diff --git a/crypto/fipsmodule/bcm.c b/crypto/fipsmodule/bcm.c index b506b43e..fb16215f 100644 --- a/crypto/fipsmodule/bcm.c +++ b/crypto/fipsmodule/bcm.c @@ -67,10 +67,10 @@ #include "ec/ec_montgomery.c" #include "ec/oct.c" #include "ec/p224-64.c" -#include "ec/p256-64.c" +#include "../../third_party/fiat/p256.c" #include "ec/p256-x86_64.c" #include "ec/simple.c" -#include "ec/util-64.c" +#include "ec/util.c" #include "ec/wnaf.c" #include "hmac/hmac.c" #include "md4/md4.c" diff --git a/crypto/fipsmodule/ec/ec.c b/crypto/fipsmodule/ec/ec.c index 41c2540b..ed54554b 100644 --- a/crypto/fipsmodule/ec/ec.c +++ b/crypto/fipsmodule/ec/ec.c @@ -215,13 +215,6 @@ static const uint8_t kP521Params[6 * 66] = { 0xB7, 0x1E, 0x91, 0x38, 0x64, 0x09, }; -// MSan appears to have a bug that causes code to be miscompiled in opt mode. -// While that is being looked at, don't run the uint128_t code under MSan. -#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \ - !defined(MEMORY_SANITIZER) -#define BORINGSSL_USE_INT128_CODE -#endif - DEFINE_METHOD_FUNCTION(struct built_in_curves, OPENSSL_built_in_curves) { // 1.3.132.0.35 static const uint8_t kOIDP521[] = {0x2b, 0x81, 0x04, 0x00, 0x23}; @@ -253,15 +246,18 @@ DEFINE_METHOD_FUNCTION(struct built_in_curves, OPENSSL_built_in_curves) { out->curves[2].param_len = 32; out->curves[2].params = kP256Params; out->curves[2].method = -#if defined(BORINGSSL_USE_INT128_CODE) +// MSan appears to have a bug that causes code to be miscompiled in opt mode. +// While that is being looked at, don't run the uint128_t code under MSan. #if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64) && \ - !defined(OPENSSL_SMALL) + !defined(OPENSSL_SMALL) && !defined(MEMORY_SANITIZER) EC_GFp_nistz256_method(); #else +#if defined(OPENSSL_32_BIT) || \ + (defined(OPENSSL_64_BIT) && !defined(MEMORY_SANITIZER)) EC_GFp_nistp256_method(); -#endif #else EC_GFp_mont_method(); +#endif #endif // 1.3.132.0.33 @@ -273,7 +269,8 @@ DEFINE_METHOD_FUNCTION(struct built_in_curves, OPENSSL_built_in_curves) { out->curves[3].param_len = 28; out->curves[3].params = kP224Params; out->curves[3].method = -#if defined(BORINGSSL_USE_INT128_CODE) && !defined(OPENSSL_SMALL) +#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) && \ + !defined(MEMORY_SANITIZER) && !defined(OPENSSL_SMALL) EC_GFp_nistp224_method(); #else EC_GFp_mont_method(); @@ -883,6 +880,24 @@ err: return ret; } +int ec_point_mul_scalar_public(const EC_GROUP *group, EC_POINT *r, + const EC_SCALAR *g_scalar, const EC_POINT *p, + const EC_SCALAR *p_scalar, BN_CTX *ctx) { + if ((g_scalar == NULL && p_scalar == NULL) || + (p == NULL) != (p_scalar == NULL)) { + OPENSSL_PUT_ERROR(EC, ERR_R_PASSED_NULL_PARAMETER); + return 0; + } + + if (EC_GROUP_cmp(group, r->group, NULL) != 0 || + (p != NULL && EC_GROUP_cmp(group, p->group, NULL) != 0)) { + OPENSSL_PUT_ERROR(EC, EC_R_INCOMPATIBLE_OBJECTS); + return 0; + } + + return group->meth->mul_public(group, r, g_scalar, p, p_scalar, ctx); +} + int ec_point_mul_scalar(const EC_GROUP *group, EC_POINT *r, const EC_SCALAR *g_scalar, const EC_POINT *p, const EC_SCALAR *p_scalar, BN_CTX *ctx) { diff --git a/crypto/fipsmodule/ec/ec_montgomery.c b/crypto/fipsmodule/ec/ec_montgomery.c index 6670b84e..898cf07a 100644 --- a/crypto/fipsmodule/ec/ec_montgomery.c +++ b/crypto/fipsmodule/ec/ec_montgomery.c @@ -270,6 +270,7 @@ DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_mont_method) { out->group_set_curve = ec_GFp_mont_group_set_curve; out->point_get_affine_coordinates = ec_GFp_mont_point_get_affine_coordinates; out->mul = ec_wNAF_mul /* XXX: Not constant time. */; + out->mul_public = ec_wNAF_mul; out->field_mul = ec_GFp_mont_field_mul; out->field_sqr = ec_GFp_mont_field_sqr; out->field_encode = ec_GFp_mont_field_encode; diff --git a/crypto/fipsmodule/ec/internal.h b/crypto/fipsmodule/ec/internal.h index 1b860c6b..145c5c40 100644 --- a/crypto/fipsmodule/ec/internal.h +++ b/crypto/fipsmodule/ec/internal.h @@ -115,6 +115,12 @@ struct ec_method_st { // non-null. int (*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); + // mul_public performs the same computation as mul. It further assumes that + // the inputs are public so there is no concern about leaking their values + // through timing. + int (*mul_public)(const EC_GROUP *group, EC_POINT *r, + const EC_SCALAR *g_scalar, const EC_POINT *p, + const EC_SCALAR *p_scalar, BN_CTX *ctx); // 'field_mul' and 'field_sqr' can be used by 'add' and 'dbl' so that the // same implementations of point operations can be used with different @@ -195,6 +201,13 @@ int ec_point_mul_scalar(const EC_GROUP *group, EC_POINT *r, const EC_SCALAR *g_scalar, const EC_POINT *p, const EC_SCALAR *p_scalar, BN_CTX *ctx); +// ec_point_mul_scalar_public performs the same computation as +// ec_point_mul_scalar. It further assumes that the inputs are public so +// there is no concern about leaking their values through timing. +int ec_point_mul_scalar_public(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 ec_wNAF_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); diff --git a/crypto/fipsmodule/ec/p224-64.c b/crypto/fipsmodule/ec/p224-64.c index ba25d22a..d0285d60 100644 --- a/crypto/fipsmodule/ec/p224-64.c +++ b/crypto/fipsmodule/ec/p224-64.c @@ -1122,6 +1122,7 @@ DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp224_method) { out->point_get_affine_coordinates = ec_GFp_nistp224_point_get_affine_coordinates; out->mul = ec_GFp_nistp224_points_mul; + out->mul_public = ec_GFp_nistp224_points_mul; out->field_mul = ec_GFp_simple_field_mul; out->field_sqr = ec_GFp_simple_field_sqr; out->field_encode = NULL; diff --git a/crypto/fipsmodule/ec/p256-64.c b/crypto/fipsmodule/ec/p256-64.c deleted file mode 100644 index d4a8ff68..00000000 --- a/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 - -#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) - -#include -#include -#include -#include - -#include - -#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 diff --git a/crypto/fipsmodule/ec/p256-x86_64.c b/crypto/fipsmodule/ec/p256-x86_64.c index a9b603ae..0e79b6dc 100644 --- a/crypto/fipsmodule/ec/p256-x86_64.c +++ b/crypto/fipsmodule/ec/p256-x86_64.c @@ -446,6 +446,7 @@ DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistz256_method) { out->group_set_curve = ec_GFp_mont_group_set_curve; out->point_get_affine_coordinates = ecp_nistz256_get_affine; out->mul = ecp_nistz256_points_mul; + out->mul_public = ecp_nistz256_points_mul; out->field_mul = ec_GFp_mont_field_mul; out->field_sqr = ec_GFp_mont_field_sqr; out->field_encode = ec_GFp_mont_field_encode; diff --git a/crypto/fipsmodule/ec/util-64.c b/crypto/fipsmodule/ec/util.c similarity index 98% rename from crypto/fipsmodule/ec/util-64.c rename to crypto/fipsmodule/ec/util.c index 0cb117b4..7303a151 100644 --- a/crypto/fipsmodule/ec/util-64.c +++ b/crypto/fipsmodule/ec/util.c @@ -14,9 +14,6 @@ #include - -#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) - #include #include "internal.h" @@ -105,5 +102,3 @@ void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, *sign = s & 1; *digit = d; } - -#endif // 64_BIT && !WINDOWS diff --git a/crypto/fipsmodule/ecdsa/ecdsa.c b/crypto/fipsmodule/ecdsa/ecdsa.c index 6571c941..9e038de1 100644 --- a/crypto/fipsmodule/ecdsa/ecdsa.c +++ b/crypto/fipsmodule/ecdsa/ecdsa.c @@ -275,7 +275,7 @@ int ECDSA_do_verify(const uint8_t *digest, size_t digest_len, OPENSSL_PUT_ERROR(ECDSA, ERR_R_MALLOC_FAILURE); goto err; } - if (!ec_point_mul_scalar(group, point, &u1, pub_key, &u2, ctx)) { + if (!ec_point_mul_scalar_public(group, point, &u1, pub_key, &u2, ctx)) { OPENSSL_PUT_ERROR(ECDSA, ERR_R_EC_LIB); goto err; } diff --git a/third_party/fiat/p256.c b/third_party/fiat/p256.c new file mode 100644 index 00000000..19a82847 --- /dev/null +++ b/third_party/fiat/p256.c @@ -0,0 +1,1725 @@ +// The MIT License (MIT) +// +// Copyright (c) 2015-2016 the fiat-crypto authors (see the AUTHORS file). +// +// Permission is hereby granted, free of charge, to any person obtaining a copy +// of this software and associated documentation files (the "Software"), to deal +// in the Software without restriction, including without limitation the rights +// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell +// copies of the Software, and to permit persons to whom the Software is +// furnished to do so, subject to the following conditions: +// +// The above copyright notice and this permission notice shall be included in all +// copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR +// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, +// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE +// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER +// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, +// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE +// SOFTWARE. + +// The field arithmetic code is generated by Fiat +// (https://github.com/mit-plv/fiat-crypto), which is MIT licensed. +// +// An implementation of the NIST P-256 elliptic curve point multiplication. +// 256-bit Montgomery form, generated using fiat-crypto, for 64 and 32-bit. +// Field operations with inputs in [0,p) return outputs in [0,p). + +#include + +// MSVC does not implement uint128_t, and crashes with intrinsics +#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) +#define BORINGSSL_NISTP256_64BIT 1 +#endif + +#include +#include +#include +#include + +#include + +#include "../../crypto/fipsmodule/delocate.h" +#include "../../crypto/internal.h" +#include "../../crypto/fipsmodule/ec/internal.h" + + +// "intrinsics" + +#if defined(BORINGSSL_NISTP256_64BIT) + +static uint64_t mulx_u64(uint64_t a, uint64_t b, uint64_t *high) { + uint128_t x = (uint128_t)a * b; + *high = (uint64_t) (x >> 64); + return (uint64_t) x; +} + +static uint64_t addcarryx_u64(uint8_t c, uint64_t a, uint64_t b, uint64_t *low) { + uint128_t x = (uint128_t)a + b + c; + *low = (uint64_t) x; + return (uint64_t) (x>>64); +} + +static uint64_t subborrow_u64(uint8_t c, uint64_t a, uint64_t b, uint64_t *low) { + uint128_t t = ((uint128_t) b + c); + uint128_t x = a-t; + *low = (uint64_t) x; + return (uint8_t) (x>>127); +} + +static uint64_t cmovznz_u64(uint64_t t, uint64_t z, uint64_t nz) { + t = -!!t; // all set if nonzero, 0 if 0 + return (t&nz) | ((~t)&z); +} + +#else + +static uint32_t mulx_u32(uint32_t a, uint32_t b, uint32_t *high) { + uint64_t x = (uint64_t)a * b; + *high = (uint32_t) (x >> 32); + return (uint32_t) x; +} + +static uint32_t addcarryx_u32(uint8_t c, uint32_t a, uint32_t b, uint32_t *low) { + uint64_t x = (uint64_t)a + b + c; + *low = (uint32_t) x; + return (uint32_t) (x>>32); +} + +static uint32_t subborrow_u32(uint8_t c, uint32_t a, uint32_t b, uint32_t *low) { + uint64_t t = ((uint64_t) b + c); + uint64_t x = a-t; + *low = (uint32_t) x; + return (uint8_t) (x>>63); +} + +static uint32_t cmovznz_u32(uint32_t t, uint32_t z, uint32_t nz) { + t = -!!t; // all set if nonzero, 0 if 0 + return (t&nz) | ((~t)&z); +} + +#endif + +// fiat-crypto generated code + +#if defined(BORINGSSL_NISTP256_64BIT) + +static void fe_add(uint64_t out[4], const uint64_t in1[4], const uint64_t in2[4]) { + { const uint64_t x8 = in1[3]; + { const uint64_t x9 = in1[2]; + { const uint64_t x7 = in1[1]; + { const uint64_t x5 = in1[0]; + { const uint64_t x14 = in2[3]; + { const uint64_t x15 = in2[2]; + { const uint64_t x13 = in2[1]; + { const uint64_t x11 = in2[0]; + { uint64_t x17; uint8_t x18 = addcarryx_u64(0x0, x5, x11, &x17); + { uint64_t x20; uint8_t x21 = addcarryx_u64(x18, x7, x13, &x20); + { uint64_t x23; uint8_t x24 = addcarryx_u64(x21, x9, x15, &x23); + { uint64_t x26; uint8_t x27 = addcarryx_u64(x24, x8, x14, &x26); + { uint64_t x29; uint8_t x30 = subborrow_u64(0x0, x17, 0xffffffffffffffffL, &x29); + { uint64_t x32; uint8_t x33 = subborrow_u64(x30, x20, 0xffffffff, &x32); + { uint64_t x35; uint8_t x36 = subborrow_u64(x33, x23, 0x0, &x35); + { uint64_t x38; uint8_t x39 = subborrow_u64(x36, x26, 0xffffffff00000001L, &x38); + { uint64_t _1; uint8_t x42 = subborrow_u64(x39, x27, 0x0, &_1); + { uint64_t x43 = cmovznz_u64(x42, x38, x26); + { uint64_t x44 = cmovznz_u64(x42, x35, x23); + { uint64_t x45 = cmovznz_u64(x42, x32, x20); + { uint64_t x46 = cmovznz_u64(x42, x29, x17); + out[0] = x46; + out[1] = x45; + out[2] = x44; + out[3] = x43; + }}}}}}}}}}}}}}}}}}}}} +} + +// fe_op sets out = -in +static void fe_opp(uint64_t out[4], const uint64_t in1[4]) { + const uint64_t x5 = in1[3]; + const uint64_t x6 = in1[2]; + const uint64_t x4 = in1[1]; + const uint64_t x2 = in1[0]; + uint64_t x8; uint8_t x9 = subborrow_u64(0x0, 0x0, x2, &x8); + uint64_t x11; uint8_t x12 = subborrow_u64(x9, 0x0, x4, &x11); + uint64_t x14; uint8_t x15 = subborrow_u64(x12, 0x0, x6, &x14); + uint64_t x17; uint8_t x18 = subborrow_u64(x15, 0x0, x5, &x17); + uint64_t x19 = (uint64_t)cmovznz_u64(x18, 0x0, 0xffffffffffffffffL); + uint64_t x20 = (x19 & 0xffffffffffffffffL); + uint64_t x22; uint8_t x23 = addcarryx_u64(0x0, x8, x20, &x22); + uint64_t x24 = (x19 & 0xffffffff); + uint64_t x26; uint8_t x27 = addcarryx_u64(x23, x11, x24, &x26); + uint64_t x29; uint8_t x30 = addcarryx_u64(x27, x14, 0x0, &x29); + uint64_t x31 = (x19 & 0xffffffff00000001L); + uint64_t x33; addcarryx_u64(x30, x17, x31, &x33); + out[0] = x22; + out[1] = x26; + out[2] = x29; + out[3] = x33; +} + +static void fe_mul(uint64_t out[4], const uint64_t in1[4], const uint64_t in2[4]) { + const uint64_t x8 = in1[3]; + const uint64_t x9 = in1[2]; + const uint64_t x7 = in1[1]; + const uint64_t x5 = in1[0]; + const uint64_t x14 = in2[3]; + const uint64_t x15 = in2[2]; + const uint64_t x13 = in2[1]; + const uint64_t x11 = in2[0]; + uint64_t x18; uint64_t x17 = mulx_u64(x5, x11, &x18); + uint64_t x21; uint64_t x20 = mulx_u64(x5, x13, &x21); + uint64_t x24; uint64_t x23 = mulx_u64(x5, x15, &x24); + uint64_t x27; uint64_t x26 = mulx_u64(x5, x14, &x27); + uint64_t x29; uint8_t x30 = addcarryx_u64(0x0, x18, x20, &x29); + uint64_t x32; uint8_t x33 = addcarryx_u64(x30, x21, x23, &x32); + uint64_t x35; uint8_t x36 = addcarryx_u64(x33, x24, x26, &x35); + uint64_t x38; addcarryx_u64(0x0, x36, x27, &x38); + uint64_t