2020-11-12 16:08:08 -08:00
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use bitvec::{array::BitArray, order::Lsb0};
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2020-08-22 13:15:39 -07:00
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use core::convert::TryInto;
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use core::fmt;
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2020-11-12 11:37:48 -08:00
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use core::ops::{Add, Mul, Neg, Sub};
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2021-01-04 15:39:52 -08:00
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use lazy_static::lazy_static;
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2020-11-12 16:08:08 -08:00
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use rand::RngCore;
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2020-08-22 13:15:39 -07:00
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use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
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2021-01-04 15:39:52 -08:00
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use crate::arithmetic::{adc, mac, sbb, FieldExt, Group, SqrtTables};
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2020-08-22 13:15:39 -07:00
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/// This represents an element of $\mathbb{F}_p$ where
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///
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2020-12-03 12:13:40 -08:00
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/// `p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001`
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2020-08-22 13:15:39 -07:00
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///
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2020-12-03 12:13:40 -08:00
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/// is the base field of the Pallas curve.
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2020-08-22 13:15:39 -07:00
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// The internal representation of this type is four 64-bit unsigned
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// integers in little-endian order. `Fp` values are always in
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// Montgomery form; i.e., Fp(a) = aR mod p, with R = 2^256.
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#[derive(Clone, Copy, Eq)]
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pub struct Fp(pub(crate) [u64; 4]);
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impl fmt::Debug for Fp {
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fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
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let tmp = self.to_bytes();
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write!(f, "0x")?;
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for &b in tmp.iter().rev() {
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write!(f, "{:02x}", b)?;
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}
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Ok(())
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}
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}
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impl From<bool> for Fp {
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fn from(bit: bool) -> Fp {
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if bit {
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Fp::one()
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} else {
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Fp::zero()
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}
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}
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}
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impl From<u64> for Fp {
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fn from(val: u64) -> Fp {
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Fp([val, 0, 0, 0]) * R2
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}
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}
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impl ConstantTimeEq for Fp {
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fn ct_eq(&self, other: &Self) -> Choice {
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self.0[0].ct_eq(&other.0[0])
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& self.0[1].ct_eq(&other.0[1])
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& self.0[2].ct_eq(&other.0[2])
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& self.0[3].ct_eq(&other.0[3])
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}
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}
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impl PartialEq for Fp {
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#[inline]
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fn eq(&self, other: &Self) -> bool {
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self.ct_eq(other).unwrap_u8() == 1
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}
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}
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2020-10-07 06:54:00 -07:00
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impl std::cmp::Ord for Fp {
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fn cmp(&self, other: &Self) -> std::cmp::Ordering {
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let left = self.to_bytes();
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let right = other.to_bytes();
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2020-10-07 21:23:16 -07:00
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left.iter()
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.zip(right.iter())
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.rev()
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.find_map(|(left_byte, right_byte)| match left_byte.cmp(right_byte) {
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std::cmp::Ordering::Equal => None,
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res => Some(res),
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})
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.unwrap_or(std::cmp::Ordering::Equal)
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2020-10-07 06:54:00 -07:00
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}
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}
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impl std::cmp::PartialOrd for Fp {
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fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
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Some(self.cmp(other))
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}
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}
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2020-08-22 13:15:39 -07:00
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impl ConditionallySelectable for Fp {
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fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {
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Fp([
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u64::conditional_select(&a.0[0], &b.0[0], choice),
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u64::conditional_select(&a.0[1], &b.0[1], choice),
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u64::conditional_select(&a.0[2], &b.0[2], choice),
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u64::conditional_select(&a.0[3], &b.0[3], choice),
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])
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}
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}
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/// Constant representing the modulus
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/// p = 0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001
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const MODULUS: Fp = Fp([
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2020-12-03 12:13:40 -08:00
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0x992d30ed00000001,
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0x224698fc094cf91b,
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0x0000000000000000,
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2020-08-22 13:15:39 -07:00
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0x4000000000000000,
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]);
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2020-11-12 16:08:08 -08:00
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/// The modulus as u32 limbs.
