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use super::element::UnsignedInteger;
pub struct MontgomeryAlgorithms;
impl MontgomeryAlgorithms {
/// Compute CIOS multiplication of `a` * `b`
/// `q` is the modulus
/// `mu` is the inverse of -q modulo 2^{64}
/// Notice CIOS stands for Coarsely Integrated Operand Scanning
/// For more information see section 2.3.2 of Tolga Acar's thesis
/// https://www.microsoft.com/en-us/research/wp-content/uploads/1998/06/97Acar.pdf
#[inline(always)]
pub const fn cios<const NUM_LIMBS: usize>(
a: &UnsignedInteger<NUM_LIMBS>,
b: &UnsignedInteger<NUM_LIMBS>,
q: &UnsignedInteger<NUM_LIMBS>,
mu: &u64,
) -> UnsignedInteger<NUM_LIMBS> {
let mut t = [0_u64; NUM_LIMBS];
let mut t_extra = [0_u64; 2];
let mut i: usize = NUM_LIMBS;
while i > 0 {
i -= 1;
// C := 0
let mut c: u128 = 0;
// for j=N-1 to 0
// (C,t[j]) := t[j] + a[j]*b[i] + C
let mut cs: u128;
let mut j: usize = NUM_LIMBS;
while j > 0 {
j -= 1;
cs = t[j] as u128 + (a.limbs[j] as u128) * (b.limbs[i] as u128) + c;
c = cs >> 64;
t[j] = cs as u64;
}
// (t_extra[0],t_extra[1]) := t_extra[1] + C
cs = (t_extra[1] as u128) + c;
t_extra[0] = (cs >> 64) as u64;
t_extra[1] = cs as u64;
let mut c: u128;
// m := t[N-1]*q'[N-1] mod D
let m = ((t[NUM_LIMBS - 1] as u128 * *mu as u128) << 64) >> 64;
// (C,_) := t[N-1] + m*q[N-1]
c = (t[NUM_LIMBS - 1] as u128 + m * (q.limbs[NUM_LIMBS - 1] as u128)) >> 64;
// for j=N-1 to 1
// (C,t[j+1]) := t[j] + m*q[j] + C
let mut j: usize = NUM_LIMBS - 1;
while j > 0 {
j -= 1;
cs = t[j] as u128 + m * (q.limbs[j] as u128) + c;
c = cs >> 64;
t[j + 1] = ((cs << 64) >> 64) as u64;
}
// (C,t[0]) := t_extra[1] + C
cs = (t_extra[1] as u128) + c;
c = cs >> 64;
t[0] = ((cs << 64) >> 64) as u64;
// t_extra[1] := t_extra[0] + C
t_extra[1] = t_extra[0] + c as u64;
}
let mut result = UnsignedInteger { limbs: t };
let overflow = t_extra[1] > 0;
if overflow || UnsignedInteger::const_le(q, &result) {
(result, _) = UnsignedInteger::sub(&result, q);
}
result
}
/// Compute CIOS multiplication of `a` * `b`
/// This is the Algorithm 2 described in the paper
/// "EdMSM: Multi-Scalar-Multiplication for SNARKs and Faster Montgomery multiplication"
/// https://eprint.iacr.org/2022/1400.pdf.
/// It is only suited for moduli with `q[0]` smaller than `2^63 - 1`.
