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Diffstat (limited to 'src/crypto/fipsmodule/ec/p256-64.c')
-rw-r--r-- | src/crypto/fipsmodule/ec/p256-64.c | 1674 |
1 files changed, 0 insertions, 1674 deletions
diff --git a/src/crypto/fipsmodule/ec/p256-64.c b/src/crypto/fipsmodule/ec/p256-64.c deleted file mode 100644 index d4a8ff68..00000000 --- a/src/crypto/fipsmodule/ec/p256-64.c +++ /dev/null @@ -1,1674 +0,0 @@ -/* Copyright (c) 2015, 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. */ - -// A 64-bit implementation of the NIST P-256 elliptic curve point -// multiplication -// -// OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c. -// Otherwise based on Emilia's P224 work, which was inspired by my curve25519 -// work which got its smarts from Daniel J. Bernstein's work on the same. - -#include <openssl/base.h> - -#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) - -#include <openssl/bn.h> -#include <openssl/ec.h> -#include <openssl/err.h> -#include <openssl/mem.h> - -#include <string.h> - -#include "../delocate.h" -#include "../../internal.h" -#include "internal.h" - - -// The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We -// can serialise an element of this field into 32 bytes. We call this an -// felem_bytearray. -typedef uint8_t felem_bytearray[32]; - -// The representation of field elements. -// ------------------------------------ -// -// We represent field elements with either four 128-bit values, eight 128-bit -// values, or four 64-bit values. The field element represented is: -// v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p) -// or: -// v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p) -// -// 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits -// apart, but are 128-bits wide, the most significant bits of each limb overlap -// with the least significant bits of the next. -// -// A field element with four limbs is an 'felem'. One with eight limbs is a -// 'longfelem' -// -// A field element with four, 64-bit values is called a 'smallfelem'. Small -// values are used as intermediate values before multiplication. - -#define NLIMBS 4 - -typedef uint128_t limb; -typedef limb felem[NLIMBS]; -typedef limb longfelem[NLIMBS * 2]; -typedef uint64_t smallfelem[NLIMBS]; - -// This is the value of the prime as four 64-bit words, little-endian. -static const uint64_t kPrime[4] = {0xfffffffffffffffful, 0xffffffff, 0, - 0xffffffff00000001ul}; -static const uint64_t bottom63bits = 0x7ffffffffffffffful; - -static uint64_t load_u64(const uint8_t in[8]) { - uint64_t ret; - OPENSSL_memcpy(&ret, in, sizeof(ret)); - return ret; -} - -static void store_u64(uint8_t out[8], uint64_t in) { - OPENSSL_memcpy(out, &in, sizeof(in)); -} - -// bin32_to_felem takes a little-endian byte array and converts it into felem -// form. This assumes that the CPU is little-endian. -static void bin32_to_felem(felem out, const uint8_t in[32]) { - out[0] = load_u64(&in[0]); - out[1] = load_u64(&in[8]); - out[2] = load_u64(&in[16]); - out[3] = load_u64(&in[24]); -} - -// smallfelem_to_bin32 takes a smallfelem and serialises into a little endian, -// 32 byte array. This assumes that the CPU is little-endian. -static void smallfelem_to_bin32(uint8_t out[32], const smallfelem in) { - store_u64(&out[0], in[0]); - store_u64(&out[8], in[1]); - store_u64(&out[16], in[2]); - store_u64(&out[24], in[3]); -} - -// To preserve endianness when using BN_bn2bin and BN_bin2bn. -static void flip_endian(uint8_t *out, const uint8_t *in, size_t len) { - for (size_t i = 0; i < len; ++i) { - out[i] = in[len - 1 - i]; - } -} - -// BN_to_felem converts an OpenSSL BIGNUM into an felem. -static int BN_to_felem(felem out, const BIGNUM *bn) { - if (BN_is_negative(bn)) { - OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); - return 0; - } - - felem_bytearray b_out; - // BN_bn2bin eats leading zeroes - OPENSSL_memset(b_out, 0, sizeof(b_out)); - size_t num_bytes = BN_num_bytes(bn); - if (num_bytes > sizeof(b_out)) { - OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); - return 0; - } - - felem_bytearray b_in; - num_bytes = BN_bn2bin(bn, b_in); - flip_endian(b_out, b_in, num_bytes); - bin32_to_felem(out, b_out); - return 1; -} - -// felem_to_BN converts an felem into an OpenSSL BIGNUM. -static BIGNUM *smallfelem_to_BN(BIGNUM *out, const smallfelem in) { - felem_bytearray b_in, b_out; - smallfelem_to_bin32(b_in, in); - flip_endian(b_out, b_in, sizeof(b_out)); - return BN_bin2bn(b_out, sizeof(b_out), out); -} - -// Field operations. - -static void felem_assign(felem out, const felem in) { - out[0] = in[0]; - out[1] = in[1]; - out[2] = in[2]; - out[3] = in[3]; -} - -// felem_sum sets out = out + in. -static void felem_sum(felem out, const felem in) { - out[0] += in[0]; - out[1] += in[1]; - out[2] += in[2]; - out[3] += in[3]; -} - -// felem_small_sum sets out = out + in. -static void felem_small_sum(felem out, const smallfelem in) { - out[0] += in[0]; - out[1] += in[1]; - out[2] += in[2]; - out[3] += in[3]; -} - -// felem_scalar sets out = out * scalar -static void felem_scalar(felem out, const uint64_t scalar) { - out[0] *= scalar; - out[1] *= scalar; - out[2] *= scalar; - out[3] *= scalar; -} - -// longfelem_scalar sets out = out * scalar -static void longfelem_scalar(longfelem out, const uint64_t scalar) { - out[0] *= scalar; - out[1] *= scalar; - out[2] *= scalar; - out[3] *= scalar; - out[4] *= scalar; - out[5] *= scalar; - out[6] *= scalar; - out[7] *= scalar; -} - -#define two105m41m9 ((((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)) -#define two105 (((limb)1) << 105) -#define two105m41p9 ((((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)) - -// zero105 is 0 mod p -static const felem zero105 = {two105m41m9, two105, two105m41p9, two105m41p9}; - -// smallfelem_neg sets |out| to |-small| -// On exit: -// out[i] < out[i] + 2^105 -static void smallfelem_neg(felem out, const smallfelem small) { - // In order to prevent underflow, we subtract from 0 mod p. - out[0] = zero105[0] - small[0]; - out[1] = zero105[1] - small[1]; - out[2] = zero105[2] - small[2]; - out[3] = zero105[3] - small[3]; -} - -// felem_diff