/* Copyright (c) 2018, Google Inc.
*
* Permission to use, copy, modify, and/or distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
* SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
* OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
* CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
#include <openssl/hrss.h>
#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
#include <openssl/bn.h>
#include <openssl/cpu.h>
#include <openssl/hmac.h>
#include <openssl/mem.h>
#include <openssl/sha.h>
#if defined(OPENSSL_X86) || defined(OPENSSL_X86_64)
#include <emmintrin.h>
#endif
#if (defined(OPENSSL_ARM) || defined(OPENSSL_AARCH64)) && \
(defined(__ARM_NEON__) || defined(__ARM_NEON))
#include <arm_neon.h>
#endif
#if defined(_MSC_VER)
#define RESTRICT
#else
#define RESTRICT restrict
#endif
#include "../internal.h"
#include "internal.h"
// This is an implementation of [HRSS], but with a KEM transformation based on
// [SXY]. The primary references are:
// HRSS: https://eprint.iacr.org/2017/667.pdf
// HRSSNIST:
// https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum-Cryptography/documents/round-1/submissions/NTRU_HRSS_KEM.zip
// SXY: https://eprint.iacr.org/2017/1005.pdf
// NTRUTN14:
// https://assets.onboardsecurity.com/static/downloads/NTRU/resources/NTRUTech014.pdf
// NTRUCOMP:
// https://eprint.iacr.org/2018/1174
// Vector operations.
//
// A couple of functions in this file can use vector operations to meaningful
// effect. If we're building for a target that has a supported vector unit,
// |HRSS_HAVE_VECTOR_UNIT| will be defined and |vec_t| will be typedefed to a
// 128-bit vector. The following functions abstract over the differences between
// NEON and SSE2 for implementing some vector operations.
// TODO: MSVC can likely also be made to work with vector operations.
#if ((defined(__SSE__) && defined(OPENSSL_X86)) || defined(OPENSSL_X86_64)) && \
(defined(__clang__) || !defined(_MSC_VER))
#define HRSS_HAVE_VECTOR_UNIT
typedef __m128i vec_t;
// vec_capable returns one iff the current platform supports SSE2.
static int vec_capable(void) {
#if defined(__SSE2__)
return 1;
#else
int has_sse2 = (OPENSSL_ia32cap_P[0] & (1 << 26)) != 0;
return has_sse2;
#endif
}
// vec_add performs a pair-wise addition of four uint16s from |a| and |b|.
static inline vec_t vec_add(vec_t a, vec_t b) { return _mm_add_epi16(a, b); }
// vec_sub performs a pair-wise subtraction of four uint16s from |a| and |b|.
static inline vec_t vec_sub(vec_t a, vec_t b) { return _mm_sub_epi16(a, b); }
// vec_mul multiplies each uint16_t in |a| by |b| and returns the resulting
// vector.
static inline vec_t vec_mul(vec_t a, uint16_t b) {
return _mm_mullo_epi16(a, _mm_set1_epi16(b));
}
// vec_fma multiplies each uint16_t in |b| by |c|, adds the result to |a|, and
// returns the resulting vector.
static inline vec_t vec_fma(vec_t a, vec_t b, uint16_t c) {
return _mm_add_epi16(a, _mm_mullo_epi16(b, _mm_set1_epi16(c)));
}
// vec3_rshift_word right-shifts the 24 uint16_t's in |v| by one uint16.
static inline void vec3_rshift_word(vec_t v[3]) {
// Intel's left and right shifting is backwards compared to the order in
// memory because they're based on little-endian order of words (and not just
// bytes). So the shifts in this function will be backwards from what one
// might expect.
const __m128i carry0 = _mm_srli_si128(v[0], 14);
v[0] = _mm_slli_si128(v[0], 2);
const __m128i carry1 = _mm_srli_si128(v[1], 14);
v[1] = _mm_slli_si128(v[1], 2);
v[1] |= carry0;
v[2] = _mm_slli_si128(v[2], 2);
v[2] |= carry1;
}
// vec4_rshift_word right-shifts the 32 uint16_t's in |v| by one uint16.
static inline void vec4_rshift_word(vec_t v[4]) {
// Intel's left and right shifting is backwards compared to the order in
// memory because they're based on little-endian order of words (and not just
// bytes). So the shifts in this function will be backwards from what one
// might expect.
const __m128i carry0 = _mm_srli_si128(v[0], 14);
v[0] = _mm_slli_si128(v[0], 2);
const __m128i carry1 = _mm_srli_si128(v[1], 14);
v[1] = _mm_slli_si128(v[1], 2);
v[1] |= carry0;
const __m128i carry2 = _mm_srli_si128(v[2], 14);
v[2] = _mm_slli_si128(v[2], 2);
v[2] |= carry1;
v[3] = _mm_slli_si128(v[3], 2);
v[3] |= carry2;
}
// vec_merge_3_5 takes the final three uint16_t's from |left|, appends the first
// five from |right|, and returns the resulting vector.
static inline vec_t vec_merge_3_5(vec_t left, vec_t right) {
return _mm_srli_si128(left, 10) | _mm_slli_si128(right, 6);
}
// poly3_vec_lshift1 left-shifts the 768 bits in |a_s|, and in |a_a|, by one
// bit.
static inline void poly3_vec_lshift1(vec_t a_s[6], vec_t a_a[6]) {
vec_t carry_s = {0};
vec_t carry_a = {0};
for (int i = 0; i < 6; i++) {
vec_t next_carry_s = _mm_srli_epi64(a_s[i], 63);
a_s[i] = _mm_slli_epi64(a_s[i], 1);
a_s[i] |= _mm_slli_si128(next_carry_s, 8);
a_s[i] |= carry_s;
carry_s = _mm_srli_si128(next_carry_s, 8);
vec_t next_carry_a = _mm_srli_epi64(a_a[i], 63);
a_a[i] = _mm_slli_epi64(a_a[i], 1);
a_a[i] |= _mm_slli_si128(next_carry_a, 8);
a_a[i] |= carry_a;
carry_a = _mm_srli_si128(next_carry_a, 8);
}
}
// poly3_vec_rshift1 right-shifts the 768 bits in |a_s|, and in |a_a|, by one
// bit.
static inline void poly3_vec_rshift1(vec_t a_s[6], vec_t a_a[6]) {
vec_t carry_s = {0};
vec_t carry_a = {0};
for (int i = 5; i >= 0; i--) {
const vec_t next_carry_s = _mm_slli_epi64(a_s[i], 63);
a_s[i] = _mm_srli_epi64(a_s[i], 1);
a_s[i] |= _mm_srli_si128(next_carry_s, 8);
a_s[i] |= carry_s;
carry_s = _mm_slli_si128(next_carry_s, 8);
const vec_t next_carry_a = _mm_slli_epi64(a_a[i], 63);
a_a[i] = _mm_srli_epi64(a_a[i], 1);
a_a[i] |= _mm_srli_si128(next_carry_a, 8);
a_a[i] |= carry_a;
carry_a = _mm_slli_si128(next_carry_a, 8);
}
}
// vec_broadcast_bit duplicates the least-significant bit in |a| to all bits in
// a vector and returns the result.
static inline vec_t vec_broadcast_bit(vec_t a) {
return _mm_shuffle_epi32(_mm_srai_epi32(_mm_slli_epi64(a, 63), 31),
0b01010101);
}
// vec_broadcast_bit15 duplicates the most-significant bit of the first word in
// |a| to all bits in a vector and returns the result.
static inline vec_t vec_broadcast_bit15(vec_t a) {
return _mm_shuffle_epi32(_mm_srai_epi32(_mm_slli_epi64(a, 63 - 15), 31),
0b01010101);
}
// vec_get_word returns the |i|th uint16_t in |v|. (This is a macro because the
// compiler requires that |i| be a compile-time constant.)
#define vec_get_word(v, i) _mm_extract_epi16(v, i)
#elif (defined(OPENSSL_ARM) || defined(OPENSSL_AARCH64)) && \
(defined(__ARM_NEON__) || defined(__ARM_NEON))
#define HRSS_HAVE_VECTOR_UNIT
typedef uint16x8_t vec_t;
// These functions perform the same actions as the SSE2 function of the same
// name, above.
static int vec_capable(void) { return CRYPTO_is_NEON_capable(); }
static inline vec_t vec_add(vec_t a, vec_t b) { return a + b; }
static inline vec_t vec_sub(vec_t a, vec_t b) { return a - b; }
static inline vec_t vec_mul(vec_t a, uint16_t b) { return vmulq_n_u16(a, b); }
static inline vec_t vec_fma(vec_t a, vec_t b, uint16_t c) {
return vmlaq_n_u16(a, b, c);
}
static inline void vec3_rshift_word(vec_t v[3]) {
const uint16x8_t kZero = {0};
v[2] = vextq_u16(v[1], v[2], 7);
v[1] = vextq_u16(v[0], v[1], 7);
v[0] = vextq_u16(kZero, v[0], 7);
}
static inline void vec4_rshift_word(vec_t v[4]) {
const uint16x8_t kZero = {0};
v[3] = vextq_u16(v[2], v[3], 7);
v[2] = vextq_u16(v[1], v[2], 7);
v[1] = vextq_u16(v[0], v[1], 7);
v[0] = vextq_u16(kZero, v[0], 7);
}
static inline vec_t vec_merge_3_5(vec_t left, vec_t right) {
return vextq_u16(left, right, 5);
}
static inline uint16_t vec_get_word(vec_t v, unsigned i) {
return v[i];
}
#if !defined(OPENSSL_AARCH64)
static inline vec_t vec_broadcast_bit(vec_t a) {
a = (vec_t)vshrq_n_s16(((int16x8_t)a) << 15, 15);
return vdupq_lane_u16(vget_low_u16(a), 0);
}
static inline vec_t vec_broadcast_bit15(vec_t a) {
a = (vec_t)vshrq_n_s16((int16x8_t)a, 15);
return vdupq_lane_u16(vget_low_u16(a), 0);
}
static inline void poly3_vec_lshift1(vec_t a_s[6], vec_t a_a[6]) {
vec_t carry_s = {0};
vec_t carry_a = {0};
const vec_t kZero = {0};
for (int i = 0; i < 6; i++) {
vec_t next_carry_s = a_s[i] >> 15;
a_s[i] <<= 1;
a_s[i] |= vextq_u16(kZero, next_carry_s, 7);
a_s[i] |= carry_s;
carry_s = vextq_u16(next_carry_s, kZero, 7);
vec_t next_carry_a = a_a[i] >> 15;
a_a[i] <<= 1;
a_a[i] |= vextq_u16(kZero, next_carry_a, 7);
a_a[i] |= carry_a;
carry_a = vextq_u16(next_carry_a, kZero, 7);
}
}
static inline void poly3_vec_rshift1(vec_t a_s[6], vec_t a_a[6]) {
vec_t carry_s = {0};
vec_t carry_a = {0};
const vec_t kZero = {0};
for (int i = 5; i >= 0; i--) {
vec_t next_carry_s = a_s[i] << 15;
a_s[i] >>= 1;
a_s[i] |= vextq_u16(next_carry_s, kZero, 1);
a_s[i] |= carry_s;
carry_s = vextq_u16(kZero, next_carry_s, 1);
vec_t next_carry_a = a_a[i] << 15;
a_a[i] >>= 1;
a_a[i] |= vextq_u16(next_carry_a, kZero, 1);
a_a[i] |= carry_a;
carry_a = vextq_u16(kZero, next_carry_a, 1);
}
}
#endif // !OPENSSL_AARCH64
#endif // (ARM || AARCH64) && NEON
// Polynomials in this scheme have N terms.