x42; uint64_t x41 = mulx_u64(x17, 0xffffffffffffffffL, &x42); + uint64_t x45; uint64_t x44 = mulx_u64(x17, 0xffffffff, &x45); + uint64_t x48; uint64_t x47 = mulx_u64(x17, 0xffffffff00000001L, &x48); + uint64_t x50; uint8_t x51 = addcarryx_u64(0x0, x42, x44, &x50); + uint64_t x53; uint8_t x54 = addcarryx_u64(x51, x45, 0x0, &x53); + uint64_t x56; uint8_t x57 = addcarryx_u64(x54, 0x0, x47, &x56); + uint64_t x59; addcarryx_u64(0x0, x57, x48, &x59); + uint64_t _2; uint8_t x63 = addcarryx_u64(0x0, x17, x41, &_2); + uint64_t x65; uint8_t x66 = addcarryx_u64(x63, x29, x50, &x65); + uint64_t x68; uint8_t x69 = addcarryx_u64(x66, x32, x53, &x68); + uint64_t x71; uint8_t x72 = addcarryx_u64(x69, x35, x56, &x71); + uint64_t x74; uint8_t x75 = addcarryx_u64(x72, x38, x59, &x74); + uint64_t x78; uint64_t x77 = mulx_u64(x7, x11, &x78); + uint64_t x81; uint64_t x80 = mulx_u64(x7, x13, &x81); + uint64_t x84; uint64_t x83 = mulx_u64(x7, x15, &x84); + uint64_t x87; uint64_t x86 = mulx_u64(x7, x14, &x87); + uint64_t x89; uint8_t x90 = addcarryx_u64(0x0, x78, x80, &x89); + uint64_t x92; uint8_t x93 = addcarryx_u64(x90, x81, x83, &x92); + uint64_t x95; uint8_t x96 = addcarryx_u64(x93, x84, x86, &x95); + uint64_t x98; addcarryx_u64(0x0, x96, x87, &x98); + uint64_t x101; uint8_t x102 = addcarryx_u64(0x0, x65, x77, &x101); + uint64_t x104; uint8_t x105 = addcarryx_u64(x102, x68, x89, &x104); + uint64_t x107; uint8_t x108 = addcarryx_u64(x105, x71, x92, &x107); + uint64_t x110; uint8_t x111 = addcarryx_u64(x108, x74, x95, &x110); + uint64_t x113; uint8_t x114 = addcarryx_u64(x111, x75, x98, &x113); + uint64_t x117; uint64_t x116 = mulx_u64(x101, 0xffffffffffffffffL, &x117); + uint64_t x120; uint64_t x119 = mulx_u64(x101, 0xffffffff, &x120); + uint64_t x123; uint64_t x122 = mulx_u64(x101, 0xffffffff00000001L, &x123); + uint64_t x125; uint8_t x126 = addcarryx_u64(0x0, x117, x119, &x125); + uint64_t x128; uint8_t x129 = addcarryx_u64(x126, x120, 0x0, &x128); + uint64_t x131; uint8_t x132 = addcarryx_u64(x129, 0x0, x122, &x131); + uint64_t x134; addcarryx_u64(0x0, x132, x123, &x134); + uint64_t _3; uint8_t x138 = addcarryx_u64(0x0, x101, x116, &_3); + uint64_t x140; uint8_t x141 = addcarryx_u64(x138, x104, x125, &x140); + uint64_t x143; uint8_t x144 = addcarryx_u64(x141, x107, x128, &x143); + uint64_t x146; uint8_t x147 = addcarryx_u64(x144, x110, x131, &x146); + uint64_t x149; uint8_t x150 = addcarryx_u64(x147, x113, x134, &x149); + uint8_t x151 = (x150 + x114); + uint64_t x154; uint64_t x153 = mulx_u64(x9, x11, &x154); + uint64_t x157; uint64_t x156 = mulx_u64(x9, x13, &x157); + uint64_t x160; uint64_t x159 = mulx_u64(x9, x15, &x160); + uint64_t x163; uint64_t x162 = mulx_u64(x9, x14, &x163); + uint64_t x165; uint8_t x166 = addcarryx_u64(0x0, x154, x156, &x165); + uint64_t x168; uint8_t x169 = addcarryx_u64(x166, x157, x159, &x168); + uint64_t x171; uint8_t x172 = addcarryx_u64(x169, x160, x162, &x171); + uint64_t x174; addcarryx_u64(0x0, x172, x163, &x174); + uint64_t x177; uint8_t x178 = addcarryx_u64(0x0, x140, x153, &x177); + uint64_t x180; uint8_t x181 = addcarryx_u64(x178, x143, x165, &x180); + uint64_t x183; uint8_t x184 = addcarryx_u64(x181, x146, x168, &x183); + uint64_t x186; uint8_t x187 = addcarryx_u64(x184, x149, x171, &x186); + uint64_t x189; uint8_t x190 = addcarryx_u64(x187, x151, x174, &x189); + uint64_t x193; uint64_t x192 = mulx_u64(x177, 0xffffffffffffffffL, &x193); + uint64_t x196; uint64_t x195 = mulx_u64(x177, 0xffffffff, &x196); + uint64_t x199; uint64_t x198 = mulx_u64(x177, 0xffffffff00000001L, &x199); + uint64_t x201; uint8_t x202 = addcarryx_u64(0x0, x193, x195, &x201); + uint64_t x204; uint8_t x205 = addcarryx_u64(x202, x196, 0x0, &x204); + uint64_t x207; uint8_t x208 = addcarryx_u64(x205, 0x0, x198, &x207); + uint64_t x210; addcarryx_u64(0x0, x208, x199, &x210); + uint64_t _4; uint8_t x214 = addcarryx_u64(0x0, x177, x192, &_4); + uint64_t x216; uint8_t x217 = addcarryx_u64(x214, x180, x201, &x216); + uint64_t x219; uint8_t x220 = addcarryx_u64(x217, x183, x204, &x219); + uint64_t x222; uint8_t x223 = addcarryx_u64(x220, x186, x207, &x222); + uint64_t x225; uint8_t x226 = addcarryx_u64(x223, x189, x210, &x225); + uint8_t x227 = (x226 + x190); + uint64_t x230; uint64_t x229 = mulx_u64(x8, x11, &x230); + uint64_t x233; uint64_t x232 = mulx_u64(x8, x13, &x233); + uint64_t x236; uint64_t x235 = mulx_u64(x8, x15, &x236); + uint64_t x239; uint64_t x238 = mulx_u64(x8, x14, &x239); + uint64_t x241; uint8_t x242 = addcarryx_u64(0x0, x230, x232, &x241); + uint64_t x244; uint8_t x245 = addcarryx_u64(x242, x233, x235, &x244); + uint64_t x247; uint8_t x248 = addcarryx_u64(x245, x236, x238, &x247); + uint64_t x250; addcarryx_u64(0x0, x248, x239, &x250); + uint64_t x253; uint8_t x254 = addcarryx_u64(0x0, x216, x229, &x253); + uint64_t x256; uint8_t x257 = addcarryx_u64(x254, x219, x241, &x256); + uint64_t x259; uint8_t x260 = addcarryx_u64(x257, x222, x244, &x259); + uint64_t x262; uint8_t x263 = addcarryx_u64(x260, x225, x247, &x262); + uint64_t x265; uint8_t x266 = addcarryx_u64(x263, x227, x250, &x265); + uint64_t x269; uint64_t x268 = mulx_u64(x253, 0xffffffffffffffffL, &x269); + uint64_t x272; uint64_t x271 = mulx_u64(x253, 0xffffffff, &x272); + uint64_t x275; uint64_t x274 = mulx_u64(x253, 0xffffffff00000001L, &x275); + uint64_t x277; uint8_t x278 = addcarryx_u64(0x0, x269, x271, &x277); + uint64_t x280; uint8_t x281 = addcarryx_u64(x278, x272, 0x0, &x280); + uint64_t x283; uint8_t x284 = addcarryx_u64(x281, 0x0, x274, &x283); + uint64_t x286; addcarryx_u64(0x0, x284, x275, &x286); + uint64_t _5; uint8_t x290 = addcarryx_u64(0x0, x253, x268, &_5); + uint64_t x292; uint8_t x293 = addcarryx_u64(x290, x256, x277, &x292); + uint64_t x295; uint8_t x296 = addcarryx_u64(x293, x259, x280, &x295); + uint64_t x298; uint8_t x299 = addcarryx_u64(x296, x262, x283, &x298); + uint64_t x301; uint8_t x302 = addcarryx_u64(x299, x265, x286, &x301); + uint8_t x303 = (x302 + x266); + uint64_t x305; uint8_t x306 = subborrow_u64(0x0, x292, 0xffffffffffffffffL, &x305); + uint64_t x308; uint8_t x309 = subborrow_u64(x306, x295, 0xffffffff, &x308); + uint64_t x311; uint8_t x312 = subborrow_u64(x309, x298, 0x0, &x311); + uint64_t x314; uint8_t x315 = subborrow_u64(x312, x301, 0xffffffff00000001L, &x314); + uint64_t _6; uint8_t x318 = subborrow_u64(x315, x303, 0x0, &_6); + uint64_t x319 = cmovznz_u64(x318, x314, x301); + uint64_t x320 = cmovznz_u64(x318, x311, x298); + uint64_t x321 = cmovznz_u64(x318, x308, x295); + uint64_t x322 = cmovznz_u64(x318, x305, x292); + out[0] = x322; + out[1] = x321; + out[2] = x320; + out[3] = x319; +} + +static void fe_sub(uint64_t out[4], const uint64_t in1[4], const uint64_t in2[4]) { + const uint64_t x8 = in1[3]; + const uint64_t x9 = in1[2]; + const uint64_t x7 = in1[1]; + const uint64_t x5 = in1[0]; + const uint64_t x14 = in2[3]; + const uint64_t x15 = in2[2]; + const uint64_t x13 = in2[1]; + const uint64_t x11 = in2[0]; + uint64_t x17; uint8_t x18 = subborrow_u64(0x0, x5, x11, &x17); + uint64_t x20; uint8_t x21 = subborrow_u64(x18, x7, x13, &x20); + uint64_t x23; uint8_t x24 = subborrow_u64(x21, x9, x15, &x23); + uint64_t x26; uint8_t x27 = subborrow_u64(x24, x8, x14, &x26); + uint64_t x28 = (uint64_t)cmovznz_u64(x27, 0x0, 0xffffffffffffffffL); + uint64_t x29 = (x28 & 0xffffffffffffffffL); + uint64_t x31; uint8_t x32 = addcarryx_u64(0x0, x17, x29, &x31); + uint64_t x33 = (x28 & 0xffffffff); + uint64_t x35; uint8_t x36 = addcarryx_u64(x32, x20, x33, &x35); + uint64_t x38; uint8_t x39 = addcarryx_u64(x36, x23, 0x0, &x38); + uint64_t x40 = (x28 & 0xffffffff00000001L); + uint64_t x42; addcarryx_u64(x39, x26, x40, &x42); + out[0] = x31; + out[1] = x35; + out[2] = x38; + out[3] = x42; +} + +#else // 64BIT, else 32BIT + +static void fe_add(uint32_t out[8], const uint32_t in1[8], const uint32_t in2[8]) { + const uint32_t x16 = in1[7]; + const uint32_t x17 = in1[6]; + const uint32_t x15 = in1[5]; + const uint32_t x13 = in1[4]; + const uint32_t x11 = in1[3]; + const uint32_t x9 = in1[2]; + const uint32_t x7 = in1[1]; + const uint32_t x5 = in1[0]; + const uint32_t x30 = in2[7]; + const uint32_t x31 = in2[6]; + const uint32_t x29 = in2[5]; + const uint32_t x27 = in2[4]; + const uint32_t x25 = in2[3]; + const uint32_t x23 = in2[2]; + const uint32_t x21 = in2[1]; + const uint32_t x19 = in2[0]; + uint32_t x33; uint8_t x34 = addcarryx_u32(0x0, x5, x19, &x33); + uint32_t x36; uint8_t x37 = addcarryx_u32(x34, x7, x21, &x36); + uint32_t x39; uint8_t x40 = addcarryx_u32(x37, x9, x23, &x39); + uint32_t x42; uint8_t x43 = addcarryx_u32(x40, x11, x25, &x42); + uint32_t x45; uint8_t x46 = addcarryx_u32(x43, x13, x27, &x45); + uint32_t x48; uint8_t x49 = addcarryx_u32(x46, x15, x29, &x48); + uint32_t x51; uint8_t x52 = addcarryx_u32(x49, x17, x31, &x51); + uint32_t x54; uint8_t x55 = addcarryx_u32(x52, x16, x30, &x54); + uint32_t x57; uint8_t x58 = subborrow_u32(0x0, x33, 0xffffffff, &x57); + uint32_t x60; uint8_t x61 = subborrow_u32(x58, x36, 0xffffffff, &x60); + uint32_t x63; uint8_t x64 = subborrow_u32(x61, x39, 0xffffffff, &x63); + uint32_t x66; uint8_t x67 = subborrow_u32(x64, x42, 0x0, &x66); + uint32_t x69; uint8_t x70 = subborrow_u32(x67, x45, 0x0, &x69); + uint32_t x72; uint8_t x73 = subborrow_u32(x70, x48, 0x0, &x72); + uint32_t x75; uint8_t x76 = subborrow_u32(x73, x51, 0x1, &x75); + uint32_t x78; uint8_t x79 = subborrow_u32(x76, x54, 0xffffffff, &x78); + uint32_t _; uint8_t x82 = subborrow_u32(x79, x55, 0x0, &_); + uint32_t x83 = cmovznz_u32(x82, x78, x54); + uint32_t x84 = cmovznz_u32(x82, x75, x51); + uint32_t x85 = cmovznz_u32(x82, x72, x48); + uint32_t x86 = cmovznz_u32(x82, x69, x45); + uint32_t x87 = cmovznz_u32(x82, x66, x42); + uint32_t x88 = cmovznz_u32(x82, x63, x39); + uint32_t x89 = cmovznz_u32(x82, x60, x36); + uint32_t x90 = cmovznz_u32(x82, x57, x33); + out[0] = x90; + out[1] = x89; + out[2] = x88; + out[3] = x87; + out[4] = x86; + out[5] = x85; + out[6] = x84; + out[7] = x83; +} + +static void fe_mul(uint32_t out[8], const uint32_t in1[8], const uint32_t in2[8]) { + const uint32_t x16 = in1[7]; + const uint32_t x17 = in1[6]; + const uint32_t x15 = in1[5]; + const uint32_t x13 = in1[4]; + const uint32_t x11 = in1[3]; + const uint32_t x9 = in1[2]; + const uint32_t x7 = in1[1]; + const uint32_t x5 = in1[0]; + const uint32_t x30 = in2[7]; + const uint32_t x31 = in2[6]; + const uint32_t x29 = in2[5]; + const uint32_t x27 = in2[4]; + const uint32_t x25 = in2[3]; + const uint32_t x23 = in2[2]; + const uint32_t x21 = in2[1]; + const uint32_t x19 = in2[0]; + uint32_t x34; uint32_t x33 = mulx_u32(x5, x19, &x34); + uint32_t x37; uint32_t x36 = mulx_u32(x5, x21, &x37); + uint32_t x40; uint32_t x39 = mulx_u32(x5, x23, &x40); + uint32_t x43; uint32_t x42 = mulx_u32(x5, x25, &x43); + uint32_t x46; uint32_t x45 = mulx_u32(x5, x27, &x46); + uint32_t x49; uint32_t x48 = mulx_u32(x5, x29, &x49); + uint32_t x52; uint32_t x51 = mulx_u32(x5, x31, &x52); + uint32_t x55; uint32_t x54 = mulx_u32(x5, x30, &x55); + uint32_t x57; uint8_t x58 = addcarryx_u32(0x0, x34, x36, &x57); + uint32_t x60; uint8_t x61 = addcarryx_u32(x58, x37, x39, &x60); + uint32_t x63; uint8_t x64 = addcarryx_u32(x61, x40, x42, &x63); + uint32_t x66; uint8_t x67 = addcarryx_u32(x64, x43, x45, &x66); + uint32_t x69; uint8_t x70 = addcarryx_u32(x67, x46, x48, &x69); + uint32_t x72; uint8_t x73 = addcarryx_u32(x70, x49, x51, &x72); + uint32_t x75; uint8_t x76 = addcarryx_u32(x73, x52, x54, &x75); + uint32_t x78; addcarryx_u32(0x0, x76, x55, &x78); + uint32_t x82; uint32_t x81 = mulx_u32(x33, 0xffffffff, &x82); + uint32_t x85; uint32_t x84 = mulx_u32(x33, 0xffffffff, &x85); + uint32_t x88; uint32_t x87 = mulx_u32(x33, 0xffffffff, &x88); + uint32_t x91; uint32_t x90 = mulx_u32(x33, 0xffffffff, &x91); + uint32_t x93; uint8_t x94 = addcarryx_u32(0x0, x82, x84, &x93); + uint32_t x96; uint8_t x97 = addcarryx_u32(x94, x85, x87, &x96); + uint32_t x99; uint8_t x100 = addcarryx_u32(x97, x88, 0x0, &x99); + uint8_t x101 = (0x0 + 0x0); + uint32_t _1; uint8_t x104 = addcarryx_u32(0x0, x33, x81, &_1); + uint32_t x106; uint8_t x107 = addcarryx_u32(x104, x57, x93, &x106); + uint32_t x109; uint8_t x110 = addcarryx_u32(x107, x60, x96, &x109); + uint32_t x112; uint8_t x113 = addcarryx_u32(x110, x63, x99, &x112); + uint32_t x115; uint8_t x116 = addcarryx_u32(x113, x66, x100, &x115); + uint32_t x118; uint8_t x119 = addcarryx_u32(x116, x69, x101, &x118); + uint32_t x121; uint8_t x122 = addcarryx_u32(x119, x72, x33, &x121); + uint32_t x124; uint8_t x125 = addcarryx_u32(x122, x75, x90, &x124); + uint32_t x127; uint8_t x128 = addcarryx_u32(x125, x78, x91, &x127); + uint8_t x129 = (x128 + 0x0); + uint32_t x132; uint32_t x131 = mulx_u32(x7, x19, &x132); + uint32_t x135; uint32_t x134 = mulx_u32(x7, x21, &x135); + uint32_t x138; uint32_t x137 = mulx_u32(x7, x23, &x138); + uint32_t x141; uint32_t x140 = mulx_u32(x7, x25, &x141); + uint32_t x144; uint32_t x143 = mulx_u32(x7, x27, &x144); + uint32_t x147; uint32_t x146 = mulx_u32(x7, x29, &x147); + uint32_t x150; uint32_t x149 = mulx_u32(x7, x31, &x150); + uint32_t x153; uint32_t x152 = mulx_u32(x7, x30, &x153); + uint32_t x155; uint8_t x156 = addcarryx_u32(0x0, x132, x134, &x155); + uint32_t x158; uint8_t x159 = addcarryx_u32(x156, x135, x137, &x158); + uint32_t x161; uint8_t x162 = addcarryx_u32(x159, x138, x140, &x161); + uint32_t x164; uint8_t x165 = addcarryx_u32(x162, x141, x143, &x164); + uint32_t x167; uint8_t x168 = addcarryx_u32(x165, x144, x146, &x167); + uint32_t x170; uint8_t x171 = addcarryx_u32(x168, x147, x149, &x170); + uint32_t x173; uint8_t x174 = addcarryx_u32(x171, x150, x152, &x173); + uint32_t x176; addcarryx_u32(0x0, x174, x153, &x176); + uint32_t x179; uint8_t x180 = addcarryx_u32(0x0, x106, x131, &x179); + uint32_t x182; uint8_t x183 = addcarryx_u32(x180, x109, x155, &x182); + uint32_t x185; uint8_t x186 = addcarryx_u32(x183, x112, x158, &x185); + uint32_t x188; uint8_t x189 = addcarryx_u32(x186, x115, x161, &x188); + uint32_t x191; uint8_t x192 = addcarryx_u32(x189, x118, x164, &x191); + uint32_t x194; uint8_t x195 = addcarryx_u32(x192, x121, x167, &x194); + uint32_t