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#[cfg(not(target_pointer_width = "64"))]
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const MODULUS_LIMBS_32: [u32; 8] = [
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0x0000_0001,
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2020-12-03 12:13:40 -08:00
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0x992d_30ed,
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0x094c_f91b,
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0x2246_98fc,
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2020-11-12 16:08:08 -08:00
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0x0000_0000,
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0x0000_0000,
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0x0000_0000,
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0x4000_0000,
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];
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2020-08-22 13:15:39 -07:00
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impl<'a> Neg for &'a Fp {
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type Output = Fp;
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#[inline]
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fn neg(self) -> Fp {
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self.neg()
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}
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}
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impl Neg for Fp {
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type Output = Fp;
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#[inline]
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fn neg(self) -> Fp {
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-&self
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}
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}
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impl<'a, 'b> Sub<&'b Fp> for &'a Fp {
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type Output = Fp;
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#[inline]
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fn sub(self, rhs: &'b Fp) -> Fp {
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self.sub(rhs)
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}
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}
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impl<'a, 'b> Add<&'b Fp> for &'a Fp {
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type Output = Fp;
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#[inline]
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fn add(self, rhs: &'b Fp) -> Fp {
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self.add(rhs)
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}
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}
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impl<'a, 'b> Mul<&'b Fp> for &'a Fp {
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type Output = Fp;
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#[inline]
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fn mul(self, rhs: &'b Fp) -> Fp {
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self.mul(rhs)
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}
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}
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impl_binops_additive!(Fp, Fp);
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impl_binops_multiplicative!(Fp, Fp);
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/// INV = -(p^{-1} mod 2^64) mod 2^64
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const INV: u64 = 0x992d30ecffffffff;
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2020-08-22 13:15:39 -07:00
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/// R = 2^256 mod p
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const R: Fp = Fp([
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2020-12-03 12:13:40 -08:00
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0x34786d38fffffffd,
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0x992c350be41914ad,
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2020-08-22 13:15:39 -07:00
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0xffffffffffffffff,
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0x3fffffffffffffff,
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]);
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/// R^2 = 2^512 mod p
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const R2: Fp = Fp([
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2020-12-03 12:13:40 -08:00
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0x8c78ecb30000000f,
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0xd7d30dbd8b0de0e7,
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0x7797a99bc3c95d18,
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2020-12-11 12:24:32 -08:00
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0x096d41af7b9cb714,
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2020-08-22 13:15:39 -07:00
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]);
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/// R^3 = 2^768 mod p
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const R3: Fp = Fp([
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2020-12-03 12:13:40 -08:00
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0xf185a5993a9e10f9,
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0xf6a68f3b6ac5b1d1,
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0xdf8d1014353fd42c,
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0x2ae309222d2d9910,
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2020-08-22 13:15:39 -07:00
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]);
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2020-12-03 12:13:40 -08:00
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/// `GENERATOR = 5 mod p` is a generator of the `p - 1` order multiplicative
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/// subgroup, or in other words a primitive root of the field.
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2020-11-12 16:08:08 -08:00
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const GENERATOR: Fp = Fp::from_raw([
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0x0000_0000_0000_0005,
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0x0000_0000_0000_0000,
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0x0000_0000_0000_0000,
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0x0000_0000_0000_0000,
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]);
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2020-12-03 12:13:40 -08:00
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const S: u32 = 32;
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2020-08-22 13:15:39 -07:00
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/// GENERATOR^t where t * 2^s + 1 = p
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/// with t odd. In other words, this
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/// is a 2^s root of unity.
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const ROOT_OF_UNITY: Fp = Fp::from_raw([
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2020-12-03 12:13:40 -08:00
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0xbdad6fabd87ea32f,
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0xea322bf2b7bb7584,
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0x362120830561f81a,
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0x2bce74deac30ebda,
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2020-08-22 13:15:39 -07:00
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]);
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2020-08-28 22:51:42 -07:00
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/// GENERATOR^{2^s} where t * 2^s + 1 = p
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/// with t odd. In other words, this
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/// is a t root of unity.
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const DELTA: Fp = Fp::from_raw([
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2020-12-03 12:13:40 -08:00
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0x6a6ccd20dd7b9ba2,
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0xf5e4f3f13eee5636,
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0xbd455b7112a5049d,
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2020-12-11 12:24:32 -08:00
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0x0a757d0f0006ab6c,
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2020-08-28 22:51:42 -07:00
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]);
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2020-08-22 13:15:39 -07:00
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impl Default for Fp {
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#[inline]
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fn default() -> Self {
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Self::zero()
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}
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}
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impl Fp {
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/// Returns zero, the additive identity.
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#[inline]
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pub const fn zero() -> Fp {
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Fp([0, 0, 0, 0])
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}
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/// Returns one, the multiplicative identity.
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#[inline]
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pub const fn one() -> Fp {
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R
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}
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/// Doubles this field element.
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#[inline]
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pub const fn double(&self) -> Fp {
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// TODO: This can be achieved more efficiently with a bitshift.
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self.add(self)
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}
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fn from_u512(limbs: [u64; 8]) -> Fp {
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// We reduce an arbitrary 512-bit number by decomposing it into two 256-bit digits
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// with the higher bits multiplied by 2^256. Thus, we perform two reductions
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//
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// 1. the lower bits are multiplied by R^2, as normal
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// 2. the upper bits are multiplied by R^2 * 2^256 = R^3
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//
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// and computing their sum in the field. It remains to see that arbitrary 256-bit
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// numbers can be placed into Montgomery form safely using the reduction. The
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2020-09-01 10:45:20 -07:00
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// reduction works so long as the product is less than R=2^256 multiplied by
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2020-08-22 13:15:39 -07:00
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// the modulus. This holds because for any `c` smaller than the modulus, we have
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// that (2^256 - 1)*c is an acceptable product for the reduction. Therefore, the
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// reduction always works so long as `c` is in the field; in this case it is either the
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// constant `R2` or `R3`.
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let d0 = Fp([limbs[0], limbs[1], limbs[2], limbs[3]]);
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let d1 = Fp([limbs[4], limbs[5], limbs[6], limbs[7]]);
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// Convert to Montgomery form
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d0 * R2 + d1 * R3
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}
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/// Converts from an integer represented in little endian
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/// into its (congruent) `Fp` representation.
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pub const fn from_raw(val: [u64; 4]) -> Self {
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(&Fp(val)).mul(&R2)
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}
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/// Squares this element.
