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//! Arithmetic mod \\(2\^{252} + 27742317777372353535851937790883648493\\)
//! with five \\(52\\)-bit unsigned limbs.
//!
//! \\(51\\)-bit limbs would cover the desired bit range (\\(253\\)
//! bits), but isn't large enough to reduce a \\(512\\)-bit number with
//! Montgomery multiplication, so \\(52\\) bits is used instead. To see
//! that this is safe for intermediate results, note that the largest
//! limb in a \\(5\times 5\\) product of \\(52\\)-bit limbs will be
//!
//! ```text
//! (0xfffffffffffff^2) * 5 = 0x4ffffffffffff60000000000005 (107 bits).
//! ```
use core::fmt::Debug;
use core::ops::{Index, IndexMut};
#[cfg(feature = "zeroize")]
use zeroize::Zeroize;
use crate::constants;
/// The `Scalar52` struct represents an element in
/// \\(\mathbb Z / \ell \mathbb Z\\) as 5 \\(52\\)-bit limbs.
#[derive(Copy, Clone)]
pub struct Scalar52(pub [u64; 5]);
impl Debug for Scalar52 {
fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
write!(f, "Scalar52: {:?}", &self.0[..])
}
}
#[cfg(feature = "zeroize")]
impl Zeroize for Scalar52 {
fn zeroize(&mut self) {
self.0.zeroize();
}
}
impl Index<usize> for Scalar52 {
type Output = u64;
fn index(&self, _index: usize) -> &u64 {
&(self.0[_index])
}
}
impl IndexMut<usize> for Scalar52 {
fn index_mut(&mut self, _index: usize) -> &mut u64 {
&mut (self.0[_index])
}
}
/// u64 * u64 = u128 multiply helper
#[inline(always)]
fn m(x: u64, y: u64) -> u128 {
(x as u128) * (y as u128)
}
impl Scalar52 {
/// The scalar \\( 0 \\).
pub const ZERO: Scalar52 = Scalar52([0, 0, 0, 0, 0]);
/// Unpack a 32 byte / 256 bit scalar into 5 52-bit limbs.
#[rustfmt::skip] // keep alignment of s[*] calculations
pub fn from_bytes(bytes: &[u8; 32]) -> Scalar52 {
let mut words = [0u64; 4];
for i in 0..4 {
for j in 0..8 {
words[i] |= (bytes[(i * 8) + j] as u64) << (j * 8);
}
}
let mask = (1u64 << 52) - 1;
let top_mask = (1u64 << 48) - 1;
let mut s = Scalar52::ZERO;
s[0] = words[0] & mask;
s[1] = ((words[0] >> 52) | (words[1] << 12)) & mask;
s[2] = ((words[1] >> 40) | (words[2] << 24)) & mask;
s[3] = ((words[2] >> 28) | (words[3] << 36)) & mask;
s[4] = (words[3] >> 16) & top_mask;
s
}
/// Reduce a 64 byte / 512 bit scalar mod l
#[rustfmt::skip] // keep alignment of lo[*] and hi[*] calculations
pub fn from_bytes_wide(bytes: &[u8; 64]) -> Scalar52 {
let mut words = [0u64; 8];
for i in 0..8 {
for j in 0..8 {
words[i] |= (bytes[(i * 8) + j] as u64) << (j * 8);
}
}
let mask = (1u64 << 52) - 1;
let mut lo = Scalar52::ZERO;
let mut hi = Scalar52::ZERO;
lo[0] = words[0] & mask;
lo[1] = ((words[0] >> 52) | (words[ 1] << 12)) & mask;
