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This commit is contained in:
Kobi Gurkan 2019-09-07 16:28:19 +03:00
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target/
**/*.rs.bk
Cargo.lock
.vscode

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Copyrights in the "pairing" library are retained by their contributors. No
copyright assignment is required to contribute to the "pairing" library.
The "pairing" library is licensed under either of
* Apache License, Version 2.0, (see ./LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
* MIT license (see ./LICENSE-MIT or http://opensource.org/licenses/MIT)
at your option.
Unless you explicitly state otherwise, any contribution intentionally
submitted for inclusion in the work by you, as defined in the Apache-2.0
license, shall be dual licensed as above, without any additional terms or
conditions.

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[package]
name = "pairing_ce"
# Remember to change version string in README.md.
version = "0.18.0"
authors = [
"Sean Bowe <ewillbefull@gmail.com>",
"Jack Grigg <jack@z.cash>",
"Alex Vlasov <alex.m.vlasov@gmail.com>",
"Alex Gluchowski <alex@gluchowski.net>"
]
license = "MIT/Apache-2.0"
description = "Pairing-friendly elliptic curve library"
documentation = "https://docs.rs/pairing/"
homepage = "https://github.com/matter-labs/pairing"
repository = "https://github.com/matter-labs/pairing"
edition = "2018"
[dependencies]
rand = "0.4"
byteorder = "1"
ff = {package = "ff_ce", version = "0.7", features = ["derive"]}
#ff = { path = '../ff', package = "ff_ce", features = ["derive"]}
[features]
expose-arith = []
default = []

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# pairing "community edition"
Now published as `pairing_ce` to `crates.io` for users convenience.
Originally developed by ZCash, with extensions from us to make it a little more pleasant.
This is a Rust crate for using pairing-friendly elliptic curves. Currently, only the [BLS12-381](https://z.cash/blog/new-snark-curve.html) and BN256 curves are implemented.
## [Documentation](https://docs.rs/pairing/)
Bring the `pairing` crate into your project just as you normally would.
## Security Warnings
This library does not make any guarantees about constant-time operations, memory access patterns, or resistance to side-channel attacks.
## License
Licensed under either of
* Apache License, Version 2.0, ([LICENSE-APACHE](LICENSE-APACHE) or http://www.apache.org/licenses/LICENSE-2.0)
* MIT license ([LICENSE-MIT](LICENSE-MIT) or http://opensource.org/licenses/MIT)
at your option.
### Contribution
Unless you explicitly state otherwise, any contribution intentionally
submitted for inclusion in the work by you, as defined in the Apache-2.0
license, shall be dual licensed as above, without any additional terms or
conditions.

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mod g1 {
use rand::{Rand, SeedableRng, XorShiftRng};
use pairing::bls12_381::*;
use pairing::CurveProjective;
#[bench]
fn bench_g1_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, Fr)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g1_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, G1)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), G1::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g1_add_assign_mixed(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, G1Affine)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), G1::rand(&mut rng).into()))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign_mixed(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
}
mod g2 {
use rand::{Rand, SeedableRng, XorShiftRng};
use pairing::bls12_381::*;
use pairing::CurveProjective;
#[bench]
fn bench_g2_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G2, Fr)> = (0..SAMPLES)
.map(|_| (G2::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g2_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G2, G2)> = (0..SAMPLES)
.map(|_| (G2::rand(&mut rng), G2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g2_add_assign_mixed(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G2, G2Affine)> = (0..SAMPLES)
.map(|_| (G2::rand(&mut rng), G2::rand(&mut rng).into()))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign_mixed(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::{Field, PrimeField, PrimeFieldRepr, SqrtField};
use pairing::bls12_381::*;
#[bench]
fn bench_fq_repr_add_nocarry(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FqRepr, FqRepr)> = (0..SAMPLES)
.map(|_| {
let mut tmp1 = FqRepr::rand(&mut rng);
let mut tmp2 = FqRepr::rand(&mut rng);
// Shave a few bits off to avoid overflow.
for _ in 0..3 {
tmp1.div2();
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_nocarry(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_sub_noborrow(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FqRepr, FqRepr)> = (0..SAMPLES)
.map(|_| {
let tmp1 = FqRepr::rand(&mut rng);
let mut tmp2 = tmp1;
// Ensure tmp2 is smaller than tmp1.
for _ in 0..10 {
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_noborrow(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_num_bits(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES).map(|_| FqRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].num_bits();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_mul2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES).map(|_| FqRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.mul2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_div2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES).map(|_| FqRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.div2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq, Fq)> = (0..SAMPLES)
.map(|_| (Fq::rand(&mut rng), Fq::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq, Fq)> = (0..SAMPLES)
.map(|_| (Fq::rand(&mut rng), Fq::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq, Fq)> = (0..SAMPLES)
.map(|_| (Fq::rand(&mut rng), Fq::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_square(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].inverse()
});
}
#[bench]
fn bench_fq_negate(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.negate();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_sqrt(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES)
.map(|_| {
let mut tmp = Fq::rand(&mut rng);
tmp.square();
tmp
})
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].sqrt()
});
}
#[bench]
fn bench_fq_into_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].into_repr()
});
}
#[bench]
fn bench_fq_from_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES)
.map(|_| Fq::rand(&mut rng).into_repr())
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
Fq::from_repr(v[count])
});
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::Field;
use pairing::bls12_381::*;
#[bench]
fn bench_fq12_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq12, Fq12)> = (0..SAMPLES)
.map(|_| (Fq12::rand(&mut rng), Fq12::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq12, Fq12)> = (0..SAMPLES)
.map(|_| (Fq12::rand(&mut rng), Fq12::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq12, Fq12)> = (0..SAMPLES)
.map(|_| (Fq12::rand(&mut rng), Fq12::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_squaring(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq12> = (0..SAMPLES).map(|_| Fq12::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq12> = (0..SAMPLES).map(|_| Fq12::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].inverse();
count = (count + 1) % SAMPLES;
tmp
});
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::{Field, SqrtField};
use pairing::bls12_381::*;
#[bench]
fn bench_fq2_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq2, Fq2)> = (0..SAMPLES)
.map(|_| (Fq2::rand(&mut rng), Fq2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq2, Fq2)> = (0..SAMPLES)
.map(|_| (Fq2::rand(&mut rng), Fq2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq2, Fq2)> = (0..SAMPLES)
.map(|_| (Fq2::rand(&mut rng), Fq2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_squaring(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq2> = (0..SAMPLES).map(|_| Fq2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq2> = (0..SAMPLES).map(|_| Fq2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].inverse();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_sqrt(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq2> = (0..SAMPLES).map(|_| Fq2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].sqrt();
count = (count + 1) % SAMPLES;
tmp
});
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::{Field, PrimeField, PrimeFieldRepr, SqrtField};
use pairing::bls12_381::*;
#[bench]
fn bench_fr_repr_add_nocarry(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FrRepr, FrRepr)> = (0..SAMPLES)
.map(|_| {
let mut tmp1 = FrRepr::rand(&mut rng);
let mut tmp2 = FrRepr::rand(&mut rng);
// Shave a few bits off to avoid overflow.
for _ in 0..3 {
tmp1.div2();
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_nocarry(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_sub_noborrow(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FrRepr, FrRepr)> = (0..SAMPLES)
.map(|_| {
let tmp1 = FrRepr::rand(&mut rng);
let mut tmp2 = tmp1;
// Ensure tmp2 is smaller than tmp1.
for _ in 0..10 {
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_noborrow(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_num_bits(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES).map(|_| FrRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].num_bits();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_mul2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES).map(|_| FrRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.mul2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_div2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES).map(|_| FrRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.div2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fr, Fr)> = (0..SAMPLES)
.map(|_| (Fr::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fr, Fr)> = (0..SAMPLES)
.map(|_| (Fr::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fr, Fr)> = (0..SAMPLES)
.map(|_| (Fr::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_square(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].inverse()
});
}
#[bench]
fn bench_fr_negate(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.negate();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_sqrt(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES)
.map(|_| {
let mut tmp = Fr::rand(&mut rng);
tmp.square();
tmp
})
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].sqrt()
});
}
#[bench]
fn bench_fr_into_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].into_repr()
});
}
#[bench]
fn bench_fr_from_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES)
.map(|_| Fr::rand(&mut rng).into_repr())
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
Fr::from_repr(v[count])
});
}

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mod ec;
mod fq;
mod fq12;
mod fq2;
mod fr;
use rand::{Rand, SeedableRng, XorShiftRng};
use pairing::bls12_381::*;
use pairing::{CurveAffine, Engine};
#[bench]
fn bench_pairing_g1_preparation(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<G1> = (0..SAMPLES).map(|_| G1::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = G1Affine::from(v[count]).prepare();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_g2_preparation(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<G2> = (0..SAMPLES).map(|_| G2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = G2Affine::from(v[count]).prepare();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_miller_loop(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1Prepared, G2Prepared)> = (0..SAMPLES)
.map(|_| {
(
G1Affine::from(G1::rand(&mut rng)).prepare(),
G2Affine::from(G2::rand(&mut rng)).prepare(),
)
})
.collect();
let mut count = 0;
b.iter(|| {
let tmp = Bls12::miller_loop(&[(&v[count].0, &v[count].1)]);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_final_exponentiation(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq12> = (0..SAMPLES)
.map(|_| {
(
G1Affine::from(G1::rand(&mut rng)).prepare(),
G2Affine::from(G2::rand(&mut rng)).prepare(),
)
})
.map(|(ref p, ref q)| Bls12::miller_loop(&[(p, q)]))
.collect();
let mut count = 0;
b.iter(|| {
let tmp = Bls12::final_exponentiation(&v[count]);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_full(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, G2)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), G2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let tmp = Bls12::pairing(v[count].0, v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}

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mod g1 {
use rand::{Rand, SeedableRng, XorShiftRng};
use pairing::bn256::*;
use pairing::CurveProjective;
#[bench]
fn bench_g1_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, Fr)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g1_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, G1)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), G1::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g1_add_assign_mixed(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, G1Affine)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), G1::rand(&mut rng).into()))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign_mixed(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
}
mod g2 {
use rand::{Rand, SeedableRng, XorShiftRng};
use pairing::bls12_381::*;
use pairing::CurveProjective;
#[bench]
fn bench_g2_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G2, Fr)> = (0..SAMPLES)
.map(|_| (G2::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g2_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G2, G2)> = (0..SAMPLES)
.map(|_| (G2::rand(&mut rng), G2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_g2_add_assign_mixed(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G2, G2Affine)> = (0..SAMPLES)
.map(|_| (G2::rand(&mut rng), G2::rand(&mut rng).into()))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign_mixed(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::{Field, PrimeField, PrimeFieldRepr, SqrtField};
use pairing::bn256::*;
#[bench]
fn bench_fq_repr_add_nocarry(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FqRepr, FqRepr)> = (0..SAMPLES)
.map(|_| {
let mut tmp1 = FqRepr::rand(&mut rng);
let mut tmp2 = FqRepr::rand(&mut rng);
// Shave a few bits off to avoid overflow.
for _ in 0..3 {
tmp1.div2();
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_nocarry(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_sub_noborrow(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FqRepr, FqRepr)> = (0..SAMPLES)
.map(|_| {
let tmp1 = FqRepr::rand(&mut rng);
let mut tmp2 = tmp1;
// Ensure tmp2 is smaller than tmp1.
for _ in 0..10 {
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_noborrow(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_num_bits(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES).map(|_| FqRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].num_bits();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_mul2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES).map(|_| FqRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.mul2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_repr_div2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES).map(|_| FqRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.div2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq, Fq)> = (0..SAMPLES)
.map(|_| (Fq::rand(&mut rng), Fq::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq, Fq)> = (0..SAMPLES)
.map(|_| (Fq::rand(&mut rng), Fq::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq, Fq)> = (0..SAMPLES)
.map(|_| (Fq::rand(&mut rng), Fq::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_square(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].inverse()
});
}
#[bench]
fn bench_fq_negate(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.negate();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq_sqrt(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES)
.map(|_| {
let mut tmp = Fq::rand(&mut rng);
tmp.square();
tmp
})
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].sqrt()
});
}
#[bench]
fn bench_fq_into_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq> = (0..SAMPLES).map(|_| Fq::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].into_repr()
});
}
#[bench]
fn bench_fq_from_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FqRepr> = (0..SAMPLES)
.map(|_| Fq::rand(&mut rng).into_repr())
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
Fq::from_repr(v[count])
});
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::Field;
use pairing::bn256::*;
#[bench]
fn bench_fq12_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq12, Fq12)> = (0..SAMPLES)
.map(|_| (Fq12::rand(&mut rng), Fq12::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq12, Fq12)> = (0..SAMPLES)
.map(|_| (Fq12::rand(&mut rng), Fq12::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq12, Fq12)> = (0..SAMPLES)
.map(|_| (Fq12::rand(&mut rng), Fq12::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_squaring(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq12> = (0..SAMPLES).map(|_| Fq12::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq12_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq12> = (0..SAMPLES).map(|_| Fq12::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].inverse();
count = (count + 1) % SAMPLES;
tmp
});
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::{Field, SqrtField};
use pairing::bn256::*;
#[bench]
fn bench_fq2_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq2, Fq2)> = (0..SAMPLES)
.map(|_| (Fq2::rand(&mut rng), Fq2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq2, Fq2)> = (0..SAMPLES)
.map(|_| (Fq2::rand(&mut rng), Fq2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fq2, Fq2)> = (0..SAMPLES)
.map(|_| (Fq2::rand(&mut rng), Fq2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_squaring(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq2> = (0..SAMPLES).map(|_| Fq2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq2> = (0..SAMPLES).map(|_| Fq2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].inverse();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fq2_sqrt(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq2> = (0..SAMPLES).map(|_| Fq2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].sqrt();
count = (count + 1) % SAMPLES;
tmp
});
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use ff::{Field, PrimeField, PrimeFieldRepr, SqrtField};
use pairing::bn256::*;
#[bench]
fn bench_fr_repr_add_nocarry(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FrRepr, FrRepr)> = (0..SAMPLES)
.map(|_| {
let mut tmp1 = FrRepr::rand(&mut rng);
let mut tmp2 = FrRepr::rand(&mut rng);
// Shave a few bits off to avoid overflow.
for _ in 0..3 {
tmp1.div2();
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_nocarry(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_sub_noborrow(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(FrRepr, FrRepr)> = (0..SAMPLES)
.map(|_| {
let tmp1 = FrRepr::rand(&mut rng);
let mut tmp2 = tmp1;
// Ensure tmp2 is smaller than tmp1.
for _ in 0..10 {
tmp2.div2();
}
(tmp1, tmp2)
})
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_noborrow(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_num_bits(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES).map(|_| FrRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = v[count].num_bits();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_mul2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES).map(|_| FrRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.mul2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_repr_div2(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES).map(|_| FrRepr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.div2();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_add_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fr, Fr)> = (0..SAMPLES)
.map(|_| (Fr::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.add_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_sub_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fr, Fr)> = (0..SAMPLES)
.map(|_| (Fr::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.sub_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_mul_assign(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(Fr, Fr)> = (0..SAMPLES)
.map(|_| (Fr::rand(&mut rng), Fr::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count].0;
tmp.mul_assign(&v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_square(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.square();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_inverse(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].inverse()
});
}
#[bench]
fn bench_fr_negate(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let mut tmp = v[count];
tmp.negate();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_fr_sqrt(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES)
.map(|_| {
let mut tmp = Fr::rand(&mut rng);
tmp.square();
tmp
})
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].sqrt()
});
}
#[bench]
fn bench_fr_into_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fr> = (0..SAMPLES).map(|_| Fr::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
v[count].into_repr()
});
}
#[bench]
fn bench_fr_from_repr(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<FrRepr> = (0..SAMPLES)
.map(|_| Fr::rand(&mut rng).into_repr())
.collect();
let mut count = 0;
b.iter(|| {
count = (count + 1) % SAMPLES;
Fr::from_repr(v[count])
});
}

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mod ec;
mod fq;
mod fq12;
mod fq2;
mod fr;
use rand::{Rand, SeedableRng, XorShiftRng};
use pairing::bn256::*;
use pairing::{CurveAffine, Engine};
#[bench]
fn bench_pairing_g1_preparation(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<G1> = (0..SAMPLES).map(|_| G1::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = G1Affine::from(v[count]).prepare();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_g2_preparation(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<G2> = (0..SAMPLES).map(|_| G2::rand(&mut rng)).collect();
let mut count = 0;
b.iter(|| {
let tmp = G2Affine::from(v[count]).prepare();
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_miller_loop(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1Prepared, G2Prepared)> = (0..SAMPLES)
.map(|_| {
(
G1Affine::from(G1::rand(&mut rng)).prepare(),
G2Affine::from(G2::rand(&mut rng)).prepare(),
)
})
.collect();
let mut count = 0;
b.iter(|| {
let tmp = Bn256::miller_loop(&[(&v[count].0, &v[count].1)]);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_final_exponentiation(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<Fq12> = (0..SAMPLES)
.map(|_| {
(
G1Affine::from(G1::rand(&mut rng)).prepare(),
G2Affine::from(G2::rand(&mut rng)).prepare(),
)
})
.map(|(ref p, ref q)| Bn256::miller_loop(&[(p, q)]))
.collect();
let mut count = 0;
b.iter(|| {
let tmp = Bn256::final_exponentiation(&v[count]);
count = (count + 1) % SAMPLES;
tmp
});
}
#[bench]
fn bench_pairing_full(b: &mut ::test::Bencher) {
const SAMPLES: usize = 1000;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let v: Vec<(G1, G2)> = (0..SAMPLES)
.map(|_| (G1::rand(&mut rng), G2::rand(&mut rng)))
.collect();
let mut count = 0;
b.iter(|| {
let tmp = Bn256::pairing(v[count].0, v[count].1);
count = (count + 1) % SAMPLES;
tmp
});
}

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#![feature(test)]
extern crate ff;
extern crate pairing_ce;
extern crate rand;
extern crate test;
mod bls12_381;
mod bn256;

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# BLS12-381
This is an implementation of the BLS12-381 pairing-friendly elliptic curve construction.
## BLS12 Parameterization
BLS12 curves are parameterized by a value *x* such that the base field modulus *q* and subgroup *r* can be computed by:
* q = (x - 1)<sup>2</sup> ((x<sup>4</sup> - x<sup>2</sup> + 1) / 3) + x
* r = (x<sup>4</sup> - x<sup>2</sup> + 1)
Given primes *q* and *r* parameterized as above, we can easily construct an elliptic curve over the prime field F<sub>*q*</sub> which contains a subgroup of order *r* such that *r* | (*q*<sup>12</sup> - 1), giving it an embedding degree of 12. Instantiating its sextic twist over an extension field F<sub>q<sup>2</sup></sub> gives rise to an efficient bilinear pairing function between elements of the order *r* subgroups of either curves, into an order *r* multiplicative subgroup of F<sub>q<sup>12</sup></sub>.
In zk-SNARK schemes, we require F<sub>r</sub> with large 2<sup>n</sup> roots of unity for performing efficient fast-fourier transforms. As such, guaranteeing that large 2<sup>n</sup> | (r - 1), or equivalently that *x* has a large 2<sup>n</sup> factor, gives rise to BLS12 curves suitable for zk-SNARKs.
Due to recent research, it is estimated by many that *q* should be approximately 384 bits to target 128-bit security. Conveniently, *r* is approximately 256 bits when *q* is approximately 384 bits, making BLS12 curves ideal for 128-bit security. It also makes them ideal for many zk-SNARK applications, as the scalar field can be used for keying material such as embedded curve constructions.
Many curves match our descriptions, but we require some extra properties for efficiency purposes:
* *q* should be smaller than 2<sup>383</sup>, and *r* should be smaller than 2<sup>255</sup>, so that the most significant bit is unset when using 64-bit or 32-bit limbs. This allows for cheap reductions.
* F<sub>q<sup>12</sup></sub> is typically constructed using towers of extension fields. As a byproduct of [research](https://eprint.iacr.org/2011/465.pdf) for BLS curves of embedding degree 24, we can identify subfamilies of BLS12 curves (for our purposes, where x mod 72 = {16, 64}) that produce efficient extension field towers and twisting isomorphisms.
* We desire *x* of small Hamming weight, to increase the performance of the pairing function.
## BLS12-381 Instantiation
The BLS12-381 construction is instantiated by `x = -0xd201000000010000`, which produces the largest `q` and smallest Hamming weight of `x` that meets the above requirements. This produces:
* q = `0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab` (381 bits)
* r = `0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001` (255 bits)
Our extension field tower is constructed as follows:
1. F<sub>q<sup>2</sup></sub> is constructed as F<sub>q</sub>(u) / (u<sup>2</sup> - β) where β = -1.
2. F<sub>q<sup>6</sup></sub> is constructed as F<sub>q<sup>2</sup></sub>(v) / (v<sup>3</sup> - ξ) where ξ = u + 1
3. F<sub>q<sup>12</sup></sub> is constructed as F<sub>q<sup>6</sup></sub>(w) / (w<sup>2</sup> - γ) where γ = v
Now, we instantiate the elliptic curve E(F<sub>q</sub>) : y<sup>2</sup> = x<sup>3</sup> + 4, and the elliptic curve E'(F<sub>q<sup>2</sup></sub>) : y<sup>2</sup> = x<sup>3</sup> + 4(u + 1).
The group G<sub>1</sub> is the *r* order subgroup of E, which has cofactor (x - 1)<sup>2</sup> / 3. The group G<sub>2</sub> is the *r* order subgroup of E', which has cofactor (x<sup>8</sup> - 4x<sup>7</sup> + 5x<sup>6</sup> - 4x<sup>4</sup> + 6x<sup>3</sup> - 4x<sup>2</sup> - 4x + 13) / 9.
### Generators
The generators of G<sub>1</sub> and G<sub>2</sub> are computed by finding the lexicographically smallest valid `x`-coordinate, and its lexicographically smallest `y`-coordinate and scaling it by the cofactor such that the result is not the point at infinity.
#### G1
```
x = 3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507
y = 1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569
```
#### G2
```
x = 3059144344244213709971259814753781636986470325476647558659373206291635324768958432433509563104347017837885763365758*u + 352701069587466618187139116011060144890029952792775240219908644239793785735715026873347600343865175952761926303160
y = 927553665492332455747201965776037880757740193453592970025027978793976877002675564980949289727957565575433344219582*u + 1985150602287291935568054521177171638300868978215655730859378665066344726373823718423869104263333984641494340347905
```
### Serialization
* Fq elements are encoded in big-endian form. They occupy 48 bytes in this form.
* Fq2 elements are encoded in big-endian form, meaning that the Fq element c0 + c1 * u is represented by the Fq element c1 followed by the Fq element c0. This means Fq2 elements occupy 96 bytes in this form.
* The group G1 uses Fq elements for coordinates. The group G2 uses Fq2 elements for coordinates.
* G1 and G2 elements can be encoded in uncompressed form (the x-coordinate followed by the y-coordinate) or in compressed form (just the x-coordinate). G1 elements occupy 96 bytes in uncompressed form, and 48 bytes in compressed form. G2 elements occupy 192 bytes in uncompressed form, and 96 bytes in compressed form.
The most-significant three bits of a G1 or G2 encoding should be masked away before the coordinate(s) are interpreted. These bits are used to unambiguously represent the underlying element:
* The most significant bit, when set, indicates that the point is in compressed form. Otherwise, the point is in uncompressed form.
* The second-most significant bit indicates that the point is at infinity. If this bit is set, the remaining bits of the group element's encoding should be set to zero.
* The third-most significant bit is set if (and only if) this point is in compressed form _and_ it is not the point at infinity _and_ its y-coordinate is the lexicographically largest of the two associated with the encoded x-coordinate.