subtracts |in| from |out| -// On entry: -// in[i] < 2^104 -// On exit: -// out[i] < out[i] + 2^105. -static void felem_diff(felem out, const felem in) { - // In order to prevent underflow, we add 0 mod p before subtracting. - out[0] += zero105[0]; - out[1] += zero105[1]; - out[2] += zero105[2]; - out[3] += zero105[3]; - - out[0] -= in[0]; - out[1] -= in[1]; - out[2] -= in[2]; - out[3] -= in[3]; -} - -#define two107m43m11 \ - ((((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)) -#define two107 (((limb)1) << 107) -#define two107m43p11 \ - ((((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)) - -// zero107 is 0 mod p -static const felem zero107 = {two107m43m11, two107, two107m43p11, two107m43p11}; - -// An alternative felem_diff for larger inputs |in| -// felem_diff_zero107 subtracts |in| from |out| -// On entry: -// in[i] < 2^106 -// On exit: -// out[i] < out[i] + 2^107. -static void felem_diff_zero107(felem out, const felem in) { - // In order to prevent underflow, we add 0 mod p before subtracting. - out[0] += zero107[0]; - out[1] += zero107[1]; - out[2] += zero107[2]; - out[3] += zero107[3]; - - out[0] -= in[0]; - out[1] -= in[1]; - out[2] -= in[2]; - out[3] -= in[3]; -} - -// longfelem_diff subtracts |in| from |out| -// On entry: -// in[i] < 7*2^67 -// On exit: -// out[i] < out[i] + 2^70 + 2^40. -static void longfelem_diff(longfelem out, const longfelem in) { - static const limb two70m8p6 = - (((limb)1) << 70) - (((limb)1) << 8) + (((limb)1) << 6); - static const limb two70p40 = (((limb)1) << 70) + (((limb)1) << 40); - static const limb two70 = (((limb)1) << 70); - static const limb two70m40m38p6 = (((limb)1) << 70) - (((limb)1) << 40) - - (((limb)1) << 38) + (((limb)1) << 6); - static const limb two70m6 = (((limb)1) << 70) - (((limb)1) << 6); - - // add 0 mod p to avoid underflow - out[0] += two70m8p6; - out[1] += two70p40; - out[2] += two70; - out[3] += two70m40m38p6; - out[4] += two70m6; - out[5] += two70m6; - out[6] += two70m6; - out[7] += two70m6; - - // in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 - out[0] -= in[0]; - out[1] -= in[1]; - out[2] -= in[2]; - out[3] -= in[3]; - out[4] -= in[4]; - out[5] -= in[5]; - out[6] -= in[6]; - out[7] -= in[7]; -} - -#define two64m0 ((((limb)1) << 64) - 1) -#define two110p32m0 ((((limb)1) << 110) + (((limb)1) << 32) - 1) -#define two64m46 ((((limb)1) << 64) - (((limb)1) << 46)) -#define two64m32 ((((limb)1) << 64) - (((limb)1) << 32)) - -// zero110 is 0 mod p. -static const felem zero110 = {two64m0, two110p32m0, two64m46, two64m32}; - -// felem_shrink converts an felem into a smallfelem. The result isn't quite -// minimal as the value may be greater than p. -// -// On entry: -// in[i] < 2^109 -// On exit: -// out[i] < 2^64. -static void felem_shrink(smallfelem out, const felem in) { - felem tmp; - uint64_t a, b, mask; - int64_t high, low; - static const uint64_t kPrime3Test = - 0x7fffffff00000001ul; // 2^63 - 2^32 + 1 - - // Carry 2->3 - tmp[3] = zero110[3] + in[3] + ((uint64_t)(in[2] >> 64)); - // tmp[3] < 2^110 - - tmp[2] = zero110[2] + (uint64_t)in[2]; - tmp[0] = zero110[0] + in[0]; - tmp[1] = zero110[1] + in[1]; - // tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 - - // We perform two partial reductions where we eliminate the high-word of - // tmp[3]. We don't update the other words till the end. - a = tmp[3] >> 64; // a < 2^46 - tmp[3] = (uint64_t)tmp[3]; - tmp[3] -= a; - tmp[3] += ((limb)a) << 32; - // tmp[3] < 2^79 - - b = a; - a = tmp[3] >> 64; // a < 2^15 - b += a; // b < 2^46 + 2^15 < 2^47 - tmp[3] = (uint64_t)tmp[3]; - tmp[3] -= a; - tmp[3] += ((limb)a) << 32; - // tmp[3] < 2^64 + 2^47 - - // This adjusts the other two words to complete the two partial - // reductions. - tmp[0] += b; - tmp[1] -= (((limb)b) << 32); - - // In order to make space in tmp[3] for the carry from 2 -> 3, we - // conditionally subtract kPrime if tmp[3] is large enough. - high = tmp[3] >> 64; - // As tmp[3] < 2^65, high is either 1 or 0 - high = ~(high - 1); - // high is: - // all ones if the high word of tmp[3] is 1 - // all zeros if the high word of tmp[3] if 0 - low = tmp[3]; - mask = low >> 63; - // mask is: - // all ones if the MSB of low is 1 - // all zeros if the MSB of low if 0 - low &= bottom63bits; - low -= kPrime3Test; - // if low was greater than kPrime3Test then the MSB is zero - low = ~low; - low >>= 63; - // low is: - // all ones if low was > kPrime3Test - // all zeros if low was <= kPrime3Test - mask = (mask & low) | high; - tmp[0] -= mask & kPrime[0]; - tmp[1] -= mask & kPrime[1]; - // kPrime[2] is zero, so omitted - tmp[3] -= mask & kPrime[3]; - // tmp[3] < 2**64 - 2**32 + 1 - - tmp[1] += ((uint64_t)(tmp[0] >> 64)); - tmp[0] = (uint64_t)tmp[0]; - tmp[2] += ((uint64_t)(tmp[1] >> 64)); - tmp[1] = (uint64_t)tmp[1]; - tmp[3] += ((uint64_t)(tmp[2] >> 64)); - tmp[2] = (uint64_t)tmp[2]; - // tmp[i] < 2^64 - - out[0] = tmp[0]; - out[1] = tmp[1]; - out[2] = tmp[2]; - out[3] = tmp[3]; -} - -// smallfelem_expand converts a smallfelem to an felem -static void smallfelem_expand(felem out, const smallfelem in) { - out[0] = in[0]; - out[1] = in[1]; - out[2] = in[2]; - out[3] = in[3]; -} - -// smallfelem_square sets |out| = |small|^2 -// On entry: -// small[i] < 2^64 -// On exit: -// out[i] < 7 * 2^64 < 2^67 -static void smallfelem_square(longfelem out, const smallfelem small) { - limb a; - uint64_t high, low; - - a = ((uint128_t)small[0]) * small[0]; - low = a; - high = a >> 64; - out[0] = low; - out[1] = high; - - a = ((uint128_t)small[0]) * small[1]; - low = a; - high = a >> 64; - out[1] += low; - out[1] += low; - out[2] = high; - - a = ((uint128_t)small[0]) * small[2]; - low = a; - high = a >> 64; - out[2] += low; - out[2] *= 2; - out[3] = high; - - a = ((uint128_t)small[0]) * small[3]; - low = a; - high = a >> 64; - out[3] += low; - out[4] = high; - - a = ((uint128_t)small[1]) * small[2]; - low = a; - high = a >> 64; - out[3] += low; - out[3] *= 2; - out[4] += high; - - a = ((uint128_t)small[1]) * small[1]; - low = a; - high = a >> 64; - out[2] += low; - out[3] += high; - - a = ((uint128_t)small[1]) * small[3]; - low = a; - high = a >> 