lo[2] = ((words[1] >> 40) | (words[ 2] << 24)) & mask;
lo[3] = ((words[2] >> 28) | (words[ 3] << 36)) & mask;
lo[4] = ((words[3] >> 16) | (words[ 4] << 48)) & mask;
hi[0] = (words[4] >> 4) & mask;
hi[1] = ((words[4] >> 56) | (words[ 5] << 8)) & mask;
hi[2] = ((words[5] >> 44) | (words[ 6] << 20)) & mask;
hi[3] = ((words[6] >> 32) | (words[ 7] << 32)) & mask;
hi[4] = words[7] >> 20 ;
lo = Scalar52::montgomery_mul(&lo, &constants::R); // (lo * R) / R = lo
hi = Scalar52::montgomery_mul(&hi, &constants::RR); // (hi * R^2) / R = hi * R
Scalar52::add(&hi, &lo)
}
/// Pack the limbs of this `Scalar52` into 32 bytes
#[rustfmt::skip] // keep alignment of s[*] calculations
#[allow(clippy::identity_op)]
pub fn as_bytes(&self) -> [u8; 32] {
let mut s = [0u8; 32];
s[ 0] = (self.0[ 0] >> 0) as u8;
s[ 1] = (self.0[ 0] >> 8) as u8;
s[ 2] = (self.0[ 0] >> 16) as u8;
s[ 3] = (self.0[ 0] >> 24) as u8;
s[ 4] = (self.0[ 0] >> 32) as u8;
s[ 5] = (self.0[ 0] >> 40) as u8;
s[ 6] = ((self.0[ 0] >> 48) | (self.0[ 1] << 4)) as u8;
s[ 7] = (self.0[ 1] >> 4) as u8;
s[ 8] = (self.0[ 1] >> 12) as u8;
s[ 9] = (self.0[ 1] >> 20) as u8;
s[10] = (self.0[ 1] >> 28) as u8;
s[11] = (self.0[ 1] >> 36) as u8;
s[12] = (self.0[ 1] >> 44) as u8;
s[13] = (self.0[ 2] >> 0) as u8;
s[14] = (self.0[ 2] >> 8) as u8;
s[15] = (self.0[ 2] >> 16) as u8;
s[16] = (self.0[ 2] >> 24) as u8;
s[17] = (self.0[ 2] >> 32) as u8;
s[18] = (self.0[ 2] >> 40) as u8;
s[19] = ((self.0[ 2] >> 48) | (self.0[ 3] << 4)) as u8;
s[20] = (self.0[ 3] >> 4) as u8;
s[21] = (self.0[ 3] >> 12) as u8;
s[22] = (self.0[ 3] >> 20) as u8;
s[23] = (self.0[ 3] >> 28) as u8;
s[24] = (self.0[ 3] >> 36) as u8;
s[25] = (self.0[ 3] >> 44) as u8;
s[26] = (self.0[ 4] >> 0) as u8;
s[27] = (self.0[ 4] >> 8) as u8;
s[28] = (self.0[ 4] >> 16) as u8;
s[29] = (self.0[ 4] >> 24) as u8;
s[30] = (self.0[ 4] >> 32) as u8;
s[31] = (self.0[ 4] >> 40) as u8;
s
}
/// Compute `a + b` (mod l)
pub fn add(a: &Scalar52, b: &Scalar52) -> Scalar52 {
let mut sum = Scalar52::ZERO;
let mask = (1u64 << 52) - 1;
// a + b
let mut carry: u64 = 0;
for i in 0..5 {
carry = a[i] + b[i] + (carry >> 52);
sum[i] = carry & mask;
}
// subtract l if the sum is >= l
Scalar52::sub(&sum, &constants::L)
}
/// Compute `a - b` (mod l)
pub fn sub(a: &Scalar52, b: &Scalar52) -> Scalar52 {
// Optimization barrier to prevent compiler from inserting branch instructions
// TODO(tarcieri): find a better home (or abstraction) for this
fn black_box(value: u64) -> u64 {
// SAFETY: `u64` is a simple integer `Copy` type and `value` lives on the stack so
// a pointer to it will be valid.
