| /* Copyright (c) 2015, Google Inc. |
| * |
| * Permission to use, copy, modify, and/or distribute this software for any |
| * purpose with or without fee is hereby granted, provided that the above |
| * copyright notice and this permission notice appear in all copies. |
| * |
| * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES |
| * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF |
| * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY |
| * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES |
| * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION |
| * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN |
| * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ |
| |
| /* A 64-bit implementation of the NIST P-256 elliptic curve point |
| * multiplication |
| * |
| * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c. |
| * Otherwise based on Emilia's P224 work, which was inspired by my curve25519 |
| * work which got its smarts from Daniel J. Bernstein's work on the same. */ |
| |
| #include <openssl/base.h> |
| |
| #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) |
| |
| #include <openssl/bn.h> |
| #include <openssl/ec.h> |
| #include <openssl/err.h> |
| #include <openssl/mem.h> |
| |
| #include <string.h> |
| |
| #include "internal.h" |
| #include "../internal.h" |
| |
| |
| typedef uint8_t u8; |
| typedef uint64_t u64; |
| typedef int64_t s64; |
| |
| /* The underlying field. P256 operates over GF(2^256-2^224+2^192+2^96-1). We |
| * can serialise an element of this field into 32 bytes. We call this an |
| * felem_bytearray. */ |
| typedef u8 felem_bytearray[32]; |
| |
| /* The representation of field elements. |
| * ------------------------------------ |
| * |
| * We represent field elements with either four 128-bit values, eight 128-bit |
| * values, or four 64-bit values. The field element represented is: |
| * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192 (mod p) |
| * or: |
| * v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512 (mod p) |
| * |
| * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits |
| * apart, but are 128-bits wide, the most significant bits of each limb overlap |
| * with the least significant bits of the next. |
| * |
| * A field element with four limbs is an 'felem'. One with eight limbs is a |
| * 'longfelem' |
| * |
| * A field element with four, 64-bit values is called a 'smallfelem'. Small |
| * values are used as intermediate values before multiplication. */ |
| |
| #define NLIMBS 4 |
| |
| typedef uint128_t limb; |
| typedef limb felem[NLIMBS]; |
| typedef limb longfelem[NLIMBS * 2]; |
| typedef u64 smallfelem[NLIMBS]; |
| |
| /* This is the value of the prime as four 64-bit words, little-endian. */ |
| static const u64 kPrime[4] = {0xfffffffffffffffful, 0xffffffff, 0, |
| 0xffffffff00000001ul}; |
| static const u64 bottom63bits = 0x7ffffffffffffffful; |
| |
| /* bin32_to_felem takes a little-endian byte array and converts it into felem |
| * form. This assumes that the CPU is little-endian. */ |
| static void bin32_to_felem(felem out, const u8 in[32]) { |
| out[0] = *((const u64 *)&in[0]); |
| out[1] = *((const u64 *)&in[8]); |
| out[2] = *((const u64 *)&in[16]); |
| out[3] = *((const u64 *)&in[24]); |
| } |
| |
| /* smallfelem_to_bin32 takes a smallfelem and serialises into a little endian, |
| * 32 byte array. This assumes that the CPU is little-endian. */ |
| static void smallfelem_to_bin32(u8 out[32], const smallfelem in) { |
| *((u64 *)&out[0]) = in[0]; |
| *((u64 *)&out[8]) = in[1]; |
| *((u64 *)&out[16]) = in[2]; |
| *((u64 *)&out[24]) = in[3]; |
| } |
| |
| /* To preserve endianness when using BN_bn2bin and BN_bin2bn. */ |
