320 lines
11 KiB
Rust
320 lines
11 KiB
Rust
//! Contains different structs and methods related to the Falcon DSA.
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//!
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//! It uses and acknowledges the work in:
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//!
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//! 1. The [reference](https://falcon-sign.info/impl/README.txt.html) implementation by Thomas Pornin.
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//! 2. The [Rust](https://github.com/aszepieniec/falcon-rust) implementation by Alan Szepieniec.
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use super::MODULUS;
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use alloc::{string::String, vec::Vec};
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use core::ops::MulAssign;
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use num::{BigInt, FromPrimitive, One, Zero};
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use num_complex::Complex64;
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use rand::Rng;
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#[cfg(not(feature = "std"))]
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use num::Float;
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mod fft;
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pub use fft::{CyclotomicFourier, FastFft};
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mod field;
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pub use field::FalconFelt;
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mod ffsampling;
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pub use ffsampling::{ffldl, ffsampling, gram, normalize_tree, LdlTree};
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mod samplerz;
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use self::samplerz::sampler_z;
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mod polynomial;
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pub use polynomial::Polynomial;
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pub trait Inverse: Copy + Zero + MulAssign + One {
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/// Gets the inverse of a, or zero if it is zero.
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fn inverse_or_zero(self) -> Self;
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/// Gets the inverses of a batch of elements, and skip over any that are zero.
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fn batch_inverse_or_zero(batch: &[Self]) -> Vec<Self> {
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let mut acc = Self::one();
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let mut rp: Vec<Self> = Vec::with_capacity(batch.len());
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for batch_item in batch {
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if !batch_item.is_zero() {
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rp.push(acc);
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acc = *batch_item * acc;
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} else {
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rp.push(Self::zero());
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}
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}
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let mut inv = Self::inverse_or_zero(acc);
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for i in (0..batch.len()).rev() {
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if !batch[i].is_zero() {
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rp[i] *= inv;
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inv *= batch[i];
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}
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}
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rp
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}
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}
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impl Inverse for Complex64 {
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fn inverse_or_zero(self) -> Self {
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let modulus = self.re * self.re + self.im * self.im;
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Complex64::new(self.re / modulus, -self.im / modulus)
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}
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fn batch_inverse_or_zero(batch: &[Self]) -> Vec<Self> {
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batch.iter().map(|&c| Complex64::new(1.0, 0.0) / c).collect()
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}
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}
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impl Inverse for f64 {
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fn inverse_or_zero(self) -> Self {
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1.0 / self
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}
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fn batch_inverse_or_zero(batch: &[Self]) -> Vec<Self> {
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batch.iter().map(|&c| 1.0 / c).collect()
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}
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}
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/// Samples 4 small polynomials f, g, F, G such that f * G - g * F = q mod (X^n + 1).
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/// Algorithm 5 (NTRUgen) of the documentation [1, p.34].
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///
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/// [1]: https://falcon-sign.info/falcon.pdf
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pub(crate) fn ntru_gen<R: Rng>(n: usize, rng: &mut R) -> [Polynomial<i16>; 4] {
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loop {
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let f = gen_poly(n, rng);
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let g = gen_poly(n, rng);
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let f_ntt = f.map(|&i| FalconFelt::new(i)).fft();
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if f_ntt.coefficients.iter().any(|e| e.is_zero()) {
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continue;
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}
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let gamma = gram_schmidt_norm_squared(&f, &g);
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if gamma > 1.3689f64 * (MODULUS as f64) {
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continue;
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}
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if let Some((capital_f, capital_g)) =
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ntru_solve(&f.map(|&i| i.into()), &g.map(|&i| i.into()))
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{
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return [
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g,
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-f,
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capital_g.map(|i| i.try_into().unwrap()),
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-capital_f.map(|i| i.try_into().unwrap()),
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];
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}
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}
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}
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/// Solves the NTRU equation. Given f, g in ZZ[X], find F, G in ZZ[X] such that:
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///
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/// f G - g F = q mod (X^n + 1)
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///
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/// Algorithm 6 of the specification [1, p.35].
