Performing reconstruction of the codebase.
This commit is contained in:
Sean Bowe
parent
3591426d44
commit
699e72ca7f
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@ -1,6 +1,6 @@
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[package]
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name = "bn"
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version = "0.1.0"
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version = "0.2.0"
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authors = ["Sean Bowe <ewillbefull@gmail.com>"]
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description = "Pairing cryptography with the Barreto-Naehrig curve"
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keywords = ["pairing","crypto","cryptography"]
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@ -9,6 +9,8 @@ homepage = "https://github.com/zcash/bn"
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repository = "https://github.com/zcash/bn"
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license = "MIT OR Apache-2.0"
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[[bench]]
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name = "fp"
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[dependencies]
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num = "0.1.32"
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rand = "0.3.14"
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@ -0,0 +1,113 @@
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// TODO: later move all these benchmarks to within the code itself.
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#![feature(test)]
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extern crate test;
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extern crate rand;
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extern crate bn;
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use bn::*;
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#[bench]
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fn fr_multiplication(b: &mut test::Bencher) {
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const SAMPLES: usize = 1000;
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let rng = &mut rand::thread_rng();
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let v1: Vec<_> = (0..SAMPLES).map(|_| Fr::random(rng)).collect();
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let v2: Vec<_> = (0..SAMPLES).map(|_| Fr::random(rng)).collect();
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let mut ctr = 0;
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b.iter(|| {
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ctr += 1;
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v1[ctr % SAMPLES] * v2[ctr % SAMPLES]
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});
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}
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#[bench]
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fn fr_inverses(b: &mut test::Bencher) {
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const SAMPLES: usize = 1000;
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let rng = &mut rand::thread_rng();
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let v1: Vec<_> = (0..SAMPLES).map(|_| Fr::random(rng)).collect();
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let mut ctr = 0;
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b.iter(|| {
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ctr += 1;
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v1[ctr % SAMPLES].inverse()
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});
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}
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#[bench]
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fn fq_multiplication(b: &mut test::Bencher) {
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const SAMPLES: usize = 1000;
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let rng = &mut rand::thread_rng();
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let v1: Vec<_> = (0..SAMPLES).map(|_| Fq::random(rng)).collect();
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let v2: Vec<_> = (0..SAMPLES).map(|_| Fq::random(rng)).collect();
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let mut ctr = 0;
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b.iter(|| {
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ctr += 1;
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v1[ctr % SAMPLES] * v2[ctr % SAMPLES]
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});
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}
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#[bench]
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fn fq_inverses(b: &mut test::Bencher) {
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const SAMPLES: usize = 1000;
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let rng = &mut rand::thread_rng();
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let v1: Vec<_> = (0..SAMPLES).map(|_| Fq::random(rng)).collect();
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let mut ctr = 0;
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b.iter(|| {
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ctr += 1;
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v1[ctr % SAMPLES].inverse()
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});
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}
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#[bench]
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fn fq2_multiplication(b: &mut test::Bencher) {
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const SAMPLES: usize = 1000;
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let rng = &mut rand::thread_rng();
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let v1: Vec<_> = (0..SAMPLES).map(|_| Fq2::random(rng)).collect();
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let v2: Vec<_> = (0..SAMPLES).map(|_| Fq2::random(rng)).collect();
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let mut ctr = 0;
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b.iter(|| {
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ctr += 1;
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v1[ctr % SAMPLES] * v2[ctr % SAMPLES]
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});
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}
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#[bench]
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fn fq2_inverses(b: &mut test::Bencher) {
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const SAMPLES: usize = 1000;
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let rng = &mut rand::thread_rng();
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let v1: Vec<_> = (0..SAMPLES).map(|_| Fq2::random(rng)).collect();
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let mut ctr = 0;
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b.iter(|| {
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ctr += 1;
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v1[ctr % SAMPLES].inverse()
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});
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}
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@ -0,0 +1,167 @@
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use rand::Rng;
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use num::{BigUint, Num};
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use std::ops::{Mul,Add,Sub,Neg};
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use std::cmp::{PartialEq, Eq};
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use std::convert::From;
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use std::fmt;
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use std::marker::PhantomData;
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use super::Field;
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pub trait PrimeFieldParams {
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fn modulus() -> BigUint;
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fn bits() -> usize;
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fn name() -> &'static str;
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}
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pub struct Fp<P: PrimeFieldParams> {
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value: BigUint,
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_marker: PhantomData<P>
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}
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impl<P: PrimeFieldParams> fmt::Debug for Fp<P> {
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fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
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write!(f, "{}({})", P::name(), self.value)
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}
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}
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impl<P: PrimeFieldParams> Field for Fp<P> {
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fn zero() -> Self {
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use num::Zero;
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Fp {
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value: BigUint::zero(),
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_marker: PhantomData
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}
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}
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fn one() -> Self {
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use num::One;
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Fp {
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value: BigUint::one(),
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_marker: PhantomData
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}
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}
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fn random<R: Rng>(rng: &mut R) -> Self {
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use num::num_bigint::RandBigInt;
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use num::Zero;
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Fp {
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value: rng.gen_biguint_range(&BigUint::zero(), &P::modulus()),
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_marker: PhantomData
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}
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}
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fn inverse(&self) -> Self {
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if self.is_zero() {
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// TODO: this should likely bleed through the abstraction layers
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panic!("cannot get the multiplicative inverse of zero")
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} else {
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let mut res = Self::one();
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let mut found_one = false;
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let exp = Self::zero() - Self::one() - Self::one();
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for i in (0..P::bits()).rev() {
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if found_one {
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res = res.squared();
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}
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if exp.test_bit(i) {
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found_one = true;
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res = self * &res;
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}
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}
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res
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}
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}
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fn neg(&self) -> Self {
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use num::Zero;
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Fp {
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value: if self.value.is_zero() {
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self.value.clone()
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} else {
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P::modulus() - &self.value
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},
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_marker: PhantomData
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}
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}
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fn mul(&self, other: &Self) -> Self {
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Fp {
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value: (&self.value * &other.value) % &P::modulus(),
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_marker: PhantomData
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}
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}
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fn sub(&self, other: &Self) -> Self {
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if other.value > self.value {
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Fp {
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value: (&self.value + P::modulus()) - &other.value,
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_marker: PhantomData
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}
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} else {
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Fp {
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value: &self.value - &other.value,
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_marker: PhantomData
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}
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}
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}
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fn add(&self, other: &Self) -> Self {
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let tmp = &self.value + &other.value;
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if tmp >= P::modulus() {
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Fp {
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value: tmp - P::modulus(),
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_marker: PhantomData
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}
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} else {
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Fp {
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value: tmp,
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_marker: PhantomData
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}
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}
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}
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fn eq(&self, other: &Self) -> bool {
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self.value == other.value
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}
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}
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impl<P: PrimeFieldParams> Fp<P> {
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pub fn test_bit(&self, bit: usize) -> bool {
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// TODO: This is a naive approach.
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use num::{One, Zero};
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let mut b = BigUint::one();
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let two = &b + &b;
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for _ in 0..bit {
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b = &b + &b;
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}
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(&self.value / b) % two != BigUint::zero()
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}
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}
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impl<'a, P: PrimeFieldParams> From<&'a str> for Fp<P> {
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fn from(s: &'a str) -> Self {
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Fp {
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value: BigUint::from_str_radix(s, 10).unwrap() % P::modulus(),
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_marker: PhantomData
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}
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}
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}
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impl<P: PrimeFieldParams> Clone for Fp<P> {
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fn clone(&self) -> Self {
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Fp {
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value: self.value.clone(),
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_marker: PhantomData
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}
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}
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}
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forward_ops_to_field_ops!(impl(P: PrimeFieldParams) Fp<P>);
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@ -0,0 +1,54 @@
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#[macro_use]
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mod macros;
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pub mod fp;
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pub mod fp2;
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pub mod fp6;
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pub mod fp12;
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#[cfg(test)]
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pub mod tests;
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use rand::Rng;
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use self::fp::{Fp, PrimeFieldParams};
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use std::fmt::Debug;
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pub trait Field: Sized + Clone + Debug {
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fn zero() -> Self;
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fn one() -> Self;
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fn random<R: Rng>(rng: &mut R) -> Self;
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fn is_zero(&self) -> bool {
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self.eq(&Self::zero())
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}
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fn inverse(&self) -> Self;
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fn squared(&self) -> Self {
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self.mul(self)
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}
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fn pow<P: PrimeFieldParams>(&self, exp: &Fp<P>) -> Self {
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let mut res = Self::one();
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let mut found_one = false;
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for i in (0..P::bits()).rev() {
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if found_one {
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res = res.squared();
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}
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if exp.test_bit(i) {
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found_one = true;
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res = self.mul(&res);
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}
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}
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res
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}
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fn eq(&self, other: &Self) -> bool;
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fn ne(&self, other: &Self) -> bool {
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!self.eq(other)
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}
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fn neg(&self) -> Self;
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fn mul(&self, other: &Self) -> Self;
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fn sub(&self, other: &Self) -> Self;
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fn add(&self, other: &Self) -> Self;
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}
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@ -0,0 +1,203 @@
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use rand::{Rng,SeedableRng,StdRng};
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use fields::Field;
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mod large_field {
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use fields::fp::*;
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use num::{BigUint, Num};
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struct Large;
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impl PrimeFieldParams for Large {
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fn modulus() -> BigUint {
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BigUint::from_str_radix("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10).unwrap()
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}
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fn bits() -> usize { 254 }
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fn name() -> &'static str { "Large" }
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}
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type Ft = Fp<Large>;
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#[test]
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fn bit_testing() {
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let a = Ft::from("13");
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assert!(a.test_bit(0) == true);
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assert!(a.test_bit(1) == false);
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assert!(a.test_bit(2) == true);
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assert!(a.test_bit(3) == true);
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let expected: Vec<bool> = [1,1,0,1,1,0,0,0,0,1,0,0,1,1,1,0,0,0,0,0,1,1,0,0,1,0,0,1,1]
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.iter().map(|a| *a == 1).rev().collect();
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let a = Ft::from("453624211");
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for (i, b) in expected.into_iter().enumerate() {
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assert!(a.test_bit(i) == b);
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}
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let expected: Vec<bool> = [1,1,1,1,0,1,0,1,1,0,1,0,0,0,1,1,1,0,1,1,1,1,0,0,0,0,1,1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,1,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,1,1,1,0,1,1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0,0,0,1,1,0,0,1,1,1,1,0,1,1,1,1,0,0,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0,1,1,0,1,1,0,0,1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,1,1,1,1,1,0,0,1,1,1,1,1,1,0,1,0,0,1,0,1,1,0,0]
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.iter().map(|a| *a == 1).rev().collect();
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let a = Ft::from("13888242871869275222244405745257275088696211157297823662689037894645226208556");
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for (i, b) in expected.into_iter().enumerate() {
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assert!(a.test_bit(i) == b);
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}
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}
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}
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mod small_field {
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use fields::fp::*;
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use fields::Field;
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use num::{BigUint, Num};
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struct Small;
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impl PrimeFieldParams for Small {
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fn modulus() -> BigUint {
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BigUint::from_str_radix("13", 10).unwrap()
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}
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fn bits() -> usize { 6 }
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fn name() -> &'static str { "Small" }
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}
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type Ft = Fp<Small>;
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#[test]
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fn field_ops() {
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fn test_field_operation<C: Fn(&Ft, &Ft) -> Ft>(a: u64, b: u64, f: C, expected: u64) {
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let af = Ft::from(format!("{}", a).as_ref());
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let bf = Ft::from(format!("{}", b).as_ref());
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let expectedf = Ft::from(format!("{}", expected).as_ref());
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let res = f(&af, &bf);
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if res != expectedf {
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panic!("res={:?} != expectedf={:?} (a={}, b={}, expected={})", res, expectedf, a, b, expected);
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}
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}
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const MODULO: u64 = 13;
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for a in 0..13u64 {
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for b in 0..13u64 {
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test_field_operation(a, b, |a,b| {a * b}, (a*b)%MODULO);
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test_field_operation(a, b, |a,b| {a + b}, (a+b)%MODULO);
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test_field_operation(a, b, |a,b| {a - b}, {
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let mut tmp = (a as i64) - (b as i64);
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if tmp < 0 {
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tmp += MODULO as i64;
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}
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tmp as u64
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});
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test_field_operation(a, b, |a,b| {a.pow(b)}, (a.pow(b as u32))%MODULO);
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}
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test_field_operation(a, 0, |a,_| {-a}, if a == 0 { 0 } else { MODULO - a });
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if a > 0 {
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test_field_operation(a, 0, |a,_| {&a.inverse() * a}, 1);
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}
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}
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}
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}
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fn can_invert<F: Field>() {
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let mut a = F::one();
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for _ in 0..1000 {
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assert!(a.ne(&F::zero()));
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let inv = a.inverse();
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assert!(a.mul(&inv).eq(&F::one()));
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a = a.add(&F::one());
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}
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}
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fn rand_element_squaring<F: Field, R: Rng>(rng: &mut R) {
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for _ in 0..100 {
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let a = F::random(rng);