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#[inline]
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pub const fn square(&self) -> Fp {
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let (r1, carry) = mac(0, self.0[0], self.0[1], 0);
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let (r2, carry) = mac(0, self.0[0], self.0[2], carry);
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let (r3, r4) = mac(0, self.0[0], self.0[3], carry);
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let (r3, carry) = mac(r3, self.0[1], self.0[2], 0);
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let (r4, r5) = mac(r4, self.0[1], self.0[3], carry);
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let (r5, r6) = mac(r5, self.0[2], self.0[3], 0);
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let r7 = r6 >> 63;
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let r6 = (r6 << 1) | (r5 >> 63);
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let r5 = (r5 << 1) | (r4 >> 63);
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let r4 = (r4 << 1) | (r3 >> 63);
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let r3 = (r3 << 1) | (r2 >> 63);
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let r2 = (r2 << 1) | (r1 >> 63);
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let r1 = r1 << 1;
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let (r0, carry) = mac(0, self.0[0], self.0[0], 0);
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let (r1, carry) = adc(0, r1, carry);
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let (r2, carry) = mac(r2, self.0[1], self.0[1], carry);
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let (r3, carry) = adc(0, r3, carry);
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let (r4, carry) = mac(r4, self.0[2], self.0[2], carry);
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let (r5, carry) = adc(0, r5, carry);
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let (r6, carry) = mac(r6, self.0[3], self.0[3], carry);
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let (r7, _) = adc(0, r7, carry);
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Fp::montgomery_reduce(r0, r1, r2, r3, r4, r5, r6, r7)
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}
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#[inline(always)]
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const fn montgomery_reduce(
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r0: u64,
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r1: u64,
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r2: u64,
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r3: u64,
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r4: u64,
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r5: u64,
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r6: u64,
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r7: u64,
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) -> Self {
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// The Montgomery reduction here is based on Algorithm 14.32 in
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// Handbook of Applied Cryptography
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// <http://cacr.uwaterloo.ca/hac/about/chap14.pdf>.
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let k = r0.wrapping_mul(INV);
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let (_, carry) = mac(r0, k, MODULUS.0[0], 0);
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let (r1, carry) = mac(r1, k, MODULUS.0[1], carry);
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let (r2, carry) = mac(r2, k, MODULUS.0[2], carry);
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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);
|
|
|
|
|
|
|
|
// Result may be within MODULUS of the correct value
|
|
|
|
(&Fp([r4, r5, r6, r7])).sub(&MODULUS)
|
|
|
|
}
|
|
|
|
|
|
|
|
/// Multiplies `rhs` by `self`, returning the result.
|
|
|
|
#[inline]
|
|
|
|
pub const fn mul(&self, rhs: &Self) -> Self {
|
|
|
|
// 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);
|
|
|
|
|
|
|
|
Fp::montgomery_reduce(r0, r1, r2, r3, r4, r5, r6, r7)
|
|
|
|
}
|
|
|
|
|
|
|
|
/// Subtracts `rhs` from `self`, returning the result.
|
|
|
|
#[inline]
|
|
|
|
pub const fn sub(&self, rhs: &Self) -> Self {
|
|
|
|
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);
|
|
|
|
|
|
|
|
Fp([d0, d1, d2, d3])
|
|
|
|
}
|
|
|
|
|
|
|
|
/// Adds `rhs` to `self`, returning the result.
|
|
|
|
#[inline]
|
|
|
|
pub const fn add(&self, rhs: &Self) -> Self {
|
|
|
|
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.
|
|
|
|
(&Fp([d0, d1, d2, d3])).sub(&MODULUS)
|
|
|
|
}
|
|
|
|
|
|
|
|
/// Negates `self`.
|
|
|
|
#[inline]
|
|
|
|
pub const fn neg(&self) -> Self {
|
|
|
|
// 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 = (((self.0[0] | self.0[1] | self.0[2] | self.0[3]) == 0) as u64).wrapping_sub(1);
|
|
|
|
|
|
|
|
Fp([d0 & mask, d1 & mask, d2 & mask, d3 & mask])
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
impl From<Fp> for [u8; 32] {
|
|
|
|
fn from(value: Fp) -> [u8; 32] {
|
|
|
|
value.to_bytes()
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
2020-08-22 13:15:39 -07:00
|
|
|
impl<'a> From<&'a Fp> for [u8; 32] {
|
|
|
|
fn from(value: &'a Fp) -> [u8; 32] {
|
|
|
|
value.to_bytes()
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
|
|
|
impl Group for Fp {
|
|
|
|
type Scalar = Fp;
|
|
|
|
|
|
|
|
fn group_zero() -> Self {
|
|
|
|
Self::zero()
|
|
|
|
}
|
|
|
|
fn group_add(&mut self, rhs: &Self) {
|
|
|
|
*self = *self + *rhs;
|
|
|
|
}
|
|
|
|
fn group_sub(&mut self, rhs: &Self) {
|
|
|
|
*self = *self - *rhs;
|
|
|
|
}
|
|
|
|
fn group_scale(&mut self, by: &Self::Scalar) {
|
|
|
|
*self = *self * (*by);
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
impl ff::Field for Fp {
|
|
|
|
fn random(mut rng: impl RngCore) -> Self {
|
|
|
|
let mut random_bytes = [0; 64];
|
|
|
|
rng.fill_bytes(&mut random_bytes[..]);
|
2020-08-22 13:15:39 -07:00
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
Self::from_bytes_wide(&random_bytes)
|
2020-08-22 13:15:39 -07:00
|
|
|
}
|
|
|
|
|
|
|
|
fn zero() -> Self {
|
|
|
|
Self::zero()
|
|
|
|
}
|
|
|
|
|
|
|
|
fn one() -> Self {
|
|
|
|
Self::one()
|
|
|
|
}
|
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
fn is_zero(&self) -> bool {
|
|
|
|
self.ct_is_zero().into()
|
2020-08-22 13:15:39 -07:00
|
|
|
}
|
|
|
|
|
|
|
|
fn double(&self) -> Self {
|
|
|
|
self.double()
|
|
|
|
}
|
|
|
|
|
|
|
|
#[inline(always)]
|
|
|
|
fn square(&self) -> Self {
|
|
|
|
self.square()
|
|
|
|
}
|
|
|
|
|
|
|
|
/// Computes the square root of this element, if it exists.
