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Use montgomery multiplication everywhere. #49

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Aug 22, 2023
Merged

Use montgomery multiplication everywhere. #49

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d70f68a
Use montgomery reduction for NTT.
xvzcf Aug 18, 2023
21dd711
cargo fmt
xvzcf Aug 21, 2023
bcaa69e
Comments and slight refactoring.
xvzcf Aug 21, 2023
c70266b
Key generation and decryption now use montgomery multiplication only.
xvzcf Aug 21, 2023
ac72615
Use montgomery_multiplication everywhere.
xvzcf Aug 22, 2023
598875d
Comments and formatting.
xvzcf Aug 22, 2023
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33 changes: 19 additions & 14 deletions src/kem/kyber768/arithmetic.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -15,20 +15,25 @@ pub(crate) fn barrett_reduce(value: i16) -> KyberFieldElement {
value - (quotient * FIELD_MODULUS)
}

pub(crate) fn fe_mul(lhs: KyberFieldElement, rhs: KyberFieldElement) -> KyberFieldElement {
// TODO: This will shortly be replaced by an implementation of
// montgomery reduction.
let product: i32 = i32::from(lhs) * i32::from(rhs);

let reduced = (product % i32::from(FIELD_MODULUS)) as i16;

if reduced > FIELD_MODULUS / 2 {
reduced - FIELD_MODULUS
} else if reduced < -FIELD_MODULUS / 2 {
reduced + FIELD_MODULUS
} else {
reduced
}
const MONTGOMERY_SHIFT: i64 = 16;
const MONTGOMERY_R: i64 = 1i64 << MONTGOMERY_SHIFT;
const INVERSE_OF_MODULUS_MOD_R: i64 = -3327; // FIELD_MODULUS^{-1} mod MONTGOMERY_R

pub(crate) fn montgomery_reduce(value: i32) -> KyberFieldElement {
let t: i64 = i64::from(value) * INVERSE_OF_MODULUS_MOD_R;
let t: i32 = (t & (MONTGOMERY_R - 1)) as i32;

let t = value - (t * i32::from(FIELD_MODULUS));

(t >> MONTGOMERY_SHIFT) as i16
}

// Given a |value|, return |value|*R mod q. Notice that montgomery_reduce
// returns a value aR^{-1} mod q, and so montgomery_reduce(|value| * R^2)
// returns |value| * R^2 & R^{-1} mod q = |value| * R mod q.
pub(crate) fn to_montgomery_domain(value: i32) -> KyberFieldElement {
// R^2 mod q = (2^16)^2 mod 3329 = 1353
montgomery_reduce(1353 * value)
}

#[derive(Debug, Clone, Copy, PartialEq, Eq)]
Expand Down
6 changes: 4 additions & 2 deletions src/kem/kyber768/ind_cpa.rs
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Expand Up @@ -180,7 +180,9 @@ pub(crate) fn generate_keypair(
))
}

fn encode_and_compress_u(input: [KyberPolynomialRingElement; RANK]) -> [u8; VECTOR_U_ENCODED_SIZE] {
fn compress_then_encode_u(
input: [KyberPolynomialRingElement; RANK],
) -> [u8; VECTOR_U_ENCODED_SIZE] {
let mut out = [0u8; VECTOR_U_ENCODED_SIZE];
for (i, re) in input.into_iter().enumerate() {
out[i * BYTES_PER_ENCODED_ELEMENT_OF_U..(i + 1) * BYTES_PER_ENCODED_ELEMENT_OF_U]
Expand Down Expand Up @@ -258,7 +260,7 @@ pub(crate) fn encrypt(
+ decompress(message_as_ring_element, 1);

// c_1 := Encode_{du}(Compress_q(u,d_u))
let c1 = encode_and_compress_u(u);
let c1 = compress_then_encode_u(u);

// c_2 := Encode_{dv}(Compress_q(v,d_v))
let c2 = serialize_little_endian_4(compress(v, VECTOR_V_COMPRESSION_FACTOR));
Expand Down
120 changes: 68 additions & 52 deletions src/kem/kyber768/ntt.rs
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Original file line number Diff line number Diff line change
@@ -1,38 +1,28 @@
use crate::kem::kyber768::{
arithmetic::{barrett_reduce, KyberPolynomialRingElement},
arithmetic::{barrett_reduce, to_montgomery_domain, KyberPolynomialRingElement},
parameters::RANK,
};

use self::kyber_polynomial_ring_element_mod::ntt_multiply;

pub(crate) mod kyber_polynomial_ring_element_mod {
use crate::kem::kyber768::{
arithmetic::{barrett_reduce, fe_mul, KyberFieldElement, KyberPolynomialRingElement},
arithmetic::{
barrett_reduce, montgomery_reduce, KyberFieldElement, KyberPolynomialRingElement,
},
parameters::COEFFICIENTS_IN_RING_ELEMENT,
};