// #define N 701
// Underlying data types and arithmetic operations.
// ------------------------------------------------
// Binary polynomials.
// poly2 represents a degree-N polynomial over GF(2). The words are in little-
// endian order, i.e. the coefficient of x^0 is the LSB of the first word. The
// final word is only partially used since N is not a multiple of the word size.
// Defined in internal.h:
// struct poly2 {
// crypto_word_t v[WORDS_PER_POLY];
// };
OPENSSL_UNUSED static void hexdump(const void *void_in, size_t len) {
const uint8_t *in = (const uint8_t *)void_in;
for (size_t i = 0; i < len; i++) {
printf("%02x", in[i]);
}
printf("\n");
}
static void poly2_zero(struct poly2 *p) {
OPENSSL_memset(&p->v[0], 0, sizeof(crypto_word_t) * WORDS_PER_POLY);
}
// poly2_cmov sets |out| to |in| iff |mov| is all ones.
static void poly2_cmov(struct poly2 *out, const struct poly2 *in,
crypto_word_t mov) {
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
out->v[i] = (out->v[i] & ~mov) | (in->v[i] & mov);
}
}
// poly2_rotr_words performs a right-rotate on |in|, writing the result to
// |out|. The shift count, |bits|, must be a non-zero multiple of the word size.
static void poly2_rotr_words(struct poly2 *out, const struct poly2 *in,
size_t bits) {
assert(bits >= BITS_PER_WORD && bits % BITS_PER_WORD == 0);
assert(out != in);
const size_t start = bits / BITS_PER_WORD;
const size_t n = (N - bits) / BITS_PER_WORD;
// The rotate is by a whole number of words so the first few words are easy:
// just move them down.
for (size_t i = 0; i < n; i++) {
out->v[i] = in->v[start + i];
}
// Since the last word is only partially filled, however, the remainder needs
// shifting and merging of words to take care of that.
crypto_word_t carry = in->v[WORDS_PER_POLY - 1];
for (size_t i = 0; i < start; i++) {
out->v[n + i] = carry | in->v[i] << BITS_IN_LAST_WORD;
carry = in->v[i] >> (BITS_PER_WORD - BITS_IN_LAST_WORD);
}
out->v[WORDS_PER_POLY - 1] = carry;
}
// poly2_rotr_bits performs a right-rotate on |in|, writing the result to |out|.
// The shift count, |bits|, must be a power of two that is less than
// |BITS_PER_WORD|.
static void poly2_rotr_bits(struct poly2 *out, const struct poly2 *in,
size_t bits) {
assert(bits <= BITS_PER_WORD / 2);
assert(bits != 0);
assert((bits & (bits - 1)) == 0);
assert(out != in);
// BITS_PER_WORD/2 is the greatest legal value of |bits|. If
// |BITS_IN_LAST_WORD| is smaller than this then the code below doesn't work
// because more than the last word needs to carry down in the previous one and
// so on.
OPENSSL_STATIC_ASSERT(
BITS_IN_LAST_WORD >= BITS_PER_WORD / 2,
"there are more carry bits than fit in BITS_IN_LAST_WORD");
crypto_word_t carry = in->v[WORDS_PER_POLY - 1] << (BITS_PER_WORD - bits);
for (size_t i = WORDS_PER_POLY - 2; i < WORDS_PER_POLY; i--) {
out->v[i] = carry | in->v[i] >> bits;
carry = in->v[i] << (BITS_PER_WORD - bits);
}
crypto_word_t last_word = carry >> (BITS_PER_WORD - BITS_IN_LAST_WORD) |
in->v[WORDS_PER_POLY - 1] >> bits;
last_word &= (UINT64_C(1) << BITS_IN_LAST_WORD) - 1;
out->v[WORDS_PER_POLY - 1] = last_word;
}
// HRSS_poly2_rotr_consttime right-rotates |p| by |bits| in constant-time.
void HRSS_poly2_rotr_consttime(struct poly2 *p, size_t bits) {
assert(bits <= N);
assert(p->v[WORDS_PER_POLY-1] >> BITS_IN_LAST_WORD == 0);
// Constant-time rotation is implemented by calculating the rotations of
// powers-of-two bits and throwing away the unneeded values. 2^9 (i.e. 512) is
// the largest power-of-two shift that we need to consider because 2^10 > N.
#define HRSS_POLY2_MAX_SHIFT 9
size_t shift = HRSS_POLY2_MAX_SHIFT;
OPENSSL_STATIC_ASSERT((1 << (HRSS_POLY2_MAX_SHIFT + 1)) > N,
"maximum shift is too small");
OPENSSL_STATIC_ASSERT((1 << HRSS_POLY2_MAX_SHIFT) <= N,
"maximum shift is too large");
struct poly2 shifted;
for (; (UINT64_C(1) << shift) >= BITS_PER_WORD; shift--) {
poly2_rotr_words(&shifted, p, UINT64_C(1) << shift);
poly2_cmov(p, &shifted, ~((1 & (bits >> shift)) - 1));
}
for (; shift < HRSS_POLY2_MAX_SHIFT; shift--) {
poly2_rotr_bits(&shifted, p, UINT64_C(1) << shift);
poly2_cmov(p, &shifted, ~((1 & (bits >> shift)) - 1));
}
#undef HRSS_POLY2_MAX_SHIFT
}
// poly2_cswap exchanges the values of |a| and |b| if |swap| is all ones.
static void poly2_cswap(struct poly2 *a, struct poly2 *b, crypto_word_t swap) {
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
const crypto_word_t sum = swap & (a->v[i] ^ b->v[i]);
a->v[i] ^= sum;
b->v[i] ^= sum;
}
}
// poly2_fmadd sets |out| to |out| + |in| * m, where m is either
// |CONSTTIME_TRUE_W| or |CONSTTIME_FALSE_W|.
static void poly2_fmadd(struct poly2 *out, const struct poly2 *in,
crypto_word_t m) {
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
out->v[i] ^= in->v[i] & m;
}
}
// poly2_lshift1 left-shifts |p| by one bit.
static void poly2_lshift1(struct poly2 *p) {
crypto_word_t carry = 0;
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
const crypto_word_t next_carry = p->v[i] >> (BITS_PER_WORD - 1);
p->v[i] <<= 1;
p->v[i] |= carry;
carry = next_carry;
}
}
// poly2_rshift1 right-shifts |p| by one bit.
static void poly2_rshift1(struct poly2 *p) {
crypto_word_t carry = 0;
for (size_t i = WORDS_PER_POLY - 1; i < WORDS_PER_POLY; i--) {
const crypto_word_t next_carry = p->v[i] & 1;
p->v[i] >>= 1;
p->v[i] |= carry << (BITS_PER_WORD - 1);
carry = next_carry;
}
}
// poly2_clear_top_bits clears the bits in the final word that are only for
// alignment.
static void poly2_clear_top_bits(struct poly2 *p) {
p->v[WORDS_PER_POLY - 1] &= (UINT64_C(1) << BITS_IN_LAST_WORD) - 1;
}
// poly2_top_bits_are_clear returns one iff the extra bits in the final words of
// |p| are zero.
static int poly2_top_bits_are_clear(const struct poly2 *p) {
return (p->v[WORDS_PER_POLY - 1] &
~((UINT64_C(1) << BITS_IN_LAST_WORD) - 1)) == 0;
}
// Ternary polynomials.
// poly3 represents a degree-N polynomial over GF(3). Each coefficient is
// bitsliced across the |s| and |a| arrays, like this:
//
// s | a | value
// -----------------
// 0 | 0 | 0
// 0 | 1 | 1
// 1 | 1 | -1 (aka 2)
// 1 | 0 | <invalid>
//
// ('s' is for sign, and 'a' is the absolute value.)
//
// Once bitsliced as such, the following circuits can be used to implement
// addition and multiplication mod 3:
//
// (s3, a3) = (s1, a1) × (s2, a2)
// a3 = a1 ∧ a2
// s3 = (s1 ⊕ s2) ∧ a3
//
// (s3, a3) = (s1, a1) + (s2, a2)
// t = s1 ⊕ a2
// s3 = t ∧ (s2 ⊕ a1)
// a3 = (a1 ⊕ a2) ∨ (t ⊕ s2)
//
// (s3, a3) = (s1, a1) - (s2, a2)
// t = a1 ⊕ a2
// s3 = (s1 ⊕ a2) ∧ (t ⊕ s2)
// a3 = t ∨ (s1 ⊕ s2)
//
// Negating a value just involves XORing s by a.
//
// struct poly3 {
// struct poly2 s, a;
// };
OPENSSL_UNUSED static void poly3_print(const struct poly3 *in) {
struct poly3 p;
OPENSSL_memcpy(&p, in, sizeof(p));
p.s.v[WORDS_PER_POLY - 1] &= ((crypto_word_t)1 << BITS_IN_LAST_WORD) - 1;
p.a.v[WORDS_PER_POLY - 1] &= ((crypto_word_t)1 << BITS_IN_LAST_WORD) - 1;
printf("{[");
for (unsigned i = 0; i < WORDS_PER_POLY; i++) {
if (i) {
printf(" ");
}
printf(BN_HEX_FMT2, p.s.v[i]);
}
printf("] [");
for (unsigned i = 0; i < WORDS_PER_POLY; i++) {
if (i) {
printf(" ");
}
printf(BN_HEX_FMT2, p.a.v[i]);
}
printf("]}\n");
}
static void poly3_zero(struct poly3 *p) {
poly2_zero(&p->s);
poly2_zero(&p->a);
}
// poly3_word_mul sets (|out_s|, |out_a) to (|s1|, |a1|) × (|s2|, |a2|).
static void poly3_word_mul(crypto_word_t *out_s, crypto_word_t *out_a,
const crypto_word_t s1, const crypto_word_t a1,
const crypto_word_t s2, const crypto_word_t a2) {
*out_a = a1 & a2;
*out_s = (s1 ^ s2) & *out_a;
}
// poly3_word_add sets (|out_s|, |out_a|) to (|s1|, |a1|) + (|s2|, |a2|).
static void poly3_word_add(crypto_word_t *out_s, crypto_word_t *out_a,
const crypto_word_t s1, const crypto_word_t a1,
const crypto_word_t s2, const crypto_word_t a2) {
const crypto_word_t t = s1 ^ a2;
*out_s = t & (s2 ^ a1);
*out_a = (a1 ^ a2) | (t ^ s2);
}
// poly3_word_sub sets (|out_s|, |out_a|) to (|s1|, |a1|) - (|s2|, |a2|).
