|
- /* 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;
- }
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