miden-crypto 0.25.0

// FFT-BASED MDS MULTIPLICATION HELPER FUNCTIONS
// ================================================================================================

//! This module contains helper functions as well as constants used to perform the vector-matrix
//! multiplication step of the Rescue prime permutation. The special form of our MDS matrix
//! i.e. being circular, allows us to reduce the vector-matrix multiplication to a Hadamard product
//! of two vectors in "frequency domain". This follows from the simple fact that every circulant
//! matrix has the columns of the discrete Fourier transform matrix as orthogonal eigenvectors.
//! The implementation also avoids the use of 3-point FFTs, and 3-point iFFTs, and substitutes that
//! with explicit expressions. It also avoids, due to the form of our matrix in the frequency
//! domain, divisions by 2 and repeated modular reductions. This is because of our explicit choice
//! of an MDS matrix that has small powers of 2 entries in frequency domain.
//! The following implementation has benefited greatly from the discussions and insights of
//! Hamish Ivey-Law and Jacqueline Nabaglo of Polygon Zero and is base on Nabaglo's Plonky2
//! implementation.

// Rescue MDS matrix in frequency domain.
//
// More precisely, this is the output of the three 4-point (real) FFTs of the first column of
// the MDS matrix i.e. just before the multiplication with the appropriate twiddle factors
// and application of the final four 3-point FFT in order to get the full 12-point FFT.
// The entries have been scaled appropriately in order to avoid divisions by 2 in iFFT2 and iFFT4.
// The code to generate the matrix in frequency domain is based on an adaptation of a code, to
// generate MDS matrices efficiently in original domain, that was developed by the Polygon Zero
// team.
const MDS_FREQ_BLOCK_ONE: [i64; 3] = [16, 8, 16];
const MDS_FREQ_BLOCK_TWO: [(i64, i64); 3] = [(-1, 2), (-1, 1), (4, 8)];
const MDS_FREQ_BLOCK_THREE: [i64; 3] = [-8, 1, 1];

// We use split 3 x 4 FFT transform in order to transform our vectors into the frequency domain.
#[inline(always)]
pub const fn mds_multiply_freq(state: [u64; 12]) -> [u64; 12] {
    let [s0, s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11] = state;

    let (u0, u1, u2) = fft4_real([s0, s3, s6, s9]);
    let (u4, u5, u6) = fft4_real([s1, s4, s7, s10]);
    let (u8, u9, u10) = fft4_real([s2, s5, s8, s11]);

    // This where the multiplication in frequency domain is done. More precisely, and with
    // the appropriate permutations in between, the sequence of
    // 3-point FFTs --> multiplication by twiddle factors --> Hadamard multiplication -->
    // 3 point iFFTs --> multiplication by (inverse) twiddle factors
    // is "squashed" into one step composed of the functions "block1", "block2" and "block3".
    // The expressions in the aforementioned functions are the result of explicit computations
    // combined with the Karatsuba trick for the multiplication of Complex numbers.

    let [v0, v4, v8] = block1([u0, u4, u8], MDS_FREQ_BLOCK_ONE);
    let [v1, v5, v9] = block2([u1, u5, u9], MDS_FREQ_BLOCK_TWO);
    let [v2, v6, v10] = block3([u2, u6, u10], MDS_FREQ_BLOCK_THREE);
    // The 4th block is not computed as it is similar to the 2nd one, up to complex conjugation,
    // and is, due to the use of the real FFT and iFFT, redundant.

    let [s0, s3, s6, s9] = ifft4_real((v0, v1, v2));
    let [s1, s4, s7, s10] = ifft4_real((v4, v5, v6));
    let [s2, s5, s8, s11] = ifft4_real((v8, v9, v10));

    [s0, s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11]
}

// We use the real FFT to avoid redundant computations. See https://www.mdpi.com/2076-3417/12/9/4700
#[inline(always)]
const fn fft2_real(x: [u64; 2]) -> [i64; 2] {
    [(x[0] as i64 + x[1] as i64), (x[0] as i64 - x[1] as i64)]
}

#[inline(always)]
const fn ifft2_real(y: [i64; 2]) -> [u64; 2] {
    // We avoid divisions by 2 by appropriately scaling the MDS matrix constants.
    [(y[0] + y[1]) as u64, (y[0] - y[1]) as u64]
}

#[inline(always)]
const fn fft4_real(x: [u64; 4]) -> (i64, (i64, i64), i64) {
    let [z0, z2] = fft2_real([x[0], x[2]]);
    let [z1, z3] = fft2_real([x[1], x[3]]);
    let y0 = z0 + z1;
    let y1 = (z2, -z3);
    let y2 = z0 - z1;
    (y0, y1, y2)
}