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///
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/// [1]: https://falcon-sign.info/falcon.pdf
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fn ntru_solve(
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f: &Polynomial<BigInt>,
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g: &Polynomial<BigInt>,
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) -> Option<(Polynomial<BigInt>, Polynomial<BigInt>)> {
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let n = f.coefficients.len();
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if n == 1 {
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let (gcd, u, v) = xgcd(&f.coefficients[0], &g.coefficients[0]);
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if gcd != BigInt::one() {
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return None;
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}
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return Some((
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(Polynomial::new(vec![-v * BigInt::from_u32(MODULUS as u32).unwrap()])),
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Polynomial::new(vec![u * BigInt::from_u32(MODULUS as u32).unwrap()]),
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));
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}
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let f_prime = f.field_norm();
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let g_prime = g.field_norm();
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let (capital_f_prime, capital_g_prime) = ntru_solve(&f_prime, &g_prime)?;
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let capital_f_prime_xsq = capital_f_prime.lift_next_cyclotomic();
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let capital_g_prime_xsq = capital_g_prime.lift_next_cyclotomic();
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let f_minx = f.galois_adjoint();
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let g_minx = g.galois_adjoint();
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let mut capital_f = (capital_f_prime_xsq.karatsuba(&g_minx)).reduce_by_cyclotomic(n);
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let mut capital_g = (capital_g_prime_xsq.karatsuba(&f_minx)).reduce_by_cyclotomic(n);
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match babai_reduce(f, g, &mut capital_f, &mut capital_g) {
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Ok(_) => Some((capital_f, capital_g)),
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Err(_e) => {
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#[cfg(test)]
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{
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panic!("{}", _e);
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}
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#[cfg(not(test))]
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{
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None
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}
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}
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}
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}
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/// Generates a polynomial of degree at most n-1 whose coefficients are distributed according
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/// to a discrete Gaussian with mu = 0 and sigma = 1.17 * sqrt(Q / (2n)).
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fn gen_poly<R: Rng>(n: usize, rng: &mut R) -> Polynomial<i16> {
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let mu = 0.0;
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let sigma_star = 1.43300980528773;
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Polynomial {
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coefficients: (0..4096)
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.map(|_| sampler_z(mu, sigma_star, sigma_star - 0.001, rng))
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.collect::<Vec<i16>>()
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.chunks(4096 / n)
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.map(|ch| ch.iter().sum())
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.collect(),
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}
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}
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/// Computes the Gram-Schmidt norm of B = [[g, -f], [G, -F]] from f and g.
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/// Corresponds to line 9 in algorithm 5 of the spec [1, p.34]
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///
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/// [1]: https://falcon-sign.info/falcon.pdf
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fn gram_schmidt_norm_squared(f: &Polynomial<i16>, g: &Polynomial<i16>) -> f64 {
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let n = f.coefficients.len();
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let norm_f_squared = f.l2_norm_squared();
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let norm_g_squared = g.l2_norm_squared();
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let gamma1 = norm_f_squared + norm_g_squared;
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let f_fft = f.map(|i| Complex64::new(*i as f64, 0.0)).fft();
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let g_fft = g.map(|i| Complex64::new(*i as f64, 0.0)).fft();
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let f_adj_fft = f_fft.map(|c| c.conj());
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let g_adj_fft = g_fft.map(|c| c.conj());
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let ffgg_fft = f_fft.hadamard_mul(&f_adj_fft) + g_fft.hadamard_mul(&g_adj_fft);
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let ffgg_fft_inverse = ffgg_fft.hadamard_inv();
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let qf_over_ffgg_fft = f_adj_fft.map(|c| c * (MODULUS as f64)).hadamard_mul(&ffgg_fft_inverse);
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let qg_over_ffgg_fft = g_adj_fft.map(|c| c * (MODULUS as f64)).hadamard_mul(&ffgg_fft_inverse);
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let norm_f_over_ffgg_squared =
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qf_over_ffgg_fft.coefficients.iter().map(|c| (c * c.conj()).re).sum::<f64>() / (n as f64);
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let norm_g_over_ffgg_squared =
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qg_over_ffgg_fft.coefficients.iter().map(|c| (c * c.conj()).re).sum::<f64>() / (n as f64);
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let gamma2 = norm_f_over_ffgg_squared + norm_g_over_ffgg_squared;
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f64::max(gamma1, gamma2)
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}
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/// Reduces the vector (F,G) relative to (f,g). This method follows the python implementation [1].
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/// Note that this algorithm can end up in an infinite loop. (It's one of the things the author
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/// would like to fix.) When this happens, control returns an error (hence the return type) and
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/// generates another keypair with fresh randomness.
