use core::fmt;
use core::ops::{Add, AddAssign, Mul, MulAssign, Neg, Sub, SubAssign};

use byteorder::{ByteOrder, LittleEndian};
use subtle::{Choice, ConditionallySelectable, ConstantTimeEq};
use crate::util::{adc, mac, sbb};

/// Represents an element of `GF(r)`.
// The internal representation of this type is four 64-bit unsigned
// integers in little-endian order. Elements of Fr are always in
// Montgomery form; i.e., Fr(a) = aR mod r, with R = 2^256.
#[derive(Clone, Copy, Eq)]
pub struct Fr(pub(crate) [u64; 4]);

impl fmt::Debug for Fr {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        let tmp = self.into_bytes();
        write!(f, "0x")?;
        for &b in tmp.iter().rev() {
            write!(f, "{:02x}", b)?;
        }
        Ok(())
    }
}

impl From<u64> for Fr {
    fn from(val: u64) -> Fr {
        Fr([val, 0, 0, 0]) * R2
    }
}

impl ConstantTimeEq for Fr {
    fn ct_eq(&self, other: &Self) -> Choice {
        self.0[0].ct_eq(&other.0[0])
            & self.0[1].ct_eq(&other.0[1])
            & self.0[2].ct_eq(&other.0[2])
            & self.0[3].ct_eq(&other.0[3])
    }
}

impl PartialEq for Fr {
    fn eq(&self, other: &Self) -> bool {
        self.ct_eq(other).unwrap_u8() == 1
    }
}

impl ConditionallySelectable for Fr {
    fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {
        Fr([
            u64::conditional_select(&a.0[0], &b.0[0], choice),
            u64::conditional_select(&a.0[1], &b.0[1], choice),
            u64::conditional_select(&a.0[2], &b.0[2], choice),
            u64::conditional_select(&a.0[3], &b.0[3], choice),
        ])
    }
}

// Constant representing the modulus
// r = 0x0e7db4ea6533afa906673b0101343b00a6682093ccc81082d0970e5ed6f72cb7
const MODULUS: Fr = Fr([
    0xd0970e5ed6f72cb7,
    0xa6682093ccc81082,
    0x06673b0101343b00,
    0x0e7db4ea6533afa9,
]);

impl<'a> Neg for &'a Fr {
    type Output = Fr;

    #[inline]
    fn neg(self) -> Fr {
        // Subtract `self` from `MODULUS` to negate. Ignore the final
        // borrow because it cannot underflow; self is guaranteed to
        // be in the field.
        let (d0, borrow) = sbb(MODULUS.0[0], self.0[0], 0);
        let (d1, borrow) = sbb(MODULUS.0[1], self.0[1], borrow);
        let (d2, borrow) = sbb(MODULUS.0[2], self.0[2], borrow);
        let (d3, _) = sbb(MODULUS.0[3], self.0[3], borrow);

        // `tmp` could be `MODULUS` if `self` was zero. Create a mask that is
        // zero if `self` was zero, and `u64::max_value()` if self was nonzero.
        let mask = u64::from((self.0[0] | self.0[1] | self.0[2] | self.0[3]) == 0).wrapping_sub(1);

        Fr([d0 & mask, d1 & mask, d2 & mask, d3 & mask])
    }
}

impl Neg for Fr {
    type Output = Fr;

    #[inline]
    fn neg(self) -> Fr {
        -&self
    }
}

impl<'a, 'b> Sub<&'b Fr> for &'a Fr {
    type Output = Fr;

    #[inline]
    fn sub(self, rhs: &'b Fr) -> Fr {
        let (d0, borrow) = sbb(self.0[0], rhs.0[0], 0);
        let (d1, borrow) = sbb(self.0[1], rhs.0[1], borrow);
        let (d2, borrow) = sbb(self.0[2], rhs.0[2], borrow);
        let (d3, borrow) = sbb(self.0[3], rhs.0[3], borrow);