static void poly3_word_sub(crypto_word_t *out_s, crypto_word_t *out_a,
const crypto_word_t s1, const crypto_word_t a1,
const crypto_word_t s2, const crypto_word_t a2) {
const crypto_word_t t = a1 ^ a2;
*out_s = (s1 ^ a2) & (t ^ s2);
*out_a = t | (s1 ^ s2);
}
// lsb_to_all replicates the least-significant bit of |v| to all bits of the
// word. This is used in bit-slicing operations to make a vector from a fixed
// value.
static crypto_word_t lsb_to_all(crypto_word_t v) { return 0u - (v & 1); }
// poly3_mul_const sets |p| to |p|×m, where m = (ms, ma).
static void poly3_mul_const(struct poly3 *p, crypto_word_t ms,
crypto_word_t ma) {
ms = lsb_to_all(ms);
ma = lsb_to_all(ma);
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
poly3_word_mul(&p->s.v[i], &p->a.v[i], p->s.v[i], p->a.v[i], ms, ma);
}
}
// poly3_rotr_consttime right-rotates |p| by |bits| in constant-time.
static void poly3_rotr_consttime(struct poly3 *p, size_t bits) {
assert(bits <= N);
HRSS_poly2_rotr_consttime(&p->s, bits);
HRSS_poly2_rotr_consttime(&p->a, bits);
}
// poly3_fmadd sets |out| to |out| - |in|×m, where m is (ms, ma).
static void poly3_fmsub(struct poly3 *RESTRICT out,
const struct poly3 *RESTRICT in, crypto_word_t ms,
crypto_word_t ma) {
crypto_word_t product_s, product_a;
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
poly3_word_mul(&product_s, &product_a, in->s.v[i], in->a.v[i], ms, ma);
poly3_word_sub(&out->s.v[i], &out->a.v[i], out->s.v[i], out->a.v[i],
product_s, product_a);
}
}
// final_bit_to_all replicates the bit in the final position of the last word to
// all the bits in the word.
static crypto_word_t final_bit_to_all(crypto_word_t v) {
return lsb_to_all(v >> (BITS_IN_LAST_WORD - 1));
}
// poly3_top_bits_are_clear returns one iff the extra bits in the final words of
// |p| are zero.
OPENSSL_UNUSED static int poly3_top_bits_are_clear(const struct poly3 *p) {
return poly2_top_bits_are_clear(&p->s) && poly2_top_bits_are_clear(&p->a);
}
// poly3_mod_phiN reduces |p| by Φ(N).
static void poly3_mod_phiN(struct poly3 *p) {
// In order to reduce by Φ(N) we subtract by the value of the greatest
// coefficient.
const crypto_word_t factor_s = final_bit_to_all(p->s.v[WORDS_PER_POLY - 1]);
const crypto_word_t factor_a = final_bit_to_all(p->a.v[WORDS_PER_POLY - 1]);
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
poly3_word_sub(&p->s.v[i], &p->a.v[i], p->s.v[i], p->a.v[i], factor_s,
factor_a);
}
poly2_clear_top_bits(&p->s);
poly2_clear_top_bits(&p->a);
}
static void poly3_cswap(struct poly3 *a, struct poly3 *b, crypto_word_t swap) {
poly2_cswap(&a->s, &b->s, swap);
poly2_cswap(&a->a, &b->a, swap);
}
static void poly3_lshift1(struct poly3 *p) {
poly2_lshift1(&p->s);
poly2_lshift1(&p->a);
}
static void poly3_rshift1(struct poly3 *p) {
poly2_rshift1(&p->s);
poly2_rshift1(&p->a);
}
// poly3_span represents a pointer into a poly3.
struct poly3_span {
crypto_word_t *s;
crypto_word_t *a;
};
// poly3_span_add adds |n| words of values from |a| and |b| and writes the
// result to |out|.
static void poly3_span_add(const struct poly3_span *out,
const struct poly3_span *a,
const struct poly3_span *b, size_t n) {
for (size_t i = 0; i < n; i++) {
poly3_word_add(&out->s[i], &out->a[i], a->s[i], a->a[i], b->s[i], b->a[i]);
}
}
// poly3_span_sub subtracts |n| words of |b| from |n| words of |a|.
static void poly3_span_sub(const struct poly3_span *a,
const struct poly3_span *b, size_t n) {
for (size_t i = 0; i < n; i++) {
poly3_word_sub(&a->s[i], &a->a[i], a->s[i], a->a[i], b->s[i], b->a[i]);
}
}
// poly3_mul_aux is a recursive function that multiplies |n| words from |a| and
// |b| and writes 2×|n| words to |out|. Each call uses 2*ceil(n/2) elements of
// |scratch| and the function recurses, except if |n| == 1, when |scratch| isn't
// used and the recursion stops. For |n| in {11, 22}, the transitive total
// amount of |scratch| needed happens to be 2n+2.
static void poly3_mul_aux(const struct poly3_span *out,
const struct poly3_span *scratch,
const struct poly3_span *a,
const struct poly3_span *b, size_t n) {
if (n == 1) {
crypto_word_t r_s_low = 0, r_s_high = 0, r_a_low = 0, r_a_high = 0;
crypto_word_t b_s = b->s[0], b_a = b->a[0];
const crypto_word_t a_s = a->s[0], a_a = a->a[0];
for (size_t i = 0; i < BITS_PER_WORD; i++) {
// Multiply (s, a) by the next value from (b_s, b_a).
crypto_word_t m_s, m_a;
poly3_word_mul(&m_s, &m_a, a_s, a_a, lsb_to_all(b_s), lsb_to_all(b_a));
b_s >>= 1;
b_a >>= 1;
if (i == 0) {
// Special case otherwise the code tries to shift by BITS_PER_WORD
// below, which is undefined.
r_s_low = m_s;
r_a_low = m_a;
continue;
}
// Shift the multiplication result to the correct position.
const crypto_word_t m_s_low = m_s << i;
const crypto_word_t m_s_high = m_s >> (BITS_PER_WORD - i);
const crypto_word_t m_a_low = m_a << i;
const crypto_word_t m_a_high = m_a >> (BITS_PER_WORD - i);
// Add into the result.
poly3_word_add(&r_s_low, &r_a_low, r_s_low, r_a_low, m_s_low, m_a_low);
poly3_word_add(&r_s_high, &r_a_high, r_s_high, r_a_high, m_s_high,
m_a_high);
}
out->s[0] = r_s_low;
out->s[1] = r_s_high;
out->a[0] = r_a_low;
out->a[1] = r_a_high;
return;
}
// Karatsuba multiplication.
// https://en.wikipedia.org/wiki/Karatsuba_algorithm
// When |n| is odd, the two "halves" will have different lengths. The first
// is always the smaller.
const size_t low_len = n / 2;
const size_t high_len = n - low_len;
const struct poly3_span a_high = {&a->s[low_len], &a->a[low_len]};
const struct poly3_span b_high = {&b->s[low_len], &b->a[low_len]};
// Store a_1 + a_0 in the first half of |out| and b_1 + b_0 in the second
// half.
const struct poly3_span a_cross_sum = *out;
const struct poly3_span b_cross_sum = {&out->s[high_len], &out->a[high_len]};
poly3_span_add(&a_cross_sum, a, &a_high, low_len);
poly3_span_add(&b_cross_sum, b, &b_high, low_len);
if (high_len != low_len) {
a_cross_sum.s[low_len] = a_high.s[low_len];
a_cross_sum.a[low_len] = a_high.a[low_len];
b_cross_sum.s[low_len] = b_high.s[low_len];
b_cross_sum.a[low_len] = b_high.a[low_len];
}
const struct poly3_span child_scratch = {&scratch->s[2 * high_len],
&scratch->a[2 * high_len]};
const struct poly3_span out_mid = {&out->s[low_len], &out->a[low_len]};
const struct poly3_span out_high = {&out->s[2 * low_len],
&out->a[2 * low_len]};
// Calculate (a_1 + a_0) × (b_1 + b_0) and write to scratch buffer.
poly3_mul_aux(scratch, &child_scratch, &a_cross_sum, &b_cross_sum, high_len);
// Calculate a_1 × b_1.
poly3_mul_aux(&out_high, &child_scratch, &a_high, &b_high, high_len);
// Calculate a_0 × b_0.
poly3_mul_aux(out, &child_scratch, a, b, low_len);
// Subtract those last two products from the first.
poly3_span_sub(scratch, out, low_len * 2);
poly3_span_sub(scratch, &out_high, high_len * 2);
// Add the middle product into the output.
poly3_span_add(&out_mid, &out_mid, scratch, high_len * 2);
}
// HRSS_poly3_mul sets |*out| to |x|×|y| mod Φ(N).
void HRSS_poly3_mul(struct poly3 *out, const struct poly3 *x,
const struct poly3 *y) {
crypto_word_t prod_s[WORDS_PER_POLY * 2];
crypto_word_t prod_a[WORDS_PER_POLY * 2];
crypto_word_t scratch_s[WORDS_PER_POLY * 2 + 2];
crypto_word_t scratch_a[WORDS_PER_POLY * 2 + 2];
const struct poly3_span prod_span = {prod_s, prod_a};
const struct poly3_span scratch_span = {scratch_s, scratch_a};
const struct poly3_span x_span = {(crypto_word_t *)x->s.v,
(crypto_word_t *)x->a.v};
const struct poly3_span y_span = {(crypto_word_t *)y->s.v,
(crypto_word_t *)y->a.v};
poly3_mul_aux(&prod_span, &scratch_span, &x_span, &y_span, WORDS_PER_POLY);
// |prod| needs to be reduced mod (𝑥^n - 1), which just involves adding the
// upper-half to the lower-half. However, N is 701, which isn't a multiple of
// BITS_PER_WORD, so the upper-half vectors all have to be shifted before
// being added to the lower-half.
for (size_t i = 0; i < WORDS_PER_POLY; i++) {
crypto_word_t v_s = prod_s[WORDS_PER_POLY + i - 1] >> BITS_IN_LAST_WORD;
v_s |= prod_s[WORDS_PER_POLY + i] << (BITS_PER_WORD - BITS_IN_LAST_WORD);
crypto_word_t v_a = prod_a[WORDS_PER_POLY + i - 1] >> BITS_IN_LAST_WORD;
v_a |= prod_a[WORDS_PER_POLY + i] << (BITS_PER_WORD - BITS_IN_LAST_WORD);
poly3_word_add(&out->s.v[i], &out->a.v[i], prod_s[i], prod_a[i], v_s, v_a);
}
poly3_mod_phiN(out);
}
#if defined(HRSS_HAVE_VECTOR_UNIT) && !defined(OPENSSL_AARCH64)
// poly3_vec_cswap swaps (|a_s|, |a_a|) and (|b_s|, |b_a|) if |swap| is
// |0xff..ff|. Otherwise, |swap| must be zero.
static inline void poly3_vec_cswap(vec_t a_s[6], vec_t a_a[6], vec_t b_s[6],
vec_t b_a[6], const vec_t swap) {
for (int i = 0; i < 6; i++) {
const vec_t sum_s = swap & (a_s[i] ^ b_s[i]);
a_s[i] ^= sum_s;
b_s[i] ^= sum_s;
const vec_t sum_a = swap & (a_a[i] ^ b_a[i]);
a_a[i] ^= sum_a;
b_a[i] ^= sum_a;
}
}
// poly3_vec_fmsub subtracts (|ms|, |ma|) × (|b_s|, |b_a|) from (|a_s|, |a_a|).