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///
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/// Algorithm 7 in the spec [2, p.35]
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///
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/// [1]: https://github.com/tprest/falcon.py
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///
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/// [2]: https://falcon-sign.info/falcon.pdf
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fn babai_reduce(
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f: &Polynomial<BigInt>,
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g: &Polynomial<BigInt>,
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capital_f: &mut Polynomial<BigInt>,
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capital_g: &mut Polynomial<BigInt>,
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) -> Result<(), String> {
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let bitsize = |bi: &BigInt| (bi.bits() + 7) & (u64::MAX ^ 7);
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let n = f.coefficients.len();
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let size = [
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f.map(bitsize).fold(0, |a, &b| u64::max(a, b)),
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g.map(bitsize).fold(0, |a, &b| u64::max(a, b)),
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53,
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]
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.into_iter()
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.max()
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.unwrap();
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let shift = (size as i64) - 53;
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let f_adjusted = f
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.map(|bi| Complex64::new(i64::try_from(bi >> shift).unwrap() as f64, 0.0))
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.fft();
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let g_adjusted = g
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.map(|bi| Complex64::new(i64::try_from(bi >> shift).unwrap() as f64, 0.0))
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.fft();
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let f_star_adjusted = f_adjusted.map(|c| c.conj());
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let g_star_adjusted = g_adjusted.map(|c| c.conj());
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let denominator_fft =
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f_adjusted.hadamard_mul(&f_star_adjusted) + g_adjusted.hadamard_mul(&g_star_adjusted);
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let mut counter = 0;
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loop {
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let capital_size = [
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capital_f.map(bitsize).fold(0, |a, &b| u64::max(a, b)),
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capital_g.map(bitsize).fold(0, |a, &b| u64::max(a, b)),
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53,
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]
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.into_iter()
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.max()
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.unwrap();
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if capital_size < size {
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break;
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}
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let capital_shift = (capital_size as i64) - 53;
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let capital_f_adjusted = capital_f
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.map(|bi| Complex64::new(i64::try_from(bi >> capital_shift).unwrap() as f64, 0.0))
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.fft();
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let capital_g_adjusted = capital_g
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.map(|bi| Complex64::new(i64::try_from(bi >> capital_shift).unwrap() as f64, 0.0))
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.fft();
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let numerator = capital_f_adjusted.hadamard_mul(&f_star_adjusted)
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+ capital_g_adjusted.hadamard_mul(&g_star_adjusted);
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let quotient = numerator.hadamard_div(&denominator_fft).ifft();
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let k = quotient.map(|f| Into::<BigInt>::into(f.re.round() as i64));
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if k.is_zero() {
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break;
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}
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let kf = (k.clone().karatsuba(f))
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.reduce_by_cyclotomic(n)
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.map(|bi| bi << (capital_size - size));
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let kg = (k.clone().karatsuba(g))
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.reduce_by_cyclotomic(n)
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.map(|bi| bi << (capital_size - size));
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*capital_f -= kf;
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*capital_g -= kg;
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counter += 1;
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if counter > 1000 {
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// If we get here, that means that (with high likelihood) we are in an
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// infinite loop. We know it happens from time to time -- seldomly, but it
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// does. It would be nice to fix that! But in order to fix it we need to be
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// able to reproduce it, and for that we need test vectors. So print them
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// and hope that one day they circle back to the implementor.
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return Err(format!("Encountered infinite loop in babai_reduce of falcon-rust.\n\\
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Please help the developer(s) fix it! You can do this by sending them the inputs to the function that caused the behavior:\n\\
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f: {:?}\n\\
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g: {:?}\n\\
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capital_f: {:?}\n\\
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capital_g: {:?}\n", f.coefficients, g.coefficients, capital_f.coefficients, capital_g.coefficients));
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}
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}
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Ok(())
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}
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/// Extended Euclidean algorithm for computing the greatest common divisor (g) and
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/// Bézout coefficients (u, v) for the relation
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///
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/// $$ u a + v b = g . $$
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///
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/// Implementation adapted from Wikipedia [1].
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///
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/// [1]: https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Pseudocode
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fn xgcd(a: &BigInt, b: &BigInt) -> (BigInt, BigInt, BigInt) {
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let (mut old_r, mut r) = (a.clone(), b.clone());
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let (mut old_s, mut s) = (BigInt::one(), BigInt::zero());
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let (mut old_t, mut t) = (BigInt::zero(), BigInt::one());
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while r != BigInt::zero() {
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let quotient = old_r.clone() / r.clone();
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(old_r, r) = (r.clone(), old_r.clone() - quotient.clone() * r);
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(old_s, s) = (s.clone(), old_s.clone() - quotient.clone() * s);
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(old_t, t) = (t.clone(), old_t.clone() - quotient * t);
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}
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(old_r, old_s, old_t)
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}
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