        // If underflow occurred on the final limb, borrow = 0xfff...fff, otherwise
        // borrow = 0x000...000. Thus, we use it as a mask to conditionally add the modulus.
        let (d0, carry) = adc(d0, MODULUS.0[0] & borrow, 0);
        let (d1, carry) = adc(d1, MODULUS.0[1] & borrow, carry);
        let (d2, carry) = adc(d2, MODULUS.0[2] & borrow, carry);
        let (d3, _) = adc(d3, MODULUS.0[3] & borrow, carry);

        Fr([d0, d1, d2, d3])
    }
}

impl<'a, 'b> Add<&'b Fr> for &'a Fr {
    type Output = Fr;

    #[inline]
    fn add(self, rhs: &'b Fr) -> Fr {
        let (d0, carry) = adc(self.0[0], rhs.0[0], 0);
        let (d1, carry) = adc(self.0[1], rhs.0[1], carry);
        let (d2, carry) = adc(self.0[2], rhs.0[2], carry);
        let (d3, _) = adc(self.0[3], rhs.0[3], carry);

        // Attempt to subtract the modulus, to ensure the value
        // is smaller than the modulus.
        Fr([d0, d1, d2, d3]) - &MODULUS
    }
}

impl<'a, 'b> Mul<&'b Fr> for &'a Fr {
    type Output = Fr;

    #[inline]
    fn mul(self, rhs: &'b Fr) -> Fr {
        // Schoolbook multiplication

        let (r0, carry) = mac(0, self.0[0], rhs.0[0], 0);
        let (r1, carry) = mac(0, self.0[0], rhs.0[1], carry);
        let (r2, carry) = mac(0, self.0[0], rhs.0[2], carry);
        let (r3, r4) = mac(0, self.0[0], rhs.0[3], carry);

        let (r1, carry) = mac(r1, self.0[1], rhs.0[0], 0);
        let (r2, carry) = mac(r2, self.0[1], rhs.0[1], carry);
        let (r3, carry) = mac(r3, self.0[1], rhs.0[2], carry);
        let (r4, r5) = mac(r4, self.0[1], rhs.0[3], carry);

        let (r2, carry) = mac(r2, self.0[2], rhs.0[0], 0);
        let (r3, carry) = mac(r3, self.0[2], rhs.0[1], carry);
        let (r4, carry) = mac(r4, self.0[2], rhs.0[2], carry);
        let (r5, r6) = mac(r5, self.0[2], rhs.0[3], carry);

        let (r3, carry) = mac(r3, self.0[3], rhs.0[0], 0);
        let (r4, carry) = mac(r4, self.0[3], rhs.0[1], carry);
        let (r5, carry) = mac(r5, self.0[3], rhs.0[2], carry);
        let (r6, r7) = mac(r6, self.0[3], rhs.0[3], carry);

        Fr::montgomery_reduce(r0, r1, r2, r3, r4, r5, r6, r7)
    }
}

impl_binops_additive!(Fr, Fr);
impl_binops_multiplicative!(Fr, Fr);

/// INV = -(r^{-1} mod 2^64) mod 2^64
const INV: u64 = 0x1ba3a358ef788ef9;

/// R = 2^256 mod r
const R: Fr = Fr([
    0x25f80bb3b99607d9,
    0xf315d62f66b6e750,
    0x932514eeeb8814f4,
    0x09a6fc6f479155c6,
]);

/// R^2 = 2^512 mod r
const R2: Fr = Fr([
    0x67719aa495e57731,
    0x51b0cef09ce3fc26,
    0x69dab7fac026e9a5,
    0x04f6547b8d127688,
]);

impl Default for Fr {
    fn default() -> Self {
        Self::zero()
    }
}

impl Fr {
    pub fn zero() -> Fr {
        Fr([0, 0, 0, 0])
    }

    pub fn one() -> Fr {
        R
    }

    #[inline]
    pub fn double(&self) -> Fr {
        self + self
    }

    /// Attempts to convert a little-endian byte representation of
    /// a field element into an element of `Fr`, failing if the input
    /// is not canonical (is not smaller than r).
    ///
    /// **This operation is variable time.**
    pub fn from_bytes_vartime(bytes: [u8; 32]) -> Option<Fr> {
        let mut tmp = Fr([0, 0, 0, 0]);

        tmp.0[0] = LittleEndian::read_u64(&bytes[0..8]);
        tmp.0[1] = LittleEndian::read_u64(&bytes[8..16]);
        tmp.0[2] = LittleEndian::read_u64(&bytes[16..24]);
        tmp.0[3] = LittleEndian::read_u64(&bytes[24..32]);