| static void flip_endian(u8 *out, const u8 *in, size_t len) { |
| size_t i; |
| for (i = 0; i < len; ++i) { |
| out[i] = in[len - 1 - i]; |
| } |
| } |
| |
| /* BN_to_felem converts an OpenSSL BIGNUM into an felem. */ |
| static int BN_to_felem(felem out, const BIGNUM *bn) { |
| if (BN_is_negative(bn)) { |
| OPENSSL_PUT_ERROR(EC, EC_R_BIGNUM_OUT_OF_RANGE); |
| return 0; |
| } |
| |
| felem_bytearray b_out; |
| /* BN_bn2bin eats leading zeroes */ |
| 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 u64 scalar) { |
| out[0] *= scalar; |
| out[1] *= scalar; |
| out[2] *= scalar; |
| out[3] *= scalar; |
| } |
| |
| /* longfelem_scalar sets out = out * scalar */ |
| static void longfelem_scalar(longfelem out, const u64 scalar) { |
| out[0] *= scalar; |
| out[1] *= scalar; |
| out[2] *= scalar; |
| out[3] *= scalar; |
| out[4] *= scalar; |
| out[5] *= scalar; |
| out[6] *= scalar; |
| out[7] *= scalar; |
| } |
| |
| #define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9) |
| #define two105 (((limb)1) << 105) |
| #define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9) |
| |
| /* zero105 is 0 mod p */ |
| static const felem zero105 = {two105m41m9, two105, two105m41p9, two105m41p9}; |
| |
| /* smallfelem_neg sets |out| to |-small| |
| * On exit: |
| * out[i] < out[i] + 2^105 */ |
| static void smallfelem_neg(felem out, const smallfelem small) { |
| /* In order to prevent underflow, we subtract from 0 mod p. */ |
| out[0] = zero105[0] - small[0]; |
| out[1] = zero105[1] - small[1]; |
| out[2] = zero105[2] - small[2]; |
| out[3] = zero105[3] - small[3]; |
| } |
| |
| /* felem_diff subtracts |in| from |out| |
| * On entry: |
| * in[i] < 2^104 |
| * On exit: |
| * out[i] < out[i] + 2^105. */ |
| static void felem_diff(felem out, const felem in) { |
| /* In order to prevent underflow, we add 0 mod p before subtracting. */ |
| out[0] += zero105[0]; |
| out[1] += zero105[1]; |
| out[2] += zero105[2]; |
| out[3] += zero105[3]; |
| |
| out[0] -= in[0]; |
| out[1] -= in[1]; |
| out[2] -= in[2]; |
| out[3] -= in[3]; |
| } |
| |
| #define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11) |
| #define two107 (((limb)1) << 107) |
| #define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11) |
| |
| /* zero107 is 0 mod p */ |
| static const felem zero107 = {two107m43m11, two107, two107m43p11, two107m43p11}; |
| |
| /* An alternative felem_diff for larger inputs |in| |
| * felem_diff_zero107 subtracts |in| from |out| |
| * On entry: |
| * in[i] < 2^106 |
| * On exit: |
| * out[i] < out[i] + 2^107. */ |
| static void felem_diff_zero107(felem out, const felem in) { |
| /* In order to prevent underflow, we add 0 mod p before subtracting. */ |
| out[0] += zero107[0]; |
| out[1] += zero107[1]; |
| out[2] += zero107[2]; |
| out[3] += zero107[3]; |
| |
| out[0] -= in[0]; |
| out[1] -= in[1]; |
| out[2] -= in[2]; |
| out[3] -= in[3]; |
| } |
| |
| /* longfelem_diff subtracts |in| from |out| |
| * On entry: |
| * in[i] < 7*2^67 |
| * On exit: |
| * out[i] < out[i] + 2^70 + 2^40. */ |
| static void longfelem_diff(longfelem out, const longfelem in) { |
| static const limb two70m8p6 = |
| (((limb)1) << 70) - (((limb)1) << 8) + (((limb)1) << 6); |
| static const limb two70p40 = (((limb)1) << 70) + (((limb)1) << 40); |
| static const limb two70 = (((limb)1) << 70); |
| static const limb two70m40m38p6 = (((limb)1) << 70) - (((limb)1) << 40) - |
| (((limb)1) << 38) + (((limb)1) << 6); |
| static const limb two70m6 = (((limb)1) << 70) - (((limb)1) << 6); |
| |
| /* add 0 mod p to avoid underflow */ |
| out[0] += two70m8p6; |
| out[1] += two70p40; |
| out[2] += two70; |
| out[3] += two70m40m38p6; |
| out[4] += two70m6; |
| out[5] += two70m6; |