64; - out[4] += low; - out[4] *= 2; - out[5] = high; - - a = ((uint128_t)small[2]) * small[3]; - low = a; - high = a >> 64; - out[5] += low; - out[5] *= 2; - out[6] = high; - out[6] += high; - - a = ((uint128_t)small[2]) * small[2]; - low = a; - high = a >> 64; - out[4] += low; - out[5] += high; - - a = ((uint128_t)small[3]) * small[3]; - low = a; - high = a >> 64; - out[6] += low; - out[7] = high; -} - -//felem_square sets |out| = |in|^2 -// On entry: -// in[i] < 2^109 -// On exit: -// out[i] < 7 * 2^64 < 2^67. -static void felem_square(longfelem out, const felem in) { - uint64_t small[4]; - felem_shrink(small, in); - smallfelem_square(out, small); -} - -// smallfelem_mul sets |out| = |small1| * |small2| -// On entry: -// small1[i] < 2^64 -// small2[i] < 2^64 -// On exit: -// out[i] < 7 * 2^64 < 2^67. -static void smallfelem_mul(longfelem out, const smallfelem small1, - const smallfelem small2) { - limb a; - uint64_t high, low; - - a = ((uint128_t)small1[0]) * small2[0]; - low = a; - high = a >> 64; - out[0] = low; - out[1] = high; - - a = ((uint128_t)small1[0]) * small2[1]; - low = a; - high = a >> 64; - out[1] += low; - out[2] = high; - - a = ((uint128_t)small1[1]) * small2[0]; - low = a; - high = a >> 64; - out[1] += low; - out[2] += high; - - a = ((uint128_t)small1[0]) * small2[2]; - low = a; - high = a >> 64; - out[2] += low; - out[3] = high; - - a = ((uint128_t)small1[1]) * small2[1]; - low = a; - high = a >> 64; - out[2] += low; - out[3] += high; - - a = ((uint128_t)small1[2]) * small2[0]; - low = a; - high = a >> 64; - out[2] += low; - out[3] += high; - - a = ((uint128_t)small1[0]) * small2[3]; - low = a; - high = a >> 64; - out[3] += low; - out[4] = high; - - a = ((uint128_t)small1[1]) * small2[2]; - low = a; - high = a >> 64; - out[3] += low; - out[4] += high; - - a = ((uint128_t)small1[2]) * small2[1]; - low = a; - high = a >> 64; - out[3] += low; - out[4] += high; - - a = ((uint128_t)small1[3]) * small2[0]; - low = a; - high = a >> 64; - out[3] += low; - out[4] += high; - - a = ((uint128_t)small1[1]) * small2[3]; - low = a; - high = a >> 64; - out[4] += low; - out[5] = high; - - a = ((uint128_t)small1[2]) * small2[2]; - low = a; - high = a >> 64; - out[4] += low; - out[5] += high; - - a = ((uint128_t)small1[3]) * small2[1]; - low = a; - high = a >> 64; - out[4] += low; - out[5] += high; - - a = ((uint128_t)small1[2]) * small2[3]; - low = a; - high = a >> 64; - out[5] += low; - out[6] = high; - - a = ((uint128_t)small1[3]) * small2[2]; - low = a; - high = a >> 64; - out[5] += low; - out[6] += high; - - a = ((uint128_t)small1[3]) * small2[3]; - low = a; - high = a >> 64; - out[6] += low; - out[7] = high; -} - -// felem_mul sets |out| = |in1| * |in2| -// On entry: -// in1[i] < 2^109 -// in2[i] < 2^109 -// On exit: -// out[i] < 7 * 2^64 < 2^67 -static void felem_mul(longfelem out, const felem in1, const felem in2) { - smallfelem small1, small2; - felem_shrink(small1, in1); - felem_shrink(small2, in2); - smallfelem_mul(out, small1, small2); -} - -// felem_small_mul sets |out| = |small1| * |in2| -// On entry: -// small1[i] < 2^64 -// in2[i] < 2^109 -// On exit: -// out[i] < 7 * 2^64 < 2^67 -static void felem_small_mul(longfelem out, const smallfelem small1, - const felem in2) { - smallfelem small2; - felem_shrink(small2, in2); - smallfelem_mul(out, small1, small2); -} - -#define two100m36m4 ((((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)) -#define two100 (((limb)1) << 100) -#define two100m36p4 ((((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)) - -// zero100 is 0 mod p -static const felem zero100 = {two100m36m4, two100, two100m36p4, two100m36p4}; - -// Internal function for the different flavours of felem_reduce. -// felem_reduce_ reduces the higher coefficients in[4]-in[7]. -// On entry: -// out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7] -// out[1] >= in[7] + 2^32*in[4] -// out[2] >= in[5] + 2^32*in[5] -// out[3] >= in[4] + 2^32*in[5] + 2^32*in[6] -// On exit: -// out[0] <= out[0] + in[4] + 2^32*in[5] -// out[1] <= out[1] + in[5] + 2^33*in[6] -// out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7] -// out[3] <= out[3] + 2^32*in[4] + 3*in[7] -static void felem_reduce_(felem out, const longfelem in) { - int128_t c; - // combine common terms from below - c = in[4] + (in[5] << 32); - out[0] += c; - out[3] -= c; - - c = in[5] - in[7]; - out[1] += c; - out[2] -= c; - - // the remaining terms - // 256: [(0,1),(96,-1),(192,-1),(224,1)] - out[1] -= (in[4] << 32); - out[3] += (in[4] << 32); - - // 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] - out[2] -= (in[5] << 32); - - // 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] - out[0] -= in[6]; - out[0] -= (in[6] << 32); - out[1] += (in[6] << 33); - out[2] += (in[6] * 2); - out[3] -= (in[6] << 32); - - // 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] - out[0] -= in[7]; - out[0] -= (in[7] << 32); - out[2] += (in[7] << 33); - out[3] += (in[7] * 3); -} - -// felem_reduce converts a longfelem into an felem. -// To be called directly after felem_square or felem_mul. -// On entry: -// in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64 -// in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64 -// On exit: -// out[i] < 2^101 -static void felem_reduce(felem out, const longfelem in) { - out[0] = zero100[0] + in[0]; - out[1] = zero100[1] + in[1]; - out[2] = zero100[2] + in[2]; - out[3] = zero100[3] + in[3]; - - felem_reduce_(out, in); - - // out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0 - // out[1] > 2^100 - 2^64 - 7*2^96 > 0 - // out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0 - // out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0 - // - // out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101 - // out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101 - // out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101 - // out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101 -} - -// felem_reduce_zero105 converts a larger longfelem into an felem. -// On entry: -// in[0] < 2^71 -// On exit: -// out[i] < 2^106 -static void felem_reduce_zero105(felem out, const