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pasta_curves/src/fields/fq.rs

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use core::fmt;
use core::ops::{Add, Mul, Neg, Sub};
use ff::{Field, FromUniformBytes, PrimeField, WithSmallOrderMulGroup};
use rand::RngCore;
use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
#[cfg(feature = "sqrt-table")]
use lazy_static::lazy_static;
#[cfg(feature = "bits")]
use ff::{FieldBits, PrimeFieldBits};
use crate::arithmetic::{adc, mac, sbb, SqrtTableHelpers};
#[cfg(feature = "sqrt-table")]
use crate::arithmetic::SqrtTables;
/// This represents an element of $\mathbb{F}_q$ where
///
/// `q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001`
///
/// is the base field of the Vesta curve.
// The internal representation of this type is four 64-bit unsigned
// integers in little-endian order. `Fq` values are always in
// Montgomery form; i.e., Fq(a) = aR mod q, with R = 2^256.
#[derive(Clone, Copy, Eq)]
#[repr(transparent)]
pub struct Fq(pub(crate) [u64; 4]);
impl fmt::Debug for Fq {
fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
let tmp = self.to_repr();
write!(f, "0x")?;
for &b in tmp.iter().rev() {
write!(f, "{:02x}", b)?;
}
Ok(())
}
}
impl From<bool> for Fq {
fn from(bit: bool) -> Fq {
if bit {
Fq::one()
} else {
Fq::zero()
}
}
}
impl From<u64> for Fq {
fn from(val: u64) -> Fq {
Fq([val, 0, 0, 0]) * R2
}
}
impl ConstantTimeEq for Fq {
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 Fq {
#[inline]
fn eq(&self, other: &Self) -> bool {
self.ct_eq(other).unwrap_u8() == 1
}
}
impl core::cmp::Ord for Fq {
fn cmp(&self, other: &Self) -> core::cmp::Ordering {
let left = self.to_repr();
let right = other.to_repr();
left.iter()
.zip(right.iter())
.rev()
.find_map(|(left_byte, right_byte)| match left_byte.cmp(right_byte) {
core::cmp::Ordering::Equal => None,
res => Some(res),
})
.unwrap_or(core::cmp::Ordering::Equal)
}
}
impl core::cmp::PartialOrd for Fq {
fn partial_cmp(&self, other: &Self) -> Option<core::cmp::Ordering> {
Some(self.cmp(other))
}
}
impl ConditionallySelectable for Fq {
fn conditional_select(a: &Self, b: &Self, choice: Choice) -> Self {
Fq([
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
/// q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001
const MODULUS: Fq = Fq([
0x8c46eb2100000001,
0x224698fc0994a8dd,
0x0,
0x4000000000000000,
]);
/// The modulus as u32 limbs.
#[cfg(not(target_pointer_width = "64"))]
const MODULUS_LIMBS_32: [u32; 8] = [
0x0000_0001,
0x8c46_eb21,
0x0994_a8dd,
0x2246_98fc,
0x0000_0000,
0x0000_0000,
0x0000_0000,
0x4000_0000,
];
impl<'a> Neg for &'a Fq {
type Output = Fq;
#[inline]
fn neg(self) -> Fq {
self.neg()
}
}
impl Neg for Fq {
type Output = Fq;
#[inline]
fn neg(self) -> Fq {
-&self
}
}
impl<'a, 'b> Sub<&'b Fq> for &'a Fq {
type Output = Fq;
#[inline]
fn sub(self, rhs: &'b Fq) -> Fq {
self.sub(rhs)
}
}
impl<'a, 'b> Add<&'b Fq> for &'a Fq {
type Output = Fq;
#[inline]
fn add(self, rhs: &'b Fq) -> Fq {
self.add(rhs)
}
}
impl<'a, 'b> Mul<&'b Fq> for &'a Fq {
type Output = Fq;
#[inline]
fn mul(self, rhs: &'b Fq) -> Fq {
self.mul(rhs)
}
}
impl_binops_additive!(Fq, Fq);
impl_binops_multiplicative!(Fq, Fq);
impl<T: ::core::borrow::Borrow<Fq>> ::core::iter::Sum<T> for Fq {
fn sum<I: Iterator<Item = T>>(iter: I) -> Self {
iter.fold(Self::ZERO, |acc, item| acc + item.borrow())
}
}
impl<T: ::core::borrow::Borrow<Fq>> ::core::iter::Product<T> for Fq {
fn product<I: Iterator<Item = T>>(iter: I) -> Self {
iter.fold(Self::ONE, |acc, item| acc * item.borrow())
}
}
/// INV = -(q^{-1} mod 2^64) mod 2^64
const INV: u64 = 0x8c46eb20ffffffff;
/// R = 2^256 mod q
const R: Fq = Fq([
0x5b2b3e9cfffffffd,
0x992c350be3420567,
0xffffffffffffffff,
0x3fffffffffffffff,
]);
/// R^2 = 2^512 mod q
const R2: Fq = Fq([
0xfc9678ff0000000f,
0x67bb433d891a16e3,
0x7fae231004ccf590,
0x096d41af7ccfdaa9,
]);
/// R^3 = 2^768 mod q
const R3: Fq = Fq([
0x008b421c249dae4c,
0xe13bda50dba41326,
0x88fececb8e15cb63,
0x07dd97a06e6792c8,
]);
/// `GENERATOR = 5 mod q` is a generator of the `q - 1` order multiplicative
/// subgroup, or in other words a primitive root of the field.