        // Check if the value is in the field
        for i in (0..4).rev() {
            if tmp.0[i] < MODULUS.0[i] {
                // Convert to Montgomery form by computing
                // (a.R^{-1} * R^2) / R = a.R
                tmp.mul_assign(&R2);

                return Some(tmp);
            }

            if tmp.0[i] > MODULUS.0[i] {
                return None;
            }
        }

        // Value is equal to the modulus
        None
    }

    /// Converts an element of `Fr` into a byte representation in
    /// little-endian byte order.
    pub fn into_bytes(&self) -> [u8; 32] {
        // Turn into canonical form by computing
        // (a.R) / R = a
        let tmp = Fr::montgomery_reduce(self.0[0], self.0[1], self.0[2], self.0[3], 0, 0, 0, 0);

        let mut res = [0; 32];
        LittleEndian::write_u64(&mut res[0..8], tmp.0[0]);
        LittleEndian::write_u64(&mut res[8..16], tmp.0[1]);
        LittleEndian::write_u64(&mut res[16..24], tmp.0[2]);
        LittleEndian::write_u64(&mut res[24..32], tmp.0[3]);

        res
    }

    /// Squares this element.
    pub fn square(&self) -> Fr {
        let (r1, carry) = mac(0, self.0[0], self.0[1], 0);
        let (r2, carry) = mac(0, self.0[0], self.0[2], carry);
        let (r3, r4) = mac(0, self.0[0], self.0[3], carry);

        let (r3, carry) = mac(r3, self.0[1], self.0[2], 0);
        let (r4, r5) = mac(r4, self.0[1], self.0[3], carry);

        let (r5, r6) = mac(r5, self.0[2], self.0[3], 0);

        let r7 = r6 >> 63;
        let r6 = (r6 << 1) | (r5 >> 63);
        let r5 = (r5 << 1) | (r4 >> 63);
        let r4 = (r4 << 1) | (r3 >> 63);
        let r3 = (r3 << 1) | (r2 >> 63);
        let r2 = (r2 << 1) | (r1 >> 63);
        let r1 = r1 << 1;

        let (r0, carry) = mac(0, self.0[0], self.0[0], 0);
        let (r1, carry) = adc(0, r1, carry);
        let (r2, carry) = mac(r2, self.0[1], self.0[1], carry);
        let (r3, carry) = adc(0, r3, carry);
        let (r4, carry) = mac(r4, self.0[2], self.0[2], carry);
        let (r5, carry) = adc(0, r5, carry);
        let (r6, carry) = mac(r6, self.0[3], self.0[3], carry);
        let (r7, _) = adc(0, r7, carry);

        Fr::montgomery_reduce(r0, r1, r2, r3, r4, r5, r6, r7)
    }

    /// Computes the square root of this element, if it exists.
    ///
    /// **This operation is variable time.**
    pub fn sqrt_vartime(&self) -> Option<Self> {
        // Because r = 3 (mod 4)
        // sqrt can be done with only one exponentiation,
        // via the computation of  self^((r + 1) // 4) (mod r)
        // a1 = self^((r - 3) // 4)
        let a1 = self.pow_vartime(&[
            0xb425c397b5bdcb2d,
            0x299a0824f3320420,
            0x4199cec0404d0ec0,
            0x039f6d3a994cebea,
        ]);
        let a0 = a1 * (a1 * self);
        if a0 == -Self::one() {
            None
        } else {
            Some(a1 * self)
        }
    }

    /// Exponentiates `self` by `by`, where `by` is a
    /// little-endian order integer exponent.
    pub fn pow(&self, by: &[u64; 4]) -> Self {
        let mut res = Self::one();
        for e in by.iter().rev() {
            for i in (0..64).rev() {
                res = res.square();
                let mut tmp = res;
                tmp.mul_assign(self);
                res.conditional_assign(&tmp, (((*e >> i) & 0x1) as u8).into());
            }
        }
        res
    }