static inline void poly3_vec_fmsub(vec_t a_s[6], vec_t a_a[6], vec_t b_s[6],
vec_t b_a[6], const vec_t ms,
const vec_t ma) {
for (int i = 0; i < 6; i++) {
// See the bitslice formula, above.
const vec_t s = b_s[i];
const vec_t a = b_a[i];
const vec_t product_a = a & ma;
const vec_t product_s = (s ^ ms) & product_a;
const vec_t out_s = a_s[i];
const vec_t out_a = a_a[i];
const vec_t t = out_a ^ product_a;
a_s[i] = (out_s ^ product_a) & (t ^ product_s);
a_a[i] = t | (out_s ^ product_s);
}
}
// poly3_invert_vec sets |*out| to |in|^-1, i.e. such that |out|×|in| == 1 mod
// Φ(N).
static void poly3_invert_vec(struct poly3 *out, const struct poly3 *in) {
// See the comment in |HRSS_poly3_invert| about this algorithm. In addition to
// the changes described there, this implementation attempts to use vector
// registers to speed up the computation. Even non-poly3 variables are held in
// vectors where possible to minimise the amount of data movement between
// the vector and general-purpose registers.
vec_t b_s[6], b_a[6], c_s[6], c_a[6], f_s[6], f_a[6], g_s[6], g_a[6];
const vec_t kZero = {0};
const vec_t kOne = {1};
static const uint8_t kOneBytes[sizeof(vec_t)] = {1};
static const uint8_t kBottomSixtyOne[sizeof(vec_t)] = {
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x1f};
memset(b_s, 0, sizeof(b_s));
memcpy(b_a, kOneBytes, sizeof(kOneBytes));
memset(&b_a[1], 0, 5 * sizeof(vec_t));
memset(c_s, 0, sizeof(c_s));
memset(c_a, 0, sizeof(c_a));
f_s[5] = kZero;
memcpy(f_s, in->s.v, WORDS_PER_POLY * sizeof(crypto_word_t));
f_a[5] = kZero;
memcpy(f_a, in->a.v, WORDS_PER_POLY * sizeof(crypto_word_t));
// Set g to all ones.
memset(g_s, 0, sizeof(g_s));
memset(g_a, 0xff, 5 * sizeof(vec_t));
memcpy(&g_a[5], kBottomSixtyOne, sizeof(kBottomSixtyOne));
vec_t deg_f = {N - 1}, deg_g = {N - 1}, rotation = kZero;
vec_t k = kOne;
vec_t f0s = {0}, f0a = {0};
vec_t still_going;
memset(&still_going, 0xff, sizeof(still_going));
for (unsigned i = 0; i < 2 * (N - 1) - 1; i++) {
const vec_t s_a = vec_broadcast_bit(still_going & (f_a[0] & g_a[0]));
const vec_t s_s =
vec_broadcast_bit(still_going & ((f_s[0] ^ g_s[0]) & s_a));
const vec_t should_swap =
(s_s | s_a) & vec_broadcast_bit15(deg_f - deg_g);
poly3_vec_cswap(f_s, f_a, g_s, g_a, should_swap);
poly3_vec_fmsub(f_s, f_a, g_s, g_a, s_s, s_a);
poly3_vec_rshift1(f_s, f_a);
poly3_vec_cswap(b_s, b_a, c_s, c_a, should_swap);
poly3_vec_fmsub(b_s, b_a, c_s, c_a, s_s, s_a);
poly3_vec_lshift1(c_s, c_a);
const vec_t deg_sum = should_swap & (deg_f ^ deg_g);
deg_f ^= deg_sum;
deg_g ^= deg_sum;
deg_f -= kOne;
still_going &= ~vec_broadcast_bit15(deg_f - kOne);
const vec_t f0_is_nonzero = vec_broadcast_bit(f_s[0] | f_a[0]);
// |f0_is_nonzero| implies |still_going|.
rotation ^= f0_is_nonzero & (k ^ rotation);
k += kOne;
const vec_t f0s_sum = f0_is_nonzero & (f_s[0] ^ f0s);
f0s ^= f0s_sum;
const vec_t f0a_sum = f0_is_nonzero & (f_a[0] ^ f0a);
f0a ^= f0a_sum;
}
crypto_word_t rotation_word = vec_get_word(rotation, 0);
rotation_word -= N & constant_time_lt_w(N, rotation_word);
memcpy(out->s.v, b_s, WORDS_PER_POLY * sizeof(crypto_word_t));
memcpy(out->a.v, b_a, WORDS_PER_POLY * sizeof(crypto_word_t));
assert(poly3_top_bits_are_clear(out));
poly3_rotr_consttime(out, rotation_word);
poly3_mul_const(out, vec_get_word(f0s, 0), vec_get_word(f0a, 0));
poly3_mod_phiN(out);
}
#endif // HRSS_HAVE_VECTOR_UNIT
// HRSS_poly3_invert sets |*out| to |in|^-1, i.e. such that |out|×|in| == 1 mod
// Φ(N).
void HRSS_poly3_invert(struct poly3 *out, const struct poly3 *in) {
// The vector version of this function seems slightly slower on AArch64, but
// is useful on ARMv7 and x86-64.
#if defined(HRSS_HAVE_VECTOR_UNIT) && !defined(OPENSSL_AARCH64)
if (vec_capable()) {
poly3_invert_vec(out, in);
return;
}
#endif
// This algorithm mostly follows algorithm 10 in the paper. Some changes:
// 1) k should start at zero, not one. In the code below k is omitted and
// the loop counter, |i|, is used instead.
// 2) The rotation count is conditionally updated to handle trailing zero
// coefficients.
// The best explanation for why it works is in the "Why it works" section of
// [NTRUTN14].
struct poly3 c, f, g;
OPENSSL_memcpy(&f, in, sizeof(f));
// Set g to all ones.
OPENSSL_memset(&g.s, 0, sizeof(struct poly2));
OPENSSL_memset(&g.a, 0xff, sizeof(struct poly2));
g.a.v[WORDS_PER_POLY - 1] >>= BITS_PER_WORD - BITS_IN_LAST_WORD;
struct poly3 *b = out;
poly3_zero(b);
poly3_zero(&c);
// Set b to one.
b->a.v[0] = 1;
crypto_word_t deg_f = N - 1, deg_g = N - 1, rotation = 0;
crypto_word_t f0s = 0, f0a = 0;
crypto_word_t still_going = CONSTTIME_TRUE_W;
for (unsigned i = 0; i < 2 * (N - 1) - 1; i++) {
const crypto_word_t s_a = lsb_to_all(
still_going & (f.a.v[0] & g.a.v[0]));
const crypto_word_t s_s = lsb_to_all(
still_going & ((f.s.v[0] ^ g.s.v[0]) & s_a));
const crypto_word_t should_swap =
(s_s | s_a) & constant_time_lt_w(deg_f, deg_g);
poly3_cswap(&f, &g, should_swap);
poly3_cswap(b, &c, should_swap);
const crypto_word_t deg_sum = should_swap & (deg_f ^ deg_g);
deg_f ^= deg_sum;
deg_g ^= deg_sum;
assert(deg_g >= 1);
poly3_fmsub(&f, &g, s_s, s_a);
poly3_fmsub(b, &c, s_s, s_a);
poly3_rshift1(&f);
poly3_lshift1(&c);
deg_f--;
const crypto_word_t f0_is_nonzero =
lsb_to_all(f.s.v[0]) | lsb_to_all(f.a.v[0]);
// |f0_is_nonzero| implies |still_going|.
assert(!(f0_is_nonzero && !still_going));
still_going &= ~constant_time_is_zero_w(deg_f);
rotation = constant_time_select_w(f0_is_nonzero, i, rotation);
f0s = constant_time_select_w(f0_is_nonzero, f.s.v[0], f0s);
f0a = constant_time_select_w(f0_is_nonzero, f.a.v[0], f0a);
}
rotation++;
rotation -= N & constant_time_lt_w(N, rotation);
assert(poly3_top_bits_are_clear(out));
poly3_rotr_consttime(out, rotation);
poly3_mul_const(out, f0s, f0a);
poly3_mod_phiN(out);
}
// Polynomials in Q.
// Coefficients are reduced mod Q. (Q is clearly not prime, therefore the
// coefficients do not form a field.)
#define Q 8192
// VECS_PER_POLY is the number of 128-bit vectors needed to represent a
// polynomial.
#define COEFFICIENTS_PER_VEC (sizeof(vec_t) / sizeof(uint16_t))
#define VECS_PER_POLY ((N + COEFFICIENTS_PER_VEC - 1) / COEFFICIENTS_PER_VEC)
// poly represents a polynomial with coefficients mod Q. Note that, while Q is a
// power of two, this does not operate in GF(Q). That would be a binary field
// but this is simply mod Q. Thus the coefficients are not a field.
//
// Coefficients are ordered little-endian, thus the coefficient of x^0 is the
// first element of the array.
struct poly {
#if defined(HRSS_HAVE_VECTOR_UNIT)
union {
// N + 3 = 704, which is a multiple of 64 and thus aligns things, esp for
// the vector code.
uint16_t v[N + 3];
vec_t vectors[VECS_PER_POLY];
};
#else
// Even if !HRSS_HAVE_VECTOR_UNIT, external assembly may be called that
// requires alignment.
alignas(16) uint16_t v[N + 3];
#endif
};
OPENSSL_UNUSED static void poly_print(const struct poly *p) {
printf("[");
for (unsigned i = 0; i < N; i++) {
if (i) {
printf(" ");
}
printf("%d", p->v[i]);
}
printf("]\n");
}
#if defined(HRSS_HAVE_VECTOR_UNIT)
// poly_mul_vec_aux is a recursive function that multiplies |n| words from |a|
// and |b| and writes 2×|n| words to |out|. Each call uses 2*ceil(n/2) elements
// of |scratch| and the function recurses, except if |n| < 3, when |scratch|
// isn't used and the recursion stops. If |n| == |VECS_PER_POLY| then |scratch|
// needs 172 elements.
static void poly_mul_vec_aux(vec_t *restrict out, vec_t *restrict scratch,
const vec_t *restrict a, const vec_t *restrict b,
const size_t n) {
// In [HRSS], the technique they used for polynomial multiplication is
// described: they start with Toom-4 at the top level and then two layers of
// Karatsuba. Karatsuba is a specific instance of the general Toom–Cook
// decomposition, which splits an input n-ways and produces 2n-1
// multiplications of those parts. So, starting with 704 coefficients (rounded
// up from 701 to have more factors of two), Toom-4 gives seven
// multiplications of degree-174 polynomials. Each round of Karatsuba (which
// is Toom-2) increases the number of multiplications by a factor of three
// while halving the size of the values being multiplied. So two rounds gives
// 63 multiplications of degree-44 polynomials. Then they (I think) form
// vectors by gathering all 63 coefficients of each power together, for each
// input, and doing more rounds of Karatsuba on the vectors until they bottom-
// out somewhere with schoolbook multiplication.
