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Diffstat (limited to 'src/crypto/ec/p256-64.c')
| -rw-r--r-- | src/crypto/ec/p256-64.c | 1936 |
1 files changed, 1936 insertions, 0 deletions
diff --git a/src/crypto/ec/p256-64.c b/src/crypto/ec/p256-64.c new file mode 100644 index 0000000..8f824de --- /dev/null +++ b/src/crypto/ec/p256-64.c @@ -0,0 +1,1936 @@ +/* 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 <openssl/obj.h> + +#include <string.h> + +#include "internal.h" + + +typedef uint8_t u8; +typedef uint64_t u64; +typedef int64_t s64; +typedef __uint128_t uint128_t; +typedef __int128_t int128_t; + +/* 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 u8 felem_bytearray[32]; + +/* These are the parameters of P256, taken from FIPS 186-3, page 86. These + * values are big-endian. */ +static const felem_bytearray nistp256_curve_params[5] = { + {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* p */ + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff}, + {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01, /* a = -3 */ + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, + 0xfc}, /* b */ + {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7, 0xb3, 0xeb, 0xbd, 0x55, + 0x76, 0x98, 0x86, 0xbc, 0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6, + 0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b}, + {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47, /* x */ + 0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2, 0x77, 0x03, 0x7d, 0x81, + 0x2d, 0xeb, 0x33, 0xa0, 0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96}, + {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b, /* y */ + 0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16, 0x2b, 0xce, 0x33, 0x57, + 0x6b, 0x31, 0x5e, 0xce, 0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}}; + +/* 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 u64 smallfelem[NLIMBS]; + +/* This is the value of the prime as four 64-bit words, little-endian. */ +static const u64 kPrime[4] = {0xfffffffffffffffful, 0xffffffff, 0, + 0xffffffff00000001ul}; +static const u64 bottom63bits = 0x7ffffffffffffffful; + +/* 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 u8 in[32]) { + out[0] = *((u64 *)&in[0]); + out[1] = *((u64 *)&in[8]); + out[2] = *((u64 *)&in[16]); + out[3] = *((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(u8 out[32], const smallfelem in) { + *((u64 *)&out[0]) = in[0]; + *((u64 *)&out[8]) = in[1]; + *((u64 *)&out[16]) = in[2]; + *((u64 *)&out[24]) = in[3]; +} + +/* To preserve endianness when using BN_bn2bin and BN_bin2bn. */ +static void flip_endian(u8 *out, const u8 *in, unsigned len) { + unsigned i; + for (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, BN_to_felem, EC_R_BIGNUM_OUT_OF_RANGE); + return 0; + } + + felem_bytearray b_out; + /* BN_bn2bin eats leading zeroes */ + memset(b_out, 0, sizeof(b_out)); + unsigned num_bytes = BN_num_bytes(bn); + if (num_bytes > sizeof(b_out)) { + OPENSSL_PUT_ERROR(EC, BN_to_felem, 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 smallfelem_one(smallfelem out) { + out[0] = 1; + out[1] = 0; + out[2] = 0; + out[3] = 0; +} + +static void smallfelem_assign(smallfelem out, const smallfelem in) { + out[0] = in[0]; + out[1] = in[1]; + out[2] = in[2]; + out[3] = in[3]; +} + +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 u64 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 u64 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; + u64 a, b, mask; + s64 high, low; + static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */ + + /* Carry 2->3 */ + tmp[3] = zero110[3] + in[3] + ((u64)(in[2] >> 64)); + /* tmp[3] < 2^110 */ + + tmp[2] = zero110[2] + (u64)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] = (u64)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] = (u64)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 <<= 63; + high >>= 63; + /* 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] += ((u64)(tmp[0] >> 64)); + tmp[0] = (u64)tmp[0]; + tmp[2] += ((u64)(tmp[1] >> 64)); + tmp[1] = (u64)tmp[1]; + tmp[3] += ((u64)(tmp[2] >> 64)); + tmp[2] = (u64)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; + u64 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) { + u64 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; + u64 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(u64 *result, u64 *carry, u64 v) { + uint128_t r = *result; + r -= v; + *carry = (r >> 64) & 1; + *result = (u64)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) { + u64 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 u64, 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. */ + unsigned i; + for (i = 3; i < 4; i--) { + u64 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 & ((u64)(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 = ((s64)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. */ + u64 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]); +} + +static void smallfelem_square_contract(smallfelem out, const smallfelem in) { + longfelem longtmp; + felem tmp; + + smallfelem_square(longtmp, in); + felem_reduce(tmp, longtmp); + felem_contract(out, tmp); +} + +static void smallfelem_mul_contract(smallfelem out, const smallfelem in1, + const smallfelem in2) { + longfelem longtmp; + felem tmp; + + smallfelem_mul(longtmp, in1, in2); + felem_reduce(tmp, longtmp); + felem_contract(out, tmp); +} + +/* 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; + u64 is_p; + + u64 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 = ((s64)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 = ((s64)is_p) >> 63; + + is_zero |= is_p; + + result = is_zero; + result |= ((limb)is_zero) << 64; + return result; +} + +static int smallfelem_is_zero_int(const smallfelem small) { + return (int)(smallfelem_is_zero(small) & ((limb)1)); +} + +/* 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; + unsigned i; + + 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 (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 (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 (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 (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 (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 (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 (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 */ +} + +static void smallfelem_inv_contract(smallfelem out, const smallfelem in) { + felem tmp; + + smallfelem_expand(tmp, in); + felem_inv(tmp, tmp); + felem_contract(out, tmp); +} + +/* 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); +} + +/* copy_conditional copies in to out iff mask is all ones. */ +static void copy_conditional(felem out, const felem in, limb mask) { + unsigned i; + for (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) { + unsigned i; + const u64 mask64 = mask; + for (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); + copy_conditional(x_out, x1, z2_is_zero); + copy_small_conditional(y_out, y2, z1_is_zero); + copy_conditional(y_out, y1, z2_is_zero); + copy_small_conditional(z_out, z2, z1_is_zero); + 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. */ + +/* gmul is the table of precomputed base points */ +static const smallfelem gmul[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 u64 idx, unsigned int size, + const smallfelem pre_comp[16][3], smallfelem out[3]) { + unsigned i, j; + u64 *outlimbs = &out[0][0]; + memset(outlimbs, 0, 3 * sizeof(smallfelem)); + + for (i = 0; i < size; i++) { + const u64 *inlimbs = (u64 *)&pre_comp[i][0][0]; + u64 mask = i ^ idx; + mask |= mask >> 4; + mask |= mask >> 2; + mask |= mask >> 1; + mask &= 1; + mask--; + for (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 pre_comp[], the scalars + * in scalars[]. 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 felem_bytearray scalars[], + const unsigned num_points, const u8 *g_scalar, + const int mixed, const smallfelem pre_comp[][17][3], + const smallfelem g_pre_comp[2][16][3]) { + int i, skip; + unsigned num, gen_mul = (g_scalar != NULL); + felem nq[3], ftmp; + smallfelem tmp[3]; + u64 bits; + u8 sign, digit; + + /* set nq to the point at infinity */ + memset(nq, 0, 3 * sizeof(felem)); + + /* Loop over all scalars msb-to-lsb, interleaving additions of multiples + * of the generator (two in each of the last 32 rounds) and additions of + * other points multiples (every 5th round). */ + + skip = 1; /* save two point operations in the first + * round */ + for (i = (num_points ? 255 : 31); i >= 0; --i) { + /* double */ + if (!skip) { + point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); + } + + /* add multiples of the generator */ + if (gen_mul && 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) { + /* Arg 1 below is for "mixed" */ + point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1, 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); + /* Arg 1 below is for "mixed" */ + point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], + tmp[2]); + } + + /* do other additions every 5 doublings */ + if (num_points && (i % 5 == 0)) { + /* loop over all scalars */ + for (num = 0; num < num_points; ++num) { + bits = get_bit(scalars[num], i + 4) << 5; + bits |= get_bit(scalars[num], i + 3) << 4; + bits |= get_bit(scalars[num], i + 2) << 3; + bits |= get_bit(scalars[num], i + 1) << 2; + bits |= get_bit(scalars[num], i) << 1; + bits |= get_bit(scalars[num], 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, pre_comp[num], 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], 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; + } + } + } + } + felem_assign(x_out, nq[0]); + felem_assign(y_out, nq[1]); + felem_assign(z_out, nq[2]); +} + +/* Precomputation