longfelem in) { - out[0] = zero105[0] + in[0]; - out[1] = zero105[1] + in[1]; - out[2] = zero105[2] + in[2]; - out[3] = zero105[3] + in[3]; - - felem_reduce_(out, in); - - // out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0 - // out[1] > 2^105 - 2^71 - 2^103 > 0 - // out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0 - // out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0 - // - // out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 - // out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 - // out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106 - // out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106 -} - -// subtract_u64 sets *result = *result - v and *carry to one if the -// subtraction underflowed. -static void subtract_u64(uint64_t *result, uint64_t *carry, uint64_t v) { - uint128_t r = *result; - r -= v; - *carry = (r >> 64) & 1; - *result = (uint64_t)r; -} - -// felem_contract converts |in| to its unique, minimal representation. On -// entry: in[i] < 2^109. -static void felem_contract(smallfelem out, const felem in) { - uint64_t all_equal_so_far = 0, result = 0; - - felem_shrink(out, in); - // small is minimal except that the value might be > p - - all_equal_so_far--; - // We are doing a constant time test if out >= kPrime. We need to compare - // each uint64_t, from most-significant to least significant. For each one, if - // all words so far have been equal (m is all ones) then a non-equal - // result is the answer. Otherwise we continue. - for (size_t i = 3; i < 4; i--) { - uint64_t equal; - uint128_t a = ((uint128_t)kPrime[i]) - out[i]; - // if out[i] > kPrime[i] then a will underflow and the high 64-bits - // will all be set. - result |= all_equal_so_far & ((uint64_t)(a >> 64)); - - // if kPrime[i] == out[i] then |equal| will be all zeros and the - // decrement will make it all ones. - equal = kPrime[i] ^ out[i]; - equal--; - equal &= equal << 32; - equal &= equal << 16; - equal &= equal << 8; - equal &= equal << 4; - equal &= equal << 2; - equal &= equal << 1; - equal = ((int64_t)equal) >> 63; - - all_equal_so_far &= equal; - } - - // if all_equal_so_far is still all ones then the two values are equal - // and so out >= kPrime is true. - result |= all_equal_so_far; - - // if out >= kPrime then we subtract kPrime. - uint64_t carry; - subtract_u64(&out[0], &carry, result & kPrime[0]); - subtract_u64(&out[1], &carry, carry); - subtract_u64(&out[2], &carry, carry); - subtract_u64(&out[3], &carry, carry); - - subtract_u64(&out[1], &carry, result & kPrime[1]); - subtract_u64(&out[2], &carry, carry); - subtract_u64(&out[3], &carry, carry); - - subtract_u64(&out[2], &carry, result & kPrime[2]); - subtract_u64(&out[3], &carry, carry); - - subtract_u64(&out[3], &carry, result & kPrime[3]); -} - -// felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0 -// otherwise. -// On entry: -// small[i] < 2^64 -static limb smallfelem_is_zero(const smallfelem small) { - limb result; - uint64_t is_p; - - uint64_t is_zero = small[0] | small[1] | small[2] | small[3]; - is_zero--; - is_zero &= is_zero << 32; - is_zero &= is_zero << 16; - is_zero &= is_zero << 8; - is_zero &= is_zero << 4; - is_zero &= is_zero << 2; - is_zero &= is_zero << 1; - is_zero = ((int64_t)is_zero) >> 63; - - is_p = (small[0] ^ kPrime[0]) | (small[1] ^ kPrime[1]) | - (small[2] ^ kPrime[2]) | (small[3] ^ kPrime[3]); - is_p--; - is_p &= is_p << 32; - is_p &= is_p << 16; - is_p &= is_p << 8; - is_p &= is_p << 4; - is_p &= is_p << 2; - is_p &= is_p << 1; - is_p = ((int64_t)is_p) >> 63; - - is_zero |= is_p; - - result = is_zero; - result |= ((limb)is_zero) << 64; - return result; -} - -// felem_inv calculates |out| = |in|^{-1} -// -// Based on Fermat's Little Theorem: -// a^p = a (mod p) -// a^{p-1} = 1 (mod p) -// a^{p-2} = a^{-1} (mod p) -static void felem_inv(felem out, const felem in) { - felem ftmp, ftmp2; - // each e_I will hold |in|^{2^I - 1} - felem e2, e4, e8, e16, e32, e64; - longfelem tmp; - - felem_square(tmp, in); - felem_reduce(ftmp, tmp); // 2^1 - felem_mul(tmp, in, ftmp); - felem_reduce(ftmp, tmp); // 2^2 - 2^0 - felem_assign(e2, ftmp); - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); // 2^3 - 2^1 - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); // 2^4 - 2^2 - felem_mul(tmp, ftmp, e2); - felem_reduce(ftmp, tmp); // 2^4 - 2^0 - felem_assign(e4, ftmp); - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); // 2^5 - 2^1 - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); // 2^6 - 2^2 - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); // 2^7 - 2^3 - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); // 2^8 - 2^4 - felem_mul(tmp, ftmp, e4); - felem_reduce(ftmp, tmp); // 2^8 - 2^0 - felem_assign(e8, ftmp); - for (size_t i = 0; i < 8; i++) { - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); - } // 2^16 - 2^8 - felem_mul(tmp, ftmp, e8); - felem_reduce(ftmp, tmp); // 2^16 - 2^0 - felem_assign(e16, ftmp); - for (size_t i = 0; i < 16; i++) { - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); - } // 2^32 - 2^16 - felem_mul(tmp, ftmp, e16); - felem_reduce(ftmp, tmp); // 2^32 - 2^0 - felem_assign(e32, ftmp); - for (size_t i = 0; i < 32; i++) { - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); - } // 2^64 - 2^32 - felem_assign(e64, ftmp); - felem_mul(tmp, ftmp, in); - felem_reduce(ftmp, tmp); // 2^64 - 2^32 + 2^0 - for (size_t i = 0; i < 192; i++) { - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); - } // 2^256 - 2^224 + 2^192 - - felem_mul(tmp, e64, e32); - felem_reduce(ftmp2, tmp); // 2^64 - 2^0 - for (size_t i = 0; i < 16; i++) { - felem_square(tmp, ftmp2); - felem_reduce(ftmp2, tmp); - } // 2^80 - 2^16 - felem_mul(tmp, ftmp2, e16); - felem_reduce(ftmp2, tmp); // 2^80 - 2^0 - for (size_t i = 0; i < 8; i++) { - felem_square(tmp, ftmp2); - felem_reduce(ftmp2, tmp); - } // 2^88 - 2^8 - felem_mul(tmp, ftmp2, e8); - felem_reduce(ftmp2, tmp); // 2^88 - 2^0 - for (size_t i = 0; i < 4; i++) { - felem_square(tmp, ftmp2); - felem_reduce(ftmp2, tmp); - } // 2^92 - 2^4 - felem_mul(tmp, ftmp2, e4); - felem_reduce(ftmp2, tmp); // 2^92 - 2^0 - felem_square(tmp, ftmp2); - felem_reduce(ftmp2, tmp); // 2^93 - 2^1 - felem_square(tmp, ftmp2); - felem_reduce(ftmp2, tmp); // 2^94 - 2^2 - felem_mul(tmp, ftmp2, e2); - felem_reduce(ftmp2, tmp); // 2^94 - 2^0 - felem_square(tmp, ftmp2); - felem_reduce(ftmp2, tmp); // 2^95 - 2^1 - felem_square(tmp, ftmp2); - felem_reduce(ftmp2, tmp); // 2^96 - 2^2 - felem_mul(tmp, ftmp2, in); - felem_reduce(ftmp2, tmp); // 2^96 - 3 - - felem_mul(tmp, ftmp2, ftmp); - felem_reduce(out, tmp); // 2^256 - 2^224 + 2^192 + 2^96 - 3 -} - -// Group operations -// ---------------- -// -// Building on top of the field operations we have the operations on the -// elliptic curve group itself. Points on the curve are represented in Jacobian -// coordinates. - -// point_double calculates 2*(x_in, y_in, z_in) -// -// The method is taken from: -// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b -// -// Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. -// while x_out == y_in is not (maybe this works, but it's not tested). -static void point_double(felem x_out, felem y_out, felem z_out, - const felem x_in, const felem y_in, const felem z_in) { - longfelem tmp, tmp2; - felem delta, gamma, beta, alpha, ftmp, ftmp2; - smallfelem small1, small2; - - felem_assign(ftmp, x_in); - // ftmp[i] < 2^106 - felem_assign(ftmp2, x_in); - // ftmp2[i] < 2^106 - - // delta = z^2 - felem_square(tmp, z_in); - felem_reduce(delta, tmp); - // delta[i] < 2^101 - - // gamma = y^2 - felem_square(tmp, y_in); - felem_reduce(gamma, tmp); - // gamma[i] < 2^101 - felem_shrink(small1, gamma); - - // beta = x*gamma - felem_small_mul(tmp, small1, x_in); - felem_reduce(beta, tmp); - // beta[i] < 2^101 - - // alpha = 3*(x-delta)*(x+delta) - felem_diff(ftmp, delta); - // ftmp[i] < 2^105 + 2^106 < 2^107 - felem_sum(ftmp2, delta); - // ftmp2[i] < 2^105 + 2^106 < 2^107 - felem_scalar(ftmp2, 3); - // ftmp2[i] < 3 * 2^107 < 2^109 - felem_mul(tmp, ftmp, ftmp2); - felem_reduce(alpha, tmp); - // alpha[i] < 2^101 - felem_shrink(small2, alpha); - - // x' = alpha^2 - 8*beta - smallfelem_square(tmp, small2); - felem_reduce(x_out, tmp); - felem_assign(ftmp, beta); - felem_scalar(ftmp, 8); - // ftmp[i] < 8 * 2^101 = 2^104 - felem_diff(x_out, ftmp); - // x_out[i] < 2^105 + 2^101 < 2^106 - - // z' = (y + z)^2 - gamma - delta - felem_sum(delta, gamma); - // delta[i] < 2^101 + 2^101 = 2^102 - felem_assign(ftmp, y_in); - felem_sum(ftmp, z_in); - // ftmp[i] < 2^106 + 2^106 = 2^107 - felem_square(tmp, ftmp); - felem_reduce(z_out, tmp); - felem_diff(z_out, delta); - // z_out[i] < 2^105 + 2^101 < 2^106 - - // y' = alpha*(4*beta - x') - 8*gamma^2 - felem_scalar(beta, 4); - // beta[i] < 4 * 2^101 = 2^103 - felem_diff_zero107(beta, x_out); - // beta[i] < 2^107 + 2^103 < 2^108 - felem_small_mul(tmp, small2, beta); - // tmp[i] < 7 * 2^64 < 2^67 - smallfelem_square(tmp2, small1); - // tmp2[i] < 7 * 2^64 - longfelem_scalar(tmp2, 8); - // tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 - longfelem_diff(tmp, tmp2); - // tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 - felem_reduce_zero105(y_out, tmp); - // y_out[i] < 2^106 -} - -// point_double_small is the same as point_double, except that it operates on -// smallfelems. -static void point_double_small(smallfelem x_out, smallfelem y_out, - smallfelem z_out, const smallfelem x_in, - const smallfelem y_in, const smallfelem z_in) { - felem felem_x_out, felem_y_out, felem_z_out; - felem felem_x_in, felem_y_in, felem_z_in; - - smallfelem_expand(felem_x_in, x_in); - smallfelem_expand(felem_y_in, y_in); - smallfelem_expand(felem_z_in, z_in); - point_double(felem_x_out, felem_y_out, felem_z_out, felem_x_in, felem_y_in, - felem_z_in); - felem_shrink(x_out, felem_x_out); - felem_shrink(y_out, felem_y_out); - felem_shrink(z_out, felem_z_out); -} - -// p256_copy_conditional copies in to out iff mask is all ones. -static void p256_copy_conditional(felem out, const felem in, limb mask) { - for (size_t i = 0; i < NLIMBS; ++i) { - const limb tmp = mask & (in[i] ^ out[i]); - out[i] ^= tmp; - } -} - -// copy_small_conditional copies in to out iff mask is all ones. -static void copy_small_conditional(felem out, const smallfelem in, limb mask) { - const uint64_t mask64 = mask; - for (size_t i = 0; i < NLIMBS; ++i) { - out[i] = ((limb)(in[i] & mask64)) | (out[i] & ~mask); - } -} - -// point_add calcuates (x1, y1, z1) + (x2, y2, z2) -// -// The method is taken from: -// http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, -// adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). -// -// This function includes a branch for checking whether the two input points -// are equal, (while not equal to the point at infinity). This case never -// happens during single point multiplication, so there is no timing leak for -// ECDH or ECDSA signing. -static void point_add(felem x3, felem y3, felem z3, const felem x1, - const felem y1, const felem z1, const int mixed, - const smallfelem x2, const smallfelem y2, - const smallfelem z2) { - felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out; - longfelem tmp, tmp2; - smallfelem small1, small2, small3, small4, small5; - limb x_equal, y_equal, z1_is_zero, z2_is_zero; - - felem_shrink(small3, z1); - - z1_is_zero = smallfelem_is_zero(small3); - z2_is_zero = smallfelem_is_zero(z2); - - // ftmp = z1z1 = z1**2 - smallfelem_square(tmp, small3); - felem_reduce(ftmp, tmp); - // ftmp[i] < 2^101 - felem_shrink(small1, ftmp); - - if (!mixed) { - // ftmp2 = z2z2 = z2**2 - smallfelem_square(tmp, z2); - felem_reduce(ftmp2, tmp); - // ftmp2[i] < 2^101 - felem_shrink(small2, ftmp2); - - felem_shrink(small5, x1); - - // u1 = ftmp3 = x1*z2z2 - smallfelem_mul(tmp, small5, small2); - felem_reduce(ftmp3, tmp); - // ftmp3[i] < 2^101 - - // ftmp5 = z1 + z2 - felem_assign(ftmp5, z1); - felem_small_sum(ftmp5, z2); - // ftmp5[i] < 2^107 - - // ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 - felem_square(tmp, ftmp5); - felem_reduce(ftmp5, tmp); - // ftmp2 = z2z2 + z1z1 - felem_sum(ftmp2, ftmp); - // ftmp2[i] < 2^101 + 2^101 = 2^102 - felem_diff(ftmp5, ftmp2); - // ftmp5[i] < 2^105 + 2^101 < 2^106 - - // ftmp2 = z2 * z2z2 - smallfelem_mul(tmp, small2, z2); - felem_reduce(ftmp2, tmp); - - // s1 = ftmp2 = y1 * z2**3 - felem_mul(tmp, y1, ftmp2); - felem_reduce(ftmp6, tmp); - // ftmp6[i] < 2^101 - } else { - // We'll assume z2 = 1 (special case z2 = 0 is handled later). - - // u1 = ftmp3 = x1*z2z2 - felem_assign(ftmp3, x1); - // ftmp3[i] < 2^106 - - // ftmp5 = 2z1z2 - felem_assign(ftmp5, z1); - felem_scalar(ftmp5, 2); - // ftmp5[i] < 2*2^106 = 2^107 - - // s1 = ftmp2 = y1 * z2**3 - felem_assign(ftmp6, y1); - // ftmp6[i] < 2^106 - } - - // u2 = x2*z1z1 - smallfelem_mul(tmp, x2, small1); - felem_reduce(ftmp4, tmp); - - // h = ftmp4 = u2 - u1 - felem_diff_zero107(ftmp4, ftmp3); - // ftmp4[i] < 2^107 + 2^101 < 2^108 - felem_shrink(small4, ftmp4); - - x_equal = smallfelem_is_zero(small4); - - // z_out = ftmp5 * h - felem_small_mul(tmp, small4, ftmp5); - felem_reduce(z_out, tmp); - // z_out[i] < 2^101 - - // ftmp = z1 * z1z1 - smallfelem_mul(tmp, small1, small3); - felem_reduce(ftmp, tmp); - - // s2 = tmp = y2 * z1**3 - felem_small_mul(tmp, y2, ftmp); - felem_reduce(ftmp5, tmp); - - // r = ftmp5 = (s2 - s1)*2 - felem_diff_zero107(ftmp5, ftmp6); - // ftmp5[i] < 2^107 + 2^107 = 2^108 - felem_scalar(ftmp5, 2); - // ftmp5[i] < 2^109 - felem_shrink(small1, ftmp5); - y_equal = smallfelem_is_zero(small1); - - if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { - point_double(x3, y3, z3, x1, y1, z1); - return; - } - - // I = ftmp = (2h)**2 - felem_assign(ftmp, ftmp4); - felem_scalar(ftmp, 2); - // ftmp[i] < 2*2^108 = 2^109 - felem_square(tmp, ftmp); - felem_reduce(ftmp, tmp); - - // J = ftmp2 = h * I - felem_mul(tmp, ftmp4, ftmp); - felem_reduce(ftmp2, tmp); - - // V = ftmp4 = U1 * I - felem_mul(tmp, ftmp3, ftmp); - felem_reduce(ftmp4, tmp); - - // x_out = r**2 - J - 2V - smallfelem_square(tmp, small1); - felem_reduce(x_out, tmp); - felem_assign(ftmp3, ftmp4); - felem_scalar(ftmp4, 2); - felem_sum(ftmp4, ftmp2); - // ftmp4[i] < 2*2^101 + 2^101 < 2^103 - felem_diff(x_out, ftmp4); - // x_out[i] < 2^105 + 2^101 - - // y_out = r(V-x_out) - 2 * s1 * J - felem_diff_zero107(ftmp3, x_out); - // ftmp3[i] < 2^107 + 2^101 < 2^108 - felem_small_mul(tmp, small1, ftmp3); - felem_mul(tmp2, ftmp6, ftmp2); - longfelem_scalar(tmp2, 2); - // tmp2[i] < 2*2^67 = 2^68 - longfelem_diff(tmp, tmp2); - // tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 - felem_reduce_zero105(y_out, tmp); - // y_out[i] < 2^106 - - copy_small_conditional(x_out, x2, z1_is_zero); - p256_copy_conditional(x_out, x1, z2_is_zero); - copy_small_conditional(y_out, y2, z1_is_zero); - p256_copy_conditional(y_out, y1, z2_is_zero); - copy_small_conditional(z_out, z2, z1_is_zero); - p256_copy_conditional(z_out, z1, z2_is_zero); - felem_assign(x3, x_out); - felem_assign(y3, y_out); - felem_assign(z3, z_out); -} - -// point_add_small is the same as point_add, except that it operates on -// smallfelems. -static void point_add_small(smallfelem x3, smallfelem y3, smallfelem z3, - smallfelem x1, smallfelem y1, smallfelem z1, - smallfelem x2, smallfelem y2, smallfelem z2) { - felem felem_x3, felem_y3, felem_z3; - felem felem_x1, felem_y1, felem_z1; - smallfelem_expand(felem_x1, x1); - smallfelem_expand(felem_y1, y1); - smallfelem_expand(felem_z1, z1); - point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0, x2, - y2, z2); - felem_shrink(x3, felem_x3); - felem_shrink(y3, felem_y3); - felem_shrink(z3, felem_z3); -} - -// Base point pre computation -// -------------------------- -// -// Two different sorts of precomputed tables are used in the following code. -// Each contain various points on the curve, where each point is three field -// elements (x, y, z). -// -// For the base point table, z is usually 1 (0 for the point at infinity). -// This table has 2 * 16 elements, starting with the following: -// index | bits | point -// ------+---------+------------------------------ -// 0 | 0 0 0 0 | 0G -// 1 | 0 0 0 1 | 1G -// 2 | 0 0 1 0 | 2^64G -// 3 | 0 0 1 1 | (2^64 + 1)G -// 4 | 0 1 0 0 | 2^128G -// 5 | 0 1 0 1 | (2^128 + 1)G -// 6 | 0 1 1 0 | (2^128 + 2^64)G -// 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G -// 8 | 1 0 0 0 | 2^192G -// 9 | 1 0 0 1 | (2^192 + 1)G -// 10 | 1 0 1 0 | (2^192 + 2^64)G -// 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G -// 12 | 1 1 0 0 | (2^192 + 2^128)G -// 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G -// 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G -// 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G -// followed by a copy of this with each element multiplied by 2^32. -// -// The reason for this is so that we can clock bits into four different -// locations when doing simple scalar multiplies against the base point, -// and then another four locations using the second 16 elements. -// -// Tables for other points have table[i] = iG for i in 0 .. 16. - -// g_pre_comp is the table of precomputed base points -static const smallfelem g_pre_comp[2][16][3] = { - {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, - {{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2, - 0x6b17d1f2e12c4247}, - {0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16, - 0x4fe342e2fe1a7f9b}, - {1, 0, 0, 0}}, - {{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de, - 0x0fa822bc2811aaa5}, - {0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b, - 0xbff44ae8f5dba80d}, - {1, 0, 0, 0}}, - {{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789, - 0x300a4bbc89d6726f}, - {0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f, - 0x72aac7e0d09b4644}, - {1, 0, 0, 0}}, - {{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e, - 0x447d739beedb5e67}, - {0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7, - 0x2d4825ab834131ee}, - {1, 0, 0, 0}}, - {{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60, - 0xef9519328a9c72ff}, - {0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c, - 0x611e9fc37dbb2c9b}, - {1, 0, 0, 0}}, - {{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf, - 0x550663797b51f5d8}, - {0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5, - 0x157164848aecb851}, - {1, 0, 0, 0}}, - {{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391, - 