|
|
|
|
fn sqrt(&self) -> CtOption<Self> {
|
2021-01-13 05:56:13 -08:00
|
|
|
let (is_square, res) = self.sqrt_alt();
|
|
|
|
CtOption::new(res, is_square)
|
2020-08-22 13:15:39 -07:00
|
|
|
}
|
|
|
|
|
|
|
|
/// Computes the multiplicative inverse of this element,
|
|
|
|
/// failing if the element is zero.
|
|
|
|
fn invert(&self) -> CtOption<Self> {
|
|
|
|
let tmp = self.pow_vartime(&[
|
2020-12-03 12:13:40 -08:00
|
|
|
0x992d30ecffffffff,
|
|
|
|
0x224698fc094cf91b,
|
2020-08-22 13:15:39 -07:00
|
|
|
0x0,
|
|
|
|
0x4000000000000000,
|
|
|
|
]);
|
|
|
|
|
|
|
|
CtOption::new(tmp, !self.ct_eq(&Self::zero()))
|
|
|
|
}
|
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
fn pow_vartime<S: AsRef<[u64]>>(&self, exp: S) -> Self {
|
|
|
|
let mut res = Self::one();
|
|
|
|
let mut found_one = false;
|
|
|
|
for e in exp.as_ref().iter().rev() {
|
|
|
|
for i in (0..64).rev() {
|
|
|
|
if found_one {
|
|
|
|
res = res.square();
|
|
|
|
}
|
|
|
|
|
|
|
|
if ((*e >> i) & 1) == 1 {
|
|
|
|
found_one = true;
|
|
|
|
res *= self;
|
|
|
|
}
|
|
|
|
}
|
|
|
|
}
|
|
|
|
res
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
|
|
|
#[cfg(not(target_pointer_width = "64"))]
|
|
|
|
type ReprBits = [u32; 8];
|
|
|
|
|
|
|
|
#[cfg(target_pointer_width = "64")]
|
|
|
|
type ReprBits = [u64; 4];
|
|
|
|
|
|
|
|
impl ff::PrimeField for Fp {
|
|
|
|
type Repr = [u8; 32];
|
|
|
|
type ReprBits = ReprBits;
|
|
|
|
|
|
|
|
const NUM_BITS: u32 = 255;
|
|
|
|
const CAPACITY: u32 = 254;
|
|
|
|
const S: u32 = S;
|
|
|
|
|
|
|
|
fn from_repr(repr: Self::Repr) -> Option<Self> {
|
|
|
|
Self::from_bytes(&repr).into()
|
|
|
|
}
|
|
|
|
|
|
|
|
fn to_repr(&self) -> Self::Repr {
|
|
|
|
self.to_bytes()
|
|
|
|
}
|
|
|
|
|
|
|
|
fn to_le_bits(&self) -> BitArray<Lsb0, Self::ReprBits> {
|
|
|
|
let bytes = self.to_bytes();
|
|
|
|
|
|
|
|
#[cfg(not(target_pointer_width = "64"))]
|
|
|
|
let limbs = [
|
|
|
|
u32::from_le_bytes(bytes[0..4].try_into().unwrap()),
|
|
|
|
u32::from_le_bytes(bytes[4..8].try_into().unwrap()),
|
|
|
|
u32::from_le_bytes(bytes[8..12].try_into().unwrap()),
|
|
|
|
u32::from_le_bytes(bytes[12..16].try_into().unwrap()),
|
|
|
|
u32::from_le_bytes(bytes[16..20].try_into().unwrap()),
|
|
|
|
u32::from_le_bytes(bytes[20..24].try_into().unwrap()),
|
|
|
|
u32::from_le_bytes(bytes[24..28].try_into().unwrap()),
|
|
|
|
u32::from_le_bytes(bytes[28..32].try_into().unwrap()),
|
|
|
|
];
|
|
|
|
|
|
|
|
#[cfg(target_pointer_width = "64")]
|
|
|
|
let limbs = [
|
|
|
|
u64::from_le_bytes(bytes[0..8].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[8..16].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[16..24].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[24..32].try_into().unwrap()),
|
|
|
|
];
|
|
|
|
|
|
|
|
BitArray::new(limbs)
|
|
|
|
}
|
|
|
|
|
|
|
|
fn is_odd(&self) -> bool {
|
|
|
|
self.to_bytes()[0] & 1 == 1
|
|
|
|
}
|
|
|
|
|
|
|
|
fn char_le_bits() -> BitArray<Lsb0, Self::ReprBits> {
|
|
|
|
#[cfg(not(target_pointer_width = "64"))]
|
|
|
|
{
|
|
|
|
BitArray::new(MODULUS_LIMBS_32)
|
|
|
|
}
|
|
|
|
|
|
|
|
#[cfg(target_pointer_width = "64")]
|
|
|
|
BitArray::new(MODULUS.0)
|
|
|
|
}
|
|
|
|
|
|
|
|
fn multiplicative_generator() -> Self {
|
|
|
|
GENERATOR
|
|
|
|
}
|
|
|
|
|
|
|
|
fn root_of_unity() -> Self {
|
|
|
|
Self::ROOT_OF_UNITY
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
2021-01-09 16:23:13 -08:00
|
|
|
lazy_static! {
|
2021-01-16 18:24:09 -08:00
|
|
|
// The perfect hash parameters are found by `squareroottab.sage` in zcash/pasta.
