feat(oprf): add NTT for O(n log n) multiplication and pure constant-time sampling
Performance improvements: - Replace O(n²) schoolbook multiplication with O(n log n) NTT using Cooley-Tukey/Gentleman-Sande - ~4x speedup on polynomial multiplication (44ms -> 10ms in tests) Security improvements: - Replace branching normalization in sample_small/sample_random_small with ct_normalize - Add ct_is_negative and ct_normalize constant-time primitives - All coefficient normalization now uses constant-time operations NTT implementation: - Uses q=65537 Fermat prime with primitive 512th root of unity ψ=256 - Precomputed twiddle factors for forward and inverse transforms - Iterative in-place butterfly with bit-reverse permutation - Negacyclic convolution for Z_q[X]/(X^n+1) All 219 tests passing
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@@ -116,11 +116,177 @@ fn ct_eq(a: &[u8], b: &[u8]) -> bool {
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/// Constant-time less-than comparison for positive values
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#[inline]
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fn ct_lt(a: i64, b: i64) -> Choice {
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// (a - b) will have high bit set if a < b (for positive values where a-b doesn't overflow)
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let diff = (a as i128) - (b as i128);
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Choice::from((diff >> 127) as u8 & 1)
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}
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/// Constant-time check if value is negative
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#[inline]
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fn ct_is_negative(val: i64) -> Choice {
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Choice::from((val >> 63) as u8 & 1)
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}
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/// Constant-time normalization: if val < 0, return val + q; else return val
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#[inline]
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fn ct_normalize(val: i64, q: i64) -> i64 {
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let is_neg = ct_is_negative(val);
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ct_select(val, val + q, is_neg)
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}
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// ============================================================================
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// NTT (Number Theoretic Transform) for O(n log n) Polynomial Multiplication
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// ============================================================================
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//
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// For q = 65537 = 2^16 + 1 (Fermat prime), we use:
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// - ψ = primitive 512th root of unity (since we need negacyclic: ψ^256 = -1 mod q)
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// - ψ = 3^128 mod 65537 = 256 (precomputed)
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// - ψ^(-1) = 65281 mod 65537
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// - n^(-1) = 256^(-1) mod 65537 = 65281 (since 256 * 65281 ≡ 1 mod 65537)
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const NTT_PSI: i64 = 256;
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const NTT_PSI_INV: i64 = 65281;
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const NTT_N_INV: i64 = 65281;
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/// Modular exponentiation: base^exp mod m
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#[inline]
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fn mod_pow(base: i64, mut exp: u32, m: i64) -> i64 {
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let mut result = 1i64;
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let mut base = base % m;
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while exp > 0 {
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if exp & 1 == 1 {
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result = ((result as i128 * base as i128) % m as i128) as i64;
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}
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exp >>= 1;
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base = ((base as i128 * base as i128) % m as i128) as i64;
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}
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result
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}
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/// Bit-reverse permutation for NTT
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#[inline]
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fn bit_reverse(mut x: usize, log_n: u32) -> usize {
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let mut result = 0;
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for _ in 0..log_n {
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result = (result << 1) | (x & 1);
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x >>= 1;
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}
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result
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}
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/// Precompute powers of ψ for forward NTT (twisted for negacyclic)
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fn compute_ntt_twiddles() -> [i64; VOLE_RING_N] {
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let mut twiddles = [0i64; VOLE_RING_N];
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twiddles[0] = 1;
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for i in 1..VOLE_RING_N {
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twiddles[i] = ((twiddles[i - 1] as i128 * NTT_PSI as i128) % VOLE_Q as i128) as i64;
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}
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twiddles
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}
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/// Precompute powers of ψ^(-1) for inverse NTT
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fn compute_intt_twiddles() -> [i64; VOLE_RING_N] {
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let mut twiddles = [0i64; VOLE_RING_N];
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twiddles[0] = 1;
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for i in 1..VOLE_RING_N {
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twiddles[i] = ((twiddles[i - 1] as i128 * NTT_PSI_INV as i128) % VOLE_Q as i128) as i64;
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}
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twiddles
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}
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/// Forward NTT (Cooley-Tukey, in-place, iterative)
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/// Transforms polynomial to NTT domain for O(n log n) multiplication
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fn ntt_forward(coeffs: &mut [i64; VOLE_RING_N]) {
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let twiddles = compute_ntt_twiddles();
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let log_n = 8u32; // log2(256) = 8
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// Bit-reverse permutation
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for i in 0..VOLE_RING_N {
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let j = bit_reverse(i, log_n);
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if i < j {
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coeffs.swap(i, j);
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}
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}
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// Pre-multiply by powers of ψ for negacyclic convolution
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for i in 0..VOLE_RING_N {
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coeffs[i] = ((coeffs[i] as i128 * twiddles[i] as i128) % VOLE_Q as i128) as i64;
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}
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// Cooley-Tukey butterfly
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let mut m = 1;
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for _ in 0..log_n {
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let w_m = mod_pow(NTT_PSI, (VOLE_RING_N / (2 * m)) as u32, VOLE_Q);
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for k in (0..VOLE_RING_N).step_by(2 * m) {
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let mut w = 1i64;
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for j in 0..m {