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use super::fq::FROBENIUS_COEFF_FQ12_C1;
use super::fq2::Fq2;
use super::fq6::Fq6;
use ff::Field;
use rand::{Rand, Rng};
/// An element of Fq12, represented by c0 + c1 * w.
#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct Fq12 {
pub c0: Fq6,
pub c1: Fq6,
}
impl ::std::fmt::Display for Fq12 {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
write!(f, "Fq12({} + {} * w)", self.c0, self.c1)
}
}
impl Rand for Fq12 {
fn rand<R: Rng>(rng: &mut R) -> Self {
Fq12 {
c0: rng.gen(),
c1: rng.gen(),
}
}
}
impl Fq12 {
pub fn conjugate(&mut self) {
self.c1.negate();
}
pub fn mul_by_014(&mut self, c0: &Fq2, c1: &Fq2, c4: &Fq2) {
let mut aa = self.c0;
aa.mul_by_01(c0, c1);
let mut bb = self.c1;
bb.mul_by_1(c4);
let mut o = *c1;
o.add_assign(c4);
self.c1.add_assign(&self.c0);
self.c1.mul_by_01(c0, &o);
self.c1.sub_assign(&aa);
self.c1.sub_assign(&bb);
self.c0 = bb;
self.c0.mul_by_nonresidue();
self.c0.add_assign(&aa);
}
}
impl Field for Fq12 {
fn zero() -> Self {
Fq12 {
c0: Fq6::zero(),
c1: Fq6::zero(),
}
}
fn one() -> Self {
Fq12 {
c0: Fq6::one(),
c1: Fq6::zero(),
}
}
fn is_zero(&self) -> bool {
self.c0.is_zero() && self.c1.is_zero()
}
fn double(&mut self) {
self.c0.double();
self.c1.double();
}
fn negate(&mut self) {
self.c0.negate();
self.c1.negate();
}
fn add_assign(&mut self, other: &Self) {
self.c0.add_assign(&other.c0);
self.c1.add_assign(&other.c1);
}
fn sub_assign(&mut self, other: &Self) {
self.c0.sub_assign(&other.c0);
self.c1.sub_assign(&other.c1);
}
fn frobenius_map(&mut self, power: usize) {
self.c0.frobenius_map(power);
self.c1.frobenius_map(power);
self.c1.c0.mul_assign(&FROBENIUS_COEFF_FQ12_C1[power % 12]);
self.c1.c1.mul_assign(&FROBENIUS_COEFF_FQ12_C1[power % 12]);
self.c1.c2.mul_assign(&FROBENIUS_COEFF_FQ12_C1[power % 12]);
}
fn square(&mut self) {
let mut ab = self.c0;
ab.mul_assign(&self.c1);
let mut c0c1 = self.c0;
c0c1.add_assign(&self.c1);
let mut c0 = self.c1;
c0.mul_by_nonresidue();
c0.add_assign(&self.c0);
c0.mul_assign(&c0c1);
c0.sub_assign(&ab);
self.c1 = ab;
self.c1.add_assign(&ab);
ab.mul_by_nonresidue();
c0.sub_assign(&ab);
self.c0 = c0;
}
fn mul_assign(&mut self, other: &Self) {
let mut aa = self.c0;
aa.mul_assign(&other.c0);
let mut bb = self.c1;
bb.mul_assign(&other.c1);
let mut o = other.c0;
o.add_assign(&other.c1);
self.c1.add_assign(&self.c0);
self.c1.mul_assign(&o);
self.c1.sub_assign(&aa);
self.c1.sub_assign(&bb);
self.c0 = bb;
self.c0.mul_by_nonresidue();
self.c0.add_assign(&aa);
}
fn inverse(&self) -> Option<Self> {
let mut c0s = self.c0;
c0s.square();
let mut c1s = self.c1;
c1s.square();
c1s.mul_by_nonresidue();
c0s.sub_assign(&c1s);
c0s.inverse().map(|t| {
let mut tmp = Fq12 { c0: t, c1: t };
tmp.c0.mul_assign(&self.c0);
tmp.c1.mul_assign(&self.c1);
tmp.c1.negate();
tmp
})
}
}
#[cfg(test)]
use rand::{SeedableRng, XorShiftRng};
#[test]
fn test_fq12_mul_by_014() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let c0 = Fq2::rand(&mut rng);
let c1 = Fq2::rand(&mut rng);
let c5 = Fq2::rand(&mut rng);
let mut a = Fq12::rand(&mut rng);
let mut b = a;
a.mul_by_014(&c0, &c1, &c5);
b.mul_assign(&Fq12 {
c0: Fq6 {
c0: c0,
c1: c1,
c2: Fq2::zero(),
},
c1: Fq6 {
c0: Fq2::zero(),
c1: c5,
c2: Fq2::zero(),
},
});
assert_eq!(a, b);
}
}
#[test]
fn fq12_field_tests() {
use ff::PrimeField;
crate::tests::field::random_field_tests::<Fq12>();
crate::tests::field::random_frobenius_tests::<Fq12, _>(super::fq::Fq::char(), 13);
}

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use super::fq::{FROBENIUS_COEFF_FQ2_C1, Fq, NEGATIVE_ONE};
use ff::{Field, SqrtField};
use rand::{Rand, Rng};
use std::cmp::Ordering;
/// An element of Fq2, represented by c0 + c1 * u.
#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct Fq2 {
pub c0: Fq,
pub c1: Fq,
}
impl ::std::fmt::Display for Fq2 {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
write!(f, "Fq2({} + {} * u)", self.c0, self.c1)
}
}
/// `Fq2` elements are ordered lexicographically.
impl Ord for Fq2 {
#[inline(always)]
fn cmp(&self, other: &Fq2) -> Ordering {
match self.c1.cmp(&other.c1) {
Ordering::Greater => Ordering::Greater,
Ordering::Less => Ordering::Less,
Ordering::Equal => self.c0.cmp(&other.c0),
}
}
}
impl PartialOrd for Fq2 {
#[inline(always)]
fn partial_cmp(&self, other: &Fq2) -> Option<Ordering> {
Some(self.cmp(other))
}
}
impl Fq2 {
/// Multiply this element by the cubic and quadratic nonresidue 1 + u.
pub fn mul_by_nonresidue(&mut self) {
let t0 = self.c0;
self.c0.sub_assign(&self.c1);
self.c1.add_assign(&t0);
}
/// Norm of Fq2 as extension field in i over Fq
pub fn norm(&self) -> Fq {
let mut t0 = self.c0;
let mut t1 = self.c1;
t0.square();
t1.square();
t1.add_assign(&t0);
t1
}
}
impl Rand for Fq2 {
fn rand<R: Rng>(rng: &mut R) -> Self {
Fq2 {
c0: rng.gen(),
c1: rng.gen(),
}
}
}
impl Field for Fq2 {
fn zero() -> Self {
Fq2 {
c0: Fq::zero(),
c1: Fq::zero(),
}
}
fn one() -> Self {
Fq2 {
c0: Fq::one(),
c1: Fq::zero(),
}
}
fn is_zero(&self) -> bool {
self.c0.is_zero() && self.c1.is_zero()
}
fn square(&mut self) {
let mut ab = self.c0;
ab.mul_assign(&self.c1);
let mut c0c1 = self.c0;
c0c1.add_assign(&self.c1);
let mut c0 = self.c1;
c0.negate();
c0.add_assign(&self.c0);
c0.mul_assign(&c0c1);
c0.sub_assign(&ab);
self.c1 = ab;
self.c1.add_assign(&ab);
c0.add_assign(&ab);
self.c0 = c0;
}
fn double(&mut self) {
self.c0.double();
self.c1.double();
}
fn negate(&mut self) {
self.c0.negate();
self.c1.negate();
}
fn add_assign(&mut self, other: &Self) {
self.c0.add_assign(&other.c0);
self.c1.add_assign(&other.c1);
}
fn sub_assign(&mut self, other: &Self) {
self.c0.sub_assign(&other.c0);
self.c1.sub_assign(&other.c1);
}
fn mul_assign(&mut self, other: &Self) {
let mut aa = self.c0;
aa.mul_assign(&other.c0);
let mut bb = self.c1;
bb.mul_assign(&other.c1);
let mut o = other.c0;
o.add_assign(&other.c1);
self.c1.add_assign(&self.c0);
self.c1.mul_assign(&o);
self.c1.sub_assign(&aa);
self.c1.sub_assign(&bb);
self.c0 = aa;
self.c0.sub_assign(&bb);
}
fn inverse(&self) -> Option<Self> {
let mut t1 = self.c1;
t1.square();
let mut t0 = self.c0;
t0.square();
t0.add_assign(&t1);
t0.inverse().map(|t| {
let mut tmp = Fq2 {
c0: self.c0,
c1: self.c1,
};
tmp.c0.mul_assign(&t);
tmp.c1.mul_assign(&t);
tmp.c1.negate();
tmp
})
}
fn frobenius_map(&mut self, power: usize) {
self.c1.mul_assign(&FROBENIUS_COEFF_FQ2_C1[power % 2]);
}
}
impl SqrtField for Fq2 {
fn legendre(&self) -> ::ff::LegendreSymbol {
self.norm().legendre()
}
fn sqrt(&self) -> Option<Self> {
// Algorithm 9, https://eprint.iacr.org/2012/685.pdf
if self.is_zero() {
Some(Self::zero())
} else {
// a1 = self^((q - 3) / 4)
let mut a1 = self.pow([
0xee7fbfffffffeaaa,
0x7aaffffac54ffff,
0xd9cc34a83dac3d89,
0xd91dd2e13ce144af,
0x92c6e9ed90d2eb35,
0x680447a8e5ff9a6,
]);
let mut alpha = a1;
alpha.square();
alpha.mul_assign(self);
let mut a0 = alpha;
a0.frobenius_map(1);
a0.mul_assign(&alpha);
let neg1 = Fq2 {
c0: NEGATIVE_ONE,
c1: Fq::zero(),
};
if a0 == neg1 {
None
} else {
a1.mul_assign(self);
if alpha == neg1 {
a1.mul_assign(&Fq2 {
c0: Fq::zero(),
c1: Fq::one(),
});
} else {
alpha.add_assign(&Fq2::one());
// alpha = alpha^((q - 1) / 2)
alpha = alpha.pow([
0xdcff7fffffffd555,
0xf55ffff58a9ffff,
0xb39869507b587b12,
0xb23ba5c279c2895f,
0x258dd3db21a5d66b,
0xd0088f51cbff34d,
]);
a1.mul_assign(&alpha);
}
Some(a1)
}
}
}
}
#[test]
fn test_fq2_ordering() {
let mut a = Fq2 {
c0: Fq::zero(),
c1: Fq::zero(),
};
let mut b = a.clone();
assert!(a.cmp(&b) == Ordering::Equal);
b.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Less);
a.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Equal);
b.c1.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Less);
a.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Less);
a.c1.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Greater);
b.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Equal);
}
#[test]
fn test_fq2_basics() {
assert_eq!(
Fq2 {
c0: Fq::zero(),
c1: Fq::zero(),
},
Fq2::zero()
);
assert_eq!(
Fq2 {
c0: Fq::one(),
c1: Fq::zero(),
},
Fq2::one()
);
assert!(Fq2::zero().is_zero());
assert!(!Fq2::one().is_zero());
assert!(
!Fq2 {
c0: Fq::zero(),
c1: Fq::one(),
}.is_zero()
);
}
#[test]
fn test_fq2_squaring() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::one(),
c1: Fq::one(),
}; // u + 1
a.square();
assert_eq!(
a,
Fq2 {
c0: Fq::zero(),
c1: Fq::from_repr(FqRepr::from(2)).unwrap(),
}
); // 2u
let mut a = Fq2 {
c0: Fq::zero(),
c1: Fq::one(),
}; // u
a.square();
assert_eq!(a, {
let mut neg1 = Fq::one();
neg1.negate();
Fq2 {
c0: neg1,
c1: Fq::zero(),
}
}); // -1
let mut a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x9c2c6309bbf8b598,
0x4eef5c946536f602,
0x90e34aab6fb6a6bd,
0xf7f295a94e58ae7c,
0x41b76dcc1c3fbe5e,
0x7080c5fa1d8e042,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x38f473b3c870a4ab,
0x6ad3291177c8c7e5,
0xdac5a4c911a4353e,
0xbfb99020604137a0,
0xfc58a7b7be815407,
0x10d1615e75250a21,
])).unwrap(),
};
a.square();
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0xf262c28c538bcf68,
0xb9f2a66eae1073ba,
0xdc46ab8fad67ae0,
0xcb674157618da176,
0x4cf17b5893c3d327,
0x7eac81369c43361
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0xc1579cf58e980cf8,
0xa23eb7e12dd54d98,
0xe75138bce4cec7aa,
0x38d0d7275a9689e1,
0x739c983042779a65,
0x1542a61c8a8db994
])).unwrap(),
}
);
}
#[test]
fn test_fq2_mul() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x85c9f989e1461f03,
0xa2e33c333449a1d6,
0x41e461154a7354a3,
0x9ee53e7e84d7532e,
0x1c202d8ed97afb45,
0x51d3f9253e2516f,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0xa7348a8b511aedcf,
0x143c215d8176b319,
0x4cc48081c09b8903,
0x9533e4a9a5158be,
0x7a5e1ecb676d65f9,
0x180c3ee46656b008,
])).unwrap(),
};
a.mul_assign(&Fq2 {
c0: Fq::from_repr(FqRepr([
0xe21f9169805f537e,
0xfc87e62e179c285d,
0x27ece175be07a531,
0xcd460f9f0c23e430,
0x6c9110292bfa409,
0x2c93a72eb8af83e,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x4b1c3f936d8992d4,
0x1d2a72916dba4c8a,
0x8871c508658d1e5f,
0x57a06d3135a752ae,
0x634cd3c6c565096d,
0x19e17334d4e93558,
])).unwrap(),
});
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x95b5127e6360c7e4,
0xde29c31a19a6937e,
0xf61a96dacf5a39bc,
0x5511fe4d84ee5f78,
0x5310a202d92f9963,
0x1751afbe166e5399
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x84af0e1bd630117a,
0x6c63cd4da2c2aa7,
0x5ba6e5430e883d40,
0xc975106579c275ee,
0x33a9ac82ce4c5083,
0x1ef1a36c201589d
])).unwrap(),
}
);
}
#[test]
fn test_fq2_inverse() {
use super::fq::FqRepr;
use ff::PrimeField;
assert!(Fq2::zero().inverse().is_none());
let a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x85c9f989e1461f03,
0xa2e33c333449a1d6,
0x41e461154a7354a3,
0x9ee53e7e84d7532e,
0x1c202d8ed97afb45,
0x51d3f9253e2516f,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0xa7348a8b511aedcf,
0x143c215d8176b319,
0x4cc48081c09b8903,
0x9533e4a9a5158be,
0x7a5e1ecb676d65f9,
0x180c3ee46656b008,
])).unwrap(),
};
let a = a.inverse().unwrap();
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x70300f9bcb9e594,
0xe5ecda5fdafddbb2,
0x64bef617d2915a8f,
0xdfba703293941c30,
0xa6c3d8f9586f2636,
0x1351ef01941b70c4
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x8c39fd76a8312cb4,
0x15d7b6b95defbff0,
0x947143f89faedee9,
0xcbf651a0f367afb2,
0xdf4e54f0d3ef15a6,
0x103bdf241afb0019
])).unwrap(),
}
);
}
#[test]
fn test_fq2_addition() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837,
])).unwrap(),
};
a.add_assign(&Fq2 {
c0: Fq::from_repr(FqRepr([
0x619a02d78dc70ef2,
0xb93adfc9119e33e8,
0x4bf0b99a9f0dca12,
0x3b88899a42a6318f,
0x986a4a62fa82a49d,
0x13ce433fa26027f5,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x66323bf80b58b9b9,
0xa1379b6facf6e596,
0x402aef1fb797e32f,
0x2236f55246d0d44d,
0x4c8c1800eb104566,
0x11d6e20e986c2085,
])).unwrap(),
});
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x8e9a7adaf6eb0eb9,
0xcb207e6b3341eaba,
0xd70b0c7b481d23ff,
0xf4ef57d604b6bca2,
0x65309427b3d5d090,
0x14c715d5553f01d2
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0xfdb032e7d9079a94,
0x35a2809d15468d83,
0xfe4b23317e0796d5,
0xd62fa51334f560fa,
0x9ad265eb46e01984,
0x1303f3465112c8bc
])).unwrap(),
}
);
}
#[test]
fn test_fq2_subtraction() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837,
])).unwrap(),
};
a.sub_assign(&Fq2 {
c0: Fq::from_repr(FqRepr([
0x619a02d78dc70ef2,
0xb93adfc9119e33e8,
0x4bf0b99a9f0dca12,
0x3b88899a42a6318f,
0x986a4a62fa82a49d,
0x13ce433fa26027f5,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x66323bf80b58b9b9,
0xa1379b6facf6e596,
0x402aef1fb797e32f,
0x2236f55246d0d44d,
0x4c8c1800eb104566,
0x11d6e20e986c2085,
])).unwrap(),
});
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x8565752bdb5c9b80,
0x7756bed7c15982e9,
0xa65a6be700b285fe,
0xe255902672ef6c43,
0x7f77a718021c342d,
0x72ba14049fe9881
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0xeb4abaf7c255d1cd,
0x11df49bc6cacc256,
0xe52617930588c69a,
0xf63905f39ad8cb1f,
0x4cd5dd9fb40b3b8f,
0x957411359ba6e4c
])).unwrap(),
}
);
}
#[test]
fn test_fq2_negation() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837,
])).unwrap(),
};
a.negate();
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x8cfe87fc96dbaae4,
0xcc6615c8fb0492d,
0xdc167fc04da19c37,
0xab107d49317487ab,
0x7e555df189f880e3,
0x19083f5486a10cbd
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x228109103250c9d0,
0x8a411ad149045812,
0xa9109e8f3041427e,
0xb07e9bc405608611,
0xfcd559cbe77bd8b8,
0x18d400b280d93e62
])).unwrap(),
}
);
}
#[test]
fn test_fq2_doubling() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837,
])).unwrap(),
};
a.double();
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x5a00f006d247ff8e,
0x23cb3d4443476da4,
0x1634a5c1521eb3da,
0x72cd9c7784211627,
0x998c938972a657e7,
0x1f1a52b65bdb3b9
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x2efbeddf9b5dc1b6,
0x28d5ca5ad09f4fdb,
0x7c4068238cdf674b,
0x67f15f81dc49195b,
0x9c8c9bd4b79fa83d,
0x25a226f714d506e
])).unwrap(),
}
);
}
#[test]
fn test_fq2_frobenius_map() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc,
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837,
])).unwrap(),
};
a.frobenius_map(0);
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837
])).unwrap(),
}
);
a.frobenius_map(1);
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x228109103250c9d0,
0x8a411ad149045812,
0xa9109e8f3041427e,
0xb07e9bc405608611,
0xfcd559cbe77bd8b8,
0x18d400b280d93e62
])).unwrap(),
}
);
a.frobenius_map(1);
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837
])).unwrap(),
}
);
a.frobenius_map(2);
assert_eq!(
a,
Fq2 {
c0: Fq::from_repr(FqRepr([
0x2d0078036923ffc7,
0x11e59ea221a3b6d2,
0x8b1a52e0a90f59ed,
0xb966ce3bc2108b13,
0xccc649c4b9532bf3,
0xf8d295b2ded9dc
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0x977df6efcdaee0db,
0x946ae52d684fa7ed,
0xbe203411c66fb3a5,
0xb3f8afc0ee248cad,
0x4e464dea5bcfd41e,
0x12d1137b8a6a837
])).unwrap(),
}
);
}
#[test]
fn test_fq2_sqrt() {
use super::fq::FqRepr;
use ff::PrimeField;
assert_eq!(
Fq2 {
c0: Fq::from_repr(FqRepr([
0x476b4c309720e227,
0x34c2d04faffdab6,
0xa57e6fc1bab51fd9,
0xdb4a116b5bf74aa1,
0x1e58b2159dfe10e2,
0x7ca7da1f13606ac
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0xfa8de88b7516d2c3,
0x371a75ed14f41629,
0x4cec2dca577a3eb6,
0x212611bca4e99121,
0x8ee5394d77afb3d,
0xec92336650e49d5
])).unwrap(),
}.sqrt()
.unwrap(),
Fq2 {
c0: Fq::from_repr(FqRepr([
0x40b299b2704258c5,
0x6ef7de92e8c68b63,
0x6d2ddbe552203e82,
0x8d7f1f723d02c1d3,
0x881b3e01b611c070,
0x10f6963bbad2ebc5
])).unwrap(),
c1: Fq::from_repr(FqRepr([
0xc099534fc209e752,
0x7670594665676447,
0x28a20faed211efe7,
0x6b852aeaf2afcb1b,
0xa4c93b08105d71a9,
0x8d7cfff94216330
])).unwrap(),
}
);
assert_eq!(
Fq2 {
c0: Fq::from_repr(FqRepr([
0xb9f78429d1517a6b,
0x1eabfffeb153ffff,
0x6730d2a0f6b0f624,
0x64774b84f38512bf,
0x4b1ba7b6434bacd7,
0x1a0111ea397fe69a
])).unwrap(),
c1: Fq::zero(),
}.sqrt()
.unwrap(),
Fq2 {
c0: Fq::zero(),
c1: Fq::from_repr(FqRepr([
0xb9fefffffd4357a3,
0x1eabfffeb153ffff,
0x6730d2a0f6b0f624,
0x64774b84f38512bf,
0x4b1ba7b6434bacd7,
0x1a0111ea397fe69a
])).unwrap(),
}
);
}
#[test]
fn test_fq2_legendre() {
use ff::LegendreSymbol::*;
assert_eq!(Zero, Fq2::zero().legendre());
// i^2 = -1
let mut m1 = Fq2::one();
m1.negate();
assert_eq!(QuadraticResidue, m1.legendre());
m1.mul_by_nonresidue();
assert_eq!(QuadraticNonResidue, m1.legendre());
}
#[cfg(test)]
use rand::{SeedableRng, XorShiftRng};
#[test]
fn test_fq2_mul_nonresidue() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let nqr = Fq2 {
c0: Fq::one(),
c1: Fq::one(),
};
for _ in 0..1000 {
let mut a = Fq2::rand(&mut rng);
let mut b = a;
a.mul_by_nonresidue();
b.mul_assign(&nqr);
assert_eq!(a, b);
}
}
#[test]
fn fq2_field_tests() {
use ff::PrimeField;
crate::tests::field::random_field_tests::<Fq2>();
crate::tests::field::random_sqrt_tests::<Fq2>();
crate::tests::field::random_frobenius_tests::<Fq2, _>(super::fq::Fq::char(), 13);
}

374
pairing/src/bls12_381/fq6.rs Normal file
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@ -0,0 +1,374 @@
use super::fq::{FROBENIUS_COEFF_FQ6_C1, FROBENIUS_COEFF_FQ6_C2};
use super::fq2::Fq2;
use ff::Field;
use rand::{Rand, Rng};
/// An element of Fq6, represented by c0 + c1 * v + c2 * v^(2).
#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct Fq6 {
pub c0: Fq2,
pub c1: Fq2,
pub c2: Fq2,
}
impl ::std::fmt::Display for Fq6 {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
write!(f, "Fq6({} + {} * v, {} * v^2)", self.c0, self.c1, self.c2)
}
}
impl Rand for Fq6 {
fn rand<R: Rng>(rng: &mut R) -> Self {
Fq6 {
c0: rng.gen(),
c1: rng.gen(),
c2: rng.gen(),
}
}
}
impl Fq6 {
/// Multiply by quadratic nonresidue v.
pub fn mul_by_nonresidue(&mut self) {
use std::mem::swap;
swap(&mut self.c0, &mut self.c1);
swap(&mut self.c0, &mut self.c2);
self.c0.mul_by_nonresidue();
}
pub fn mul_by_1(&mut self, c1: &Fq2) {
let mut b_b = self.c1;
b_b.mul_assign(c1);
let mut t1 = *c1;
{
let mut tmp = self.c1;
tmp.add_assign(&self.c2);
t1.mul_assign(&tmp);
t1.sub_assign(&b_b);
t1.mul_by_nonresidue();
}
let mut t2 = *c1;
{
let mut tmp = self.c0;
tmp.add_assign(&self.c1);
t2.mul_assign(&tmp);
t2.sub_assign(&b_b);
}
self.c0 = t1;
self.c1 = t2;
self.c2 = b_b;
}
pub fn mul_by_01(&mut self, c0: &Fq2, c1: &Fq2) {
let mut a_a = self.c0;
let mut b_b = self.c1;
a_a.mul_assign(c0);
b_b.mul_assign(c1);
let mut t1 = *c1;
{
let mut tmp = self.c1;
tmp.add_assign(&self.c2);
t1.mul_assign(&tmp);
t1.sub_assign(&b_b);
t1.mul_by_nonresidue();
t1.add_assign(&a_a);
}
let mut t3 = *c0;
{
let mut tmp = self.c0;
tmp.add_assign(&self.c2);
t3.mul_assign(&tmp);
t3.sub_assign(&a_a);
t3.add_assign(&b_b);
}
let mut t2 = *c0;
t2.add_assign(c1);
{
let mut tmp = self.c0;
tmp.add_assign(&self.c1);
t2.mul_assign(&tmp);
t2.sub_assign(&a_a);
t2.sub_assign(&b_b);
}
self.c0 = t1;
self.c1 = t2;
self.c2 = t3;
}
}
impl Field for Fq6 {
fn zero() -> Self {
Fq6 {
c0: Fq2::zero(),
c1: Fq2::zero(),
c2: Fq2::zero(),
}
}
fn one() -> Self {
Fq6 {
c0: Fq2::one(),
c1: Fq2::zero(),
c2: Fq2::zero(),
}
}
fn is_zero(&self) -> bool {
self.c0.is_zero() && self.c1.is_zero() && self.c2.is_zero()
}
fn double(&mut self) {
self.c0.double();
self.c1.double();
self.c2.double();
}
fn negate(&mut self) {
self.c0.negate();
self.c1.negate();
self.c2.negate();
}
fn add_assign(&mut self, other: &Self) {
self.c0.add_assign(&other.c0);
self.c1.add_assign(&other.c1);
self.c2.add_assign(&other.c2);
}
fn sub_assign(&mut self, other: &Self) {
self.c0.sub_assign(&other.c0);
self.c1.sub_assign(&other.c1);
self.c2.sub_assign(&other.c2);
}
fn frobenius_map(&mut self, power: usize) {
self.c0.frobenius_map(power);
self.c1.frobenius_map(power);
self.c2.frobenius_map(power);
self.c1.mul_assign(&FROBENIUS_COEFF_FQ6_C1[power % 6]);
self.c2.mul_assign(&FROBENIUS_COEFF_FQ6_C2[power % 6]);
}
fn square(&mut self) {
let mut s0 = self.c0;
s0.square();
let mut ab = self.c0;
ab.mul_assign(&self.c1);
let mut s1 = ab;
s1.double();
let mut s2 = self.c0;
s2.sub_assign(&self.c1);
s2.add_assign(&self.c2);
s2.square();
let mut bc = self.c1;
bc.mul_assign(&self.c2);
let mut s3 = bc;
s3.double();
let mut s4 = self.c2;
s4.square();
self.c0 = s3;
self.c0.mul_by_nonresidue();
self.c0.add_assign(&s0);
self.c1 = s4;
self.c1.mul_by_nonresidue();
self.c1.add_assign(&s1);
self.c2 = s1;
self.c2.add_assign(&s2);
self.c2.add_assign(&s3);
self.c2.sub_assign(&s0);
self.c2.sub_assign(&s4);
}
fn mul_assign(&mut self, other: &Self) {
let mut a_a = self.c0;
let mut b_b = self.c1;
let mut c_c = self.c2;
a_a.mul_assign(&other.c0);
b_b.mul_assign(&other.c1);
c_c.mul_assign(&other.c2);
let mut t1 = other.c1;
t1.add_assign(&other.c2);
{
let mut tmp = self.c1;
tmp.add_assign(&self.c2);
t1.mul_assign(&tmp);
t1.sub_assign(&b_b);
t1.sub_assign(&c_c);
t1.mul_by_nonresidue();
t1.add_assign(&a_a);
}
let mut t3 = other.c0;
t3.add_assign(&other.c2);
{
let mut tmp = self.c0;
tmp.add_assign(&self.c2);
t3.mul_assign(&tmp);
t3.sub_assign(&a_a);
t3.add_assign(&b_b);
t3.sub_assign(&c_c);
}
let mut t2 = other.c0;
t2.add_assign(&other.c1);
{
let mut tmp = self.c0;
tmp.add_assign(&self.c1);
t2.mul_assign(&tmp);
t2.sub_assign(&a_a);
t2.sub_assign(&b_b);
c_c.mul_by_nonresidue();
t2.add_assign(&c_c);
}
self.c0 = t1;
self.c1 = t2;
self.c2 = t3;
}
fn inverse(&self) -> Option<Self> {
let mut c0 = self.c2;
c0.mul_by_nonresidue();
c0.mul_assign(&self.c1);
c0.negate();
{
let mut c0s = self.c0;
c0s.square();
c0.add_assign(&c0s);
}
let mut c1 = self.c2;
c1.square();
c1.mul_by_nonresidue();
{
let mut c01 = self.c0;
c01.mul_assign(&self.c1);
c1.sub_assign(&c01);
}
let mut c2 = self.c1;
c2.square();
{
let mut c02 = self.c0;
c02.mul_assign(&self.c2);
c2.sub_assign(&c02);
}
let mut tmp1 = self.c2;
tmp1.mul_assign(&c1);
let mut tmp2 = self.c1;
tmp2.mul_assign(&c2);
tmp1.add_assign(&tmp2);
tmp1.mul_by_nonresidue();
tmp2 = self.c0;
tmp2.mul_assign(&c0);
tmp1.add_assign(&tmp2);
match tmp1.inverse() {
Some(t) => {
let mut tmp = Fq6 {
c0: t,
c1: t,
c2: t,
};
tmp.c0.mul_assign(&c0);
tmp.c1.mul_assign(&c1);
tmp.c2.mul_assign(&c2);
Some(tmp)
}
None => None,
}
}
}
#[cfg(test)]
use rand::{SeedableRng, XorShiftRng};
#[test]
fn test_fq6_mul_nonresidue() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let nqr = Fq6 {
c0: Fq2::zero(),
c1: Fq2::one(),
c2: Fq2::zero(),
};
for _ in 0..1000 {
let mut a = Fq6::rand(&mut rng);
let mut b = a;
a.mul_by_nonresidue();
b.mul_assign(&nqr);
assert_eq!(a, b);
}
}
#[test]
fn test_fq6_mul_by_1() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let c1 = Fq2::rand(&mut rng);
let mut a = Fq6::rand(&mut rng);
let mut b = a;
a.mul_by_1(&c1);
b.mul_assign(&Fq6 {
c0: Fq2::zero(),
c1: c1,
c2: Fq2::zero(),
});
assert_eq!(a, b);
}
}
#[test]
fn test_fq6_mul_by_01() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let c0 = Fq2::rand(&mut rng);
let c1 = Fq2::rand(&mut rng);
let mut a = Fq6::rand(&mut rng);
let mut b = a;
a.mul_by_01(&c0, &c1);
b.mul_assign(&Fq6 {
c0: c0,
c1: c1,
c2: Fq2::zero(),
});
assert_eq!(a, b);
}
}
#[test]
fn fq6_field_tests() {
use ff::PrimeField;
crate::tests::field::random_field_tests::<Fq6>();
crate::tests::field::random_frobenius_tests::<Fq6, _>(super::fq::Fq::char(), 13);
}