/// `q` is the modulus
/// `mu` is the inverse of -q modulo 2^{64}
#[inline(always)]
pub fn cios_optimized_for_moduli_with_one_spare_bit<const NUM_LIMBS: usize>(
a: &UnsignedInteger<NUM_LIMBS>,
b: &UnsignedInteger<NUM_LIMBS>,
q: &UnsignedInteger<NUM_LIMBS>,
mu: &u64,
) -> UnsignedInteger<NUM_LIMBS> {
let mut t = [0_u64; NUM_LIMBS];
let mut t_extra;
let mut i: usize = NUM_LIMBS;
while i > 0 {
i -= 1;
// C := 0
let mut c: u128 = 0;
// for j=N-1 to 0
// (C,t[j]) := t[j] + a[j]*b[i] + C
let mut cs: u128;
let mut j: usize = NUM_LIMBS;
while j > 0 {
j -= 1;
cs = t[j] as u128 + (a.limbs[j] as u128) * (b.limbs[i] as u128) + c;
c = cs >> 64;
t[j] = cs as u64;
}
t_extra = c as u64;
let mut c: u128;
// m := t[N-1]*q'[N-1] mod D
let m = ((t[NUM_LIMBS - 1] as u128 * *mu as u128) << 64) >> 64;
// (C,_) := t[0] + m*q[0]
c = (t[NUM_LIMBS - 1] as u128 + m * (q.limbs[NUM_LIMBS - 1] as u128)) >> 64;
// for j=N-1 to 1
// (C,t[j+1]) := t[j] + m*q[j] + C
let mut j: usize = NUM_LIMBS - 1;
while j > 0 {
j -= 1;
cs = t[j] as u128 + m * (q.limbs[j] as u128) + c;
c = cs >> 64;
t[j + 1] = ((cs << 64) >> 64) as u64;
}
// (C,t[0]) := t_extra + C
cs = (t_extra as u128) + c;
t[0] = ((cs << 64) >> 64) as u64;
}
let mut result = UnsignedInteger { limbs: t };
if UnsignedInteger::const_le(q, &result) {
(result, _) = UnsignedInteger::sub(&result, q);
}
result
}
// Separated Operand Scanning Method (2.3.1)
#[inline(always)]
pub fn sos_square<const NUM_LIMBS: usize>(
a: &UnsignedInteger<NUM_LIMBS>,
q: &UnsignedInteger<NUM_LIMBS>,
mu: &u64,
) -> UnsignedInteger<NUM_LIMBS> {
// NOTE: we use explicit `while` loops in this function because profiling pointed
// at iterators of the form `(<x>..<y>).rev()` as the main performance bottleneck.
// Step 1: Compute `(hi, lo) = a * a`
let (mut hi, mut lo) = UnsignedInteger::square(a);
// Step 2: Add terms to `(hi, lo)` until multiple it
// is a multiple of both `2^{NUM_LIMBS * 64}` and
// `q`.
let mut c: u128 = 0;
let mut i = NUM_LIMBS;
let mut overflow = false;
while i > 0 {
i -= 1;
c = 0;
let m = (lo.limbs[i] as u128 * *mu as u128) as u64;
let mut j = NUM_LIMBS;
while j > 0 {
j -= 1;
if i + j >= NUM_LIMBS - 1 {
let index = i + j - (NUM_LIMBS - 1);
let cs = lo.limbs[index] as u128 + m as u128 * (q.limbs[j] as u128) + c;
c = cs >> 64;
lo.limbs[index] = cs as u64;
} else {
let index = i + j + 1;
let cs = hi.limbs[index] as u128 + m as u128 * (q.limbs[j] as u128) + c;
c = cs >> 64;
hi.limbs[index] = cs as u64;
}
}
// Carry propagation to `hi`
let mut t = 0;
while c > 0 && i >= t {
let cs = hi.limbs[i - t] as u128 + c;
c = cs >> 64;
hi.limbs[i - t] = cs as u64;
t += 1;
}
overflow |= c > 0;
}
// Step 3: At this point `overflow * 2^{2 * NUM_LIMBS * 64} + (hi, lo)` is a multiple
// of `2^{NUM_LIMBS * 64}` and the result is obtained by dividing it by `2^{NUM_LIMBS * 64}`.
// In other words, `lo` is zero and the result is
// `overflow * 2^{NUM_LIMBS * 64} + hi`.
// That number is always strictly smaller than `2 * q`. To normalize it we substract
// `q` whenever it is larger than `q`.