x197; uint8_t x198 = addcarryx_u32(x195, x124, x170, &x197); + uint32_t x200; uint8_t x201 = addcarryx_u32(x198, x127, x173, &x200); + uint32_t x203; uint8_t x204 = addcarryx_u32(x201, x129, x176, &x203); + uint32_t x207; uint32_t x206 = mulx_u32(x179, 0xffffffff, &x207); + uint32_t x210; uint32_t x209 = mulx_u32(x179, 0xffffffff, &x210); + uint32_t x213; uint32_t x212 = mulx_u32(x179, 0xffffffff, &x213); + uint32_t x216; uint32_t x215 = mulx_u32(x179, 0xffffffff, &x216); + uint32_t x218; uint8_t x219 = addcarryx_u32(0x0, x207, x209, &x218); + uint32_t x221; uint8_t x222 = addcarryx_u32(x219, x210, x212, &x221); + uint32_t x224; uint8_t x225 = addcarryx_u32(x222, x213, 0x0, &x224); + uint8_t x226 = (0x0 + 0x0); + uint32_t _2; uint8_t x229 = addcarryx_u32(0x0, x179, x206, &_2); + uint32_t x231; uint8_t x232 = addcarryx_u32(x229, x182, x218, &x231); + uint32_t x234; uint8_t x235 = addcarryx_u32(x232, x185, x221, &x234); + uint32_t x237; uint8_t x238 = addcarryx_u32(x235, x188, x224, &x237); + uint32_t x240; uint8_t x241 = addcarryx_u32(x238, x191, x225, &x240); + uint32_t x243; uint8_t x244 = addcarryx_u32(x241, x194, x226, &x243); + uint32_t x246; uint8_t x247 = addcarryx_u32(x244, x197, x179, &x246); + uint32_t x249; uint8_t x250 = addcarryx_u32(x247, x200, x215, &x249); + uint32_t x252; uint8_t x253 = addcarryx_u32(x250, x203, x216, &x252); + uint8_t x254 = (x253 + x204); + uint32_t x257; uint32_t x256 = mulx_u32(x9, x19, &x257); + uint32_t x260; uint32_t x259 = mulx_u32(x9, x21, &x260); + uint32_t x263; uint32_t x262 = mulx_u32(x9, x23, &x263); + uint32_t x266; uint32_t x265 = mulx_u32(x9, x25, &x266); + uint32_t x269; uint32_t x268 = mulx_u32(x9, x27, &x269); + uint32_t x272; uint32_t x271 = mulx_u32(x9, x29, &x272); + uint32_t x275; uint32_t x274 = mulx_u32(x9, x31, &x275); + uint32_t x278; uint32_t x277 = mulx_u32(x9, x30, &x278); + uint32_t x280; uint8_t x281 = addcarryx_u32(0x0, x257, x259, &x280); + uint32_t x283; uint8_t x284 = addcarryx_u32(x281, x260, x262, &x283); + uint32_t x286; uint8_t x287 = addcarryx_u32(x284, x263, x265, &x286); + uint32_t x289; uint8_t x290 = addcarryx_u32(x287, x266, x268, &x289); + uint32_t x292; uint8_t x293 = addcarryx_u32(x290, x269, x271, &x292); + uint32_t x295; uint8_t x296 = addcarryx_u32(x293, x272, x274, &x295); + uint32_t x298; uint8_t x299 = addcarryx_u32(x296, x275, x277, &x298); + uint32_t x301; addcarryx_u32(0x0, x299, x278, &x301); + uint32_t x304; uint8_t x305 = addcarryx_u32(0x0, x231, x256, &x304); + uint32_t x307; uint8_t x308 = addcarryx_u32(x305, x234, x280, &x307); + uint32_t x310; uint8_t x311 = addcarryx_u32(x308, x237, x283, &x310); + uint32_t x313; uint8_t x314 = addcarryx_u32(x311, x240, x286, &x313); + uint32_t x316; uint8_t x317 = addcarryx_u32(x314, x243, x289, &x316); + uint32_t x319; uint8_t x320 = addcarryx_u32(x317, x246, x292, &x319); + uint32_t x322; uint8_t x323 = addcarryx_u32(x320, x249, x295, &x322); + uint32_t x325; uint8_t x326 = addcarryx_u32(x323, x252, x298, &x325); + uint32_t x328; uint8_t x329 = addcarryx_u32(x326, x254, x301, &x328); + uint32_t x332; uint32_t x331 = mulx_u32(x304, 0xffffffff, &x332); + uint32_t x335; uint32_t x334 = mulx_u32(x304, 0xffffffff, &x335); + uint32_t x338; uint32_t x337 = mulx_u32(x304, 0xffffffff, &x338); + uint32_t x341; uint32_t x340 = mulx_u32(x304, 0xffffffff, &x341); + uint32_t x343; uint8_t x344 = addcarryx_u32(0x0, x332, x334, &x343); + uint32_t x346; uint8_t x347 = addcarryx_u32(x344, x335, x337, &x346); + uint32_t x349; uint8_t x350 = addcarryx_u32(x347, x338, 0x0, &x349); + uint8_t x351 = (0x0 + 0x0); + uint32_t _3; uint8_t x354 = addcarryx_u32(0x0, x304, x331, &_3); + uint32_t x356; uint8_t x357 = addcarryx_u32(x354, x307, x343, &x356); + uint32_t x359; uint8_t x360 = addcarryx_u32(x357, x310, x346, &x359); + uint32_t x362; uint8_t x363 = addcarryx_u32(x360, x313, x349, &x362); + uint32_t x365; uint8_t x366 = addcarryx_u32(x363, x316, x350, &x365); + uint32_t x368; uint8_t x369 = addcarryx_u32(x366, x319, x351, &x368); + uint32_t x371; uint8_t x372 = addcarryx_u32(x369, x322, x304, &x371); + uint32_t x374; uint8_t x375 = addcarryx_u32(x372, x325, x340, &x374); + uint32_t x377; uint8_t x378 = addcarryx_u32(x375, x328, x341, &x377); + uint8_t x379 = (x378 + x329); + uint32_t x382; uint32_t x381 = mulx_u32(x11, x19, &x382); + uint32_t x385; uint32_t x384 = mulx_u32(x11, x21, &x385); + uint32_t x388; uint32_t x387 = mulx_u32(x11, x23, &x388); + uint32_t x391; uint32_t x390 = mulx_u32(x11, x25, &x391); + uint32_t x394; uint32_t x393 = mulx_u32(x11, x27, &x394); + uint32_t x397; uint32_t x396 = mulx_u32(x11, x29, &x397); + uint32_t x400; uint32_t x399 = mulx_u32(x11, x31, &x400); + uint32_t x403; uint32_t x402 = mulx_u32(x11, x30, &x403); + uint32_t x405; uint8_t x406 = addcarryx_u32(0x0, x382, x384, &x405); + uint32_t x408; uint8_t x409 = addcarryx_u32(x406, x385, x387, &x408); + uint32_t x411; uint8_t x412 = addcarryx_u32(x409, x388, x390, &x411); + uint32_t x414; uint8_t x415 = addcarryx_u32(x412, x391, x393, &x414); + uint32_t x417; uint8_t x418 = addcarryx_u32(x415, x394, x396, &x417); + uint32_t x420; uint8_t x421 = addcarryx_u32(x418, x397, x399, &x420); + uint32_t x423; uint8_t x424 = addcarryx_u32(x421, x400, x402, &x423); + uint32_t x426; addcarryx_u32(0x0, x424, x403, &x426); + uint32_t x429; uint8_t x430 = addcarryx_u32(0x0, x356, x381, &x429); + uint32_t x432; uint8_t x433 = addcarryx_u32(x430, x359, x405, &x432); + uint32_t x435; uint8_t x436 = addcarryx_u32(x433, x362, x408, &x435); + uint32_t x438; uint8_t x439 = addcarryx_u32(x436, x365, x411, &x438); + uint32_t x441; uint8_t x442 = addcarryx_u32(x439, x368, x414, &x441); + uint32_t x444; uint8_t x445 = addcarryx_u32(x442, x371, x417, &x444); + uint32_t x447; uint8_t x448 = addcarryx_u32(x445, x374, x420, &x447); + uint32_t x450; uint8_t x451 = addcarryx_u32(x448, x377, x423, &x450); + uint32_t x453; uint8_t x454 = addcarryx_u32(x451, x379, x426, &x453); + uint32_t x457; uint32_t x456 = mulx_u32(x429, 0xffffffff, &x457); + uint32_t x460; uint32_t x459 = mulx_u32(x429, 0xffffffff, &x460); + uint32_t x463; uint32_t x462 = mulx_u32(x429, 0xffffffff, &x463); + uint32_t x466; uint32_t x465 = mulx_u32(x429, 0xffffffff, &x466); + uint32_t x468; uint8_t x469 = addcarryx_u32(0x0, x457, x459, &x468); + uint32_t x471; uint8_t x472 = addcarryx_u32(x469, x460, x462, &x471); + uint32_t x474; uint8_t x475 = addcarryx_u32(x472, x463, 0x0, &x474); + uint8_t x476 = (0x0 + 0x0); + uint32_t _4; uint8_t x479 = addcarryx_u32(0x0, x429, x456, &_4); + uint32_t x481; uint8_t x482 = addcarryx_u32(x479, x432, x468, &x481); + uint32_t x484; uint8_t x485 = addcarryx_u32(x482, x435, x471, &x484); + uint32_t x487; uint8_t x488 = addcarryx_u32(x485, x438, x474, &x487); + uint32_t x490; uint8_t x491 = addcarryx_u32(x488, x441, x475, &x490); + uint32_t x493; uint8_t x494 = addcarryx_u32(x491, x444, x476, &x493); + uint32_t x496; uint8_t x497 = addcarryx_u32(x494, x447, x429, &x496); + uint32_t x499; uint8_t x500 = addcarryx_u32(x497, x450, x465, &x499); + uint32_t