//
// I tried something like that for NEON. NEON vectors are 128 bits so hold
// eight coefficients. I wrote a function that did Karatsuba on eight
// multiplications at the same time, using such vectors, and a Go script that
// decomposed from degree-704, with Karatsuba in non-transposed form, until it
// reached multiplications of degree-44. It batched up those 81
// multiplications into lots of eight with a single one left over (which was
// handled directly).
//
// It worked, but it was significantly slower than the dumb algorithm used
// below. Potentially that was because I misunderstood how [HRSS] did it, or
// because Clang is bad at generating good code from NEON intrinsics on ARMv7.
// (Which is true: the code generated by Clang for the below is pretty crap.)
//
// This algorithm is much simpler. It just does Karatsuba decomposition all
// the way down and never transposes. When it gets down to degree-16 or
// degree-24 values, they are multiplied using schoolbook multiplication and
// vector intrinsics. The vector operations form each of the eight phase-
// shifts of one of the inputs, point-wise multiply, and then add into the
// result at the correct place. This means that 33% (degree-16) or 25%
// (degree-24) of the multiplies and adds are wasted, but it does ok.
if (n == 2) {
vec_t result[4];
vec_t vec_a[3];
static const vec_t kZero = {0};
vec_a[0] = a[0];
vec_a[1] = a[1];
vec_a[2] = kZero;
result[0] = vec_mul(vec_a[0], vec_get_word(b[0], 0));
result[1] = vec_mul(vec_a[1], vec_get_word(b[0], 0));
result[1] = vec_fma(result[1], vec_a[0], vec_get_word(b[1], 0));
result[2] = vec_mul(vec_a[1], vec_get_word(b[1], 0));
result[3] = kZero;
vec3_rshift_word(vec_a);
#define BLOCK(x, y) \
do { \
result[x + 0] = \
vec_fma(result[x + 0], vec_a[0], vec_get_word(b[y / 8], y % 8)); \
result[x + 1] = \
vec_fma(result[x + 1], vec_a[1], vec_get_word(b[y / 8], y % 8)); \
result[x + 2] = \
vec_fma(result[x + 2], vec_a[2], vec_get_word(b[y / 8], y % 8)); \
} while (0)
BLOCK(0, 1);
BLOCK(1, 9);
vec3_rshift_word(vec_a);
BLOCK(0, 2);
BLOCK(1, 10);
vec3_rshift_word(vec_a);
BLOCK(0, 3);
BLOCK(1, 11);
vec3_rshift_word(vec_a);
BLOCK(0, 4);
BLOCK(1, 12);
vec3_rshift_word(vec_a);
BLOCK(0, 5);
BLOCK(1, 13);
vec3_rshift_word(vec_a);
BLOCK(0, 6);
BLOCK(1, 14);
vec3_rshift_word(vec_a);
BLOCK(0, 7);
BLOCK(1, 15);
#undef BLOCK
memcpy(out, result, sizeof(result));
return;
}
if (n == 3) {
vec_t result[6];
vec_t vec_a[4];
static const vec_t kZero = {0};
vec_a[0] = a[0];
vec_a[1] = a[1];
vec_a[2] = a[2];
vec_a[3] = kZero;
result[0] = vec_mul(a[0], vec_get_word(b[0], 0));
result[1] = vec_mul(a[1], vec_get_word(b[0], 0));
result[2] = vec_mul(a[2], vec_get_word(b[0], 0));
#define BLOCK_PRE(x, y) \
do { \
result[x + 0] = \
vec_fma(result[x + 0], vec_a[0], vec_get_word(b[y / 8], y % 8)); \
result[x + 1] = \
vec_fma(result[x + 1], vec_a[1], vec_get_word(b[y / 8], y % 8)); \
result[x + 2] = vec_mul(vec_a[2], vec_get_word(b[y / 8], y % 8)); \
} while (0)
BLOCK_PRE(1, 8);
BLOCK_PRE(2, 16);
result[5] = kZero;
vec4_rshift_word(vec_a);
#define BLOCK(x, y) \
do { \
result[x + 0] = \
vec_fma(result[x + 0], vec_a[0], vec_get_word(b[y / 8], y % 8)); \
result[x + 1] = \
vec_fma(result[x + 1], vec_a[1], vec_get_word(b[y / 8], y % 8)); \
result[x + 2] = \
vec_fma(result[x + 2], vec_a[2], vec_get_word(b[y / 8], y % 8)); \
result[x + 3] = \
vec_fma(result[x + 3], vec_a[3], vec_get_word(b[y / 8], y % 8)); \
} while (0)
BLOCK(0, 1);
BLOCK(1, 9);
BLOCK(2, 17);
vec4_rshift_word(vec_a);
BLOCK(0, 2);
BLOCK(1, 10);
BLOCK(2, 18);
vec4_rshift_word(vec_a);
BLOCK(0, 3);
BLOCK(1, 11);
BLOCK(2, 19);
vec4_rshift_word(vec_a);
BLOCK(0, 4);
BLOCK(1, 12);
BLOCK(2, 20);
vec4_rshift_word(vec_a);
BLOCK(0, 5);
BLOCK(1, 13);
BLOCK(2, 21);
vec4_rshift_word(vec_a);
BLOCK(0, 6);
BLOCK(1, 14);
BLOCK(2, 22);
vec4_rshift_word(vec_a);
BLOCK(0, 7);
BLOCK(1, 15);
BLOCK(2, 23);
#undef BLOCK
#undef BLOCK_PRE
memcpy(out, result, sizeof(result));
return;
}
// Karatsuba multiplication.
// https://en.wikipedia.org/wiki/Karatsuba_algorithm
// When |n| is odd, the two "halves" will have different lengths. The first is
// always the smaller.
const size_t low_len = n / 2;
const size_t high_len = n - low_len;
const vec_t *a_high = &a[low_len];
const vec_t *b_high = &b[low_len];
// Store a_1 + a_0 in the first half of |out| and b_1 + b_0 in the second
// half.
for (size_t i = 0; i < low_len; i++) {
out[i] = vec_add(a_high[i], a[i]);
out[high_len + i] = vec_add(b_high[i], b[i]);
}
if (high_len != low_len) {
out[low_len] = a_high[low_len];
out[high_len + low_len] = b_high[low_len];
}
vec_t *const child_scratch = &scratch[2 * high_len];
// Calculate (a_1 + a_0) × (b_1 + b_0) and write to scratch buffer.
poly_mul_vec_aux(scratch, child_scratch, out, &out[high_len], high_len);
// Calculate a_1 × b_1.
poly_mul_vec_aux(&out[low_len * 2], child_scratch, a_high, b_high, high_len);
// Calculate a_0 × b_0.
poly_mul_vec_aux(out, child_scratch, a, b, low_len);
// Subtract those last two products from the first.
for (size_t i = 0; i < low_len * 2; i++) {
scratch[i] = vec_sub(scratch[i], vec_add(out[i], out[low_len * 2 + i]));
}
if (low_len != high_len) {
scratch[low_len * 2] = vec_sub(scratch[low_len * 2], out[low_len * 4]);
scratch[low_len * 2 + 1] =
vec_sub(scratch[low_len * 2 + 1], out[low_len * 4 + 1]);
}
// Add the middle product into the output.
for (size_t i = 0; i < high_len * 2; i++) {
out[low_len + i] = vec_add(out[low_len + i], scratch[i]);
}
}
// poly_mul_vec sets |*out| to |x|×|y| mod (𝑥^n - 1).
static void poly_mul_vec(struct poly *out, const struct poly *x,
const struct poly *y) {
OPENSSL_memset((uint16_t *)&x->v[N], 0, 3 * sizeof(uint16_t));
OPENSSL_memset((uint16_t *)&y->v[N], 0, 3 * sizeof(uint16_t));
OPENSSL_STATIC_ASSERT(sizeof(out->v) == sizeof(vec_t) * VECS_PER_POLY,
"struct poly is the wrong size");
OPENSSL_STATIC_ASSERT(alignof(struct poly) == alignof(vec_t),
"struct poly has incorrect alignment");
vec_t prod[VECS_PER_POLY * 2];
vec_t scratch[172];
poly_mul_vec_aux(prod, scratch, x->vectors, y->vectors, VECS_PER_POLY);
// |prod| needs to be reduced mod (𝑥^n - 1), which just involves adding the
// upper-half to the lower-half. However, N is 701, which isn't a multiple of
// the vector size, so the upper-half vectors all have to be shifted before
// being added to the lower-half.
vec_t *out_vecs = (vec_t *)out->v;
for (size_t i = 0; i < VECS_PER_POLY; i++) {
const vec_t prev = prod[VECS_PER_POLY - 1 + i];
const vec_t this = prod[VECS_PER_POLY + i];
out_vecs[i] = vec_add(prod[i], vec_merge_3_5(prev, this));
}
OPENSSL_memset(&out->v[N], 0, 3 * sizeof(uint16_t));
}
#endif // HRSS_HAVE_VECTOR_UNIT
// poly_mul_novec_aux writes the product of |a| and |b| to |out|, using
// |scratch| as scratch space. It'll use Karatsuba if the inputs are large
// enough to warrant it. Each call uses 2*ceil(n/2) elements of |scratch| and
// the function recurses, except if |n| < 64, when |scratch| isn't used and the
// recursion stops. If |n| == |N| then |scratch| needs 1318 elements.
static void poly_mul_novec_aux(uint16_t *out, uint16_t *scratch,
const uint16_t *a, const uint16_t *b, size_t n) {
static const size_t kSchoolbookLimit = 64;
if (n < kSchoolbookLimit) {
OPENSSL_memset(out, 0, sizeof(uint16_t) * n * 2);
for (size_t i = 0; i < n; i++) {
for (size_t j = 0; j < n; j++) {
out[i + j] += (unsigned) a[i] * b[j];
}
}
return;
}
// Karatsuba multiplication.