const ZETAS: [i16; 128] = [
1, -1600, -749, -40, -687, 630, -1432, 848, 1062, -1410, 193, 797, -543, -69, 569, -1583,
296, -882, 1339, 1476, -283, 56, -1089, 1333, 1426, -1235, 535, -447, -936, -450, -1355,
821, 289, 331, -76, -1573, 1197, -1025, -1052, -1274, 650, -1352, -816, 632, -464, 33,
1320, -1414, -1010, 1435, 807, 452, 1438, -461, 1534, -927, -682, -712, 1481, 648, -855,
-219, 1227, 910, 17, -568, 583, -680, 1637, 723, -1041, 1100, 1409, -667, -48, 233, 756,
-1173, -314, -279, -1626, 1651, -540, -1540, -1482, 952, 1461, -642, 939, -1021, -892,
-941, 733, -992, 268, 641, 1584, -1031, -1292, -109, 375, -780, -1239, 1645, 1063, 319,
-556, 757, -1230, 561, -863, -735, -525, 1092, 403, 1026, 1143, -1179, -554, 886, -1607,
1212, -1455, 1029, -1219, -394, 885, -1175,
];

const MOD_ROOTS: [i16; 128] = [
17, -17, -568, 568, 583, -583, -680, 680, 1637, -1637, 723, -723, -1041, 1041, 1100, -1100,
1409, -1409, -667, 667, -48, 48, 233, -233, 756, -756, -1173, 1173, -314, 314, -279, 279,
-1626, 1626, 1651, -1651, -540, 540, -1540, 1540, -1482, 1482, 952, -952, 1461, -1461,
-642, 642, 939, -939, -1021, 1021, -892, 892, -941, 941, 733, -733, -992, 992, 268, -268,
641, -641, 1584, -1584, -1031, 1031, -1292, 1292, -109, 109, 375, -375, -780, 780, -1239,
1239, 1645, -1645, 1063, -1063, 319, -319, -556, 556, 757, -757, -1230, 1230, 561, -561,
-863, 863, -735, 735, -525, 525, 1092, -1092, 403, -403, 1026, -1026, 1143, -1143, -1179,
1179, -554, 554, 886, -886, -1607, 1607, 1212, -1212, -1455, 1455, 1029, -1029, -1219,
1219, -394, 394, 885, -885, -1175, 1175,
const ZETAS_MONTGOMERY_DOMAIN: [i32; 128] = [
-1044, -758, -359, -1517, 1493, 1422, 287, 202, -171, 622, 1577, 182, 962, -1202, -1474,
1468, 573, -1325, 264, 383, -829, 1458, -1602, -130, -681, 1017, 732, 608, -1542, 411,
-205, -1571, 1223, 652, -552, 1015, -1293, 1491, -282, -1544, 516, -8, -320, -666, -1618,
-1162, 126, 1469, -853, -90, -271, 830, 107, -1421, -247, -951, -398, 961, -1508, -725,
448, -1065, 677, -1275, -1103, 430, 555, 843, -1251, 871, 1550, 105, 422, 587, 177, -235,
-291, -460, 1574, 1653, -246, 778, 1159, -147, -777, 1483, -602, 1119, -1590, 644, -872,
349, 418, 329, -156, -75, 817, 1097, 603, 610, 1322, -1285, -1465, 384, -1215, -136, 1218,
-1335, -874, 220, -1187, -1659, -1185, -1530, -1278, 794, -1510, -854, -870, 478, -108,
-308, 996, 991, 958, -1460, 1522, 1628,
];

const NTT_LAYERS: [usize; 7] = [2, 4, 8, 16, 32, 64, 128];
Expand All @@ -44,7 +34,9 @@ pub(crate) mod kyber_polynomial_ring_element_mod {
zeta_i += 1;

for j in offset..offset + layer {
let t = fe_mul(re[j + layer], ZETAS[zeta_i]);
let t = montgomery_reduce(
i32::from(re[j + layer]) * ZETAS_MONTGOMERY_DOMAIN[zeta_i],
);
re[j + layer] = re[j] - t;
re[j] += t;
}
Expand All @@ -56,34 +48,48 @@ pub(crate) mod kyber_polynomial_ring_element_mod {
}

pub fn invert_ntt(mut re: KyberPolynomialRingElement) -> KyberPolynomialRingElement {
let inverse_of_2: i16 = -1664;

let mut zeta_i = COEFFICIENTS_IN_RING_ELEMENT / 2;

for layer in NTT_LAYERS {
for offset in (0..(COEFFICIENTS_IN_RING_ELEMENT - layer)).step_by(2 * layer) {
zeta_i -= 1;

for j in offset..offset + layer {
let a_minus_b = re[j + layer] - re[j];
re[j] = fe_mul(re[j] + re[j + layer], inverse_of_2);
re[j + layer] = fe_mul(fe_mul(a_minus_b, ZETAS[zeta_i]), inverse_of_2);
let a_minus_b = (re[j + layer] - re[j]) as i32;

