pairing_ce/bls12_381/

fr.rs

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>();
}