// The easy case is when `overflow` is zero. We just use the `sub` function.
// If `overflow` is 1, then `hi` is smaller than `q`. The function `sub(hi, q)` wraps
// around `2^{NUM_LIMBS * 64}`. This is the result we need.
overflow |= c > 0;
if overflow || UnsignedInteger::const_le(q, &hi) {
(hi, _) = UnsignedInteger::sub(&hi, q);
}
hi
}
}
#[cfg(test)]
mod tests {
use crate::unsigned_integer::{element::U384, montgomery::MontgomeryAlgorithms};
use proptest::prelude::*;
proptest! {
#[test]
fn cios_vs_cios_optimized(a in any::<[u64; 6]>(), b in any::<[u64; 6]>()) {
let x = U384::from_limbs(a);
let y = U384::from_limbs(b);
let m = U384::from_hex_unchecked("cdb061954fdd36e5176f50dbdcfd349570a29ce1"); // this is prime
let mu: u64 = 16085280245840369887; // negative of the inverse of `m` modulo 2^{64}
assert_eq!(
MontgomeryAlgorithms::cios(&x, &y, &m, &mu),
MontgomeryAlgorithms::cios_optimized_for_moduli_with_one_spare_bit(&x, &y, &m, &mu)
);
}
#[test]
fn cios_vs_sos_square(a in any::<[u64; 6]>()) {
let x = U384::from_limbs(a);
let m = U384::from_hex_unchecked("cdb061954fdd36e5176f50dbdcfd349570a29ce1"); // this is prime
let mu: u64 = 16085280245840369887; // negative of the inverse of `m` modulo 2^{64}
assert_eq!(
MontgomeryAlgorithms::cios(&x, &x, &m, &mu),
MontgomeryAlgorithms::sos_square(&x, &m, &mu)
);
}
}
#[test]
fn montgomery_multiplication_works_0() {
let x = U384::from_u64(11_u64);
let y = U384::from_u64(10_u64);
let m = U384::from_u64(23_u64); //
let mu: u64 = 3208129404123400281; // negative of the inverse of `m` modulo 2^{64}.
let c = U384::from_u64(13_u64); // x * y * (r^{-1}) % m, where r = 2^{64 * 6} and r^{-1} mod m = 2.
assert_eq!(MontgomeryAlgorithms::cios(&x, &y, &m, &mu), c);
}
#[test]
fn montgomery_multiplication_works_1() {
let x = U384::from_hex_unchecked("05ed176deb0e80b4deb7718cdaa075165f149c");
let y = U384::from_hex_unchecked("5f103b0bd4397d4df560eb559f38353f80eeb6");
let m = U384::from_hex_unchecked("cdb061954fdd36e5176f50dbdcfd349570a29ce1"); // this is prime
let mu: u64 = 16085280245840369887; // negative of the inverse of `m` modulo 2^{64}
let c = U384::from_hex_unchecked("8d65cdee621682815d59f465d2641eea8a1274dc"); // x * y * (r^{-1}) % m, where r = 2^{64 * 6}
assert_eq!(MontgomeryAlgorithms::cios(&x, &y, &m, &mu), c);
}
#[test]
fn montgomery_multiplication_works_2() {
let x = U384::from_hex_unchecked("8d65cdee621682815d59f465d2641eea8a1274dc");
let m = U384::from_hex_unchecked("cdb061954fdd36e5176f50dbdcfd349570a29ce1"); // this is prime
let r_mod_m = U384::from_hex_unchecked("58dfb0e1b3dd5e674bdcde4f42eb5533b8759d33");
let mu: u64 = 16085280245840369887; // negative of the inverse of `m` modulo 2^{64}
let c = U384::from_hex_unchecked("8d65cdee621682815d59f465d2641eea8a1274dc");
assert_eq!(MontgomeryAlgorithms::cios(&x, &r_mod_m, &m, &mu), c);
}
}