unsafe { core::ptr::read_volatile(&value) }
}
let mut difference = Scalar52::ZERO;
let mask = (1u64 << 52) - 1;
// a - b
let mut borrow: u64 = 0;
for i in 0..5 {
borrow = a[i].wrapping_sub(b[i] + (borrow >> 63));
difference[i] = borrow & mask;
}
// conditionally add l if the difference is negative
let underflow_mask = ((borrow >> 63) ^ 1).wrapping_sub(1);
let mut carry: u64 = 0;
for i in 0..5 {
// SECURITY: `black_box` prevents LLVM from inserting a `jns` conditional on x86(_64)
// which can be used to bypass this section when `underflow_mask` is zero.
carry = (carry >> 52) + difference[i] + (constants::L[i] & black_box(underflow_mask));
difference[i] = carry & mask;
}
difference
}
/// Compute `a * b`
#[inline(always)]
#[rustfmt::skip] // keep alignment of z[*] calculations
pub (crate) fn mul_internal(a: &Scalar52, b: &Scalar52) -> [u128; 9] {
let mut z = [0u128; 9];
z[0] = m(a[0], b[0]);
z[1] = m(a[0], b[1]) + m(a[1], b[0]);
z[2] = m(a[0], b[2]) + m(a[1], b[1]) + m(a[2], b[0]);
z[3] = m(a[0], b[3]) + m(a[1], b[2]) + m(a[2], b[1]) + m(a[3], b[0]);
z[4] = m(a[0], b[4]) + m(a[1], b[3]) + m(a[2], b[2]) + m(a[3], b[1]) + m(a[4], b[0]);
z[5] = m(a[1], b[4]) + m(a[2], b[3]) + m(a[3], b[2]) + m(a[4], b[1]);
z[6] = m(a[2], b[4]) + m(a[3], b[3]) + m(a[4], b[2]);
z[7] = m(a[3], b[4]) + m(a[4], b[3]);
z[8] = m(a[4], b[4]);
z
}
/// Compute `a^2`
#[inline(always)]
#[rustfmt::skip] // keep alignment of return calculations
fn square_internal(a: &Scalar52) -> [u128; 9] {
let aa = [
a[0] * 2,
a[1] * 2,
a[2] * 2,
a[3] * 2,
];
[
m( a[0], a[0]),
m(aa[0], a[1]),
m(aa[0], a[2]) + m( a[1], a[1]),
m(aa[0], a[3]) + m(aa[1], a[2]),
m(aa[0], a[4]) + m(aa[1], a[3]) + m( a[2], a[2]),
m(aa[1], a[4]) + m(aa[2], a[3]),
m(aa[2], a[4]) + m( a[3], a[3]),
m(aa[3], a[4]),
m(a[4], a[4])
]
}
/// Compute `limbs/R` (mod l), where R is the Montgomery modulus 2^260
#[inline(always)]
#[rustfmt::skip] // keep alignment of n* and r* calculations
pub (crate) fn montgomery_reduce(limbs: &[u128; 9]) -> Scalar52 {
#[inline(always)]
fn part1(sum: u128) -> (u128, u64) {
let p = (sum as u64).wrapping_mul(constants::LFACTOR) & ((1u64 << 52) - 1);
((sum + m(p, constants::L[0])) >> 52, p)
}
#[inline(always)]
fn part2(sum: u128) -> (u128, u64) {
let w = (sum as u64) & ((1u64 << 52) - 1);
(sum >> 52, w)
}
// note: l[3] is zero, so its multiples can be skipped
let l = &constants::L;
// the first half computes the Montgomery adjustment factor n, and begins adding n*l to make limbs divisible by R
let (carry, n0) = part1( limbs[0]);
let (carry, n1) = part1(carry + limbs[1] + m(n0, l[1]));
let (carry, n2) = part1(carry + limbs[2] + m(n0, l[2]) + m(n1, l[1]));
let (carry, n3) = part1(carry + limbs[3] + m(n1, l[2]) + m(n2, l[1]));
let (carry, n4) = part1(carry + limbs[4] + m(n0, l[4]) + m(n2, l[2]) + m(n3, l[1]));