0xeb5d7745b21141ea}, - {0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee, - 0xeafd72ebdbecc17b}, - {1, 0, 0, 0}}, - {{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5, - 0xa6d39677a7849276}, - {0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf, - 0x674f84749b0b8816}, - {1, 0, 0, 0}}, - {{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb, - 0x4e769e7672c9ddad}, - {0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281, - 0x42b99082de830663}, - {1, 0, 0, 0}}, - {{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478, - 0x78878ef61c6ce04d}, - {0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def, - 0xb6cb3f5d7b72c321}, - {1, 0, 0, 0}}, - {{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae, - 0x0c88bc4d716b1287}, - {0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa, - 0xdd5ddea3f3901dc6}, - {1, 0, 0, 0}}, - {{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3, - 0x68f344af6b317466}, - {0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3, - 0x31b9c405f8540a20}, - {1, 0, 0, 0}}, - {{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0, - 0x4052bf4b6f461db9}, - {0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8, - 0xfecf4d5190b0fc61}, - {1, 0, 0, 0}}, - {{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a, - 0x1eddbae2c802e41a}, - {0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0, - 0x43104d86560ebcfc}, - {1, 0, 0, 0}}, - {{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a, - 0xb48e26b484f7a21c}, - {0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668, - 0xfac015404d4d3dab}, - {1, 0, 0, 0}}}, - {{{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}, - {{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da, - 0x7fe36b40af22af89}, - {0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1, - 0xe697d45825b63624}, - {1, 0, 0, 0}}, - {{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902, - 0x4a5b506612a677a6}, - {0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40, - 0xeb13461ceac089f1}, - {1, 0, 0, 0}}, - {{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857, - 0x0781b8291c6a220a}, - {0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434, - 0x690cde8df0151593}, - {1, 0, 0, 0}}, - {{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326, - 0x8a535f566ec73617}, - {0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf, - 0x0455c08468b08bd7}, - {1, 0, 0, 0}}, - {{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279, - 0x06bada7ab77f8276}, - {0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70, - 0x5b476dfd0e6cb18a}, - {1, 0, 0, 0}}, - {{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8, - 0x3e29864e8a2ec908}, - {0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed, - 0x239b90ea3dc31e7e}, - {1, 0, 0, 0}}, - {{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4, - 0x820f4dd949f72ff7}, - {0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3, - 0x140406ec783a05ec}, - {1, 0, 0, 0}}, - {{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe, - 0x68f6b8542783dfee}, - {0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028, - 0xcbe1feba92e40ce6}, - {1, 0, 0, 0}}, - {{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927, - 0xd0b2f94d2f420109}, - {0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a, - 0x971459828b0719e5}, - {1, 0, 0, 0}}, - {{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687, - 0x961610004a866aba}, - {0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c, - 0x7acb9fadcee75e44}, - {1, 0, 0, 0}}, - {{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea, - 0x24eb9acca333bf5b}, - {0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d, - 0x69f891c5acd079cc}, - {1, 0, 0, 0}}, - {{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514, - 0xe51f547c5972a107}, - {0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06, - 0x1c309a2b25bb1387}, - {1, 0, 0, 0}}, - {{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828, - 0x20b87b8aa2c4e503}, - {0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044, - 0xf5c6fa49919776be}, - {1, 0, 0, 0}}, - {{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56, - 0x1ed7d1b9332010b9}, - {0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24, - 0x3a2b03f03217257a}, - {1, 0, 0, 0}}, - {{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b, - 0x15fee545c78dd9f6}, - {0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb, - 0x4ab5b6b2b8753f81}, - {1, 0, 0, 0}}}}; - -// select_point selects the |idx|th point from a precomputation table and -// copies it to out. -static void select_point(const uint64_t idx, size_t size, - const smallfelem pre_comp[/*size*/][3], - smallfelem out[3]) { - uint64_t *outlimbs = &out[0][0]; - OPENSSL_memset(outlimbs, 0, 3 * sizeof(smallfelem)); - - for (size_t i = 0; i < size; i++) { - const uint64_t *inlimbs = (const uint64_t *)&pre_comp[i][0][0]; - uint64_t mask = i ^ idx; - mask |= mask >> 4; - mask |= mask >> 2; - mask |= mask >> 1; - mask &= 1; - mask--; - for (size_t j = 0; j < NLIMBS * 3; j++) { - outlimbs[j] |= inlimbs[j] & mask; - } - } -} - -// get_bit returns the |i|th bit in |in| -static char get_bit(const felem_bytearray in, int i) { - if (i < 0 || i >= 256) { - return 0; - } - return (in[i >> 3] >> (i & 7)) & 1; -} - -// Interleaved point multiplication using precomputed point multiples: The -// small point multiples 0*P, 1*P, ..., 17*P are in p_pre_comp, the scalar -// in p_scalar, if non-NULL. If g_scalar is non-NULL, we also add this multiple -// of the generator, using certain (large) precomputed multiples in g_pre_comp. -// Output point (X, Y, Z) is stored in x_out, y_out, z_out. -static void batch_mul(felem x_out, felem y_out, felem z_out, - const uint8_t *p_scalar, const uint8_t *g_scalar, - const smallfelem p_pre_comp[17][3]) { - felem nq[3], ftmp; - smallfelem tmp[3]; - uint64_t bits; - uint8_t sign, digit; - - // set nq to the point at infinity - OPENSSL_memset(nq, 0, 3 * sizeof(felem)); - - // Loop over both scalars msb-to-lsb, interleaving additions of multiples - // of the generator (two in each of the last 32 rounds) and additions of p - // (every 5th round). - - int skip = 1; // save two point operations in the first round - size_t i = p_scalar != NULL ? 