// https://en.wikipedia.org/wiki/Karatsuba_algorithm
// When |n| is odd, the two "halves" will have different lengths. The
// first is always the smaller.
const size_t low_len = n / 2;
const size_t high_len = n - low_len;
const uint16_t *const a_high = &a[low_len];
const uint16_t *const b_high = &b[low_len];
for (size_t i = 0; i < low_len; i++) {
out[i] = a_high[i] + a[i];
out[high_len + i] = b_high[i] + b[i];
}
if (high_len != low_len) {
out[low_len] = a_high[low_len];
out[high_len + low_len] = b_high[low_len];
}
uint16_t *const child_scratch = &scratch[2 * high_len];
poly_mul_novec_aux(scratch, child_scratch, out, &out[high_len], high_len);
poly_mul_novec_aux(&out[low_len * 2], child_scratch, a_high, b_high,
high_len);
poly_mul_novec_aux(out, child_scratch, a, b, low_len);
for (size_t i = 0; i < low_len * 2; i++) {
scratch[i] -= out[i] + out[low_len * 2 + i];
}
if (low_len != high_len) {
scratch[low_len * 2] -= out[low_len * 4];
assert(out[low_len * 4 + 1] == 0);
}
for (size_t i = 0; i < high_len * 2; i++) {
out[low_len + i] += scratch[i];
}
}
// poly_mul_novec sets |*out| to |x|×|y| mod (𝑥^n - 1).
static void poly_mul_novec(struct poly *out, const struct poly *x,
const struct poly *y) {
uint16_t prod[2 * N];
uint16_t scratch[1318];
poly_mul_novec_aux(prod, scratch, x->v, y->v, N);
for (size_t i = 0; i < N; i++) {
out->v[i] = prod[i] + prod[i + N];
}
OPENSSL_memset(&out->v[N], 0, 3 * sizeof(uint16_t));
}
static void poly_mul(struct poly *r, const struct poly *a,
const struct poly *b) {
#if defined(POLY_RQ_MUL_ASM)
const int has_avx2 = (OPENSSL_ia32cap_P[2] & (1 << 5)) != 0;
if (has_avx2) {
poly_Rq_mul(r->v, a->v, b->v);
return;
}
#endif
#if defined(HRSS_HAVE_VECTOR_UNIT)
if (vec_capable()) {
poly_mul_vec(r, a, b);
return;
}
#endif
// Fallback, non-vector case.
poly_mul_novec(r, a, b);
}
// poly_mul_x_minus_1 sets |p| to |p|×(𝑥 - 1) mod (𝑥^n - 1).
static void poly_mul_x_minus_1(struct poly *p) {
// Multiplying by (𝑥 - 1) means negating each coefficient and adding in
// the value of the previous one.
const uint16_t orig_final_coefficient = p->v[N - 1];
for (size_t i = N - 1; i > 0; i--) {
p->v[i] = p->v[i - 1] - p->v[i];
}
p->v[0] = orig_final_coefficient - p->v[0];
}
// poly_mod_phiN sets |p| to |p| mod Φ(N).
static void poly_mod_phiN(struct poly *p) {
const uint16_t coeff700 = p->v[N - 1];
for (unsigned i = 0; i < N; i++) {
p->v[i] -= coeff700;
}
}
// poly_clamp reduces each coefficient mod Q.
static void poly_clamp(struct poly *p) {
for (unsigned i = 0; i < N; i++) {
p->v[i] &= Q - 1;
}
}
// Conversion functions
// --------------------
// poly2_from_poly sets |*out| to |in| mod 2.
static void poly2_from_poly(struct poly2 *out, const struct poly *in) {
crypto_word_t *words = out->v;
unsigned shift = 0;
crypto_word_t word = 0;
for (unsigned i = 0; i < N; i++) {
word >>= 1;
word |= (crypto_word_t)(in->v[i] & 1) << (BITS_PER_WORD - 1);
shift++;
if (shift == BITS_PER_WORD) {
*words = word;
words++;
word = 0;
shift = 0;
}
}
word >>= BITS_PER_WORD - shift;
*words = word;
}
// mod3 treats |a| as a signed number and returns |a| mod 3.
static uint16_t mod3(int16_t a) {
const int16_t q = ((int32_t)a * 21845) >> 16;
int16_t ret = a - 3 * q;
// At this point, |ret| is in {0, 1, 2, 3} and that needs to be mapped to {0,
// 1, 2, 0}.
return ret & ((ret & (ret >> 1)) - 1);
}
// poly3_from_poly sets |*out| to |in|.
static void poly3_from_poly(struct poly3 *out, const struct poly *in) {
crypto_word_t *words_s = out->s.v;
crypto_word_t *words_a = out->a.v;
crypto_word_t s = 0;
crypto_word_t a = 0;
unsigned shift = 0;
for (unsigned i = 0; i < N; i++) {
// This duplicates the 13th bit upwards to the top of the uint16,
// essentially treating it as a sign bit and converting into a signed int16.
// The signed value is reduced mod 3, yielding {0, 1, 2}.
const uint16_t v = mod3((int16_t)(in->v[i] << 3) >> 3);
s >>= 1;
const crypto_word_t s_bit = (crypto_word_t)(v & 2) << (BITS_PER_WORD - 2);
s |= s_bit;
a >>= 1;
a |= s_bit | (crypto_word_t)(v & 1) << (BITS_PER_WORD - 1);
shift++;
if (shift == BITS_PER_WORD) {
*words_s = s;
words_s++;
*words_a = a;
words_a++;
s = a = 0;
shift = 0;
}
}
s >>= BITS_PER_WORD - shift;
a >>= BITS_PER_WORD - shift;
*words_s = s;
*words_a = a;
}
// poly3_from_poly_checked sets |*out| to |in|, which has coefficients in {0, 1,
// Q-1}. It returns a mask indicating whether all coefficients were found to be
// in that set.
static crypto_word_t poly3_from_poly_checked(struct poly3 *out,
const struct poly *in) {
crypto_word_t *words_s = out->s.v;
crypto_word_t *words_a = out->a.v;
crypto_word_t s = 0;
crypto_word_t a = 0;
unsigned shift = 0;
crypto_word_t ok = CONSTTIME_TRUE_W;
for (unsigned i = 0; i < N; i++) {
const uint16_t v = in->v[i];
// Maps {0, 1, Q-1} to {0, 1, 2}.
uint16_t mod3 = v & 3;
mod3 ^= mod3 >> 1;
const uint16_t expected = (uint16_t)((~((mod3 >> 1) - 1)) | mod3) % Q;
ok &= constant_time_eq_w(v, expected);
s >>= 1;
const crypto_word_t s_bit = (crypto_word_t)(mod3 & 2)
<< (BITS_PER_WORD - 2);
s |= s_bit;
a >>= 1;
a |= s_bit | (crypto_word_t)(mod3 & 1) << (BITS_PER_WORD - 1);
shift++;
if (shift == BITS_PER_WORD) {
*words_s = s;
words_s++;
*words_a = a;
words_a++;
s = a = 0;
shift = 0;
}
}
s >>= BITS_PER_WORD - shift;
a >>= BITS_PER_WORD - shift;
*words_s = s;
*words_a = a;
return ok;
}
static void poly_from_poly2(struct poly *out, const struct poly2 *in) {
const crypto_word_t *words = in->v;
unsigned shift = 0;
crypto_word_t word = *words;
for (unsigned i = 0; i < N; i++) {
out->v[i] = word & 1;
word >>= 1;
shift++;
if (shift == BITS_PER_WORD) {
words++;
word = *words;
shift = 0;
}
}
}
static void poly_from_poly3(struct poly *out, const struct poly3 *in) {
const crypto_word_t *words_s = in->s.v;
const crypto_word_t *words_a = in->a.v;
crypto_word_t word_s = ~(*words_s);
crypto_word_t word_a = *words_a;
unsigned shift = 0;
for (unsigned i = 0; i < N; i++) {
out->v[i] = (uint16_t)(word_s & 1) - 1;
out->v[i] |= word_a & 1;
word_s >>= 1;
word_a >>= 1;
shift++;
if (shift == BITS_PER_WORD) {
words_s++;
words_a++;
word_s = ~(*words_s);
word_a = *words_a;
shift = 0;
}
}
}
// Polynomial inversion
// --------------------
// poly_invert_mod2 sets |*out| to |in^-1| (i.e. such that |*out|×|in| = 1 mod
// Φ(N)), all mod 2. This isn't useful in itself, but is part of doing inversion
// mod Q.
static void poly_invert_mod2(struct poly *out, const struct poly *in) {
// This algorithm follows algorithm 10 in the paper. (Although, in contrast to
// the paper, k should start at zero, not one, and the rotation count is needs
// to handle trailing zero coefficients.) The best explanation for why it
// works is in the "Why it works" section of [NTRUTN14].
struct poly2 b, c, f, g;
poly2_from_poly(&f, in);
OPENSSL_memset(&b, 0, sizeof(b));
b.v[0] = 1;
OPENSSL_memset(&c, 0, sizeof(c));
// Set g to all ones.
OPENSSL_memset(&g, 0xff, sizeof(struct poly2));
g.v[WORDS_PER_POLY - 1] >>= BITS_PER_WORD - BITS_IN_LAST_WORD;
crypto_word_t deg_f = N - 1, deg_g = N - 1, rotation = 0;
crypto_word_t still_going = CONSTTIME_TRUE_W;
for (unsigned i = 0; i < 2 * (N - 1) - 1; i++) {
const crypto_word_t s = still_going & lsb_to_all(f.v[0]);
const crypto_word_t should_swap = s & constant_time_lt_w(deg_f, deg_g);
poly2_cswap(&f, &g, should_swap);
poly2_cswap(&b, &c, should_swap);
const crypto_word_t deg_sum = should_swap & (deg_f ^ deg_g);
deg_f ^= deg_sum;
deg_g ^= deg_sum;
assert(deg_g >= 1);
poly2_fmadd(&f, &g, s);
poly2_fmadd(&b, &c, s);
poly2_rshift1(&f);
poly2_lshift1(&c);
deg_f--;
const crypto_word_t f0_is_nonzero = lsb_to_all(f.v[0]);
// |f0_is_nonzero| implies |still_going|.
assert(!(f0_is_nonzero && !still_going));
rotation = constant_time_select_w(f0_is_nonzero, i, rotation);
still_going &= ~constant_time_is_zero_w(deg_f);
}
rotation++;
rotation -= N & constant_time_lt_w(N, rotation);
assert(poly2_top_bits_are_clear(&b));
HRSS_poly2_rotr_consttime(&b, rotation);
poly_from_poly2(out, &b);
}
// poly_invert sets |*out| to |in^-1| (i.e. such that |*out|×|in| = 1 mod Φ(N)).
static void poly_invert(struct poly *out, const struct poly *in) {
// Inversion mod Q, which is done based on the result of inverting mod
// 2. See [NTRUTN14] paper, bottom of page two.
struct poly a, *b, tmp;
// a = -in.
for (unsigned i = 0; i < N; i++) {
a.v[i] = -in->v[i];
}
// b = in^-1 mod 2.
b = out;
poly_invert_mod2(b, in);
// We are working mod Q=2**13 and we need to iterate ceil(log_2(13))
// times, which is four.
for (unsigned i = 0; i < 4; i++) {
poly_mul(&tmp, &a, b);
tmp.v[0] += 2;
poly_mul(b, b, &tmp);
}
}
// Marshal and unmarshal functions for various basic types.