986
pairing/src/bls12_381/fr.rs Normal file
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@ -0,0 +1,986 @@
use ff::{Field, PrimeField, PrimeFieldRepr};
#[derive(PrimeField)]
#[PrimeFieldModulus = "52435875175126190479447740508185965837690552500527637822603658699938581184513"]
#[PrimeFieldGenerator = "7"]
pub struct Fr(FrRepr);
#[cfg(test)]
use rand::{Rand, SeedableRng, XorShiftRng};
#[test]
fn test_fr_repr_ordering() {
fn assert_equality(a: FrRepr, b: FrRepr) {
assert_eq!(a, b);
assert!(a.cmp(&b) == ::std::cmp::Ordering::Equal);
}
fn assert_lt(a: FrRepr, b: FrRepr) {
assert!(a < b);
assert!(b > a);
}
assert_equality(
FrRepr([9999, 9999, 9999, 9999]),
FrRepr([9999, 9999, 9999, 9999]),
);
assert_equality(
FrRepr([9999, 9998, 9999, 9999]),
FrRepr([9999, 9998, 9999, 9999]),
);
assert_equality(
FrRepr([9999, 9999, 9999, 9997]),
FrRepr([9999, 9999, 9999, 9997]),
);
assert_lt(
FrRepr([9999, 9997, 9999, 9998]),
FrRepr([9999, 9997, 9999, 9999]),
);
assert_lt(
FrRepr([9999, 9997, 9998, 9999]),
FrRepr([9999, 9997, 9999, 9999]),
);
assert_lt(
FrRepr([9, 9999, 9999, 9997]),
FrRepr([9999, 9999, 9999, 9997]),
);
}
#[test]
fn test_fr_repr_from() {
assert_eq!(FrRepr::from(100), FrRepr([100, 0, 0, 0]));
}
#[test]
fn test_fr_repr_is_odd() {
assert!(!FrRepr::from(0).is_odd());
assert!(FrRepr::from(0).is_even());
assert!(FrRepr::from(1).is_odd());
assert!(!FrRepr::from(1).is_even());
assert!(!FrRepr::from(324834872).is_odd());
assert!(FrRepr::from(324834872).is_even());
assert!(FrRepr::from(324834873).is_odd());
assert!(!FrRepr::from(324834873).is_even());
}
#[test]
fn test_fr_repr_is_zero() {
assert!(FrRepr::from(0).is_zero());
assert!(!FrRepr::from(1).is_zero());
assert!(!FrRepr([0, 0, 1, 0]).is_zero());
}
#[test]
fn test_fr_repr_div2() {
let mut a = FrRepr([
0xbd2920b19c972321,
0x174ed0466a3be37e,
0xd468d5e3b551f0b5,
0xcb67c072733beefc,
]);
a.div2();
assert_eq!(
a,
FrRepr([
0x5e949058ce4b9190,
0x8ba76823351df1bf,
0x6a346af1daa8f85a,
0x65b3e039399df77e
])
);
for _ in 0..10 {
a.div2();
}
assert_eq!(
a,
FrRepr([
0x6fd7a524163392e4,
0x16a2e9da08cd477c,
0xdf9a8d1abc76aa3e,
0x196cf80e4e677d
])
);
for _ in 0..200 {
a.div2();
}
assert_eq!(a, FrRepr([0x196cf80e4e67, 0x0, 0x0, 0x0]));
for _ in 0..40 {
a.div2();
}
assert_eq!(a, FrRepr([0x19, 0x0, 0x0, 0x0]));
for _ in 0..4 {
a.div2();
}
assert_eq!(a, FrRepr([0x1, 0x0, 0x0, 0x0]));
a.div2();
assert!(a.is_zero());
}
#[test]
fn test_fr_repr_shr() {
let mut a = FrRepr([
0xb33fbaec482a283f,
0x997de0d3a88cb3df,
0x9af62d2a9a0e5525,
0x36003ab08de70da1,
]);
a.shr(0);
assert_eq!(
a,
FrRepr([
0xb33fbaec482a283f,
0x997de0d3a88cb3df,
0x9af62d2a9a0e5525,
0x36003ab08de70da1
])
);
a.shr(1);
assert_eq!(
a,
FrRepr([
0xd99fdd762415141f,
0xccbef069d44659ef,
0xcd7b16954d072a92,
0x1b001d5846f386d0
])
);
a.shr(50);
assert_eq!(
a,
FrRepr([
0xbc1a7511967bf667,
0xc5a55341caa4b32f,
0x75611bce1b4335e,
0x6c0
])
);
a.shr(130);
assert_eq!(a, FrRepr([0x1d5846f386d0cd7, 0x1b0, 0x0, 0x0]));
a.shr(64);
assert_eq!(a, FrRepr([0x1b0, 0x0, 0x0, 0x0]));
}
#[test]
fn test_fr_repr_mul2() {
let mut a = FrRepr::from(23712937547);
a.mul2();
assert_eq!(a, FrRepr([0xb0acd6c96, 0x0, 0x0, 0x0]));
for _ in 0..60 {
a.mul2();
}
assert_eq!(a, FrRepr([0x6000000000000000, 0xb0acd6c9, 0x0, 0x0]));
for _ in 0..128 {
a.mul2();
}
assert_eq!(a, FrRepr([0x0, 0x0, 0x6000000000000000, 0xb0acd6c9]));
for _ in 0..60 {
a.mul2();
}
assert_eq!(a, FrRepr([0x0, 0x0, 0x0, 0x9600000000000000]));
for _ in 0..7 {
a.mul2();
}
assert!(a.is_zero());
}
#[test]
fn test_fr_repr_num_bits() {
let mut a = FrRepr::from(0);
assert_eq!(0, a.num_bits());
a = FrRepr::from(1);
for i in 1..257 {
assert_eq!(i, a.num_bits());
a.mul2();
}
assert_eq!(0, a.num_bits());
}
#[test]
fn test_fr_repr_sub_noborrow() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let mut t = FrRepr([
0x8e62a7e85264e2c3,
0xb23d34c1941d3ca,
0x5976930b7502dd15,
0x600f3fb517bf5495,
]);
t.sub_noborrow(&FrRepr([
0xd64f669809cbc6a4,
0xfa76cb9d90cf7637,
0xfefb0df9038d43b3,
0x298a30c744b31acf,
]));
assert!(
t == FrRepr([
0xb813415048991c1f,
0x10ad07ae88725d92,
0x5a7b851271759961,
0x36850eedd30c39c5
])
);
for _ in 0..1000 {
let mut a = FrRepr::rand(&mut rng);
a.0[3] >>= 30;
let mut b = a;
for _ in 0..10 {
b.mul2();
}
let mut c = b;
for _ in 0..10 {
c.mul2();
}
assert!(a < b);
assert!(b < c);
let mut csub_ba = c;
csub_ba.sub_noborrow(&b);
csub_ba.sub_noborrow(&a);
let mut csub_ab = c;
csub_ab.sub_noborrow(&a);
csub_ab.sub_noborrow(&b);
assert_eq!(csub_ab, csub_ba);
}
// Subtracting r+1 from r should produce -1 (mod 2**256)
let mut qplusone = FrRepr([
0xffffffff00000001,
0x53bda402fffe5bfe,
0x3339d80809a1d805,
0x73eda753299d7d48,
]);
qplusone.sub_noborrow(&FrRepr([
0xffffffff00000002,
0x53bda402fffe5bfe,
0x3339d80809a1d805,
0x73eda753299d7d48,
]));
assert_eq!(
qplusone,
FrRepr([
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff
])
);
}
#[test]
fn test_fr_legendre() {
use ff::LegendreSymbol::*;
use ff::SqrtField;
assert_eq!(QuadraticResidue, Fr::one().legendre());
assert_eq!(Zero, Fr::zero().legendre());
let e = FrRepr([
0x0dbc5349cd5664da,
0x8ac5b6296e3ae29d,
0x127cb819feceaa3b,
0x3a6b21fb03867191,
]);
assert_eq!(QuadraticResidue, Fr::from_repr(e).unwrap().legendre());
let e = FrRepr([
0x96341aefd047c045,
0x9b5f4254500a4d65,
0x1ee08223b68ac240,
0x31d9cd545c0ec7c6,
]);
assert_eq!(QuadraticNonResidue, Fr::from_repr(e).unwrap().legendre());
}
#[test]
fn test_fr_repr_add_nocarry() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let mut t = FrRepr([
0xd64f669809cbc6a4,
0xfa76cb9d90cf7637,
0xfefb0df9038d43b3,
0x298a30c744b31acf,
]);
t.add_nocarry(&FrRepr([
0x8e62a7e85264e2c3,
0xb23d34c1941d3ca,
0x5976930b7502dd15,
0x600f3fb517bf5495,
]));
assert_eq!(
t,
FrRepr([
0x64b20e805c30a967,
0x59a9ee9aa114a02,
0x5871a104789020c9,
0x8999707c5c726f65
])
);
// Test for the associativity of addition.
for _ in 0..1000 {
let mut a = FrRepr::rand(&mut rng);
let mut b = FrRepr::rand(&mut rng);
let mut c = FrRepr::rand(&mut rng);
// Unset the first few bits, so that overflow won't occur.
a.0[3] >>= 3;
b.0[3] >>= 3;
c.0[3] >>= 3;
let mut abc = a;
abc.add_nocarry(&b);
abc.add_nocarry(&c);
let mut acb = a;
acb.add_nocarry(&c);
acb.add_nocarry(&b);
let mut bac = b;
bac.add_nocarry(&a);
bac.add_nocarry(&c);
let mut bca = b;
bca.add_nocarry(&c);
bca.add_nocarry(&a);
let mut cab = c;
cab.add_nocarry(&a);
cab.add_nocarry(&b);
let mut cba = c;
cba.add_nocarry(&b);
cba.add_nocarry(&a);
assert_eq!(abc, acb);
assert_eq!(abc, bac);
assert_eq!(abc, bca);
assert_eq!(abc, cab);
assert_eq!(abc, cba);
}
// Adding 1 to (2^256 - 1) should produce zero
let mut x = FrRepr([
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
]);
x.add_nocarry(&FrRepr::from(1));
assert!(x.is_zero());
}
#[test]
fn test_fr_is_valid() {
let mut a = Fr(MODULUS);
assert!(!a.is_valid());
a.0.sub_noborrow(&FrRepr::from(1));
assert!(a.is_valid());
assert!(Fr(FrRepr::from(0)).is_valid());
assert!(
Fr(FrRepr([
0xffffffff00000000,
0x53bda402fffe5bfe,
0x3339d80809a1d805,
0x73eda753299d7d48
])).is_valid()
);
assert!(
!Fr(FrRepr([
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff
])).is_valid()
);
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let a = Fr::rand(&mut rng);
assert!(a.is_valid());
}
}
#[test]
fn test_fr_add_assign() {
{
// Random number
let mut tmp = Fr(FrRepr([
0x437ce7616d580765,
0xd42d1ccb29d1235b,
0xed8f753821bd1423,
0x4eede1c9c89528ca,
]));
assert!(tmp.is_valid());
// Test that adding zero has no effect.
tmp.add_assign(&Fr(FrRepr::from(0)));
assert_eq!(
tmp,
Fr(FrRepr([
0x437ce7616d580765,
0xd42d1ccb29d1235b,
0xed8f753821bd1423,
0x4eede1c9c89528ca
]))
);
// Add one and test for the result.
tmp.add_assign(&Fr(FrRepr::from(1)));
assert_eq!(
tmp,
Fr(FrRepr([
0x437ce7616d580766,
0xd42d1ccb29d1235b,
0xed8f753821bd1423,
0x4eede1c9c89528ca
]))
);
// Add another random number that exercises the reduction.
tmp.add_assign(&Fr(FrRepr([
0x946f435944f7dc79,
0xb55e7ee6533a9b9b,
0x1e43b84c2f6194ca,
0x58717ab525463496,
])));
assert_eq!(
tmp,
Fr(FrRepr([
0xd7ec2abbb24fe3de,
0x35cdf7ae7d0d62f7,
0xd899557c477cd0e9,
0x3371b52bc43de018
]))
);
// Add one to (r - 1) and test for the result.
tmp = Fr(FrRepr([
0xffffffff00000000,
0x53bda402fffe5bfe,
0x3339d80809a1d805,
0x73eda753299d7d48,
]));
tmp.add_assign(&Fr(FrRepr::from(1)));
assert!(tmp.0.is_zero());
// Add a random number to another one such that the result is r - 1
tmp = Fr(FrRepr([
0xade5adacdccb6190,
0xaa21ee0f27db3ccd,
0x2550f4704ae39086,
0x591d1902e7c5ba27,
]));
tmp.add_assign(&Fr(FrRepr([
0x521a525223349e70,
0xa99bb5f3d8231f31,
0xde8e397bebe477e,
0x1ad08e5041d7c321,
])));
assert_eq!(
tmp,
Fr(FrRepr([
0xffffffff00000000,
0x53bda402fffe5bfe,
0x3339d80809a1d805,
0x73eda753299d7d48
]))
);
// Add one to the result and test for it.
tmp.add_assign(&Fr(FrRepr::from(1)));
assert!(tmp.0.is_zero());
}
// Test associativity
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
// Generate a, b, c and ensure (a + b) + c == a + (b + c).
let a = Fr::rand(&mut rng);
let b = Fr::rand(&mut rng);
let c = Fr::rand(&mut rng);
let mut tmp1 = a;
tmp1.add_assign(&b);
tmp1.add_assign(&c);
let mut tmp2 = b;
tmp2.add_assign(&c);
tmp2.add_assign(&a);
assert!(tmp1.is_valid());
assert!(tmp2.is_valid());
assert_eq!(tmp1, tmp2);
}
}
#[test]
fn test_fr_sub_assign() {
{
// Test arbitrary subtraction that tests reduction.
let mut tmp = Fr(FrRepr([
0x6a68c64b6f735a2b,
0xd5f4d143fe0a1972,
0x37c17f3829267c62,
0xa2f37391f30915c,
]));
tmp.sub_assign(&Fr(FrRepr([
0xade5adacdccb6190,
0xaa21ee0f27db3ccd,
0x2550f4704ae39086,
0x591d1902e7c5ba27,
])));
assert_eq!(
tmp,
Fr(FrRepr([
0xbc83189d92a7f89c,
0x7f908737d62d38a3,
0x45aa62cfe7e4c3e1,
0x24ffc5896108547d
]))
);
// Test the opposite subtraction which doesn't test reduction.
tmp = Fr(FrRepr([
0xade5adacdccb6190,
0xaa21ee0f27db3ccd,
0x2550f4704ae39086,
0x591d1902e7c5ba27,
]));
tmp.sub_assign(&Fr(FrRepr([
0x6a68c64b6f735a2b,
0xd5f4d143fe0a1972,
0x37c17f3829267c62,
0xa2f37391f30915c,
])));
assert_eq!(
tmp,
Fr(FrRepr([
0x437ce7616d580765,
0xd42d1ccb29d1235b,
0xed8f753821bd1423,
0x4eede1c9c89528ca
]))
);
// Test for sensible results with zero
tmp = Fr(FrRepr::from(0));
tmp.sub_assign(&Fr(FrRepr::from(0)));
assert!(tmp.is_zero());
tmp = Fr(FrRepr([
0x437ce7616d580765,
0xd42d1ccb29d1235b,
0xed8f753821bd1423,
0x4eede1c9c89528ca,
]));
tmp.sub_assign(&Fr(FrRepr::from(0)));
assert_eq!(
tmp,
Fr(FrRepr([
0x437ce7616d580765,
0xd42d1ccb29d1235b,
0xed8f753821bd1423,
0x4eede1c9c89528ca
]))
);
}
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
// Ensure that (a - b) + (b - a) = 0.
let a = Fr::rand(&mut rng);
let b = Fr::rand(&mut rng);
let mut tmp1 = a;
tmp1.sub_assign(&b);
let mut tmp2 = b;
tmp2.sub_assign(&a);
tmp1.add_assign(&tmp2);
assert!(tmp1.is_zero());
}
}
#[test]
fn test_fr_mul_assign() {
let mut tmp = Fr(FrRepr([
0x6b7e9b8faeefc81a,
0xe30a8463f348ba42,
0xeff3cb67a8279c9c,
0x3d303651bd7c774d,
]));
tmp.mul_assign(&Fr(FrRepr([
0x13ae28e3bc35ebeb,
0xa10f4488075cae2c,
0x8160e95a853c3b5d,
0x5ae3f03b561a841d,
])));
assert!(
tmp == Fr(FrRepr([
0x23717213ce710f71,
0xdbee1fe53a16e1af,
0xf565d3e1c2a48000,
0x4426507ee75df9d7
]))
);
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000000 {
// Ensure that (a * b) * c = a * (b * c)
let a = Fr::rand(&mut rng);
let b = Fr::rand(&mut rng);
let c = Fr::rand(&mut rng);
let mut tmp1 = a;
tmp1.mul_assign(&b);
tmp1.mul_assign(&c);
let mut tmp2 = b;
tmp2.mul_assign(&c);
tmp2.mul_assign(&a);
assert_eq!(tmp1, tmp2);
}
for _ in 0..1000000 {
// Ensure that r * (a + b + c) = r*a + r*b + r*c
let r = Fr::rand(&mut rng);
let mut a = Fr::rand(&mut rng);
let mut b = Fr::rand(&mut rng);
let mut c = Fr::rand(&mut rng);
let mut tmp1 = a;
tmp1.add_assign(&b);
tmp1.add_assign(&c);
tmp1.mul_assign(&r);
a.mul_assign(&r);
b.mul_assign(&r);
c.mul_assign(&r);
a.add_assign(&b);
a.add_assign(&c);
assert_eq!(tmp1, a);
}
}
#[test]
fn test_fr_squaring() {
let mut a = Fr(FrRepr([
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
0x73eda753299d7d47,
]));
assert!(a.is_valid());
a.square();
assert_eq!(
a,
Fr::from_repr(FrRepr([
0xc0d698e7bde077b8,
0xb79a310579e76ec2,
0xac1da8d0a9af4e5f,
0x13f629c49bf23e97
])).unwrap()
);
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000000 {
// Ensure that (a * a) = a^2
let a = Fr::rand(&mut rng);
let mut tmp = a;
tmp.square();
let mut tmp2 = a;
tmp2.mul_assign(&a);
assert_eq!(tmp, tmp2);
}
}
#[test]
fn test_fr_inverse() {
assert!(Fr::zero().inverse().is_none());
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let one = Fr::one();
for _ in 0..1000 {
// Ensure that a * a^-1 = 1
let mut a = Fr::rand(&mut rng);
let ainv = a.inverse().unwrap();
a.mul_assign(&ainv);
assert_eq!(a, one);
}
}
#[test]
fn test_fr_double() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
// Ensure doubling a is equivalent to adding a to itself.
let mut a = Fr::rand(&mut rng);
let mut b = a;
b.add_assign(&a);
a.double();
assert_eq!(a, b);
}
}
#[test]
fn test_fr_negate() {
{
let mut a = Fr::zero();
a.negate();
assert!(a.is_zero());
}
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
// Ensure (a - (-a)) = 0.
let mut a = Fr::rand(&mut rng);
let mut b = a;
b.negate();
a.add_assign(&b);
assert!(a.is_zero());
}
}
#[test]
fn test_fr_pow() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for i in 0..1000 {
// Exponentiate by various small numbers and ensure it consists with repeated
// multiplication.
let a = Fr::rand(&mut rng);
let target = a.pow(&[i]);
let mut c = Fr::one();
for _ in 0..i {
c.mul_assign(&a);
}
assert_eq!(c, target);
}
for _ in 0..1000 {
// Exponentiating by the modulus should have no effect in a prime field.
let a = Fr::rand(&mut rng);
assert_eq!(a, a.pow(Fr::char()));
}
}
#[test]
fn test_fr_sqrt() {
use ff::SqrtField;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
assert_eq!(Fr::zero().sqrt().unwrap(), Fr::zero());
for _ in 0..1000 {
// Ensure sqrt(a^2) = a or -a
let a = Fr::rand(&mut rng);
let mut nega = a;
nega.negate();
let mut b = a;
b.square();
let b = b.sqrt().unwrap();
assert!(a == b || nega == b);
}
for _ in 0..1000 {
// Ensure sqrt(a)^2 = a for random a
let a = Fr::rand(&mut rng);
if let Some(mut tmp) = a.sqrt() {
tmp.square();
assert_eq!(a, tmp);
}
}
}
#[test]
fn test_fr_from_into_repr() {
// r + 1 should not be in the field
assert!(
Fr::from_repr(FrRepr([
0xffffffff00000002,
0x53bda402fffe5bfe,
0x3339d80809a1d805,
0x73eda753299d7d48
])).is_err()
);
// r should not be in the field
assert!(Fr::from_repr(Fr::char()).is_err());
// Multiply some arbitrary representations to see if the result is as expected.
let a = FrRepr([
0x25ebe3a3ad3c0c6a,
0x6990e39d092e817c,
0x941f900d42f5658e,
0x44f8a103b38a71e0,
]);
let mut a_fr = Fr::from_repr(a).unwrap();
let b = FrRepr([
0x264e9454885e2475,
0x46f7746bb0308370,
0x4683ef5347411f9,
0x58838d7f208d4492,
]);
let b_fr = Fr::from_repr(b).unwrap();
let c = FrRepr([
0x48a09ab93cfc740d,
0x3a6600fbfc7a671,
0x838567017501d767,
0x7161d6da77745512,
]);
a_fr.mul_assign(&b_fr);
assert_eq!(a_fr.into_repr(), c);
// Zero should be in the field.
assert!(Fr::from_repr(FrRepr::from(0)).unwrap().is_zero());
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
// Try to turn Fr elements into representations and back again, and compare.
let a = Fr::rand(&mut rng);
let a_repr = a.into_repr();
let b_repr = FrRepr::from(a);
assert_eq!(a_repr, b_repr);
let a_again = Fr::from_repr(a_repr).unwrap();
assert_eq!(a, a_again);
}
}
#[test]
fn test_fr_repr_display() {
assert_eq!(
format!(
"{}",
FrRepr([
0x2829c242fa826143,
0x1f32cf4dd4330917,
0x932e4e479d168cd9,
0x513c77587f563f64
])
),
"0x513c77587f563f64932e4e479d168cd91f32cf4dd43309172829c242fa826143".to_string()
);
assert_eq!(
format!(
"{}",
FrRepr([
0x25ebe3a3ad3c0c6a,
0x6990e39d092e817c,
0x941f900d42f5658e,
0x44f8a103b38a71e0
])
),
"0x44f8a103b38a71e0941f900d42f5658e6990e39d092e817c25ebe3a3ad3c0c6a".to_string()
);
assert_eq!(
format!(
"{}",
FrRepr([
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff
])
),
"0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff".to_string()
);
assert_eq!(
format!("{}", FrRepr([0, 0, 0, 0])),
"0x0000000000000000000000000000000000000000000000000000000000000000".to_string()
);
}
#[test]
fn test_fr_display() {
assert_eq!(
format!(
"{}",
Fr::from_repr(FrRepr([
0xc3cae746a3b5ecc7,
0x185ec8eb3f5b5aee,
0x684499ffe4b9dd99,
0x7c9bba7afb68faa
])).unwrap()
),
"Fr(0x07c9bba7afb68faa684499ffe4b9dd99185ec8eb3f5b5aeec3cae746a3b5ecc7)".to_string()
);
assert_eq!(
format!(
"{}",
Fr::from_repr(FrRepr([
0x44c71298ff198106,
0xb0ad10817df79b6a,
0xd034a80a2b74132b,
0x41cf9a1336f50719
])).unwrap()
),
"Fr(0x41cf9a1336f50719d034a80a2b74132bb0ad10817df79b6a44c71298ff198106)".to_string()
);
}
#[test]
fn test_fr_num_bits() {
assert_eq!(Fr::NUM_BITS, 255);
assert_eq!(Fr::CAPACITY, 254);
}
#[test]
fn test_fr_root_of_unity() {
use ff::SqrtField;
assert_eq!(Fr::S, 32);
assert_eq!(
Fr::multiplicative_generator(),
Fr::from_repr(FrRepr::from(7)).unwrap()
);
assert_eq!(
Fr::multiplicative_generator().pow([
0xfffe5bfeffffffff,
0x9a1d80553bda402,
0x299d7d483339d808,
0x73eda753
]),
Fr::root_of_unity()
);
assert_eq!(Fr::root_of_unity().pow([1 << Fr::S]), Fr::one());
assert!(Fr::multiplicative_generator().sqrt().is_none());
}
#[test]
fn fr_field_tests() {
crate::tests::field::random_field_tests::<Fr>();
crate::tests::field::random_sqrt_tests::<Fr>();
crate::tests::field::random_frobenius_tests::<Fr, _>(Fr::char(), 13);
crate::tests::field::from_str_tests::<Fr>();
}
#[test]
fn fr_repr_tests() {
crate::tests::repr::random_repr_tests::<FrRepr>();
}