|
2021-01-09 16:23:13 -08:00
|
|
|
static ref FP_TABLES: SqrtTables<Fp> = SqrtTables::new(0x11BE, 1098);
|
|
|
|
}
|
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
impl FieldExt for Fp {
|
|
|
|
const ROOT_OF_UNITY: Self = ROOT_OF_UNITY;
|
|
|
|
const ROOT_OF_UNITY_INV: Self = Fp::from_raw([
|
2020-12-03 12:13:40 -08:00
|
|
|
0xf0b87c7db2ce91f6,
|
|
|
|
0x84a0a1d8859f066f,
|
|
|
|
0xb4ed8e647196dad1,
|
|
|
|
0x2cd5282c53116b5c,
|
2020-11-12 16:08:08 -08:00
|
|
|
]);
|
2021-01-04 15:39:52 -08:00
|
|
|
const T_MINUS1_OVER2: [u64; 4] = [
|
|
|
|
0x04a67c8dcc969876,
|
|
|
|
0x0000000011234c7e,
|
|
|
|
0x0000000000000000,
|
|
|
|
0x20000000,
|
|
|
|
];
|
2020-11-12 16:08:08 -08:00
|
|
|
const DELTA: Self = DELTA;
|
|
|
|
const TWO_INV: Self = Fp::from_raw([
|
2020-12-03 12:13:40 -08:00
|
|
|
0xcc96987680000001,
|
|
|
|
0x11234c7e04a67c8d,
|
2020-11-12 16:08:08 -08:00
|
|
|
0x0000000000000000,
|
|
|
|
0x2000000000000000,
|
|
|
|
]);
|
|
|
|
const RESCUE_ALPHA: u64 = 5;
|
|
|
|
const RESCUE_INVALPHA: [u64; 4] = [
|
2020-12-03 12:13:40 -08:00
|
|
|
0xe0f0f3f0cccccccd,
|
|
|
|
0x4e9ee0c9a10a60e2,
|
2020-11-12 16:08:08 -08:00
|
|
|
0x3333333333333333,
|
|
|
|
0x3333333333333333,
|
|
|
|
];
|
|
|
|
const ZETA: Self = Fp::from_raw([
|
2020-12-07 08:42:33 -08:00
|
|
|
0x1dad5ebdfdfe4ab9,
|
|
|
|
0x1d1f8bd237ad3149,
|
|
|
|
0x2caad5dc57aab1b0,
|
|
|
|
0x12ccca834acdba71,
|
2020-11-12 16:08:08 -08:00
|
|
|
]);
|
|
|
|
|
2021-01-09 16:23:13 -08:00
|
|
|
fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {
|
|
|
|
FP_TABLES.sqrt_ratio(num, div)
|
2021-01-04 15:39:52 -08:00
|
|
|
}
|
|
|
|
|
2021-01-10 08:51:18 -08:00
|
|
|
fn sqrt_alt(&self) -> (Choice, Self) {
|
|
|
|
FP_TABLES.sqrt_alt(self)
|
|
|
|
}
|
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
fn ct_is_zero(&self) -> Choice {
|
|
|
|
self.ct_eq(&Self::zero())
|
|
|
|
}
|
|
|
|
|
|
|
|
fn from_u64(v: u64) -> Self {
|
|
|
|
Fp::from_raw([v as u64, 0, 0, 0])
|
|
|
|
}
|
|
|
|
|
|
|
|
fn from_u128(v: u128) -> Self {
|
|
|
|
Fp::from_raw([v as u64, (v >> 64) as u64, 0, 0])
|
|
|
|
}
|
|
|
|
|
2020-08-22 13:15:39 -07:00
|
|
|
/// Attempts to convert a little-endian byte representation of
|
|
|
|
/// a scalar into a `Fp`, failing if the input is not canonical.