#[inline(always)]
const fn ifft4_real(y: (i64, (i64, i64), i64)) -> [u64; 4] {
    // In calculating 'z0' and 'z1', division by 2 is avoided by appropriately scaling
    // the MDS matrix constants.
    let z0 = y.0 + y.2;
    let z1 = y.0 - y.2;
    let z2 = y.1.0;
    let z3 = -y.1.1;

    let [x0, x2] = ifft2_real([z0, z2]);
    let [x1, x3] = ifft2_real([z1, z3]);

    [x0, x1, x2, x3]
}

#[inline(always)]
const fn block1(x: [i64; 3], y: [i64; 3]) -> [i64; 3] {
    let [x0, x1, x2] = x;
    let [y0, y1, y2] = y;
    let z0 = x0 * y0 + x1 * y2 + x2 * y1;
    let z1 = x0 * y1 + x1 * y0 + x2 * y2;
    let z2 = x0 * y2 + x1 * y1 + x2 * y0;

    [z0, z1, z2]
}

#[inline(always)]
const fn block2(x: [(i64, i64); 3], y: [(i64, i64); 3]) -> [(i64, i64); 3] {
    let [(x0r, x0i), (x1r, x1i), (x2r, x2i)] = x;
    let [(y0r, y0i), (y1r, y1i), (y2r, y2i)] = y;
    let x0s = x0r + x0i;
    let x1s = x1r + x1i;
    let x2s = x2r + x2i;
    let y0s = y0r + y0i;
    let y1s = y1r + y1i;
    let y2s = y2r + y2i;

    // Compute x0​y0 ​− ix1​y2​ − ix2​y1​ using Karatsuba for complex numbers multiplication
    let m0 = (x0r * y0r, x0i * y0i);
    let m1 = (x1r * y2r, x1i * y2i);
    let m2 = (x2r * y1r, x2i * y1i);
    let z0r = (m0.0 - m0.1) + (x1s * y2s - m1.0 - m1.1) + (x2s * y1s - m2.0 - m2.1);
    let z0i = (x0s * y0s - m0.0 - m0.1) + (-m1.0 + m1.1) + (-m2.0 + m2.1);
    let z0 = (z0r, z0i);

    // Compute x0​y1​ + x1​y0​ − ix2​y2 using Karatsuba for complex numbers multiplication
    let m0 = (x0r * y1r, x0i * y1i);
    let m1 = (x1r * y0r, x1i * y0i);
    let m2 = (x2r * y2r, x2i * y2i);
    let z1r = (m0.0 - m0.1) + (m1.0 - m1.1) + (x2s * y2s - m2.0 - m2.1);
    let z1i = (x0s * y1s - m0.0 - m0.1) + (x1s * y0s - m1.0 - m1.1) + (-m2.0 + m2.1);
    let z1 = (z1r, z1i);

    // Compute x0​y2​ + x1​y1 ​+ x2​y0​ using Karatsuba for complex numbers multiplication
    let m0 = (x0r * y2r, x0i * y2i);
    let m1 = (x1r * y1r, x1i * y1i);
    let m2 = (x2r * y0r, x2i * y0i);
    let z2r = (m0.0 - m0.1) + (m1.0 - m1.1) + (m2.0 - m2.1);
    let z2i = (x0s * y2s - m0.0 - m0.1) + (x1s * y1s - m1.0 - m1.1) + (x2s * y0s - m2.0 - m2.1);
    let z2 = (z2r, z2i);

    [z0, z1, z2]
}

#[inline(always)]
const fn block3(x: [i64; 3], y: [i64; 3]) -> [i64; 3] {
    let [x0, x1, x2] = x;
    let [y0, y1, y2] = y;
    let z0 = x0 * y0 - x1 * y2 - x2 * y1;
    let z1 = x0 * y1 + x1 * y0 - x2 * y2;
    let z2 = x0 * y2 + x1 * y1 + x2 * y0;

    [z0, z1, z2]
}

// TESTS
// ================================================================================================

#[cfg(test)]
mod tests {
    use proptest::prelude::*;

    use super::super::{Felt, MDS, ZERO, apply_mds};

    const STATE_WIDTH: usize = 12;

    #[inline(always)]
    fn apply_mds_naive(state: &mut [Felt; STATE_WIDTH]) {
        let mut result = [ZERO; STATE_WIDTH];
        result.iter_mut().zip(MDS.iter()).for_each(|(r, mds_row)| {
            state.iter().zip(mds_row).for_each(|(&s, &m)| {
                *r += m * s;
            });
        });
        *state = result;
    }

    proptest! {
        #[test]
        fn mds_freq_proptest(a in any::<[u64; STATE_WIDTH]>()) {

            let mut v1 = [ZERO; STATE_WIDTH];
            let mut v2;

            for i in 0..STATE_WIDTH {
                v1[i] = Felt::new_unchecked(a[i]);
            }
            v2 = v1;

            apply_mds_naive(&mut v1);
            apply_mds(&mut v2);

            prop_assert_eq!(v1, v2);
        }
    }
}