x502; uint8_t x503 = addcarryx_u32(x500, x453, x466, &x502); + uint8_t x504 = (x503 + x454); + uint32_t x507; uint32_t x506 = mulx_u32(x13, x19, &x507); + uint32_t x510; uint32_t x509 = mulx_u32(x13, x21, &x510); + uint32_t x513; uint32_t x512 = mulx_u32(x13, x23, &x513); + uint32_t x516; uint32_t x515 = mulx_u32(x13, x25, &x516); + uint32_t x519; uint32_t x518 = mulx_u32(x13, x27, &x519); + uint32_t x522; uint32_t x521 = mulx_u32(x13, x29, &x522); + uint32_t x525; uint32_t x524 = mulx_u32(x13, x31, &x525); + uint32_t x528; uint32_t x527 = mulx_u32(x13, x30, &x528); + uint32_t x530; uint8_t x531 = addcarryx_u32(0x0, x507, x509, &x530); + uint32_t x533; uint8_t x534 = addcarryx_u32(x531, x510, x512, &x533); + uint32_t x536; uint8_t x537 = addcarryx_u32(x534, x513, x515, &x536); + uint32_t x539; uint8_t x540 = addcarryx_u32(x537, x516, x518, &x539); + uint32_t x542; uint8_t x543 = addcarryx_u32(x540, x519, x521, &x542); + uint32_t x545; uint8_t x546 = addcarryx_u32(x543, x522, x524, &x545); + uint32_t x548; uint8_t x549 = addcarryx_u32(x546, x525, x527, &x548); + uint32_t x551; addcarryx_u32(0x0, x549, x528, &x551); + uint32_t x554; uint8_t x555 = addcarryx_u32(0x0, x481, x506, &x554); + uint32_t x557; uint8_t x558 = addcarryx_u32(x555, x484, x530, &x557); + uint32_t x560; uint8_t x561 = addcarryx_u32(x558, x487, x533, &x560); + uint32_t x563; uint8_t x564 = addcarryx_u32(x561, x490, x536, &x563); + uint32_t x566; uint8_t x567 = addcarryx_u32(x564, x493, x539, &x566); + uint32_t x569; uint8_t x570 = addcarryx_u32(x567, x496, x542, &x569); + uint32_t x572; uint8_t x573 = addcarryx_u32(x570, x499, x545, &x572); + uint32_t x575; uint8_t x576 = addcarryx_u32(x573, x502, x548, &x575); + uint32_t x578; uint8_t x579 = addcarryx_u32(x576, x504, x551, &x578); + uint32_t x582; uint32_t x581 = mulx_u32(x554, 0xffffffff, &x582); + uint32_t x585; uint32_t x584 = mulx_u32(x554, 0xffffffff, &x585); + uint32_t x588; uint32_t x587 = mulx_u32(x554, 0xffffffff, &x588); + uint32_t x591; uint32_t x590 = mulx_u32(x554, 0xffffffff, &x591); + uint32_t x593; uint8_t x594 = addcarryx_u32(0x0, x582, x584, &x593); + uint32_t x596; uint8_t x597 = addcarryx_u32(x594, x585, x587, &x596); + uint32_t x599; uint8_t x600 = addcarryx_u32(x597, x588, 0x0, &x599); + uint8_t x601 = (0x0 + 0x0); + uint32_t _5; uint8_t x604 = addcarryx_u32(0x0, x554, x581, &_5); + uint32_t x606; uint8_t x607 = addcarryx_u32(x604, x557, x593, &x606); + uint32_t x609; uint8_t x610 = addcarryx_u32(x607, x560, x596, &x609); + uint32_t x612; uint8_t x613 = addcarryx_u32(x610, x563, x599, &x612); + uint32_t x615; uint8_t x616 = addcarryx_u32(x613, x566, x600, &x615); + uint32_t x618; uint8_t x619 = addcarryx_u32(x616, x569, x601, &x618); + uint32_t x621; uint8_t x622 = addcarryx_u32(x619, x572, x554, &x621); + uint32_t x624; uint8_t x625 = addcarryx_u32(x622, x575, x590, &x624); + uint32_t x627; uint8_t x628 = addcarryx_u32(x625, x578, x591, &x627); + uint8_t x629 = (x628 + x579); + uint32_t x632; uint32_t x631 = mulx_u32(x15, x19, &x632); + uint32_t x635; uint32_t x634 = mulx_u32(x15, x21, &x635); + uint32_t x638; uint32_t x637 = mulx_u32(x15, x23, &x638); + uint32_t x641; uint32_t x640 = mulx_u32(x15, x25, &x641); + uint32_t x644; uint32_t x643 = mulx_u32(x15, x27, &x644); + uint32_t x647; uint32_t x646 = mulx_u32(x15, x29, &x647); + uint32_t x650; uint32_t x649 = mulx_u32(x15, x31, &x650); + uint32_t x653; uint32_t x652 = mulx_u32(x15, x30, &x653); + uint32_t x655; uint8_t x656 = addcarryx_u32(0x0, x632, x634, &x655); + uint32_t x658; uint8_t x659 = addcarryx_u32(x656, x635, x637, &x658); + uint32_t x661; uint8_t x662 = addcarryx_u32(x659, x638, x640, &x661); + uint32_t x664; uint8_t x665 = addcarryx_u32(x662, x641, x643, &x664); + uint32_t x667; uint8_t x668 = addcarryx_u32(x665, x644, x646, &x667); + uint32_t x670; uint8_t x671 = addcarryx_u32(x668, x647, x649, &x670); + uint32_t x673; uint8_t x674 = addcarryx_u32(x671, x650, x652, &x673); + uint32_t x676; addcarryx_u32(0x0, x674, x653, &x676); + uint32_t x679; uint8_t x680 = addcarryx_u32(0x0, x606, x631, &x679); + uint32_t x682; uint8_t x683 = addcarryx_u32(x680, x609, x655, &x682); + uint32_t x685; uint8_t x686 = addcarryx_u32(x683, x612, x658, &x685); + uint32_t x688; uint8_t x689 = addcarryx_u32(x686, x615, x661, &x688); + uint32_t x691; uint8_t x692 = addcarryx_u32(x689, x618, x664, &x691); + uint32_t x694; uint8_t x695 = addcarryx_u32(x692, x621, x667, &x694); + uint32_t x697; uint8_t x698 = addcarryx_u32(x695, x624, x670, &x697); + uint32_t x700; uint8_t x701 = addcarryx_u32(x698, x627, x673, &x700); + uint32_t x703; uint8_t x704 = addcarryx_u32(x701, x629, x676, &x703); + uint32_t x707; uint32_t x706 = mulx_u32(x679, 0xffffffff, &x707); + uint32_t x710; uint32_t x709 = mulx_u32(x679, 0xffffffff, &x710); + uint32_t x713; uint32_t x712 = mulx_u32(x679, 0xffffffff, &x713); + uint32_t x716; uint32_t x715 = mulx_u32(x679, 0xffffffff, &x716); + uint32_t x718; uint8_t x719 = addcarryx_u32(0x0, x707, x709, &x718); + uint32_t x721; uint8_t x722 = addcarryx_u32(x719, x710, x712, &x721); + uint32_t x724; uint8_t x725 = addcarryx_u32(x722, x713, 0x0, &x724); + uint8_t x726 = (0x0 + 0x0); + uint32_t _6; uint8_t x729 = addcarryx_u32(0x0, x679, x706, &_6); + uint32_t x731; uint8_t x732 = addcarryx_u32(x729, x682, x718, &x731); + uint32_t x734; uint8_t x735 = addcarryx_u32(x732, x685, x721, &x734); + uint32_t x737; uint8_t x738 = addcarryx_u32(x735, x688, x724, &x737); + uint32_t x740; uint8_t x741 = addcarryx_u32(x738, x691, x725, &x740); + uint32_t x743; uint8_t x744 = addcarryx_u32(x741, x694, x726, &x743); + uint32_t x746; uint8_t x747 = addcarryx_u32(x744, x697, x679, &x746); + uint32_t x749; uint8_t x750 = addcarryx_u32(x747, x700, x715, &x749); + uint32_t x752; uint8_t x753 = addcarryx_u32(x750, x703, x716, &x752); + uint8_t x754 = (x753 + x704); + uint32_t x757; uint32_t x756 = mulx_u32(x17, x19, &x757); + uint32_t x760; uint32_t x759 = mulx_u32(x17, x21, &x760); + uint32_t x763; uint32_t x762 = mulx_u32(x17, x23, &x763); + uint32_t x766; uint32_t x765 = mulx_u32(x17, x25, &x766); + uint32_t x769; uint32_t x768 = mulx_u32(x17, x27, &x769); + uint32_t x772; uint32_t x771 = mulx_u32(x17, x29, &x772); + uint32_t x775; uint32_t x774 = mulx_u32(x17, x31, &x775); + uint32_t x778; uint32_t x777 = mulx_u32(x17, x30, &x778); + uint32_t x780; uint8_t x781 = addcarryx_u32(0x0, x757, x759, &x780); + uint32_t x783; uint8_t x784 = addcarryx_u32(x781, x760, x762, &x783); + uint32_t x786; uint8_t x787 = addcarryx_u32(x784, x763, x765, &x786); + uint32_t x789; uint8_t x790 = addcarryx_u32(x787, x766, x768, &x789); + uint32_t x792; uint8_t x793 = addcarryx_u32(x790, x769, x771, &x792); + uint32_t x795; uint8_t x796 = addcarryx_u32(x793, x772, x774, &x795); + uint32_t x798; uint8_t x799 = addcarryx_u32(x796, x775, x777, &x798); + uint32_t x801; addcarryx_u32(0x0, x799, x778, &x801); + uint32_t x804; uint8_t x805 = addcarryx_u32(0x0, x731, x756, &x804); + uint32_t x807; uint8_t x808 = addcarryx_u32(x805, x734, x780, &x807); + uint32_t x810; uint8_t x811 = addcarryx_u32(x808, x737, x783, &x810); + uint32_t x813; uint8_t x814 = addcarryx_u32(x811, x740, x786, &x813); + uint32_t x816; uint8_t x817 = addcarryx_u32(x814, x743, x789, &x816); + uint32_t x819; uint8_t x820 = addcarryx_u32(x817, x746, x792, &x819); + uint32_t x822; uint8_t x823 = addcarryx_u32(x820, x749, x795, &x822); + uint32_t x825; uint8_t x826 = addcarryx_u32(x823, x752, x798, &x825); + uint32_t x828; uint8_t x829 = addcarryx_u32(x826, x754, x801, &x828); + uint32_t x832; uint32_t x831 = mulx_u32(x804, 0xffffffff, &x832); + uint32_t x835; uint32_t x834 = mulx_u32(x804, 0xffffffff, &x835); + uint32_t x838; uint32_t x837 = mulx_u32(x804, 0xffffffff, &x838); + uint32_t x841; uint32_t x840 = mulx_u32(x804, 0xffffffff, &x841); + uint32_t x843; uint8_t x844 = addcarryx_u32(0x0, x832, x834, &x843); + uint32_t x846; uint8_t x847 = addcarryx_u32(x844, x835, x837, &x846); + uint32_t x849; uint8_t x850 = addcarryx_u32(x847, x838, 0x0, &x849); + uint8_t x851 = (0x0 + 0x0); + uint32_t _7; uint8_t x854 = addcarryx_u32(0x0, x804, x831, &_7); + uint32_t x856; uint8_t x857 = addcarryx_u32(x854, x807, x843, &x856); + uint32_t x859; uint8_t x860 = addcarryx_u32(x857, x810, x846, &x859); + uint32_t x862; uint8_t x863 = addcarryx_u32(x860, x813, x849, &x862); + uint32_t x865; uint8_t x866 = addcarryx_u32(x863, x816, x850, &x865); + uint32_t x868; uint8_t x869 = addcarryx_u32(x866, x819, x851, &x868); + uint32_t x871; uint8_t x872 = addcarryx_u32(x869, x822, x804, &x871); + uint32_t x874; uint8_t x875 = addcarryx_u32(x872, x825, x840, &x874); + uint32_t x877; uint8_t x878 = addcarryx_u32(x875, x828, x841, &x877); + uint8_t x879 = (x878 + x829); + uint32_t x882; uint32_t x881 = mulx_u32(x16, x19, &x882); + uint32_t x885; uint32_t x884 = mulx_u32(x16, x21, &x885); + uint32_t x888; uint32_t x887 = mulx_u32(x16, x23, &x888); + uint32_t x891; uint32_t x890 = mulx_u32(x16, x25, &x891); + uint32_t x894; uint32_t x893 = mulx_u32(x16, x27, &x894); + uint32_t x897; uint32_t x896 = mulx_u32(x16, x29, &x897); + uint32_t x900; uint32_t x899 = mulx_u32(x16, x31, &x900); + uint32_t x903; uint32_t x902 = mulx_u32(x16, x30, &x903); + uint32_t x905; uint8_t x906 = addcarryx_u32(0x0, x882, x884, &x905); + uint32_t x908; uint8_t x909 = addcarryx_u32(x906, x885, x887, &x908); + uint32_t x911; uint8_t x912 = addcarryx_u32(x909, x888, x890, &x911); + uint32_t x914; uint8_t x915 = addcarryx_u32(x912, x891, x893, &x914); + uint32_t x917; uint8_t x918 = addcarryx_u32(x915, x894, x896, &x917); + uint32_t x920; uint8_t x921 = addcarryx_u32(x918, x897, x899, &x920); + uint32_t x923; uint8_t x924 = addcarryx_u32(x921, x900, x902, &x923); + uint32_t x926; addcarryx_u32(0x0, x924, x903, &x926); + uint32_t x929; uint8_t x930 = addcarryx_u32(0x0, x856, x881, &x929); + uint32_t x932; uint8_t x933 = addcarryx_u32(x930, x859, x905, &x932); + uint32_t x935; uint8_t x936 = addcarryx_u32(x933, x862, x908, &x935); + uint32_t x938; uint8_t x939 = addcarryx_u32(x936, x865, x911, &x938); + uint32_t x941; uint8_t x942 = addcarryx_u32(x939, x868, x914, &x941); + uint32_t x944; uint8_t x945 = addcarryx_u32(x942, x871, x917, &x944); + uint32_t x947; uint8_t x948 = addcarryx_u32(x945, x874, x920, &x947); + uint32_t x950; uint8_t x951 = addcarryx_u32(x948, x877, x923, &x950); + uint32_t x953; uint8_t x954 = addcarryx_u32(x951, x879, x926, &x953); + uint32_t x957; uint32_t x956 = mulx_u32(x929, 0xffffffff, &x957); + uint32_t x960; uint32_t x959 = mulx_u32(x929, 0xffffffff, &x960); + uint32_t x963; uint32_t x962 = mulx_u32(x929, 0xffffffff, &x963); + uint32_t x966; uint32_t x965 = mulx_u32(x929, 0xffffffff, &x966); + uint32_t x968; uint8_t x969 = addcarryx_u32(0x0, x957, x959, &x968); + uint32_t x971; uint8_t x972 = addcarryx_u32(x969, x960, x962, &x971); + uint32_t x974; uint8_t x975 = addcarryx_u32(x972, x963, 0x0, &x974); + uint8_t x976 = (0x0 + 0x0); + uint32_t _8; uint8_t x979 = addcarryx_u32(0x0, x929, x956, &_8); + uint32_t x981; uint8_t x982 = addcarryx_u32(x979, x932, x968, &x981); + uint32_t x984; uint8_t x985 = addcarryx_u32(x982, x935, x971, &x984); + uint32_t x987; uint8_t x988 = addcarryx_u32(x985, x938, x974, &x987); + uint32_t x990; uint8_t x991 = addcarryx_u32(x988, x941, x975, &x990); + uint32_t x993; uint8_t x994 = addcarryx_u32(x991, x944, x976, &x993); + uint32_t x996; uint8_t x997 = addcarryx_u32(x994, x947, x929, &x996); + uint32_t x999; uint8_t x1000 = addcarryx_u32(x997, x950, x965, &x999); + uint32_t x1002; uint8_t x1003 = addcarryx_u32(x1000, x953, x966, &x1002); + uint8_t x1004 = (x1003 + x954); + uint32_t x1006; uint8_t x1007 = subborrow_u32(0x0, x981, 0xffffffff, &x1006); + uint32_t x1009; uint8_t x1010 = subborrow_u32(x1007, x984, 0xffffffff, &x1009); + uint32_t x1012; uint8_t x1013 = subborrow_u32(x1010, x987, 0xffffffff, &x1012); + uint32_t x1015; uint8_t x1016 = subborrow_u32(x1013, x990, 0x0, &x1015); + uint32_t x1018; uint8_t x1019 = subborrow_u32(x1016, x993, 0x0, &x1018); + uint32_t x1021; uint8_t x1022 = subborrow_u32(x1019, x996, 0x0, &x1021); + uint32_t x1024; uint8_t x1025 = subborrow_u32(x1022, x999, 0x1, &x1024); + uint32_t x1027; uint8_t x1028 = subborrow_u32(x1025, x1002, 0xffffffff, &x1027); + uint32_t _9; uint8_t x1031 = subborrow_u32(x1028, x1004, 0x0, &_9); + uint32_t x1032 = cmovznz_u32(x1031, x1027, x1002); + uint32_t x1033 = cmovznz_u32(x1031, x1024, x999); + uint32_t x1034 = cmovznz_u32(x1031, x1021, x996); + uint32_t x1035 = cmovznz_u32(x1031, x1018, x993); + uint32_t x1036 = cmovznz_u32(x1031, x1015, x990); + uint32_t x1037 = cmovznz_u32(x1031, x1012, x987); + uint32_t x1038 = cmovznz_u32(x1031, x1009, x984); + uint32_t x1039 = cmovznz_u32(x1031, x1006, x981); + out[0] = x1039; + out[1] = x1038; + out[2] = x1037; + out[3] = x1036; + out[4] = x1035; + out[5] = x1034; + out[6] = x1033; + out[7] = x1032; +} + +// NOTE: the following functions are generated from fiat-crypto, from the same +// template as their 64-bit counterparts above, but the correctness proof of +// the template was not composed with the correctness proof of the +// specialization pipeline. This is because Coq unexplainedly loops on trying +// to synthesize opp and sub using the normal pipeline. + +static void fe_sub(uint32_t out[8], const uint32_t in1[8], const uint32_t in2[8]) { + const uint32_t x14 = in1[7]; + const uint32_t x15 = in1[6]; + const uint32_t x13 = in1[5]; + const uint32_t x11 = in1[4]; + const uint32_t x9 = in1[3]; + const uint32_t x7 = in1[2]; + const uint32_t x5 = in1[1]; + const uint32_t x3 = in1[0]; + const uint32_t x28 = in2[7]; + const uint32_t x29 = in2[6]; + const uint32_t x27 = in2[5]; + const uint32_t x25 = in2[4]; + const uint32_t x23 = in2[3]; + const uint32_t x21 = in2[2]; + const uint32_t x19 = in2[1]; + const uint32_t x17 = in2[0]; + uint32_t x31; uint8_t x32 = subborrow_u32(0x0, x3, x17, &x31); + uint32_t x34; uint8_t x35 = subborrow_u32(x32, x5, x19, &x34); + uint32_t x37; uint8_t x38 = subborrow_u32(x35, x7, x21, &x37); + uint32_t x40; uint8_t x41 = subborrow_u32(x38, x9, x23, &x40); + uint32_t x43; uint8_t x44 = subborrow_u32(x41, x11, x25, &x43); + uint32_t