for the group generator. */ +typedef struct { + smallfelem g_pre_comp[2][16][3]; + int references; +} NISTP256_PRE_COMP; + +/******************************************************************************/ +/* + * OPENSSL EC_METHOD FUNCTIONS + */ + +int ec_GFp_nistp256_group_init(EC_GROUP *group) { + int ret = ec_GFp_simple_group_init(group); + group->a_is_minus3 = 1; + return ret; +} + +int ec_GFp_nistp256_group_set_curve(EC_GROUP *group, const BIGNUM *p, + const BIGNUM *a, const BIGNUM *b, + BN_CTX *ctx) { + int ret = 0; + BN_CTX *new_ctx = NULL; + BIGNUM *curve_p, *curve_a, *curve_b; + + if (ctx == NULL) { + if ((ctx = new_ctx = BN_CTX_new()) == NULL) { + return 0; + } + } + BN_CTX_start(ctx); + if (((curve_p = BN_CTX_get(ctx)) == NULL) || + ((curve_a = BN_CTX_get(ctx)) == NULL) || + ((curve_b = BN_CTX_get(ctx)) == NULL)) { + goto err; + } + BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p); + BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a); + BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b); + if (BN_cmp(curve_p, p) || + BN_cmp(curve_a, a) || + BN_cmp(curve_b, b)) { + OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_group_set_curve, + EC_R_WRONG_CURVE_PARAMETERS); + goto err; + } + ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx); + +err: + BN_CTX_end(ctx); + BN_CTX_free(new_ctx); + return ret; +} + +/* Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') = + * (X/Z^2, Y/Z^3). */ +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_GFp_nistp256_point_get_affine_coordinates, + 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); + felem_mul(tmp, x_in, z1); + felem_reduce(x_in, tmp); + felem_contract(x_out, x_in); + if (x != NULL && !smallfelem_to_BN(x, x_out)) { + OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates, + ERR_R_BN_LIB); + return 0; + } + 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 (y != NULL && !smallfelem_to_BN(y, y_out)) { + OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_point_get_affine_coordinates, + ERR_R_BN_LIB); + return 0; + } + return 1; +} + +/* points below is of size |num|, and tmp_smallfelems is of size |num+1| */ +static void make_points_affine(size_t num, smallfelem points[][3], + smallfelem tmp_smallfelems[]) { + /* Runs in constant time, unless an input is the point at infinity (which + * normally shouldn't happen). */ + ec_GFp_nistp_points_make_affine_internal( + num, points, sizeof(smallfelem), tmp_smallfelems, + (void (*)(void *))smallfelem_one, + (int (*)(const void *))smallfelem_is_zero_int, + (void (*)(void *, const void *))smallfelem_assign, + (void (*)(void *, const void *))smallfelem_square_contract, + (void (*)(void *, const void *, const void *))smallfelem_mul_contract, + (void (*)(void *, const void *))smallfelem_inv_contract, + /* nothing to contract */ + (void (*)(void *, const void *))smallfelem_assign); +} + +/* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL + * values Result is stored in r (r can equal one of the inputs). */ +int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r, + const BIGNUM *scalar, size_t num, + const EC_POINT *points[], + const BIGNUM *scalars[], BN_CTX *ctx) { + int ret = 0; + int j; + int mixed = 0; + BN_CTX *new_ctx = NULL; + BIGNUM *x, *y, *z, *tmp_scalar; + felem_bytearray g_secret; + felem_bytearray *secrets = NULL; + smallfelem(*pre_comp)[17][3] = NULL; + smallfelem *tmp_smallfelems = NULL; + felem_bytearray tmp; + unsigned i, num_bytes; + int have_pre_comp = 0; + size_t num_points = num; + smallfelem x_in, y_in, z_in; + felem x_out, y_out, z_out; + const smallfelem(*g_pre_comp)[16][3] = NULL; + EC_POINT *generator = NULL; + const EC_POINT *p = NULL; + const BIGNUM *p_scalar = NULL; + + 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 (scalar != NULL) { + /* try to use the standard precomputation */ + g_pre_comp = &gmul[0]; + generator = EC_POINT_new(group); + if (generator == NULL) { + goto err; + } + /* get the generator from precomputation */ + if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) || + !smallfelem_to_BN(y, g_pre_comp[0][1][1]) || + !smallfelem_to_BN(z, g_pre_comp[0][1][2])) { + OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); + goto err; + } + if (!ec_point_set_Jprojective_coordinates_GFp(group, generator, x, y, z, + ctx)) { + goto err; + } + if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) { + /* precomputation matches generator */ + have_pre_comp = 1; + } else { + /* we don't have valid precomputation: treat the generator as a + * random point. */ + num_points++; + } + } + + if (num_points > 0) { + if (num_points >= 3) { + /* unless we precompute multiples for just one or two points, + * converting those into affine form is time well spent */ + mixed = 1; + } + secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray)); + pre_comp = OPENSSL_malloc(num_points * 17 * 3 * sizeof(smallfelem)); + if (mixed) { + tmp_smallfelems = + OPENSSL_malloc((num_points * 17 + 1) * sizeof(smallfelem)); + } + if (secrets == NULL || pre_comp == NULL || + (mixed && tmp_smallfelems == NULL)) { + OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_MALLOC_FAILURE); + goto err; + } + + /* we treat NULL scalars as 0, and NULL points as points at infinity, + * i.e., they contribute nothing to the linear combination. */ + memset(secrets, 0, num_points * sizeof(felem_bytearray)); + memset(pre_comp, 0, num_points * 17 * 3 * sizeof(smallfelem)); + for (i = 0; i < num_points; ++i) { + if (i == num) { + /* we didn't have a valid precomputation, so we pick the generator. */ + p = EC_GROUP_get0_generator(group); + p_scalar = scalar; + } else { + /* the i^th point */ + p = points[i]; + p_scalar = scalars[i]; + } + if (p_scalar != NULL && p != NULL) { + /* reduce scalar to 0 <= scalar < 2^256 */ + if (BN_num_bits(p_scalar) > 256 || BN_is_negative(p_scalar)) { + /* this is an unusual input, and we don't guarantee + * constant-timeness. */ + if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) { + OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); + goto err; + } + num_bytes = BN_bn2bin(tmp_scalar, tmp); + } else { + num_bytes = BN_bn2bin(p_scalar, tmp); + } + flip_endian(secrets[i], tmp, num_bytes); + /* 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(pre_comp[i][1][0], x_out); + felem_shrink(pre_comp[i][1][1], y_out); + felem_shrink(pre_comp[i][1][2], z_out); + for (j = 2; j <= 16; ++j) { + if (j & 1) { + point_add_small(pre_comp[i][j][0], pre_comp[i][j][1], + pre_comp[i][j][2], pre_comp[i][1][0], + pre_comp[i][1][1], pre_comp[i][1][2], + pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], + pre_comp[i][j - 1][2]); + } else { + point_double_small(pre_comp[i][j][0], pre_comp[i][j][1], + pre_comp[i][j][2], pre_comp[i][j / 2][0], + pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]); + } + } + } + } + if (mixed) { + make_points_affine(num_points * 17, pre_comp[0], tmp_smallfelems); + } + } + + /* the scalar for the generator */ + if (scalar != NULL && have_pre_comp) { + memset(g_secret, 0, sizeof(g_secret)); + /* reduce scalar to 0 <= scalar < 2^256 */ + if (BN_num_bits(scalar) > 256 || BN_is_negative(scalar)) { + /* this is an unusual input, and we don't guarantee + * constant-timeness. */ + if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx)) { + OPENSSL_PUT_ERROR(EC, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); + goto err; + } + num_bytes = BN_bn2bin(tmp_scalar, tmp); + } else { + num_bytes = BN_bn2bin(scalar, tmp); + } + flip_endian(g_secret, tmp, num_bytes); + /* do the multiplication with generator precomputation */ + batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, + num_points, g_secret, mixed, (const smallfelem(*)[17][3])pre_comp, + g_pre_comp); + } else { + /* do the multiplication without generator precomputation */ + batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, + num_points, NULL, mixed, (const smallfelem(*)[17][3])pre_comp, + NULL); + } + + /* 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, ec_GFp_nistp256_points_mul, ERR_R_BN_LIB); + goto err; + } + ret = ec_point_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx); + +err: + BN_CTX_end(ctx); + EC_POINT_free(generator); + BN_CTX_free(new_ctx); + OPENSSL_free(secrets); + OPENSSL_free(pre_comp); + OPENSSL_free(tmp_smallfelems); + return ret; +} + +const EC_METHOD *EC_GFp_nistp256_method(void) { + static const EC_METHOD ret = { + EC_FLAGS_DEFAULT_OCT, + ec_GFp_nistp256_group_init, + ec_GFp_simple_group_finish, + ec_GFp_simple_group_clear_finish, + ec_GFp_simple_group_copy, ec_GFp_nistp256_group_set_curve, + ec_GFp_simple_group_get_curve, ec_GFp_simple_group_get_degree, + ec_GFp_simple_group_check_discriminant, ec_GFp_simple_point_init, + ec_GFp_simple_point_finish, ec_GFp_simple_point_clear_finish, + ec_GFp_simple_point_copy, ec_GFp_simple_point_set_to_infinity, + ec_GFp_simple_set_Jprojective_coordinates_GFp, + ec_GFp_simple_get_Jprojective_coordinates_GFp, + ec_GFp_simple_point_set_affine_coordinates, + ec_GFp_nistp256_point_get_affine_coordinates, + 0 /* point_set_compressed_coordinates */, 0 /* point2oct */, + 0 /* oct2point */, ec_GFp_simple_add, ec_GFp_simple_dbl, + ec_GFp_simple_invert, ec_GFp_simple_is_at_infinity, + ec_GFp_simple_is_on_curve, ec_GFp_simple_cmp, ec_GFp_simple_make_affine, + ec_GFp_simple_points_make_affine, ec_GFp_nistp256_points_mul, + 0 /* precompute_mult */, 0 /* have_precompute_mult */, + ec_GFp_simple_field_mul, ec_GFp_simple_field_sqr, 0 /* field_div */, + 0 /* field_encode */, 0 /* field_decode */, 0 /* field_set_to_one */ + }; + + return &ret; +} + +#endif /* 64_BIT && !WINDOWS */ |