const GENERATOR: Fq = Fq::from_raw([
0x0000_0000_0000_0005,
0x0000_0000_0000_0000,
0x0000_0000_0000_0000,
0x0000_0000_0000_0000,
]);
const S: u32 = 32;
/// GENERATOR^t where t * 2^s + 1 = q
/// with t odd. In other words, this
/// is a 2^s root of unity.
const ROOT_OF_UNITY: Fq = Fq::from_raw([
0xa70e2c1102b6d05f,
0x9bb97ea3c106f049,
0x9e5c4dfd492ae26e,
0x2de6a9b8746d3f58,
]);
/// GENERATOR^{2^s} where t * 2^s + 1 = q
/// with t odd. In other words, this
/// is a t root of unity.
const DELTA: Fq = Fq::from_raw([
0x8494392472d1683c,
0xe3ac3376541d1140,
0x06f0a88e7f7949f8,
0x2237d54423724166,
]);
/// `(t - 1) // 2` where t * 2^s + 1 = p with t odd.
#[cfg(any(test, not(feature = "sqrt-table")))]
const T_MINUS1_OVER2: [u64; 4] = [
0x04ca_546e_c623_7590,
0x0000_0000_1123_4c7e,
0x0000_0000_0000_0000,
0x0000_0000_2000_0000,
];
impl Default for Fq {
#[inline]
fn default() -> Self {
Self::zero()
}
}
impl Fq {
/// Returns zero, the additive identity.
#[inline]
pub const fn zero() -> Fq {
Fq([0, 0, 0, 0])
}
/// Returns one, the multiplicative identity.
#[inline]
pub const fn one() -> Fq {
R
}
/// Doubles this field element.
#[inline]
pub const fn double(&self) -> Fq {
// TODO: This can be achieved more efficiently with a bitshift.
self.add(self)
}
fn from_u512(limbs: [u64; 8]) -> Fq {
// We reduce an arbitrary 512-bit number by decomposing it into two 256-bit digits
// with the higher bits multiplied by 2^256. Thus, we perform two reductions
//
// 1. the lower bits are multiplied by R^2, as normal
// 2. the upper bits are multiplied by R^2 * 2^256 = R^3
//
// and computing their sum in the field. It remains to see that arbitrary 256-bit
// numbers can be placed into Montgomery form safely using the reduction. The
// reduction works so long as the product is less than R=2^256 multiplied by
// the modulus. This holds because for any `c` smaller than the modulus, we have
// that (2^256 - 1)*c is an acceptable product for the reduction. Therefore, the
// reduction always works so long as `c` is in the field; in this case it is either the
// constant `R2` or `R3`.
let d0 = Fq([limbs[0], limbs[1], limbs[2], limbs[3]]);
let d1 = Fq([limbs[4], limbs[5], limbs[6], limbs[7]]);
// Convert to Montgomery form
d0 * R2 + d1 * R3
}
/// Converts from an integer represented in little endian
/// into its (congruent) `Fq` representation.
pub const fn from_raw(val: [u64; 4]) -> Self {
(&Fq(val)).mul(&R2)
}
/// Squares this element.
