in Fp2 and are complex conjugates
/// x_1 * x_3 * x_5, x_0 * x_2 * x_4
/// Thus phi = norm(x_1 * x_3 * x_5), and hence the inverse is given by
/// normalize((x_1 * x_3) * x_5) * (x_1 * x_3)_1
impl Div for Fp6 {
type Output = Self;
@ -315,20 +319,24 @@ impl Mul for Fp12 {
/// By Galois Theory, for x: Fp12, the product
/// phi = Prod_{i=0}^11 x_i
/// lands in Fp, and hence the inverse of x (= x_0) is given by
/// lands in Fp, and hence the inverse of x is given by
/// (Prod_{i=1}^11 x_i) / phi
/// We note that the 6th Frobenius map gives the Fp12 conjugate:
/// x_6 = (a + bz)_6 = a + b(z^(p^6)) = a - bz
/// 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_13579b, prod_02468a
/// Thus phi = norm(prod_13579b), and hence the inverse is given by
/// conj_fp12(x) * normalize([(prod_17) * (prod_17)_2] * (prod_17)_4) * [(prod_17) * (prod_17)_2]_1
///
/// Note that in the variable names below, we use a and b to denote 10 and 11
impl Div for Fp12 {
type Output = Self;
fn div(self, rhs: Self) -> Self::Output {
let prod_17 = (frob_fp12(1, rhs) * frob_fp12(7, rhs)).z0;
let prod_1379= prod_17 * frob_fp6(2, prod_17);
let prod_1379 = prod_17 * frob_fp6(2, prod_17);
let prod_13579b = (prod_1379 * frob_fp6(4, prod_17)).t0;
let prod_odds_over_phi = normalize_fp2(prod_13579b);
let prod_248a = frob_fp6(1, prod_1379);