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mod ec;
mod fq;
mod fq12;
mod fq2;
mod fq6;
mod fr;
#[cfg(test)]
mod tests;
pub use self::ec::{
G1, G1Affine, G1Compressed, G1Prepared, G1Uncompressed, G2, G2Affine, G2Compressed, G2Prepared,
G2Uncompressed,
};
pub use self::fq::{Fq, FqRepr};
pub use self::fq12::Fq12;
pub use self::fq2::Fq2;
pub use self::fq6::Fq6;
pub use self::fr::{Fr, FrRepr};
use super::{CurveAffine, Engine};
use ff::{BitIterator, Field, ScalarEngine};
// The BLS parameter x for BLS12-381 is -0xd201000000010000
const BLS_X: u64 = 0xd201000000010000;
const BLS_X_IS_NEGATIVE: bool = true;
#[derive(Clone, Debug)]
pub struct Bls12;
impl ScalarEngine for Bls12 {
type Fr = Fr;
}
impl Engine for Bls12 {
type G1 = G1;
type G1Affine = G1Affine;
type G2 = G2;
type G2Affine = G2Affine;
type Fq = Fq;
type Fqe = Fq2;
type Fqk = Fq12;
fn miller_loop<'a, I>(i: I) -> Self::Fqk
where
I: IntoIterator<
Item = &'a (
&'a <Self::G1Affine as CurveAffine>::Prepared,
&'a <Self::G2Affine as CurveAffine>::Prepared,
),
>,
{
let mut pairs = vec![];
for &(p, q) in i {
if !p.is_zero() && !q.is_zero() {
pairs.push((p, q.coeffs.iter()));
}
}
// Twisting isomorphism from E to E'
fn ell(f: &mut Fq12, coeffs: &(Fq2, Fq2, Fq2), p: &G1Affine) {
let mut c0 = coeffs.0;
let mut c1 = coeffs.1;
c0.c0.mul_assign(&p.y);
c0.c1.mul_assign(&p.y);
c1.c0.mul_assign(&p.x);
c1.c1.mul_assign(&p.x);
// Sparse multiplication in Fq12
f.mul_by_014(&coeffs.2, &c1, &c0);
}
let mut f = Fq12::one();
let mut found_one = false;
for i in BitIterator::new(&[BLS_X >> 1]) {
if !found_one {
found_one = i;
continue;
}
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
if i {
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
}
f.square();
}
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
if BLS_X_IS_NEGATIVE {
f.conjugate();
}
f
}
fn final_exponentiation(r: &Fq12) -> Option<Fq12> {
let mut f1 = *r;
f1.conjugate();
match r.inverse() {
Some(mut f2) => {
let mut r = f1;
r.mul_assign(&f2);
f2 = r;
r.frobenius_map(2);
r.mul_assign(&f2);
fn exp_by_x(f: &mut Fq12, x: u64) {
*f = f.pow(&[x]);
if BLS_X_IS_NEGATIVE {
f.conjugate();
}
}
let mut x = BLS_X;
let mut y0 = r;
y0.square();
let mut y1 = y0;
exp_by_x(&mut y1, x);
x >>= 1;
let mut y2 = y1;
exp_by_x(&mut y2, x);
x <<= 1;
let mut y3 = r;
y3.conjugate();
y1.mul_assign(&y3);
y1.conjugate();
y1.mul_assign(&y2);
y2 = y1;
exp_by_x(&mut y2, x);
y3 = y2;
exp_by_x(&mut y3, x);
y1.conjugate();
y3.mul_assign(&y1);
y1.conjugate();
y1.frobenius_map(3);
y2.frobenius_map(2);
y1.mul_assign(&y2);
y2 = y3;
exp_by_x(&mut y2, x);
y2.mul_assign(&y0);
y2.mul_assign(&r);
y1.mul_assign(&y2);
y2 = y3;
y2.frobenius_map(1);
y1.mul_assign(&y2);
Some(y1)
}
None => None,
}
}
}
impl G2Prepared {
pub fn is_zero(&self) -> bool {
self.infinity
}
pub fn from_affine(q: G2Affine) -> Self {
if q.is_zero() {
return G2Prepared {
coeffs: vec![],
infinity: true,
};
}
fn doubling_step(r: &mut G2) -> (Fq2, Fq2, Fq2) {
// Adaptation of Algorithm 26, https://eprint.iacr.org/2010/354.pdf
let mut tmp0 = r.x;
tmp0.square();
let mut tmp1 = r.y;
tmp1.square();
let mut tmp2 = tmp1;
tmp2.square();
let mut tmp3 = tmp1;
tmp3.add_assign(&r.x);
tmp3.square();
tmp3.sub_assign(&tmp0);
tmp3.sub_assign(&tmp2);
tmp3.double();
let mut tmp4 = tmp0;
tmp4.double();
tmp4.add_assign(&tmp0);
let mut tmp6 = r.x;
tmp6.add_assign(&tmp4);
let mut tmp5 = tmp4;
tmp5.square();
let mut zsquared = r.z;
zsquared.square();
r.x = tmp5;
r.x.sub_assign(&tmp3);
r.x.sub_assign(&tmp3);
r.z.add_assign(&r.y);
r.z.square();
r.z.sub_assign(&tmp1);
r.z.sub_assign(&zsquared);
r.y = tmp3;
r.y.sub_assign(&r.x);
r.y.mul_assign(&tmp4);
tmp2.double();
tmp2.double();
tmp2.double();
r.y.sub_assign(&tmp2);
tmp3 = tmp4;
tmp3.mul_assign(&zsquared);
tmp3.double();
tmp3.negate();
tmp6.square();
tmp6.sub_assign(&tmp0);
tmp6.sub_assign(&tmp5);
tmp1.double();
tmp1.double();
tmp6.sub_assign(&tmp1);
tmp0 = r.z;
tmp0.mul_assign(&zsquared);
tmp0.double();
(tmp0, tmp3, tmp6)
}
fn addition_step(r: &mut G2, q: &G2Affine) -> (Fq2, Fq2, Fq2) {
// Adaptation of Algorithm 27, https://eprint.iacr.org/2010/354.pdf
let mut zsquared = r.z;
zsquared.square();
let mut ysquared = q.y;
ysquared.square();
let mut t0 = zsquared;
t0.mul_assign(&q.x);
let mut t1 = q.y;
t1.add_assign(&r.z);
t1.square();
t1.sub_assign(&ysquared);
t1.sub_assign(&zsquared);
t1.mul_assign(&zsquared);
let mut t2 = t0;
t2.sub_assign(&r.x);
let mut t3 = t2;
t3.square();
let mut t4 = t3;
t4.double();
t4.double();
let mut t5 = t4;
t5.mul_assign(&t2);
let mut t6 = t1;
t6.sub_assign(&r.y);
t6.sub_assign(&r.y);
let mut t9 = t6;
t9.mul_assign(&q.x);
let mut t7 = t4;
t7.mul_assign(&r.x);
r.x = t6;
r.x.square();
r.x.sub_assign(&t5);
r.x.sub_assign(&t7);
r.x.sub_assign(&t7);
r.z.add_assign(&t2);
r.z.square();
r.z.sub_assign(&zsquared);
r.z.sub_assign(&t3);
let mut t10 = q.y;
t10.add_assign(&r.z);
let mut t8 = t7;
t8.sub_assign(&r.x);
t8.mul_assign(&t6);
t0 = r.y;
t0.mul_assign(&t5);
t0.double();
r.y = t8;
r.y.sub_assign(&t0);
t10.square();
t10.sub_assign(&ysquared);
let mut ztsquared = r.z;
ztsquared.square();
t10.sub_assign(&ztsquared);
t9.double();
t9.sub_assign(&t10);
t10 = r.z;
t10.double();
t6.negate();
t1 = t6;
t1.double();
(t10, t1, t9)
}
let mut coeffs = vec![];
let mut r: G2 = q.into();
let mut found_one = false;
for i in BitIterator::new([BLS_X >> 1]) {
if !found_one {
found_one = i;
continue;
}
coeffs.push(doubling_step(&mut r));
if i {
coeffs.push(addition_step(&mut r, &q));
}
}
coeffs.push(doubling_step(&mut r));
G2Prepared {
coeffs,
infinity: false,
}
}
}
#[test]
fn bls12_engine_tests() {
crate::tests::engine::engine_tests::<Bls12>();
}

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use super::*;
use crate::*;
#[test]
fn test_pairing_result_against_relic() {
/*
Sent to me from Diego Aranha (author of RELIC library):
1250EBD871FC0A92 A7B2D83168D0D727 272D441BEFA15C50 3DD8E90CE98DB3E7 B6D194F60839C508 A84305AACA1789B6
089A1C5B46E5110B 86750EC6A5323488 68A84045483C92B7 AF5AF689452EAFAB F1A8943E50439F1D 59882A98EAA0170F
1368BB445C7C2D20 9703F239689CE34C 0378A68E72A6B3B2 16DA0E22A5031B54 DDFF57309396B38C 881C4C849EC23E87
193502B86EDB8857 C273FA075A505129 37E0794E1E65A761 7C90D8BD66065B1F FFE51D7A579973B1 315021EC3C19934F
01B2F522473D1713 91125BA84DC4007C FBF2F8DA752F7C74 185203FCCA589AC7 19C34DFFBBAAD843 1DAD1C1FB597AAA5
018107154F25A764 BD3C79937A45B845 46DA634B8F6BE14A 8061E55CCEBA478B 23F7DACAA35C8CA7 8BEAE9624045B4B6
19F26337D205FB46 9CD6BD15C3D5A04D C88784FBB3D0B2DB DEA54D43B2B73F2C BB12D58386A8703E 0F948226E47EE89D
06FBA23EB7C5AF0D 9F80940CA771B6FF D5857BAAF222EB95 A7D2809D61BFE02E 1BFD1B68FF02F0B8 102AE1C2D5D5AB1A
11B8B424CD48BF38 FCEF68083B0B0EC5 C81A93B330EE1A67 7D0D15FF7B984E89 78EF48881E32FAC9 1B93B47333E2BA57
03350F55A7AEFCD3 C31B4FCB6CE5771C C6A0E9786AB59733 20C806AD36082910 7BA810C5A09FFDD9 BE2291A0C25A99A2
04C581234D086A99 02249B64728FFD21 A189E87935A95405 1C7CDBA7B3872629 A4FAFC05066245CB 9108F0242D0FE3EF
0F41E58663BF08CF 068672CBD01A7EC7 3BACA4D72CA93544 DEFF686BFD6DF543 D48EAA24AFE47E1E FDE449383B676631
*/
assert_eq!(Bls12::pairing(G1::one(), G2::one()), Fq12 {
c0: Fq6 {
c0: Fq2 {
c0: Fq::from_str("2819105605953691245277803056322684086884703000473961065716485506033588504203831029066448642358042597501014294104502").unwrap(),
c1: Fq::from_str("1323968232986996742571315206151405965104242542339680722164220900812303524334628370163366153839984196298685227734799").unwrap()
},
c1: Fq2 {
c0: Fq::from_str("2987335049721312504428602988447616328830341722376962214011674875969052835043875658579425548512925634040144704192135").unwrap(),
c1: Fq::from_str("3879723582452552452538684314479081967502111497413076598816163759028842927668327542875108457755966417881797966271311").unwrap()
},
c2: Fq2 {
c0: Fq::from_str("261508182517997003171385743374653339186059518494239543139839025878870012614975302676296704930880982238308326681253").unwrap(),
c1: Fq::from_str("231488992246460459663813598342448669854473942105054381511346786719005883340876032043606739070883099647773793170614").unwrap()
}
},
c1: Fq6 {
c0: Fq2 {
c0: Fq::from_str("3993582095516422658773669068931361134188738159766715576187490305611759126554796569868053818105850661142222948198557").unwrap(),
c1: Fq::from_str("1074773511698422344502264006159859710502164045911412750831641680783012525555872467108249271286757399121183508900634").unwrap()
},
c1: Fq2 {
c0: Fq::from_str("2727588299083545686739024317998512740561167011046940249988557419323068809019137624943703910267790601287073339193943").unwrap(),
c1: Fq::from_str("493643299814437640914745677854369670041080344349607504656543355799077485536288866009245028091988146107059514546594").unwrap()
},
c2: Fq2 {
c0: Fq::from_str("734401332196641441839439105942623141234148957972407782257355060229193854324927417865401895596108124443575283868655").unwrap(),
c1: Fq::from_str("2348330098288556420918672502923664952620152483128593484301759394583320358354186482723629999370241674973832318248497").unwrap()
}
}
});
}
fn test_vectors<G: CurveProjective, E: EncodedPoint<Affine = G::Affine>>(expected: &[u8]) {
let mut e = G::zero();
let mut v = vec![];
{
let mut expected = expected;
for _ in 0..1000 {
let e_affine = e.into_affine();
let encoded = E::from_affine(e_affine);
v.extend_from_slice(encoded.as_ref());
let mut decoded = E::empty();
decoded.as_mut().copy_from_slice(&expected[0..E::size()]);
expected = &expected[E::size()..];
let decoded = decoded.into_affine().unwrap();
assert_eq!(e_affine, decoded);
e.add_assign(&G::one());
}
}
assert_eq!(&v[..], expected);
}
#[test]
fn test_g1_uncompressed_valid_vectors() {
test_vectors::<G1, G1Uncompressed>(include_bytes!("g1_uncompressed_valid_test_vectors.dat"));
}
#[test]
fn test_g1_compressed_valid_vectors() {
test_vectors::<G1, G1Compressed>(include_bytes!("g1_compressed_valid_test_vectors.dat"));
}
#[test]
fn test_g2_uncompressed_valid_vectors() {
test_vectors::<G2, G2Uncompressed>(include_bytes!("g2_uncompressed_valid_test_vectors.dat"));
}
#[test]
fn test_g2_compressed_valid_vectors() {
test_vectors::<G2, G2Compressed>(include_bytes!("g2_compressed_valid_test_vectors.dat"));
}
#[test]
fn test_g1_uncompressed_invalid_vectors() {
{
let z = G1Affine::zero().into_uncompressed();
{
let mut z = z;
z.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected an uncompressed point");
}
}
{
let mut z = z;
z.as_mut()[0] |= 0b0010_0000;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
}
}
for i in 0..G1Uncompressed::size() {
let mut z = z;
z.as_mut()[i] |= 0b0000_0001;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
}
}
}
let o = G1Affine::one().into_uncompressed();
{
let mut o = o;
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected an uncompressed point");
}
}
let m = Fq::char();
{
let mut o = o;
m.write_be(&mut o.as_mut()[0..]).unwrap();
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "x coordinate");
} else {
panic!("should have rejected the point")
}
}
{
let mut o = o;
m.write_be(&mut o.as_mut()[48..]).unwrap();
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "y coordinate");
} else {
panic!("should have rejected the point")
}
}
{
let m = Fq::zero().into_repr();
let mut o = o;
m.write_be(&mut o.as_mut()[0..]).unwrap();
if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
// :)
} else {
panic!("should have rejected the point because it isn't on the curve")
}
}
{
let mut o = o;
let mut x = Fq::one();
loop {
let mut x3b = x;
x3b.square();
x3b.mul_assign(&x);
x3b.add_assign(&Fq::from_repr(FqRepr::from(4)).unwrap()); // TODO: perhaps expose coeff_b through API?
if let Some(y) = x3b.sqrt() {
// We know this is on the curve, but it's likely not going to be in the correct subgroup.
x.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
y.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
break;
} else {
panic!(
"should have rejected the point because it isn't in the correct subgroup"
)
}
} else {
x.add_assign(&Fq::one());
}
}
}
}
#[test]
fn test_g2_uncompressed_invalid_vectors() {
{
let z = G2Affine::zero().into_uncompressed();
{
let mut z = z;
z.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected an uncompressed point");
}
}
{
let mut z = z;
z.as_mut()[0] |= 0b0010_0000;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
}
}
for i in 0..G2Uncompressed::size() {
let mut z = z;
z.as_mut()[i] |= 0b0000_0001;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
}
}
}
let o = G2Affine::one().into_uncompressed();
{
let mut o = o;
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected an uncompressed point");
}
}
let m = Fq::char();
{
let mut o = o;
m.write_be(&mut o.as_mut()[0..]).unwrap();
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "x coordinate (c1)");
} else {
panic!("should have rejected the point")
}
}
{
let mut o = o;
m.write_be(&mut o.as_mut()[48..]).unwrap();
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "x coordinate (c0)");
} else {
panic!("should have rejected the point")
}
}
{
let mut o = o;
m.write_be(&mut o.as_mut()[96..]).unwrap();
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "y coordinate (c1)");
} else {
panic!("should have rejected the point")
}
}
{
let mut o = o;
m.write_be(&mut o.as_mut()[144..]).unwrap();
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "y coordinate (c0)");
} else {
panic!("should have rejected the point")
}
}
{
let m = Fq::zero().into_repr();
let mut o = o;
m.write_be(&mut o.as_mut()[0..]).unwrap();
m.write_be(&mut o.as_mut()[48..]).unwrap();
if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
// :)
} else {
panic!("should have rejected the point because it isn't on the curve")
}
}
{
let mut o = o;
let mut x = Fq2::one();
loop {
let mut x3b = x;
x3b.square();
x3b.mul_assign(&x);
x3b.add_assign(&Fq2 {
c0: Fq::from_repr(FqRepr::from(4)).unwrap(),
c1: Fq::from_repr(FqRepr::from(4)).unwrap(),
}); // TODO: perhaps expose coeff_b through API?
if let Some(y) = x3b.sqrt() {
// We know this is on the curve, but it's likely not going to be in the correct subgroup.
x.c1.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
x.c0.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
y.c1.into_repr().write_be(&mut o.as_mut()[96..]).unwrap();
y.c0.into_repr().write_be(&mut o.as_mut()[144..]).unwrap();
if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
break;
} else {
panic!(
"should have rejected the point because it isn't in the correct subgroup"
)
}
} else {
x.add_assign(&Fq2::one());
}
}
}
}
#[test]
fn test_g1_compressed_invalid_vectors() {
{
let z = G1Affine::zero().into_compressed();
{
let mut z = z;
z.as_mut()[0] &= 0b0111_1111;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected a compressed point");
}
}
{
let mut z = z;
z.as_mut()[0] |= 0b0010_0000;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
}
}
for i in 0..G1Compressed::size() {
let mut z = z;
z.as_mut()[i] |= 0b0000_0001;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
}
}
}
let o = G1Affine::one().into_compressed();
{
let mut o = o;
o.as_mut()[0] &= 0b0111_1111;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected a compressed point");
}
}
let m = Fq::char();
{
let mut o = o;
m.write_be(&mut o.as_mut()[0..]).unwrap();
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "x coordinate");
} else {
panic!("should have rejected the point")
}
}
{
let mut o = o;
let mut x = Fq::one();
loop {
let mut x3b = x;
x3b.square();
x3b.mul_assign(&x);
x3b.add_assign(&Fq::from_repr(FqRepr::from(4)).unwrap()); // TODO: perhaps expose coeff_b through API?
if let Some(_) = x3b.sqrt() {
x.add_assign(&Fq::one());
} else {
x.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
break;
} else {
panic!("should have rejected the point because it isn't on the curve")
}
}
}
}
{
let mut o = o;
let mut x = Fq::one();
loop {
let mut x3b = x;
x3b.square();
x3b.mul_assign(&x);
x3b.add_assign(&Fq::from_repr(FqRepr::from(4)).unwrap()); // TODO: perhaps expose coeff_b through API?
if let Some(_) = x3b.sqrt() {
// We know this is on the curve, but it's likely not going to be in the correct subgroup.
x.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
break;
} else {
panic!(
"should have rejected the point because it isn't in the correct subgroup"
)
}
} else {
x.add_assign(&Fq::one());
}
}
}
}
#[test]
fn test_g2_compressed_invalid_vectors() {
{
let z = G2Affine::zero().into_compressed();
{
let mut z = z;
z.as_mut()[0] &= 0b0111_1111;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected a compressed point");
}
}
{
let mut z = z;
z.as_mut()[0] |= 0b0010_0000;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the parity bit should not be set if the point is at infinity");
}
}
for i in 0..G2Compressed::size() {
let mut z = z;
z.as_mut()[i] |= 0b0000_0001;
if let Err(GroupDecodingError::UnexpectedInformation) = z.into_affine() {
// :)
} else {
panic!("should have rejected the point because the coordinates should be zeroes at the point at infinity");
}
}
}
let o = G2Affine::one().into_compressed();
{
let mut o = o;
o.as_mut()[0] &= 0b0111_1111;
if let Err(GroupDecodingError::UnexpectedCompressionMode) = o.into_affine() {
// :)
} else {
panic!("should have rejected the point because we expected a compressed point");
}
}
let m = Fq::char();
{
let mut o = o;
m.write_be(&mut o.as_mut()[0..]).unwrap();
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "x coordinate (c1)");
} else {
panic!("should have rejected the point")
}
}
{
let mut o = o;
m.write_be(&mut o.as_mut()[48..]).unwrap();
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::CoordinateDecodingError(coordinate, _)) = o.into_affine() {
assert_eq!(coordinate, "x coordinate (c0)");
} else {
panic!("should have rejected the point")
}
}
{
let mut o = o;
let mut x = Fq2 {
c0: Fq::one(),
c1: Fq::one(),
};
loop {
let mut x3b = x;
x3b.square();
x3b.mul_assign(&x);
x3b.add_assign(&Fq2 {
c0: Fq::from_repr(FqRepr::from(4)).unwrap(),
c1: Fq::from_repr(FqRepr::from(4)).unwrap(),
}); // TODO: perhaps expose coeff_b through API?
if let Some(_) = x3b.sqrt() {
x.add_assign(&Fq2::one());
} else {
x.c1.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
x.c0.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::NotOnCurve) = o.into_affine() {
break;
} else {
panic!("should have rejected the point because it isn't on the curve")
}
}
}
}
{
let mut o = o;
let mut x = Fq2 {
c0: Fq::one(),
c1: Fq::one(),
};
loop {
let mut x3b = x;
x3b.square();
x3b.mul_assign(&x);
x3b.add_assign(&Fq2 {
c0: Fq::from_repr(FqRepr::from(4)).unwrap(),
c1: Fq::from_repr(FqRepr::from(4)).unwrap(),
}); // TODO: perhaps expose coeff_b through API?
if let Some(_) = x3b.sqrt() {
// We know this is on the curve, but it's likely not going to be in the correct subgroup.
x.c1.into_repr().write_be(&mut o.as_mut()[0..]).unwrap();
x.c0.into_repr().write_be(&mut o.as_mut()[48..]).unwrap();
o.as_mut()[0] |= 0b1000_0000;
if let Err(GroupDecodingError::NotInSubgroup) = o.into_affine() {
break;
} else {
panic!(
"should have rejected the point because it isn't in the correct subgroup"
)
}
} else {
x.add_assign(&Fq2::one());
}
}
}
}

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pairing/src/bn256/README.md Normal file
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# BN256
This is an implementation of the BN256 pairing-friendly elliptic curve construction.
## BN256 Parameterization
Follows go-ethereum parametrization.
## Notes
- I couldn't find an easy wat of getting random G2 for BN256 curve (also have no idea why just scaling by cofactor works for BLS12), so don't use it. Make random sccalar and multiply by generator.
- For this reason tests had to be copied and modified for some cases.