#[cfg_attr(not(feature = "uninline-portable"), inline)]
pub const fn square(&self) -> Fq {
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);
Fq::montgomery_reduce(r0, r1, r2, r3, r4, r5, r6, r7)
}
#[allow(clippy::too_many_arguments)]
#[cfg_attr(not(feature = "uninline-portable"), inline(always))]
const 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);
// Result may be within MODULUS of the correct value
(&Fq([r4, r5, r6, r7])).sub(&MODULUS)
}
/// Multiplies `rhs` by `self`, returning the result.
#[cfg_attr(not(feature = "uninline-portable"), 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);
Fq::montgomery_reduce(r0, r1, r2, r3, r4, r5, r6, r7)
}
/// Subtracts `rhs` from `self`, returning the result.
#[cfg_attr(not(feature = "uninline-portable"), 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);
Fq([d0, d1, d2, d3])
}
/// Adds `rhs` to `self`, returning the result.
#[cfg_attr(not(feature = "uninline-portable"), 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.
(&Fq([d0, d1, d2, d3])).sub(&MODULUS)
}
/// Negates `self`.
#[cfg_attr(not(feature = "uninline-portable"), 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);
Fq([d0 & mask, d1 & mask, d2 & mask, d3 & mask])
}
}
impl From<Fq> for [u8; 32] {
fn from(value: Fq) -> [u8; 32] {
value.to_repr()
}
}
impl<'a> From<&'a Fq> for [u8; 32] {
fn from(value: &'a Fq) -> [u8; 32] {
value.to_repr()
}
}
impl ff::Field for Fq {
const ZERO: Self = Self::zero();
const ONE: Self = Self::one();
fn random(mut rng: impl RngCore) -> Self {
Self::from_u512([
rng.next_u64(),
rng.next_u64(),
rng.next_u64(),
rng.next_u64(),
rng.next_u64(),
rng.next_u64(),
rng.next_u64(),
rng.next_u64(),
])
}
fn double(&self) -> Self {
self.double()
}
#[inline(always)]
fn square(&self) -> Self {
self.square()
}
fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {
#[cfg(feature = "sqrt-table")]
{
FQ_TABLES.sqrt_ratio(num, div)
}
#[cfg(not(feature = "sqrt-table"))]
ff::helpers::sqrt_ratio_generic(num, div)
}
#[cfg(feature = "sqrt-table")]
fn sqrt_alt(&self) -> (Choice, Self) {
FQ_TABLES.sqrt_alt(self)
}
/// Computes the square root of this element, if it exists.
fn sqrt(&self) -> CtOption<Self> {
#[cfg(feature = "sqrt-table")]
{
let (is_square, res) = FQ_TABLES.sqrt_alt(self);
CtOption::new(res, is_square)
}
#[cfg(not(feature = "sqrt-table"))]
ff::helpers::sqrt_tonelli_shanks(self, &T_MINUS1_OVER2)
}
/// Computes the multiplicative inverse of this element,
/// failing if the element is zero.
fn invert(&self) -> CtOption<Self> {
let tmp = self.pow_vartime(&[
0x8c46eb20ffffffff,
0x224698fc0994a8dd,
0x0,
0x4000000000000000,
]);
CtOption::new(tmp, !self.ct_eq(&Self::zero()))
}
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
}
}
impl ff::PrimeField for Fq {
type Repr = [u8; 32];
const MODULUS: &'static str =
"0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001";
const NUM_BITS: u32 = 255;
const CAPACITY: u32 = 254;
const TWO_INV: Self = Fq::from_raw([
0xc623759080000001,
0x11234c7e04ca546e,
0x0000000000000000,
0x2000000000000000,
]);
const MULTIPLICATIVE_GENERATOR: Self = GENERATOR;
const S: u32 = S;
const ROOT_OF_UNITY: Self = ROOT_OF_UNITY;
const ROOT_OF_UNITY_INV: Self = Fq::from_raw([
0x57eecda0a84b6836,
0x4ad38b9084b8a80c,
0xf4c8f353124086c1,
0x2235e1a7415bf936,
]);
const DELTA: Self = DELTA;
fn from_u128(v: u128) -> Self {
Fq::from_raw([v as u64, (v >> 64) as u64, 0, 0])