|
|
|
|
fn from_bytes(bytes: &[u8; 32]) -> CtOption<Fp> {
|
|
|
|
let mut tmp = Fp([0, 0, 0, 0]);
|
|
|
|
|
|
|
|
tmp.0[0] = u64::from_le_bytes(bytes[0..8].try_into().unwrap());
|
|
|
|
tmp.0[1] = u64::from_le_bytes(bytes[8..16].try_into().unwrap());
|
|
|
|
tmp.0[2] = u64::from_le_bytes(bytes[16..24].try_into().unwrap());
|
|
|
|
tmp.0[3] = u64::from_le_bytes(bytes[24..32].try_into().unwrap());
|
|
|
|
|
|
|
|
// Try to subtract the modulus
|
|
|
|
let (_, borrow) = sbb(tmp.0[0], MODULUS.0[0], 0);
|
|
|
|
let (_, borrow) = sbb(tmp.0[1], MODULUS.0[1], borrow);
|
|
|
|
let (_, borrow) = sbb(tmp.0[2], MODULUS.0[2], borrow);
|
|
|
|
let (_, borrow) = sbb(tmp.0[3], MODULUS.0[3], borrow);
|
|
|
|
|
|
|
|
// If the element is smaller than MODULUS then the
|
|
|
|
// subtraction will underflow, producing a borrow value
|
|
|
|
// of 0xffff...ffff. Otherwise, it'll be zero.
|
|
|
|
let is_some = (borrow as u8) & 1;
|
|
|
|
|
|
|
|
// Convert to Montgomery form by computing
|
|
|
|
// (a.R^0 * R^2) / R = a.R
|
|
|
|
tmp *= &R2;
|
|
|
|
|
|
|
|
CtOption::new(tmp, Choice::from(is_some))
|
|
|
|
}
|
|
|
|
|
|
|
|
/// Converts an element of `Fp` into a byte representation in
|
|
|
|
/// little-endian byte order.
|
|
|
|
fn to_bytes(&self) -> [u8; 32] {
|
|
|
|
// Turn into canonical form by computing
|
|
|
|
// (a.R) / R = a
|
|
|
|
let tmp = Fp::montgomery_reduce(self.0[0], self.0[1], self.0[2], self.0[3], 0, 0, 0, 0);
|
|
|
|
|
|
|
|
let mut res = [0; 32];
|
|
|
|
res[0..8].copy_from_slice(&tmp.0[0].to_le_bytes());
|
|
|
|
res[8..16].copy_from_slice(&tmp.0[1].to_le_bytes());
|
|
|
|
res[16..24].copy_from_slice(&tmp.0[2].to_le_bytes());
|
|
|
|
res[24..32].copy_from_slice(&tmp.0[3].to_le_bytes());
|
|
|
|
|
|
|
|
res
|
|
|
|
}
|
|
|
|
|
|
|
|
/// Converts a 512-bit little endian integer into
|
|
|
|
/// a `Fp` by reducing by the modulus.
|
|
|
|
fn from_bytes_wide(bytes: &[u8; 64]) -> Fp {
|
|
|
|
Fp::from_u512([
|
|
|
|
u64::from_le_bytes(bytes[0..8].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[8..16].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[16..24].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[24..32].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[32..40].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[40..48].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[48..56].try_into().unwrap()),
|
|
|
|
u64::from_le_bytes(bytes[56..64].try_into().unwrap()),
|
|
|
|
])
|
|
|
|
}
|
|
|
|
|
|
|
|
fn get_lower_128(&self) -> u128 {
|
|
|
|
let tmp = Fp::montgomery_reduce(self.0[0], self.0[1], self.0[2], self.0[3], 0, 0, 0, 0);
|
|
|
|
|
|
|
|
u128::from(tmp.0[0]) | (u128::from(tmp.0[1]) << 64)
|
|
|
|
}
|
2021-01-04 15:39:52 -08:00
|
|
|
|
|
|
|
fn get_lower_32(&self) -> u32 {
|
|
|
|
// TODO: don't reduce, just hash the Montgomery form. (Requires rebuilding perfect hash table.)
|
|
|
|
let tmp = Fp::montgomery_reduce(self.0[0], self.0[1], self.0[2], self.0[3], 0, 0, 0, 0);
|
|
|
|
|
|
|
|
tmp.0[0] as u32
|
|
|
|
}
|
2021-01-10 07:22:24 -08:00
|
|
|
|
|
|
|
fn pow_by_t_minus1_over2(&self) -> Self {
|
|
|
|
let sqr = |x: Fp, i: u32| (0..i).fold(x, |x, _| x.square());
|
|
|
|
|
|
|
|
let r10 = self.square();
|
|
|
|
let r11 = r10 * self;
|
|
|
|
let r110 = r11.square();
|
|
|
|
let r111 = r110 * self;
|
|
|
|