// Instead of dividing by 2 here, we just divide by
// 2^7 at once in the end.
re[j] = barrett_reduce(re[j] + re[j + layer]);
re[j + layer] = montgomery_reduce(a_minus_b * ZETAS_MONTGOMERY_DOMAIN[zeta_i]);
}
}
}
re.coefficients = re.coefficients.map(barrett_reduce);

// We first convert (2^7)^-1 = (128)^-1 into the montgomery domain by
// computing (2^7)^-1 * 2^16 mod 3329. This is equal to 512 = 2^9, and
// so we can just left-shift by 9 bits.
re.coefficients = re
.coefficients
.map(|coefficient| montgomery_reduce((coefficient as i32) << 9));

re
}

fn ntt_multiply_binomials(
(a0, a1): (KyberFieldElement, KyberFieldElement),
(b0, b1): (KyberFieldElement, KyberFieldElement),
zeta: i16,
zeta: i32,
) -> (KyberFieldElement, KyberFieldElement) {
let a0 = a0 as i32;
let a1 = a1 as i32;

let b0 = b0 as i32;
let b1 = b1 as i32;

(
fe_mul(a0, b0) + fe_mul(fe_mul(a1, b1), zeta),
fe_mul(a0, b1) + fe_mul(a1, b0),
montgomery_reduce(a0 * b0)
+ montgomery_reduce((montgomery_reduce(a1 * b1) as i32) * zeta),
montgomery_reduce(a0 * b1) + montgomery_reduce(a1 * b0),
)
}

Expand All @@ -97,25 +103,44 @@ pub(crate) mod kyber_polynomial_ring_element_mod {
let product = ntt_multiply_binomials(
(left[i], left[i + 1]),
(right[i], right[i + 1]),
MOD_ROOTS[i / 2],
ZETAS_MONTGOMERY_DOMAIN[64 + (i / 4)],
);
out[i] = product.0;
out[i + 1] = product.1;

let product = ntt_multiply_binomials(
(left[i + 2], left[i + 3]),
(right[i + 2], right[i + 3]),
MOD_ROOTS[(i + 2) / 2],
-ZETAS_MONTGOMERY_DOMAIN[64 + (i / 4)],
);
out[i + 2] = product.0;
out[i + 3] = product.1;
}
out.coefficients = out.coefficients.map(barrett_reduce);

out
}
}

pub(crate) fn multiply_row_by_column(
row_vector: &[KyberPolynomialRingElement; RANK],
column_vector: &[KyberPolynomialRingElement; RANK],
) -> KyberPolynomialRingElement {
let mut result = KyberPolynomialRingElement::ZERO;

for (row_element, column_element) in row_vector.iter().zip(column_vector.iter()) {
result = result + ntt_multiply(row_element, column_element);
}

// The coefficients of the form aR^{-1} mod q, which means
// calling to_montgomery_domain() on them should return a mod q.
result.coefficients = result
.coefficients
.map(|coefficient| to_montgomery_domain(coefficient as i32))
.map(barrett_reduce);

result
}

pub(crate) fn multiply_matrix_by_column(
matrix: &[[KyberPolynomialRingElement; RANK]; RANK],
vector: &[KyberPolynomialRingElement; RANK],
Expand All @@ -127,23 +152,14 @@ pub(crate) fn multiply_matrix_by_column(
let product = ntt_multiply(matrix_element, &vector[j]);
result[i] = result[i] + product;
}
result[i].coefficients = result[i].coefficients.map(barrett_reduce);
}

result
}

pub(crate) fn multiply_row_by_column(
row_vector: &[KyberPolynomialRingElement; RANK],
column_vector: &[KyberPolynomialRingElement; RANK],
) -> KyberPolynomialRingElement {
let mut result = KyberPolynomialRingElement::ZERO;

for (row_element, column_element) in row_vector.iter().zip(column_vector.iter()) {
result = result + ntt_multiply(row_element, column_element);
// The coefficients of the form aR^{-1} mod q, which means
// calling to_montgomery_domain() on them should return a mod q.
result[i].coefficients = result[i]
.coefficients
.map(|coefficient| to_montgomery_domain(coefficient as i32))
.map(barrett_reduce);
}

result.coefficients = result.coefficients.map(barrett_reduce);

result
}
6 changes: 6 additions & 0 deletions src/kem/kyber768/serialize.rs
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Expand Up @@ -32,6 +32,12 @@ use crate::kem::kyber768::{
///
/// Otherwise `deserialize_little_endian` is not injective and therefore has
/// no left inverse.
///
/// N.B.: All the `serialize_little_endian_{n}` functions work on the canonical
/// unsigned representative of each coefficient in the polynomial ring.
/// Only `serialize_little_endian_12` actually performs this conversion in the
/// function itself; the rest don't since they are called only after `compress_q`
/// is called, and `compress_q` also performs this conversion.

pub fn serialize_little_endian_1(
re: KyberPolynomialRingElement,
Expand Down