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let mul = a.mul(&a);
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let sq = a.squared();
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assert!(sq.eq(&mul));
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}
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let mut cur = F::zero();
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for _ in 0..100 {
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let mul = cur.mul(&cur);
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let sq = cur.squared();
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||||
assert!(sq.eq(&mul));
|
||||
|
||||
cur = cur.add(&F::one());
|
||||
}
|
||||
}
|
||||
|
||||
fn rand_element_addition_and_negation<F: Field, R: Rng>(rng: &mut R) {
|
||||
for _ in 0..10 {
|
||||
let mut a = F::random(rng);
|
||||
let r = F::random(rng);
|
||||
let mut b = a.add(&r);
|
||||
|
||||
for _ in 0..10 {
|
||||
let r = F::random(rng);
|
||||
a = a.add(&r);
|
||||
b = b.add(&r);
|
||||
|
||||
let r = F::random(rng);
|
||||
a = a.sub(&r);
|
||||
b = b.sub(&r);
|
||||
|
||||
let r = F::random(rng);
|
||||
a = a.add(&r);
|
||||
b = b.add(&r);
|
||||
}
|
||||
|
||||
b = b.sub(&r);
|
||||
assert!(a.eq(&b));
|
||||
}
|
||||
}
|
||||
|
||||
fn rand_element_inverse<F: Field, R: Rng>(rng: &mut R) {
|
||||
for _ in 0..100 {
|
||||
let mut n = F::random(rng);
|
||||
n = n.inverse().mul(&n);
|
||||
assert!(n.eq(&F::one()));
|
||||
}
|
||||
for _ in 0..100 {
|
||||
let a = F::random(rng);
|
||||
let b = F::random(rng);
|
||||
assert!(a.mul(&b).mul(&a.inverse()).eq(&b));
|
||||
}
|
||||
}
|
||||
|
||||
fn rand_element_multiplication<F: Field, R: Rng>(rng: &mut R) {
|
||||
// If field is not associative under multiplication, 1/8 of all triplets a, b, c
|
||||
// will fail the test (a*b)*c = a*(b*c).
|
||||
|
||||
for _ in 0..250 {
|
||||
let a = F::random(rng);
|
||||
let b = F::random(rng);
|
||||
let c = F::random(rng);
|
||||
|
||||
assert!(a.mul(&b).mul(&c).eq(&b.mul(&c).mul(&a)));
|
||||
}
|
||||
}
|
||||
|
||||
pub fn field_trials<F: Field>() {
|
||||
can_invert::<F>();
|
||||
|
||||
let seed: [usize; 4] = [103245, 191922, 1293, 192103];
|
||||
let mut rng = StdRng::from_seed(&seed);
|
||||
|
||||
rand_element_squaring::<F, StdRng>(&mut rng);
|
||||
rand_element_addition_and_negation::<F, StdRng>(&mut rng);
|
||||
rand_element_multiplication::<F, StdRng>(&mut rng);
|
||||
rand_element_inverse::<F, StdRng>(&mut rng);
|
||||
}
|
||||
|
||||
|
||||
|
|
@ -0,0 +1,406 @@
|
|||
extern crate num;
|
||||
extern crate rand;
|
||||
|
||||
mod fields;
|
||||
mod params;
|
||||
|
||||
mod groups;
|
||||
|
||||
pub use groups::Gt;
|
||||
pub use fields::fp::Fp;
|
||||
pub use fields::fp2::Fp2;
|
||||
pub use fields::fp6::Fp6;
|
||||
pub use fields::fp12::Fp12;
|
||||
pub use params::{FrParams,FqParams,Fq2Params,G1Params,G2Params,Fq6Params,Fq12Params,ate_loop_count,final_exponent_z};
|
||||
pub use fields::Field;
|
||||
pub use groups::{Jacobian};
|
||||
use groups::{EllCoeffs,Affine};
|
||||
use fields::fp::PrimeFieldParams;
|
||||
|
||||
pub type Fr = Fp<FrParams>;
|
||||
pub type Fq = Fp<FqParams>;
|
||||
pub type Fq2 = Fp2<Fq2Params>;
|
||||
pub type Fq6 = Fp6<Fq6Params>;
|
||||
pub type Fq12 = Fp12<Fq12Params>;
|
||||
|
||||
pub type Scalar = Fr;
|
||||
pub type G1 = Jacobian<G1Params>;
|
||||
pub type G2 = Jacobian<G2Params>;
|
||||
|
||||
#[derive(PartialEq,Eq)]
|
||||
struct G2Precomp {
|
||||
q: Affine<G2Params>,
|
||||
coeffs: Vec<EllCoeffs>
|
||||
}
|
||||
|
||||
impl G2Precomp {
|
||||
fn new(from: &G2) -> G2Precomp {
|
||||
let g2 = from.to_affine();
|
||||
|
||||
let mut r = g2.to_jacobian();
|
||||
|
||||
let mut result = G2Precomp {
|
||||
q: g2.clone(),
|
||||
coeffs: Vec::new()
|
||||
};
|
||||
|
||||
let two_inv = Fq::from("2").inverse();
|
||||
|
||||
let mut found_one = false;
|
||||
let ate = ate_loop_count();
|
||||
for i in (0..FrParams::bits()).rev() {
|
||||
let bit = ate.test_bit(i);
|
||||
|
||||
if !found_one {
|
||||
found_one = bit;
|
||||
continue;
|
||||
}
|
||||
|
||||
result.coeffs.push(r.doubling_step_for_flipped_miller_loop(&two_inv));
|
||||
|
||||
if bit {
|
||||
result.coeffs.push(r.mixed_addition_step_for_flipped_miller_loop(&g2));
|
||||
}
|
||||
}
|
||||
|
||||
let q1 = g2.mul_by_q();
|
||||
let q2 = q1.mul_by_q().neg();
|
||||
|
||||
result.coeffs.push(r.mixed_addition_step_for_flipped_miller_loop(&q1));
|
||||
result.coeffs.push(r.mixed_addition_step_for_flipped_miller_loop(&q2));
|
||||
|
||||
result
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_prepared_g2() {
|
||||
let g2 = G2::one() * Fr::from("20390255904278144451778773028944684152769293537511418234311120800877067946");
|
||||
|
||||
let g2_p = G2Precomp::new(&g2);
|
||||
|
||||
let expected_g2_p = G2Precomp {
|
||||
q: G2::new(
|
||||
Fq2::new(
|
||||
Fq::from("13936578204263895229092967414825041569724079982537616431212348844388899776640"),
|
||||
Fq::from("6372636725635053773371996212293600406870925440022386078671828127711809436031")
|
||||
),
|
||||
Fq2::new(
|
||||
Fq::from("19293035970010898452454381709939058714495082898872914526540420247178075881697"),
|
||||
Fq::from("13822349107533275437553410197128434338071760794202849712402800746887549494974"),
|
||||
),
|
||||
Fq2::new(
|
||||
Fq::from("1"),
|
||||
Fq::from("0"),
|
||||
)
|
||||
).unwrap().to_affine(),
|
||||
coeffs: vec![
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("2627043964130257285960335798481049684473505235282434252362906013168360965985"), Fq::from("14787221188041042838526170263203159226721816513593198738527261794343476005673")), ell_vw: Fq2::new(Fq::from("5190413803656753539584048070636432748402456516849818272297235294934300653772"), Fq::from("16131787528611999569385991096257681501249100726189947900572474295515353427218")), ell_vv: Fq2::new(Fq::from("11284217811624345285836951206193951701052480344824103057273798441033858118424"), Fq::from("6392116536365389562188363539617434898082427085812013656312597845585909267447")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("827617134098165717451808940080463277390770457691666780560712143809003953598"), Fq::from("6776229088211374446530917353066321938640163548858035832637439231634790575465")), ell_vw: Fq2::new(Fq::from("987776078024262725561041258416387561158070255475504730561661362421251696401"), Fq::from("15312963471998242334683179861466148222641884112991952428739813077336923189144")), ell_vv: Fq2::new(Fq::from("2813988028633040066320201189843971639620433430176492766961373503539074898364"), Fq::from("17055167212030988864288747645634552775658830115224987200613220012554982651578")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17093477194591041266397380404112224367919571835678040889250664706433487493043"), Fq::from("8643160753646550179100165428401120938543691942215161396502118430923535820500")), ell_vw: Fq2::new(Fq::from("10933962898922943120024964690690003920835888964592980369975615298196656183772"), Fq::from("10054435552662455933652209211749007190981947542684285471884164039871580864235")), ell_vv: Fq2::new(Fq::from("1475406195060586931258578851599932495282912871465827155842007043242102046659"), Fq::from("5049830924161187876168845800328902567850325970867534538120475909389784841990")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("10434276065336457961840942984114655645641355797054067910262735814196303889171"), Fq::from("13878009844584745291193244072670264656592436467816383615606044714978763054890")), ell_vw: Fq2::new(Fq::from("20993505060568459085534388708128029516059470166174503494622391683433030357130"), Fq::from("10192769806017258272908841309051731389378033999625803563624881649401237928876")), ell_vv: Fq2::new(Fq::from("17377091829483421118284926147077357677507183290871250621761583281560333718550"), Fq::from("18297780639039540893260206902113850213293978821982237672423968999898103045256")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("5064242330961655837810472366173802140673816516016739528934082190598759216129"), Fq::from("20172028727708469864987113767379594545870343866584998839699240724281670882460")), ell_vw: Fq2::new(Fq::from("2996734698689069564052128450078797792056313730363946953355442668966438200951"), Fq::from("9910941594900797404370917094355311738210439203708671856236954583027355115302")), ell_vv: Fq2::new(Fq::from("6899792008443933570863502403567742315007264221354931185809349393350618040985"), Fq::from("18692424201235977456836009406318755884773471215777102117879530727096807383696")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9406196439119227017197200697795682187217614676635965071601916278141238949664"), Fq::from("1156397052245190909533108891675012415349316096091404012699655652677336827483")), ell_vw: Fq2::new(Fq::from("9249275523859165912800910211783530541095037634552893633043938333737241478198"), Fq::from("13674086724537885208439774394455021050268654286498757861381516198880504360984")), ell_vv: Fq2::new(Fq::from("5760486636714317173624828119078349011897522431614835257025421978626292083787"), Fq::from("19195594701877562770296730558597966521343920474874926443609709880727520577918")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12166638768862632131967787101186130559865354321458713212316891357299196109517"), Fq::from("1517854601414954363921968422436855204960697222979994177630673862982254831193")), ell_vw: Fq2::new(Fq::from("13366104107979832799705928625277388583673352228059815944738106504080223007902"), Fq::from("21648874073566751720910466086303831459661520354253552089758049591613143727506")), ell_vv: Fq2::new(Fq::from("3488776451414334656597670851583995239177248056342375163461521452312663358741"), Fq::from("20333414955874213710027365371143864644434939642550352414403887754960702108637")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20468104499950674906991599591024893438007901046236752159815194628487603192867"), Fq::from("19069685413250623899891139234721987960230148622397590130148098858949057649761")), ell_vw: Fq2::new(Fq::from("2458162095258922865086779772179745380154704066745856173311580123496018896409"), Fq::from("14594824564551859025266692529845827840804005950672166057231487026855067744741")), ell_vv: Fq2::new(Fq::from("9689655654286496198845571066967567466951511863880400774464590553022163229348"), Fq::from("10870127206120874488753221889350215535903993427218589071224774335669836789433")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15334990572515495832170458963151379997993357347608954164027751164770129172212"), Fq::from("13576640273125978757652414940234881692674718901710647917847939367156260249620")), ell_vw: Fq2::new(Fq::from("17441399177222030197667035179458353890777336233312342937564354849059149975908"), Fq::from("17007488955907409465543686050546106251553784366237996525801373957978511005278")), ell_vv: Fq2::new(Fq::from("11907792503811346855438657769533223907267837220830333730410940125854558637383"), Fq::from("19490060942462197243379937108210110931787757078893580540054690416131017931738")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17315779415820661602591970083599829709759601467527578787441772726289774228451"), Fq::from("12080802446892182171725856453369856169039939341293608060423916935382017714476")), ell_vw: Fq2::new(Fq::from("12912774602739952889552025090054804346758888115608888766725054474682313950190"), Fq::from("18204955971588024842905918028829132053489100674596042021111813303048407250440")), ell_vv: Fq2::new(Fq::from("15721987682182104464109212111163313043868345920774975073102123159159010623181"), Fq::from("12041652399509652544977783565303273845883245418094457529808044120827920548972")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19999070717670362479390656255660288732744476951677182489244240784338456791622"), Fq::from("3906002755238658676658974095984902840159437088441472633765497396881255783772")), ell_vw: Fq2::new(Fq::from("4998301920940599524538129746790252820575786197710804708434384224972978627969"), Fq::from("13740819604543965458992763357705315276969968429480411530007322182403523242226")), ell_vv: Fq2::new(Fq::from("12357083524385039711813704948190809675818592911528835736620903716727226601953"), Fq::from("13949012942996636079549854439740332990496218971788334041233648192605019948979")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19805154539184543843735550309592886829271314363908537964547876413114433734624"), Fq::from("16481416862400234230001935040590314552205484054708993091993487440981092053599")), ell_vw: Fq2::new(Fq::from("2850933690997557614680452825756662353227291804228932023648526608593113870536"), Fq::from("4786423507519240726599235257249783193682886531662683554713943115802063984530")), ell_vv: Fq2::new(Fq::from("1023533030117941985730558522296044345509721849960856436490999820443312747347"), Fq::from("16431548648882316869128893343209930754508425074299553782883149795599907406302")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17203075478724975791441470425150871365957382183648220405350599958349478966969"), Fq::from("19566853320706390536799238411444563555185156853468371794865496424286288424312")), ell_vw: Fq2::new(Fq::from("15310908404682220539815850064952575682898427316407655213706791187971849703724"), Fq::from("16518878079785062271298378680376701241016697782308466806532677867880353948389")), ell_vv: Fq2::new(Fq::from("14461573034674366293217641735721285869692106578982161360427147953320122006893"), Fq::from("6682535969691272372531048932989583075363217464304983455197465867679730319705")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20895553497690200905762157235991552162755238828234592755315911339400881373107"), Fq::from("11947919061415103173593352950044808416347417810515010536678086292686107222044")), ell_vw: Fq2::new(Fq::from("6277981470671734560229149673222400771590408919719217743512900463425397921908"), Fq::from("857212248193599195410711267373529614085739023676782457331058533755337294689")), ell_vv: Fq2::new(Fq::from("13885916052404740835214284721162700704805447448922857617672445286139626987191"), Fq::from("19585389042100086858633036225209396447924441966778662250950999864988284098182")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12748401205319976481200663793287165427761341490879316890008250070038707706903"), Fq::from("9347025175062646190962067344106887221752685406563421042329129538495481178145")), ell_vw: Fq2::new(Fq::from("12819473645128401077507234830331609583133696659610406468927174080626676619197"), Fq::from("9039389340404634934394726812078086316924048399159604642340339967187032834011")), ell_vv: Fq2::new(Fq::from("3576972655652865483567741810151575278761470911419381093571973994483543095285"), Fq::from("15017468236028111406921228193688973260631093975412858847563400987354453569177")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15407570760857043664675198351546637734249070940748070513828892036078049928562"), Fq::from("16864837887933576789187120028615006826049916557295792205508706545038103706967")), ell_vw: Fq2::new(Fq::from("10721360465919602144653520112889689786555051462198759164582568961994647563952"), Fq::from("15294101887395462362997873520785213441488786364828649600395509678143054744021")), ell_vv: Fq2::new(Fq::from("338877711237691732068166553384070140091720428235937017002925264628733553275"), Fq::from("8565770517833678396194963042166078986348581926120249815567974171840711179000")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17302401852599488824262121640328139426135883695023154134865582322858078206108"), Fq::from("17762524466294762819341377108361075499043287383142581974755577861093917129883")), ell_vw: Fq2::new(Fq::from("13015933066581331586057031655765394156438563571733887223640331819027233831808"), Fq::from("758435331870656231667722761984400968773108978847112431665229252181053186864")), ell_vv: Fq2::new(Fq::from("1969703955244816291771524536928710541897907643982046151796341732075930805350"), Fq::from("7031491051442716548063495812915693048061021502544985435130728834982309403017")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("3256431897334799395000264533777543923057110731812322608781781329228902421476"), Fq::from("12181683857561584905762347955319259913388176042023651776593530168946356566719")), ell_vw: Fq2::new(Fq::from("796109596241493969908761241268545233387446802589790513588935996990189190052"), Fq::from("1881011435205668659920588004752042238985141500261993259857887774967416952583")), ell_vv: Fq2::new(Fq::from("8939978626202472531630769965940628317376245215184468794030552730424109408120"), Fq::from("6876296539554825957959953448623382464100292373640362636397165101276891176372")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9977776755404508431321449793657366203354342168643747704006143818738023991417"), Fq::from("17287266532802977377832339048078093713694407321084566908200498605546793435820")), ell_vw: Fq2::new(Fq::from("13828304001931491210449888619926027655755205497065268858009259221646932328015"), Fq::from("15160507616608365341639459841002920786281000303114113782821091199951204875277")), ell_vv: Fq2::new(Fq::from("11161349646944933007552237093278324942110971730799870343491465916610232191523"), Fq::from("1406353163795290818414111649956135144227974028417320470897284179379367096127")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7051926909684750402992796647032741235565177570654429988875094870973533133596"), Fq::from("20900584116079647608136401104271956963096697793208792824067775068966626057321")), ell_vw: Fq2::new(Fq::from("16845876804097000415961425779514720605919619149076212452284059758659650969766"), Fq::from("20806461795944565303534619813753388058736049320841472653130287521152836019826")), ell_vv: Fq2::new(Fq::from("14700937697004489523945953560825193972493636932341330040072362595384517591325"), Fq::from("17137955424118909932502168320776219042080643724609838664247554484115994974522")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("11997142079850902731018491830701934107296522353910410046841604519639199995406"), Fq::from("7846127392296015970710836702134527490160588060582987110380879354952036676012")), ell_vw: Fq2::new(Fq::from("17663414534578185845724108244425572237371794484114570058559740097895281553299"), Fq::from("5029140669755495687983399191536061323171057684732073778423303267496666431971")), ell_vv: Fq2::new(Fq::from("18428442704115060890708987869503078260482829714024061977805185634781850796860"), Fq::from("10997089106320530533400287751108615584976629751702164406696900094998369084403")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13049677819639682137107272322562642308503411465986079606334189062279441432203"), Fq::from("8874624972677935217588093490885486721673594121072678233714072789659128555367")), ell_vw: Fq2::new(Fq::from("17057994029556488440657870420250293132874472170733557198308184169150330844407"), Fq::from("21633949388482190843211557537195287403106536528012825991981214057356654097223")), ell_vv: Fq2::new(Fq::from("6897044784874980042628318562068623805084522888972873438474139199226581764756"), Fq::from("2692625602037608101594849144393949134766066123857076012132350588113519149572")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9195356099870032305655798667367006686897880523242566329014149856866611060003"), Fq::from("8052880750949660720559492348565167987253311417692073778040482170195175946428")), ell_vw: Fq2::new(Fq::from("14126308928695836758114496610102001680995565158909643521688860213965125059872"), Fq::from("671075569038912473223544948981506373517307914345286644404839238190008163502")), ell_vv: Fq2::new(Fq::from("16828103633974871724641544770525140436938333764966095252502003672344759889143"), Fq::from("11294529501746593948507624028584472676389760897716457892342249916318954501723")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21124500310722447577044753724306337899524073055454053171030128408285122310058"), Fq::from("6266274689105996173186441194758121417834460925587732781640173398761521329055")), ell_vw: Fq2::new(Fq::from("21576331134605070996263178912402854747994495442070419140713471256922277734125"), Fq::from("10074747339717657657921727946300324710951646905879034242434327125816600264379")), ell_vv: Fq2::new(Fq::from("21765519811276677250031796638148457542841530645055777846258335421409598000534"), Fq::from("21113224637771003677145667454854210585751245642107143223999187554888833454243")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("3972300323627467772646468544189830113523911494046612234673787196935654523692"), Fq::from("4678843490808368914310764558657240771944434980659329233409943752329617161210")), ell_vw: Fq2::new(Fq::from("17273599953138772488805612764323870175561796764345770950040891840763740358904"), Fq::from("20415406312940209525918367703821373463952436739154646433333601347381723237136")), ell_vv: Fq2::new(Fq::from("9160987523541118769404859219229455872777126347530590315596894278463599930633"), Fq::from("9054614112960826755739076858442558227570393423663838494067070216861352843202")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9994954547022379603539679290217274617475455164007187108524364097751858323148"), Fq::from("20207771379440433457553782484784849880846158790602128386495229840158065235194")), ell_vw: Fq2::new(Fq::from("4256518944443059194982602310083745017699415852751800911480635583088649250060"), Fq::from("15878038811235111024146051545935959211923223574364333740058604258589674044428")), ell_vv: Fq2::new(Fq::from("15953919860849853924955813198899405295748673040941543212644580663271517228860"), Fq::from("19594437991878043589671466710739298733201896072505590020656735731744884281871")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("1333543700232406379832215757617068657339748039338978081735192495239856176346"), Fq::from("13583045269132775344737980756385257860562307101313068058661941588822340692712")), ell_vw: Fq2::new(Fq::from("11316045827391404343818336116749364562466461792138476673985290066961360847098"), Fq::from("6683447190424452097068562814518005185732181765401062334661636064120276671717")), ell_vv: Fq2::new(Fq::from("21483741541431414142042872578822487244163622589552506298253453995156009765340"), Fq::from("16502358172728062550923345759488126380665850742066670591562640787191536290296")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4727679401549029925466046980042210063407620223804057104870816459356196798234"), Fq::from("4245208253099058296636688301996648265313738760450861302514451065904449375848")), ell_vw: Fq2::new(Fq::from("19589426516750929424306774984752060788246605243046082989256140076730177865962"), Fq::from("19202599366770569221216501935740579180019961851730854119509268985533547631061")), ell_vv: Fq2::new(Fq::from("18329650333197074546830734977650617140322812110454751288122555715911145072440"), Fq::from("18685828888894718887886440900247948352873037333999376257561751598045361934940")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15348453159719894937627693446890053385899104841918429189733580400902437025934"), Fq::from("7501190780205720975723852235835733848687299193764526538831345783610137314719")), ell_vw: Fq2::new(Fq::from("12341649147852680664345690534373468938440543264264387581538764313149640739302"), Fq::from("140543540021318882234026314588789476198722270107997413913717074968501481488")), ell_vv: Fq2::new(Fq::from("5960374673747504135132782820167542238733653290730217772559545641742563125282"), Fq::from("14507139191276441762644240084427079430815677404266073216544260783868466836469")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("1463974716527100203803811366059855506152810619047460592789012516365557382809"), Fq::from("12894560712946121716686238503640546420854178898471895472462918225441749701869")), ell_vw: Fq2::new(Fq::from("10508120321944855490271402587060090544503473558782513226881398929302598320386"), Fq::from("20659878091446285231719184399846880718538938504006216923931157918871132219202")), ell_vv: Fq2::new(Fq::from("9314687360620406591930278900314522432964075856747694070384017245464130048974"), Fq::from("17694675150016443475335083504221747995503842086249559620223686305992007745569")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14196539467892272085884829545587228956041818801561947743113049357924737022135"), Fq::from("11319357657686312684765696405805456576475636756706867678717962076341276658626")), ell_vw: Fq2::new(Fq::from("2758949814978246411081695103954624266457693267107844100147125044169545866221"), Fq::from("9492100316900274072027501035053797502489796059345864111677781413985743864383")), ell_vv: Fq2::new(Fq::from("4704107896591496617393915690357581449785399000813194972371050438639491862314"), Fq::from("11531365345870313813093010071065540963321572589131340821665512078366854738990")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18802465664727774773666122120775907428094487475952383150269507875840089541246"), Fq::from("1731637922302665944958348222164323910730404298281499149800917211427254836819")), ell_vw: Fq2::new(Fq::from("9297609456558817283944780415839637083190742770917810313181849518385379162642"), Fq::from("16377246183633200781091121197835476043913744319877154056563766308564474143489")), ell_vv: Fq2::new(Fq::from("9542461764772800142773586971354345045722377861760780709590923740133551871717"), Fq::from("21640615613954317564353959148225302612886621120325752435425437529806262143694")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19906049876176293236988182778475214525096978055900789654464603831282713330781"), Fq::from("20362580925091945825523087376754169675844247448742703441518744743887060798158")), ell_vw: Fq2::new(Fq::from("12267599301269062964822189844562856172284191535014283588478875529832316401461"), Fq::from("16669825137620567269982790548542800570592942517936126877686519256685036586668")), ell_vv: Fq2::new(Fq::from("8883573260261802165780406246211290131940260376316892224137508921505842308805"), Fq::from("19231108274886583687672953546761020040720197580241947146286750565108059707992")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("634182853515641884871095612114627494589971356340786857463490962898713112220"), Fq::from("11430884471074068345749191262196070058384500365559620896062382995797164195815")), ell_vw: Fq2::new(Fq::from("10550902705393315932525976596381459439846414130661652807183357995272197708172"), Fq::from("3961587743443277010160905820636488892023891053433292002931434257476626412826")), ell_vv: Fq2::new(Fq::from("12451440882697904111698352784988882971851697245540249016187908252126835129000"), Fq::from("11782592623486603129029758483031965320434451423371375539247662743178943477576")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("1404156345033137762434737426689373534725204733157905834242027185649572912742"), Fq::from("19570890322851668915358612375948367153948206621537633922343603465980161569078")), ell_vw: Fq2::new(Fq::from("8686503710040621382075118769097735385052600877432244428197116763766844918162"), Fq::from("17223447941612107204326938316335236704702567123017447936901253775530936851086")), ell_vv: Fq2::new(Fq::from("18767158861904777443587157807982315566545548385817262926280782765269843879565"), Fq::from("21334543677892438295079840326280352051766440588072368601899791201007143326750")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20453014818098756686381445228623803935714316636974320420742351477862354368007"), Fq::from("11004429848286161246832723709453157691786114066215975460914111961981461575367")), ell_vw: Fq2::new(Fq::from("14444837779919119651023367195508370508399423510380962187645910198528103345317"), Fq::from("4021193867366929691652413553892868612329317391162172266658368471326882495507")), ell_vv: Fq2::new(Fq::from("21098047915839588616030399663527845846589272989005835932893550785408588982098"), Fq::from("16796487305049888740077005894512139332211008298712590271618863352993144873711")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8799932768264091269394526725962837485495458036153083882004277055769655900574"), Fq::from("18596145973176446599121174893654309094121045316907408604913712044917019537350")), ell_vw: Fq2::new(Fq::from("5727901458562943136998396728651681550262281371313580881804977990767435431080"), Fq::from("16665276274186006945492338822028666786935953718794303569570538737692828117635")), ell_vv: Fq2::new(Fq::from("10910280368195212990489805322100030141483989387720003214110076576431959738476"), Fq::from("9606982070265587786427675480691869167035030735018706315315080889814741903522")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14453637744147393731588213406970287317078500183808667567110703622821377404467"), Fq::from("2399421543794348043991999124897133983828094348779941685913680103324762089779")), ell_vw: Fq2::new(Fq::from("9949473626857777965864312230577133154733097780222892008806269288857107115742"), Fq::from("3174442934451895428948522429276679446276043250911620596846605340399481680631")), ell_vv: Fq2::new(Fq::from("10773554875563236900680111254025179423521083612506388720894431016952344645908"), Fq::from("1791500946161395076321848651097476041018873800532919907790655022966446958329")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13228018442560564090007964181563674306630092611719086476342149061864561185280"), Fq::from("12610285327624889222715450538150491139050620893563766277510163987972586901233")), ell_vw: Fq2::new(Fq::from("17479662858773093405375538474745376030990506794065246603293683266124925174043"), Fq::from("2084404960859519365733972471606438409123459865402210971655148862692485946826")), ell_vv: Fq2::new(Fq::from("9675635536108496330861750800277390856149950003475979013330722219857375990089"), Fq::from("17889926389980731198337394126857435112726851886206769936779651682292365620935")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19205463777127243508919644186723680978776945615422050305472500178064412123473"), Fq::from("10530028762478985738084339660443655112791893212694172883230610760959423970096")), ell_vw: Fq2::new(Fq::from("16515559472874148379011472911124765449097459867226728232468301906190158644000"), Fq::from("17379548905015313648731271429222704537289418711815257252434537403381161039222")), ell_vv: Fq2::new(Fq::from("13018298656134311744614296809909600459918460239498208291306800551968894517759"), Fq::from("16285303269049929546880902543736204779249639088136468614535789316688521663976")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9506592875949693201584225006409064359870098448027454395748349563972724759341"), Fq::from("1756212124305090096775291444252901778765176938185769777272403090820692916832")), ell_vw: Fq2::new(Fq::from("8608069305722185875126845710591577689851258364137915830961192516727536204958"), Fq::from("7879742654857051185278955345176850926952262018841407228150061698899939295057")), ell_vv: Fq2::new(Fq::from("2311752966405194522531977544398652994349075269275918521122768201037004674541"), Fq::from("8056192790171072186146581305753657840809572079825584316343698126369167589177")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16894171323623324294071017663897158494527746239981369238071300334653959886935"), Fq::from("21839557351524624846710837291698003965911117746801809883425361529677608757736")), ell_vw: Fq2::new(Fq::from("7921995242367245298209685641602469741817826911044923939882608549077724733892"), Fq::from("16195366803093832645900011617380218833645406154510352140817411408898365560241")), ell_vv: Fq2::new(Fq::from("12718951609280728995721445896408716663776093135974310357265585518411722037823"), Fq::from("14775730569800772809191320148533632897226925930462946258120395650683306784222")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18324740767049924394294219212760164185884801967765967078427354810675386933254"), Fq::from("18510525798971302215942990229348890627301667188256739950256532108744423958467")), ell_vw: Fq2::new(Fq::from("392130232730851920199052091589353916248087256056205523658797958171832234380"), Fq::from("21095230708713942590602820178155041621291598326436725983218983799781802054278")), ell_vv: Fq2::new(Fq::from("3496296009938486548567151348321691013571513425434397040346440937975425900960"), Fq::from("5982055369526470853931810147684497423036063399292908540001447206525852512531")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19247021831371751344058078378442480867649167715937305884736113857856820190448"), Fq::from("3844115161163747856814522390569661892955450798273343821401275758096863392646")), ell_vw: Fq2::new(Fq::from("6827864046826742026186179073427918229064686953470940699125130948982461542396"), Fq::from("17615043663229174389408922909891358248158892597678816385681324344031892930590")), ell_vv: Fq2::new(Fq::from("17143020152984715373298117189870583206488981021188881359521310100992509008220"), Fq::from("2174519777022093979454116656397980458948553322768446892932246705483799164016")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7787149350876246869417695770873628344850130125888658086308716316049428077539"), Fq::from("3645376584593179034263104864050174094110419584940335745722492256564582048708")), ell_vw: Fq2::new(Fq::from("11834206166934740935662681278907472701573638397628522018255718023777042308993"), Fq::from("11564024956967266505944728035150285591786947709753433733529805379472238406883")), ell_vv: Fq2::new(Fq::from("15009525058621581543328262019401183048318643177731888421709672368574056014500"), Fq::from("13573139489477783981960389988307434036015536834494019264597795654229345703051")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16059660214342726778635842789601639915164060169646706775268999439826160969051"), Fq::from("3592864998692836817549974752747753891215991801035643051485066473571282986623")), ell_vw: Fq2::new(Fq::from("16014086539360826224382965134738311010906671679277894939278016943520015391650"), Fq::from("8080101227643664909861133876167034094240488311720250596061359317310944652491")), ell_vv: Fq2::new(Fq::from("3252331663236937425057834000082561720577180169369353108561319513563871033743"), Fq::from("11096433584385810520286571872297862049422429286608206525566845219548956753724")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19106172305007554707393707438702629275907204152148571359255256006833275609105"), Fq::from("19930525972818199875851038877132322547334829658949554806801108404941414101103")), ell_vw: Fq2::new(Fq::from("6679449617056144648556013688092503718528081567224358397706944806960707450866"), Fq::from("11834339054115289581728384446510154030614495257639549098914576992681792644317")), ell_vv: Fq2::new(Fq::from("1103971970080878735020200145931068396548255816344141900100578642050167835756"), Fq::from("17442014287618549059909689032626628911431121434337187918457502534017974292771")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13076805492928497748141329405065621544173515733010216253264862326898999186011"), Fq::from("19346998537378535293491878821381642632408077348855336790184043880601230192418")), ell_vw: Fq2::new(Fq::from("2545953185107430710067715026283426197979102699516436299879612082830465064387"), Fq::from("658480144944520773976742499043453183928200527625254756411898212059863027220")), ell_vv: Fq2::new(Fq::from("10599092041977168661053673645861514974472253862403262653806335823439984547520"), Fq::from("17162893574285047796735108280435654463748265136801600105571927738662909784323")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7085319511141515608176313666085289847826695130590757892136226409211984372940"), Fq::from("4640142023499829801003091193578633557246618475866500623404375352964089302137")), ell_vw: Fq2::new(Fq::from("1378711095367589966539356394028311641404402942969200900226036867512926984525"), Fq::from("20227188036459114276006891788262806649925222994898646338447622388573680807347")), ell_vv: Fq2::new(Fq::from("10909351851886459521617902618958119972478357132140936115187854789408351897716"), Fq::from("4270359828288113001419972416654771076387260764523167288282363565700959902034")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("10664974041382313497332907041293813168318012018754207087217051246266695094381"), Fq::from("7224602128989495563505314962160716046811485526332201590284671097356858321790")), ell_vw: Fq2::new(Fq::from("3934655983500232748821065857705540333423053242147172231346384664554964864345"), Fq::from("11454752219285395886386371667205283572988335397450041936477253847740701967591")), ell_vv: Fq2::new(Fq::from("8613140983903038954690786727135433848146709090100659195246608245554768285578"), Fq::from("11683437859485859733100235618517710813205908191940301821880216475492534630964")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9260311336006363265451970156091146121161647548618044834929651955256342089015"), Fq::from("13937163080647677520680042908210664749449075029135461146729576039103333691760")), ell_vw: Fq2::new(Fq::from("17902331142943185999552054453100857674678810044830315483268693319046882556841"), Fq::from("1544882902322119030227919062381439448920928851067359863906156836436873153173")), ell_vv: Fq2::new(Fq::from("16995331248700240765237826468302734126993772776070338276810983166039248498016"), Fq::from("14969634758504522458179971123440531767135906355134596040598769037804654587702")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9691457346326988270440356562380914538881384898220972529069554878570918299675"), Fq::from("4212749851190062070411861546581519573149862920541429109749681953529770836138")), ell_vw: Fq2::new(Fq::from("10890917529063563080488636605724946896609903254351684402546841801646043334166"), Fq::from("17064796604657821037034221187126605208939910043822292844975397354283757360178")), ell_vv: Fq2::new(Fq::from("10053764157025690527213061522668163518874554311426109722786780729745637570153"), Fq::from("12784239716790558610229592219566905741360808124013978959238298462172061891010")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18178610037180803777372696468852749823139836169202607246443680867419301660001"), Fq::from("2203553150397509140176760042861998991151797164185565920046676990283238338489")), ell_vw: Fq2::new(Fq::from("20994248737185978263276750124680792452522185817020429107744966692073225126536"), Fq::from("6349838002511478763817816496894821255234760390494115897123421446389963162622")), ell_vv: Fq2::new(Fq::from("1263928993764270656816573911502202456032289420152723320064785370753332332564"), Fq::from("3046827108505433561924898780283267608115953987844810798949064271834105292424")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8956928906602747002792065663685826131559810003765184492377190892700326559543"), Fq::from("9256558483224710959389211180674778886236784676421453067453985103303008910100")), ell_vw: Fq2::new(Fq::from("19258483215045500292709619645410473292886118206783068420253408398264757067719"), Fq::from("4451865829956913287893327659699467549695832858717473490066802599791690802339")), ell_vv: Fq2::new(Fq::from("2764660330371796211645912465865514165647712230266940655139955982219479135487"), Fq::from("13240152986607724183470740198192310673584785226612774164092563912390438227662")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("681767682327814473892054649470454096768283412403007527891528077929769397010"), Fq::from("7668193584311505399758378551993462119934308304509766677185196010163712481402")), ell_vw: Fq2::new(Fq::from("5655243760535315520783349710797795360471266639432050916814563868068635911845"), Fq::from("6563255051651012450772847952915771842363624186837705815170827122750163703821")), ell_vv: Fq2::new(Fq::from("649940975249127008407571724305690292046357719924151085455652491670338045916"), Fq::from("21579311737676937819033214124612551530127167774002946532496743344036322675381")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21175778737847401044104466077056088822729568143966863788586142950892805137666"), Fq::from("5595589116740977138584720715075423123216685509908153451278598090500673116809")), ell_vw: Fq2::new(Fq::from("10584200644032922607821101570277578933623023426596682657073390742969073652895"), Fq::from("12837953697025504547863231066503572653998301486985427806275987315834659812986")), ell_vv: Fq2::new(Fq::from("8117107548208718171682922744254403704391859020172151233388364110731129041045"), Fq::from("13049999463955485268550904857465746325902043660001708636685369259450564610454")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8364961604780202742303921161081604160922838515374423608219736048018999860853"), Fq::from("2351019119583633406807532653396626109486234484023597064733508979159842183382")), ell_vw: Fq2::new(Fq::from("21028432880575926523752559333294402827459370060933557257525967464997077946986"), Fq::from("8046872296156140750811104526560806223677002170113323621491280498583760192616")), ell_vv: Fq2::new(Fq::from("12898024833491487058567514428449740829256150305937499595589780150928565842102"), Fq::from("4285065448422006185829060466560686578745712460867295748911655362509144300948")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8072232359889306936722303478835558056241116613774315547062190240728474923834"), Fq::from("11177008054227634237663439894104913366867843529726469624208391026893438331544")), ell_vw: Fq2::new(Fq::from("6809080759521318546573252139200749353665236974149238674422052122591837770337"), Fq::from("17038522274461048605370004543286094433565235245947661777349160189041706387246")), ell_vv: Fq2::new(Fq::from("14024834693814357899802394991122512208020612837103453479716792821268632856005"), Fq::from("12162253519201182595066522680875457472054933146636012038079782236991342232964")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7517621744472550238729147112582763290619696603888340562320661653468259785459"), Fq::from("5775958628431902009928696648706206677051589697801230225207961079701351805991")), ell_vw: Fq2::new(Fq::from("17101537508347637393046612945060881733341630882242907674489426818282232192765"), Fq::from("11439924192045730134857855444951867487673481853061712676505751821713592417848")), ell_vv: Fq2::new(Fq::from("16524831954193049226680237913906512849861747525312574297009556588583016003485"), Fq::from("17961833510100001957206410577764797351002903290115563676683242640020206248673")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16912575460065719860154998920476780598787708817030774617128864437198545819929"), Fq::from("18523390726102623209642393197047309169880798198922216024709291880134883505164")), ell_vw: Fq2::new(Fq::from("8711744315121016096444521009902336544777137803294883395587631628756743226403"), Fq::from("384665796631718061484470820770858149414181395445222225443039170963661807931")), ell_vv: Fq2::new(Fq::from("15774538322747462731385931838340979144243491164741595437453112329195248618714"), Fq::from("11632275451601229685994821619837954138410931366643826111939350416539675376582")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4193777370819241345875098285464776579390947200963349948619027946434227330979"), Fq::from("509866391215626792725431227507051906680902316754753085884521978793646270092")), ell_vw: Fq2::new(Fq::from("6037175194865804429042789680410737585033925490367655507316663717551980954540"), Fq::from("18041663784750649636833746105782119188697038394822653258671508993118228507622")), ell_vv: Fq2::new(Fq::from("5823434994217288412340362524966578492063846889148854293277163714066988854109"), Fq::from("5672396008289604057966364197930061596003079754530820851713923211431895221182")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4038166397773785982077422827894979131906966567246129821324959462204170431412"), Fq::from("18437550501813119699199412531646289403507307693226141098858673836744295275291")), ell_vw: Fq2::new(Fq::from("14284923653983275338299458156785711701885868498245540450770121796221847779083"), Fq::from("12602897944901625024952695280664364105484904418233840369524699062991946110320")), ell_vv: Fq2::new(Fq::from("5486498814295375008109094800710678785628460899485721363575459728835468025031"), Fq::from("16200691130190012476008796365701719097160136918178762563260333528187553865070")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12314546787967340174315157919495563500551846522430770700093005979883034971431"), Fq::from("3526449836226747514409509674529965207620312182683788709671366453353950282439")), ell_vw: Fq2::new(Fq::from("5093273668274685698542824313074411944254604209522407110762874543698544425100"), Fq::from("14413934995969972893982204347979972725062293812449032149868616703831967954701")), ell_vv: Fq2::new(Fq::from("5926816361020227072352913556957354039762675747717706660663155084741002986207"), Fq::from("11598493987133778453993847174276604684683441101397418085992950965674684594056")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14605016886275487293555440005295724730930746064070181829857167303747503806723"), Fq::from("125488028578519699988648864181559879827394253209756997049515479109363959656")), ell_vw: Fq2::new(Fq::from("13618798589781916917754240943924749450694915714095829562214241999969133303075"), Fq::from("7919521676249934186275057369006254600339974917319129664598969307290994883444")), ell_vv: Fq2::new(Fq::from("13618646464444173552701873510284352084626546810748938598145778229202447557390"), Fq::from("9476574581262975213029808591741428214940842553330900266594833123385626035464")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20330569420634759863618802200434379756882301920599312733161738334139846403423"), Fq::from("16941424680188140264426604507849243567044731815188670894126312258606055646954")), ell_vw: Fq2::new(Fq::from("4276245652046353558826181153633997046897244969830610112857888938076187872093"), Fq::from("5218134511825218024578456539758678791071455837562669069391138178014247652283")), ell_vv: Fq2::new(Fq::from("5074559277335429630740829563679793620243538875458585401599810469948320707914"), Fq::from("8399181532479234777915440768158248538072911084393374644710065479068982458679")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("228456190514225889641604150077307010131630607092226273786181288134202785827"), Fq::from("6829691501727142726895142554058769478742020003540473851444113314854839779896")), ell_vw: Fq2::new(Fq::from("7239124420778199901449006013253700083679198792863456824897986348225520969318"), Fq::from("9612815502366829000381428453387330590336692138445456863004983078118510731327")), ell_vv: Fq2::new(Fq::from("3032095333965417803301146592958869325973325547652138997619203744855910423626"), Fq::from("9140900126707960555227810341717821370276432039323462635381934661102939064717")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15183121839690585327918066578070528787848569635478652970262274682157344344126"), Fq::from("17032257654469593612895387885450704824985120918227803531218149228038247116993")), ell_vw: Fq2::new(Fq::from("4449946720064992614521464039848704811573349967863291193814384877976473528180"), Fq::from("16213473115750238158840972181958630393893069174331658126086552358850995045944")), ell_vv: Fq2::new(Fq::from("14820364816206160787218030695278797219740217739100995460398841348849807343630"), Fq::from("13391831978082769659226335015618034703517552702532093599836056107584855920091")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19376350172596821938106071755601309991977917109030961898807003698880067029751"), Fq::from("8432677549750770318577385075581043130355856390772751886981628666318547719067")), ell_vw: Fq2::new(Fq::from("19907801542387995008974808935818119635245812040205527769417146511798917761987"), Fq::from("5161102563798536355551540455475021097419898845343559944869423371507274303876")), ell_vv: Fq2::new(Fq::from("8934093116311670296805867189709376457342708648536176030347455800639124491789"), Fq::from("2703243001408501823489300103176986467031566088207792046257025791153593204110")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14246576411085900561418149065736187542613160942023368670593924207716148262147"), Fq::from("3894319734417282749436553474309402307762162253367446669496582090013706055610")), ell_vw: Fq2::new(Fq::from("8375748526029577467798965722696242475050467009610481042928314348040697238273"), Fq::from("5150547143250446746041060583106242553976637956855972603954741388536316657102")), ell_vv: Fq2::new(Fq::from("15907790590424017252473639606980913201030067954178224590289142061847439584806"), Fq::from("11159987404206640511954116561264499629862056981304725791275082386203982274572")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8534750176510034607784585561677263909151653570221826711790070724427621935471"), Fq::from("7817152482056867378682253092041139659293041868731996264204246731388352069129")), ell_vw: Fq2::new(Fq::from("16961113237531514134058762308816541864350643273034658785830240248900850618232"), Fq::from("1179078000730113993104369721112852911420394064162245727296751799866128074554")), ell_vv: Fq2::new(Fq::from("20457374714339105220680185601375259716132143552962630535477097129353077443649"), Fq::from("4422525926709336878773825319073188719230418589270084994878618648788354576429")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7627847586963893527076371866753343482026765205726665650559945926181618692545"), Fq::from("11854660093187915385325641424610609952992715573834872602540776698350931163771")), ell_vw: Fq2::new(Fq::from("12061802008303461437760636908585485318885385024493098952040511326354153801540"), Fq::from("2483328707802038203960641185873859652602098605879879702928670691083351376828")), ell_vv: Fq2::new(Fq::from("3179938981834043637934998748622377818001614400201120097155888429286598162788"), Fq::from("3387712903817602592588061982323804658595985412237324988504760476608661908669")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18935668687475417752670528265758371887227072908926123795139093572891409166177"), Fq::from("7137948322161355685471108734802366001522057971184080351365620151904917529667")), ell_vw: Fq2::new(Fq::from("4339156032094542567838794153972725732752104151129467898887880650450609600794"), Fq::from("12048671216064877807276853910375839173096391682288251978695349073893797560105")), ell_vv: Fq2::new(Fq::from("1950772803812480571427275248471639178971170644950730072787717123474197730954"), Fq::from("9181624035531055165333621510604231877029437698520448252367489746796255738472")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("11063654615400954717295218153285806614049542407631184476400061600226642090639"), Fq::from("10322234663627758865726091502227262577963762851829427143908452375568424940506")), ell_vw: Fq2::new(Fq::from("5088512375045637070831200431021199523774202712519349524231646029565850243137"), Fq::from("13830164350727008423053294968318643994288249825064419288902919319598527803018")), ell_vv: Fq2::new(Fq::from("5664114633031170805308124108402506179037402200165762737765348606356467981435"), Fq::from("6420500289322096738785425211380588437700105870048213919996474830908364662080")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12089017351045163205876906516764262596252750496306969979802152087250767442655"), Fq::from("10210558808216547366164047615754858102665922901025755697993978799086515582216")), ell_vw: Fq2::new(Fq::from("5805411494371431706302702029555168082350433630263841912576207114214821637903"), Fq::from("13417649473832328299363309530328161775713579734460687495589323668315481789186")), ell_vv: Fq2::new(Fq::from("7100161023424716303226449735068569399024504614273633218656928626719841360262"), Fq::from("4608780226874321006375565633722017610093139802788639970961143211665122970425")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7402656885283645637844399867437832414708002274719716809314550750025200002816"), Fq::from("3359755418859898733774484791668766343777859472186478368952701397946858170923")), ell_vw: Fq2::new(Fq::from("13972633490402551449646647488711159435500678729477306484155919447807727623768"), Fq::from("7493072689132118410451346570771015894599566093753023472904593233187191189290")), ell_vv: Fq2::new(Fq::from("10553632499535499938294059057022861371325444554911689769145218759723376859522"), Fq::from("15031724096959569889887883166930429112400023614133986918505200177949730145621")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19614666780105423133718014776333000423285968913902974939878172716992608453809"), Fq::from("11603995082765755677014543451017101337084691386341657702599446172204728703168")), ell_vw: Fq2::new(Fq::from("3640285626277218751703647194759635465446306209869384174139659856814138047153"), Fq::from("19852827152161121673942769427296662914445783006570890136096053338872958018957")), ell_vv: Fq2::new(Fq::from("15133277841235179795557173122463348349264603847086033764926472096701308210280"), Fq::from("11477545358800195084620539652687203979589770655971679975137907269158992047904")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14110112013826130627308137057440352762673176006101512326319632538666268095190"), Fq::from("11939502323738533476551128890683182628994629048369580067812328846012456410750")), ell_vw: Fq2::new(Fq::from("4903570904021379863440412383519324112841733022665060696302943089410552325193"), Fq::from("14266803156893138980425811506240396635009593456307449490727186191181005152878")), ell_vv: Fq2::new(Fq::from("14881072700587137091207448172180518390500891251946414292329369965686915854422"), Fq::from("15926908436628594399583920130944357569706863129497064862365363758969852348774")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12558706151877569296451844961986237406953575964353502214928234165049347203902"), Fq::from("20986351119648698768747274174278231882270962658108929786748672594878002377020")), ell_vw: Fq2::new(Fq::from("19049669232824485247328064933680299726722169816787419902393190195363675139966"), Fq::from("9038064204016446065702766858179381867420605337773795708435102200352238225734")), ell_vv: Fq2::new(Fq::from("20935836360010341957040867041308346409399314465762274744792154001186771274330"), Fq::from("10203406114258970224185400536984459120400171034941205452745624846762632193745")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("5273485321155899320017031787320374433252287213043632779466667239152835002841"), Fq::from("3053056741460414900892697870729818263336633719763669407853703091660209931803")), ell_vw: Fq2::new(Fq::from("14340825980720147789753139378133767339849529468405152905403690972933175947523"), Fq::from("10774533501875920615470512566140848784460524907884011188458749902565676606519")), ell_vv: Fq2::new(Fq::from("3964278893337662256819152335242286127722672026565541866237058714197501521533"), Fq::from("18022244127367262334687067361628890804799640514175862613915842388966614841200")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16332502873726000727943290608233946425803707374189230473690022986938005644703"), Fq::from("1567962947294031055122165158764344785171646063465144808831306664719326769256")), ell_vw: Fq2::new(Fq::from("20252166217271346908492216031557669988560255907738941815783926955730505937199"), Fq::from("17117796187826532609679851805915085724672631310980423533710032156677468384570")), ell_vv: Fq2::new(Fq::from("17447467886258986370787137061379217598255780655216428459333948972673151012677"), Fq::from("5126879980313476901131854669269919146638602687291477011061931179530480305128")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("6088970897125028663727223015788851329277811231049906821835210021445230531470"), Fq::from("8269042454443245208032034843063473106607747629991910601236499943382389383909")), ell_vw: Fq2::new(Fq::from("9390988514296458582044517134794855627538985266829573142624063333360969055582"), Fq::from("14809192622791971819542419564616657251564340618078775692251491022692953050806")), ell_vv: Fq2::new(Fq::from("12392403118986217155446218801253632414719473065843423820454574775060624572223"), Fq::from("17315648550786131512187415565477299679361071890287921321719420341945848525344")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8509704393616883918389461491307625357351608849562781376448108143686881397874"), Fq::from("20273788844812669168701278525056990767412251703786041383351127813548056060175")), ell_vw: Fq2::new(Fq::from("13587733380957601917144020513037833512715927912961534242274129611676082179670"), Fq::from("18219007782547538891755131684432302965729501555658190323551992733355994409648")), ell_vv: Fq2::new(Fq::from("15212235563035911012497734239429958536448576200220918241945992532007455729638"), Fq::from("15868967159924843704138451124239001351258676891119605450618238262537266256000")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("566642944339341838872659990670775497018813579200118662738575067585620262683"), Fq::from("80633561422247315402489148783136836267866433112423246780090434639448081501")), ell_vw: Fq2::new(Fq::from("16776314135335488452712466347201660236874408701359052824185479604688604443386"), Fq::from("9397286871750513195379912820117368990110423949006393550333044521067916303673")), ell_vv: Fq2::new(Fq::from("11213241389292792759942014406027965770431479614859139543821417488303149467800"), Fq::from("13968940593285273772185768016764839518710968380183853737025067101492115101555")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("6582524148524166281651049223972241418275829952930531957133089376081226599310"), Fq::from("12912796945237135113112612780707994091197804899944634993937602107108817617069")), ell_vw: Fq2::new(Fq::from("5915143763477343907746278070838860419057837978194254349309930636588762054019"), Fq::from("16829580672314341143130263740133292836202318062773617917928807522786048686905")), ell_vv: Fq2::new(Fq::from("6048262046277135430490078693604752857941808479983722633615831914037376701617"), Fq::from("11339265196645696685132629437938382050130007247287625382019491127986659567508")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16221040390720130701106818492787718054167551387591190581820867019772199748452"), Fq::from("13816650124538024294513769170531808824213976236314395631188393480750356347832")), ell_vw: Fq2::new(Fq::from("17884725815671017389514271919761437310258619463085199936932965564030171526767"), Fq::from("18248973796984942533601080062907385035195180110558182226420098600644022965345")), ell_vv: Fq2::new(Fq::from("3137416660392142651716799414483848222789075294982695366740419822188364577896"), Fq::from("9694367271685190541749430052152010428005565838702388077250686840743397580564")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("6357610272973630149840225505274539351850867930627803566656124705077309138236"), Fq::from("6532064867969562545750502093085673423036025281166230602025667552530979558092")), ell_vw: Fq2::new(Fq::from("16786670094068652274677750111730144379312178408732829954083928254919389983983"), Fq::from("6341529030893580474995202784328098673009525246309131907462803539077246776781")), ell_vv: Fq2::new(Fq::from("1974498926273096258802796702119259182018579009158122203612017565775578915815"), Fq::from("16202371700345289299248140495735793174324620209466889736695203996551421837181")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7645596703863049949671625140189098533219782764649960681401839437305310089346"), Fq::from("19662686877039093771787921401379726144714150453645437909640334762978238819386")), ell_vw: Fq2::new(Fq::from("19675517233991845002128733409028273554556593365253876728982552859262257722391"), Fq::from("17810553502824017140555114404262914434390762266054795883352268646885364159410")), ell_vv: Fq2::new(Fq::from("18665072545928024509881282829034535118797458683193169885406548536617293330990"), Fq::from("19411595053923298782146703680748498557982136082135126508584153240357549362078")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14470514611309746071605255274946119202484366658141275541087020842175954067559"), Fq::from("2594897518684998941907190849745518961756699273476562463311001408776339658780")), ell_vw: Fq2::new(Fq::from("935873152179726234311073616163680460286310013909076125956427156452613025181"), Fq::from("1833293684395168387765260233178732135932817355166697645828433012629591012612")), ell_vv: Fq2::new(Fq::from("15506344864201849985196152671178663605696906693236430756412539843011256851197"), Fq::from("21249626787247514663459670711353542343321937134901366527096486234949436840608")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14844298589371811580114643149994926627935566419885127313767249060440840193460"), Fq::from("21865492910219121705480174226763323783866193080747061232287801827306146286946")), ell_vw: Fq2::new(Fq::from("17151100973274235243640792655189642956800303990602164052684336481398298289411"), Fq::from("14392190221562986523826107157981466944764379735735544220135185056395768909930")), ell_vv: Fq2::new(Fq::from("4350255565215767538405433120041486620343663937357318680211203103336875636557"), Fq::from("9926081950735179732148156026738993553156731124330466866635273595752973758855")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13218761129163938072615510267349114476398845757988353548581904005855607494827"), Fq::from("5599121822325202341068442750237639791575579072233857524815548719976323285948")), ell_vw: Fq2::new(Fq::from("15971120376538832228790474763995956330325241100841265267806125625593561200292"), Fq::from("12523288625433720594713119420870235433458343202664663107083051275205911332632")), ell_vv: Fq2::new(Fq::from("16334648365696506264242655102949242813244449705193389318653840069223698505218"), Fq::from("5656148484258838154513173637616383061988381726003070282394323376702835298685")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17813738829828434290335491956960173922453516626523775773406434062574864455163"), Fq::from("20602369742562270094694203910247000789225384961205618501257582838246099644397")), ell_vw: Fq2::new(Fq::from("19878549521928480892919729627745688905048802531306612605469432942519616386562"), Fq::from("5017457857497268504225948371697060380418757800650743056471872023476645133890")), ell_vv: Fq2::new(Fq::from("11253669115821132651967277438747600032147012598164512191026558281212405907059"), Fq::from("17527101633359473938082679669763197242520773041737623688955597341205608093817")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4215545173967763533303120784928042872427929334572670490629936371429427871071"), Fq::from("12294795716682648009432653500746413304750644076589949908249345884220200424705")), ell_vw: Fq2::new(Fq::from("18825720171660950734491494575282684507875213824417353005009465489701558195915"), Fq::from("16705960611772713893529695679517111622754789317070946155202186944060021844706")), ell_vv: Fq2::new(Fq::from("727717707174551758531261221179545916153780087179478433501944451338651779512"), Fq::from("8112271980906996458859937804680221894935591766524365848766530295751121904748")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("10233024162783159141952008177866660607852287510058437429941412641351833243474"), Fq::from("9594838646894429694778293318641489517645344438331800828882574864439372872574")), ell_vw: Fq2::new(Fq::from("16645682756797429615195184700295338945089747965333104334845226609564841400023"), Fq::from("4162352035394629024812623559649906911351409190415492302470361657064711946691")), ell_vv: Fq2::new(Fq::from("6032119528092518381867266123093895943802103316999621961287699634448441322294"), Fq::from("16978590002913514123461500015923670605519520926487110857783653291799516315656")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8218196524386055233961282665657821183883422738571989211183480610420112668380"), Fq::from("10415506866745570162620028320970394286106067432626512901854031549920339603997")), ell_vw: Fq2::new(Fq::from("5210044150482547744429104832217210823770891140961861953610847155896342519957"), Fq::from("7249576834325689075837879557249227252903732278842727723303356548006299693332")), ell_vv: Fq2::new(Fq::from("10566250213397026997176223676662431109660830193211672199901829008963029461821"), Fq::from("10109726446308488117042758961222375921166426648339199001670753736584698407831")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14958658301977599538003236672511296104101995894564376511283101560012949683335"), Fq::from("3048608469033125363311089547916169290720115268350540694596412955569118898494")), ell_vw: Fq2::new(Fq::from("574489666015527527356241753990038643397861257676315289742492198353125347063"), Fq::from("20135743140323312729475654806927894483883158119600963784497647679670289292384")), ell_vv: Fq2::new(Fq::from("429037282750459454100296565403648201904295120071497517313998879109362521423"), Fq::from("15853233349247701874380259741343021520320591334343875702627292292088300425145")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18700030680590279184945769912899822403041358764323477965514340815228877402331"), Fq::from("10683998632701796443517414309938660369292269662497518738662465524924019738910")), ell_vw: Fq2::new(Fq::from("1310694906110452150099267122613851606459600617516350757492763828252192371857"), Fq::from("4783076385012654125132398886450190041207227400254467276271580769333335151361")), ell_vv: Fq2::new(Fq::from("19588709046778297839342456362815786766378666302331732601204786295010565114246"), Fq::from("15768094978512846661820502043664035262466161785979259300748269497441797675546")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13060364876687824737412369705619641318084803694180161837383771909738942399322"), Fq::from("18217552047682533290011237332328856250732338018560605204821714348687071938690")), ell_vw: Fq2::new(Fq::from("12869661643727171099080271622741099816758098286424339832028838960253051261267"), Fq::from("9771005613420764951919925720405408601257471427085946188171574196917272408972")), ell_vv: Fq2::new(Fq::from("50909581144623642750375885935872654509592339539711209410079054797052409354"), Fq::from("18926528403466649263993967111065644144239906211315737390219036362821972145669")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20549804669307980165961565026567668900175984309510667407934326603524068980568"), Fq::from("2860891967505015044048360806986650576604923893849408592665407858213623973073")), ell_vw: Fq2::new(Fq::from("15425380359872501526721873883293811604657689327716748473720120299487795555723"), Fq::from("19476618728903347368164774764554105623361814261939990815870207909384053172994")), ell_vv: Fq2::new(Fq::from("1391900673866549134877070137078952075347059393013450066057338968238992587955"), Fq::from("4196316861965319041187484774757427170509633759722370813890904092023663955821")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17275387403896977778123657295201685204905269368879700703438840824610915049467"), Fq::from("19208898840130004899976179132541534722147305748950927170013798354273078356439")), ell_vw: Fq2::new(Fq::from("2806476769978795611970161315323438770892084130100332531802098991493644141407"), Fq::from("12027685137795878129532114449809592645364590672303716160923491816560315933093")), ell_vv: Fq2::new(Fq::from("8236588507133996535585623995380369531792969451558421380293164596516832310024"), Fq::from("7194816002224802847462171990762514812770222507636791853795854276216118633532")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8988630595018387496849477785798469553390006328427704582362113137927154840355"), Fq::from("2295777524525303845671711839319946449143831320846022233249545363172788331149")), ell_vw: Fq2::new(Fq::from("19856063720990800344827304970869772811632532243020577718537161398914360273978"), Fq::from("19822454791015240925652351664190735476954892251509376470361602440577337517739")), ell_vv: Fq2::new(Fq::from("16437024240432855113321845721989731630224984534156413407578941758236945886728"), Fq::from("10915573623321985107029933965796058231484519510670830600808943167010577314334")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21286975582879196598650077668567283554136994229408922053641846042051746388886"), Fq::from("5421529658065143080970166226046014500038127295988164061860891929414040927615")), ell_vw: Fq2::new(Fq::from("5555485154537896512520569992616892769357525491868135294451583184259218193278"), Fq::from("4831809465565304340310880868620257258305484211486308136368207144444325564275")), ell_vv: Fq2::new(Fq::from("2583089702666220608189326901976548985583885001543445628170288810575076369034"), Fq::from("7097788099664798415333498763272677734135137637990913349021964143797057593727")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21356455346146359274071336488740257199607341582407258426849457339859865454145"), Fq::from("19512280220246743343068375192508909480135195681362473260894831078332945004959")), ell_vw: Fq2::new(Fq::from("15435677241338788169971902812985017500898533325271813422525238374746874853456"), Fq::from("6382067580057986766020919077216175344977116334133202456947134195629074647989")), ell_vv: Fq2::new(Fq::from("1150360546643207487285835972411275438312694180812115547021545502055712764110"), Fq::from("2194960903427600986795419731825236689333014397679688012527722471213485607323")) }
|
||||
]
|
||||
};
|
||||
|
||||
assert!(expected_g2_p.eq(&g2_p));
|
||||
assert!(expected_g2_p.coeffs.len() == 102);
|
||||
}
|
||||
|
||||
fn miller_loop(p: &Affine<G1Params>, q: &G2Precomp) -> Fq12 {
|
||||
let mut f = Fq12::one();
|
||||
|
||||
let mut idx = 0;
|
||||
|
||||
let mut found_one = false;
|
||||
let ate = ate_loop_count();
|
||||
|
||||
for i in (0..FrParams::bits()).rev() {
|
||||
let bit = ate.test_bit(i);
|
||||
|
||||
if !found_one {
|
||||
found_one = bit;
|
||||
continue;
|
||||
}
|
||||
|
||||
let c = &q.coeffs[idx];
|
||||
idx += 1;
|
||||
f = f.squared().mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
|
||||
if bit {
|
||||
let c = &q.coeffs[idx];
|
||||
idx += 1;
|
||||
f = f.mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
}
|
||||
}
|
||||
|
||||
let c = &q.coeffs[idx];
|
||||
idx += 1;
|
||||
f = f.mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
|
||||
let c = &q.coeffs[idx];
|
||||
//idx += 1;
|
||||
f = f.mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
|
||||
f
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_miller_loop() {
|
||||
let g1 = G1::one() * Fr::from("18097487326282793650237947474982649264364522469319914492172746413872781676");
|
||||
let g2 = G2::one() * Fr::from("20390255904278144451778773028944684152769293537511418234311120800877067946");
|
||||
|
||||
let g1_pre = g1.to_affine();
|
||||
let g2_pre = G2Precomp::new(&g2);
|
||||
|
||||
let gt = miller_loop(&g1_pre, &g2_pre);
|
||||
|
||||
assert_eq!(gt,
|
||||
Fq12::new(Fq6::new(
|
||||
Fq2::new(Fq::from("14551901853310307118181117653102171756020286507151693083446930124375536995872"), Fq::from("9312135802322424742640599513015426415694425842442244572104764725304978020017")),
|
||||
Fq2::new(Fq::from("2008578374540014049115224515107136454624926345291695498760935593377832328658"), Fq::from("19401931167387470703307774451905975977586101231060812348184567722817888018105")),
|
||||
Fq2::new(Fq::from("15835061253582829097893482726334173316772697321004871665993836763948321578465"), Fq::from("2434436628082562384254182545550914004674636606111293955202388712261962820365"))
|
||||
),
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("2874440054453559166574356420729655370224872280550180463983603224123901706537"), Fq::from("21199736323249863378180814900160978651989782296293186487853700340281870105680")),
|
||||
Fq2::new(Fq::from("19165582755854282767090326095669835261356341739532443976394958023142879015770"), Fq::from("1381947898997178910398427566832118260186305708991760706544743699683050330259")),
|
||||
Fq2::new(Fq::from("282285618133171001983721596014922591835675934808772882476123488581876545578"), Fq::from("9533292755262567365755835323107174518472361243562718718917822947506880920117"))
|
||||
)
|
||||
)
|
||||
);
|
||||
}
|
||||
|
||||
pub fn final_exponentiation_first_chunk(elt: &Fq12) -> Fq12 {
|
||||
let a = elt.unitary_inverse();
|
||||
let b = elt.inverse();
|
||||
let c = a * b;
|
||||
let d = c.frobenius_map(2);
|
||||
|
||||
d * c
|
||||
}
|
||||
|
||||
pub fn exp_by_neg_z(elt: &Fq12) -> Fq12 {
|
||||
elt.cyclotomic_exp(&final_exponent_z()).unitary_inverse()
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_cyclotomic_exp() {
|
||||
let orig = Fq12::new(
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("2259924035228092997691937637688451143058635253053054071159756458902878894295"), Fq::from("13145690032701362144460254305183927872683620413225364127064863863535255135244")),
|
||||
Fq2::new(Fq::from("9910063591662383599552477067956819406417086889312288278252482503717089428441"), Fq::from("537414042055419261990282459138081732565514913399498746664966841152381183961")),
|
||||
Fq2::new(Fq::from("15311812409497308894370893420777496684951030254049554818293571309705780605004"), Fq::from("13657107176064455789881282546557276003626320193974643644160350907227082365810"))
|
||||
),
|
||||
Fq6::new(Fq2::new(Fq::from("4913017949003742946864670837361832856526234260447029873580022776602534856819"), Fq::from("7834351480852267338070670220119081676575418514182895774094743209915633114041")),
|
||||
Fq2::new(Fq::from("12837298223308203788092748646758194441270207338661891973231184407371206766993"), Fq::from("12756474445699147370503225379431475413909971718057034061593007812727141391799")),
|
||||
Fq2::new(Fq::from("9473802207170192255373153510655867502408045964296373712891954747252332944018"), Fq::from("4583089109360519374075173304035813179013579459429335467869926761027310749713"))
|
||||
)
|
||||
);
|
||||
|
||||
let expected = Fq12::new(
|
||||
Fq6::new(
|
||||
Fp2::new(Fq::from("14722956046055152398903846391223329501345567382234608299399030576415080188350"), Fq::from("14280703280777926697010730619606819467080027543707671882210769811674790473417")),
|
||||
Fp2::new(Fq::from("19969875076083990244184003223190771301761436396530543002586073549972410735411"), Fq::from("10717335566913889643303549252432531178405520196706173198634734518494041323243")),
|
||||
Fp2::new(Fq::from("6063612626166484870786832843320782567259894784043383626084549455432890717937"), Fq::from("17089783040131779205038789608891431427943860868115199598200376195935079808729"))
|
||||
),
|
||||
Fq6::new(
|
||||
Fp2::new(Fq::from("10029863438921507421569931792104023129735006154272482043027653425575205672906"), Fq::from("6406252222753462799887280578845937185621081001436094637606245493619821542775")),
|
||||
Fp2::new(Fq::from("1048245462913506652602966692378792381004227332967846949234978073448561848050"), Fq::from("1444281375189053827455518242624554285012408033699861764136810522738182087554")),
|
||||
Fp2::new(Fq::from("8839610992666735109106629514135300820412539620261852250193684883379364789120"), Fq::from("11347360242067273846784836674906058940820632082713814508736182487171407730718"))
|
||||
)
|
||||
);
|
||||
|
||||
let e = exp_by_neg_z(&orig);
|
||||
|
||||
assert_eq!(e, expected);
|
||||
}
|
||||
|
||||
pub fn final_exponentiation_last_chunk(elt: &Fq12) -> Fq12 {
|
||||
let a = exp_by_neg_z(elt);
|
||||
let b = a.cyclotomic_squared();
|
||||
let c = b.cyclotomic_squared();
|
||||
let d = c.mul(&b);
|
||||
|
||||
let e = exp_by_neg_z(&d);
|
||||
let f = e.cyclotomic_squared();
|
||||
let g = exp_by_neg_z(&f);
|
||||
let h = d.unitary_inverse();
|
||||
let i = g.unitary_inverse();
|
||||
|
||||
let j = i.mul(&e);
|
||||
let k = j.mul(&h);
|
||||
let l = k.mul(&b);
|
||||
let m = k.mul(&e);
|
||||
let n = m.mul(elt);
|
||||
|
||||
let o = l.frobenius_map(1);
|
||||
let p = o.mul(&n);
|
||||
|
||||
let q = k.frobenius_map(2);
|
||||
let r = q.mul(&p);
|
||||
|
||||
let s = elt.unitary_inverse();
|
||||
let t = s.mul(&l);
|
||||
let u = t.frobenius_map(3);
|
||||
let v = u.mul(&r);
|
||||
|
||||
v
|
||||
}
|
||||
|
||||
pub fn final_exponentiation(elt: &Fq12) -> Fq12 {
|
||||
final_exponentiation_last_chunk(&final_exponentiation_first_chunk(elt))
|
||||
}
|
||||
|
||||
pub fn pairing(p: &G1, q: &G2) -> Gt {
|
||||
let p = p.to_affine();
|
||||
let q = G2Precomp::new(q);
|
||||
|
||||
Gt::new(final_exponentiation(&miller_loop(&p, &q)))
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_reduced_pairing() {
|
||||
let g1 = G1::one() * Fr::from("18097487326282793650237947474982649264364522469319914492172746413872781676");
|
||||
let g2 = G2::one() * Fr::from("20390255904278144451778773028944684152769293537511418234311120800877067946");
|
||||
|
||||
let gt = pairing(&g1, &g2);
|
||||
|
||||
let expected = Gt::new(Fq12::new(
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("7520311483001723614143802378045727372643587653754534704390832890681688842501"), Fq::from("20265650864814324826731498061022229653175757397078253377158157137251452249882")),
|
||||
Fq2::new(Fq::from("11942254371042183455193243679791334797733902728447312943687767053513298221130"), Fq::from("759657045325139626991751731924144629256296901790485373000297868065176843620")),
|
||||
Fq2::new(Fq::from("16045761475400271697821392803010234478356356448940805056528536884493606035236"), Fq::from("4715626119252431692316067698189337228571577552724976915822652894333558784086"))
|
||||
),
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("14901948363362882981706797068611719724999331551064314004234728272909570402962"), Fq::from("11093203747077241090565767003969726435272313921345853819385060670210834379103")),
|
||||
Fq2::new(Fq::from("17897835398184801202802503586172351707502775171934235751219763553166796820753"), Fq::from("1344517825169318161285758374052722008806261739116142912817807653057880346554")),
|
||||
Fq2::new(Fq::from("11123896897251094532909582772961906225000817992624500900708432321664085800838"), Fq::from("17453370448280081813275586256976217762629631160552329276585874071364454854650"))
|
||||
)
|
||||
));
|
||||
|
||||
assert_eq!(expected, gt);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_binlinearity() {
|
||||
use rand::{SeedableRng,StdRng};
|
||||
let seed: [usize; 4] = [103245, 191922, 1293, 192103];
|
||||
let mut rng = StdRng::from_seed(&seed);
|
||||
|
||||
for _ in 0..50 {
|
||||
let p = G1::random(&mut rng);
|
||||
let q = G2::random(&mut rng);
|
||||
let s = Fr::random(&mut rng);
|
||||
let sp = &p * &s;
|
||||
let sq = &q * &s;
|
||||
|
||||
let a = pairing(&p, &q) ^ &s;
|
||||
let b = pairing(&sp, &q);
|
||||
let c = pairing(&p, &sq);
|
||||
|
||||
assert_eq!(a, b);
|
||||
assert_eq!(b, c);
|
||||
|
||||
let t = Fr::zero().sub(&Fr::one());
|
||||
|
||||
assert!(a != Gt::new(Fq12::one()));
|
||||
assert_eq!((&a ^ t) * &a, Gt::new(Fq12::one()));
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,244 @@
|
|||
use std::cmp::Ordering;
|
||||
use rand::Rng;
|
||||
|
||||
mod primitives;
|
||||
use self::primitives::*;
|
||||
|
||||
/// 256-bit, stack allocated biginteger for use in prime field
|
||||
/// arithmetic.
|
||||
#[derive(Copy, Clone, Debug, PartialEq, Eq)]
|
||||
pub struct U256([Digit; LIMBS]);
|
||||
|
||||
impl Ord for U256 {
|
||||
#[inline]
|
||||
fn cmp(&self, other: &U256) -> Ordering {
|
||||
for (a, b) in self.0.iter().zip(other.0.iter()).rev() {
|
||||
if *a < *b {
|
||||
return Ordering::Less;
|
||||
} else if *a > *b {
|
||||
return Ordering::Greater;
|
||||
}
|
||||
}
|
||||
|
||||
return Ordering::Equal;
|
||||
}
|
||||
}
|
||||
|
||||
impl PartialOrd for U256 {
|
||||
fn partial_cmp(&self, other: &U256) -> Option<Ordering> {
|
||||
Some(self.cmp(other))
|
||||
}
|
||||
}
|
||||
|
||||
impl From<u32> for U256 {
|
||||
#[inline]
|
||||
fn from(f: u32) -> U256 {
|
||||
U256([f, 0, 0, 0, 0, 0, 0, 0])
|
||||
}
|
||||
}
|
||||
|
||||
impl From<[u32; 8]> for U256 {
|
||||
#[inline]
|
||||
fn from(a: [u32; 8]) -> U256 {
|
||||
U256(a)
|
||||
}
|
||||
}
|
||||
|
||||
impl U256 {
|
||||
#[inline]
|
||||
pub fn zero() -> U256 {
|
||||
0.into()
|
||||
}
|
||||
|
||||
#[inline]
|
||||
pub fn one() -> U256 {
|
||||
1.into()
|
||||
}
|
||||
|
||||
/// Produce a random number (mod `modulo`)
|
||||
pub fn rand<R: Rng>(
|
||||
rng: &mut R,
|
||||
modulo: &U256
|
||||
) -> U256
|
||||
{
|
||||
let mut res;
|
||||
|
||||
loop {
|
||||
res = U256(rng.gen());
|
||||
|
||||
for (i, x) in modulo.bits().enumerate() {
|
||||
if !x {
|
||||
res.set_bit(255 - i, false);
|
||||
} else {
|
||||
break;
|
||||
}
|
||||
}
|
||||
|
||||
if &res < modulo { break; }
|
||||
}
|
||||
|
||||
res
|
||||
}
|
||||
|
||||
pub fn is_zero(&self) -> bool {
|
||||
for a in self.0.iter() {
|
||||
if *a != 0 {
|
||||
return false;
|
||||
}
|
||||
}
|
||||
|
||||
return true;
|
||||
}
|
||||
|
||||
pub fn set_bit(&mut self, n: usize, to: bool)
|
||||
{
|
||||
assert!(n < 256);
|
||||
|
||||
let part = n / 32;
|
||||
let bit = n - (32 * part);
|
||||
|
||||
if to {
|
||||
self.0[part] |= 1 << bit;
|
||||
} else {
|
||||
self.0[part] &= !(1 << bit);
|
||||
}
|
||||
}
|
||||
|
||||
/// Add `other` to `self` (mod `modulo`)
|
||||
pub fn add(&mut self, other: &U256, modulo: &U256) {
|
||||
add_nocarry(&mut self.0, &other.0);
|
||||
|
||||
if *self >= *modulo {
|
||||
sub_noborrow(&mut self.0, &modulo.0);
|
||||
}
|
||||
}
|
||||
|
||||
/// Subtract `other` from `self` (mod `modulo`)
|
||||
pub fn sub(&mut self, other: &U256, modulo: &U256) {
|
||||
if *self < *other {
|
||||
add_nocarry(&mut self.0, &modulo.0);
|
||||
}
|
||||
|
||||
sub_noborrow(&mut self.0, &other.0);
|
||||
}
|
||||
|
||||
/// Multiply `self` by `other` (mod `modulo`) via the Montgomery
|
||||
/// multiplication method.
|
||||
pub fn mul(&mut self, other: &U256, modulo: &U256, inv: u32) {
|
||||
mul_reduce(&mut self.0, &other.0, &modulo.0, inv);
|
||||
|
||||
if *self >= *modulo {
|
||||
sub_noborrow(&mut self.0, &modulo.0);
|
||||
}
|
||||
}
|
||||
|
||||
/// Turn `self` into its additive inverse (mod `modulo`)
|
||||
pub fn neg(&mut self, modulo: &U256) {
|
||||
if *self > Self::zero() {
|
||||
let mut tmp = modulo.0;
|
||||
sub_noborrow(&mut tmp, &self.0);
|
||||
|
||||
self.0 = tmp;
|
||||
}
|
||||
}
|
||||
|
||||
#[inline]
|
||||
pub fn is_even(&self) -> bool {
|
||||
self.0[0] & 1 == 0
|
||||
}
|
||||
|
||||
/// Turn `self` into its multiplicative inverse (mod `modulo`)
|
||||
pub fn invert(&mut self, modulo: &U256) {
|
||||
// Guajardo Kumar Paar Pelzl
|
||||
// Efficient Software-Implementation of Finite Fields with Applications to Cryptography
|
||||
// Algorithm 16 (BEA for Inversion in Fp)
|
||||
|
||||
let mut u = *self;
|
||||
let mut v = *modulo;
|
||||
let mut b = U256::one();
|
||||
let mut c = U256::zero();
|
||||
|
||||
while u != U256::one() && v != U256::one() {
|
||||
while u.is_even() {
|
||||
div2(&mut u.0);
|
||||
|
||||
if b.is_even() {
|
||||
div2(&mut b.0);
|
||||
} else {
|
||||
add_nocarry(&mut b.0, &modulo.0);
|
||||
div2(&mut b.0);
|
||||
}
|
||||
}
|
||||
while v.is_even() {
|
||||
div2(&mut v.0);
|
||||
|
||||
if c.is_even() {
|
||||
div2(&mut c.0);
|
||||
} else {
|
||||
add_nocarry(&mut c.0, &modulo.0);
|
||||
div2(&mut c.0);
|
||||
}
|
||||
}
|
||||
|
||||
if u >= v {
|
||||
sub_noborrow(&mut u.0, &v.0);
|
||||
b.sub(&c, modulo);
|
||||
} else {
|
||||
sub_noborrow(&mut v.0, &u.0);
|
||||
c.sub(&b, modulo);
|
||||
}
|
||||
}
|
||||
|
||||
if u == U256::one() {
|
||||
self.0 = b.0;
|
||||
} else {
|
||||
self.0 = c.0;
|
||||
}
|
||||
}
|
||||
|
||||
/// Return an Iterator<Item=bool> over all bits from
|
||||
/// MSB to LSB.
|
||||
pub fn bits(&self) -> BitIterator {
|
||||
BitIterator {
|
||||
int: &self,
|
||||
n: 256
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
pub struct BitIterator<'a> {
|
||||
int: &'a U256,
|
||||
n: usize
|
||||
}
|
||||
|
||||
impl<'a> Iterator for BitIterator<'a> {
|
||||
type Item = bool;
|
||||
|
||||
fn next(&mut self) -> Option<bool> {
|
||||
if self.n == 0 {
|
||||
None
|
||||
}
|
||||
else {
|
||||
self.n -= 1;
|
||||
|
||||
let part = self.n / 32;
|
||||
let bit = self.n - (32 * part);
|
||||
|
||||
Some(self.int.0[part] & (1 << bit) > 0)
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn setting_bits() {
|
||||
let rng = &mut ::rand::thread_rng();
|
||||
let modulo = U256([0xffffffff; LIMBS]);
|
||||
|
||||
let a = U256::rand(rng, &modulo);
|
||||
let mut e = U256::zero();
|
||||
for (i, b) in a.bits().enumerate() {
|
||||
e.set_bit(255 - i, b);
|
||||
}
|
||||
|
||||
assert_eq!(a, e);
|
||||
}
|
||||
|
|
@ -0,0 +1,130 @@
|
|||
pub type Digit = u32;
|
||||
pub const LIMBS: usize = 8;
|
||||
type DoubleDigit = u64;
|
||||
const BITS: usize = 32;
|
||||
const BASE: DoubleDigit = 1 << BITS;
|
||||
const LO_MASK: DoubleDigit = (-1i32 as DoubleDigit) >> BITS;
|
||||
|
||||
#[inline]
|
||||
fn get_hi(n: DoubleDigit) -> Digit {
|
||||
(n >> BITS) as Digit
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn get_lo(n: DoubleDigit) -> Digit {
|
||||
(n & LO_MASK) as Digit
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn split(n: DoubleDigit) -> (Digit, Digit) {
|
||||
(get_hi(n), get_lo(n))
|
||||
}
|
||||
|
||||
/// Divide by two
|
||||
#[inline]
|
||||
pub fn div2(a: &mut [Digit; LIMBS]) {
|
||||
let mut b = 0;
|
||||
for a in a.iter_mut().rev() {
|
||||
let t = *a << 31;
|
||||
*a = *a >> 1;
|
||||
*a = *a | b;
|
||||
b = t;
|
||||
}
|
||||
}
|
||||
|
||||
#[inline]
|
||||
pub fn add_nocarry(a: &mut [Digit; LIMBS], b: &[Digit; LIMBS]) {
|
||||
#[inline]
|
||||
fn adc(a: Digit, b: Digit, carry: &mut Digit) -> Digit {
|
||||
let (hi, lo) = split((a as DoubleDigit) + (b as DoubleDigit) +
|
||||
(*carry as DoubleDigit));
|
||||
|
||||
*carry = hi;
|
||||
lo
|
||||
}
|
||||
|
||||
let mut carry = 0;
|
||||
|
||||
for (a, b) in a.into_iter().zip(b.iter()) {
|
||||
*a = adc(*a, *b, &mut carry);
|
||||
}
|
||||
|
||||
assert!(0 == carry);
|
||||
}
|
||||
|
||||
#[inline]
|
||||
pub fn sub_noborrow(a: &mut [Digit; LIMBS], b: &[Digit; LIMBS]) {
|
||||
#[inline]
|
||||
fn sbb(a: Digit, b: Digit, borrow: &mut Digit) -> Digit {
|
||||
let (hi, lo) = split(BASE + (a as DoubleDigit) -
|
||||
(b as DoubleDigit) -
|
||||
(*borrow as DoubleDigit));
|
||||
*borrow = (hi == 0) as Digit;
|
||||
lo
|
||||
}
|
||||
|
||||
let mut borrow = 0;
|
||||
|
||||
for (a, b) in a.into_iter().zip(b.iter()) {
|
||||
*a = sbb(*a, *b, &mut borrow);
|
||||
}
|
||||
|
||||
assert!(0 == borrow);
|
||||
}
|
||||
|
||||
|
||||
pub fn mul_reduce(
|
||||
this: &mut [Digit; LIMBS],
|
||||
by: &[Digit; LIMBS],
|
||||
modulus: &[Digit; LIMBS],
|
||||
inv: Digit
|
||||
)
|
||||
{
|
||||
#[inline]
|
||||
fn mac_digit(acc: &mut [Digit], b: &[Digit], c: Digit)
|
||||
{
|
||||
#[inline]
|
||||
fn mac_with_carry(a: Digit, b: Digit, c: Digit, carry: &mut Digit) -> Digit {
|
||||
let (hi, lo) = split((a as DoubleDigit) +
|
||||
(b as DoubleDigit) * (c as DoubleDigit) +
|
||||
(*carry as DoubleDigit));
|
||||
*carry = hi;
|
||||
lo
|
||||
}
|
||||
|
||||
if c == 0 {
|
||||
return;
|
||||
}
|
||||
|
||||
let mut b_iter = b.iter();
|
||||
let mut carry = 0;
|
||||
|
||||
for ai in acc.iter_mut() {
|
||||
if let Some(bi) = b_iter.next() {
|
||||
*ai = mac_with_carry(*ai, *bi, c, &mut carry);
|
||||
} else if carry != 0 {
|
||||
*ai = mac_with_carry(*ai, 0, c, &mut carry);
|
||||
} else {
|
||||
break;
|
||||
}
|
||||
}
|
||||
|
||||
assert!(carry == 0);
|
||||
}
|
||||
|
||||
// The Montgomery reduction here is based on Algorithm 14.32 in
|
||||
// Handbook of Applied Cryptography
|
||||
// <http://cacr.uwaterloo.ca/hac/about/chap14.pdf>.