}
fn from_repr(repr: Self::Repr) -> CtOption<Self> {
let mut tmp = Fq([0, 0, 0, 0]);
tmp.0[0] = u64::from_le_bytes(repr[0..8].try_into().unwrap());
tmp.0[1] = u64::from_le_bytes(repr[8..16].try_into().unwrap());
tmp.0[2] = u64::from_le_bytes(repr[16..24].try_into().unwrap());
tmp.0[3] = u64::from_le_bytes(repr[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))
}
fn to_repr(&self) -> Self::Repr {
// Turn into canonical form by computing
// (a.R) / R = a
let tmp = Fq::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
}
fn is_odd(&self) -> Choice {
Choice::from(self.to_repr()[0] & 1)
}
}
#[cfg(all(feature = "bits", not(target_pointer_width = "64")))]
type ReprBits = [u32; 8];
#[cfg(all(feature = "bits", target_pointer_width = "64"))]
type ReprBits = [u64; 4];
#[cfg(feature = "bits")]
impl PrimeFieldBits for Fq {
type ReprBits = ReprBits;
fn to_le_bits(&self) -> FieldBits<Self::ReprBits> {
let bytes = self.to_repr();
#[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()),
];
FieldBits::new(limbs)
}
fn char_le_bits() -> FieldBits<Self::ReprBits> {
#[cfg(not(target_pointer_width = "64"))]
{
FieldBits::new(MODULUS_LIMBS_32)
}
#[cfg(target_pointer_width = "64")]
FieldBits::new(MODULUS.0)
}
}
#[cfg(feature = "sqrt-table")]
lazy_static! {
// The perfect hash parameters are found by `squareroottab.sage` in zcash/pasta.
#[cfg_attr(docsrs, doc(cfg(feature = "sqrt-table")))]
static ref FQ_TABLES: SqrtTables<Fq> = SqrtTables::new(0x116A9E, 1206);
}
impl SqrtTableHelpers for Fq {
fn pow_by_t_minus1_over2(&self) -> Self {
let sqr = |x: Fq, i: u32| (0..i).fold(x, |x, _| x.square());
let s10 = self.square();
let s11 = s10 * self;
let s111 = s11.square() * self;
let s1001 = s111 * s10;
let s1011 = s1001 * s10;
let s1101 = s1011 * s10;
let sa = sqr(*self, 129) * self;
let sb = sqr(sa, 7) * s1001;
let sc = sqr(sb, 7) * s1101;
let sd = sqr(sc, 4) * s11;
let se = sqr(sd, 6) * s111;
let sf = sqr(se, 3) * s111;
let sg = sqr(sf, 10) * s1001;
let sh = sqr(sg, 4) * s1001;
let si = sqr(sh, 5) * s1001;
let sj = sqr(si, 5) * s1001;
let sk = sqr(sj, 3) * s1001;
let sl = sqr(sk, 4) * s1011;
let sm = sqr(sl, 4) * s1011;
let sn = sqr(sm, 5) * s11;
let so = sqr(sn, 4) * self;
let sp = sqr(so, 5) * s11;
let sq = sqr(sp, 4) * s111;
let sr = sqr(sq, 5) * s1011;
let ss = sqr(sr, 3) * self;
sqr(ss, 4) // st
}
fn get_lower_32(&self) -> u32 {
// TODO: don't reduce, just hash the Montgomery form. (Requires rebuilding perfect hash table.)
let tmp = Fq::montgomery_reduce(self.0[0], self.0[1], self.0[2], self.0[3], 0, 0, 0, 0);
tmp.0[0] as u32
}
}
impl WithSmallOrderMulGroup<3> for Fq {
const ZETA: Self = Fq::from_raw([
0x2aa9d2e050aa0e4f,
0x0fed467d47c033af,
0x511db4d81cf70f5a,
0x06819a58283e528e,
]);
}
impl FromUniformBytes<64> for Fq {
/// Converts a 512-bit little endian integer into
/// a `Fq` by reducing by the modulus.
fn from_uniform_bytes(bytes: &[u8; 64]) -> Fq {
Fq::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()),
])
}
}
#[cfg(feature = "gpu")]
impl ec_gpu::GpuName for Fq {
fn name() -> alloc::string::String {
ec_gpu::name!()
}
}
#[cfg(feature = "gpu")]
impl ec_gpu::GpuField for Fq {
fn one() -> alloc::vec::Vec<u32> {
crate::fields::u64_to_u32(&R.0[..])
}
fn r2() -> alloc::vec::Vec<u32> {
crate::fields::u64_to_u32(&R2.0[..])
}
fn modulus() -> alloc::vec::Vec<u32> {
crate::fields::u64_to_u32(&MODULUS.0[..])