    /// Exponentiates `self` by `by`, where `by` is a
    /// little-endian order integer exponent.
    ///
    /// **This operation is variable time with respect
    /// to the exponent.** If the exponent is fixed,
    /// this operation is effectively constant time.
    pub fn pow_vartime(&self, by: &[u64; 4]) -> Self {
        let mut res = Self::one();
        for e in by.iter().rev() {
            for i in (0..64).rev() {
                res = res.square();

                if ((*e >> i) & 1) == 1 {
                    res.mul_assign(self);
                }
            }
        }
        res
    }

    /// Exponentiates `self` by r - 2, which has the
    /// effect of inverting the element if it is
    /// nonzero.
    pub fn invert_nonzero(&self) -> Self {
        #[inline(always)]
        fn square_assign_multi(n: &mut Fr, num_times: usize) {
            for _ in 0..num_times {
                *n = n.square();
            }
        }
        // found using https://github.com/kwantam/addchain
        let t10 = *self;
        let t1 = t10.square();
        let t0 = t1.square();
        let t3 = t0 * &t1;
        let t6 = t3 * &t10;
        let t7 = t6 * &t1;
        let t12 = t7 * &t3;
        let t13 = t12 * &t0;
        let t16 = t12 * &t3;
        let t2 = t13 * &t3;
        let t15 = t16 * &t3;
        let t19 = t2 * &t0;
        let t9 = t15 * &t3;
        let t18 = t9 * &t3;
        let t14 = t18 * &t1;
        let t4 = t18 * &t0;
        let t8 = t18 * &t3;
        let t17 = t14 * &t3;
        let t11 = t8 * &t3;
        let t1 = t17 * &t3;
        let t5 = t11 * &t3;
        let t3 = t5 * &t0;
        let mut t0 = t5.square();
        square_assign_multi(&mut t0, 5);
        t0.mul_assign(&t3);
        square_assign_multi(&mut t0, 6);
        t0.mul_assign(&t8);
        square_assign_multi(&mut t0, 7);
        t0.mul_assign(&t19);
        square_assign_multi(&mut t0, 6);
        t0.mul_assign(&t13);
        square_assign_multi(&mut t0, 8);
        t0.mul_assign(&t14);
        square_assign_multi(&mut t0, 6);
        t0.mul_assign(&t18);
        square_assign_multi(&mut t0, 7);
        t0.mul_assign(&t17);
        square_assign_multi(&mut t0, 5);
        t0.mul_assign(&t16);
        square_assign_multi(&mut t0, 3);
        t0.mul_assign(&t10);
        square_assign_multi(&mut t0, 11);
        t0.mul_assign(&t11);
        square_assign_multi(&mut t0, 8);
        t0.mul_assign(&t5);
        square_assign_multi(&mut t0, 5);
        t0.mul_assign(&t15);
        square_assign_multi(&mut t0, 8);
        t0.mul_assign(&t10);
        square_assign_multi(&mut t0, 12);
        t0.mul_assign(&t13);
        square_assign_multi(&mut t0, 7);
        t0.mul_assign(&t9);
        square_assign_multi(&mut t0, 5);
        t0.mul_assign(&t15);
        square_assign_multi(&mut t0, 14);
        t0.mul_assign(&t14);
        square_assign_multi(&mut t0, 5);
        t0.mul_assign(&t13);
        square_assign_multi(&mut t0, 2);
        t0.mul_assign(&t10);
        square_assign_multi(&mut t0, 6);
        t0.mul_assign(&t10);
        square_assign_multi(&mut t0, 9);
        t0.mul_assign(&t7);
        square_assign_multi(&mut t0, 6);
        t0.mul_assign(&t12);
        square_assign_multi(&mut t0, 8);
        t0.mul_assign(&t11);
        square_assign_multi(&mut t0, 3);
        t0.mul_assign(&t10);
        square_assign_multi(&mut t0, 12);
        t0.mul_assign(&t9);
        square_assign_multi(&mut t0, 11);
        t0.mul_assign(&t8);
        square_assign_multi(&mut t0, 8);
        t0.mul_assign(&t7);
        square_assign_multi(&mut t0, 4);
        t0.mul_assign(&t6);
        square_assign_multi(&mut t0, 10);
        t0.mul_assign(&t5);
        square_assign_multi(&mut t0, 7);
        t0.mul_assign(&t3);
        square_assign_multi(&mut t0, 6);
        t0.mul_assign(&t4);
        square_assign_multi(&mut t0, 7);
        t0.mul_assign(&t3);
        square_assign_multi(&mut t0, 5);
        t0.mul_assign(&t2);
        square_assign_multi(&mut t0, 6);
        t0.mul_assign(&t2);
        square_assign_multi(&mut t0, 7);
        t0.mul_assign(&t1);