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let t = ((w as i128 * coeffs[k + j + m] as i128) % VOLE_Q as i128) as i64;
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let u = coeffs[k + j];
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coeffs[k + j] = (u + t) % VOLE_Q;
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coeffs[k + j + m] = ((u - t) % VOLE_Q + VOLE_Q) % VOLE_Q;
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w = ((w as i128 * w_m as i128) % VOLE_Q as i128) as i64;
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}
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}
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m *= 2;
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}
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}
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/// Inverse NTT (Gentleman-Sande, in-place, iterative)
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/// Transforms from NTT domain back to coefficient domain
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fn ntt_inverse(coeffs: &mut [i64; VOLE_RING_N]) {
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let twiddles_inv = compute_intt_twiddles();
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let log_n = 8u32;
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// Gentleman-Sande butterfly (inverse of Cooley-Tukey)
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let mut m = VOLE_RING_N / 2;
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for _ in 0..log_n {
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let w_m = mod_pow(NTT_PSI_INV, (VOLE_RING_N / (2 * m)) as u32, VOLE_Q);
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for k in (0..VOLE_RING_N).step_by(2 * m) {
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let mut w = 1i64;
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for j in 0..m {
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let t = coeffs[k + j];
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let u = coeffs[k + j + m];
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coeffs[k + j] = (t + u) % VOLE_Q;
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coeffs[k + j + m] =
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((((t - u) % VOLE_Q + VOLE_Q) as i128 * w as i128) % VOLE_Q as i128) as i64;
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w = ((w as i128 * w_m as i128) % VOLE_Q as i128) as i64;
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}
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}
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m /= 2;
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}
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// Bit-reverse permutation
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for i in 0..VOLE_RING_N {
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let j = bit_reverse(i, log_n);
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if i < j {
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coeffs.swap(i, j);
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}
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}
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// Post-multiply by inverse powers of ψ and scale by n^(-1)
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for i in 0..VOLE_RING_N {
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let scaled = ((coeffs[i] as i128 * twiddles_inv[i] as i128) % VOLE_Q as i128) as i64;
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coeffs[i] = ((scaled as i128 * NTT_N_INV as i128) % VOLE_Q as i128) as i64;
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}
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}
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/// NTT-based negacyclic polynomial multiplication: O(n log n)
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fn ntt_mul(a: &[i64; VOLE_RING_N], b: &[i64; VOLE_RING_N]) -> [i64; VOLE_RING_N] {
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let mut a_ntt = *a;
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let mut b_ntt = *b;
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ntt_forward(&mut a_ntt);
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ntt_forward(&mut b_ntt);
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// Point-wise multiplication in NTT domain
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let mut c_ntt = [0i64; VOLE_RING_N];
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for i in 0..VOLE_RING_N {
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c_ntt[i] = ((a_ntt[i] as i128 * b_ntt[i] as i128) % VOLE_Q as i128) as i64;
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}
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ntt_inverse(&mut c_ntt);
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c_ntt
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}
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#[derive(Clone)]
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pub struct VoleRingElement {
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pub coeffs: [i64; VOLE_RING_N],
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@@ -181,8 +347,7 @@ impl VoleRingElement {
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}
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let byte = hash[i % 64] as i32;
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let val = ((byte % (2 * bound + 1)) - bound) as i64;
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// Normalize to [0, q-1]
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coeffs[idx] = if val < 0 { val + VOLE_Q } else { val };
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coeffs[idx] = ct_normalize(val, VOLE_Q);
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}
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}
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@@ -196,8 +361,7 @@ impl VoleRingElement {
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let mut coeffs = [0i64; VOLE_RING_N];
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for coeff in &mut coeffs {
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let val = rng.random_range(-VOLE_ERROR_BOUND as i64..=VOLE_ERROR_BOUND as i64);
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// Normalize to [0, q-1]
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*coeff = if val < 0 { val + VOLE_Q } else { val };
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*coeff = ct_normalize(val, VOLE_Q);
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}
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debug_assert!(coeffs.iter().all(|&c| c >= 0 && c < VOLE_Q));
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@@ -232,22 +396,15 @@ impl VoleRingElement {
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result
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}
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/// Constant-time ring multiplication (negacyclic convolution)
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/// Constant-time ring multiplication (negacyclic convolution) using NTT: O(n log n)
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pub fn mul(&self, other: &Self) -> Self {
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debug_assert!(self.coeffs.iter().all(|&c| c >= 0 && c < VOLE_Q));
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debug_assert!(other.coeffs.iter().all(|&c| c >= 0 && c < VOLE_Q));
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let mut result = [0i128; 2 * VOLE_RING_N];
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for i in 0..VOLE_RING_N {
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for j in 0..VOLE_RING_N {
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result[i + j] += (self.coeffs[i] as i128) * (other.coeffs[j] as i128);
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}
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}
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let mut out = Self::zero();
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for i in 0..VOLE_RING_N {
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let combined = result[i] - result[i + VOLE_RING_N];
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out.coeffs[i] = ct_reduce(combined, VOLE_Q);
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}
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let result_coeffs = ntt_mul(&self.coeffs, &other.coeffs);
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let out = Self {
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coeffs: result_coeffs,
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};
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debug_assert!(out.coeffs.iter().all(|&c| c >= 0 && c < VOLE_Q));
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out
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