// limbs is divisible by R now, so we can divide by R by simply storing the upper half as the result
let (carry, r0) = part2(carry + limbs[5] + m(n1, l[4]) + m(n3, l[2]) + m(n4, l[1]));
let (carry, r1) = part2(carry + limbs[6] + m(n2,l[4]) + m(n4, l[2]));
let (carry, r2) = part2(carry + limbs[7] + m(n3, l[4]) );
let (carry, r3) = part2(carry + limbs[8] + m(n4, l[4]));
let r4 = carry as u64;
// result may be >= l, so attempt to subtract l
Scalar52::sub(&Scalar52([r0, r1, r2, r3, r4]), l)
}
/// Compute `a * b` (mod l)
#[inline(never)]
pub fn mul(a: &Scalar52, b: &Scalar52) -> Scalar52 {
let ab = Scalar52::montgomery_reduce(&Scalar52::mul_internal(a, b));
Scalar52::montgomery_reduce(&Scalar52::mul_internal(&ab, &constants::RR))
}
/// Compute `a^2` (mod l)
#[inline(never)]
#[allow(dead_code)] // XXX we don't expose square() via the Scalar API
pub fn square(&self) -> Scalar52 {
let aa = Scalar52::montgomery_reduce(&Scalar52::square_internal(self));
Scalar52::montgomery_reduce(&Scalar52::mul_internal(&aa, &constants::RR))
}
/// Compute `(a * b) / R` (mod l), where R is the Montgomery modulus 2^260
#[inline(never)]
pub fn montgomery_mul(a: &Scalar52, b: &Scalar52) -> Scalar52 {
Scalar52::montgomery_reduce(&Scalar52::mul_internal(a, b))
}
/// Compute `(a^2) / R` (mod l) in Montgomery form, where R is the Montgomery modulus 2^260
#[inline(never)]
pub fn montgomery_square(&self) -> Scalar52 {
Scalar52::montgomery_reduce(&Scalar52::square_internal(self))
}
/// Puts a Scalar52 in to Montgomery form, i.e. computes `a*R (mod l)`
#[inline(never)]
pub fn as_montgomery(&self) -> Scalar52 {
Scalar52::montgomery_mul(self, &constants::RR)
}
/// Takes a Scalar52 out of Montgomery form, i.e. computes `a/R (mod l)`
#[allow(clippy::wrong_self_convention)]
#[inline(never)]
pub fn from_montgomery(&self) -> Scalar52 {
let mut limbs = [0u128; 9];
for i in 0..5 {
limbs[i] = self[i] as u128;
}
Scalar52::montgomery_reduce(&limbs)
}
}
#[cfg(test)]
mod test {
use super::*;
/// Note: x is 2^253-1 which is slightly larger than the largest scalar produced by
/// this implementation (l-1), and should show there are no overflows for valid scalars
///
/// x = 14474011154664524427946373126085988481658748083205070504932198000989141204991
/// x = 7237005577332262213973186563042994240801631723825162898930247062703686954002 mod l
/// x = 3057150787695215392275360544382990118917283750546154083604586903220563173085*R mod l in Montgomery form
pub static X: Scalar52 = Scalar52([
0x000fffffffffffff,
0x000fffffffffffff,
0x000fffffffffffff,
0x000fffffffffffff,
0x00001fffffffffff,
]);
/// x^2 = 3078544782642840487852506753550082162405942681916160040940637093560259278169 mod l
pub static XX: Scalar52 = Scalar52([
0x0001668020217559,
0x000531640ffd0ec0,
0x00085fd6f9f38a31,
0x000c268f73bb1cf4,
0x000006ce65046df0,
]);
/// x^2 = 4413052134910308800482070043710297189082115023966588301924965890668401540959*R mod l in Montgomery form
