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Dmitry Vagner 2023-01-19 00:56:18 +07:00
parent d6167a630d
commit 54676487e1
1 changed files with 9 additions and 11 deletions
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20
evm/src/bn254_arithmetic.rs
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@ -321,26 +321,24 @@ impl Mul for Fp12 {
/// 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
/// We note that the 6th Frobenius map gives the Fp12 conjugate:
/// x_6 = (a + bz)_6 = a + b(z^(p^6)) = a - bz
/// The 6th Frob map is nontrivial but leaves Fp6 fixed and hence must be the conjugate:
/// x_6 = (a + bz)_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
/// prod_odds, prod_evens
/// Thus phi = norm(prod_odds), and hence the inverse is given by
/// normalize(prod_odds) * prod_evens_except_six * conj_fp12(x)
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_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);
let prod_penultimate = mul_fp2_fp6(prod_odds_over_phi, prod_248a);
let prod_odds = (prod_1379 * frob_fp6(4, prod_17)).t0;
let prod_odds_over_phi = normalize_fp2(prod_odds);
let prod_evens_except_six = frob_fp6(1, prod_1379);
let prod_penultimate = mul_fp2_fp6(prod_odds_over_phi, prod_evens_except_six);
let inv = mul_fp6_fp12(prod_penultimate, conj_fp12(rhs));
self * inv
}