let r1001 = r111 * r10;
|
|
|
|
let r1101 = r111 * r110;
|
|
|
|
let ra = sqr(*self, 129) * self;
|
|
|
|
let rb = sqr(ra, 7) * r1001;
|
|
|
|
let rc = sqr(rb, 7) * r1101;
|
|
|
|
let rd = sqr(rc, 4) * r11;
|
|
|
|
let re = sqr(rd, 6) * r111;
|
|
|
|
let rf = sqr(re, 3) * r111;
|
|
|
|
let rg = sqr(rf, 10) * r1001;
|
|
|
|
let rh = sqr(rg, 5) * r1001;
|
|
|
|
let ri = sqr(rh, 4) * r1001;
|
|
|
|
let rj = sqr(ri, 3) * r111;
|
|
|
|
let rk = sqr(rj, 4) * r1001;
|
|
|
|
let rl = sqr(rk, 5) * r11;
|
|
|
|
let rm = sqr(rl, 4) * r111;
|
|
|
|
let rn = sqr(rm, 4) * r11;
|
|
|
|
let ro = sqr(rn, 6) * r1001;
|
|
|
|
let rp = sqr(ro, 5) * r1101;
|
|
|
|
let rq = sqr(rp, 4) * r11;
|
|
|
|
let rr = sqr(rq, 7) * r111;
|
|
|
|
let rs = sqr(rr, 3) * r11;
|
|
|
|
let rt = rs.square();
|
|
|
|
rt
|
|
|
|
}
|
2020-08-22 13:15:39 -07:00
|
|
|
}
|
|
|
|
|
2020-11-12 16:08:08 -08:00
|
|
|
#[cfg(test)]
|
|
|
|
use ff::{Field, PrimeField};
|
|
|
|
|
2020-08-22 13:15:39 -07:00
|
|
|
#[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);
|
|
|
|
}
|
|
|
|
|
2020-12-03 12:13:40 -08:00
|
|
|
#[test]
|
|
|
|
fn test_rescue() {
|
|
|
|
// NB: TWO_INV is standing in as a "random" field element
|
|
|
|
assert_eq!(
|
|
|
|
Fp::TWO_INV
|
|
|
|
.pow_vartime(&[Fp::RESCUE_ALPHA, 0, 0, 0])
|
|
|
|
.pow_vartime(&Fp::RESCUE_INVALPHA),
|
|
|
|
Fp::TWO_INV
|
|
|
|
);
|
|
|
|
}
|
|
|
|
|
|
|
|
#[test]
|
|
|
|
fn test_sqrt() {
|
|
|
|
// NB: TWO_INV is standing in as a "random" field element
|
|
|
|
let v = (Fp::TWO_INV).square().sqrt().unwrap();
|
|
|
|
assert!(v == Fp::TWO_INV || (-v) == Fp::TWO_INV);
|
|
|
|
}
|
|
|
|
|
2021-01-12 16:12:12 -08:00
|
|
|
#[test]
|
|
|
|
fn test_pow_by_t_minus1_over2() {
|
|
|
|
// NB: TWO_INV is standing in as a "random" field element
|
|
|
|
let v = (Fp::TWO_INV).pow_by_t_minus1_over2();
|
|
|
|
assert!(v == ff::Field::pow_vartime(&Fp::TWO_INV, &Fp::T_MINUS1_OVER2));
|
|
|
|
}
|
|
|
|
|
2021-01-04 15:39:52 -08:00
|
|
|
#[test]
|
2021-01-10 08:51:18 -08:00
|
|
|
fn test_sqrt_ratio_and_alt() {
|
2021-01-04 15:39:52 -08:00
|
|
|
// (true, sqrt(num/div)), if num and div are nonzero and num/div is a square in the field
|
|
|
|
let num = (Fp::TWO_INV).square();
|
|
|
|
let div = Fp::from_u64(25);
|
2021-01-10 08:51:18 -08:00
|
|
|
let div_inverse = div.invert().unwrap();
|
2021-01-04 15:39:52 -08:00
|
|
|
let expected = Fp::TWO_INV * Fp::from_u64(5).invert().unwrap();
|
|
|
|
let (is_square, v) = Fp::sqrt_ratio(&num, &div);
|
|
|
|
assert!(bool::from(is_square));
|
|
|
|
assert!(v == expected || (-v) == expected);
|
|
|
|
|
2021-01-10 08:51:18 -08:00
|
|
|
let (is_square_alt, v_alt) = Fp::sqrt_alt(&(num * div_inverse));
|
|
|
|
assert!(bool::from(is_square_alt));
|
|
|
|
assert!(v_alt == v);
|
|
|
|
|
2021-01-04 15:39:52 -08:00
|
|
|
// (false, sqrt(ROOT_OF_UNITY * num/div)), if num and div are nonzero and num/div is a nonsquare in the field
|
|
|
|
let num = num * Fp::ROOT_OF_UNITY;
|
|
|
|
let expected = Fp::TWO_INV * Fp::ROOT_OF_UNITY * Fp::from_u64(5).invert().unwrap();
|
|
|
|
let (is_square, v) = Fp::sqrt_ratio(&num, &div);
|
|
|
|
assert!(!bool::from(is_square));
|
|
|
|
assert!(v == expected || (-v) == expected);
|
|
|
|
|
2021-01-10 08:51:18 -08:00
|
|
|
let (is_square_alt, v_alt) = Fp::sqrt_alt(&(num * div_inverse));
|
|
|
|
assert!(!bool::from(is_square_alt));