| out[6] += two70m6; |
| out[7] += two70m6; |
| |
| /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */ |
| out[0] -= in[0]; |
| out[1] -= in[1]; |
| out[2] -= in[2]; |
| out[3] -= in[3]; |
| out[4] -= in[4]; |
| out[5] -= in[5]; |
| out[6] -= in[6]; |
| out[7] -= in[7]; |
| } |
| |
| #define two64m0 (((limb)1) << 64) - 1 |
| #define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1 |
| #define two64m46 (((limb)1) << 64) - (((limb)1) << 46) |
| #define two64m32 (((limb)1) << 64) - (((limb)1) << 32) |
| |
| /* zero110 is 0 mod p. */ |
| static const felem zero110 = {two64m0, two110p32m0, two64m46, two64m32}; |
| |
| /* felem_shrink converts an felem into a smallfelem. The result isn't quite |
| * minimal as the value may be greater than p. |
| * |
| * On entry: |
| * in[i] < 2^109 |
| * On exit: |
| * out[i] < 2^64. */ |
| static void felem_shrink(smallfelem out, const felem in) { |
| felem tmp; |
| u64 a, b, mask; |
| s64 high, low; |
| static const u64 kPrime3Test = 0x7fffffff00000001ul; /* 2^63 - 2^32 + 1 */ |
| |
| /* Carry 2->3 */ |
| tmp[3] = zero110[3] + in[3] + ((u64)(in[2] >> 64)); |
| /* tmp[3] < 2^110 */ |
| |
| tmp[2] = zero110[2] + (u64)in[2]; |
| tmp[0] = zero110[0] + in[0]; |
| tmp[1] = zero110[1] + in[1]; |
| /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */ |
| |
| /* We perform two partial reductions where we eliminate the high-word of |
| * tmp[3]. We don't update the other words till the end. */ |
| a = tmp[3] >> 64; /* a < 2^46 */ |
| tmp[3] = (u64)tmp[3]; |
| tmp[3] -= a; |
| tmp[3] += ((limb)a) << 32; |
| /* tmp[3] < 2^79 */ |
| |
| b = a; |
| a = tmp[3] >> 64; /* a < 2^15 */ |
| b += a; /* b < 2^46 + 2^15 < 2^47 */ |
| tmp[3] = (u64)tmp[3]; |
| tmp[3] -= a; |
| tmp[3] += ((limb)a) << 32; |
| /* tmp[3] < 2^64 + 2^47 */ |
| |
| /* This adjusts the other two words to complete the two partial |
| * reductions. */ |
| tmp[0] += b; |
| tmp[1] -= (((limb)b) << 32); |
| |
| /* In order to make space in tmp[3] for the carry from 2 -> 3, we |
| * conditionally subtract kPrime if tmp[3] is large enough. */ |
| high = tmp[3] >> 64; |
| /* As tmp[3] < 2^65, high is either 1 or 0 */ |
| high = ~(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] += ((u64)(tmp[0] >> 64)); |
| tmp[0] = (u64)tmp[0]; |
| tmp[2] += ((u64)(tmp[1] >> 64)); |
| tmp[1] = (u64)tmp[1]; |
| tmp[3] += ((u64)(tmp[2] >> 64)); |
| tmp[2] = (u64)tmp[2]; |
| /* tmp[i] < 2^64 */ |
| |
| out[0] = tmp[0]; |
| out[1] = tmp[1]; |
| out[2] = tmp[2]; |
| out[3] = tmp[3]; |
| } |
| |
| /* smallfelem_expand converts a smallfelem to an felem */ |
| static void smallfelem_expand(felem out, const smallfelem in) { |
| out[0] = in[0]; |
| out[1] = in[1]; |
| out[2] = in[2]; |
| out[3] = in[3]; |
| } |
| |
| /* smallfelem_square sets |out| = |small|^2 |
| * On entry: |
| * small[i] < 2^64 |
| * On exit: |
| * out[i] < 7 * 2^64 < 2^67 */ |
| static void smallfelem_square(longfelem out, const smallfelem small) { |
| limb a; |
| u64 high, low; |
| |
| a = ((uint128_t)small[0]) * small[0]; |
| low = a; |
| high = a >> 64; |
| out[0] = low; |
| out[1] = high; |
| |
| a = ((uint128_t)small[0]) * small[1]; |
| low = a; |
| high = a >> 64; |
| out[1] += low; |
| out[1] += low; |
| out[2] = high; |
| |
| a = ((uint128_t)small[0]) * small[2]; |
| low = a; |
| high = a >> 64; |
| out[2] += low; |
| out[2] *= 2; |
| out[3] = high; |
| |
| a = ((uint128_t)small[0]) * small[3]; |
| low = a; |
| high = a >> 64; |
| out[3] += low; |
| out[4] = high; |
| |
| a = ((uint128_t)small[1]) * small[2]; |
| low = a; |
| high = a >> 64; |
| out[3] += low; |
| out[3] *= 2; |
| out[4] += high; |
| |
| a = ((uint128_t)small[1]) * small[1]; |
| low = a; |