}
}
#[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);
}
#[test]
fn test_sqrt() {
// NB: TWO_INV is standing in as a "random" field element
let v = (Fq::TWO_INV).square().sqrt().unwrap();
assert!(v == Fq::TWO_INV || (-v) == Fq::TWO_INV);
}
#[test]
fn test_sqrt_32bit_overflow() {
assert!((Fq::from(5)).sqrt().is_none().unwrap_u8() == 1);
}
#[test]
fn test_pow_by_t_minus1_over2() {
// NB: TWO_INV is standing in as a "random" field element
let v = (Fq::TWO_INV).pow_by_t_minus1_over2();
assert!(v == ff::Field::pow_vartime(&Fq::TWO_INV, &T_MINUS1_OVER2));
}
#[test]
fn test_sqrt_ratio_and_alt() {
// (true, sqrt(num/div)), if num and div are nonzero and num/div is a square in the field
let num = (Fq::TWO_INV).square();
let div = Fq::from(25);
let div_inverse = div.invert().unwrap();
let expected = Fq::TWO_INV * Fq::from(5).invert().unwrap();
let (is_square, v) = Fq::sqrt_ratio(&num, &div);
assert!(bool::from(is_square));
assert!(v == expected || (-v) == expected);
let (is_square_alt, v_alt) = Fq::sqrt_alt(&(num * div_inverse));
assert!(bool::from(is_square_alt));
assert!(v_alt == v);
// (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 * Fq::ROOT_OF_UNITY;
let expected = Fq::TWO_INV * Fq::ROOT_OF_UNITY * Fq::from(5).invert().unwrap();
let (is_square, v) = Fq::sqrt_ratio(&num, &div);
assert!(!bool::from(is_square));
assert!(v == expected || (-v) == expected);
let (is_square_alt, v_alt) = Fq::sqrt_alt(&(num * div_inverse));
assert!(!bool::from(is_square_alt));
assert!(v_alt == v);
// (true, 0), if num is zero
let num = Fq::zero();
let expected = Fq::zero();
let (is_square, v) = Fq::sqrt_ratio(&num, &div);
assert!(bool::from(is_square));
assert!(v == expected);
let (is_square_alt, v_alt) = Fq::sqrt_alt(&(num * div_inverse));
assert!(bool::from(is_square_alt));
assert!(v_alt == v);
// (false, 0), if num is nonzero and div is zero
let num = (Fq::TWO_INV).square();
let div = Fq::zero();
let expected = Fq::zero();
let (is_square, v) = Fq::sqrt_ratio(&num, &div);
assert!(!bool::from(is_square));
assert!(v == expected);
}
#[test]
fn test_zeta() {
assert_eq!(
format!("{:?}", Fq::ZETA),
"0x06819a58283e528e511db4d81cf70f5a0fed467d47c033af2aa9d2e050aa0e4f"
);
let a = Fq::ZETA;
assert!(a != Fq::one());
let b = a * a;
assert!(b != Fq::one());
let c = b * a;
assert!(c == Fq::one());
}
#[test]
fn test_root_of_unity() {
assert_eq!(
Fq::ROOT_OF_UNITY.pow_vartime(&[1 << Fq::S, 0, 0, 0]),
Fq::one()
);
}
#[test]
fn test_inv_root_of_unity() {
assert_eq!(Fq::ROOT_OF_UNITY_INV, Fq::ROOT_OF_UNITY.invert().unwrap());
}
#[test]
fn test_inv_2() {
assert_eq!(Fq::TWO_INV, Fq::from(2).invert().unwrap());
}
#[test]
fn test_delta() {
assert_eq!(Fq::DELTA, GENERATOR.pow(&[1u64 << Fq::S, 0, 0, 0]));
assert_eq!(
Fq::DELTA,
Fq::MULTIPLICATIVE_GENERATOR.pow(&[1u64 << Fq::S, 0, 0, 0])
);
}
#[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!(
Fq::from_raw([
0xe22bd0d1b22cc43e,
0x6b84e5b52490a7c8,
0x264262941ac9e229,
0x27dcfdf361ce4254
]),
Fq::from_u512([
0x64a80cce0b5a2369,
0x84f2ef0501bc783c,
0x696e5e63c86bbbde,
0x924072f52dc6cc62,
0x8288a507c8d61128,
0x3b2efb1ef697e3fe,
0x75a4998d06855f27,
0x52ea589e69712cc0
])
);
}
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