        t0
    }

    #[inline]
    fn montgomery_reduce(
        r0: u64,
        r1: u64,
        r2: u64,
        r3: u64,
        r4: u64,
        r5: u64,
        r6: u64,
        r7: u64,
    ) -> Self {
        // The Montgomery reduction here is based on Algorithm 14.32 in
        // Handbook of Applied Cryptography
        // <http://cacr.uwaterloo.ca/hac/about/chap14.pdf>.

        let k = r0.wrapping_mul(INV);
        let (_, carry) = mac(r0, k, MODULUS.0[0], 0);
        let (r1, carry) = mac(r1, k, MODULUS.0[1], carry);
        let (r2, carry) = mac(r2, k, MODULUS.0[2], carry);
        let (r3, carry) = mac(r3, k, MODULUS.0[3], carry);
        let (r4, carry2) = adc(r4, 0, carry);

        let k = r1.wrapping_mul(INV);
        let (_, carry) = mac(r1, k, MODULUS.0[0], 0);
        let (r2, carry) = mac(r2, k, MODULUS.0[1], carry);
        let (r3, carry) = mac(r3, k, MODULUS.0[2], carry);
        let (r4, carry) = mac(r4, k, MODULUS.0[3], carry);
        let (r5, carry2) = adc(r5, carry2, carry);

        let k = r2.wrapping_mul(INV);
        let (_, carry) = mac(r2, k, MODULUS.0[0], 0);
        let (r3, carry) = mac(r3, k, MODULUS.0[1], carry);
        let (r4, carry) = mac(r4, k, MODULUS.0[2], carry);
        let (r5, carry) = mac(r5, k, MODULUS.0[3], carry);
        let (r6, carry2) = adc(r6, carry2, carry);

        let k = r3.wrapping_mul(INV);
        let (_, carry) = mac(r3, k, MODULUS.0[0], 0);
        let (r4, carry) = mac(r4, k, MODULUS.0[1], carry);
        let (r5, carry) = mac(r5, k, MODULUS.0[2], carry);
        let (r6, carry) = mac(r6, k, MODULUS.0[3], carry);
        let (r7, _) = adc(r7, carry2, carry);

        let mut tmp = Fr([r4, r5, r6, r7]);

        // Result may be within MODULUS of the correct value
        tmp.sub_assign(&MODULUS);

        tmp
    }
}

impl<'a> From<&'a Fr> for [u8; 32] {
    fn from(value: &'a Fr) -> [u8; 32] {
        value.into_bytes()
    }
}

#[test]
fn test_inv() {
    // Compute -(r^{-1} mod 2^64) mod 2^64 by exponentiating
    // by totient(2**64) - 1

    let mut inv = 1u64;
    for _ in 0..63 {
        inv = inv.wrapping_mul(inv);
        inv = inv.wrapping_mul(MODULUS.0[0]);
    }
    inv = inv.wrapping_neg();

    assert_eq!(inv, INV);
}

#[cfg(feature = "std")]
#[test]
fn test_debug() {
    assert_eq!(
        format!("{:?}", Fr::zero()),
        "0x0000000000000000000000000000000000000000000000000000000000000000"
    );
    assert_eq!(
        format!("{:?}", Fr::one()),
        "0x0000000000000000000000000000000000000000000000000000000000000001"
    );
    assert_eq!(
        format!("{:?}", R2),
        "0x09a6fc6f479155c6932514eeeb8814f4f315d62f66b6e75025f80bb3b99607d9"
    );
}