|
||||
|
||||
let mut res = [0; 2*LIMBS];
|
||||
for (i, xi) in this.iter().enumerate() {
|
||||
mac_digit(&mut res[i..], by, *xi);
|
||||
}
|
||||
|
||||
for i in 0..LIMBS {
|
||||
let k = inv.wrapping_mul(res[i]);
|
||||
mac_digit(&mut res[i..], modulus, k);
|
||||
}
|
||||
|
||||
this.copy_from_slice(&res[LIMBS..]);
|
||||
}
|
||||
353
src/fields/fp.rs
353
src/fields/fp.rs
|
|
@ -1,167 +1,234 @@
|
|||
use rand::Rng;
|
||||
use num::{BigUint, Num};
|
||||
use std::ops::{Mul,Add,Sub,Neg};
|
||||
use std::cmp::{PartialEq, Eq};
|
||||
use std::convert::From;
|
||||
use std::ops::{Add, Sub, Mul, Neg};
|
||||
use std::fmt;
|
||||
use std::marker::PhantomData;
|
||||
use super::Field;
|
||||
use super::FieldElement;
|
||||
|
||||
pub trait PrimeFieldParams {
|
||||
fn modulus() -> BigUint;
|
||||
fn bits() -> usize;
|
||||
use arith::U256;
|
||||
|
||||
pub trait FpParams {
|
||||
fn name() -> &'static str;
|
||||
fn modulus() -> U256;
|
||||
fn inv() -> u32;
|
||||
fn rsquared() -> U256;
|
||||
fn rcubed() -> U256;
|
||||
fn one() -> U256;
|
||||
}
|
||||
|
||||
pub struct Fp<P: PrimeFieldParams> {
|
||||
value: BigUint,
|
||||
_marker: PhantomData<P>
|
||||
pub struct Fp<P: FpParams>(U256, PhantomData<P>);
|
||||
impl<P: FpParams> Copy for Fp<P> { }
|
||||
impl<P: FpParams> Clone for Fp<P> {
|
||||
fn clone(&self) -> Self { *self }
|
||||
}
|
||||
|
||||
impl<P: PrimeFieldParams> fmt::Debug for Fp<P> {
|
||||
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
|
||||
write!(f, "{}({})", P::name(), self.value)
|
||||
}
|
||||
}
|
||||
|
||||
impl<P: PrimeFieldParams> Field for Fp<P> {
|
||||
fn zero() -> Self {
|
||||
use num::Zero;
|
||||
|
||||
Fp {
|
||||
value: BigUint::zero(),
|
||||
_marker: PhantomData
|
||||
}
|
||||
}
|
||||
fn one() -> Self {
|
||||
use num::One;
|
||||
|
||||
Fp {
|
||||
value: BigUint::one(),
|
||||
_marker: PhantomData
|
||||
}
|
||||
}
|
||||
fn random<R: Rng>(rng: &mut R) -> Self {
|
||||
use num::num_bigint::RandBigInt;
|
||||
use num::Zero;
|
||||
|
||||
Fp {
|
||||
value: rng.gen_biguint_range(&BigUint::zero(), &P::modulus()),
|
||||
_marker: PhantomData
|
||||
}
|
||||
}
|
||||
|
||||
fn inverse(&self) -> Self {
|
||||
if self.is_zero() {
|
||||
// TODO: this should likely bleed through the abstraction layers
|
||||
panic!("cannot get the multiplicative inverse of zero")
|
||||
} else {
|
||||
let mut res = Self::one();
|
||||
|
||||
let mut found_one = false;
|
||||
|
||||
let exp = Self::zero() - Self::one() - Self::one();
|
||||
|
||||
for i in (0..P::bits()).rev() {
|
||||
if found_one {
|
||||
res = res.squared();
|
||||
}
|
||||
|
||||
if exp.test_bit(i) {
|
||||
found_one = true;
|
||||
res = self * &res;
|
||||
}
|
||||
}
|
||||
|
||||
res
|
||||
}
|
||||
}
|
||||
|
||||
fn neg(&self) -> Self {
|
||||
use num::Zero;
|
||||
|
||||
Fp {
|
||||
value: if self.value.is_zero() {
|
||||
self.value.clone()
|
||||
} else {
|
||||
P::modulus() - &self.value
|
||||
},
|
||||
_marker: PhantomData
|
||||
}
|
||||
}
|
||||
|
||||
fn mul(&self, other: &Self) -> Self {
|
||||
Fp {
|
||||
value: (&self.value * &other.value) % &P::modulus(),
|
||||
_marker: PhantomData
|
||||
}
|
||||
}
|
||||
|
||||
fn sub(&self, other: &Self) -> Self {
|
||||
if other.value > self.value {
|
||||
Fp {
|
||||
value: (&self.value + P::modulus()) - &other.value,
|
||||
_marker: PhantomData
|
||||
}
|
||||
} else {
|
||||
Fp {
|
||||
value: &self.value - &other.value,
|
||||
_marker: PhantomData
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
fn add(&self, other: &Self) -> Self {
|
||||
let tmp = &self.value + &other.value;
|
||||
if tmp >= P::modulus() {
|
||||
Fp {
|
||||
value: tmp - P::modulus(),
|
||||
_marker: PhantomData
|
||||
}
|
||||
} else {
|
||||
Fp {
|
||||
value: tmp,
|
||||
_marker: PhantomData
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
impl<P: FpParams> PartialEq for Fp<P> {
|
||||
fn eq(&self, other: &Self) -> bool {
|
||||
self.value == other.value
|
||||
self.0 == other.0
|
||||
}
|
||||
}
|
||||
impl<P: FpParams> Eq for Fp<P> { }
|
||||
impl<P: FpParams> fmt::Debug for Fp<P> {
|
||||
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
|
||||
write!(f, "{}({:?})", P::name(), self.0)
|
||||
}
|
||||
}
|
||||
impl<P: FpParams> From<Fp<P>> for U256 {
|
||||
fn from(mut a: Fp<P>) -> Self {
|
||||
a.0.mul(&U256::one(), &P::modulus(), P::inv());
|
||||
|
||||
a.0
|
||||
}
|
||||
}
|
||||
|
||||
impl<P: PrimeFieldParams> Fp<P> {
|
||||
pub fn test_bit(&self, bit: usize) -> bool {
|
||||
// TODO: This is a naive approach.
|
||||
use num::{One, Zero};
|
||||
#[inline]
|
||||
pub fn const_fp<P: FpParams, I: Into<U256>>(i: I) -> Fp<P> {
|
||||
Fp(i.into(), PhantomData)
|
||||
}
|
||||
|
||||
let mut b = BigUint::one();
|
||||
let two = &b + &b;
|
||||
for _ in 0..bit {
|
||||
b = &b + &b;
|
||||
impl<P: FpParams> Fp<P> {
|
||||
pub fn from_str(s: &str) -> Option<Self> {
|
||||
let ints: Vec<_> = {
|
||||
let mut acc = Self::zero();
|
||||
(0..11).map(|_| {let tmp = acc; acc = acc + Self::one(); tmp}).collect()
|
||||
};
|
||||
|
||||
let mut res = Self::zero();
|
||||
for c in s.chars() {
|
||||
match c.to_digit(10) {
|
||||
Some(d) => {
|
||||
res = res * ints[10];
|
||||
res = res + ints[d as usize];
|
||||
},
|
||||
None => {
|
||||
return None;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
(&self.value / b) % two != BigUint::zero()
|
||||
Some(res)
|
||||
}
|
||||
}
|
||||
|
||||
impl<'a, P: PrimeFieldParams> From<&'a str> for Fp<P> {
|
||||
fn from(s: &'a str) -> Self {
|
||||
Fp {
|
||||
value: BigUint::from_str_radix(s, 10).unwrap() % P::modulus(),
|
||||
_marker: PhantomData
|
||||
}
|
||||
impl<P: FpParams> Fp<P> {
|
||||
/// Assumes input is mod p, not exposed publicly
|
||||
fn new_checked(mut a: U256) -> Self {
|
||||
a.mul(&P::rsquared(), &P::modulus(), P::inv());
|
||||
|
||||
Fp(a, PhantomData)
|
||||
}
|
||||
}
|
||||
|
||||
impl<P: PrimeFieldParams> Clone for Fp<P> {
|
||||
fn clone(&self) -> Self {
|
||||
Fp {
|
||||
value: self.value.clone(),
|
||||
_marker: PhantomData
|
||||
}
|
||||
impl<P: FpParams> FieldElement for Fp<P> {
|
||||
fn zero() -> Self {
|
||||
const_fp(U256::zero())
|
||||
}
|
||||
|
||||
fn one() -> Self {
|
||||
const_fp(P::one())
|
||||
}
|
||||
|
||||
fn random<R: Rng>(rng: &mut R) -> Self {
|
||||
Fp::new_checked(U256::rand(rng, &P::modulus()))
|
||||
}
|
||||
|
||||
fn is_zero(&self) -> bool {
|
||||
self.0.is_zero()
|
||||
}
|
||||
|
||||
fn inverse(mut self) -> Self {
|
||||
assert!(!self.is_zero());
|
||||
|
||||
self.0.invert(&P::modulus());
|
||||
self.0.mul(&P::rcubed(), &P::modulus(), P::inv());
|
||||
|
||||
self
|
||||
}
|
||||
}
|
||||
|
||||
forward_ops_to_field_ops!(impl(P: PrimeFieldParams) Fp<P>);
|
||||
impl<P: FpParams> Add for Fp<P> {
|
||||
type Output = Fp<P>;
|
||||
|
||||
fn add(mut self, other: Fp<P>) -> Fp<P> {
|
||||
self.0.add(&other.0, &P::modulus());
|
||||
|
||||
self
|
||||
}
|
||||
}
|
||||
|
||||
impl<P: FpParams> Sub for Fp<P> {
|
||||
type Output = Fp<P>;
|
||||
|
||||
fn sub(mut self, other: Fp<P>) -> Fp<P> {
|
||||
self.0.sub(&other.0, &P::modulus());
|
||||
|
||||
self
|
||||
}
|
||||
}
|
||||
|
||||
impl<P: FpParams> Mul for Fp<P> {
|
||||
type Output = Fp<P>;
|
||||
|
||||
fn mul(mut self, other: Fp<P>) -> Fp<P> {
|
||||
self.0.mul(&other.0, &P::modulus(), P::inv());
|
||||
|
||||
self
|
||||
}
|
||||
}
|
||||
|
||||
impl<P: FpParams> Neg for Fp<P> {
|
||||
type Output = Fp<P>;
|
||||
|
||||
fn neg(mut self) -> Fp<P> {
|
||||
self.0.neg(&P::modulus());
|
||||
|
||||
self
|
||||
}
|
||||
}
|
||||
|
||||
pub struct FrParams;
|
||||
pub type Fr = Fp<FrParams>;
|
||||
impl FpParams for FrParams {
|
||||
fn name() -> &'static str { "Fr" }
|
||||
|
||||
#[inline]
|
||||
fn modulus() -> U256 {
|
||||
// 21888242871839275222246405745257275088548364400416034343698204186575808495617
|
||||
[0xf0000001, 0x43e1f593, 0x79b97091, 0x2833e848, 0x8181585d, 0xb85045b6, 0xe131a029, 0x30644e72].into()
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn inv() -> u32 {
|
||||
0xefffffff
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn rsquared() -> U256 {
|
||||
// 944936681149208446651664254269745548490766851729442924617792859073125903783
|
||||
[0xae216da7, 0x1bb8e645, 0xe35c59e3, 0x53fe3ab1, 0x53bb8085, 0x8c49833d, 0x7f4e44a5, 0x0216d0b1].into()
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn rcubed() -> U256 {
|
||||
// 5866548545943845227489894872040244720403868105578784105281690076696998248512
|
||||
[0xb4bf0040, 0x5e94d8e1, 0x1cfbb6b8, 0x2a489cbe, 0xa19fcfed, 0x893cc664, 0x7fcc657c, 0x0cf8594b].into()
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn one() -> U256 {
|
||||
[0x4ffffffb, 0xac96341c, 0x9f60cd29, 0x36fc7695, 0x7879462e, 0x666ea36f, 0x9a07df2f, 0x0e0a77c1].into()
|
||||
}
|
||||
}
|
||||
|
||||
pub struct FqParams;
|
||||
pub type Fq = Fp<FqParams>;
|
||||
impl FpParams for FqParams {
|
||||
fn name() -> &'static str { "Fq" }
|
||||
|
||||
#[inline]
|
||||
fn modulus() -> U256 {
|
||||
// 21888242871839275222246405745257275088696311157297823662689037894645226208583
|
||||
[0xd87cfd47, 0x3c208c16, 0x6871ca8d, 0x97816a91, 0x8181585d, 0xb85045b6, 0xe131a029, 0x30644e72].into()
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn inv() -> u32 {
|
||||
0xe4866389
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn rsquared() -> U256 {
|
||||
// 3096616502983703923843567936837374451735540968419076528771170197431451843209
|
||||
[0x538afa89, 0xf32cfc5b, 0xd44501fb, 0xb5e71911, 0x0a417ff6, 0x47ab1eff, 0xcab8351f, 0x06d89f71].into()
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn rcubed() -> U256 {
|
||||
// 14921786541159648185948152738563080959093619838510245177710943249661917737183
|
||||
[0xda1530df, 0xb1cd6daf, 0xa7283db6, 0x62f210e6, 0x0ada0afb, 0xef7f0b0c, 0x2d592544, 0x20fd6e90].into()
|
||||
}
|
||||
|
||||
#[inline]
|
||||
fn one() -> U256 {
|
||||
[0xc58f0d9d, 0xd35d438d, 0xf5c70b3d, 0x0a78eb28, 0x7879462c, 0x666ea36f, 0x9a07df2f, 0x0e0a77c1].into()
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_rsquared() {
|
||||
let rng = &mut ::rand::thread_rng();
|
||||
|
||||
for _ in 0..1000 {
|
||||
let a = Fr::random(rng);
|
||||
let b: U256 = a.into();
|
||||
let c = Fr::new_checked(b);
|
||||
|
||||
assert_eq!(a, c);
|
||||
}
|
||||
|
||||
for _ in 0..1000 {
|
||||
let a = Fq::random(rng);
|
||||
let b: U256 = a.into();
|
||||
let c = Fq::new_checked(b);
|
||||
|
||||
assert_eq!(a, c);
|
||||
}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -0,0 +1,128 @@
|
|||
use fields::{FieldElement, const_fp, Fq};
|
||||
use std::ops::{Add, Sub, Mul, Neg};
|
||||
use rand::Rng;
|
||||
|
||||
#[inline]
|
||||
fn non_residue() -> Fq {
|
||||
// (q - 1) is a quadratic nonresidue in Fq
|
||||
// 21888242871839275222246405745257275088696311157297823662689037894645226208582
|
||||
const_fp([317583274, 1757628553, 1923792719, 2366144360, 151523889, 1373741639, 1193918714, 576313009])
|
||||
}
|
||||
|
||||
#[derive(Copy, Clone, Debug, PartialEq, Eq)]
|
||||
pub struct Fq2 {
|
||||
c0: Fq,
|
||||
c1: Fq
|
||||
}
|
||||
|
||||
impl Fq2 {
|
||||
pub fn new(c0: Fq, c1: Fq) -> Self {
|
||||
Fq2 {
|
||||
c0: c0,
|
||||
c1: c1
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
impl FieldElement for Fq2 {
|
||||
fn zero() -> Self {
|
||||
Fq2 {
|
||||
c0: Fq::zero(),
|
||||
c1: Fq::zero()
|
||||
}
|
||||
}
|
||||
|
||||
fn one() -> Self {
|
||||
Fq2 {
|
||||
c0: Fq::one(),
|
||||
c1: Fq::zero()
|
||||
}
|
||||
}
|
||||
|
||||
fn random<R: Rng>(rng: &mut R) -> Self {
|
||||
Fq2 {
|
||||
c0: Fq::random(rng),
|
||||
c1: Fq::random(rng)
|
||||
}
|
||||
}
|
||||
|
||||
fn is_zero(&self) -> bool {
|
||||
self.c0.is_zero() && self.c1.is_zero()
|
||||
}
|
||||
|
||||
fn squared(&self) -> Self {
|
||||
// Devegili OhEig Scott Dahab
|
||||
// Multiplication and Squaring on Pairing-Friendly Fields.pdf
|
||||
// Section 3 (Complex squaring)
|
||||
|
||||
let ab = self.c0 * self.c1;
|
||||
|
||||
Fq2 {
|
||||
c0: (self.c1 * non_residue() + self.c0) * (self.c0 + self.c1) - ab - ab * non_residue(),
|
||||
c1: ab + ab
|
||||
}
|
||||
}
|
||||
|
||||
fn inverse(self) -> Self {
|
||||
// "High-Speed Software Implementation of the Optimal Ate Pairing
|
||||
// over Barreto–Naehrig Curves"; Algorithm 8
|
||||
|
||||
let t = (self.c0.squared() - (self.c1.squared() * non_residue())).inverse();
|
||||
|
||||
Fq2 {
|
||||
c0: self.c0 * t,
|
||||
c1: -(self.c1 * t)
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
impl Mul for Fq2 {
|
||||
type Output = Fq2;
|
||||
|
||||
fn mul(self, other: Fq2) -> Fq2 {
|
||||
// Devegili OhEig Scott Dahab
|
||||
// Multiplication and Squaring on Pairing-Friendly Fields.pdf
|
||||
// Section 3 (Karatsuba)
|
||||
|
||||
let aa = self.c0 * other.c0;
|
||||
let bb = self.c1 * other.c1;
|
||||
|
||||
Fq2 {
|
||||
c0: bb * non_residue() + aa,
|
||||
c1: (self.c0 + self.c1) * (other.c0 + other.c1) - aa - bb
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
impl Sub for Fq2 {
|
||||
type Output = Fq2;
|
||||
|
||||
fn sub(self, other: Fq2) -> Fq2 {
|
||||
Fq2 {
|
||||
c0: self.c0 - other.c0,
|
||||
c1: self.c1 - other.c1
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
impl Add for Fq2 {
|
||||
type Output = Fq2;
|
||||
|
||||
fn add(self, other: Fq2) -> Fq2 {
|
||||
Fq2 {
|
||||
c0: self.c0 + other.c0,
|
||||
c1: self.c1 + other.c1
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
impl Neg for Fq2 {
|
||||
type Output = Fq2;
|
||||
|
||||
fn neg(self) -> Fq2 {
|
||||
Fq2 {
|
||||
c0: -self.c0,
|
||||
c1: -self.c1
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
@ -1,54 +1,95 @@
|
|||
#[macro_use]
|
||||
mod macros;
|
||||
|
||||
pub mod fp;
|
||||
pub mod fp2;
|
||||
pub mod fp6;
|
||||
pub mod fp12;
|
||||
|
||||
#[cfg(test)]
|
||||
pub mod tests;
|
||||
mod fp;
|
||||
mod fq2;
|
||||
|
||||
use arith::U256;
|
||||
use rand::Rng;
|
||||
use self::fp::{Fp, PrimeFieldParams};
|
||||
use std::ops::{Add, Sub, Mul, Neg};
|
||||
use std::fmt::Debug;
|
||||
|
||||
pub trait Field: Sized + Clone + Debug {
|
||||
pub use self::fp::{Fq,Fr,const_fp};
|
||||
pub use self::fq2::Fq2;
|
||||
|
||||
pub trait FieldElement: Sized
|
||||
+ Copy
|
||||
+ Clone
|
||||
+ Add<Output=Self>
|
||||
+ Sub<Output=Self>
|
||||
+ Mul<Output=Self>
|
||||
+ Neg<Output=Self>
|
||||
+ PartialEq
|
||||
+ Eq
|
||||
+ Debug
|
||||
{
|
||||
fn zero() -> Self;
|
||||
fn one() -> Self;
|
||||
fn random<R: Rng>(rng: &mut R) -> Self;
|
||||
fn is_zero(&self) -> bool {
|
||||
self.eq(&Self::zero())
|
||||
}
|
||||
fn inverse(&self) -> Self;
|
||||
fn random<R: Rng>(&mut R) -> Self;
|
||||
fn is_zero(&self) -> bool;
|
||||
fn squared(&self) -> Self {
|
||||
self.mul(self)
|
||||
(*self) * (*self)
|
||||
}
|
||||
fn pow<P: PrimeFieldParams>(&self, exp: &Fp<P>) -> Self {
|
||||
fn inverse(self) -> Self;
|
||||
fn pow<I: Into<U256>>(&self, by: I) -> Self {
|
||||
let mut res = Self::one();
|
||||
|
||||
let mut found_one = false;
|
||||
|
||||
for i in (0..P::bits()).rev() {
|
||||
if found_one {
|
||||
res = res.squared();
|
||||
}
|
||||
|
||||
if exp.test_bit(i) {
|
||||
found_one = true;
|
||||
res = self.mul(&res);
|
||||
for i in by.into().bits() {
|
||||
res = res.squared();
|
||||
if i {
|
||||
res = *self * res;
|
||||
}
|
||||
}
|
||||
|
||||
res
|
||||
}
|
||||
fn eq(&self, other: &Self) -> bool;
|
||||
fn ne(&self, other: &Self) -> bool {
|
||||
!self.eq(other)
|
||||
}
|
||||
|
||||
fn neg(&self) -> Self;
|
||||
fn mul(&self, other: &Self) -> Self;
|
||||
fn sub(&self, other: &Self) -> Self;
|
||||
fn add(&self, other: &Self) -> Self;
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests;
|
||||
|
||||
#[test]
|
||||
fn test_fr() {
|
||||
tests::field_trials::<Fr>();
|
||||
|
||||
assert_eq!(
|
||||
const_fp::<self::fp::FrParams, _>([0xf0000000, 0x43e1f593, 0x79b97091, 0x2833e848, 0x8181585d, 0xb85045b6, 0xe131a029, 0x30644e72]).inverse(),
|
||||
const_fp::<self::fp::FrParams, _>([1105105498, 673779534, 2522683054, 3560287638, 767940567, 738640505, 1642290052, 776830401])
|
||||
);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_fq() {
|
||||
tests::field_trials::<Fq>();
|
||||
|
||||
assert_eq!(
|
||||
const_fp::<self::fp::FqParams, _>([0xd87cfd46, 0x3c208c16, 0x6871ca8d, 0x97816a91, 0x8181585d, 0xb85045b6, 0xe131a029, 0x30644e72]).inverse(),
|
||||
const_fp::<self::fp::FqParams, _>([2230452926, 1223921595, 2485962897, 3784987007, 2000672870, 1889871543, 377056010, 697020161])
|
||||
);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_fq2() {
|
||||
tests::field_trials::<Fq2>();
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_str() {
|
||||
assert_eq!(-Fr::one(), Fr::from_str("21888242871839275222246405745257275088548364400416034343698204186575808495616").unwrap());
|
||||
assert_eq!(-Fq::one(), Fq::from_str("21888242871839275222246405745257275088696311157297823662689037894645226208582").unwrap());
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_fr_squaring() {
|
||||
// Test vector.
|
||||
assert_eq!(
|
||||
Fr::from_str("21888242871839275222246405745257275088548364400416034343698204186575808495610").unwrap().pow([0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff]),
|
||||
const_fp([2572677806, 4136501712, 2816341579, 1079830324, 2386666211, 2796237710, 2115470140, 673230670])
|
||||
);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_fq_squaring() {
|
||||
// Test vector
|
||||
assert_eq!(
|
||||
Fq::from_str("21888242871839275222246405745257275088696311157297823662689037894645226208540").unwrap().pow([0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff]),
|
||||
const_fp([1128196772, 3636718463, 2555618142, 559034227, 1481851987, 3987127805, 2583602581, 88292310])
|
||||
);
|
||||
}
|
||||
|
|
|
|||
|
|
@ -1,181 +1,81 @@
|
|||
use rand::{Rng,SeedableRng,StdRng};
|
||||
use fields::Field;
|
||||
use super::FieldElement;
|
||||
|
||||
mod large_field {
|
||||
use fields::fp::*;
|
||||
use num::{BigUint, Num};
|
||||
|
||||
struct Large;
|
||||
|
||||
impl PrimeFieldParams for Large {
|
||||
fn modulus() -> BigUint {
|
||||
BigUint::from_str_radix("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10).unwrap()
|
||||
}
|
||||
|
||||
fn bits() -> usize { 254 }
|
||||
fn name() -> &'static str { "Large" }
|
||||
}
|
||||
|
||||
type Ft = Fp<Large>;
|
||||
|
||||
#[test]
|
||||
fn bit_testing() {
|
||||
let a = Ft::from("13");
|
||||
assert!(a.test_bit(0) == true);
|
||||
assert!(a.test_bit(1) == false);
|
||||
assert!(a.test_bit(2) == true);
|
||||
assert!(a.test_bit(3) == true);
|
||||
|
||||
let expected: Vec<bool> = [1,1,0,1,1,0,0,0,0,1,0,0,1,1,1,0,0,0,0,0,1,1,0,0,1,0,0,1,1]
|
||||
.iter().map(|a| *a == 1).rev().collect();
|
||||
|
||||
let a = Ft::from("453624211");
|
||||
|
||||
for (i, b) in expected.into_iter().enumerate() {
|
||||
assert!(a.test_bit(i) == b);
|
||||
}
|
||||
|
||||
let expected: Vec<bool> = [1,1,1,1,0,1,0,1,1,0,1,0,0,0,1,1,1,0,1,1,1,1,0,0,0,0,1,1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,1,1,1,1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,1,1,1,0,1,1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0,0,0,1,1,0,0,1,1,1,1,0,1,1,1,1,0,0,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0,1,1,0,1,1,0,0,1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,1,1,1,1,1,0,0,1,1,1,1,1,1,0,1,0,0,1,0,1,1,0,0]
|
||||
.iter().map(|a| *a == 1).rev().collect();
|
||||
let a = Ft::from("13888242871869275222244405745257275088696211157297823662689037894645226208556");
|
||||
|
||||
for (i, b) in expected.into_iter().enumerate() {
|
||||
assert!(a.test_bit(i) == b);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
mod small_field {
|
||||
use fields::fp::*;
|
||||
use fields::Field;
|
||||
use num::{BigUint, Num};
|
||||
|
||||
struct Small;
|
||||
|
||||
impl PrimeFieldParams for Small {
|
||||
fn modulus() -> BigUint {
|
||||
BigUint::from_str_radix("13", 10).unwrap()
|
||||
}
|
||||
|
||||
fn bits() -> usize { 6 }
|
||||
fn name() -> &'static str { "Small" }
|
||||
}
|
||||
|
||||
type Ft = Fp<Small>;
|
||||
|
||||
#[test]
|
||||
fn field_ops() {
|
||||
fn test_field_operation<C: Fn(&Ft, &Ft) -> Ft>(a: u64, b: u64, f: C, expected: u64) {
|
||||
let af = Ft::from(format!("{}", a).as_ref());
|
||||
let bf = Ft::from(format!("{}", b).as_ref());
|
||||
let expectedf = Ft::from(format!("{}", expected).as_ref());
|
||||
|
||||
let res = f(&af, &bf);
|
||||
|
||||
if res != expectedf {
|
||||
panic!("res={:?} != expectedf={:?} (a={}, b={}, expected={})", res, expectedf, a, b, expected);
|
||||
}
|
||||
}
|
||||
|
||||
const MODULO: u64 = 13;
|
||||
|
||||
for a in 0..13u64 {
|
||||
for b in 0..13u64 {
|
||||
test_field_operation(a, b, |a,b| {a * b}, (a*b)%MODULO);
|
||||
test_field_operation(a, b, |a,b| {a + b}, (a+b)%MODULO);
|
||||
test_field_operation(a, b, |a,b| {a - b}, {
|
||||
let mut tmp = (a as i64) - (b as i64);
|
||||
if tmp < 0 {
|
||||
tmp += MODULO as i64;
|
||||
}
|
||||
|
||||
tmp as u64
|
||||
});
|
||||
test_field_operation(a, b, |a,b| {a.pow(b)}, (a.pow(b as u32))%MODULO);
|
||||
}
|
||||
test_field_operation(a, 0, |a,_| {-a}, if a == 0 { 0 } else { MODULO - a });
|
||||
if a > 0 {
|
||||
test_field_operation(a, 0, |a,_| {&a.inverse() * a}, 1);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
fn can_invert<F: Field>() {
|
||||
fn can_invert<F: FieldElement>() {
|
||||
let mut a = F::one();
|
||||
for _ in 0..1000 {
|
||||
assert!(a.ne(&F::zero()));
|
||||
|
||||
let inv = a.inverse();
|
||||
for _ in 0..10000 {
|
||||
assert_eq!(a * a.inverse(), F::one());
|
||||
|
||||
assert!(a.mul(&inv).eq(&F::one()));
|
||||
a = a + F::one();
|
||||
}
|
||||
|
||||
a = a.add(&F::one());
|
||||
a = -F::one();
|
||||
for _ in 0..10000 {
|
||||
assert_eq!(a * a.inverse(), F::one());
|
||||
|
||||
a = a - F::one();
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
fn rand_element_squaring<F: Field, R: Rng>(rng: &mut R) {
|
||||
fn rand_element_squaring<F: FieldElement, R: Rng>(rng: &mut R) {
|
||||
for _ in 0..100 {
|
||||
let a = F::random(rng);
|
||||
|
||||
let mul = a.mul(&a);
|
||||
let sq = a.squared();
|
||||
|
||||
assert!(sq.eq(&mul));
|
||||
assert!(a * a == a.squared());
|
||||
}
|
||||
|
||||
let mut cur = F::zero();
|
||||
for _ in 0..100 {
|
||||
let mul = cur.mul(&cur);
|
||||
let sq = cur.squared();
|
||||
assert_eq!(cur.squared(), cur * cur);
|
||||
|
||||
assert!(sq.eq(&mul));
|
||||
|
||||
cur = cur.add(&F::one());
|
||||
cur = cur + F::one();
|
||||
}
|
||||
}
|
||||
|
||||
fn rand_element_addition_and_negation<F: Field, R: Rng>(rng: &mut R) {
|
||||
for _ in 0..10 {
|
||||
fn rand_element_addition_and_negation<F: FieldElement, R: Rng>(rng: &mut R) {
|
||||
for _ in 0..100 {
|
||||
let mut a = F::random(rng);
|
||||
let r = F::random(rng);
|
||||
let mut b = a.add(&r);
|
||||
let mut b = a + r;
|
||||
|
||||
for _ in 0..10 {
|
||||
let r = F::random(rng);
|
||||
a = a.add(&r);
|
||||
b = b.add(&r);
|
||||
a = a + r;
|
||||
b = b + r;
|
||||
|
||||
let r = F::random(rng);
|
||||
a = a.sub(&r);
|
||||
b = b.sub(&r);
|
||||
a = a - r;
|
||||
b = b - r;
|
||||
|
||||
let r = F::random(rng);
|
||||
a = a.add(&r);
|
||||
b = b.add(&r);
|
||||
a = a + (-(-r));
|
||||
b = b + (-(-r));
|
||||
|
||||
let r = F::random(rng);
|
||||
a = a - r;
|
||||
b = b + (-r);
|
||||
|
||||
let r = F::random(rng);
|
||||
a = a + (-r);
|
||||
b = b - r;
|
||||
}
|
||||
|
||||
b = b.sub(&r);
|
||||
assert!(a.eq(&b));
|
||||
b = b - r;
|
||||
assert_eq!(a, b);
|
||||
}
|
||||
}
|
||||
|
||||
fn rand_element_inverse<F: Field, R: Rng>(rng: &mut R) {
|
||||
for _ in 0..100 {
|
||||
let mut n = F::random(rng);
|
||||
n = n.inverse().mul(&n);
|
||||
assert!(n.eq(&F::one()));
|
||||
}
|
||||
for _ in 0..100 {
|
||||
fn rand_element_inverse<F: FieldElement, R: Rng>(rng: &mut R) {
|
||||
for _ in 0..10000 {
|
||||
let a = F::random(rng);
|
||||
assert!(a.inverse() * a == F::one());
|
||||
let b = F::random(rng);
|
||||
assert!(a.mul(&b).mul(&a.inverse()).eq(&b));
|
||||
assert_eq!((a * b) * (a.inverse()), b);
|
||||
}
|
||||
}
|
||||
|
||||
fn rand_element_multiplication<F: Field, R: Rng>(rng: &mut R) {
|
||||
fn rand_element_multiplication<F: FieldElement, R: Rng>(rng: &mut R) {
|
||||
// If field is not associative under multiplication, 1/8 of all triplets a, b, c
|
||||
// will fail the test (a*b)*c = a*(b*c).