pub static XX_MONT: Scalar52 = Scalar52([
0x000c754eea569a5c,
0x00063b6ed36cb215,
0x0008ffa36bf25886,
0x000e9183614e7543,
0x0000061db6c6f26f,
]);
/// y = 6145104759870991071742105800796537629880401874866217824609283457819451087098
pub static Y: Scalar52 = Scalar52([
0x000b75071e1458fa,
0x000bf9d75e1ecdac,
0x000433d2baf0672b,
0x0005fffcc11fad13,
0x00000d96018bb825,
]);
/// x*y = 36752150652102274958925982391442301741 mod l
pub static XY: Scalar52 = Scalar52([
0x000ee6d76ba7632d,
0x000ed50d71d84e02,
0x00000000001ba634,
0x0000000000000000,
0x0000000000000000,
]);
/// x*y = 658448296334113745583381664921721413881518248721417041768778176391714104386*R mod l in Montgomery form
pub static XY_MONT: Scalar52 = Scalar52([
0x0006d52bf200cfd5,
0x00033fb1d7021570,
0x000f201bc07139d8,
0x0001267e3e49169e,
0x000007b839c00268,
]);
/// a = 2351415481556538453565687241199399922945659411799870114962672658845158063753
pub static A: Scalar52 = Scalar52([
0x0005236c07b3be89,
0x0001bc3d2a67c0c4,
0x000a4aa782aae3ee,
0x0006b3f6e4fec4c4,
0x00000532da9fab8c,
]);
/// b = 4885590095775723760407499321843594317911456947580037491039278279440296187236
pub static B: Scalar52 = Scalar52([
0x000d3fae55421564,
0x000c2df24f65a4bc,
0x0005b5587d69fb0b,
0x00094c091b013b3b,
0x00000acd25605473,
]);
/// a+b = 0
/// a-b = 4702830963113076907131374482398799845891318823599740229925345317690316127506
pub static AB: Scalar52 = Scalar52([
0x000a46d80f677d12,
0x0003787a54cf8188,
0x0004954f0555c7dc,
0x000d67edc9fd8989,
0x00000a65b53f5718,
]);
// c = (2^512 - 1) % l = 1627715501170711445284395025044413883736156588369414752970002579683115011840
pub static C: Scalar52 = Scalar52([
0x000611e3449c0f00,
0x000a768859347a40,
0x0007f5be65d00e1b,
0x0009a3dceec73d21,
0x00000399411b7c30,
]);
#[test]
fn mul_max() {
let res = Scalar52::mul(&X, &X);
for i in 0..5 {
assert!(res[i] == XX[i]);
}
}
#[test]
fn square_max() {
let res = X.square();
for i in 0..5 {
assert!(res[i] == XX[i]);
}
}
#[test]
fn montgomery_mul_max() {
let res = Scalar52::montgomery_mul(&X, &X);
for i in 0..5 {
assert!(res[i] == XX_MONT[i]);
}
}
#[test]
fn montgomery_square_max() {
let res = X.montgomery_square();
for i in 0..5 {
assert!(res[i] == XX_MONT[i]);
}
}
#[test]
fn mul() {
let res = Scalar52::mul(&X, &Y);
for i in 0..5 {
assert!(res[i] == XY[i]);
}
}
#[test]
fn montgomery_mul() {
let res = Scalar52::montgomery_mul(&X, &Y);
for i in 0..5 {
assert!(res[i] == XY_MONT[i]);
}
}
#[test]
fn add() {
let res = Scalar52::add(&A, &B);
let zero = Scalar52::ZERO;
for i in 0..5 {
assert!(res[i] == zero[i]);
}
}
#[test]
fn sub() {
let res = Scalar52::sub(&A, &B);
for i in 0..5 {
assert!(res[i] == AB[i]);
}
}
#[test]
fn from_bytes_wide() {
let bignum = [255u8; 64]; // 2^512 - 1
let reduced = Scalar52::from_bytes_wide(&bignum);
for i in 0..5 {
assert!(reduced[i] == C[i]);
}
}
}