1717
pairing/src/bn256/ec.rs Normal file
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579
pairing/src/bn256/fq.rs Normal file
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use super::fq2::Fq2;
use ff::{Field, PrimeField, PrimeFieldRepr};
#[derive(PrimeField)]
#[PrimeFieldModulus = "21888242871839275222246405745257275088696311157297823662689037894645226208583"]
#[PrimeFieldGenerator = "2"]
pub struct Fq(FqRepr);
// B coefficient of BN256 curve, B = 3
// In Montgommery form with R = 2^256
pub const B_COEFF: Fq = Fq(FqRepr([
0x7a17caa950ad28d7,
0x1f6ac17ae15521b9,
0x334bea4e696bd284,
0x2a1f6744ce179d8e,
]));
pub const B_COEFF_FQ2: Fq2 = Fq2 {
c0: Fq(FqRepr([
0x3bf938e377b802a8,
0x020b1b273633535d,
0x26b7edf049755260,
0x2514c6324384a86d,
])),
c1: Fq(FqRepr([
0x38e7ecccd1dcff67,
0x65f0b37d93ce0d3e,
0xd749d0dd22ac00aa,
0x0141b9ce4a688d4d,
])),
};
// The generators of G1/G2
// Generator of G1
// x = 1
// y = 2
pub const G1_GENERATOR_X: Fq = Fq(FqRepr([
0xd35d438dc58f0d9d,
0x0a78eb28f5c70b3d,
0x666ea36f7879462c,
0x0e0a77c19a07df2f,
]));
pub const G1_GENERATOR_Y: Fq = Fq(FqRepr([
0xa6ba871b8b1e1b3a,
0x14f1d651eb8e167b,
0xccdd46def0f28c58,
0x1c14ef83340fbe5e,
]));
// Generator of G2
//
// x = 11559732032986387107991004021392285783925812861821192530917403151452391805634*u
// + 10857046999023057135944570762232829481370756359578518086990519993285655852781
//
// y = 4082367875863433681332203403145435568316851327593401208105741076214120093531*u
// + 8495653923123431417604973247489272438418190587263600148770280649306958101930
pub const G2_GENERATOR_X_C0: Fq = Fq(FqRepr([
0x8e83b5d102bc2026,
0xdceb1935497b0172,
0xfbb8264797811adf,
0x19573841af96503b,
]));
pub const G2_GENERATOR_X_C1: Fq = Fq(FqRepr([
0xafb4737da84c6140,
0x6043dd5a5802d8c4,
0x09e950fc52a02f86,
0x14fef0833aea7b6b,
]));
pub const G2_GENERATOR_Y_C0: Fq = Fq(FqRepr([
0x619dfa9d886be9f6,
0xfe7fd297f59e9b78,
0xff9e1a62231b7dfe,
0x28fd7eebae9e4206,
]));
pub const G2_GENERATOR_Y_C1: Fq = Fq(FqRepr([
0x64095b56c71856ee,
0xdc57f922327d3cbb,
0x55f935be33351076,
0x0da4a0e693fd6482,
]));
// Coefficients for the Frobenius automorphism.
pub const FROBENIUS_COEFF_FQ2_C1: [Fq; 2] = [
// Fq(-1)**(((q^0) - 1) / 2)
// it's 1 in Montgommery form
Fq(FqRepr([
0xd35d438dc58f0d9d,
0x0a78eb28f5c70b3d,
0x666ea36f7879462c,
0x0e0a77c19a07df2f,
])),
// Fq(-1)**(((q^1) - 1) / 2)
Fq(FqRepr([
0x68c3488912edefaa,
0x8d087f6872aabf4f,
0x51e1a24709081231,
0x2259d6b14729c0fa,
])),
];
// Fq2(u + 9)**(((q^1) - 1) / 2)
pub const XI_TO_Q_MINUS_1_OVER_2: Fq2 = Fq2 {
c0: Fq(FqRepr([
0xe4bbdd0c2936b629,
0xbb30f162e133bacb,
0x31a9d1b6f9645366,
0x253570bea500f8dd,
])),
c1: Fq(FqRepr([
0xa1d77ce45ffe77c7,
0x07affd117826d1db,
0x6d16bd27bb7edc6b,
0x2c87200285defecc,
])),
};
pub const FROBENIUS_COEFF_FQ6_C1: [Fq2; 6] = [
// Fq2(u + 9)**(((q^0) - 1) / 3)
Fq2 {
c0: Fq(FqRepr([
0xd35d438dc58f0d9d,
0x0a78eb28f5c70b3d,
0x666ea36f7879462c,
0x0e0a77c19a07df2f,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0])),
},
// Fq2(u + 9)**(((q^1) - 1) / 3)
// taken from go-ethereum and also re-calculated manually
Fq2 {
c0: Fq(FqRepr([
0xb5773b104563ab30,
0x347f91c8a9aa6454,
0x7a007127242e0991,
0x1956bcd8118214ec,
])),
c1: Fq(FqRepr([
0x6e849f1ea0aa4757,
0xaa1c7b6d89f89141,
0xb6e713cdfae0ca3a,
0x26694fbb4e82ebc3,
])),
},
// Fq2(u + 9)**(((q^2) - 1) / 3)
// this one and other below are recalculated manually
Fq2 {
c0: Fq(FqRepr([
0x3350c88e13e80b9c,
0x7dce557cdb5e56b9,
0x6001b4b8b615564a,
0x2682e617020217e0,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0])),
},
// Fq2(u + 9)**(((q^3) - 1) / 3)
Fq2 {
c0: Fq(FqRepr([
0xc9af22f716ad6bad,
0xb311782a4aa662b2,
0x19eeaf64e248c7f4,
0x20273e77e3439f82,
])),
c1: Fq(FqRepr([
0xacc02860f7ce93ac,
0x3933d5817ba76b4c,
0x69e6188b446c8467,
0x0a46036d4417cc55,
])),
},
// Fq2(u + 9)**(((q^4) - 1) / 3)
Fq2 {
c0: Fq(FqRepr([
0x71930c11d782e155,
0xa6bb947cffbe3323,
0xaa303344d4741444,
0x2c3b3f0d26594943,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0,])),
},
// Fq2(u + 9)**(((q^5) - 1) / 3)
Fq2 {
c0: Fq(FqRepr([
0xf91aba2654e8e3b1,
0x4771cb2fdc92ce12,
0xdcb16ae0fc8bdf35,
0x274aa195cd9d8be4,
])),
c1: Fq(FqRepr([
0x5cfc50ae18811f8b,
0x4bb28433cb43988c,
0x4fd35f13c3b56219,
0x301949bd2fc8883a,
])),
},
];
pub const FROBENIUS_COEFF_FQ6_C2: [Fq2; 6] = [
// Fq2(u + 1)**(((2q^0) - 2) / 3)
Fq2 {
c0: Fq(FqRepr([
0xd35d438dc58f0d9d,
0x0a78eb28f5c70b3d,
0x666ea36f7879462c,
0x0e0a77c19a07df2f,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0])),
},
// Fq2(u + 1)**(((2q^1) - 2) / 3)
Fq2 {
c0: Fq(FqRepr([
0x7361d77f843abe92,
0xa5bb2bd3273411fb,
0x9c941f314b3e2399,
0x15df9cddbb9fd3ec,
])),
c1: Fq(FqRepr([
0x5dddfd154bd8c949,
0x62cb29a5a4445b60,
0x37bc870a0c7dd2b9,
0x24830a9d3171f0fd,
])),
},
// Fq2(u + 1)**(((2q^2) - 2) / 3)
Fq2 {
c0: Fq(FqRepr([
0x71930c11d782e155,
0xa6bb947cffbe3323,
0xaa303344d4741444,
0x2c3b3f0d26594943,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0,])),
},
// Fq2(u + 1)**(((2q^3) - 2) / 3)
Fq2 {
c0: Fq(FqRepr([
0x448a93a57b6762df,
0xbfd62df528fdeadf,
0xd858f5d00e9bd47a,
0x06b03d4d3476ec58,
])),
c1: Fq(FqRepr([
0x2b19daf4bcc936d1,
0xa1a54e7a56f4299f,
0xb533eee05adeaef1,
0x170c812b84dda0b2,
])),
},
// Fq2(u + 1)**(((2q^4) - 2) / 3)
Fq2 {
c0: Fq(FqRepr([
0x3350c88e13e80b9c,
0x7dce557cdb5e56b9,
0x6001b4b8b615564a,
0x2682e617020217e0,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0,])),
},
// Fq2(u + 1)**(((2q^5) - 2) / 3)
Fq2 {
c0: Fq(FqRepr([
0x843420f1d8dadbd6,
0x31f010c9183fcdb2,
0x436330b527a76049,
0x13d47447f11adfe4,
])),
c1: Fq(FqRepr([
0xef494023a857fa74,
0x2a925d02d5ab101a,
0x83b015829ba62f10,
0x2539111d0c13aea3,
])),
},
];
// non_residue^((modulus^i-1)/6) for i=0,...,11
pub const FROBENIUS_COEFF_FQ12_C1: [Fq2; 12] = [
// Fq2(u + 1)**(((q^0) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0xd35d438dc58f0d9d,
0x0a78eb28f5c70b3d,
0x666ea36f7879462c,
0x0e0a77c19a07df2f,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0])),
},
// Fq2(u + 1)**(((q^1) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0xaf9ba69633144907,
0xca6b1d7387afb78a,
0x11bded5ef08a2087,
0x02f34d751a1f3a7c,
])),
c1: Fq(FqRepr([
0xa222ae234c492d72,
0xd00f02a4565de15b,
0xdc2ff3a253dfc926,
0x10a75716b3899551,
])),
},
// Fq2(u + 1)**(((q^2) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0xca8d800500fa1bf2,
0xf0c5d61468b39769,
0x0e201271ad0d4418,
0x04290f65bad856e6,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0])),
},
// Fq2(u + 1)**(((q^3) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x365316184e46d97d,
0x0af7129ed4c96d9f,
0x659da72fca1009b5,
0x08116d8983a20d23,
])),
c1: Fq(FqRepr([
0xb1df4af7c39c1939,
0x3d9f02878a73bf7f,
0x9b2220928caf0ae0,
0x26684515eff054a6,
])),
},
// Fq2(u + 1)**(((q^4) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x3350c88e13e80b9c,
0x7dce557cdb5e56b9,
0x6001b4b8b615564a,
0x2682e617020217e0,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0,])),
},
// Fq2(u + 1)**(((q^5) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x86b76f821b329076,
0x408bf52b4d19b614,
0x53dfb9d0d985e92d,
0x051e20146982d2a7,
])),
c1: Fq(FqRepr([
0x0fbc9cd47752ebc7,
0x6d8fffe33415de24,
0xbef22cf038cf41b9,
0x15c0edff3c66bf54,
])),
},
// Fq2(u + 1)**(((q^6) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x68c3488912edefaa,
0x8d087f6872aabf4f,
0x51e1a24709081231,
0x2259d6b14729c0fa,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0,])),
},
// Fq2(u + 1)**(((q^7) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x8c84e580a568b440,
0xcd164d1de0c21302,
0xa692585790f737d5,
0x2d7100fdc71265ad,
])),
c1: Fq(FqRepr([
0x99fdddf38c33cfd5,
0xc77267ed1213e931,
0xdc2052142da18f36,
0x1fbcf75c2da80ad7,
])),
},
// Fq2(u + 1)**(((q^8) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x71930c11d782e155,
0xa6bb947cffbe3323,
0xaa303344d4741444,
0x2c3b3f0d26594943,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0,])),
},
// Fq2(u + 1)**(((q^9) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x05cd75fe8a3623ca,
0x8c8a57f293a85cee,
0x52b29e86b7714ea8,
0x2852e0e95d8f9306,
])),
c1: Fq(FqRepr([
0x8a41411f14e0e40e,
0x59e26809ddfe0b0d,
0x1d2e2523f4d24d7d,
0x09fc095cf1414b83,
])),
},
// Fq2(u + 1)**(((q^10) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0x08cfc388c494f1ab,
0x19b315148d1373d4,
0x584e90fdcb6c0213,
0x09e1685bdf2f8849,
])),
c1: Fq(FqRepr([0x0, 0x0, 0x0, 0x0,])),
},
// Fq2(u + 1)**(((q^11) - 1) / 6)
Fq2 {
c0: Fq(FqRepr([
0xb5691c94bd4a6cd1,
0x56f575661b581478,
0x64708be5a7fb6f30,
0x2b462e5e77aecd82,
])),
c1: Fq(FqRepr([
0x2c63ef42612a1180,
0x29f16aae345bec69,
0xf95e18c648b216a4,
0x1aa36073a4cae0d4,
])),
},
];
// -((2**256) mod q) mod q
pub const NEGATIVE_ONE: Fq = Fq(FqRepr([
0x974bc177a0000006,
0xf13771b2da58a367,
0x51e1a2470908122e,
0x2259d6b14729c0fa,
]));
#[cfg(test)]
use rand::{Rand, SeedableRng, XorShiftRng};
#[test]
fn test_fq_repr_from() {
assert_eq!(FqRepr::from(100), FqRepr([100, 0, 0, 0]));
assert_eq!(FqRepr::from(3), FqRepr([3, 0, 0, 0]));
}
#[test]
fn test_fq_repr_is_odd() {
assert!(!FqRepr::from(0).is_odd());
assert!(FqRepr::from(0).is_even());
assert!(FqRepr::from(1).is_odd());
assert!(!FqRepr::from(1).is_even());
assert!(!FqRepr::from(324834872).is_odd());
assert!(FqRepr::from(324834872).is_even());
assert!(FqRepr::from(324834873).is_odd());
assert!(!FqRepr::from(324834873).is_even());
}
#[test]
fn test_fq_repr_num_bits() {
let mut a = FqRepr::from(0);
assert_eq!(0, a.num_bits());
a = FqRepr::from(1);
for i in 1..257 {
assert_eq!(i, a.num_bits());
a.mul2();
}
assert_eq!(0, a.num_bits());
}
#[test]
fn test_fq_is_valid() {
print!("modulus = {}\n", MODULUS);
print!("R = {}\n", R);
let mut a = Fq(MODULUS);
assert!(!a.is_valid());
a.0.sub_noborrow(&FqRepr::from(1));
assert!(a.is_valid());
assert!(Fq(FqRepr::from(0)).is_valid());
assert!(
Fq(FqRepr([
0xdf4671abd14dab3e,
0xe2dc0c9f534fbd33,
0x31ca6c880cc444a6,
0x257a67e70ef33359
])).is_valid()
);
assert!(
!Fq(FqRepr([
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
0xffffffffffffffff,
])).is_valid()
);
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let a = Fq::rand(&mut rng);
assert!(a.is_valid());
}
}
#[test]
fn test_fq_repr_display() {
assert_eq!(
format!("{}", Fq::into_repr(&Fq::one())),
"0x0000000000000000000000000000000000000000000000000000000000000001".to_string()
);
assert_eq!(
format!("{}", FqRepr([0, 0, 0, 0])),
"0x0000000000000000000000000000000000000000000000000000000000000000".to_string()
);
}
#[test]
fn test_fq_num_bits() {
assert_eq!(Fq::NUM_BITS, 254);
assert_eq!(Fq::CAPACITY, 253);
}
#[test]
fn test_fq_sqrt() {
use ff::SqrtField;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
assert_eq!(Fq::zero().sqrt().unwrap(), Fq::zero());
for _ in 0..1000 {
// Ensure sqrt(a^2) = a or -a
let a = Fq::rand(&mut rng);
let mut nega = a;
nega.negate();
let mut b = a;
b.square();
let b = b.sqrt().unwrap();
assert!(a == b || nega == b);
}
for _ in 0..1000 {
// Ensure sqrt(a)^2 = a for random a
let a = Fq::rand(&mut rng);
if let Some(mut tmp) = a.sqrt() {
tmp.square();
assert_eq!(a, tmp);
}
}
}
#[test]
fn test_fq_sqrt_2() {
use ff::SqrtField;
let x = Fq::from_str("4").unwrap();
print!("x = {}\n", x);
if let Some(y) = x.sqrt() {
print!("y = {}\n", y);
let mut y_other = y;
y_other.negate();
print!("y' = {}\n", y_other);
}
}
#[test]
fn fq_field_tests() {
crate::tests::field::random_field_tests::<Fq>();
crate::tests::field::random_sqrt_tests::<Fq>();
crate::tests::field::random_frobenius_tests::<Fq, _>(Fq::char(), 13);
crate::tests::field::from_str_tests::<Fq>();
}

221
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use super::fq::FROBENIUS_COEFF_FQ12_C1;
use super::fq2::Fq2;
use super::fq6::Fq6;
use ff::Field;
use rand::{Rand, Rng};
/// An element of Fq12, represented by c0 + c1 * w.
#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct Fq12 {
pub c0: Fq6,
pub c1: Fq6,
}
impl ::std::fmt::Display for Fq12 {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
write!(f, "Fq12({} + {} * w)", self.c0, self.c1)
}
}
impl Rand for Fq12 {
fn rand<R: Rng>(rng: &mut R) -> Self {
Fq12 {
c0: rng.gen(),
c1: rng.gen(),
}
}
}
// BN256 and BLS12 implementations should be the same
// Defined over w^2 - v = 0
impl Fq12 {
pub fn conjugate(&mut self) {
self.c1.negate();
}
pub fn mul_by_014(&mut self, c0: &Fq2, c1: &Fq2, c4: &Fq2) {
let mut aa = self.c0;
aa.mul_by_01(c0, c1);
let mut bb = self.c1;
bb.mul_by_1(c4);
let mut o = *c1;
o.add_assign(c4);
self.c1.add_assign(&self.c0);
self.c1.mul_by_01(c0, &o);
self.c1.sub_assign(&aa);
self.c1.sub_assign(&bb);
self.c0 = bb;
self.c0.mul_by_nonresidue();
self.c0.add_assign(&aa);
}
// TODO make it hand optimized
// // multiply by (c0, c1, c2) + (c3, c4, c5)*w where only c0, c3 and c4 are non-zero
pub fn mul_by_034(&mut self, c0: &Fq2, c3: &Fq2, c4: &Fq2) {
self.mul_assign(&Fq12 {
c0: Fq6 {
c0: *c0,
c1: Fq2::zero(),
c2: Fq2::zero(),
},
c1: Fq6 {
c0: *c3,
c1: *c4,
c2: Fq2::zero(),
},
});
}
}
impl Field for Fq12 {
fn zero() -> Self {
Fq12 {
c0: Fq6::zero(),
c1: Fq6::zero(),
}
}
fn one() -> Self {
Fq12 {
c0: Fq6::one(),
c1: Fq6::zero(),
}
}
fn is_zero(&self) -> bool {
self.c0.is_zero() && self.c1.is_zero()
}
fn double(&mut self) {
self.c0.double();
self.c1.double();
}
fn negate(&mut self) {
self.c0.negate();
self.c1.negate();
}
fn add_assign(&mut self, other: &Self) {
self.c0.add_assign(&other.c0);
self.c1.add_assign(&other.c1);
}
fn sub_assign(&mut self, other: &Self) {
self.c0.sub_assign(&other.c0);
self.c1.sub_assign(&other.c1);
}
fn frobenius_map(&mut self, power: usize) {
self.c0.frobenius_map(power);
self.c1.frobenius_map(power);
self.c1.c0.mul_assign(&FROBENIUS_COEFF_FQ12_C1[power % 12]);
self.c1.c1.mul_assign(&FROBENIUS_COEFF_FQ12_C1[power % 12]);
self.c1.c2.mul_assign(&FROBENIUS_COEFF_FQ12_C1[power % 12]);
}
fn square(&mut self) {
let mut ab = self.c0;
ab.mul_assign(&self.c1);
let mut c0c1 = self.c0;
c0c1.add_assign(&self.c1);
let mut c0 = self.c1;
c0.mul_by_nonresidue();
c0.add_assign(&self.c0);
c0.mul_assign(&c0c1);
c0.sub_assign(&ab);
self.c1 = ab;
self.c1.add_assign(&ab);
ab.mul_by_nonresidue();
c0.sub_assign(&ab);
self.c0 = c0;
}
fn mul_assign(&mut self, other: &Self) {
let mut aa = self.c0;
aa.mul_assign(&other.c0);
let mut bb = self.c1;
bb.mul_assign(&other.c1);
let mut o = other.c0;
o.add_assign(&other.c1);
self.c1.add_assign(&self.c0);
self.c1.mul_assign(&o);
self.c1.sub_assign(&aa);
self.c1.sub_assign(&bb);
self.c0 = bb;
self.c0.mul_by_nonresidue();
self.c0.add_assign(&aa);
}
fn inverse(&self) -> Option<Self> {
let mut c0s = self.c0;
c0s.square();
let mut c1s = self.c1;
c1s.square();
c1s.mul_by_nonresidue();
c0s.sub_assign(&c1s);
c0s.inverse().map(|t| {
let mut tmp = Fq12 { c0: t, c1: t };
tmp.c0.mul_assign(&self.c0);
tmp.c1.mul_assign(&self.c1);
tmp.c1.negate();
tmp
})
}
}
#[cfg(test)]
use rand::{SeedableRng, XorShiftRng};
#[test]
fn test_fq12_mul_by_014() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let c0 = Fq2::rand(&mut rng);
let c1 = Fq2::rand(&mut rng);
let c5 = Fq2::rand(&mut rng);
let mut a = Fq12::rand(&mut rng);
let mut b = a;
a.mul_by_014(&c0, &c1, &c5);
b.mul_assign(&Fq12 {
c0: Fq6 {
c0: c0,
c1: c1,
c2: Fq2::zero(),
},
c1: Fq6 {
c0: Fq2::zero(),
c1: c5,
c2: Fq2::zero(),
},
});
assert_eq!(a, b);
}
}
#[test]
fn test_squaring() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let mut a = Fq12::rand(&mut rng);
let mut b = a;
b.mul_assign(&a);
a.square();
assert_eq!(a, b);
}
}
#[test]
fn fq12_field_tests() {
use ff::PrimeField;
crate::tests::field::random_field_tests::<Fq12>();
crate::tests::field::random_frobenius_tests::<Fq12, _>(super::fq::Fq::char(), 13);
}