#[test]
fn test_equality() {
    assert_eq!(Fr::zero(), Fr::zero());
    assert_eq!(Fr::one(), Fr::one());
    assert_eq!(R2, R2);

    assert!(Fr::zero() != Fr::one());
    assert!(Fr::one() != R2);
}

#[test]
fn test_into_bytes() {
    assert_eq!(
        Fr::zero().into_bytes(),
        [
            0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
            0, 0, 0
        ]
    );

    assert_eq!(
        Fr::one().into_bytes(),
        [
            1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
            0, 0, 0
        ]
    );

    assert_eq!(
        R2.into_bytes(),
        [
            217, 7, 150, 185, 179, 11, 248, 37, 80, 231, 182, 102, 47, 214, 21, 243, 244, 20, 136,
            235, 238, 20, 37, 147, 198, 85, 145, 71, 111, 252, 166, 9
        ]
    );

    assert_eq!(
        (-&Fr::one()).into_bytes(),
        [
            182, 44, 247, 214, 94, 14, 151, 208, 130, 16, 200, 204, 147, 32, 104, 166, 0, 59, 52,
            1, 1, 59, 103, 6, 169, 175, 51, 101, 234, 180, 125, 14
        ]
    );
}

#[test]
fn test_from_bytes_vartime() {
    assert_eq!(
        Fr::from_bytes_vartime([
            0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
            0, 0, 0
        ]).unwrap(),
        Fr::zero()
    );

    assert_eq!(
        Fr::from_bytes_vartime([
            1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
            0, 0, 0
        ]).unwrap(),
        Fr::one()
    );

    assert_eq!(
        Fr::from_bytes_vartime([
            217, 7, 150, 185, 179, 11, 248, 37, 80, 231, 182, 102, 47, 214, 21, 243, 244, 20, 136,
            235, 238, 20, 37, 147, 198, 85, 145, 71, 111, 252, 166, 9
        ]).unwrap(),
        R2
    );

    // -1 should work
    assert!(
        Fr::from_bytes_vartime([
            182, 44, 247, 214, 94, 14, 151, 208, 130, 16, 200, 204, 147, 32, 104, 166, 0, 59, 52,
            1, 1, 59, 103, 6, 169, 175, 51, 101, 234, 180, 125, 14
        ]).is_some()
    );

    // modulus is invalid
    assert!(
        Fr::from_bytes_vartime([
            183, 44, 247, 214, 94, 14, 151, 208, 130, 16, 200, 204, 147, 32, 104, 166, 0, 59, 52,
            1, 1, 59, 103, 6, 169, 175, 51, 101, 234, 180, 125, 14
        ]).is_none()
    );

    // Anything larger than the modulus is invalid
    assert!(
        Fr::from_bytes_vartime([
            184, 44, 247, 214, 94, 14, 151, 208, 130, 16, 200, 204, 147, 32, 104, 166, 0, 59, 52,
            1, 1, 59, 103, 6, 169, 175, 51, 101, 234, 180, 125, 14
        ]).is_none()
    );

    assert!(
        Fr::from_bytes_vartime([
            183, 44, 247, 214, 94, 14, 151, 208, 130, 16, 200, 204, 147, 32, 104, 166, 0, 59, 52,
            1, 1, 59, 104, 6, 169, 175, 51, 101, 234, 180, 125, 14
        ]).is_none()
    );

    assert!(
        Fr::from_bytes_vartime([
            183, 44, 247, 214, 94, 14, 151, 208, 130, 16, 200, 204, 147, 32, 104, 166, 0, 59, 52,
            1, 1, 59, 103, 6, 169, 175, 51, 101, 234, 180, 125, 15
        ]).is_none()
    );
}

#[cfg(test)]
const LARGEST: Fr = Fr([
    0xd0970e5ed6f72cb6,
    0xa6682093ccc81082,
    0x06673b0101343b00,
    0x0e7db4ea6533afa9,
]);