|
||||
|
||||
|
|
@ -184,13 +84,17 @@ fn rand_element_multiplication<F: Field, R: Rng>(rng: &mut R) {
|
|||
let b = F::random(rng);
|
||||
let c = F::random(rng);
|
||||
|
||||
assert!(a.mul(&b).mul(&c).eq(&b.mul(&c).mul(&a)));
|
||||
assert_eq!((a * b) * c, a * (b * c));
|
||||
}
|
||||
}
|
||||
|
||||
pub fn field_trials<F: Field>() {
|
||||
pub fn field_trials<F: FieldElement>() {
|
||||
can_invert::<F>();
|
||||
|
||||
assert_eq!(-F::zero(), F::zero());
|
||||
assert_eq!(-F::one() + F::one(), F::zero());
|
||||
assert_eq!(F::zero() - F::zero(), F::zero());
|
||||
|
||||
let seed: [usize; 4] = [103245, 191922, 1293, 192103];
|
||||
let mut rng = StdRng::from_seed(&seed);
|
||||
|
||||
|
|
@ -199,5 +103,3 @@ pub fn field_trials<F: Field>() {
|
|||
rand_element_multiplication::<F, StdRng>(&mut rng);
|
||||
rand_element_inverse::<F, StdRng>(&mut rng);
|
||||
}
|
||||
|
||||
|
||||
|
|
|
|||
405
src/lib.rs
405
src/lib.rs
|
|
@ -1,406 +1,7 @@
|
|||
extern crate num;
|
||||
extern crate rand;
|
||||
|
||||
mod arith;
|
||||
mod fields;
|
||||
mod params;
|
||||
|
||||
mod groups;
|
||||
|
||||
pub use groups::Gt;
|
||||
pub use fields::fp::Fp;
|
||||
pub use fields::fp2::Fp2;
|
||||
pub use fields::fp6::Fp6;
|
||||
pub use fields::fp12::Fp12;
|
||||
pub use params::{FrParams,FqParams,Fq2Params,G1Params,G2Params,Fq6Params,Fq12Params,ate_loop_count,final_exponent_z};
|
||||
pub use fields::Field;
|
||||
pub use groups::{Jacobian};
|
||||
use groups::{EllCoeffs,Affine};
|
||||
use fields::fp::PrimeFieldParams;
|
||||
|
||||
pub type Fr = Fp<FrParams>;
|
||||
pub type Fq = Fp<FqParams>;
|
||||
pub type Fq2 = Fp2<Fq2Params>;
|
||||
pub type Fq6 = Fp6<Fq6Params>;
|
||||
pub type Fq12 = Fp12<Fq12Params>;
|
||||
|
||||
pub type Scalar = Fr;
|
||||
pub type G1 = Jacobian<G1Params>;
|
||||
pub type G2 = Jacobian<G2Params>;
|
||||
|
||||
#[derive(PartialEq,Eq)]
|
||||
struct G2Precomp {
|
||||
q: Affine<G2Params>,
|
||||
coeffs: Vec<EllCoeffs>
|
||||
}
|
||||
|
||||
impl G2Precomp {
|
||||
fn new(from: &G2) -> G2Precomp {
|
||||
let g2 = from.to_affine();
|
||||
|
||||
let mut r = g2.to_jacobian();
|
||||
|
||||
let mut result = G2Precomp {
|
||||
q: g2.clone(),
|
||||
coeffs: Vec::new()
|
||||
};
|
||||
|
||||
let two_inv = Fq::from("2").inverse();
|
||||
|
||||
let mut found_one = false;
|
||||
let ate = ate_loop_count();
|
||||
for i in (0..FrParams::bits()).rev() {
|
||||
let bit = ate.test_bit(i);
|
||||
|
||||
if !found_one {
|
||||
found_one = bit;
|
||||
continue;
|
||||
}
|
||||
|
||||
result.coeffs.push(r.doubling_step_for_flipped_miller_loop(&two_inv));
|
||||
|
||||
if bit {
|
||||
result.coeffs.push(r.mixed_addition_step_for_flipped_miller_loop(&g2));
|
||||
}
|
||||
}
|
||||
|
||||
let q1 = g2.mul_by_q();
|
||||
let q2 = q1.mul_by_q().neg();
|
||||
|
||||
result.coeffs.push(r.mixed_addition_step_for_flipped_miller_loop(&q1));
|
||||
result.coeffs.push(r.mixed_addition_step_for_flipped_miller_loop(&q2));
|
||||
|
||||
result
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_prepared_g2() {
|
||||
let g2 = G2::one() * Fr::from("20390255904278144451778773028944684152769293537511418234311120800877067946");
|
||||
|
||||
let g2_p = G2Precomp::new(&g2);
|
||||
|
||||
let expected_g2_p = G2Precomp {
|
||||
q: G2::new(
|
||||
Fq2::new(
|
||||
Fq::from("13936578204263895229092967414825041569724079982537616431212348844388899776640"),
|
||||
Fq::from("6372636725635053773371996212293600406870925440022386078671828127711809436031")
|
||||
),
|
||||
Fq2::new(
|
||||
Fq::from("19293035970010898452454381709939058714495082898872914526540420247178075881697"),
|
||||
Fq::from("13822349107533275437553410197128434338071760794202849712402800746887549494974"),
|
||||
),
|
||||
Fq2::new(
|
||||
Fq::from("1"),
|
||||
Fq::from("0"),
|
||||
)
|
||||
).unwrap().to_affine(),
|
||||
coeffs: vec![
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("2627043964130257285960335798481049684473505235282434252362906013168360965985"), Fq::from("14787221188041042838526170263203159226721816513593198738527261794343476005673")), ell_vw: Fq2::new(Fq::from("5190413803656753539584048070636432748402456516849818272297235294934300653772"), Fq::from("16131787528611999569385991096257681501249100726189947900572474295515353427218")), ell_vv: Fq2::new(Fq::from("11284217811624345285836951206193951701052480344824103057273798441033858118424"), Fq::from("6392116536365389562188363539617434898082427085812013656312597845585909267447")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("827617134098165717451808940080463277390770457691666780560712143809003953598"), Fq::from("6776229088211374446530917353066321938640163548858035832637439231634790575465")), ell_vw: Fq2::new(Fq::from("987776078024262725561041258416387561158070255475504730561661362421251696401"), Fq::from("15312963471998242334683179861466148222641884112991952428739813077336923189144")), ell_vv: Fq2::new(Fq::from("2813988028633040066320201189843971639620433430176492766961373503539074898364"), Fq::from("17055167212030988864288747645634552775658830115224987200613220012554982651578")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17093477194591041266397380404112224367919571835678040889250664706433487493043"), Fq::from("8643160753646550179100165428401120938543691942215161396502118430923535820500")), ell_vw: Fq2::new(Fq::from("10933962898922943120024964690690003920835888964592980369975615298196656183772"), Fq::from("10054435552662455933652209211749007190981947542684285471884164039871580864235")), ell_vv: Fq2::new(Fq::from("1475406195060586931258578851599932495282912871465827155842007043242102046659"), Fq::from("5049830924161187876168845800328902567850325970867534538120475909389784841990")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("10434276065336457961840942984114655645641355797054067910262735814196303889171"), Fq::from("13878009844584745291193244072670264656592436467816383615606044714978763054890")), ell_vw: Fq2::new(Fq::from("20993505060568459085534388708128029516059470166174503494622391683433030357130"), Fq::from("10192769806017258272908841309051731389378033999625803563624881649401237928876")), ell_vv: Fq2::new(Fq::from("17377091829483421118284926147077357677507183290871250621761583281560333718550"), Fq::from("18297780639039540893260206902113850213293978821982237672423968999898103045256")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("5064242330961655837810472366173802140673816516016739528934082190598759216129"), Fq::from("20172028727708469864987113767379594545870343866584998839699240724281670882460")), ell_vw: Fq2::new(Fq::from("2996734698689069564052128450078797792056313730363946953355442668966438200951"), Fq::from("9910941594900797404370917094355311738210439203708671856236954583027355115302")), ell_vv: Fq2::new(Fq::from("6899792008443933570863502403567742315007264221354931185809349393350618040985"), Fq::from("18692424201235977456836009406318755884773471215777102117879530727096807383696")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9406196439119227017197200697795682187217614676635965071601916278141238949664"), Fq::from("1156397052245190909533108891675012415349316096091404012699655652677336827483")), ell_vw: Fq2::new(Fq::from("9249275523859165912800910211783530541095037634552893633043938333737241478198"), Fq::from("13674086724537885208439774394455021050268654286498757861381516198880504360984")), ell_vv: Fq2::new(Fq::from("5760486636714317173624828119078349011897522431614835257025421978626292083787"), Fq::from("19195594701877562770296730558597966521343920474874926443609709880727520577918")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12166638768862632131967787101186130559865354321458713212316891357299196109517"), Fq::from("1517854601414954363921968422436855204960697222979994177630673862982254831193")), ell_vw: Fq2::new(Fq::from("13366104107979832799705928625277388583673352228059815944738106504080223007902"), Fq::from("21648874073566751720910466086303831459661520354253552089758049591613143727506")), ell_vv: Fq2::new(Fq::from("3488776451414334656597670851583995239177248056342375163461521452312663358741"), Fq::from("20333414955874213710027365371143864644434939642550352414403887754960702108637")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20468104499950674906991599591024893438007901046236752159815194628487603192867"), Fq::from("19069685413250623899891139234721987960230148622397590130148098858949057649761")), ell_vw: Fq2::new(Fq::from("2458162095258922865086779772179745380154704066745856173311580123496018896409"), Fq::from("14594824564551859025266692529845827840804005950672166057231487026855067744741")), ell_vv: Fq2::new(Fq::from("9689655654286496198845571066967567466951511863880400774464590553022163229348"), Fq::from("10870127206120874488753221889350215535903993427218589071224774335669836789433")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15334990572515495832170458963151379997993357347608954164027751164770129172212"), Fq::from("13576640273125978757652414940234881692674718901710647917847939367156260249620")), ell_vw: Fq2::new(Fq::from("17441399177222030197667035179458353890777336233312342937564354849059149975908"), Fq::from("17007488955907409465543686050546106251553784366237996525801373957978511005278")), ell_vv: Fq2::new(Fq::from("11907792503811346855438657769533223907267837220830333730410940125854558637383"), Fq::from("19490060942462197243379937108210110931787757078893580540054690416131017931738")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17315779415820661602591970083599829709759601467527578787441772726289774228451"), Fq::from("12080802446892182171725856453369856169039939341293608060423916935382017714476")), ell_vw: Fq2::new(Fq::from("12912774602739952889552025090054804346758888115608888766725054474682313950190"), Fq::from("18204955971588024842905918028829132053489100674596042021111813303048407250440")), ell_vv: Fq2::new(Fq::from("15721987682182104464109212111163313043868345920774975073102123159159010623181"), Fq::from("12041652399509652544977783565303273845883245418094457529808044120827920548972")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19999070717670362479390656255660288732744476951677182489244240784338456791622"), Fq::from("3906002755238658676658974095984902840159437088441472633765497396881255783772")), ell_vw: Fq2::new(Fq::from("4998301920940599524538129746790252820575786197710804708434384224972978627969"), Fq::from("13740819604543965458992763357705315276969968429480411530007322182403523242226")), ell_vv: Fq2::new(Fq::from("12357083524385039711813704948190809675818592911528835736620903716727226601953"), Fq::from("13949012942996636079549854439740332990496218971788334041233648192605019948979")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19805154539184543843735550309592886829271314363908537964547876413114433734624"), Fq::from("16481416862400234230001935040590314552205484054708993091993487440981092053599")), ell_vw: Fq2::new(Fq::from("2850933690997557614680452825756662353227291804228932023648526608593113870536"), Fq::from("4786423507519240726599235257249783193682886531662683554713943115802063984530")), ell_vv: Fq2::new(Fq::from("1023533030117941985730558522296044345509721849960856436490999820443312747347"), Fq::from("16431548648882316869128893343209930754508425074299553782883149795599907406302")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17203075478724975791441470425150871365957382183648220405350599958349478966969"), Fq::from("19566853320706390536799238411444563555185156853468371794865496424286288424312")), ell_vw: Fq2::new(Fq::from("15310908404682220539815850064952575682898427316407655213706791187971849703724"), Fq::from("16518878079785062271298378680376701241016697782308466806532677867880353948389")), ell_vv: Fq2::new(Fq::from("14461573034674366293217641735721285869692106578982161360427147953320122006893"), Fq::from("6682535969691272372531048932989583075363217464304983455197465867679730319705")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20895553497690200905762157235991552162755238828234592755315911339400881373107"), Fq::from("11947919061415103173593352950044808416347417810515010536678086292686107222044")), ell_vw: Fq2::new(Fq::from("6277981470671734560229149673222400771590408919719217743512900463425397921908"), Fq::from("857212248193599195410711267373529614085739023676782457331058533755337294689")), ell_vv: Fq2::new(Fq::from("13885916052404740835214284721162700704805447448922857617672445286139626987191"), Fq::from("19585389042100086858633036225209396447924441966778662250950999864988284098182")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12748401205319976481200663793287165427761341490879316890008250070038707706903"), Fq::from("9347025175062646190962067344106887221752685406563421042329129538495481178145")), ell_vw: Fq2::new(Fq::from("12819473645128401077507234830331609583133696659610406468927174080626676619197"), Fq::from("9039389340404634934394726812078086316924048399159604642340339967187032834011")), ell_vv: Fq2::new(Fq::from("3576972655652865483567741810151575278761470911419381093571973994483543095285"), Fq::from("15017468236028111406921228193688973260631093975412858847563400987354453569177")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15407570760857043664675198351546637734249070940748070513828892036078049928562"), Fq::from("16864837887933576789187120028615006826049916557295792205508706545038103706967")), ell_vw: Fq2::new(Fq::from("10721360465919602144653520112889689786555051462198759164582568961994647563952"), Fq::from("15294101887395462362997873520785213441488786364828649600395509678143054744021")), ell_vv: Fq2::new(Fq::from("338877711237691732068166553384070140091720428235937017002925264628733553275"), Fq::from("8565770517833678396194963042166078986348581926120249815567974171840711179000")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17302401852599488824262121640328139426135883695023154134865582322858078206108"), Fq::from("17762524466294762819341377108361075499043287383142581974755577861093917129883")), ell_vw: Fq2::new(Fq::from("13015933066581331586057031655765394156438563571733887223640331819027233831808"), Fq::from("758435331870656231667722761984400968773108978847112431665229252181053186864")), ell_vv: Fq2::new(Fq::from("1969703955244816291771524536928710541897907643982046151796341732075930805350"), Fq::from("7031491051442716548063495812915693048061021502544985435130728834982309403017")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("3256431897334799395000264533777543923057110731812322608781781329228902421476"), Fq::from("12181683857561584905762347955319259913388176042023651776593530168946356566719")), ell_vw: Fq2::new(Fq::from("796109596241493969908761241268545233387446802589790513588935996990189190052"), Fq::from("1881011435205668659920588004752042238985141500261993259857887774967416952583")), ell_vv: Fq2::new(Fq::from("8939978626202472531630769965940628317376245215184468794030552730424109408120"), Fq::from("6876296539554825957959953448623382464100292373640362636397165101276891176372")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9977776755404508431321449793657366203354342168643747704006143818738023991417"), Fq::from("17287266532802977377832339048078093713694407321084566908200498605546793435820")), ell_vw: Fq2::new(Fq::from("13828304001931491210449888619926027655755205497065268858009259221646932328015"), Fq::from("15160507616608365341639459841002920786281000303114113782821091199951204875277")), ell_vv: Fq2::new(Fq::from("11161349646944933007552237093278324942110971730799870343491465916610232191523"), Fq::from("1406353163795290818414111649956135144227974028417320470897284179379367096127")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7051926909684750402992796647032741235565177570654429988875094870973533133596"), Fq::from("20900584116079647608136401104271956963096697793208792824067775068966626057321")), ell_vw: Fq2::new(Fq::from("16845876804097000415961425779514720605919619149076212452284059758659650969766"), Fq::from("20806461795944565303534619813753388058736049320841472653130287521152836019826")), ell_vv: Fq2::new(Fq::from("14700937697004489523945953560825193972493636932341330040072362595384517591325"), Fq::from("17137955424118909932502168320776219042080643724609838664247554484115994974522")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("11997142079850902731018491830701934107296522353910410046841604519639199995406"), Fq::from("7846127392296015970710836702134527490160588060582987110380879354952036676012")), ell_vw: Fq2::new(Fq::from("17663414534578185845724108244425572237371794484114570058559740097895281553299"), Fq::from("5029140669755495687983399191536061323171057684732073778423303267496666431971")), ell_vv: Fq2::new(Fq::from("18428442704115060890708987869503078260482829714024061977805185634781850796860"), Fq::from("10997089106320530533400287751108615584976629751702164406696900094998369084403")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13049677819639682137107272322562642308503411465986079606334189062279441432203"), Fq::from("8874624972677935217588093490885486721673594121072678233714072789659128555367")), ell_vw: Fq2::new(Fq::from("17057994029556488440657870420250293132874472170733557198308184169150330844407"), Fq::from("21633949388482190843211557537195287403106536528012825991981214057356654097223")), ell_vv: Fq2::new(Fq::from("6897044784874980042628318562068623805084522888972873438474139199226581764756"), Fq::from("2692625602037608101594849144393949134766066123857076012132350588113519149572")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9195356099870032305655798667367006686897880523242566329014149856866611060003"), Fq::from("8052880750949660720559492348565167987253311417692073778040482170195175946428")), ell_vw: Fq2::new(Fq::from("14126308928695836758114496610102001680995565158909643521688860213965125059872"), Fq::from("671075569038912473223544948981506373517307914345286644404839238190008163502")), ell_vv: Fq2::new(Fq::from("16828103633974871724641544770525140436938333764966095252502003672344759889143"), Fq::from("11294529501746593948507624028584472676389760897716457892342249916318954501723")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21124500310722447577044753724306337899524073055454053171030128408285122310058"), Fq::from("6266274689105996173186441194758121417834460925587732781640173398761521329055")), ell_vw: Fq2::new(Fq::from("21576331134605070996263178912402854747994495442070419140713471256922277734125"), Fq::from("10074747339717657657921727946300324710951646905879034242434327125816600264379")), ell_vv: Fq2::new(Fq::from("21765519811276677250031796638148457542841530645055777846258335421409598000534"), Fq::from("21113224637771003677145667454854210585751245642107143223999187554888833454243")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("3972300323627467772646468544189830113523911494046612234673787196935654523692"), Fq::from("4678843490808368914310764558657240771944434980659329233409943752329617161210")), ell_vw: Fq2::new(Fq::from("17273599953138772488805612764323870175561796764345770950040891840763740358904"), Fq::from("20415406312940209525918367703821373463952436739154646433333601347381723237136")), ell_vv: Fq2::new(Fq::from("9160987523541118769404859219229455872777126347530590315596894278463599930633"), Fq::from("9054614112960826755739076858442558227570393423663838494067070216861352843202")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9994954547022379603539679290217274617475455164007187108524364097751858323148"), Fq::from("20207771379440433457553782484784849880846158790602128386495229840158065235194")), ell_vw: Fq2::new(Fq::from("4256518944443059194982602310083745017699415852751800911480635583088649250060"), Fq::from("15878038811235111024146051545935959211923223574364333740058604258589674044428")), ell_vv: Fq2::new(Fq::from("15953919860849853924955813198899405295748673040941543212644580663271517228860"), Fq::from("19594437991878043589671466710739298733201896072505590020656735731744884281871")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("1333543700232406379832215757617068657339748039338978081735192495239856176346"), Fq::from("13583045269132775344737980756385257860562307101313068058661941588822340692712")), ell_vw: Fq2::new(Fq::from("11316045827391404343818336116749364562466461792138476673985290066961360847098"), Fq::from("6683447190424452097068562814518005185732181765401062334661636064120276671717")), ell_vv: Fq2::new(Fq::from("21483741541431414142042872578822487244163622589552506298253453995156009765340"), Fq::from("16502358172728062550923345759488126380665850742066670591562640787191536290296")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4727679401549029925466046980042210063407620223804057104870816459356196798234"), Fq::from("4245208253099058296636688301996648265313738760450861302514451065904449375848")), ell_vw: Fq2::new(Fq::from("19589426516750929424306774984752060788246605243046082989256140076730177865962"), Fq::from("19202599366770569221216501935740579180019961851730854119509268985533547631061")), ell_vv: Fq2::new(Fq::from("18329650333197074546830734977650617140322812110454751288122555715911145072440"), Fq::from("18685828888894718887886440900247948352873037333999376257561751598045361934940")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15348453159719894937627693446890053385899104841918429189733580400902437025934"), Fq::from("7501190780205720975723852235835733848687299193764526538831345783610137314719")), ell_vw: Fq2::new(Fq::from("12341649147852680664345690534373468938440543264264387581538764313149640739302"), Fq::from("140543540021318882234026314588789476198722270107997413913717074968501481488")), ell_vv: Fq2::new(Fq::from("5960374673747504135132782820167542238733653290730217772559545641742563125282"), Fq::from("14507139191276441762644240084427079430815677404266073216544260783868466836469")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("1463974716527100203803811366059855506152810619047460592789012516365557382809"), Fq::from("12894560712946121716686238503640546420854178898471895472462918225441749701869")), ell_vw: Fq2::new(Fq::from("10508120321944855490271402587060090544503473558782513226881398929302598320386"), Fq::from("20659878091446285231719184399846880718538938504006216923931157918871132219202")), ell_vv: Fq2::new(Fq::from("9314687360620406591930278900314522432964075856747694070384017245464130048974"), Fq::from("17694675150016443475335083504221747995503842086249559620223686305992007745569")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14196539467892272085884829545587228956041818801561947743113049357924737022135"), Fq::from("11319357657686312684765696405805456576475636756706867678717962076341276658626")), ell_vw: Fq2::new(Fq::from("2758949814978246411081695103954624266457693267107844100147125044169545866221"), Fq::from("9492100316900274072027501035053797502489796059345864111677781413985743864383")), ell_vv: Fq2::new(Fq::from("4704107896591496617393915690357581449785399000813194972371050438639491862314"), Fq::from("11531365345870313813093010071065540963321572589131340821665512078366854738990")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18802465664727774773666122120775907428094487475952383150269507875840089541246"), Fq::from("1731637922302665944958348222164323910730404298281499149800917211427254836819")), ell_vw: Fq2::new(Fq::from("9297609456558817283944780415839637083190742770917810313181849518385379162642"), Fq::from("16377246183633200781091121197835476043913744319877154056563766308564474143489")), ell_vv: Fq2::new(Fq::from("9542461764772800142773586971354345045722377861760780709590923740133551871717"), Fq::from("21640615613954317564353959148225302612886621120325752435425437529806262143694")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19906049876176293236988182778475214525096978055900789654464603831282713330781"), Fq::from("20362580925091945825523087376754169675844247448742703441518744743887060798158")), ell_vw: Fq2::new(Fq::from("12267599301269062964822189844562856172284191535014283588478875529832316401461"), Fq::from("16669825137620567269982790548542800570592942517936126877686519256685036586668")), ell_vv: Fq2::new(Fq::from("8883573260261802165780406246211290131940260376316892224137508921505842308805"), Fq::from("19231108274886583687672953546761020040720197580241947146286750565108059707992")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("634182853515641884871095612114627494589971356340786857463490962898713112220"), Fq::from("11430884471074068345749191262196070058384500365559620896062382995797164195815")), ell_vw: Fq2::new(Fq::from("10550902705393315932525976596381459439846414130661652807183357995272197708172"), Fq::from("3961587743443277010160905820636488892023891053433292002931434257476626412826")), ell_vv: Fq2::new(Fq::from("12451440882697904111698352784988882971851697245540249016187908252126835129000"), Fq::from("11782592623486603129029758483031965320434451423371375539247662743178943477576")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("1404156345033137762434737426689373534725204733157905834242027185649572912742"), Fq::from("19570890322851668915358612375948367153948206621537633922343603465980161569078")), ell_vw: Fq2::new(Fq::from("8686503710040621382075118769097735385052600877432244428197116763766844918162"), Fq::from("17223447941612107204326938316335236704702567123017447936901253775530936851086")), ell_vv: Fq2::new(Fq::from("18767158861904777443587157807982315566545548385817262926280782765269843879565"), Fq::from("21334543677892438295079840326280352051766440588072368601899791201007143326750")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20453014818098756686381445228623803935714316636974320420742351477862354368007"), Fq::from("11004429848286161246832723709453157691786114066215975460914111961981461575367")), ell_vw: Fq2::new(Fq::from("14444837779919119651023367195508370508399423510380962187645910198528103345317"), Fq::from("4021193867366929691652413553892868612329317391162172266658368471326882495507")), ell_vv: Fq2::new(Fq::from("21098047915839588616030399663527845846589272989005835932893550785408588982098"), Fq::from("16796487305049888740077005894512139332211008298712590271618863352993144873711")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8799932768264091269394526725962837485495458036153083882004277055769655900574"), Fq::from("18596145973176446599121174893654309094121045316907408604913712044917019537350")), ell_vw: Fq2::new(Fq::from("5727901458562943136998396728651681550262281371313580881804977990767435431080"), Fq::from("16665276274186006945492338822028666786935953718794303569570538737692828117635")), ell_vv: Fq2::new(Fq::from("10910280368195212990489805322100030141483989387720003214110076576431959738476"), Fq::from("9606982070265587786427675480691869167035030735018706315315080889814741903522")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14453637744147393731588213406970287317078500183808667567110703622821377404467"), Fq::from("2399421543794348043991999124897133983828094348779941685913680103324762089779")), ell_vw: Fq2::new(Fq::from("9949473626857777965864312230577133154733097780222892008806269288857107115742"), Fq::from("3174442934451895428948522429276679446276043250911620596846605340399481680631")), ell_vv: Fq2::new(Fq::from("10773554875563236900680111254025179423521083612506388720894431016952344645908"), Fq::from("1791500946161395076321848651097476041018873800532919907790655022966446958329")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13228018442560564090007964181563674306630092611719086476342149061864561185280"), Fq::from("12610285327624889222715450538150491139050620893563766277510163987972586901233")), ell_vw: Fq2::new(Fq::from("17479662858773093405375538474745376030990506794065246603293683266124925174043"), Fq::from("2084404960859519365733972471606438409123459865402210971655148862692485946826")), ell_vv: Fq2::new(Fq::from("9675635536108496330861750800277390856149950003475979013330722219857375990089"), Fq::from("17889926389980731198337394126857435112726851886206769936779651682292365620935")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19205463777127243508919644186723680978776945615422050305472500178064412123473"), Fq::from("10530028762478985738084339660443655112791893212694172883230610760959423970096")), ell_vw: Fq2::new(Fq::from("16515559472874148379011472911124765449097459867226728232468301906190158644000"), Fq::from("17379548905015313648731271429222704537289418711815257252434537403381161039222")), ell_vv: Fq2::new(Fq::from("13018298656134311744614296809909600459918460239498208291306800551968894517759"), Fq::from("16285303269049929546880902543736204779249639088136468614535789316688521663976")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9506592875949693201584225006409064359870098448027454395748349563972724759341"), Fq::from("1756212124305090096775291444252901778765176938185769777272403090820692916832")), ell_vw: Fq2::new(Fq::from("8608069305722185875126845710591577689851258364137915830961192516727536204958"), Fq::from("7879742654857051185278955345176850926952262018841407228150061698899939295057")), ell_vv: Fq2::new(Fq::from("2311752966405194522531977544398652994349075269275918521122768201037004674541"), Fq::from("8056192790171072186146581305753657840809572079825584316343698126369167589177")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16894171323623324294071017663897158494527746239981369238071300334653959886935"), Fq::from("21839557351524624846710837291698003965911117746801809883425361529677608757736")), ell_vw: Fq2::new(Fq::from("7921995242367245298209685641602469741817826911044923939882608549077724733892"), Fq::from("16195366803093832645900011617380218833645406154510352140817411408898365560241")), ell_vv: Fq2::new(Fq::from("12718951609280728995721445896408716663776093135974310357265585518411722037823"), Fq::from("14775730569800772809191320148533632897226925930462946258120395650683306784222")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18324740767049924394294219212760164185884801967765967078427354810675386933254"), Fq::from("18510525798971302215942990229348890627301667188256739950256532108744423958467")), ell_vw: Fq2::new(Fq::from("392130232730851920199052091589353916248087256056205523658797958171832234380"), Fq::from("21095230708713942590602820178155041621291598326436725983218983799781802054278")), ell_vv: Fq2::new(Fq::from("3496296009938486548567151348321691013571513425434397040346440937975425900960"), Fq::from("5982055369526470853931810147684497423036063399292908540001447206525852512531")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19247021831371751344058078378442480867649167715937305884736113857856820190448"), Fq::from("3844115161163747856814522390569661892955450798273343821401275758096863392646")), ell_vw: Fq2::new(Fq::from("6827864046826742026186179073427918229064686953470940699125130948982461542396"), Fq::from("17615043663229174389408922909891358248158892597678816385681324344031892930590")), ell_vv: Fq2::new(Fq::from("17143020152984715373298117189870583206488981021188881359521310100992509008220"), Fq::from("2174519777022093979454116656397980458948553322768446892932246705483799164016")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7787149350876246869417695770873628344850130125888658086308716316049428077539"), Fq::from("3645376584593179034263104864050174094110419584940335745722492256564582048708")), ell_vw: Fq2::new(Fq::from("11834206166934740935662681278907472701573638397628522018255718023777042308993"), Fq::from("11564024956967266505944728035150285591786947709753433733529805379472238406883")), ell_vv: Fq2::new(Fq::from("15009525058621581543328262019401183048318643177731888421709672368574056014500"), Fq::from("13573139489477783981960389988307434036015536834494019264597795654229345703051")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16059660214342726778635842789601639915164060169646706775268999439826160969051"), Fq::from("3592864998692836817549974752747753891215991801035643051485066473571282986623")), ell_vw: Fq2::new(Fq::from("16014086539360826224382965134738311010906671679277894939278016943520015391650"), Fq::from("8080101227643664909861133876167034094240488311720250596061359317310944652491")), ell_vv: Fq2::new(Fq::from("3252331663236937425057834000082561720577180169369353108561319513563871033743"), Fq::from("11096433584385810520286571872297862049422429286608206525566845219548956753724")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19106172305007554707393707438702629275907204152148571359255256006833275609105"), Fq::from("19930525972818199875851038877132322547334829658949554806801108404941414101103")), ell_vw: Fq2::new(Fq::from("6679449617056144648556013688092503718528081567224358397706944806960707450866"), Fq::from("11834339054115289581728384446510154030614495257639549098914576992681792644317")), ell_vv: Fq2::new(Fq::from("1103971970080878735020200145931068396548255816344141900100578642050167835756"), Fq::from("17442014287618549059909689032626628911431121434337187918457502534017974292771")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13076805492928497748141329405065621544173515733010216253264862326898999186011"), Fq::from("19346998537378535293491878821381642632408077348855336790184043880601230192418")), ell_vw: Fq2::new(Fq::from("2545953185107430710067715026283426197979102699516436299879612082830465064387"), Fq::from("658480144944520773976742499043453183928200527625254756411898212059863027220")), ell_vv: Fq2::new(Fq::from("10599092041977168661053673645861514974472253862403262653806335823439984547520"), Fq::from("17162893574285047796735108280435654463748265136801600105571927738662909784323")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7085319511141515608176313666085289847826695130590757892136226409211984372940"), Fq::from("4640142023499829801003091193578633557246618475866500623404375352964089302137")), ell_vw: Fq2::new(Fq::from("1378711095367589966539356394028311641404402942969200900226036867512926984525"), Fq::from("20227188036459114276006891788262806649925222994898646338447622388573680807347")), ell_vv: Fq2::new(Fq::from("10909351851886459521617902618958119972478357132140936115187854789408351897716"), Fq::from("4270359828288113001419972416654771076387260764523167288282363565700959902034")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("10664974041382313497332907041293813168318012018754207087217051246266695094381"), Fq::from("7224602128989495563505314962160716046811485526332201590284671097356858321790")), ell_vw: Fq2::new(Fq::from("3934655983500232748821065857705540333423053242147172231346384664554964864345"), Fq::from("11454752219285395886386371667205283572988335397450041936477253847740701967591")), ell_vv: Fq2::new(Fq::from("8613140983903038954690786727135433848146709090100659195246608245554768285578"), Fq::from("11683437859485859733100235618517710813205908191940301821880216475492534630964")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9260311336006363265451970156091146121161647548618044834929651955256342089015"), Fq::from("13937163080647677520680042908210664749449075029135461146729576039103333691760")), ell_vw: Fq2::new(Fq::from("17902331142943185999552054453100857674678810044830315483268693319046882556841"), Fq::from("1544882902322119030227919062381439448920928851067359863906156836436873153173")), ell_vv: Fq2::new(Fq::from("16995331248700240765237826468302734126993772776070338276810983166039248498016"), Fq::from("14969634758504522458179971123440531767135906355134596040598769037804654587702")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("9691457346326988270440356562380914538881384898220972529069554878570918299675"), Fq::from("4212749851190062070411861546581519573149862920541429109749681953529770836138")), ell_vw: Fq2::new(Fq::from("10890917529063563080488636605724946896609903254351684402546841801646043334166"), Fq::from("17064796604657821037034221187126605208939910043822292844975397354283757360178")), ell_vv: Fq2::new(Fq::from("10053764157025690527213061522668163518874554311426109722786780729745637570153"), Fq::from("12784239716790558610229592219566905741360808124013978959238298462172061891010")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18178610037180803777372696468852749823139836169202607246443680867419301660001"), Fq::from("2203553150397509140176760042861998991151797164185565920046676990283238338489")), ell_vw: Fq2::new(Fq::from("20994248737185978263276750124680792452522185817020429107744966692073225126536"), Fq::from("6349838002511478763817816496894821255234760390494115897123421446389963162622")), ell_vv: Fq2::new(Fq::from("1263928993764270656816573911502202456032289420152723320064785370753332332564"), Fq::from("3046827108505433561924898780283267608115953987844810798949064271834105292424")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8956928906602747002792065663685826131559810003765184492377190892700326559543"), Fq::from("9256558483224710959389211180674778886236784676421453067453985103303008910100")), ell_vw: Fq2::new(Fq::from("19258483215045500292709619645410473292886118206783068420253408398264757067719"), Fq::from("4451865829956913287893327659699467549695832858717473490066802599791690802339")), ell_vv: Fq2::new(Fq::from("2764660330371796211645912465865514165647712230266940655139955982219479135487"), Fq::from("13240152986607724183470740198192310673584785226612774164092563912390438227662")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("681767682327814473892054649470454096768283412403007527891528077929769397010"), Fq::from("7668193584311505399758378551993462119934308304509766677185196010163712481402")), ell_vw: Fq2::new(Fq::from("5655243760535315520783349710797795360471266639432050916814563868068635911845"), Fq::from("6563255051651012450772847952915771842363624186837705815170827122750163703821")), ell_vv: Fq2::new(Fq::from("649940975249127008407571724305690292046357719924151085455652491670338045916"), Fq::from("21579311737676937819033214124612551530127167774002946532496743344036322675381")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21175778737847401044104466077056088822729568143966863788586142950892805137666"), Fq::from("5595589116740977138584720715075423123216685509908153451278598090500673116809")), ell_vw: Fq2::new(Fq::from("10584200644032922607821101570277578933623023426596682657073390742969073652895"), Fq::from("12837953697025504547863231066503572653998301486985427806275987315834659812986")), ell_vv: Fq2::new(Fq::from("8117107548208718171682922744254403704391859020172151233388364110731129041045"), Fq::from("13049999463955485268550904857465746325902043660001708636685369259450564610454")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8364961604780202742303921161081604160922838515374423608219736048018999860853"), Fq::from("2351019119583633406807532653396626109486234484023597064733508979159842183382")), ell_vw: Fq2::new(Fq::from("21028432880575926523752559333294402827459370060933557257525967464997077946986"), Fq::from("8046872296156140750811104526560806223677002170113323621491280498583760192616")), ell_vv: Fq2::new(Fq::from("12898024833491487058567514428449740829256150305937499595589780150928565842102"), Fq::from("4285065448422006185829060466560686578745712460867295748911655362509144300948")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8072232359889306936722303478835558056241116613774315547062190240728474923834"), Fq::from("11177008054227634237663439894104913366867843529726469624208391026893438331544")), ell_vw: Fq2::new(Fq::from("6809080759521318546573252139200749353665236974149238674422052122591837770337"), Fq::from("17038522274461048605370004543286094433565235245947661777349160189041706387246")), ell_vv: Fq2::new(Fq::from("14024834693814357899802394991122512208020612837103453479716792821268632856005"), Fq::from("12162253519201182595066522680875457472054933146636012038079782236991342232964")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7517621744472550238729147112582763290619696603888340562320661653468259785459"), Fq::from("5775958628431902009928696648706206677051589697801230225207961079701351805991")), ell_vw: Fq2::new(Fq::from("17101537508347637393046612945060881733341630882242907674489426818282232192765"), Fq::from("11439924192045730134857855444951867487673481853061712676505751821713592417848")), ell_vv: Fq2::new(Fq::from("16524831954193049226680237913906512849861747525312574297009556588583016003485"), Fq::from("17961833510100001957206410577764797351002903290115563676683242640020206248673")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16912575460065719860154998920476780598787708817030774617128864437198545819929"), Fq::from("18523390726102623209642393197047309169880798198922216024709291880134883505164")), ell_vw: Fq2::new(Fq::from("8711744315121016096444521009902336544777137803294883395587631628756743226403"), Fq::from("384665796631718061484470820770858149414181395445222225443039170963661807931")), ell_vv: Fq2::new(Fq::from("15774538322747462731385931838340979144243491164741595437453112329195248618714"), Fq::from("11632275451601229685994821619837954138410931366643826111939350416539675376582")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4193777370819241345875098285464776579390947200963349948619027946434227330979"), Fq::from("509866391215626792725431227507051906680902316754753085884521978793646270092")), ell_vw: Fq2::new(Fq::from("6037175194865804429042789680410737585033925490367655507316663717551980954540"), Fq::from("18041663784750649636833746105782119188697038394822653258671508993118228507622")), ell_vv: Fq2::new(Fq::from("5823434994217288412340362524966578492063846889148854293277163714066988854109"), Fq::from("5672396008289604057966364197930061596003079754530820851713923211431895221182")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4038166397773785982077422827894979131906966567246129821324959462204170431412"), Fq::from("18437550501813119699199412531646289403507307693226141098858673836744295275291")), ell_vw: Fq2::new(Fq::from("14284923653983275338299458156785711701885868498245540450770121796221847779083"), Fq::from("12602897944901625024952695280664364105484904418233840369524699062991946110320")), ell_vv: Fq2::new(Fq::from("5486498814295375008109094800710678785628460899485721363575459728835468025031"), Fq::from("16200691130190012476008796365701719097160136918178762563260333528187553865070")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12314546787967340174315157919495563500551846522430770700093005979883034971431"), Fq::from("3526449836226747514409509674529965207620312182683788709671366453353950282439")), ell_vw: Fq2::new(Fq::from("5093273668274685698542824313074411944254604209522407110762874543698544425100"), Fq::from("14413934995969972893982204347979972725062293812449032149868616703831967954701")), ell_vv: Fq2::new(Fq::from("5926816361020227072352913556957354039762675747717706660663155084741002986207"), Fq::from("11598493987133778453993847174276604684683441101397418085992950965674684594056")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14605016886275487293555440005295724730930746064070181829857167303747503806723"), Fq::from("125488028578519699988648864181559879827394253209756997049515479109363959656")), ell_vw: Fq2::new(Fq::from("13618798589781916917754240943924749450694915714095829562214241999969133303075"), Fq::from("7919521676249934186275057369006254600339974917319129664598969307290994883444")), ell_vv: Fq2::new(Fq::from("13618646464444173552701873510284352084626546810748938598145778229202447557390"), Fq::from("9476574581262975213029808591741428214940842553330900266594833123385626035464")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20330569420634759863618802200434379756882301920599312733161738334139846403423"), Fq::from("16941424680188140264426604507849243567044731815188670894126312258606055646954")), ell_vw: Fq2::new(Fq::from("4276245652046353558826181153633997046897244969830610112857888938076187872093"), Fq::from("5218134511825218024578456539758678791071455837562669069391138178014247652283")), ell_vv: Fq2::new(Fq::from("5074559277335429630740829563679793620243538875458585401599810469948320707914"), Fq::from("8399181532479234777915440768158248538072911084393374644710065479068982458679")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("228456190514225889641604150077307010131630607092226273786181288134202785827"), Fq::from("6829691501727142726895142554058769478742020003540473851444113314854839779896")), ell_vw: Fq2::new(Fq::from("7239124420778199901449006013253700083679198792863456824897986348225520969318"), Fq::from("9612815502366829000381428453387330590336692138445456863004983078118510731327")), ell_vv: Fq2::new(Fq::from("3032095333965417803301146592958869325973325547652138997619203744855910423626"), Fq::from("9140900126707960555227810341717821370276432039323462635381934661102939064717")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("15183121839690585327918066578070528787848569635478652970262274682157344344126"), Fq::from("17032257654469593612895387885450704824985120918227803531218149228038247116993")), ell_vw: Fq2::new(Fq::from("4449946720064992614521464039848704811573349967863291193814384877976473528180"), Fq::from("16213473115750238158840972181958630393893069174331658126086552358850995045944")), ell_vv: Fq2::new(Fq::from("14820364816206160787218030695278797219740217739100995460398841348849807343630"), Fq::from("13391831978082769659226335015618034703517552702532093599836056107584855920091")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19376350172596821938106071755601309991977917109030961898807003698880067029751"), Fq::from("8432677549750770318577385075581043130355856390772751886981628666318547719067")), ell_vw: Fq2::new(Fq::from("19907801542387995008974808935818119635245812040205527769417146511798917761987"), Fq::from("5161102563798536355551540455475021097419898845343559944869423371507274303876")), ell_vv: Fq2::new(Fq::from("8934093116311670296805867189709376457342708648536176030347455800639124491789"), Fq::from("2703243001408501823489300103176986467031566088207792046257025791153593204110")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14246576411085900561418149065736187542613160942023368670593924207716148262147"), Fq::from("3894319734417282749436553474309402307762162253367446669496582090013706055610")), ell_vw: Fq2::new(Fq::from("8375748526029577467798965722696242475050467009610481042928314348040697238273"), Fq::from("5150547143250446746041060583106242553976637956855972603954741388536316657102")), ell_vv: Fq2::new(Fq::from("15907790590424017252473639606980913201030067954178224590289142061847439584806"), Fq::from("11159987404206640511954116561264499629862056981304725791275082386203982274572")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8534750176510034607784585561677263909151653570221826711790070724427621935471"), Fq::from("7817152482056867378682253092041139659293041868731996264204246731388352069129")), ell_vw: Fq2::new(Fq::from("16961113237531514134058762308816541864350643273034658785830240248900850618232"), Fq::from("1179078000730113993104369721112852911420394064162245727296751799866128074554")), ell_vv: Fq2::new(Fq::from("20457374714339105220680185601375259716132143552962630535477097129353077443649"), Fq::from("4422525926709336878773825319073188719230418589270084994878618648788354576429")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7627847586963893527076371866753343482026765205726665650559945926181618692545"), Fq::from("11854660093187915385325641424610609952992715573834872602540776698350931163771")), ell_vw: Fq2::new(Fq::from("12061802008303461437760636908585485318885385024493098952040511326354153801540"), Fq::from("2483328707802038203960641185873859652602098605879879702928670691083351376828")), ell_vv: Fq2::new(Fq::from("3179938981834043637934998748622377818001614400201120097155888429286598162788"), Fq::from("3387712903817602592588061982323804658595985412237324988504760476608661908669")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18935668687475417752670528265758371887227072908926123795139093572891409166177"), Fq::from("7137948322161355685471108734802366001522057971184080351365620151904917529667")), ell_vw: Fq2::new(Fq::from("4339156032094542567838794153972725732752104151129467898887880650450609600794"), Fq::from("12048671216064877807276853910375839173096391682288251978695349073893797560105")), ell_vv: Fq2::new(Fq::from("1950772803812480571427275248471639178971170644950730072787717123474197730954"), Fq::from("9181624035531055165333621510604231877029437698520448252367489746796255738472")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("11063654615400954717295218153285806614049542407631184476400061600226642090639"), Fq::from("10322234663627758865726091502227262577963762851829427143908452375568424940506")), ell_vw: Fq2::new(Fq::from("5088512375045637070831200431021199523774202712519349524231646029565850243137"), Fq::from("13830164350727008423053294968318643994288249825064419288902919319598527803018")), ell_vv: Fq2::new(Fq::from("5664114633031170805308124108402506179037402200165762737765348606356467981435"), Fq::from("6420500289322096738785425211380588437700105870048213919996474830908364662080")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12089017351045163205876906516764262596252750496306969979802152087250767442655"), Fq::from("10210558808216547366164047615754858102665922901025755697993978799086515582216")), ell_vw: Fq2::new(Fq::from("5805411494371431706302702029555168082350433630263841912576207114214821637903"), Fq::from("13417649473832328299363309530328161775713579734460687495589323668315481789186")), ell_vv: Fq2::new(Fq::from("7100161023424716303226449735068569399024504614273633218656928626719841360262"), Fq::from("4608780226874321006375565633722017610093139802788639970961143211665122970425")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7402656885283645637844399867437832414708002274719716809314550750025200002816"), Fq::from("3359755418859898733774484791668766343777859472186478368952701397946858170923")), ell_vw: Fq2::new(Fq::from("13972633490402551449646647488711159435500678729477306484155919447807727623768"), Fq::from("7493072689132118410451346570771015894599566093753023472904593233187191189290")), ell_vv: Fq2::new(Fq::from("10553632499535499938294059057022861371325444554911689769145218759723376859522"), Fq::from("15031724096959569889887883166930429112400023614133986918505200177949730145621")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("19614666780105423133718014776333000423285968913902974939878172716992608453809"), Fq::from("11603995082765755677014543451017101337084691386341657702599446172204728703168")), ell_vw: Fq2::new(Fq::from("3640285626277218751703647194759635465446306209869384174139659856814138047153"), Fq::from("19852827152161121673942769427296662914445783006570890136096053338872958018957")), ell_vv: Fq2::new(Fq::from("15133277841235179795557173122463348349264603847086033764926472096701308210280"), Fq::from("11477545358800195084620539652687203979589770655971679975137907269158992047904")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14110112013826130627308137057440352762673176006101512326319632538666268095190"), Fq::from("11939502323738533476551128890683182628994629048369580067812328846012456410750")), ell_vw: Fq2::new(Fq::from("4903570904021379863440412383519324112841733022665060696302943089410552325193"), Fq::from("14266803156893138980425811506240396635009593456307449490727186191181005152878")), ell_vv: Fq2::new(Fq::from("14881072700587137091207448172180518390500891251946414292329369965686915854422"), Fq::from("15926908436628594399583920130944357569706863129497064862365363758969852348774")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("12558706151877569296451844961986237406953575964353502214928234165049347203902"), Fq::from("20986351119648698768747274174278231882270962658108929786748672594878002377020")), ell_vw: Fq2::new(Fq::from("19049669232824485247328064933680299726722169816787419902393190195363675139966"), Fq::from("9038064204016446065702766858179381867420605337773795708435102200352238225734")), ell_vv: Fq2::new(Fq::from("20935836360010341957040867041308346409399314465762274744792154001186771274330"), Fq::from("10203406114258970224185400536984459120400171034941205452745624846762632193745")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("5273485321155899320017031787320374433252287213043632779466667239152835002841"), Fq::from("3053056741460414900892697870729818263336633719763669407853703091660209931803")), ell_vw: Fq2::new(Fq::from("14340825980720147789753139378133767339849529468405152905403690972933175947523"), Fq::from("10774533501875920615470512566140848784460524907884011188458749902565676606519")), ell_vv: Fq2::new(Fq::from("3964278893337662256819152335242286127722672026565541866237058714197501521533"), Fq::from("18022244127367262334687067361628890804799640514175862613915842388966614841200")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16332502873726000727943290608233946425803707374189230473690022986938005644703"), Fq::from("1567962947294031055122165158764344785171646063465144808831306664719326769256")), ell_vw: Fq2::new(Fq::from("20252166217271346908492216031557669988560255907738941815783926955730505937199"), Fq::from("17117796187826532609679851805915085724672631310980423533710032156677468384570")), ell_vv: Fq2::new(Fq::from("17447467886258986370787137061379217598255780655216428459333948972673151012677"), Fq::from("5126879980313476901131854669269919146638602687291477011061931179530480305128")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("6088970897125028663727223015788851329277811231049906821835210021445230531470"), Fq::from("8269042454443245208032034843063473106607747629991910601236499943382389383909")), ell_vw: Fq2::new(Fq::from("9390988514296458582044517134794855627538985266829573142624063333360969055582"), Fq::from("14809192622791971819542419564616657251564340618078775692251491022692953050806")), ell_vv: Fq2::new(Fq::from("12392403118986217155446218801253632414719473065843423820454574775060624572223"), Fq::from("17315648550786131512187415565477299679361071890287921321719420341945848525344")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8509704393616883918389461491307625357351608849562781376448108143686881397874"), Fq::from("20273788844812669168701278525056990767412251703786041383351127813548056060175")), ell_vw: Fq2::new(Fq::from("13587733380957601917144020513037833512715927912961534242274129611676082179670"), Fq::from("18219007782547538891755131684432302965729501555658190323551992733355994409648")), ell_vv: Fq2::new(Fq::from("15212235563035911012497734239429958536448576200220918241945992532007455729638"), Fq::from("15868967159924843704138451124239001351258676891119605450618238262537266256000")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("566642944339341838872659990670775497018813579200118662738575067585620262683"), Fq::from("80633561422247315402489148783136836267866433112423246780090434639448081501")), ell_vw: Fq2::new(Fq::from("16776314135335488452712466347201660236874408701359052824185479604688604443386"), Fq::from("9397286871750513195379912820117368990110423949006393550333044521067916303673")), ell_vv: Fq2::new(Fq::from("11213241389292792759942014406027965770431479614859139543821417488303149467800"), Fq::from("13968940593285273772185768016764839518710968380183853737025067101492115101555")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("6582524148524166281651049223972241418275829952930531957133089376081226599310"), Fq::from("12912796945237135113112612780707994091197804899944634993937602107108817617069")), ell_vw: Fq2::new(Fq::from("5915143763477343907746278070838860419057837978194254349309930636588762054019"), Fq::from("16829580672314341143130263740133292836202318062773617917928807522786048686905")), ell_vv: Fq2::new(Fq::from("6048262046277135430490078693604752857941808479983722633615831914037376701617"), Fq::from("11339265196645696685132629437938382050130007247287625382019491127986659567508")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("16221040390720130701106818492787718054167551387591190581820867019772199748452"), Fq::from("13816650124538024294513769170531808824213976236314395631188393480750356347832")), ell_vw: Fq2::new(Fq::from("17884725815671017389514271919761437310258619463085199936932965564030171526767"), Fq::from("18248973796984942533601080062907385035195180110558182226420098600644022965345")), ell_vv: Fq2::new(Fq::from("3137416660392142651716799414483848222789075294982695366740419822188364577896"), Fq::from("9694367271685190541749430052152010428005565838702388077250686840743397580564")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("6357610272973630149840225505274539351850867930627803566656124705077309138236"), Fq::from("6532064867969562545750502093085673423036025281166230602025667552530979558092")), ell_vw: Fq2::new(Fq::from("16786670094068652274677750111730144379312178408732829954083928254919389983983"), Fq::from("6341529030893580474995202784328098673009525246309131907462803539077246776781")), ell_vv: Fq2::new(Fq::from("1974498926273096258802796702119259182018579009158122203612017565775578915815"), Fq::from("16202371700345289299248140495735793174324620209466889736695203996551421837181")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("7645596703863049949671625140189098533219782764649960681401839437305310089346"), Fq::from("19662686877039093771787921401379726144714150453645437909640334762978238819386")), ell_vw: Fq2::new(Fq::from("19675517233991845002128733409028273554556593365253876728982552859262257722391"), Fq::from("17810553502824017140555114404262914434390762266054795883352268646885364159410")), ell_vv: Fq2::new(Fq::from("18665072545928024509881282829034535118797458683193169885406548536617293330990"), Fq::from("19411595053923298782146703680748498557982136082135126508584153240357549362078")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14470514611309746071605255274946119202484366658141275541087020842175954067559"), Fq::from("2594897518684998941907190849745518961756699273476562463311001408776339658780")), ell_vw: Fq2::new(Fq::from("935873152179726234311073616163680460286310013909076125956427156452613025181"), Fq::from("1833293684395168387765260233178732135932817355166697645828433012629591012612")), ell_vv: Fq2::new(Fq::from("15506344864201849985196152671178663605696906693236430756412539843011256851197"), Fq::from("21249626787247514663459670711353542343321937134901366527096486234949436840608")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14844298589371811580114643149994926627935566419885127313767249060440840193460"), Fq::from("21865492910219121705480174226763323783866193080747061232287801827306146286946")), ell_vw: Fq2::new(Fq::from("17151100973274235243640792655189642956800303990602164052684336481398298289411"), Fq::from("14392190221562986523826107157981466944764379735735544220135185056395768909930")), ell_vv: Fq2::new(Fq::from("4350255565215767538405433120041486620343663937357318680211203103336875636557"), Fq::from("9926081950735179732148156026738993553156731124330466866635273595752973758855")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13218761129163938072615510267349114476398845757988353548581904005855607494827"), Fq::from("5599121822325202341068442750237639791575579072233857524815548719976323285948")), ell_vw: Fq2::new(Fq::from("15971120376538832228790474763995956330325241100841265267806125625593561200292"), Fq::from("12523288625433720594713119420870235433458343202664663107083051275205911332632")), ell_vv: Fq2::new(Fq::from("16334648365696506264242655102949242813244449705193389318653840069223698505218"), Fq::from("5656148484258838154513173637616383061988381726003070282394323376702835298685")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17813738829828434290335491956960173922453516626523775773406434062574864455163"), Fq::from("20602369742562270094694203910247000789225384961205618501257582838246099644397")), ell_vw: Fq2::new(Fq::from("19878549521928480892919729627745688905048802531306612605469432942519616386562"), Fq::from("5017457857497268504225948371697060380418757800650743056471872023476645133890")), ell_vv: Fq2::new(Fq::from("11253669115821132651967277438747600032147012598164512191026558281212405907059"), Fq::from("17527101633359473938082679669763197242520773041737623688955597341205608093817")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("4215545173967763533303120784928042872427929334572670490629936371429427871071"), Fq::from("12294795716682648009432653500746413304750644076589949908249345884220200424705")), ell_vw: Fq2::new(Fq::from("18825720171660950734491494575282684507875213824417353005009465489701558195915"), Fq::from("16705960611772713893529695679517111622754789317070946155202186944060021844706")), ell_vv: Fq2::new(Fq::from("727717707174551758531261221179545916153780087179478433501944451338651779512"), Fq::from("8112271980906996458859937804680221894935591766524365848766530295751121904748")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("10233024162783159141952008177866660607852287510058437429941412641351833243474"), Fq::from("9594838646894429694778293318641489517645344438331800828882574864439372872574")), ell_vw: Fq2::new(Fq::from("16645682756797429615195184700295338945089747965333104334845226609564841400023"), Fq::from("4162352035394629024812623559649906911351409190415492302470361657064711946691")), ell_vv: Fq2::new(Fq::from("6032119528092518381867266123093895943802103316999621961287699634448441322294"), Fq::from("16978590002913514123461500015923670605519520926487110857783653291799516315656")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8218196524386055233961282665657821183883422738571989211183480610420112668380"), Fq::from("10415506866745570162620028320970394286106067432626512901854031549920339603997")), ell_vw: Fq2::new(Fq::from("5210044150482547744429104832217210823770891140961861953610847155896342519957"), Fq::from("7249576834325689075837879557249227252903732278842727723303356548006299693332")), ell_vv: Fq2::new(Fq::from("10566250213397026997176223676662431109660830193211672199901829008963029461821"), Fq::from("10109726446308488117042758961222375921166426648339199001670753736584698407831")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("14958658301977599538003236672511296104101995894564376511283101560012949683335"), Fq::from("3048608469033125363311089547916169290720115268350540694596412955569118898494")), ell_vw: Fq2::new(Fq::from("574489666015527527356241753990038643397861257676315289742492198353125347063"), Fq::from("20135743140323312729475654806927894483883158119600963784497647679670289292384")), ell_vv: Fq2::new(Fq::from("429037282750459454100296565403648201904295120071497517313998879109362521423"), Fq::from("15853233349247701874380259741343021520320591334343875702627292292088300425145")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("18700030680590279184945769912899822403041358764323477965514340815228877402331"), Fq::from("10683998632701796443517414309938660369292269662497518738662465524924019738910")), ell_vw: Fq2::new(Fq::from("1310694906110452150099267122613851606459600617516350757492763828252192371857"), Fq::from("4783076385012654125132398886450190041207227400254467276271580769333335151361")), ell_vv: Fq2::new(Fq::from("19588709046778297839342456362815786766378666302331732601204786295010565114246"), Fq::from("15768094978512846661820502043664035262466161785979259300748269497441797675546")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("13060364876687824737412369705619641318084803694180161837383771909738942399322"), Fq::from("18217552047682533290011237332328856250732338018560605204821714348687071938690")), ell_vw: Fq2::new(Fq::from("12869661643727171099080271622741099816758098286424339832028838960253051261267"), Fq::from("9771005613420764951919925720405408601257471427085946188171574196917272408972")), ell_vv: Fq2::new(Fq::from("50909581144623642750375885935872654509592339539711209410079054797052409354"), Fq::from("18926528403466649263993967111065644144239906211315737390219036362821972145669")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("20549804669307980165961565026567668900175984309510667407934326603524068980568"), Fq::from("2860891967505015044048360806986650576604923893849408592665407858213623973073")), ell_vw: Fq2::new(Fq::from("15425380359872501526721873883293811604657689327716748473720120299487795555723"), Fq::from("19476618728903347368164774764554105623361814261939990815870207909384053172994")), ell_vv: Fq2::new(Fq::from("1391900673866549134877070137078952075347059393013450066057338968238992587955"), Fq::from("4196316861965319041187484774757427170509633759722370813890904092023663955821")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("17275387403896977778123657295201685204905269368879700703438840824610915049467"), Fq::from("19208898840130004899976179132541534722147305748950927170013798354273078356439")), ell_vw: Fq2::new(Fq::from("2806476769978795611970161315323438770892084130100332531802098991493644141407"), Fq::from("12027685137795878129532114449809592645364590672303716160923491816560315933093")), ell_vv: Fq2::new(Fq::from("8236588507133996535585623995380369531792969451558421380293164596516832310024"), Fq::from("7194816002224802847462171990762514812770222507636791853795854276216118633532")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("8988630595018387496849477785798469553390006328427704582362113137927154840355"), Fq::from("2295777524525303845671711839319946449143831320846022233249545363172788331149")), ell_vw: Fq2::new(Fq::from("19856063720990800344827304970869772811632532243020577718537161398914360273978"), Fq::from("19822454791015240925652351664190735476954892251509376470361602440577337517739")), ell_vv: Fq2::new(Fq::from("16437024240432855113321845721989731630224984534156413407578941758236945886728"), Fq::from("10915573623321985107029933965796058231484519510670830600808943167010577314334")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21286975582879196598650077668567283554136994229408922053641846042051746388886"), Fq::from("5421529658065143080970166226046014500038127295988164061860891929414040927615")), ell_vw: Fq2::new(Fq::from("5555485154537896512520569992616892769357525491868135294451583184259218193278"), Fq::from("4831809465565304340310880868620257258305484211486308136368207144444325564275")), ell_vv: Fq2::new(Fq::from("2583089702666220608189326901976548985583885001543445628170288810575076369034"), Fq::from("7097788099664798415333498763272677734135137637990913349021964143797057593727")) },
|
||||
EllCoeffs { ell_0: Fq2::new(Fq::from("21356455346146359274071336488740257199607341582407258426849457339859865454145"), Fq::from("19512280220246743343068375192508909480135195681362473260894831078332945004959")), ell_vw: Fq2::new(Fq::from("15435677241338788169971902812985017500898533325271813422525238374746874853456"), Fq::from("6382067580057986766020919077216175344977116334133202456947134195629074647989")), ell_vv: Fq2::new(Fq::from("1150360546643207487285835972411275438312694180812115547021545502055712764110"), Fq::from("2194960903427600986795419731825236689333014397679688012527722471213485607323")) }
|
||||
]
|
||||
};
|
||||
|
||||
assert!(expected_g2_p.eq(&g2_p));
|
||||
assert!(expected_g2_p.coeffs.len() == 102);
|
||||
}
|
||||
|
||||
fn miller_loop(p: &Affine<G1Params>, q: &G2Precomp) -> Fq12 {
|
||||
let mut f = Fq12::one();
|
||||
|
||||
let mut idx = 0;
|
||||
|
||||
let mut found_one = false;
|
||||
let ate = ate_loop_count();
|
||||
|
||||
for i in (0..FrParams::bits()).rev() {
|
||||
let bit = ate.test_bit(i);
|
||||
|
||||
if !found_one {
|
||||
found_one = bit;
|
||||
continue;
|
||||
}
|
||||
|
||||
let c = &q.coeffs[idx];
|
||||
idx += 1;
|
||||
f = f.squared().mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
|
||||
if bit {
|
||||
let c = &q.coeffs[idx];
|
||||
idx += 1;
|
||||
f = f.mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
}
|
||||
}
|
||||
|
||||
let c = &q.coeffs[idx];
|
||||
idx += 1;
|
||||
f = f.mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
|
||||
let c = &q.coeffs[idx];
|
||||
//idx += 1;
|
||||
f = f.mul_by_024(&c.ell_0, &c.ell_vw * &p.get_y(), &c.ell_vv * &p.get_x());
|
||||
|
||||
f
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_miller_loop() {
|
||||
let g1 = G1::one() * Fr::from("18097487326282793650237947474982649264364522469319914492172746413872781676");
|
||||
let g2 = G2::one() * Fr::from("20390255904278144451778773028944684152769293537511418234311120800877067946");
|
||||
|
||||
let g1_pre = g1.to_affine();
|
||||
let g2_pre = G2Precomp::new(&g2);
|
||||
|
||||
let gt = miller_loop(&g1_pre, &g2_pre);
|
||||
|
||||
assert_eq!(gt,
|
||||
Fq12::new(Fq6::new(
|
||||
Fq2::new(Fq::from("14551901853310307118181117653102171756020286507151693083446930124375536995872"), Fq::from("9312135802322424742640599513015426415694425842442244572104764725304978020017")),
|
||||
Fq2::new(Fq::from("2008578374540014049115224515107136454624926345291695498760935593377832328658"), Fq::from("19401931167387470703307774451905975977586101231060812348184567722817888018105")),
|
||||
Fq2::new(Fq::from("15835061253582829097893482726334173316772697321004871665993836763948321578465"), Fq::from("2434436628082562384254182545550914004674636606111293955202388712261962820365"))
|
||||
),
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("2874440054453559166574356420729655370224872280550180463983603224123901706537"), Fq::from("21199736323249863378180814900160978651989782296293186487853700340281870105680")),
|
||||
Fq2::new(Fq::from("19165582755854282767090326095669835261356341739532443976394958023142879015770"), Fq::from("1381947898997178910398427566832118260186305708991760706544743699683050330259")),
|
||||
Fq2::new(Fq::from("282285618133171001983721596014922591835675934808772882476123488581876545578"), Fq::from("9533292755262567365755835323107174518472361243562718718917822947506880920117"))
|
||||
)
|
||||
)
|
||||
);
|
||||
}
|
||||
|
||||
pub fn final_exponentiation_first_chunk(elt: &Fq12) -> Fq12 {
|
||||
let a = elt.unitary_inverse();
|
||||
let b = elt.inverse();
|
||||
let c = a * b;
|
||||
let d = c.frobenius_map(2);
|
||||
|
||||
d * c
|
||||
}
|
||||
|
||||
pub fn exp_by_neg_z(elt: &Fq12) -> Fq12 {
|
||||
elt.cyclotomic_exp(&final_exponent_z()).unitary_inverse()
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_cyclotomic_exp() {
|
||||
let orig = Fq12::new(
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("2259924035228092997691937637688451143058635253053054071159756458902878894295"), Fq::from("13145690032701362144460254305183927872683620413225364127064863863535255135244")),
|
||||
Fq2::new(Fq::from("9910063591662383599552477067956819406417086889312288278252482503717089428441"), Fq::from("537414042055419261990282459138081732565514913399498746664966841152381183961")),
|
||||
Fq2::new(Fq::from("15311812409497308894370893420777496684951030254049554818293571309705780605004"), Fq::from("13657107176064455789881282546557276003626320193974643644160350907227082365810"))
|
||||
),
|
||||
Fq6::new(Fq2::new(Fq::from("4913017949003742946864670837361832856526234260447029873580022776602534856819"), Fq::from("7834351480852267338070670220119081676575418514182895774094743209915633114041")),
|
||||
Fq2::new(Fq::from("12837298223308203788092748646758194441270207338661891973231184407371206766993"), Fq::from("12756474445699147370503225379431475413909971718057034061593007812727141391799")),
|
||||
Fq2::new(Fq::from("9473802207170192255373153510655867502408045964296373712891954747252332944018"), Fq::from("4583089109360519374075173304035813179013579459429335467869926761027310749713"))
|
||||
)
|
||||
);
|
||||
|
||||
let expected = Fq12::new(
|
||||
Fq6::new(
|
||||
Fp2::new(Fq::from("14722956046055152398903846391223329501345567382234608299399030576415080188350"), Fq::from("14280703280777926697010730619606819467080027543707671882210769811674790473417")),
|
||||
Fp2::new(Fq::from("19969875076083990244184003223190771301761436396530543002586073549972410735411"), Fq::from("10717335566913889643303549252432531178405520196706173198634734518494041323243")),
|
||||
Fp2::new(Fq::from("6063612626166484870786832843320782567259894784043383626084549455432890717937"), Fq::from("17089783040131779205038789608891431427943860868115199598200376195935079808729"))
|
||||
),
|
||||
Fq6::new(
|
||||
Fp2::new(Fq::from("10029863438921507421569931792104023129735006154272482043027653425575205672906"), Fq::from("6406252222753462799887280578845937185621081001436094637606245493619821542775")),
|
||||
Fp2::new(Fq::from("1048245462913506652602966692378792381004227332967846949234978073448561848050"), Fq::from("1444281375189053827455518242624554285012408033699861764136810522738182087554")),
|
||||
Fp2::new(Fq::from("8839610992666735109106629514135300820412539620261852250193684883379364789120"), Fq::from("11347360242067273846784836674906058940820632082713814508736182487171407730718"))
|
||||
)
|
||||
);
|
||||
|
||||
let e = exp_by_neg_z(&orig);
|
||||
|
||||
assert_eq!(e, expected);
|
||||
}
|
||||
|
||||
pub fn final_exponentiation_last_chunk(elt: &Fq12) -> Fq12 {
|
||||
let a = exp_by_neg_z(elt);
|
||||
let b = a.cyclotomic_squared();
|
||||
let c = b.cyclotomic_squared();
|
||||
let d = c.mul(&b);
|
||||
|
||||
let e = exp_by_neg_z(&d);
|
||||
let f = e.cyclotomic_squared();
|
||||
let g = exp_by_neg_z(&f);
|
||||
let h = d.unitary_inverse();
|
||||
let i = g.unitary_inverse();
|
||||
|
||||
let j = i.mul(&e);
|
||||
let k = j.mul(&h);
|
||||
let l = k.mul(&b);
|
||||
let m = k.mul(&e);
|
||||
let n = m.mul(elt);
|
||||
|
||||
let o = l.frobenius_map(1);
|
||||
let p = o.mul(&n);
|
||||
|
||||
let q = k.frobenius_map(2);
|
||||
let r = q.mul(&p);
|
||||
|
||||
let s = elt.unitary_inverse();
|
||||
let t = s.mul(&l);
|
||||
let u = t.frobenius_map(3);
|
||||
let v = u.mul(&r);
|
||||
|
||||
v
|
||||
}
|
||||
|
||||
pub fn final_exponentiation(elt: &Fq12) -> Fq12 {
|
||||
final_exponentiation_last_chunk(&final_exponentiation_first_chunk(elt))
|
||||
}
|
||||
|
||||
pub fn pairing(p: &G1, q: &G2) -> Gt {
|
||||
let p = p.to_affine();
|
||||
let q = G2Precomp::new(q);
|
||||
|
||||
Gt::new(final_exponentiation(&miller_loop(&p, &q)))
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_reduced_pairing() {
|
||||
let g1 = G1::one() * Fr::from("18097487326282793650237947474982649264364522469319914492172746413872781676");
|
||||
let g2 = G2::one() * Fr::from("20390255904278144451778773028944684152769293537511418234311120800877067946");
|
||||
|
||||
let gt = pairing(&g1, &g2);
|
||||
|
||||
let expected = Gt::new(Fq12::new(
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("7520311483001723614143802378045727372643587653754534704390832890681688842501"), Fq::from("20265650864814324826731498061022229653175757397078253377158157137251452249882")),
|
||||
Fq2::new(Fq::from("11942254371042183455193243679791334797733902728447312943687767053513298221130"), Fq::from("759657045325139626991751731924144629256296901790485373000297868065176843620")),
|
||||
Fq2::new(Fq::from("16045761475400271697821392803010234478356356448940805056528536884493606035236"), Fq::from("4715626119252431692316067698189337228571577552724976915822652894333558784086"))
|
||||
),
|
||||
Fq6::new(
|
||||
Fq2::new(Fq::from("14901948363362882981706797068611719724999331551064314004234728272909570402962"), Fq::from("11093203747077241090565767003969726435272313921345853819385060670210834379103")),
|
||||
Fq2::new(Fq::from("17897835398184801202802503586172351707502775171934235751219763553166796820753"), Fq::from("1344517825169318161285758374052722008806261739116142912817807653057880346554")),
|
||||
Fq2::new(Fq::from("11123896897251094532909582772961906225000817992624500900708432321664085800838"), Fq::from("17453370448280081813275586256976217762629631160552329276585874071364454854650"))
|
||||
)
|
||||
));
|
||||
|
||||
assert_eq!(expected, gt);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_binlinearity() {
|
||||
use rand::{SeedableRng,StdRng};
|
||||
let seed: [usize; 4] = [103245, 191922, 1293, 192103];
|
||||
let mut rng = StdRng::from_seed(&seed);
|
||||
|
||||
for _ in 0..50 {
|
||||
let p = G1::random(&mut rng);
|
||||
let q = G2::random(&mut rng);
|
||||
let s = Fr::random(&mut rng);
|
||||
let sp = &p * &s;
|
||||
let sq = &q * &s;
|
||||
|
||||
let a = pairing(&p, &q) ^ &s;
|
||||
let b = pairing(&sp, &q);
|
||||
let c = pairing(&p, &sq);
|
||||
|
||||
assert_eq!(a, b);
|
||||
assert_eq!(b, c);
|
||||
|
||||
let t = Fr::zero().sub(&Fr::one());
|
||||
|
||||
assert!(a != Gt::new(Fq12::one()));
|
||||
assert_eq!((&a ^ t) * &a, Gt::new(Fq12::one()));
|
||||
}
|
||||
}
|
||||
pub use arith::U256;
|
||||
pub use fields::{Fq, Fr, Fq2, FieldElement};
|
||||
|
|
|
|||
Loading…
Reference in New Issue