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use super::fq::{FROBENIUS_COEFF_FQ2_C1, Fq, NEGATIVE_ONE};
use ff::{Field, SqrtField};
use rand::{Rand, Rng};
use std::cmp::Ordering;
/// An element of Fq2, represented by c0 + c1 * u.
#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct Fq2 {
pub c0: Fq,
pub c1: Fq,
}
impl ::std::fmt::Display for Fq2 {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
write!(f, "Fq2({} + {} * u)", self.c0, self.c1)
}
}
/// `Fq2` elements are ordered lexicographically.
impl Ord for Fq2 {
#[inline(always)]
fn cmp(&self, other: &Fq2) -> Ordering {
match self.c1.cmp(&other.c1) {
Ordering::Greater => Ordering::Greater,
Ordering::Less => Ordering::Less,
Ordering::Equal => self.c0.cmp(&other.c0),
}
}
}
impl PartialOrd for Fq2 {
#[inline(always)]
fn partial_cmp(&self, other: &Fq2) -> Option<Ordering> {
Some(self.cmp(other))
}
}
impl Fq2 {
// This is very confusing part, cause depends not of Fq2 itself, but form Fq6 construction
/// Multiply this element by quadratic nonresidue 9 + u.
pub fn mul_by_nonresidue(&mut self) {
// (xi+y)(i+9) = (9x+y)i+(9y-x)
let t0 = self.c0;
let t1 = self.c1;
// 8*x*i + 8*y
self.double();
self.double();
self.double();
// 9*y
self.c0.add_assign(&t0);
// (9*y - x)
self.c0.sub_assign(&t1);
// (9*x)i
self.c1.add_assign(&t1);
// (9*x + y)
self.c1.add_assign(&t0);
}
// Multiply this element by ξ where ξ=i+9
pub fn mul_by_xi(&mut self) {
// (xi+y)(i+9) = (9x+y)i+(9y-x)
let t0 = self.c0;
let t1 = self.c1;
// 8*x*i + 8*y
self.double();
self.double();
self.double();
// 9*y
self.c0.add_assign(&t0);
// (9*y - x)
self.c0.sub_assign(&t1);
// (9*x)i
self.c1.add_assign(&t1);
// (9*x + y)
self.c1.add_assign(&t0);
}
/// Norm of Fq2 as extension field in i over Fq
pub fn norm(&self) -> Fq {
let mut t0 = self.c0;
let mut t1 = self.c1;
t0.square();
t1.square();
t1.add_assign(&t0);
t1
}
// conjucate by negating c1
pub fn conjugate(&mut self) {
self.c1.negate();
}
}
impl Rand for Fq2 {
fn rand<R: Rng>(rng: &mut R) -> Self {
Fq2 {
c0: rng.gen(),
c1: rng.gen(),
}
}
}
impl Field for Fq2 {
fn zero() -> Self {
Fq2 {
c0: Fq::zero(),
c1: Fq::zero(),
}
}
fn one() -> Self {
Fq2 {
c0: Fq::one(),
c1: Fq::zero(),
}
}
fn is_zero(&self) -> bool {
self.c0.is_zero() && self.c1.is_zero()
}
fn square(&mut self) {
let mut ab = self.c0;
ab.mul_assign(&self.c1);
let mut c0c1 = self.c0;
c0c1.add_assign(&self.c1);
let mut c0 = self.c1;
c0.negate();
c0.add_assign(&self.c0);
c0.mul_assign(&c0c1);
c0.sub_assign(&ab);
self.c1 = ab;
self.c1.add_assign(&ab);
c0.add_assign(&ab);
self.c0 = c0;
}
fn double(&mut self) {
self.c0.double();
self.c1.double();
}
fn negate(&mut self) {
self.c0.negate();
self.c1.negate();
}
fn add_assign(&mut self, other: &Self) {
self.c0.add_assign(&other.c0);
self.c1.add_assign(&other.c1);
}
fn sub_assign(&mut self, other: &Self) {
self.c0.sub_assign(&other.c0);
self.c1.sub_assign(&other.c1);
}
fn mul_assign(&mut self, other: &Self) {
let mut aa = self.c0;
aa.mul_assign(&other.c0);
let mut bb = self.c1;
bb.mul_assign(&other.c1);
let mut o = other.c0;
o.add_assign(&other.c1);
self.c1.add_assign(&self.c0);
self.c1.mul_assign(&o);
self.c1.sub_assign(&aa);
self.c1.sub_assign(&bb);
self.c0 = aa;
self.c0.sub_assign(&bb);
}
fn inverse(&self) -> Option<Self> {
let mut t1 = self.c1;
t1.square();
let mut t0 = self.c0;
t0.square();
t0.add_assign(&t1);
t0.inverse().map(|t| {
let mut tmp = Fq2 {
c0: self.c0,
c1: self.c1,
};
tmp.c0.mul_assign(&t);
tmp.c1.mul_assign(&t);
tmp.c1.negate();
tmp
})
}
fn frobenius_map(&mut self, power: usize) {
self.c1.mul_assign(&FROBENIUS_COEFF_FQ2_C1[power % 2]);
}
}
impl SqrtField for Fq2 {
fn legendre(&self) -> ::ff::LegendreSymbol {
self.norm().legendre()
}
fn sqrt(&self) -> Option<Self> {
// Algorithm 9, https://eprint.iacr.org/2012/685.pdf
if self.is_zero() {
Some(Self::zero())
} else {
// a1 = self^((q - 3) / 4)
let mut a1 = self.pow([
0x4f082305b61f3f51,
0x65e05aa45a1c72a3,
0x6e14116da0605617,
0x0c19139cb84c680a,
]);
let mut alpha = a1;
alpha.square();
alpha.mul_assign(self);
let mut a0 = alpha;
a0.frobenius_map(1);
a0.mul_assign(&alpha);
let neg1 = Fq2 {
c0: NEGATIVE_ONE,
c1: Fq::zero(),
};
if a0 == neg1 {
None
} else {
a1.mul_assign(self);
if alpha == neg1 {
a1.mul_assign(&Fq2 {
c0: Fq::zero(),
c1: Fq::one(),
});
} else {
alpha.add_assign(&Fq2::one());
// alpha = alpha^((q - 1) / 2)
alpha = alpha.pow([
0x9e10460b6c3e7ea3,
0xcbc0b548b438e546,
0xdc2822db40c0ac2e,
0x183227397098d014,
]);
a1.mul_assign(&alpha);
}
Some(a1)
}
}
}
}
#[test]
fn test_fq2_get_b() {
use ff::Field;
let mut a = Fq2::one();
a.mul_by_nonresidue();
let mut b = a.inverse().unwrap();
let c = b;
b.double();
b.add_assign(&c);
print!("B coeff in Fq2 = {}\n", b);
}
#[test]
fn test_fq2_frobc1() {
use ff::Field;
let mut a = Fq2::one();
a.mul_by_nonresidue();
let res1 = a.pow([
0x69602eb24829a9c2,
0xdd2b2385cd7b4384,
0xe81ac1e7808072c9,
0x10216f7ba065e00d,
]);
print!("Frob1 = {}\n", res1);
let res2 = a.pow([
0x691c1d8b62747890,
0x8cab57b9adf8eb00,
0x18c55d8979dcee49,
0x56cd8a31d35b6b98,
0xb7a4a8c966ece684,
0xe5592c705cbd1cac,
0x1dde2529566d9b5e,
0x030c96e827699534,
]);
print!("Frob2 = {}\n", res2);
let res3 = a.pow([
0x3de6332b975d69b2,
0x587b5b2bd890e101,
0x16677d4bec77bbeb,
0x3fdfdba3309dd645,
0xdfd4137cd943954b,
0xcfb035047f38c226,
0x01b5daf7ac73104c,
0x4cce8699d63e4f06,
0x40c0b41264a4b9f4,
0x7806da9ba1f6d7fb,
0x110a40708107d53a,
0x00938e25ae57c88f,
]);
print!("Frob3 = {}\n", res3);
let res4 = a.pow([
0xabb30419f6bee420,
0x6ce183e5f2f8d3b9,
0x9db42a441998ac99,
0xf74b04aa96e3852f,
0x64de4542a9807c06,
0x41f83258fd90abd1,
0x5ecb5383626aeca3,
0xb60804ce8f24ca82,
0xd4b3aadc1344e8bb,
0x436b70833cb2615b,
0x1a87eeb627861611,
0x4e155ea3e5090666,
0xacfcff9291a10112,
0x1cba0005b295d5bc,
0x319c8e7f94b31729,
0x001be477ceef2455,
]);
print!("Frob4 = {}\n", res4);
let res5 = a.pow([
0x7501aa71de0e8ea2,
0x97516fd7ca83b8fe,
0x7da14ac0c03d4182,
0xaf5d35dc7f80498d,
0xb257f7f84fb899e0,
0x372cb1bd547dbe69,
0xb6696efbf52d5146,
0x03b6707d4a42574c,
0xeae6c62cf1670269,
0xfe70626cbbb760e9,
0xfa9d12d01fb42086,
0xc85218d5a7af23b7,
0x0a70a73464ed35fb,
0x878713d44d9a2aca,
0xc81d8fc5cdfe15ee,
0xa3ebe919611e544d,
0xfe46bd734126775c,
0x06f8a7579371f67f,
0xa94a371ceb68884c,
0x000545c441ba73d6,
]);
print!("Frob5 = {}\n", res5);
let res6 = a.pow([
0xfc4dae0d07a152b0,
0x3f383f79f9859a0a,
0x261f0da312f72ab2,
0x9cc6b2e6efb101d8,
0xf45a236f76e806da,
0x7158062cd79d6743,
0x8adabccc870f23db,
0x24428ff02b7988c1,
0x8f55fa0a7ecfa21d,
0xd5574a8dc73fdcc2,
0xb86f06772524e5ca,
0xb4b11653b762bd0f,
0xb84ec7291c154c58,
0x2a095f1259f99fb5,
0x6ccb38fbc9f54a74,
0x3a3f77faca5c2ea0,
0x21a469bdd36b9656,
0x0fa2e41314b53258,
0xf8ca5207cb9f028e,
0x489fbf415ec8104e,
0x711aafe44a1ab611,
0xfb508020969bab31,
0xb8b71e4e258cf968,
0x0000ff25aa9c2350,
]);
print!("Frob6 = {}\n", res6);
}
#[test]
fn test_fq2_frobc2() {
use ff::Field;
let mut a = Fq2::one();
a.mul_by_nonresidue();
let res1 = a.pow([
0xd2c05d6490535384,
0xba56470b9af68708,
0xd03583cf0100e593,
0x2042def740cbc01b,
]);
print!("Frob1 = {}\n", res1);
let res2 = a.pow([
0xd2383b16c4e8f120,
0x1956af735bf1d600,
0x318abb12f3b9dc93,
0xad9b1463a6b6d730,
0x6f495192cdd9cd08,
0xcab258e0b97a3959,
0x3bbc4a52acdb36bd,
0x06192dd04ed32a68,
]);
print!("Frob2 = {}\n", res2);
let res3 = a.pow([
0x7bcc66572ebad364,
0xb0f6b657b121c202,
0x2ccefa97d8ef77d6,
0x7fbfb746613bac8a,
0xbfa826f9b2872a96,
0x9f606a08fe71844d,
0x036bb5ef58e62099,
0x999d0d33ac7c9e0c,
0x81816824c94973e8,
0xf00db53743edaff6,
0x221480e1020faa74,
0x01271c4b5caf911e,
]);
print!("Frob3 = {}\n", res3);
let res4 = a.pow([
0x57660833ed7dc840,
0xd9c307cbe5f1a773,
0x3b68548833315932,
0xee9609552dc70a5f,
0xc9bc8a855300f80d,
0x83f064b1fb2157a2,
0xbd96a706c4d5d946,
0x6c10099d1e499504,
0xa96755b82689d177,
0x86d6e1067964c2b7,
0x350fdd6c4f0c2c22,
0x9c2abd47ca120ccc,
0x59f9ff2523420224,
0x3974000b652bab79,
0x63391cff29662e52,
0x0037c8ef9dde48aa,
]);
print!("Frob4 = {}\n", res4);
let res5 = a.pow([
0xea0354e3bc1d1d44,
0x2ea2dfaf950771fc,
0xfb429581807a8305,
0x5eba6bb8ff00931a,
0x64afeff09f7133c1,
0x6e59637aa8fb7cd3,
0x6cd2ddf7ea5aa28c,
0x076ce0fa9484ae99,
0xd5cd8c59e2ce04d2,
0xfce0c4d9776ec1d3,
0xf53a25a03f68410d,
0x90a431ab4f5e476f,
0x14e14e68c9da6bf7,
0x0f0e27a89b345594,
0x903b1f8b9bfc2bdd,
0x47d7d232c23ca89b,
0xfc8d7ae6824ceeb9,
0x0df14eaf26e3ecff,
0x52946e39d6d11098,
0x000a8b888374e7ad,
]);
print!("Frob5 = {}\n", res5);
let res6 = a.pow([
0xf89b5c1a0f42a560,
0x7e707ef3f30b3415,
0x4c3e1b4625ee5564,
0x398d65cddf6203b0,
0xe8b446deedd00db5,
0xe2b00c59af3ace87,
0x15b579990e1e47b6,
0x48851fe056f31183,
0x1eabf414fd9f443a,
0xaaae951b8e7fb985,
0x70de0cee4a49cb95,
0x69622ca76ec57a1f,
0x709d8e52382a98b1,
0x5412be24b3f33f6b,
0xd99671f793ea94e8,
0x747eeff594b85d40,
0x4348d37ba6d72cac,
0x1f45c826296a64b0,
0xf194a40f973e051c,
0x913f7e82bd90209d,
0xe2355fc894356c22,
0xf6a100412d375662,
0x716e3c9c4b19f2d1,
0x0001fe4b553846a1,
]);
print!("Frob6 = {}\n", res6);
}
#[test]
fn test_fq2_frob12() {
use ff::Field;
let mut a = Fq2::one();
a.mul_by_nonresidue();
let res1 = a.pow([
0x34b017592414d4e1,
0xee9591c2e6bda1c2,
0xf40d60f3c0403964,
0x0810b7bdd032f006,
]);
print!("Frob1 = {}\n", res1);
let res2 = a.pow([
0x348e0ec5b13a3c48,
0xc655abdcd6fc7580,
0x0c62aec4bcee7724,
0x2b66c518e9adb5cc,
0x5bd25464b3767342,
0x72ac96382e5e8e56,
0x0eef1294ab36cdaf,
0x01864b7413b4ca9a,
]);
print!("Frob2 = {}\n", res2);
let res3 = a.pow([
0x9ef31995cbaeb4d9,
0xac3dad95ec487080,
0x8b33bea5f63bddf5,
0x9fefedd1984eeb22,
0x6fea09be6ca1caa5,
0x67d81a823f9c6113,
0x00daed7bd6398826,
0x2667434ceb1f2783,
0xa0605a0932525cfa,
0x3c036d4dd0fb6bfd,
0x888520384083ea9d,
0x0049c712d72be447,
]);
print!("Frob3 = {}\n", res3);
let res4 = a.pow([
0xd5d9820cfb5f7210,
0xb670c1f2f97c69dc,
0xceda15220ccc564c,
0x7ba582554b71c297,
0xb26f22a154c03e03,
0xa0fc192c7ec855e8,
0x2f65a9c1b1357651,
0xdb04026747926541,
0xea59d56e09a2745d,
0xa1b5b8419e5930ad,
0x0d43f75b13c30b08,
0x270aaf51f2848333,
0x567e7fc948d08089,
0x8e5d0002d94aeade,
0x98ce473fca598b94,
0x000df23be777922a,
]);
print!("Frob4 = {}\n", res4);
let res5 = a.pow([
0x3a80d538ef074751,
0x4ba8b7ebe541dc7f,
0xbed0a560601ea0c1,
0x57ae9aee3fc024c6,
0xd92bfbfc27dc4cf0,
0x1b9658deaa3edf34,
0x5b34b77dfa96a8a3,
0x81db383ea5212ba6,
0xf573631678b38134,
0x7f3831365ddbb074,
0xfd4e89680fda1043,
0xe4290c6ad3d791db,
0x0538539a32769afd,
0x43c389ea26cd1565,
0xe40ec7e2e6ff0af7,
0x51f5f48cb08f2a26,
0xff235eb9a0933bae,
0x037c53abc9b8fb3f,
0x54a51b8e75b44426,
0x0002a2e220dd39eb,
]);
print!("Frob5 = {}\n", res5);
let res6 = a.pow([
0x7e26d70683d0a958,
0x1f9c1fbcfcc2cd05,
0x130f86d1897b9559,
0x4e63597377d880ec,
0xfa2d11b7bb74036d,
0xb8ac03166bceb3a1,
0xc56d5e66438791ed,
0x922147f815bcc460,
0x47aafd053f67d10e,
0x6aaba546e39fee61,
0xdc37833b929272e5,
0x5a588b29dbb15e87,
0xdc2763948e0aa62c,
0x1504af892cfccfda,
0x36659c7de4faa53a,
0x1d1fbbfd652e1750,
0x10d234dee9b5cb2b,
0x07d172098a5a992c,
0x7c652903e5cf8147,
0xa44fdfa0af640827,
0xb88d57f2250d5b08,
0x7da840104b4dd598,
0x5c5b8f2712c67cb4,
0x00007f92d54e11a8,
]);
print!("Frob6 = {}\n", res6);
let res7 = a.pow([
0x75e1bff130efc449,
0xfb4f505fb284ee15,
0x57b30efd96c492f5,
0xfdcb862e4e948b59,
0x3def467dae8887e2,
0xa47e3f76b755ca8c,
0xa63f6ea3debc563a,
0x115d1111a4fc4be2,
0xc3b2ece674d74549,
0xb2099a8141cb8830,
0x120dc0b8ac63867a,
0xae0245267985fe96,
0xed38ce9a40128f1d,
0xeba67d5d8ffa4939,
0xbff55f706b0de0f5,
0xb4f3f86e6f982aed,
0x062675f8a89bd61d,
0xa1098fb006a9726c,
0xe974fc0c7b0e5d9c,
0x0f10af0bdc56fe9e,
0x628ca855d5d4ac87,
0x7bd59e7101d9d82d,
0xed98625bf5dc71aa,
0x7ab9b78fdc8558f4,
0x489b4c8564d6f8d2,
0xd055177a2fbfcd94,
0x059f68dd1e0cb392,
0x0000181d8471f268,
]);
print!("Frob7 = {}\n", res7);
let res8 = a.pow([
0x742e7ca156ec6a20,
0x3fee59e5c3e8de2e,
0xfdef69cd295152ef,
0xe4ad8aece2ce3640,
0x05308778897ea5eb,
0x0a4fc046ae1c2e50,
0xb16faf17473bba4a,
0xd106751c900aadfa,
0x115301a6c43ba345,
0x19f012d49d8a716c,
0x6b9d91b1c2a56cc5,
0xb77230690204b675,
0xf6d68e7229980805,
0xf4263d3b11784a87,
0x24bb64e5adeaa33d,
0x684c4ff325fa1c4d,
0x79a8c6430472e684,
0x823af8186da5609c,
0x2087966741a30941,
0x1876205eaf407912,
0xa614d3f14990435e,
0xd405328435bcc8df,
0x5afac38bad541421,
0x0706fb9d17dec3d8,
0xecc747832c3f5f69,
0xe231b0ffd6651ed5,
0x45fa8e7ff2a80f15,
0xdce48166a2ee0170,
0x305fc72544895a12,
0x516ac4b20d800019,
0x826e9ab28689a4d3,
0x0000048efbc0eaac,
]);
print!("Frob8 = {}\n", res8);
let res9 = a.pow(&[
0x64984dce4c07e3c1,
0x2e2096f441339496,
0xd50c9bd49d279670,
0xd52ead3ce3a93422,
0x426dad5fc6a6779a,
0x3f9dd6b6f19bc638,
0x6be503d3981b0db5,
0x0b222e7512412d2c,
0x484bd275e77ff0bf,
0xb357542fb851205b,
0xd8c995246bf492ff,
0xc6b92fc3bf2887bc,
0xcd27cfd0d4499277,
0x967aa0012f40dcf9,
0x312baab0f5bc64e3,
0xe465b3c98a822e05,
0x3133d12c8828f7b8,
0x357a20a6a8a244ca,
0xd40b61719905e5b9,
0xcc4f1d5e2aed7a75,
0x7895032e16409563,
0x536db2a17eb54630,
0xdd66ae0d2d5ac57e,
0xe150b5a7f229f541,
0xd882dbabee789616,
0x1f380eb8775416ca,
0x73eca6c1c0abcd02,
0x8bd4f78c2fe1861e,
0xc53f421003b18ea2,
0xcae3f7b5d0591ecb,
0xbebe6ab21737113e,
0x838f0df2a5f7f26d,
0xbc2aa2593b06d88f,
0x0cb02b95a74a8a0a,
0x74bd9a7b50725838,
0x000000dc98741fbf,
][..]);
print!("Frob9 = {}\n", res9);
let res10 = a.pow(&[
0xed4127472fd6bc68,
0x72748872e11c4b47,
0x9c84e64776edc3f8,
0x008119b96d78b386,
0xfb0fbff1c5556968,
0x5009c51f998020be,
0xd6e688613527a368,
0xbe4f27942823152c,
0xd0f09d15c45fe09e,
0x7eb531158d2bbea5,
0x51bbe8e71be2cfd1,
0xbab37561b8c0c7c4,
0xd9173b5ab551b267,
0x05fafd9be4c78781,
0x61883bc8a78540ee,
0x7fe7aee3dcb694fb,
0x0e4e85b12b4ac8a8,
0x9a0aa13a9ab47a86,
0xd5a3bd591ae12d4b,
0x5865cbfabbe53b4d,
0xf98188a9b0cd490f,
0x3985ef4af715da43,
0x573661cd006ced38,
0x95853a6aaa77d5c1,
0x165d538f0628b55e,
0x583e75f890f32cac,
0x5becf43a08a490b9,
0x63ed4071c1a8087a,
0x151d41c7701faa25,
0x1c661c8e4900b051,
0x581aa0f552590875,
0x31bf39ff43375aca,
0xe27c0f3d11310329,
0x04071459ef3a42c2,
0x59a2b029be2d6a1f,
0x30ef71f271cdbf61,
0xf3774b177f326e78,
0x976d79b23e8501c5,
0x9ed0e138633123c9,
0x00000029b304ecc1,
][..]);
print!("Frob10 = {}\n", res10);
let res11 = a.pow(&[
0xf4e8c249a335ddb9,
0x965085c9440aef70,
0xc16d84a741174aef,
0xbe1a366b81fe0680,
0x1c65508409269d2f,
0x185861e9cd07fb21,
0x26b682d951220b7a,
0x09f189f5a7b75876,
0x0f7133ab3ecff7f0,
0xbf7d1ada5df0b2fd,
0x4b0df5207414a4b6,
0xbf6a6941b58966d3,
0x6a15cc7b6bb0483a,
0xc338843b8a236597,
0xc8d724986bc0856f,
0x1dcb8b084e928e52,
0x3645ba97c4af9161,
0x7d257d1abed180d3,
0x0a66e85068416bdb,
0x8b745a2aeb2bd27e,
0xe34f87ec4949ec06,
0x6ba47fa06f902fd6,
0x225cd33864121ed2,
0xea5d91e41a3b068b,
0x35d2fbc8b7a05f5c,
0xe5b1e22f3dcbc837,
0xa9f7bdbee44d8301,
0xbb7a57512450e143,
0x2e2ca4188fd4eb5b,
0x9d512b5d1e158636,
0xdd18753b03f38ee8,
0xbbe44db3214b380e,
0x4534f7b060cca3d2,
0xcbb0309736f9df06,
0xfcb01aba828f0678,
0xe2e4d5dac5cc7917,
0x6631e85c4224e136,
0xb6c334bbd109d480,
0x2608e9c50edc2cdf,
0x959dba8288258d16,
0x00d895fc73e207c8,
0x6b5ce08dc4a7bf13,
0xb02a4f252d6a301f,
0x00000007e1e7a192,
][..]);
print!("Frob11 = {}\n", res11);
}
#[test]
fn test_calculate_frob_1() {
let mut a = Fq2::one();
a.mul_by_nonresidue();
// Fq2(u + 9)**(((q^1) - 1) / 3)
print!("(i + 9) = {}\n", a);
}
#[test]
fn test_fq2_ordering() {
let mut a = Fq2 {
c0: Fq::zero(),
c1: Fq::zero(),
};
let mut b = a.clone();
assert!(a.cmp(&b) == Ordering::Equal);
b.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Less);
a.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Equal);
b.c1.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Less);
a.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Less);
a.c1.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Greater);
b.c0.add_assign(&Fq::one());
assert!(a.cmp(&b) == Ordering::Equal);
}
#[test]
fn test_fq2_basics() {
assert_eq!(
Fq2 {
c0: Fq::zero(),
c1: Fq::zero(),
},
Fq2::zero()
);
assert_eq!(
Fq2 {
c0: Fq::one(),
c1: Fq::zero(),
},
Fq2::one()
);
assert!(Fq2::zero().is_zero());
assert!(!Fq2::one().is_zero());
assert!(
!Fq2 {
c0: Fq::zero(),
c1: Fq::one(),
}.is_zero()
);
}
#[test]
fn test_fq2_squaring() {
use super::fq::FqRepr;
use ff::PrimeField;
let mut a = Fq2 {
c0: Fq::one(),
c1: Fq::one(),
}; // u + 1
a.square();
assert_eq!(
a,
Fq2 {
c0: Fq::zero(),
c1: Fq::from_repr(FqRepr::from(2)).unwrap(),
}
); // 2u
let mut a = Fq2 {
c0: Fq::zero(),
c1: Fq::one(),
}; // u
a.square();
assert_eq!(a, {
let mut neg1 = Fq::one();
neg1.negate();
Fq2 {
c0: neg1,
c1: Fq::zero(),
}
}); // -1
}
#[test]
fn test_fq2_legendre() {
use ff::LegendreSymbol::*;
assert_eq!(Zero, Fq2::zero().legendre());
// i^2 = -1
let mut m1 = Fq2::one();
m1.negate();
assert_eq!(QuadraticResidue, m1.legendre());
m1.mul_by_nonresidue();
assert_eq!(QuadraticNonResidue, m1.legendre());
}
#[cfg(test)]
use rand::{SeedableRng, XorShiftRng};
#[test]
fn test_fq2_mul_nonresidue() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let mut nine = Fq::one();
nine.double();
nine.double();
nine.double();
nine.add_assign(&Fq::one());
let nqr = Fq2 {
c0: nine,
c1: Fq::one(),
};
for _ in 0..1000 {
let mut a = Fq2::rand(&mut rng);
let mut b = a;
a.mul_by_nonresidue();
b.mul_assign(&nqr);
assert_eq!(a, b);
}
}
#[test]
fn fq2_field_tests() {
use ff::PrimeField;
crate::tests::field::random_field_tests::<Fq2>();
crate::tests::field::random_sqrt_tests::<Fq2>();
crate::tests::field::random_frobenius_tests::<Fq2, _>(super::fq::Fq::char(), 13);
}

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use super::fq::{FROBENIUS_COEFF_FQ6_C1, FROBENIUS_COEFF_FQ6_C2};
use super::fq2::Fq2;
use ff::Field;
use rand::{Rand, Rng};
/// An element of Fq6, represented by c0 + c1 * v + c2 * v^(2).
#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct Fq6 {
pub c0: Fq2,
pub c1: Fq2,
pub c2: Fq2,
}
impl ::std::fmt::Display for Fq6 {
fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
write!(f, "Fq6({} + {} * v, {} * v^2)", self.c0, self.c1, self.c2)
}
}
impl Rand for Fq6 {
fn rand<R: Rng>(rng: &mut R) -> Self {
Fq6 {
c0: rng.gen(),
c1: rng.gen(),
c2: rng.gen(),
}
}
}
// Here it's getting tough, because extension tower diverges with BLS12
// BLS12 (v^3 - ξ) where ξ = u + 1
// BN256 (v^3 - ξ) where ξ = u + 9
impl Fq6 {
/// Multiply by cubic nonresidue v.
pub fn mul_by_nonresidue(&mut self) {
use std::mem::swap;
swap(&mut self.c0, &mut self.c1);
swap(&mut self.c0, &mut self.c2);
// c0, c1, c2 -> c2, c0, c1
self.c0.mul_by_nonresidue();
}
/// Multiply by cubic nonresidue v.
pub fn mul_by_v(&mut self) {
use std::mem::swap;
swap(&mut self.c0, &mut self.c1);
swap(&mut self.c0, &mut self.c2);
self.c0.mul_by_xi();
}
pub fn mul_by_1(&mut self, c1: &Fq2) {
let mut b_b = self.c1;
b_b.mul_assign(c1);
let mut t1 = *c1;
{
let mut tmp = self.c1;
tmp.add_assign(&self.c2);
t1.mul_assign(&tmp);
t1.sub_assign(&b_b);
t1.mul_by_nonresidue();
}
let mut t2 = *c1;
{
let mut tmp = self.c0;
tmp.add_assign(&self.c1);
t2.mul_assign(&tmp);
t2.sub_assign(&b_b);
}
self.c0 = t1;
self.c1 = t2;
self.c2 = b_b;
}
pub fn mul_by_01(&mut self, c0: &Fq2, c1: &Fq2) {
let mut a_a = self.c0;
let mut b_b = self.c1;
a_a.mul_assign(c0);
b_b.mul_assign(c1);
let mut t1 = *c1;
{
let mut tmp = self.c1;
tmp.add_assign(&self.c2);
t1.mul_assign(&tmp);
t1.sub_assign(&b_b);
t1.mul_by_nonresidue();
t1.add_assign(&a_a);
}
let mut t3 = *c0;
{
let mut tmp = self.c0;
tmp.add_assign(&self.c2);
t3.mul_assign(&tmp);
t3.sub_assign(&a_a);
t3.add_assign(&b_b);
}
let mut t2 = *c0;
t2.add_assign(c1);
{
let mut tmp = self.c0;
tmp.add_assign(&self.c1);
t2.mul_assign(&tmp);
t2.sub_assign(&a_a);
t2.sub_assign(&b_b);
}
self.c0 = t1;
self.c1 = t2;
self.c2 = t3;
}
}
impl Field for Fq6 {
fn zero() -> Self {
Fq6 {
c0: Fq2::zero(),
c1: Fq2::zero(),
c2: Fq2::zero(),
}
}
fn one() -> Self {
Fq6 {
c0: Fq2::one(),
c1: Fq2::zero(),
c2: Fq2::zero(),
}
}
fn is_zero(&self) -> bool {
self.c0.is_zero() && self.c1.is_zero() && self.c2.is_zero()
}
fn double(&mut self) {
self.c0.double();
self.c1.double();
self.c2.double();
}
fn negate(&mut self) {
self.c0.negate();
self.c1.negate();
self.c2.negate();
}
fn add_assign(&mut self, other: &Self) {
self.c0.add_assign(&other.c0);
self.c1.add_assign(&other.c1);
self.c2.add_assign(&other.c2);
}
fn sub_assign(&mut self, other: &Self) {
self.c0.sub_assign(&other.c0);
self.c1.sub_assign(&other.c1);
self.c2.sub_assign(&other.c2);
}
fn frobenius_map(&mut self, power: usize) {
self.c0.frobenius_map(power);
self.c1.frobenius_map(power);
self.c2.frobenius_map(power);
self.c1.mul_assign(&FROBENIUS_COEFF_FQ6_C1[power % 6]);
self.c2.mul_assign(&FROBENIUS_COEFF_FQ6_C2[power % 6]);
}
fn square(&mut self) {
// s0 = a^2
let mut s0 = self.c0;
s0.square();
// s1 = 2ab
let mut ab = self.c0;
ab.mul_assign(&self.c1);
let mut s1 = ab;
s1.double();
// s2 = (a - b + c)^2
let mut s2 = self.c0;
s2.sub_assign(&self.c1);
s2.add_assign(&self.c2);
s2.square();
// bc
let mut bc = self.c1;
bc.mul_assign(&self.c2);
// s3 = 2bc
let mut s3 = bc;
s3.double();
// s4 = c^2
let mut s4 = self.c2;
s4.square();
// new c0 = 2bc.mul_by_xi + a^2
self.c0 = s3;
self.c0.mul_by_nonresidue();
// self.c0.mul_by_xi();
self.c0.add_assign(&s0);
// new c1 = (c^2).mul_by_xi + 2ab
self.c1 = s4;
self.c1.mul_by_nonresidue();
// self.c1.mul_by_xi();
self.c1.add_assign(&s1);
// new c2 = 2ab + (a - b + c)^2 + 2bc - a^2 - c^2 = b^2 + 2ac
self.c2 = s1;
self.c2.add_assign(&s2);
self.c2.add_assign(&s3);
self.c2.sub_assign(&s0);
self.c2.sub_assign(&s4);
}
fn mul_assign(&mut self, other: &Self) {
let mut a_a = self.c0;
let mut b_b = self.c1;
let mut c_c = self.c2;
a_a.mul_assign(&other.c0);
b_b.mul_assign(&other.c1);
c_c.mul_assign(&other.c2);
let mut t1 = other.c1;
t1.add_assign(&other.c2);
{
let mut tmp = self.c1;
tmp.add_assign(&self.c2);
t1.mul_assign(&tmp);
t1.sub_assign(&b_b);
t1.sub_assign(&c_c);
t1.mul_by_nonresidue();
t1.add_assign(&a_a);
}
let mut t3 = other.c0;
t3.add_assign(&other.c2);
{
let mut tmp = self.c0;
tmp.add_assign(&self.c2);
t3.mul_assign(&tmp);
t3.sub_assign(&a_a);
t3.add_assign(&b_b);
t3.sub_assign(&c_c);
}
let mut t2 = other.c0;
t2.add_assign(&other.c1);
{
let mut tmp = self.c0;
tmp.add_assign(&self.c1);
t2.mul_assign(&tmp);
t2.sub_assign(&a_a);
t2.sub_assign(&b_b);
c_c.mul_by_nonresidue();
t2.add_assign(&c_c);
}
self.c0 = t1;
self.c1 = t2;
self.c2 = t3;
}
fn inverse(&self) -> Option<Self> {
let mut c0 = self.c2;
c0.mul_by_nonresidue();
c0.mul_assign(&self.c1);
c0.negate();
{
let mut c0s = self.c0;
c0s.square();
c0.add_assign(&c0s);
}
let mut c1 = self.c2;
c1.square();
c1.mul_by_nonresidue();
{
let mut c01 = self.c0;
c01.mul_assign(&self.c1);
c1.sub_assign(&c01);
}
let mut c2 = self.c1;
c2.square();
{
let mut c02 = self.c0;
c02.mul_assign(&self.c2);
c2.sub_assign(&c02);
}
let mut tmp1 = self.c2;
tmp1.mul_assign(&c1);
let mut tmp2 = self.c1;
tmp2.mul_assign(&c2);
tmp1.add_assign(&tmp2);
tmp1.mul_by_nonresidue();
tmp2 = self.c0;
tmp2.mul_assign(&c0);
tmp1.add_assign(&tmp2);
match tmp1.inverse() {
Some(t) => {
let mut tmp = Fq6 {
c0: t,
c1: t,
c2: t,
};
tmp.c0.mul_assign(&c0);
tmp.c1.mul_assign(&c1);
tmp.c2.mul_assign(&c2);
Some(tmp)
}
None => None,
}
}
}
#[cfg(test)]
use rand::{SeedableRng, XorShiftRng};
#[test]
fn test_fq6_mul_nonresidue() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
let nqr = Fq6 {
c0: Fq2::zero(),
c1: Fq2::one(),
c2: Fq2::zero(),
};
for _ in 0..1000 {
let mut a = Fq6::rand(&mut rng);
let mut b = a;
a.mul_by_nonresidue();
b.mul_assign(&nqr);
assert_eq!(a, b);
}
}
#[test]
fn test_fq6_mul_by_1() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let c1 = Fq2::rand(&mut rng);
let mut a = Fq6::rand(&mut rng);
let mut b = a;
a.mul_by_1(&c1);
b.mul_assign(&Fq6 {
c0: Fq2::zero(),
c1: c1,
c2: Fq2::zero(),
});
assert_eq!(a, b);
}
}
#[test]
fn test_fq6_mul_by_01() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let c0 = Fq2::rand(&mut rng);
let c1 = Fq2::rand(&mut rng);
let mut a = Fq6::rand(&mut rng);
let mut b = a;
a.mul_by_01(&c0, &c1);
b.mul_assign(&Fq6 {
c0: c0,
c1: c1,
c2: Fq2::zero(),
});
assert_eq!(a, b);
}
}
#[test]
fn fq6_field_tests() {
use ff::PrimeField;
crate::tests::field::random_field_tests::<Fq6>();
crate::tests::field::random_frobenius_tests::<Fq6, _>(super::fq::Fq::char(), 13);
}