255 : 31; - for (;;) { - // double - if (!skip) { - point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); - } - - // add multiples of the generator - if (g_scalar != NULL && i <= 31) { - // first, look 32 bits upwards - bits = get_bit(g_scalar, i + 224) << 3; - bits |= get_bit(g_scalar, i + 160) << 2; - bits |= get_bit(g_scalar, i + 96) << 1; - bits |= get_bit(g_scalar, i + 32); - // select the point to add, in constant time - select_point(bits, 16, g_pre_comp[1], tmp); - - if (!skip) { - point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, - tmp[0], tmp[1], tmp[2]); - } else { - smallfelem_expand(nq[0], tmp[0]); - smallfelem_expand(nq[1], tmp[1]); - smallfelem_expand(nq[2], tmp[2]); - skip = 0; - } - - // second, look at the current position - bits = get_bit(g_scalar, i + 192) << 3; - bits |= get_bit(g_scalar, i + 128) << 2; - bits |= get_bit(g_scalar, i + 64) << 1; - bits |= get_bit(g_scalar, i); - // select the point to add, in constant time - select_point(bits, 16, g_pre_comp[0], tmp); - point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */, tmp[0], - tmp[1], tmp[2]); - } - - // do other additions every 5 doublings - if (p_scalar != NULL && i % 5 == 0) { - bits = get_bit(p_scalar, i + 4) << 5; - bits |= get_bit(p_scalar, i + 3) << 4; - bits |= get_bit(p_scalar, i + 2) << 3; - bits |= get_bit(p_scalar, i + 1) << 2; - bits |= get_bit(p_scalar, i) << 1; - bits |= get_bit(p_scalar, i - 1); - ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); - - // select the point to add or subtract, in constant time. - select_point(digit, 17, p_pre_comp, tmp); - smallfelem_neg(ftmp, tmp[1]); // (X, -Y, Z) is the negative - // point - copy_small_conditional(ftmp, tmp[1], (((limb)sign) - 1)); - felem_contract(tmp[1], ftmp); - - if (!skip) { - point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 0 /* mixed */, - tmp[0], tmp[1], tmp[2]); - } else { - smallfelem_expand(nq[0], tmp[0]); - smallfelem_expand(nq[1], tmp[1]); - smallfelem_expand(nq[2], tmp[2]); - skip = 0; - } - } - - if (i == 0) { - break; - } - --i; - } - felem_assign(x_out, nq[0]); - felem_assign(y_out, nq[1]); - felem_assign(z_out, nq[2]); -} - -// OPENSSL EC_METHOD FUNCTIONS - -// Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = -// (X/Z^2, Y/Z^3). -static int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP *group, - const EC_POINT *point, - BIGNUM *x, BIGNUM *y, - BN_CTX *ctx) { - felem z1, z2, x_in, y_in; - smallfelem x_out, y_out; - longfelem tmp; - - if (EC_POINT_is_at_infinity(group, point)) { - OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); - return 0; - } - if (!BN_to_felem(x_in, &point->X) || - !BN_to_felem(y_in, &point->Y) || - !BN_to_felem(z1, &point->Z)) { - return 0; - } - felem_inv(z2, z1); - felem_square(tmp, z2); - felem_reduce(z1, tmp); - - if (x != NULL) { - felem_mul(tmp, x_in, z1); - felem_reduce(x_in, tmp); - felem_contract(x_out, x_in); - if (!smallfelem_to_BN(x, x_out)) { - OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); - return 0; - } - } - - if (y != NULL) { - felem_mul(tmp, z1, z2); - felem_reduce(z1, tmp); - felem_mul(tmp, y_in, z1); - felem_reduce(y_in, tmp); - felem_contract(y_out, y_in); - if (!smallfelem_to_BN(y, y_out)) { - OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); - return 0; - } - } - - return 1; -} - -static int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r, - const EC_SCALAR *g_scalar, - const EC_POINT *p, - const EC_SCALAR *p_scalar, BN_CTX *ctx) { - int ret = 0; - BN_CTX *new_ctx = NULL; - BIGNUM *x, *y, *z, *tmp_scalar; - smallfelem p_pre_comp[17][3]; - smallfelem x_in, y_in, z_in; - felem x_out, y_out, z_out; - - if (ctx == NULL) { - ctx = new_ctx = BN_CTX_new(); - if (ctx == NULL) { - return 0; - } - } - - BN_CTX_start(ctx); - if ((x = BN_CTX_get(ctx)) == NULL || - (y = BN_CTX_get(ctx)) == NULL || - (z = BN_CTX_get(ctx)) == NULL || - (tmp_scalar = BN_CTX_get(ctx)) == NULL) { - goto err; - } - - if (p != NULL && p_scalar != NULL) { - // We treat NULL scalars as 0, and NULL points as points at infinity, i.e., - // they contribute nothing to the linear combination. - OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp)); - // Precompute multiples. - if (!BN_to_felem(x_out, &p->X) || - !BN_to_felem(y_out, &p->Y) || - !BN_to_felem(z_out, &p->Z)) { - goto err; - } - felem_shrink(p_pre_comp[1][0], x_out); - felem_shrink(p_pre_comp[1][1], y_out); - felem_shrink(p_pre_comp[1][2], z_out); - for (size_t j = 2; j <= 16; ++j) { - if (j & 1) { - point_add_small(p_pre_comp[j][0], p_pre_comp[j][1], - p_pre_comp[j][2], p_pre_comp[1][0], - p_pre_comp[1][1], p_pre_comp[1][2], - p_pre_comp[j - 1][0], p_pre_comp[j - 1][1], - p_pre_comp[j - 1][2]); - } else { - point_double_small(p_pre_comp[j][0], p_pre_comp[j][1], - p_pre_comp[j][2], p_pre_comp[j / 2][0], - p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]); - } - } - } - - batch_mul(x_out, y_out, z_out, - (p != NULL && p_scalar != NULL) ? p_scalar->bytes : NULL, - g_scalar != NULL ? g_scalar->bytes : NULL, - (const smallfelem(*)[3]) & p_pre_comp); - - // reduce the output to its unique minimal representation - felem_contract(x_in, x_out); - felem_contract(y_in, y_out); - felem_contract(z_in, z_out); - if (!smallfelem_to_BN(x, x_in) || - !smallfelem_to_BN(y, y_in) || - !smallfelem_to_BN(z, z_in)) { - OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); - goto err; - } - ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); - -err: - BN_CTX_end(ctx); - BN_CTX_free(new_ctx); - return ret; -} - -DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp256_method) { - out->group_init = ec_GFp_simple_group_init; - out->group_finish = ec_GFp_simple_group_finish; - out->group_set_curve = ec_GFp_simple_group_set_curve; - out->point_get_affine_coordinates = - ec_GFp_nistp256_point_get_affine_coordinates; - out->mul = ec_GFp_nistp256_points_mul; - out->field_mul = ec_GFp_simple_field_mul; - out->field_sqr = ec_GFp_simple_field_sqr; - out->field_encode = NULL; - out->field_decode = NULL; -}; - -#endif // 64_BIT && !WINDOWS |