|
|
|
|
assert!(v_alt == v);
|
|
|
|
|
2021-01-04 15:39:52 -08:00
|
|
|
// (true, 0), if num is zero
|
|
|
|
let num = Fp::zero();
|
|
|
|
let expected = Fp::zero();
|
|
|
|
let (is_square, v) = Fp::sqrt_ratio(&num, &div);
|
|
|
|
assert!(bool::from(is_square));
|
|
|
|
assert!(v == expected);
|
|
|
|
|
2021-01-10 08:51:18 -08:00
|
|
|
let (is_square_alt, v_alt) = Fp::sqrt_alt(&(num * div_inverse));
|
|
|
|
assert!(bool::from(is_square_alt));
|
|
|
|
assert!(v_alt == v);
|
|
|
|
|
2021-01-04 15:39:52 -08:00
|
|
|
// (false, 0), if num is nonzero and div is zero
|
|
|
|
let num = (Fp::TWO_INV).square();
|
|
|
|
let div = Fp::zero();
|
|
|
|
let expected = Fp::zero();
|
|
|
|
let (is_square, v) = Fp::sqrt_ratio(&num, &div);
|
|
|
|
assert!(!bool::from(is_square));
|
|
|
|
assert!(v == expected);
|
|
|
|
}
|
|
|
|
|
2020-08-22 13:15:39 -07:00
|
|
|
#[test]
|
|
|
|
fn test_zeta() {
|
|
|
|
assert_eq!(
|
|
|
|
format!("{:?}", Fp::ZETA),
|
2020-12-07 08:42:33 -08:00
|
|
|
"0x12ccca834acdba712caad5dc57aab1b01d1f8bd237ad31491dad5ebdfdfe4ab9"
|
2020-08-22 13:15:39 -07:00
|
|
|
);
|
|
|
|
|
|
|
|
let a = Fp::ZETA;
|
2020-11-12 00:09:26 -08:00
|
|
|
assert!(a != Fp::one());
|
2020-08-22 13:15:39 -07:00
|
|
|
let b = a * a;
|
2020-11-12 00:09:26 -08:00
|
|
|
assert!(b != Fp::one());
|
2020-08-22 13:15:39 -07:00
|
|
|
let c = b * a;
|
2020-11-12 00:09:26 -08:00
|
|
|
assert!(c == Fp::one());
|
2020-08-22 13:15:39 -07:00
|
|
|
}
|
|
|
|
|
2020-12-03 12:13:40 -08:00
|
|
|
#[test]
|
|
|
|
fn test_root_of_unity() {
|
|
|
|
assert_eq!(
|
2020-12-11 10:54:32 -08:00
|
|
|
Fp::ROOT_OF_UNITY.pow_vartime(&[1 << Fp::S, 0, 0, 0]),
|
2020-12-03 12:13:40 -08:00
|
|
|
Fp::one()
|
|
|
|
);
|
|
|
|
}
|
|
|
|
|
2020-08-22 13:15:39 -07:00
|
|
|
#[test]
|
|
|
|
fn test_inv_root_of_unity() {
|
|
|
|
assert_eq!(Fp::ROOT_OF_UNITY_INV, Fp::ROOT_OF_UNITY.invert().unwrap());
|
|
|
|
}
|
|
|
|
|
|
|
|
#[test]
|
|
|
|
fn test_inv_2() {
|
|
|
|
assert_eq!(Fp::TWO_INV, Fp::from(2).invert().unwrap());
|
|
|
|
}
|
2020-08-31 09:10:05 -07:00
|
|
|
|
|
|
|
#[test]
|
|
|
|
fn test_delta() {
|
2020-12-11 10:54:32 -08:00
|
|
|
assert_eq!(Fp::DELTA, GENERATOR.pow(&[1u64 << Fp::S, 0, 0, 0]));
|
2020-12-13 09:37:32 -08:00
|
|
|
assert_eq!(
|
|
|
|
Fp::DELTA,
|
|
|
|
Fp::multiplicative_generator().pow(&[1u64 << Fp::S, 0, 0, 0])
|
|
|
|
);
|
2020-08-31 09:10:05 -07:00
|
|
|
}
|
2020-12-03 12:13:40 -08:00
|
|
|
|
|
|
|
#[cfg(not(target_pointer_width = "64"))]
|
|
|
|
#[test]
|
|
|
|
fn consistent_modulus_limbs() {
|
|
|
|
for (a, &b) in MODULUS
|
|
|
|
.0
|
|
|
|
.iter()
|
|
|
|
.flat_map(|&limb| {
|
|
|
|
Some(limb as u32)
|
|
|
|
.into_iter()
|
|
|
|
.chain(Some((limb >> 32) as u32))
|
|
|
|
})
|
|
|
|
.zip(MODULUS_LIMBS_32.iter())
|
|
|
|
{
|
|
|
|
assert_eq!(a, b);
|
|
|
|
}
|
|
|
|
}
|
|
|
|
|
|
|
|
#[test]
|
|
|
|
fn test_from_u512() {
|
|
|
|
assert_eq!(
|
|
|
|
Fp::from_raw([
|
|
|
|
0x3daec14d565241d9,
|
|
|
|
0x0b7af45b6073944b,
|
|
|
|
0xea5b8bd611a5bd4c,
|
|
|
|
0x150160330625db3d
|
|
|
|
]),
|
|
|
|
Fp::from_u512([
|
|
|
|
0xee155641297678a1,
|
|
|
|
0xd83e156bdbfdbe65,
|
|
|
|
0xd9ccd834c68ba0b5,
|
|
|
|
0xf508ede312272758,
|
|
|
|
0x038df7cbf8228e89,
|
|
|
|
0x3505a1e4a3c74b41,
|
|
|
|
0xbfa46f775eb82db3,
|
|
|
|
0x26ebe27e262f471d
|
|
|
|
])
|
|
|
|
);
|
|
|
|
}
|