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use ff::{Field, PrimeField, PrimeFieldRepr};
#[derive(PrimeField)]
#[PrimeFieldModulus = "21888242871839275222246405745257275088548364400416034343698204186575808495617"]
#[PrimeFieldGenerator = "7"]
pub struct Fr(FrRepr);
#[test]
fn test_to_hex() {
assert_eq!(Fr::one().to_hex(), "0000000000000000000000000000000000000000000000000000000000000001");
}
#[test]
fn test_fr_from_hex() {
let fr = Fr::from_hex("0000000000000000000000000000000000000000000000000000000000000001").unwrap();
assert_eq!(fr, Fr::one());
let fr = Fr::from_hex("0x0000000000000000000000000000000000000000000000000000000000000001").unwrap();
assert_eq!(fr, Fr::one());
let fr = Fr::from_hex("0x01").unwrap();
assert_eq!(fr, Fr::one());
let fr = Fr::from_hex("0x00").unwrap();
assert_eq!(fr, Fr::zero());
let fr = Fr::from_hex("00").unwrap();
assert_eq!(fr, Fr::zero());
}
#[test]
fn test_roots_of_unity() {
assert_eq!(Fr::S, 28);
}
#[test]
fn test_default() {
assert_eq!(Fr::default(), Fr::zero());
}

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mod ec;
mod fq;
mod fq12;
mod fq2;
mod fq6;
mod fr;
// #[cfg(test)]
// mod tests;
pub use self::ec::{
G1, G1Affine, G1Compressed, G1Prepared, G1Uncompressed,
G2, G2Affine, G2Compressed, G2Prepared, G2Uncompressed,
};
pub use self::fq::{Fq, FqRepr, FROBENIUS_COEFF_FQ6_C1, XI_TO_Q_MINUS_1_OVER_2};
pub use self::fq12::Fq12;
pub use self::fq2::Fq2;
pub use self::fq6::Fq6;
pub use self::fr::{Fr, FrRepr};
use super::{CurveAffine, Engine};
use ff::{Field, ScalarEngine};
#[derive(Clone, Debug)]
pub struct Bn256;
// U value that originates this particular curve
pub const BN_U: u64 = 4965661367192848881;
// // 6U+2 for in NAF form
pub const SIX_U_PLUS_2_NAF : [i8; 65] = [
0, 0, 0, 1, 0, 1, 0, -1,
0, 0, 1, -1, 0, 0, 1, 0,
0, 1, 1, 0, -1, 0, 0, 1,
0, -1, 0, 0, 0, 0, 1, 1,
1, 0, 0, -1, 0, 0, 1, 0,
0, 0, 0, 0, -1, 0, 0, 1,
1, 0, 0, -1, 0, 0, 0, 1,
1, 0, -1, 0, 0, 1, 0, 1,
1];
impl ScalarEngine for Bn256 {
type Fr = Fr;
}
impl Engine for Bn256 {
type G1 = G1;
type G1Affine = G1Affine;
type G2 = G2;
type G2Affine = G2Affine;
type Fq = Fq;
type Fqe = Fq2;
type Fqk = Fq12;
fn miller_loop<'a, I>(i: I) -> Self::Fqk
where
I: IntoIterator<
Item = &'a (
&'a <Self::G1Affine as CurveAffine>::Prepared,
&'a <Self::G2Affine as CurveAffine>::Prepared,
),
>,
{
let mut pairs = vec![];
for &(p, q) in i {
if !p.is_zero() && !q.is_zero() {
pairs.push((p, q.coeffs.iter()));
}
}
// Final steps of the line function on prepared coefficients
fn ell(f: &mut Fq12, coeffs: &(Fq2, Fq2, Fq2), p: &G1Affine) {
let mut c0 = coeffs.0;
let mut c1 = coeffs.1;
c0.c0.mul_assign(&p.y);
c0.c1.mul_assign(&p.y);
c1.c0.mul_assign(&p.x);
c1.c1.mul_assign(&p.x);
// Sparse multiplication in Fq12
f.mul_by_034(&c0, &c1, &coeffs.2);
}
let mut f = Fq12::one();
for i in (1..SIX_U_PLUS_2_NAF.len()).rev() {
if i != SIX_U_PLUS_2_NAF.len() - 1 {
f.square();
}
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
let x = SIX_U_PLUS_2_NAF[i-1];
match x {
1 => {
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
}
-1 => {
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
}
_ => {
continue
}
}
}
// two additional steps: for q1 and minus q2
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
for &mut (p, ref mut coeffs) in &mut pairs {
ell(&mut f, coeffs.next().unwrap(), &p.0);
}
for &mut (_p, ref mut coeffs) in &mut pairs {
assert_eq!(coeffs.next(), None);
}
f
}
fn final_exponentiation(r: &Fq12) -> Option<Fq12> {
let mut f1 = *r;
f1.conjugate();
match r.inverse() {
Some(mut f2) => {
let mut r = f1;
r.mul_assign(&f2);
f2 = r;
r.frobenius_map(2);
r.mul_assign(&f2);
fn exp_by_x(f: &mut Fq12, x: u64) {
*f = f.pow(&[x]);
}
let x = BN_U;
let mut fp = r;
fp.frobenius_map(1);
let mut fp2 = r;
fp2.frobenius_map(2);
let mut fp3 = fp2;
fp3.frobenius_map(1);
let mut fu = r;
exp_by_x(&mut fu, x);
let mut fu2 = fu;
exp_by_x(&mut fu2, x);
let mut fu3 = fu2;
exp_by_x(&mut fu3, x);
let mut y3 = fu;
y3.frobenius_map(1);
let mut fu2p = fu2;
fu2p.frobenius_map(1);
let mut fu3p = fu3;
fu3p.frobenius_map(1);
let mut y2 = fu2;
y2.frobenius_map(2);
let mut y0 = fp;
y0.mul_assign(&fp2);
y0.mul_assign(&fp3);
let mut y1 = r;
y1.conjugate();
let mut y5 = fu2;
y5.conjugate();
y3.conjugate();
let mut y4 = fu;
y4.mul_assign(&fu2p);
y4.conjugate();
let mut y6 = fu3;
y6.mul_assign(&fu3p);
y6.conjugate();
y6.square();
y6.mul_assign(&y4);
y6.mul_assign(&y5);
let mut t1 = y3;
t1.mul_assign(&y5);
t1.mul_assign(&y6);
y6.mul_assign(&y2);
t1.square();
t1.mul_assign(&y6);
t1.square();
let mut t0 = t1;
t0.mul_assign(&y1);
t1.mul_assign(&y0);
t0.square();
t0.mul_assign(&t1);
Some(t0)
}
None => None,
}
}
}
impl G2Prepared {
pub fn is_zero(&self) -> bool {
self.infinity
}
pub fn from_affine(q: G2Affine) -> Self {
if q.is_zero() {
return G2Prepared {
coeffs: vec![],
infinity: true,
};
}
fn doubling_step(r: &mut G2) -> (Fq2, Fq2, Fq2) {
// Adaptation of Algorithm 26, https://eprint.iacr.org/2010/354.pdf
let mut tmp0 = r.x;
tmp0.square();
let mut tmp1 = r.y;
tmp1.square();
let mut tmp2 = tmp1;
tmp2.square();
let mut tmp3 = tmp1;
tmp3.add_assign(&r.x);
tmp3.square();
tmp3.sub_assign(&tmp0);
tmp3.sub_assign(&tmp2);
tmp3.double();
let mut tmp4 = tmp0;
tmp4.double();
tmp4.add_assign(&tmp0);
let mut tmp6 = r.x;
tmp6.add_assign(&tmp4);
let mut tmp5 = tmp4;
tmp5.square();
let mut zsquared = r.z;
zsquared.square();
r.x = tmp5;
r.x.sub_assign(&tmp3);
r.x.sub_assign(&tmp3);
r.z.add_assign(&r.y);
r.z.square();
r.z.sub_assign(&tmp1);
r.z.sub_assign(&zsquared);
r.y = tmp3;
r.y.sub_assign(&r.x);
r.y.mul_assign(&tmp4);
tmp2.double();
tmp2.double();
tmp2.double();
r.y.sub_assign(&tmp2);
// up to here everything was by algorith, line 11
// use R instead of new T
// tmp3 is the first part of line 12
tmp3 = tmp4;
tmp3.mul_assign(&zsquared);
tmp3.double();
tmp3.negate();
// tmp6 is from line 14
tmp6.square();
tmp6.sub_assign(&tmp0);
tmp6.sub_assign(&tmp5);
tmp1.double();
tmp1.double();
tmp6.sub_assign(&tmp1);
// tmp0 is the first part of line 16
tmp0 = r.z;
tmp0.mul_assign(&zsquared);
tmp0.double();
(tmp0, tmp3, tmp6)
}
fn addition_step(r: &mut G2, q: &G2Affine) -> (Fq2, Fq2, Fq2) {
// Adaptation of Algorithm 27, https://eprint.iacr.org/2010/354.pdf
let mut zsquared = r.z;
zsquared.square();
let mut ysquared = q.y;
ysquared.square();
// t0 corresponds to line 1
let mut t0 = zsquared;
t0.mul_assign(&q.x);
// t1 corresponds to lines 2 and 3
let mut t1 = q.y;
t1.add_assign(&r.z);
t1.square();
t1.sub_assign(&ysquared);
t1.sub_assign(&zsquared);
t1.mul_assign(&zsquared);
// t2 corresponds to line 4
let mut t2 = t0;
t2.sub_assign(&r.x);
// t3 corresponds to line 5
let mut t3 = t2;
t3.square();
// t4 corresponds to line 6
let mut t4 = t3;
t4.double();
t4.double();
// t5 corresponds to line 7
let mut t5 = t4;
t5.mul_assign(&t2);
// t6 corresponds to line 8
let mut t6 = t1;
t6.sub_assign(&r.y);
t6.sub_assign(&r.y);
// t9 corresponds to line 9
let mut t9 = t6;
t9.mul_assign(&q.x);
// corresponds to line 10
let mut t7 = t4;
t7.mul_assign(&r.x);
// corresponds to line 11, but assigns to r.x instead of T.x
r.x = t6;
r.x.square();
r.x.sub_assign(&t5);
r.x.sub_assign(&t7);
r.x.sub_assign(&t7);
// corresponds to line 12, but assigns to r.z instead of T.z
r.z.add_assign(&t2);
r.z.square();
r.z.sub_assign(&zsquared);
r.z.sub_assign(&t3);
// corresponds to line 13
let mut t10 = q.y;
t10.add_assign(&r.z);
// corresponds to line 14
let mut t8 = t7;
t8.sub_assign(&r.x);
t8.mul_assign(&t6);
// corresponds to line 15
t0 = r.y;
t0.mul_assign(&t5);
t0.double();
// corresponds to line 12, but assigns to r.y instead of T.y
r.y = t8;
r.y.sub_assign(&t0);
// corresponds to line 17
t10.square();
t10.sub_assign(&ysquared);
let mut ztsquared = r.z;
ztsquared.square();
t10.sub_assign(&ztsquared);
// corresponds to line 18
t9.double();
t9.sub_assign(&t10);
// t10 = 2*Zt from Algo 27, line 19
t10 = r.z;
t10.double();
// t1 = first multiplicator of line 21
t6.negate();
t1 = t6;
t1.double();
// t9 corresponds to t9 from Algo 27
(t10, t1, t9)
}
let mut coeffs = vec![];
let mut r: G2 = q.into();
let mut negq = q;
negq.negate();
for i in (1..SIX_U_PLUS_2_NAF.len()).rev() {
coeffs.push(doubling_step(& mut r));
let x = SIX_U_PLUS_2_NAF[i-1];
match x {
1 => {
coeffs.push(addition_step(&mut r, &q));
}
-1 => {
coeffs.push(addition_step(&mut r, &negq));
}
_ => continue,
}
}
let mut q1 = q;
q1.x.c1.negate();
q1.x.mul_assign(&FROBENIUS_COEFF_FQ6_C1[1]);
q1.y.c1.negate();
q1.y.mul_assign(&XI_TO_Q_MINUS_1_OVER_2);
coeffs.push(addition_step(&mut r, &q1));
let mut minusq2 = q;
minusq2.x.mul_assign(&FROBENIUS_COEFF_FQ6_C1[2]);
coeffs.push(addition_step(&mut r, &minusq2));
G2Prepared {
coeffs,
infinity: false,
}
}
}
#[cfg(test)]
use rand::{Rand, SeedableRng, XorShiftRng};
#[test]
fn test_pairing() {
use crate::{CurveProjective};
let mut g1 = G1::one();
let mut g2 = G2::one();
g2.double();
let pair12 = Bn256::pairing(g1, g2);
g1 = G1::one();
g1.double();
g2 = G2::one();
let pair21 = Bn256::pairing(g1, g2);
assert_eq!(pair12, pair21);
// print!("GT = {}\n", pair12);
g1 = G1::one();
g1.double();
g1.double();
let pair41 = Bn256::pairing(g1, g2);
g1 = G1::one();
g1.double();
g2.double();
let pair22 = Bn256::pairing(g1, g2);
assert_eq!(pair41, pair22);
g1 = G1::one();
g1.double();
g1.add_assign(&G1::one());
g2 = G2::one();
g2.double();
let pair32 = Bn256::pairing(g1, g2);
g2 = G2::one();
g2.double();
g2.add_assign(&G2::one());
g1 = G1::one();
g1.double();
let pair23 = Bn256::pairing(g1, g2);
assert_eq!(pair23, pair32);
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let a = Fr::rand(&mut rng);
let b = Fr::rand(&mut rng);
let mut g1 = G1::one();
g1.mul_assign(a);
let mut g2 = G2::one();
g1.mul_assign(b);
let pair_ab = Bn256::pairing(g1, g2);
g1 = G1::one();
g1.mul_assign(b);
g2 = G2::one();
g1.mul_assign(a);
let pair_ba = Bn256::pairing(g1, g2);
assert_eq!(pair_ab, pair_ba);
}
}
#[test]
fn random_bilinearity_tests() {
use crate::{CurveProjective};
use ff::PrimeField;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let mut a = G1::one();
let ka = Fr::rand(&mut rng);
a.mul_assign(ka);
let mut b = G2::one();
let kb = Fr::rand(&mut rng);
b.mul_assign(kb);
let c = Fr::rand(&mut rng);
let d = Fr::rand(&mut rng);
let mut ac = a;
ac.mul_assign(c);
let mut ad = a;
ad.mul_assign(d);
let mut bc = b;
bc.mul_assign(c);
let mut bd = b;
bd.mul_assign(d);
let acbd = Bn256::pairing(ac, bd);
let adbc = Bn256::pairing(ad, bc);
let mut cd = c;
cd.mul_assign(&d);
let abcd = Bn256::pairing(a, b).pow(cd.into_repr());
assert_eq!(acbd, adbc);
assert_eq!(acbd, abcd);
}
}
#[test]
fn bn256_engine_tests() {
crate::tests::engine::engine_tests::<Bn256>();
}

316
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// `clippy` is a code linting tool for improving code quality by catching
// common mistakes or strange code patterns. If the `cargo-clippy` feature
// is provided, all compiler warnings are prohibited.
#![cfg_attr(feature = "cargo-clippy", deny(warnings))]
// #![cfg_attr(feature = "cargo-clippy", allow(inline_always))]
// #![cfg_attr(feature = "cargo-clippy", allow(too_many_arguments))]
// #![cfg_attr(feature = "cargo-clippy", allow(unreadable_literal))]
// #![cfg_attr(feature = "cargo-clippy", allow(many_single_char_names))]
// #![cfg_attr(feature = "cargo-clippy", allow(new_without_default_derive))]
// #![cfg_attr(feature = "cargo-clippy", allow(write_literal))]
// Force public structures to implement Debug
#![deny(missing_debug_implementations)]
extern crate byteorder;
extern crate rand;
#[cfg(test)]
pub mod tests;
pub extern crate ff;
pub use ff::*;
pub mod bls12_381;
pub mod bn256;
mod wnaf;
pub use self::wnaf::Wnaf;
use ff::{Field, PrimeField, PrimeFieldDecodingError, PrimeFieldRepr, ScalarEngine, SqrtField};
use std::error::Error;
use std::fmt;
/// An "engine" is a collection of types (fields, elliptic curve groups, etc.)
/// with well-defined relationships. In particular, the G1/G2 curve groups are
/// of prime order `r`, and are equipped with a bilinear pairing function.
pub trait Engine: ScalarEngine {
/// The projective representation of an element in G1.
type G1: CurveProjective<
Engine = Self,
Base = Self::Fq,
Scalar = Self::Fr,
Affine = Self::G1Affine,
>
+ From<Self::G1Affine>;
/// The affine representation of an element in G1.
type G1Affine: CurveAffine<
Engine = Self,
Base = Self::Fq,
Scalar = Self::Fr,
Projective = Self::G1,
Pair = Self::G2Affine,
PairingResult = Self::Fqk,
>
+ From<Self::G1> + RawEncodable;
/// The projective representation of an element in G2.
type G2: CurveProjective<
Engine = Self,
Base = Self::Fqe,
Scalar = Self::Fr,
Affine = Self::G2Affine,
>
+ From<Self::G2Affine>;
/// The affine representation of an element in G2.
type G2Affine: CurveAffine<
Engine = Self,
Base = Self::Fqe,
Scalar = Self::Fr,
Projective = Self::G2,
Pair = Self::G1Affine,
PairingResult = Self::Fqk,
>
+ From<Self::G2>;
/// The base field that hosts G1.
type Fq: PrimeField + SqrtField;
/// The extension field that hosts G2.
type Fqe: SqrtField;
/// The extension field that hosts the target group of the pairing.
type Fqk: Field;
/// Perform a miller loop with some number of (G1, G2) pairs.
fn miller_loop<'a, I>(i: I) -> Self::Fqk
where
I: IntoIterator<
Item = &'a (
&'a <Self::G1Affine as CurveAffine>::Prepared,
&'a <Self::G2Affine as CurveAffine>::Prepared,
),
>;
/// Perform final exponentiation of the result of a miller loop.
fn final_exponentiation(r: &Self::Fqk) -> Option<Self::Fqk>;
/// Performs a complete pairing operation `(p, q)`.
fn pairing<G1, G2>(p: G1, q: G2) -> Self::Fqk
where
G1: Into<Self::G1Affine>,
G2: Into<Self::G2Affine>,
{
Self::final_exponentiation(&Self::miller_loop(
[(&(p.into().prepare()), &(q.into().prepare()))].into_iter(),
)).unwrap()
}
}
/// Projective representation of an elliptic curve point guaranteed to be
/// in the correct prime order subgroup.
pub trait CurveProjective:
PartialEq
+ Eq
+ Sized
+ Copy
+ Clone
+ Send
+ Sync
+ fmt::Debug
+ fmt::Display
+ rand::Rand
+ 'static
{
type Engine: Engine<Fr = Self::Scalar>;
type Scalar: PrimeField + SqrtField;
type Base: SqrtField;
type Affine: CurveAffine<Projective = Self, Scalar = Self::Scalar>;
/// Returns the additive identity.
fn zero() -> Self;
/// Returns a fixed generator of unknown exponent.
fn one() -> Self;
/// Determines if this point is the point at infinity.
fn is_zero(&self) -> bool;
/// Normalizes a slice of projective elements so that
/// conversion to affine is cheap.
fn batch_normalization(v: &mut [Self]);
/// Checks if the point is already "normalized" so that
/// cheap affine conversion is possible.
fn is_normalized(&self) -> bool;
/// Doubles this element.
fn double(&mut self);
/// Adds another element to this element.
fn add_assign(&mut self, other: &Self);
/// Subtracts another element from this element.
fn sub_assign(&mut self, other: &Self) {
let mut tmp = *other;
tmp.negate();
self.add_assign(&tmp);
}
/// Adds an affine element to this element.
fn add_assign_mixed(&mut self, other: &Self::Affine);
/// Negates this element.
fn negate(&mut self);
/// Performs scalar multiplication of this element.
fn mul_assign<S: Into<<Self::Scalar as PrimeField>::Repr>>(&mut self, other: S);
/// Converts this element into its affine representation.
fn into_affine(&self) -> Self::Affine;
/// Recommends a wNAF window table size given a scalar. Always returns a number
/// between 2 and 22, inclusive.
fn recommended_wnaf_for_scalar(scalar: <Self::Scalar as PrimeField>::Repr) -> usize;
/// Recommends a wNAF window size given the number of scalars you intend to multiply
/// a base by. Always returns a number between 2 and 22, inclusive.
fn recommended_wnaf_for_num_scalars(num_scalars: usize) -> usize;
}
/// Affine representation of an elliptic curve point guaranteed to be
/// in the correct prime order subgroup.
pub trait CurveAffine:
Copy + Clone + Sized + Send + Sync + fmt::Debug + fmt::Display + PartialEq + Eq + 'static
{
type Engine: Engine<Fr = Self::Scalar>;
type Scalar: PrimeField + SqrtField;
type Base: SqrtField;
type Projective: CurveProjective<Affine = Self, Scalar = Self::Scalar>;
type Prepared: Clone + Send + Sync + 'static;
type Uncompressed: EncodedPoint<Affine = Self>;
type Compressed: EncodedPoint<Affine = Self>;
type Pair: CurveAffine<Pair = Self>;
type PairingResult: Field;
/// Returns the additive identity.
fn zero() -> Self;
/// Returns a fixed generator of unknown exponent.
fn one() -> Self;
/// Determines if this point represents the point at infinity; the
/// additive identity.
fn is_zero(&self) -> bool;
/// Negates this element.
fn negate(&mut self);
/// Performs scalar multiplication of this element with mixed addition.
fn mul<S: Into<<Self::Scalar as PrimeField>::Repr>>(&self, other: S) -> Self::Projective;
/// Prepares this element for pairing purposes.
fn prepare(&self) -> Self::Prepared;
/// Perform a pairing
fn pairing_with(&self, other: &Self::Pair) -> Self::PairingResult;
/// Converts this element into its affine representation.
fn into_projective(&self) -> Self::Projective;
/// Converts this element into its compressed encoding, so long as it's not
/// the point at infinity.
fn into_compressed(&self) -> Self::Compressed {
<Self::Compressed as EncodedPoint>::from_affine(*self)
}
/// Converts this element into its uncompressed encoding, so long as it's not
/// the point at infinity.
fn into_uncompressed(&self) -> Self::Uncompressed {
<Self::Uncompressed as EncodedPoint>::from_affine(*self)
}
}
pub trait RawEncodable: CurveAffine {
/// Converts this element into its uncompressed encoding, so long as it's not
/// the point at infinity. Leaves coordinates in Montgommery form
fn into_raw_uncompressed_le(&self) -> Self::Uncompressed;
/// Creates a point from raw encoded coordinates without checking on curve
fn from_raw_uncompressed_le_unchecked(encoded: &Self::Uncompressed, infinity: bool) -> Result<Self, GroupDecodingError>;
/// Creates a point from raw encoded coordinates
fn from_raw_uncompressed_le(encoded: &Self::Uncompressed, infinity: bool) -> Result<Self, GroupDecodingError>;
}
/// An encoded elliptic curve point, which should essentially wrap a `[u8; N]`.
pub trait EncodedPoint:
Sized + Send + Sync + AsRef<[u8]> + AsMut<[u8]> + Clone + Copy + 'static
{
type Affine: CurveAffine;
/// Creates an empty representation.
fn empty() -> Self;
/// Returns the number of bytes consumed by this representation.
fn size() -> usize;
/// Converts an `EncodedPoint` into a `CurveAffine` element,
/// if the encoding represents a valid element.
fn into_affine(&self) -> Result<Self::Affine, GroupDecodingError>;
/// Converts an `EncodedPoint` into a `CurveAffine` element,
/// without guaranteeing that the encoding represents a valid
/// element. This is useful when the caller knows the encoding is
/// valid already.
///
/// If the encoding is invalid, this can break API invariants,
/// so caution is strongly encouraged.
fn into_affine_unchecked(&self) -> Result<Self::Affine, GroupDecodingError>;
/// Creates an `EncodedPoint` from an affine point, as long as the
/// point is not the point at infinity.
fn from_affine(affine: Self::Affine) -> Self;
}
/// An error that may occur when trying to decode an `EncodedPoint`.
#[derive(Debug)]
pub enum GroupDecodingError {
/// The coordinate(s) do not lie on the curve.
NotOnCurve,
/// The element is not part of the r-order subgroup.
NotInSubgroup,
/// One of the coordinates could not be decoded
CoordinateDecodingError(&'static str, PrimeFieldDecodingError),
/// The compression mode of the encoded element was not as expected
UnexpectedCompressionMode,
/// The encoding contained bits that should not have been set
UnexpectedInformation,
}
impl Error for GroupDecodingError {
fn description(&self) -> &str {
match *self {
GroupDecodingError::NotOnCurve => "coordinate(s) do not lie on the curve",
GroupDecodingError::NotInSubgroup => "the element is not part of an r-order subgroup",
GroupDecodingError::CoordinateDecodingError(..) => "coordinate(s) could not be decoded",
GroupDecodingError::UnexpectedCompressionMode => {
"encoding has unexpected compression mode"
}
GroupDecodingError::UnexpectedInformation => "encoding has unexpected information",
}
}
}
impl fmt::Display for GroupDecodingError {
fn fmt(&self, f: &mut fmt::Formatter) -> Result<(), fmt::Error> {
match *self {
GroupDecodingError::CoordinateDecodingError(description, ref err) => {
write!(f, "{} decoding error: {}", description, err)
}
_ => write!(f, "{}", self.description()),
}
}
}