// --------------------------------------------------------
#define POLY_BYTES 1138
// poly_marshal serialises all but the final coefficient of |in| to |out|.
static void poly_marshal(uint8_t out[POLY_BYTES], const struct poly *in) {
const uint16_t *p = in->v;
for (size_t i = 0; i < N / 8; i++) {
out[0] = p[0];
out[1] = (0x1f & (p[0] >> 8)) | ((p[1] & 0x07) << 5);
out[2] = p[1] >> 3;
out[3] = (3 & (p[1] >> 11)) | ((p[2] & 0x3f) << 2);
out[4] = (0x7f & (p[2] >> 6)) | ((p[3] & 0x01) << 7);
out[5] = p[3] >> 1;
out[6] = (0xf & (p[3] >> 9)) | ((p[4] & 0x0f) << 4);
out[7] = p[4] >> 4;
out[8] = (1 & (p[4] >> 12)) | ((p[5] & 0x7f) << 1);
out[9] = (0x3f & (p[5] >> 7)) | ((p[6] & 0x03) << 6);
out[10] = p[6] >> 2;
out[11] = (7 & (p[6] >> 10)) | ((p[7] & 0x1f) << 3);
out[12] = p[7] >> 5;
p += 8;
out += 13;
}
// There are four remaining values.
out[0] = p[0];
out[1] = (0x1f & (p[0] >> 8)) | ((p[1] & 0x07) << 5);
out[2] = p[1] >> 3;
out[3] = (3 & (p[1] >> 11)) | ((p[2] & 0x3f) << 2);
out[4] = (0x7f & (p[2] >> 6)) | ((p[3] & 0x01) << 7);
out[5] = p[3] >> 1;
out[6] = 0xf & (p[3] >> 9);
}
// poly_unmarshal parses the output of |poly_marshal| and sets |out| such that
// all but the final coefficients match, and the final coefficient is calculated
// such that evaluating |out| at one results in zero. It returns one on success
// or zero if |in| is an invalid encoding.
static int poly_unmarshal(struct poly *out, const uint8_t in[POLY_BYTES]) {
uint16_t *p = out->v;
for (size_t i = 0; i < N / 8; i++) {
p[0] = (uint16_t)(in[0]) | (uint16_t)(in[1] & 0x1f) << 8;
p[1] = (uint16_t)(in[1] >> 5) | (uint16_t)(in[2]) << 3 |
(uint16_t)(in[3] & 3) << 11;
p[2] = (uint16_t)(in[3] >> 2) | (uint16_t)(in[4] & 0x7f) << 6;
p[3] = (uint16_t)(in[4] >> 7) | (uint16_t)(in[5]) << 1 |
(uint16_t)(in[6] & 0xf) << 9;
p[4] = (uint16_t)(in[6] >> 4) | (uint16_t)(in[7]) << 4 |
(uint16_t)(in[8] & 1) << 12;
p[5] = (uint16_t)(in[8] >> 1) | (uint16_t)(in[9] & 0x3f) << 7;
p[6] = (uint16_t)(in[9] >> 6) | (uint16_t)(in[10]) << 2 |
(uint16_t)(in[11] & 7) << 10;
p[7] = (uint16_t)(in[11] >> 3) | (uint16_t)(in[12]) << 5;
p += 8;
in += 13;
}
// There are four coefficients remaining.
p[0] = (uint16_t)(in[0]) | (uint16_t)(in[1] & 0x1f) << 8;
p[1] = (uint16_t)(in[1] >> 5) | (uint16_t)(in[2]) << 3 |
(uint16_t)(in[3] & 3) << 11;
p[2] = (uint16_t)(in[3] >> 2) | (uint16_t)(in[4] & 0x7f) << 6;
p[3] = (uint16_t)(in[4] >> 7) | (uint16_t)(in[5]) << 1 |
(uint16_t)(in[6] & 0xf) << 9;
for (unsigned i = 0; i < N - 1; i++) {
out->v[i] = (int16_t)(out->v[i] << 3) >> 3;
}
// There are four unused bits in the last byte. We require them to be zero.
if ((in[6] & 0xf0) != 0) {
return 0;
}
// Set the final coefficient as specifed in [HRSSNIST] 1.9.2 step 6.
uint32_t sum = 0;
for (size_t i = 0; i < N - 1; i++) {
sum += out->v[i];
}
out->v[N - 1] = (uint16_t)(0u - sum);
return 1;
}
// mod3_from_modQ maps {0, 1, Q-1, 65535} -> {0, 1, 2, 2}. Note that |v| may
// have an invalid value when processing attacker-controlled inputs.
static uint16_t mod3_from_modQ(uint16_t v) {
v &= 3;
return v ^ (v >> 1);
}
// poly_marshal_mod3 marshals |in| to |out| where the coefficients of |in| are
// all in {0, 1, Q-1, 65535} and |in| is mod Φ(N). (Note that coefficients may
// have invalid values when processing attacker-controlled inputs.)
static void poly_marshal_mod3(uint8_t out[HRSS_POLY3_BYTES],
const struct poly *in) {
const uint16_t *coeffs = in->v;
// Only 700 coefficients are marshaled because in[700] must be zero.
assert(coeffs[N-1] == 0);
for (size_t i = 0; i < HRSS_POLY3_BYTES; i++) {
const uint16_t coeffs0 = mod3_from_modQ(coeffs[0]);
const uint16_t coeffs1 = mod3_from_modQ(coeffs[1]);
const uint16_t coeffs2 = mod3_from_modQ(coeffs[2]);
const uint16_t coeffs3 = mod3_from_modQ(coeffs[3]);
const uint16_t coeffs4 = mod3_from_modQ(coeffs[4]);
out[i] = coeffs0 + coeffs1 * 3 + coeffs2 * 9 + coeffs3 * 27 + coeffs4 * 81;
coeffs += 5;
}
}
// HRSS-specific functions
// -----------------------
// poly_short_sample samples a vector of values in {0xffff (i.e. -1), 0, 1}.
// This is the same action as the algorithm in [HRSSNIST] section 1.8.1, but
// with HRSS-SXY the sampling algorithm is now a private detail of the
// implementation (previously it had to match between two parties). This
// function uses that freedom to implement a flatter distribution of values.
static void poly_short_sample(struct poly *out,
const uint8_t in[HRSS_SAMPLE_BYTES]) {
OPENSSL_STATIC_ASSERT(HRSS_SAMPLE_BYTES == N - 1,
"HRSS_SAMPLE_BYTES incorrect");
for (size_t i = 0; i < N - 1; i++) {
uint16_t v = mod3(in[i]);
// Map {0, 1, 2} -> {0, 1, 0xffff}
v |= ((v >> 1) ^ 1) - 1;
out->v[i] = v;
}
out->v[N - 1] = 0;
}
// poly_short_sample_plus performs the T+ sample as defined in [HRSSNIST],
// section 1.8.2.
static void poly_short_sample_plus(struct poly *out,
const uint8_t in[HRSS_SAMPLE_BYTES]) {
poly_short_sample(out, in);
// sum (and the product in the for loop) will overflow. But that's fine
// because |sum| is bound by +/- (N-2), and N < 2^15 so it works out.
uint16_t sum = 0;
for (unsigned i = 0; i < N - 2; i++) {
sum += (unsigned) out->v[i] * out->v[i + 1];
}
// If the sum is negative, flip the sign of even-positioned coefficients. (See
// page 8 of [HRSS].)
sum = ((int16_t) sum) >> 15;
const uint16_t scale = sum | (~sum & 1);
for (unsigned i = 0; i < N; i += 2) {
out->v[i] = (unsigned) out->v[i] * scale;
}
}
// poly_lift computes the function discussed in [HRSS], appendix B.
static void poly_lift(struct poly *out, const struct poly *a) {
// We wish to calculate a/(𝑥-1) mod Φ(N) over GF(3), where Φ(N) is the
// Nth cyclotomic polynomial, i.e. 1 + 𝑥 + … + 𝑥^700 (since N is prime).
// 1/(𝑥-1) has a fairly basic structure that we can exploit to speed this up:
//
// R.<x> = PolynomialRing(GF(3)…)
// inv = R.cyclotomic_polynomial(1).inverse_mod(R.cyclotomic_polynomial(n))
// list(inv)[:15]
// [1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 2]
//
// This three-element pattern of coefficients repeats for the whole
// polynomial.
//
// Next define the overbar operator such that z̅ = z[0] +
// reverse(z[1:]). (Index zero of a polynomial here is the coefficient
// of the constant term. So index one is the coefficient of 𝑥 and so
// on.)
//
// A less odd way to define this is to see that z̅ negates the indexes,
// so z̅[0] = z[-0], z̅[1] = z[-1] and so on.
//
// The use of z̅ is that, when working mod (𝑥^701 - 1), vz[0] = <v,
// z̅>, vz[1] = <v, 𝑥z̅>, …. (Where <a, b> is the inner product: the sum
// of the point-wise products.) Although we calculated the inverse mod
// Φ(N), we can work mod (𝑥^N - 1) and reduce mod Φ(N) at the end.
// (That's because (𝑥^N - 1) is a multiple of Φ(N).)
//
// When working mod (𝑥^N - 1), multiplication by 𝑥 is a right-rotation
// of the list of coefficients.
//
// Thus we can consider what the pattern of z̅, 𝑥z̅, 𝑥^2z̅, … looks like:
//
// def reverse(xs):
// suffix = list(xs[1:])
// suffix.reverse()
// return [xs[0]] + suffix
//
// def rotate(xs):
// return [xs[-1]] + xs[:-1]
//
// zoverbar = reverse(list(inv) + [0])
// xzoverbar = rotate(reverse(list(inv) + [0]))
// x2zoverbar = rotate(rotate(reverse(list(inv) + [0])))
//
// zoverbar[:15]
// [1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]
// xzoverbar[:15]
// [0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]
// x2zoverbar[:15]
// [2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]
//
// (For a formula for z̅, see lemma two of appendix B.)
//
// After the first three elements have been taken care of, all then have
// a repeating three-element cycle. The next value (𝑥^3z̅) involves
// three rotations of the first pattern, thus the three-element cycle
// lines up. However, the discontinuity in the first three elements
// obviously moves to a different position. Consider the difference
// between 𝑥^3z̅ and z̅:
//
// [x-y for (x,y) in zip(zoverbar, x3zoverbar)][:15]
// [0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
//
// This pattern of differences is the same for all elements, although it
// obviously moves right with the rotations.
//
// From this, we reach algorithm eight of appendix B.
// Handle the first three elements of the inner products.
out->v[0] = a->v[0] + a->v[2];
out->v[1] = a->v[1];
out->v[2] = -a->v[0] + a->v[2];
// s0, s1, s2 are added into out->v[0], out->v[1], and out->v[2],
// respectively. We do not compute s1 because it's just -(s0 + s1).
uint16_t s0 = 0, s2 = 0;
for (size_t i = 3; i < 699; i += 3) {
s0 += -a->v[i] + a->v[i + 2];
// s1 += a->v[i] - a->v[i + 1];
s2 += a->v[i + 1] - a->v[i + 2];
}
// Handle the fact that the three-element pattern doesn't fill the
// polynomial exactly (since 701 isn't a multiple of three).
s0 -= a->v[699];
// s1 += a->v[699] - a->v[700];
s2 += a->v[700];
// Note that s0 + s1 + s2 = 0.
out->v[0] += s0;
out->v[1] -= (s0 + s2); // = s1
out->v[2] += s2;
// Calculate the remaining inner products by taking advantage of the
// fact that the pattern repeats every three cycles and the pattern of
// differences moves with the rotation.
for (size_t i = 3; i < N; i++) {
out->v[i] = (out->v[i - 3] - (a->v[i - 2] + a->v[i - 1] + a->v[i]));
}
// Reduce mod Φ(N) by subtracting a multiple of out[700] from every
// element and convert to mod Q. (See above about adding twice as
// subtraction.)
const crypto_word_t v = out->v[700];
for (unsigned i = 0; i < N; i++) {
const uint16_t vi_mod3 = mod3(out->v[i] - v);
// Map {0, 1, 2} to {0, 1, 0xffff}.