x46; uint8_t x47 = subborrow_u32(x44, x13, x27, &x46); + uint32_t x49; uint8_t x50 = subborrow_u32(x47, x15, x29, &x49); + uint32_t x52; uint8_t x53 = subborrow_u32(x50, x14, x28, &x52); + uint32_t x54 = cmovznz_u32(x53, 0x0, 0xffffffff); + uint32_t x56; uint8_t x57 = addcarryx_u32(0x0, x31, (x54 & 0xffffffff), &x56); + uint32_t x59; uint8_t x60 = addcarryx_u32(x57, x34, (x54 & 0xffffffff), &x59); + uint32_t x62; uint8_t x63 = addcarryx_u32(x60, x37, (x54 & 0xffffffff), &x62); + uint32_t x65; uint8_t x66 = addcarryx_u32(x63, x40, 0x0, &x65); + uint32_t x68; uint8_t x69 = addcarryx_u32(x66, x43, 0x0, &x68); + uint32_t x71; uint8_t x72 = addcarryx_u32(x69, x46, 0x0, &x71); + uint32_t x74; uint8_t x75 = addcarryx_u32(x72, x49, ((uint8_t)x54 & 0x1), &x74); + uint32_t x77; addcarryx_u32(x75, x52, (x54 & 0xffffffff), &x77); + out[0] = x56; + out[1] = x59; + out[2] = x62; + out[3] = x65; + out[4] = x68; + out[5] = x71; + out[6] = x74; + out[7] = x77; +} + +// fe_op sets out = -in +static void fe_opp(uint32_t out[8], const uint32_t in1[8]) { + const uint32_t x12 = in1[7]; + const uint32_t x13 = in1[6]; + const uint32_t x11 = in1[5]; + const uint32_t x9 = in1[4]; + const uint32_t x7 = in1[3]; + const uint32_t x5 = in1[2]; + const uint32_t x3 = in1[1]; + const uint32_t x1 = in1[0]; + uint32_t x15; uint8_t x16 = subborrow_u32(0x0, 0x0, x1, &x15); + uint32_t x18; uint8_t x19 = subborrow_u32(x16, 0x0, x3, &x18); + uint32_t x21; uint8_t x22 = subborrow_u32(x19, 0x0, x5, &x21); + uint32_t x24; uint8_t x25 = subborrow_u32(x22, 0x0, x7, &x24); + uint32_t x27; uint8_t x28 = subborrow_u32(x25, 0x0, x9, &x27); + uint32_t x30; uint8_t x31 = subborrow_u32(x28, 0x0, x11, &x30); + uint32_t x33; uint8_t x34 = subborrow_u32(x31, 0x0, x13, &x33); + uint32_t x36; uint8_t x37 = subborrow_u32(x34, 0x0, x12, &x36); + uint32_t x38 = cmovznz_u32(x37, 0x0, 0xffffffff); + uint32_t x40; uint8_t x41 = addcarryx_u32(0x0, x15, (x38 & 0xffffffff), &x40); + uint32_t x43; uint8_t x44 = addcarryx_u32(x41, x18, (x38 & 0xffffffff), &x43); + uint32_t x46; uint8_t x47 = addcarryx_u32(x44, x21, (x38 & 0xffffffff), &x46); + uint32_t x49; uint8_t x50 = addcarryx_u32(x47, x24, 0x0, &x49); + uint32_t x52; uint8_t x53 = addcarryx_u32(x50, x27, 0x0, &x52); + uint32_t x55; uint8_t x56 = addcarryx_u32(x53, x30, 0x0, &x55); + uint32_t x58; uint8_t x59 = addcarryx_u32(x56, x33, ((uint8_t)x38 & 0x1), &x58); + uint32_t x61; addcarryx_u32(x59, x36, (x38 & 0xffffffff), &x61); + out[0] = x40; + out[1] = x43; + out[2] = x46; + out[3] = x49; + out[4] = x52; + out[5] = x55; + out[6] = x58; + out[7] = x61; +} + +#endif + +// utility functions, handwritten + +#define NBYTES 32 + +#if defined(BORINGSSL_NISTP256_64BIT) + +#define NLIMBS 4 +typedef uint64_t limb_t; +#define cmovznz_limb cmovznz_u64 +typedef uint64_t fe[NLIMBS]; +#else // 64BIT; else 32BIT + +#define NLIMBS 8 +typedef uint32_t limb_t; +#define cmovznz_limb cmovznz_u32 +typedef uint32_t fe[NLIMBS]; + +#endif // 64BIT + +static limb_t fe_nz(const limb_t in1[NLIMBS]) { + limb_t ret = 0; + for (int i = 0; i < NLIMBS; i++) { + ret |= in1[i]; + } + return ret; +} + +static void fe_copy(limb_t out[NLIMBS], const limb_t in1[NLIMBS]) { + for (int i = 0; i < NLIMBS; i++) { + out[i] = in1[i]; + } +} + +static void fe_cmovznz(limb_t out[NLIMBS], limb_t t, const limb_t z[NLIMBS], + const limb_t nz[NLIMBS]) { + for (int i = 0; i < NLIMBS; i++) { + out[i] = cmovznz_limb(t, z[i], nz[i]); + } +} + +static void fe_sqr(fe out, const fe in) { + fe_mul(out, in, in); +} + +static void fe_tobytes(uint8_t out[NBYTES], const fe in) { + for (int i = 0; i> (8*(i%sizeof(in[0])))); + } +} + +static void fe_frombytes(fe out, const uint8_t in[NBYTES]) { + for (int 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(fe x_out, fe y_out, fe z_out, + const uint8_t *p_scalar, const uint8_t *g_scalar, + const fe p_pre_comp[17][3]) { + // set nq to the point at infinity + fe nq[3] = {{0},{0},{0}}, ftmp, tmp[3]; + uint64_t bits; + uint8_t sign, digit; + + // 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 { + fe_copy(nq[0], tmp[0]); + fe_copy(nq[1], tmp[1]); + fe_copy(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); + fe_opp(ftmp, tmp[1]); // (X, -Y, Z) is the negative point. + fe_cmovznz(tmp[1], sign, 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 { + fe_copy(nq[0], tmp[0]); + fe_copy(nq[1], tmp[1]); + fe_copy(nq[2], tmp[2]); + skip = 0; + } + } + + if (i == 0) { + break; + } + --i; + } + fe_copy(x_out, nq[0]); + fe_copy(y_out, nq[1]); + fe_copy(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_out, + BIGNUM *y_out, + BN_CTX *ctx) { + fe x, y, z1, z2; + + if (EC_POINT_is_at_infinity(group, point)) { + OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); + return 0; + } + if (!BN_to_fe(x, &point->X) || + !BN_to_fe(y, &point->Y) || + !BN_to_fe(z1, &point->Z)) { + return 0; + } + + fe_inv(z2, z1); + fe_sqr(z1, z2); + + if (x_out != NULL) { + fe_mul(x, x, z1); + fe_from_montgomery(x); + if (!fe_to_BN(x_out, x)) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + return 0; + } + } + + if (y_out != NULL) { + fe_mul(z1, z1, z2); + fe_mul(y, y, z1); + fe_from_montgomery(y); + if (!fe_to_BN(y_out, y)) { + 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 *unused_ctx) { + fe p_pre_comp[17][3]; + fe x_out, y_out, z_out; + + 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_fe(p_pre_comp[1][0], &p->X) || + !BN_to_fe(p_pre_comp[1][1], &p->Y) || + !BN_to_fe(p_pre_comp[1][2], &p->Z)) { + return 0; + } + for (size_t j = 2; j <= 16; ++j) { + if (j & 1) { + point_add(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], + 0, + p_pre_comp[j - 1][0], p_pre_comp[j - 1][1], + p_pre_comp[j - 1][2]); + } else { + point_double(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 fe (*) [3])p_pre_comp); + + if (!fe_to_BN(&r->X, x_out) || + !fe_to_BN(&r->Y, y_out) || + !fe_to_BN(&r->Z, z_out)) { + OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); + return 0; + } + return 1; +} + +DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_nistp256_method) { + out->group_init = ec_GFp_mont_group_init; + out->group_finish = ec_GFp_mont_group_finish; + out->group_set_curve = ec_GFp_mont_group_set_curve; + out->point_get_affine_coordinates = + ec_GFp_nistp256_point_get_affine_coordinates; + out->mul = ec_GFp_nistp256_points_mul; +// The variable-time wNAF point multiplication uses fewer field operations than +// the constant-time implementation here, but the 64-bit field arithmetic in +// this file is much faster than the generic BIGNUM-based field arithmetic used +// by wNAF. For 32-bit, the wNAF code is overall ~60% faster on non-precomputed +// points, so we use it for public inputs. +#if defined(BORINGSSL_NISTP256_64BIT) + out->mul_public = ec_GFp_nistp256_points_mul; +#else + out->mul_public = ec_wNAF_mul; +#endif + out->field_mul = ec_GFp_mont_field_mul; + out->field_sqr = ec_GFp_mont_field_sqr; + out->field_encode = ec_GFp_mont_field_encode; + out->field_decode = ec_GFp_mont_field_decode; +}; + +#undef BORINGSSL_NISTP256_64BIT