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use ff::Field;
use rand::{Rand, Rng, SeedableRng, XorShiftRng};
use crate::{CurveAffine, CurveProjective, EncodedPoint};
pub fn curve_tests<G: CurveProjective>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
// Negation edge case with zero.
{
let mut z = G::zero();
z.negate();
assert!(z.is_zero());
}
// Doubling edge case with zero.
{
let mut z = G::zero();
z.double();
assert!(z.is_zero());
}
// Addition edge cases with zero
{
let mut r = G::rand(&mut rng);
let rcopy = r;
r.add_assign(&G::zero());
assert_eq!(r, rcopy);
r.add_assign_mixed(&G::Affine::zero());
assert_eq!(r, rcopy);
let mut z = G::zero();
z.add_assign(&G::zero());
assert!(z.is_zero());
z.add_assign_mixed(&G::Affine::zero());
assert!(z.is_zero());
let mut z2 = z;
z2.add_assign(&r);
z.add_assign_mixed(&r.into_affine());
assert_eq!(z, z2);
assert_eq!(z, r);
}
// Transformations
{
let a = G::rand(&mut rng);
let b = a.into_affine().into_projective();
let c = a
.into_affine()
.into_projective()
.into_affine()
.into_projective();
assert_eq!(a, b);
assert_eq!(b, c);
}
random_addition_tests::<G>();
random_multiplication_tests::<G>();
random_doubling_tests::<G>();
random_negation_tests::<G>();
random_wnaf_tests::<G>();
random_encoding_tests::<G::Affine>();
}
fn random_wnaf_tests<G: CurveProjective>() {
use ff::PrimeField;
use crate::wnaf::*;
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
{
let mut table = vec![];
let mut wnaf = vec![];
for w in 2..14 {
for _ in 0..100 {
let g = G::rand(&mut rng);
let s = G::Scalar::rand(&mut rng).into_repr();
let mut g1 = g;
g1.mul_assign(s);
wnaf_table(&mut table, g, w);
wnaf_form(&mut wnaf, s, w);
let g2 = wnaf_exp(&table, &wnaf);
assert_eq!(g1, g2);
}
}
}
{
fn only_compiles_if_send<S: Send>(_: &S) {}
for _ in 0..100 {
let g = G::rand(&mut rng);
let s = G::Scalar::rand(&mut rng).into_repr();
let mut g1 = g;
g1.mul_assign(s);
let g2 = {
let mut wnaf = Wnaf::new();
wnaf.base(g, 1).scalar(s)
};
let g3 = {
let mut wnaf = Wnaf::new();
wnaf.scalar(s).base(g)
};
let g4 = {
let mut wnaf = Wnaf::new();
let mut shared = wnaf.base(g, 1).shared();
only_compiles_if_send(&shared);
shared.scalar(s)
};
let g5 = {
let mut wnaf = Wnaf::new();
let mut shared = wnaf.scalar(s).shared();
only_compiles_if_send(&shared);
shared.base(g)
};
let g6 = {
let mut wnaf = Wnaf::new();
{
// Populate the vectors.
wnaf.base(rng.gen(), 1).scalar(rng.gen());
}
wnaf.base(g, 1).scalar(s)
};
let g7 = {
let mut wnaf = Wnaf::new();
{
// Populate the vectors.
wnaf.base(rng.gen(), 1).scalar(rng.gen());
}
wnaf.scalar(s).base(g)
};
let g8 = {
let mut wnaf = Wnaf::new();
{
// Populate the vectors.
wnaf.base(rng.gen(), 1).scalar(rng.gen());
}
let mut shared = wnaf.base(g, 1).shared();
only_compiles_if_send(&shared);
shared.scalar(s)
};
let g9 = {
let mut wnaf = Wnaf::new();
{
// Populate the vectors.
wnaf.base(rng.gen(), 1).scalar(rng.gen());
}
let mut shared = wnaf.scalar(s).shared();
only_compiles_if_send(&shared);
shared.base(g)
};
assert_eq!(g1, g2);
assert_eq!(g1, g3);
assert_eq!(g1, g4);
assert_eq!(g1, g5);
assert_eq!(g1, g6);
assert_eq!(g1, g7);
assert_eq!(g1, g8);
assert_eq!(g1, g9);
}
}
}
fn random_negation_tests<G: CurveProjective>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let r = G::rand(&mut rng);
let s = G::Scalar::rand(&mut rng);
let mut sneg = s;
sneg.negate();
let mut t1 = r;
t1.mul_assign(s);
let mut t2 = r;
t2.mul_assign(sneg);
let mut t3 = t1;
t3.add_assign(&t2);
assert!(t3.is_zero());
let mut t4 = t1;
t4.add_assign_mixed(&t2.into_affine());
assert!(t4.is_zero());
t1.negate();
assert_eq!(t1, t2);
}
}
fn random_doubling_tests<G: CurveProjective>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let mut a = G::rand(&mut rng);
let mut b = G::rand(&mut rng);
// 2(a + b)
let mut tmp1 = a;
tmp1.add_assign(&b);
tmp1.double();
// 2a + 2b
a.double();
b.double();
let mut tmp2 = a;
tmp2.add_assign(&b);
let mut tmp3 = a;
tmp3.add_assign_mixed(&b.into_affine());
assert_eq!(tmp1, tmp2);
assert_eq!(tmp1, tmp3);
}
}
fn random_multiplication_tests<G: CurveProjective>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let mut a = G::rand(&mut rng);
let mut b = G::rand(&mut rng);
let a_affine = a.into_affine();
let b_affine = b.into_affine();
let s = G::Scalar::rand(&mut rng);
// s ( a + b )
let mut tmp1 = a;
tmp1.add_assign(&b);
tmp1.mul_assign(s);
// sa + sb
a.mul_assign(s);
b.mul_assign(s);
let mut tmp2 = a;
tmp2.add_assign(&b);
// Affine multiplication
let mut tmp3 = a_affine.mul(s);
tmp3.add_assign(&b_affine.mul(s));
assert_eq!(tmp1, tmp2);
assert_eq!(tmp1, tmp3);
}
}
fn random_addition_tests<G: CurveProjective>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let a = G::rand(&mut rng);
let b = G::rand(&mut rng);
let c = G::rand(&mut rng);
let a_affine = a.into_affine();
let b_affine = b.into_affine();
let c_affine = c.into_affine();
// a + a should equal the doubling
{
let mut aplusa = a;
aplusa.add_assign(&a);
let mut aplusamixed = a;
aplusamixed.add_assign_mixed(&a.into_affine());
let mut adouble = a;
adouble.double();
assert_eq!(aplusa, adouble);
assert_eq!(aplusa, aplusamixed);
}
let mut tmp = vec![G::zero(); 6];
// (a + b) + c
tmp[0] = a;
tmp[0].add_assign(&b);
tmp[0].add_assign(&c);
// a + (b + c)
tmp[1] = b;
tmp[1].add_assign(&c);
tmp[1].add_assign(&a);
// (a + c) + b
tmp[2] = a;
tmp[2].add_assign(&c);
tmp[2].add_assign(&b);
// Mixed addition
// (a + b) + c
tmp[3] = a_affine.into_projective();
tmp[3].add_assign_mixed(&b_affine);
tmp[3].add_assign_mixed(&c_affine);
// a + (b + c)
tmp[4] = b_affine.into_projective();
tmp[4].add_assign_mixed(&c_affine);
tmp[4].add_assign_mixed(&a_affine);
// (a + c) + b
tmp[5] = a_affine.into_projective();
tmp[5].add_assign_mixed(&c_affine);
tmp[5].add_assign_mixed(&b_affine);
// Comparisons
for i in 0..6 {
for j in 0..6 {
assert_eq!(tmp[i], tmp[j]);
assert_eq!(tmp[i].into_affine(), tmp[j].into_affine());
}
assert!(tmp[i] != a);
assert!(tmp[i] != b);
assert!(tmp[i] != c);
assert!(a != tmp[i]);
assert!(b != tmp[i]);
assert!(c != tmp[i]);
}
}
}
pub fn random_transformation_tests<G: CurveProjective>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let g = G::rand(&mut rng);
let g_affine = g.into_affine();
let g_projective = g_affine.into_projective();
assert_eq!(g, g_projective);
}
// Batch normalization
for _ in 0..10 {
let mut v = (0..1000).map(|_| G::rand(&mut rng)).collect::<Vec<_>>();
use rand::distributions::{IndependentSample, Range};
let between = Range::new(0, 1000);
// Sprinkle in some normalized points
for _ in 0..5 {
v[between.ind_sample(&mut rng)] = G::zero();
}
for _ in 0..5 {
let s = between.ind_sample(&mut rng);
v[s] = v[s].into_affine().into_projective();
}
let expected_v = v
.iter()
.map(|v| v.into_affine().into_projective())
.collect::<Vec<_>>();
G::batch_normalization(&mut v);
for i in &v {
assert!(i.is_normalized());
}
assert_eq!(v, expected_v);
}
}
pub fn random_transformation_tests_with_cofactor<G: CurveProjective>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let g = G::rand(&mut rng);
let g_affine = g.into_affine();
let g_projective = g_affine.into_projective();
assert_eq!(g, g_projective);
}
// Batch normalization
for _ in 0..10 {
let mut v = (0..1000).map(|_| G::rand(&mut rng)).collect::<Vec<_>>();
for i in &v {
assert!(!i.is_normalized());
}
use rand::distributions::{IndependentSample, Range};
let between = Range::new(0, 1000);
// Sprinkle in some normalized points
for _ in 0..5 {
v[between.ind_sample(&mut rng)] = G::zero();
}
for _ in 0..5 {
let s = between.ind_sample(&mut rng);
v[s] = v[s].into_affine().into_projective();
}
let expected_v = v
.iter()
.map(|v| v.into_affine().into_projective())
.collect::<Vec<_>>();
G::batch_normalization(&mut v);
for i in &v {
assert!(i.is_normalized());
}
assert_eq!(v, expected_v);
}
}
fn random_encoding_tests<G: CurveAffine>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
assert_eq!(
G::zero().into_uncompressed().into_affine().unwrap(),
G::zero()
);
assert_eq!(
G::zero().into_compressed().into_affine().unwrap(),
G::zero()
);
for _ in 0..1000 {
let mut r = G::Projective::rand(&mut rng).into_affine();
let uncompressed = r.into_uncompressed();
let de_uncompressed = uncompressed.into_affine().unwrap();
assert_eq!(de_uncompressed, r);
let compressed = r.into_compressed();
let de_compressed = compressed.into_affine().unwrap();
assert_eq!(de_compressed, r);
r.negate();
let compressed = r.into_compressed();
let de_compressed = compressed.into_affine().unwrap();
assert_eq!(de_compressed, r);
}
}

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use rand::{Rand, SeedableRng, XorShiftRng};
use crate::{CurveAffine, CurveProjective, Engine, Field, PrimeField};
pub fn engine_tests<E: Engine>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..10 {
let a = E::G1::rand(&mut rng).into_affine();
let b = E::G2::rand(&mut rng).into_affine();
assert!(a.pairing_with(&b) == b.pairing_with(&a));
assert!(a.pairing_with(&b) == E::pairing(a, b));
}
for _ in 0..1000 {
let z1 = E::G1Affine::zero().prepare();
let z2 = E::G2Affine::zero().prepare();
let a = E::G1::rand(&mut rng).into_affine().prepare();
let b = E::G2::rand(&mut rng).into_affine().prepare();
let c = E::G1::rand(&mut rng).into_affine().prepare();
let d = E::G2::rand(&mut rng).into_affine().prepare();
assert_eq!(
E::Fqk::one(),
E::final_exponentiation(&E::miller_loop(&[(&z1, &b)])).unwrap()
);
assert_eq!(
E::Fqk::one(),
E::final_exponentiation(&E::miller_loop(&[(&a, &z2)])).unwrap()
);
assert_eq!(
E::final_exponentiation(&E::miller_loop(&[(&z1, &b), (&c, &d)])).unwrap(),
E::final_exponentiation(&E::miller_loop(&[(&a, &z2), (&c, &d)])).unwrap()
);
assert_eq!(
E::final_exponentiation(&E::miller_loop(&[(&a, &b), (&z1, &d)])).unwrap(),
E::final_exponentiation(&E::miller_loop(&[(&a, &b), (&c, &z2)])).unwrap()
);
}
random_bilinearity_tests::<E>();
random_miller_loop_tests::<E>();
}
fn random_miller_loop_tests<E: Engine>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
// Exercise the miller loop for a reduced pairing
for _ in 0..1000 {
let a = E::G1::rand(&mut rng);
let b = E::G2::rand(&mut rng);
let p2 = E::pairing(a, b);
let a = a.into_affine().prepare();
let b = b.into_affine().prepare();
let p1 = E::final_exponentiation(&E::miller_loop(&[(&a, &b)])).unwrap();
assert_eq!(p1, p2);
}
// Exercise a double miller loop
for _ in 0..1000 {
let a = E::G1::rand(&mut rng);
let b = E::G2::rand(&mut rng);
let c = E::G1::rand(&mut rng);
let d = E::G2::rand(&mut rng);
let ab = E::pairing(a, b);
let cd = E::pairing(c, d);
let mut abcd = ab;
abcd.mul_assign(&cd);
let a = a.into_affine().prepare();
let b = b.into_affine().prepare();
let c = c.into_affine().prepare();
let d = d.into_affine().prepare();
let abcd_with_double_loop =
E::final_exponentiation(&E::miller_loop(&[(&a, &b), (&c, &d)])).unwrap();
assert_eq!(abcd, abcd_with_double_loop);
}
}
fn random_bilinearity_tests<E: Engine>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let a = E::G1::rand(&mut rng);
let b = E::G2::rand(&mut rng);
let c = E::Fr::rand(&mut rng);
let d = E::Fr::rand(&mut rng);
let mut ac = a;
ac.mul_assign(c);
let mut ad = a;
ad.mul_assign(d);
let mut bc = b;
bc.mul_assign(c);
let mut bd = b;
bd.mul_assign(d);
let acbd = E::pairing(ac, bd);
let adbc = E::pairing(ad, bc);
let mut cd = c;
cd.mul_assign(&d);
let abcd = E::pairing(a, b).pow(cd.into_repr());
assert_eq!(acbd, adbc);
assert_eq!(acbd, abcd);
}
}

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use ff::{Field, LegendreSymbol, PrimeField, SqrtField};
use rand::{Rng, SeedableRng, XorShiftRng};
pub fn random_frobenius_tests<F: Field, C: AsRef<[u64]>>(characteristic: C, maxpower: usize) {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..100 {
for i in 0..(maxpower + 1) {
let mut a = F::rand(&mut rng);
let mut b = a;
for _ in 0..i {
a = a.pow(&characteristic);
}
b.frobenius_map(i);
assert_eq!(a, b);
}
}
}
pub fn random_sqrt_tests<F: SqrtField>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..10000 {
let a = F::rand(&mut rng);
let mut b = a;
b.square();
assert_eq!(b.legendre(), LegendreSymbol::QuadraticResidue);
let b = b.sqrt().unwrap();
let mut negb = b;
negb.negate();
assert!(a == b || a == negb);
}
let mut c = F::one();
for _ in 0..10000 {
let mut b = c;
b.square();
assert_eq!(b.legendre(), LegendreSymbol::QuadraticResidue);
b = b.sqrt().unwrap();
if b != c {
b.negate();
}
assert_eq!(b, c);
c.add_assign(&F::one());
}
}
pub fn random_field_tests<F: Field>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
random_multiplication_tests::<F, _>(&mut rng);
random_addition_tests::<F, _>(&mut rng);
random_subtraction_tests::<F, _>(&mut rng);
random_negation_tests::<F, _>(&mut rng);
random_doubling_tests::<F, _>(&mut rng);
random_squaring_tests::<F, _>(&mut rng);
random_inversion_tests::<F, _>(&mut rng);
random_expansion_tests::<F, _>(&mut rng);
assert!(F::zero().is_zero());
{
let mut z = F::zero();
z.negate();
assert!(z.is_zero());
}
assert!(F::zero().inverse().is_none());
// Multiplication by zero
{
let mut a = F::rand(&mut rng);
a.mul_assign(&F::zero());
assert!(a.is_zero());
}
// Addition by zero
{
let mut a = F::rand(&mut rng);
let copy = a;
a.add_assign(&F::zero());
assert_eq!(a, copy);
}
}
pub fn from_str_tests<F: PrimeField>() {
{
let a = "84395729384759238745923745892374598234705297301958723458712394587103249587213984572934750213947582345792304758273458972349582734958273495872304598234";
let b = "38495729084572938457298347502349857029384609283450692834058293405982304598230458230495820394850293845098234059823049582309485203948502938452093482039";
let c = "3248875134290623212325429203829831876024364170316860259933542844758450336418538569901990710701240661702808867062612075657861768196242274635305077449545396068598317421057721935408562373834079015873933065667961469731886739181625866970316226171512545167081793907058686908697431878454091011239990119126";
let mut a = F::from_str(a).unwrap();
let b = F::from_str(b).unwrap();
let c = F::from_str(c).unwrap();
a.mul_assign(&b);
assert_eq!(a, c);
}
{
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let n: u64 = rng.gen();
let a = F::from_str(&format!("{}", n)).unwrap();
let b = F::from_repr(n.into()).unwrap();
assert_eq!(a, b);
}
}
assert!(F::from_str("").is_none());
assert!(F::from_str("0").unwrap().is_zero());
assert!(F::from_str("00").is_none());
assert!(F::from_str("00000000000").is_none());
}
fn random_multiplication_tests<F: Field, R: Rng>(rng: &mut R) {
for _ in 0..10000 {
let a = F::rand(rng);
let b = F::rand(rng);
let c = F::rand(rng);
let mut t0 = a; // (a * b) * c
t0.mul_assign(&b);
t0.mul_assign(&c);
let mut t1 = a; // (a * c) * b
t1.mul_assign(&c);
t1.mul_assign(&b);
let mut t2 = b; // (b * c) * a
t2.mul_assign(&c);
t2.mul_assign(&a);
assert_eq!(t0, t1);
assert_eq!(t1, t2);
}
}
fn random_addition_tests<F: Field, R: Rng>(rng: &mut R) {
for _ in 0..10000 {
let a = F::rand(rng);
let b = F::rand(rng);
let c = F::rand(rng);
let mut t0 = a; // (a + b) + c
t0.add_assign(&b);
t0.add_assign(&c);
let mut t1 = a; // (a + c) + b
t1.add_assign(&c);
t1.add_assign(&b);
let mut t2 = b; // (b + c) + a
t2.add_assign(&c);
t2.add_assign(&a);
assert_eq!(t0, t1);
assert_eq!(t1, t2);
}
}
fn random_subtraction_tests<F: Field, R: Rng>(rng: &mut R) {
for _ in 0..10000 {
let a = F::rand(rng);
let b = F::rand(rng);
let mut t0 = a; // (a - b)
t0.sub_assign(&b);
let mut t1 = b; // (b - a)
t1.sub_assign(&a);
let mut t2 = t0; // (a - b) + (b - a) = 0
t2.add_assign(&t1);
assert!(t2.is_zero());
}
}
fn random_negation_tests<F: Field, R: Rng>(rng: &mut R) {
for _ in 0..10000 {
let a = F::rand(rng);
let mut b = a;
b.negate();
b.add_assign(&a);
assert!(b.is_zero());
}
}
fn random_doubling_tests<F: Field, R: Rng>(rng: &mut R) {
for _ in 0..10000 {
let mut a = F::rand(rng);
let mut b = a;
a.add_assign(&b);
b.double();
assert_eq!(a, b);
}
}
fn random_squaring_tests<F: Field, R: Rng>(rng: &mut R) {
for _ in 0..10000 {
let mut a = F::rand(rng);
let mut b = a;
a.mul_assign(&b);
b.square();
assert_eq!(a, b);
}
}
fn random_inversion_tests<F: Field, R: Rng>(rng: &mut R) {
assert!(F::zero().inverse().is_none());
for _ in 0..10000 {
let mut a = F::rand(rng);
let b = a.inverse().unwrap(); // probablistically nonzero
a.mul_assign(&b);
assert_eq!(a, F::one());
}
}
fn random_expansion_tests<F: Field, R: Rng>(rng: &mut R) {
for _ in 0..10000 {
// Compare (a + b)(c + d) and (a*c + b*c + a*d + b*d)
let a = F::rand(rng);
let b = F::rand(rng);
let c = F::rand(rng);
let d = F::rand(rng);
let mut t0 = a;
t0.add_assign(&b);
let mut t1 = c;
t1.add_assign(&d);
t0.mul_assign(&t1);
let mut t2 = a;
t2.mul_assign(&c);
let mut t3 = b;
t3.mul_assign(&c);
let mut t4 = a;
t4.mul_assign(&d);
let mut t5 = b;
t5.mul_assign(&d);
t2.add_assign(&t3);
t2.add_assign(&t4);
t2.add_assign(&t5);
assert_eq!(t0, t2);
}
}

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pub mod curve;
pub mod engine;
pub mod field;
pub mod repr;

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use ff::PrimeFieldRepr;
use rand::{SeedableRng, XorShiftRng};
pub fn random_repr_tests<R: PrimeFieldRepr>() {
random_encoding_tests::<R>();
random_shl_tests::<R>();
random_shr_tests::<R>();
}
fn random_encoding_tests<R: PrimeFieldRepr>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..1000 {
let r = R::rand(&mut rng);
// Big endian
{
let mut rdecoded = R::default();
let mut v: Vec<u8> = vec![];
r.write_be(&mut v).unwrap();
rdecoded.read_be(&v[0..]).unwrap();
assert_eq!(r, rdecoded);
}
// Little endian
{
let mut rdecoded = R::default();
let mut v: Vec<u8> = vec![];
r.write_le(&mut v).unwrap();
rdecoded.read_le(&v[0..]).unwrap();
assert_eq!(r, rdecoded);
}
{
let mut rdecoded_le = R::default();
let mut rdecoded_be_flip = R::default();
let mut v: Vec<u8> = vec![];
r.write_le(&mut v).unwrap();
// This reads in little-endian, so we are done.
rdecoded_le.read_le(&v[..]).unwrap();
// This reads in big-endian, so we perform a swap of the
// bytes beforehand.
let v: Vec<u8> = v.into_iter().rev().collect();
rdecoded_be_flip.read_be(&v[..]).unwrap();
assert_eq!(rdecoded_le, rdecoded_be_flip);
}
}
}
fn random_shl_tests<R: PrimeFieldRepr>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..100 {
let r = R::rand(&mut rng);
for shift in 0..(r.num_bits() + 1) {
let mut r1 = r;
let mut r2 = r;
for _ in 0..shift {
r1.mul2();
}
r2.shl(shift);
assert_eq!(r1, r2);
}
}
}
fn random_shr_tests<R: PrimeFieldRepr>() {
let mut rng = XorShiftRng::from_seed([0x5dbe6259, 0x8d313d76, 0x3237db17, 0xe5bc0654]);
for _ in 0..100 {
let r = R::rand(&mut rng);
for shift in 0..(r.num_bits() + 1) {
let mut r1 = r;
let mut r2 = r;
for _ in 0..shift {
r1.div2();
}
r2.shr(shift);
assert_eq!(r1, r2);
}
}
}

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use super::{CurveProjective, PrimeField, PrimeFieldRepr};
/// Replaces the contents of `table` with a w-NAF window table for the given window size.
pub(crate) fn wnaf_table<G: CurveProjective>(table: &mut Vec<G>, mut base: G, window: usize) {
table.truncate(0);
table.reserve(1 << (window - 1));
let mut dbl = base;
dbl.double();
for _ in 0..(1 << (window - 1)) {
table.push(base);
base.add_assign(&dbl);
}
}
/// Replaces the contents of `wnaf` with the w-NAF representation of a scalar.
pub(crate) fn wnaf_form<S: PrimeFieldRepr>(wnaf: &mut Vec<i64>, mut c: S, window: usize) {
wnaf.truncate(0);
while !c.is_zero() {
let mut u;
if c.is_odd() {
u = (c.as_ref()[0] % (1 << (window + 1))) as i64;
if u > (1 << window) {
u -= 1 << (window + 1);
}
if u > 0 {
c.sub_noborrow(&S::from(u as u64));
} else {
c.add_nocarry(&S::from((-u) as u64));
}
} else {
u = 0;
}
wnaf.push(u);
c.div2();
}
}
/// Performs w-NAF exponentiation with the provided window table and w-NAF form scalar.
///
/// This function must be provided a `table` and `wnaf` that were constructed with
/// the same window size; otherwise, it may panic or produce invalid results.
pub(crate) fn wnaf_exp<G: CurveProjective>(table: &[G], wnaf: &[i64]) -> G {
let mut result = G::zero();
let mut found_one = false;
for n in wnaf.iter().rev() {
if found_one {
result.double();
}
if *n != 0 {
found_one = true;
if *n > 0 {
result.add_assign(&table[(n / 2) as usize]);
} else {
result.sub_assign(&table[((-n) / 2) as usize]);
}
}
}
result
}
/// A "w-ary non-adjacent form" exponentiation context.
#[derive(Debug)]
pub struct Wnaf<W, B, S> {
base: B,
scalar: S,
window_size: W,
}
impl<G: CurveProjective> Wnaf<(), Vec<G>, Vec<i64>> {
/// Construct a new wNAF context without allocating.
pub fn new() -> Self {
Wnaf {
base: vec![],
scalar: vec![],
window_size: (),
}
}
/// Given a base and a number of scalars, compute a window table and return a `Wnaf` object that
/// can perform exponentiations with `.scalar(..)`.
pub fn base(&mut self, base: G, num_scalars: usize) -> Wnaf<usize, &[G], &mut Vec<i64>> {
// Compute the appropriate window size based on the number of scalars.
let window_size = G::recommended_wnaf_for_num_scalars(num_scalars);
// Compute a wNAF table for the provided base and window size.
wnaf_table(&mut self.base, base, window_size);
// Return a Wnaf object that immutably borrows the computed base storage location,
// but mutably borrows the scalar storage location.
Wnaf {
base: &self.base[..],
scalar: &mut self.scalar,
window_size,
}
}
/// Given a scalar, compute its wNAF representation and return a `Wnaf` object that can perform
/// exponentiations with `.base(..)`.
pub fn scalar(
&mut self,
scalar: <<G as CurveProjective>::Scalar as PrimeField>::Repr,
) -> Wnaf<usize, &mut Vec<G>, &[i64]> {
// Compute the appropriate window size for the scalar.
let window_size = G::recommended_wnaf_for_scalar(scalar);
// Compute the wNAF form of the scalar.
wnaf_form(&mut self.scalar, scalar, window_size);
// Return a Wnaf object that mutably borrows the base storage location, but
// immutably borrows the computed wNAF form scalar location.
Wnaf {
base: &mut self.base,
scalar: &self.scalar[..],
window_size,
}
}
}
impl<'a, G: CurveProjective> Wnaf<usize, &'a [G], &'a mut Vec<i64>> {
/// Constructs new space for the scalar representation while borrowing
/// the computed window table, for sending the window table across threads.
pub fn shared(&self) -> Wnaf<usize, &'a [G], Vec<i64>> {
Wnaf {
base: self.base,
scalar: vec![],
window_size: self.window_size,
}
}
}
impl<'a, G: CurveProjective> Wnaf<usize, &'a mut Vec<G>, &'a [i64]> {
/// Constructs new space for the window table while borrowing
/// the computed scalar representation, for sending the scalar representation
/// across threads.
pub fn shared(&self) -> Wnaf<usize, Vec<G>, &'a [i64]> {
Wnaf {
base: vec![],
scalar: self.scalar,
window_size: self.window_size,
}
}
}
impl<B, S: AsRef<[i64]>> Wnaf<usize, B, S> {
/// Performs exponentiation given a base.
pub fn base<G: CurveProjective>(&mut self, base: G) -> G
where
B: AsMut<Vec<G>>,
{
wnaf_table(self.base.as_mut(), base, self.window_size);
wnaf_exp(self.base.as_mut(), self.scalar.as_ref())
}
}
impl<B, S: AsMut<Vec<i64>>> Wnaf<usize, B, S> {
/// Performs exponentiation given a scalar.
pub fn scalar<G: CurveProjective>(
&mut self,
scalar: <<G as CurveProjective>::Scalar as PrimeField>::Repr,
) -> G
where
B: AsRef<[G]>,
{
wnaf_form(self.scalar.as_mut(), scalar, self.window_size);
wnaf_exp(self.base.as_ref(), self.scalar.as_mut())
}
}