out->v[i] = (~((vi_mod3 >> 1) - 1)) | vi_mod3;
}
poly_mul_x_minus_1(out);
}
struct public_key {
struct poly ph;
};
struct private_key {
struct poly3 f, f_inverse;
struct poly ph_inverse;
uint8_t hmac_key[32];
};
// public_key_from_external converts an external public key pointer into an
// internal one. Externally the alignment is only specified to be eight bytes
// but we need 16-byte alignment. We could annotate the external struct with
// that alignment but we can only assume that malloced pointers are 8-byte
// aligned in any case. (Even if the underlying malloc returns values with
// 16-byte alignment, |OPENSSL_malloc| will store an 8-byte size prefix and mess
// that up.)
static struct public_key *public_key_from_external(
struct HRSS_public_key *ext) {
OPENSSL_STATIC_ASSERT(
sizeof(struct HRSS_public_key) >= sizeof(struct public_key) + 15,
"HRSS public key too small");
uintptr_t p = (uintptr_t)ext;
p = (p + 15) & ~15;
return (struct public_key *)p;
}
// private_key_from_external does the same thing as |public_key_from_external|,
// but for private keys. See the comment on that function about alignment
// issues.
static struct private_key *private_key_from_external(
struct HRSS_private_key *ext) {
OPENSSL_STATIC_ASSERT(
sizeof(struct HRSS_private_key) >= sizeof(struct private_key) + 15,
"HRSS private key too small");
uintptr_t p = (uintptr_t)ext;
p = (p + 15) & ~15;
return (struct private_key *)p;
}
void HRSS_generate_key(
struct HRSS_public_key *out_pub, struct HRSS_private_key *out_priv,
const uint8_t in[HRSS_SAMPLE_BYTES + HRSS_SAMPLE_BYTES + 32]) {
struct public_key *pub = public_key_from_external(out_pub);
struct private_key *priv = private_key_from_external(out_priv);
OPENSSL_memcpy(priv->hmac_key, in + 2 * HRSS_SAMPLE_BYTES,
sizeof(priv->hmac_key));
struct poly f;
poly_short_sample_plus(&f, in);
poly3_from_poly(&priv->f, &f);
HRSS_poly3_invert(&priv->f_inverse, &priv->f);
// pg_phi1 is p (i.e. 3) × g × Φ(1) (i.e. 𝑥-1).
struct poly pg_phi1;
poly_short_sample_plus(&pg_phi1, in + HRSS_SAMPLE_BYTES);
for (unsigned i = 0; i < N; i++) {
pg_phi1.v[i] *= 3;
}
poly_mul_x_minus_1(&pg_phi1);
struct poly pfg_phi1;
poly_mul(&pfg_phi1, &f, &pg_phi1);
struct poly pfg_phi1_inverse;
poly_invert(&pfg_phi1_inverse, &pfg_phi1);
poly_mul(&pub->ph, &pfg_phi1_inverse, &pg_phi1);
poly_mul(&pub->ph, &pub->ph, &pg_phi1);
poly_clamp(&pub->ph);
poly_mul(&priv->ph_inverse, &pfg_phi1_inverse, &f);
poly_mul(&priv->ph_inverse, &priv->ph_inverse, &f);
poly_clamp(&priv->ph_inverse);
}
static const char kSharedKey[] = "shared key";
void HRSS_encap(uint8_t out_ciphertext[POLY_BYTES],
uint8_t out_shared_key[32],
const struct HRSS_public_key *in_pub,
const uint8_t in[HRSS_SAMPLE_BYTES + HRSS_SAMPLE_BYTES]) {
const struct public_key *pub =
public_key_from_external((struct HRSS_public_key *)in_pub);
struct poly m, r, m_lifted;
poly_short_sample(&m, in);
poly_short_sample(&r, in + HRSS_SAMPLE_BYTES);
poly_lift(&m_lifted, &m);
struct poly prh_plus_m;
poly_mul(&prh_plus_m, &r, &pub->ph);
for (unsigned i = 0; i < N; i++) {
prh_plus_m.v[i] += m_lifted.v[i];
}
poly_marshal(out_ciphertext, &prh_plus_m);
uint8_t m_bytes[HRSS_POLY3_BYTES], r_bytes[HRSS_POLY3_BYTES];
poly_marshal_mod3(m_bytes, &m);
poly_marshal_mod3(r_bytes, &r);
SHA256_CTX hash_ctx;
SHA256_Init(&hash_ctx);
SHA256_Update(&hash_ctx, kSharedKey, sizeof(kSharedKey));
SHA256_Update(&hash_ctx, m_bytes, sizeof(m_bytes));
SHA256_Update(&hash_ctx, r_bytes, sizeof(r_bytes));
SHA256_Update(&hash_ctx, out_ciphertext, POLY_BYTES);
SHA256_Final(out_shared_key, &hash_ctx);
}
void HRSS_decap(uint8_t out_shared_key[HRSS_KEY_BYTES],
const struct HRSS_private_key *in_priv,
const uint8_t *ciphertext, size_t ciphertext_len) {
const struct private_key *priv =
private_key_from_external((struct HRSS_private_key *)in_priv);
// This is HMAC, expanded inline rather than using the |HMAC| function so that
// we can avoid dealing with possible allocation failures and so keep this
// function infallible.
uint8_t masked_key[SHA256_CBLOCK];
OPENSSL_STATIC_ASSERT(sizeof(priv->hmac_key) <= sizeof(masked_key),
"HRSS HMAC key larger than SHA-256 block size");
for (size_t i = 0; i < sizeof(priv->hmac_key); i++) {
masked_key[i] = priv->hmac_key[i] ^ 0x36;
}
OPENSSL_memset(masked_key + sizeof(priv->hmac_key), 0x36,
sizeof(masked_key) - sizeof(priv->hmac_key));
SHA256_CTX hash_ctx;
SHA256_Init(&hash_ctx);
SHA256_Update(&hash_ctx, masked_key, sizeof(masked_key));
SHA256_Update(&hash_ctx, ciphertext, ciphertext_len);
uint8_t inner_digest[SHA256_DIGEST_LENGTH];
SHA256_Final(inner_digest, &hash_ctx);
for (size_t i = 0; i < sizeof(priv->hmac_key); i++) {
masked_key[i] ^= (0x5c ^ 0x36);
}
OPENSSL_memset(masked_key + sizeof(priv->hmac_key), 0x5c,
sizeof(masked_key) - sizeof(priv->hmac_key));
SHA256_Init(&hash_ctx);
SHA256_Update(&hash_ctx, masked_key, sizeof(masked_key));
SHA256_Update(&hash_ctx, inner_digest, sizeof(inner_digest));
OPENSSL_STATIC_ASSERT(HRSS_KEY_BYTES == SHA256_DIGEST_LENGTH,
"HRSS shared key length incorrect");
SHA256_Final(out_shared_key, &hash_ctx);
struct poly c;
// If the ciphertext is publicly invalid then a random shared key is still
// returned to simply the logic of the caller, but this path is not constant
// time.
if (ciphertext_len != HRSS_CIPHERTEXT_BYTES ||
!poly_unmarshal(&c, ciphertext)) {
return;
}
struct poly f, cf;
struct poly3 cf3, m3;
poly_from_poly3(&f, &priv->f);
poly_mul(&cf, &c, &f);
poly3_from_poly(&cf3, &cf);
// Note that cf3 is not reduced mod Φ(N). That reduction is deferred.
HRSS_poly3_mul(&m3, &cf3, &priv->f_inverse);
struct poly m, m_lifted;
poly_from_poly3(&m, &m3);
poly_lift(&m_lifted, &m);
struct poly r;
for (unsigned i = 0; i < N; i++) {
r.v[i] = c.v[i] - m_lifted.v[i];
}
poly_mul(&r, &r, &priv->ph_inverse);
poly_mod_phiN(&r);
poly_clamp(&r);
struct poly3 r3;
crypto_word_t ok = poly3_from_poly_checked(&r3, &r);
// [NTRUCOMP] section 5.1 includes ReEnc2 and a proof that it's valid. Rather
// than do an expensive |poly_mul|, it rebuilds |c'| from |c - lift(m)|
// (called |b|) with:
// t = (−b(1)/N) mod Q
// c' = b + tΦ(N) + lift(m) mod Q
//
// When polynomials are transmitted, the final coefficient is omitted and
// |poly_unmarshal| sets it such that f(1) == 0. Thus c(1) == 0. Also,
// |poly_lift| multiplies the result by (x-1) and therefore evaluating a
// lifted polynomial at 1 is also zero. Thus lift(m)(1) == 0 and so
// (c - lift(m))(1) == 0.
//
// Although we defer the reduction above, |b| is conceptually reduced mod
// Φ(N). In order to do that reduction one subtracts |c[N-1]| from every
// coefficient. Therefore b(1) = -c[N-1]×N. The value of |t|, above, then is
// just recovering |c[N-1]|, and adding tΦ(N) is simply undoing the reduction.
// Therefore b + tΦ(N) + lift(m) = c by construction and we don't need to
// recover |c| at all so long as we do the checks in
// |poly3_from_poly_checked|.
//
// The |poly_marshal| here then is just confirming that |poly_unmarshal| is
// strict and could be omitted.
uint8_t expected_ciphertext[HRSS_CIPHERTEXT_BYTES];
OPENSSL_STATIC_ASSERT(HRSS_CIPHERTEXT_BYTES == POLY_BYTES,
"ciphertext is the wrong size");
assert(ciphertext_len == sizeof(expected_ciphertext));
poly_marshal(expected_ciphertext, &c);
uint8_t m_bytes[HRSS_POLY3_BYTES];
uint8_t r_bytes[HRSS_POLY3_BYTES];
poly_marshal_mod3(m_bytes, &m);
poly_marshal_mod3(r_bytes, &r);
ok &= constant_time_is_zero_w(CRYPTO_memcmp(ciphertext, expected_ciphertext,
sizeof(expected_ciphertext)));
uint8_t shared_key[32];
SHA256_Init(&hash_ctx);
SHA256_Update(&hash_ctx, kSharedKey, sizeof(kSharedKey));
SHA256_Update(&hash_ctx, m_bytes, sizeof(m_bytes));
SHA256_Update(&hash_ctx, r_bytes, sizeof(r_bytes));
SHA256_Update(&hash_ctx, expected_ciphertext, sizeof(expected_ciphertext));
SHA256_Final(shared_key, &hash_ctx);
for (unsigned i = 0; i < sizeof(shared_key); i++) {
out_shared_key[i] =
constant_time_select_8(ok, shared_key[i], out_shared_key[i]);
}
}
void HRSS_marshal_public_key(uint8_t out[HRSS_PUBLIC_KEY_BYTES],
const struct HRSS_public_key *in_pub) {
const struct public_key *pub =
public_key_from_external((struct HRSS_public_key *)in_pub);
poly_marshal(out, &pub->ph);
}
int HRSS_parse_public_key(struct HRSS_public_key *out,
const uint8_t in[HRSS_PUBLIC_KEY_BYTES]) {
struct public_key *pub = public_key_from_external(out);
if (!poly_unmarshal(&pub->ph, in)) {
return 0;
}
OPENSSL_memset(&pub->ph.v[N], 0, 3 * sizeof(uint16_t));
return 1;
}