#[test]
fn test_addition() {
    let mut tmp = LARGEST;
    tmp += &LARGEST;

    assert_eq!(
        tmp,
        Fr([
            0xd0970e5ed6f72cb5,
            0xa6682093ccc81082,
            0x06673b0101343b00,
            0x0e7db4ea6533afa9
        ])
    );

    let mut tmp = LARGEST;
    tmp += &Fr([1, 0, 0, 0]);

    assert_eq!(tmp, Fr::zero());
}

#[test]
fn test_negation() {
    let tmp = -&LARGEST;

    assert_eq!(tmp, Fr([1, 0, 0, 0]));

    let tmp = -&Fr::zero();
    assert_eq!(tmp, Fr::zero());
    let tmp = -&Fr([1, 0, 0, 0]);
    assert_eq!(tmp, LARGEST);
}

#[test]
fn test_subtraction() {
    let mut tmp = LARGEST;
    tmp -= &LARGEST;

    assert_eq!(tmp, Fr::zero());

    let mut tmp = Fr::zero();
    tmp -= &LARGEST;

    let mut tmp2 = MODULUS;
    tmp2 -= &LARGEST;

    assert_eq!(tmp, tmp2);
}

#[test]
fn test_multiplication() {
    let mut cur = LARGEST;

    for _ in 0..100 {
        let mut tmp = cur;
        tmp *= &cur;

        let mut tmp2 = Fr::zero();
        for b in cur
            .into_bytes()
            .iter()
            .rev()
            .flat_map(|byte| (0..8).rev().map(move |i| ((byte >> i) & 1u8) == 1u8))
        {
            let tmp3 = tmp2;
            tmp2.add_assign(&tmp3);

            if b {
                tmp2.add_assign(&cur);
            }
        }

        assert_eq!(tmp, tmp2);

        cur.add_assign(&LARGEST);
    }
}

#[test]
fn test_squaring() {
    let mut cur = LARGEST;

    for _ in 0..100 {
        let mut tmp = cur;
        tmp = tmp.square();

        let mut tmp2 = Fr::zero();
        for b in cur
            .into_bytes()
            .iter()
            .rev()
            .flat_map(|byte| (0..8).rev().map(move |i| ((byte >> i) & 1u8) == 1u8))
        {
            let tmp3 = tmp2;
            tmp2.add_assign(&tmp3);

            if b {
                tmp2.add_assign(&cur);
            }
        }

        assert_eq!(tmp, tmp2);

        cur.add_assign(&LARGEST);
    }
}

#[test]
fn test_inversion() {
    assert_eq!(Fr::one().invert_nonzero(), Fr::one());
    assert_eq!((-&Fr::one()).invert_nonzero(), -&Fr::one());

    let mut tmp = R2;

    for _ in 0..100 {
        let mut tmp2 = tmp.invert_nonzero();
        tmp2.mul_assign(&tmp);

        assert_eq!(tmp2, Fr::one());

        tmp.add_assign(&R2);
    }
}

#[test]
fn test_invert_nonzero_is_pow() {
    let r_minus_2 = [
        0xd0970e5ed6f72cb5,
        0xa6682093ccc81082,
        0x06673b0101343b00,
        0x0e7db4ea6533afa9,
    ];

    let mut r1 = R;
    let mut r2 = R;
    let mut r3 = R;

    for _ in 0..100 {
        r1 = r1.invert_nonzero();
        r2 = r2.pow_vartime(&r_minus_2);
        r3 = r3.pow(&r_minus_2);

        assert_eq!(r1, r2);
        assert_eq!(r2, r3);
        // Add R so we check something different next time around
        r1.add_assign(&R);
        r2 = r1;
        r3 = r1;
    }
}

#[test]
fn test_sqrt() {
    let mut square = Fr([
        // r - 2
        0xd0970e5ed6f72cb5,
        0xa6682093ccc81082,
        0x06673b0101343b00,
        0x0e7db4ea6533afa9,
    ]);

    let mut none_count = 0;

    for _ in 0..100 {
        let square_root = square.sqrt_vartime();
        if square_root.is_none() {
            none_count += 1;
        } else {
            assert_eq!(square_root.unwrap() * square_root.unwrap(), square);
        }
        square -= Fr::one();
    }

    assert_eq!(47, none_count);
}