| high = a >> 64; |
| out[2] += low; |
| out[3] += high; |
| |
| a = ((uint128_t)small[1]) * small[3]; |
| low = a; |
| high = a >> 64; |
| out[4] += low; |
| out[4] *= 2; |
| out[5] = high; |
| |
| a = ((uint128_t)small[2]) * small[3]; |
| low = a; |
| high = a >> 64; |
| out[5] += low; |
| out[5] *= 2; |
| out[6] = high; |
| out[6] += high; |
| |
| a = ((uint128_t)small[2]) * small[2]; |
| low = a; |
| high = a >> 64; |
| out[4] += low; |
| out[5] += high; |
| |
| a = ((uint128_t)small[3]) * small[3]; |
| low = a; |
| high = a >> 64; |
| out[6] += low; |
| out[7] = high; |
| } |
| |
| /*felem_square sets |out| = |in|^2 |
| * On entry: |
| * in[i] < 2^109 |
| * On exit: |
| * out[i] < 7 * 2^64 < 2^67. */ |
| static void felem_square(longfelem out, const felem in) { |
| u64 small[4]; |
| felem_shrink(small, in); |
| smallfelem_square(out, small); |
| } |
| |
| /* smallfelem_mul sets |out| = |small1| * |small2| |
| * On entry: |
| * small1[i] < 2^64 |
| * small2[i] < 2^64 |
| * On exit: |
| * out[i] < 7 * 2^64 < 2^67. */ |
| static void smallfelem_mul(longfelem out, const smallfelem small1, |
| const smallfelem small2) { |
| limb a; |
| u64 high, low; |
| |
| a = ((uint128_t)small1[0]) * small2[0]; |
| low = a; |
| high = a >> 64; |
| out[0] = low; |
| out[1] = high; |
| |
| a = ((uint128_t)small1[0]) * small2[1]; |
| low = a; |
| high = a >> 64; |
| out[1] += low; |
| out[2] = high; |
| |
| a = ((uint128_t)small1[1]) * small2[0]; |
| low = a; |
| high = a >> 64; |
| out[1] += low; |
| out[2] += high; |
| |
| a = ((uint128_t)small1[0]) * small2[2]; |
| low = a; |
| high = a >> 64; |
| out[2] += low; |
| out[3] = high; |
| |
| a = ((uint128_t)small1[1]) * small2[1]; |
| low = a; |
| high = a >> 64; |
| out[2] += low; |
| out[3] += high; |
| |
| a = ((uint128_t)small1[2]) * small2[0]; |
| low = a; |
| high = a >> 64; |
| out[2] += low; |
| out[3] += high; |
| |
| a = ((uint128_t)small1[0]) * small2[3]; |
| low = a; |
| high = a >> 64; |
| out[3] += low; |
| out[4] = high; |
| |
| a = ((uint128_t)small1[1]) * small2[2]; |
| low = a; |
| high = a >> 64; |
| out[3] += low; |
| out[4] += high; |
| |
| a = ((uint128_t)small1[2]) * small2[1]; |
| low = a; |
| high = a >> 64; |
| out[3] += low; |
| out[4] += high; |
| |
| a = ((uint128_t)small1[3]) * small2[0]; |
| low = a; |
| high = a >> 64; |
| out[3] += low; |
| out[4] += high; |
| |
| a = ((uint128_t)small1[1]) * small2[3]; |
| low = a; |
| high = a >> 64; |
| out[4] += low; |
| out[5] = high; |
| |
| a = ((uint128_t)small1[2]) * small2[2]; |
| low = a; |
| high = a >> 64; |
| out[4] += low; |
| out[5] += high; |
| |
| a = ((uint128_t)small1[3]) * small2[1]; |
| low = a; |
| high = a >> 64; |
| out[4] += low; |
| out[5] += high; |
| |
| a = ((uint128_t)small1[2]) * small2[3]; |
| low = a; |
| high = a >> 64; |
| out[5] += low; |
| out[6] = high; |
| |
| a = ((uint128_t)small1[3]) * small2[2]; |
| low = a; |
| high = a >> 64; |
| out[5] += low; |
| out[6] += high; |
| |
| a = ((uint128_t)small1[3]) * small2[3]; |
| low = a; |
| high = a >> 64; |
| out[6] += low; |
| out[7] = high; |
| } |
| |
| /* felem_mul sets |out| = |in1| * |in2| |
| * On entry: |
| * in1[i] < 2^109 |
| * in2[i] < 2^109 |
| * On exit: |
| * out[i] < 7 * 2^64 < 2^67 */ |
| static void felem_mul(longfelem out, const felem in1, const felem in2) { |
| smallfelem small1, small2; |
| felem_shrink(small1, in1); |
| felem_shrink(small2, in2); |
| smallfelem_mul(out, small1, small2); |
| } |
| |
| /* felem_small_mul sets |out| = |small1| * |in2| |
| * On entry: |
| * small1[i] < 2^64 |
| * in2[i] < 2^109 |
| * On exit: |
| * out[i] < 7 * 2^64 < 2^67 */ |
| static void felem_small_mul(longfelem out, const smallfelem small1, |
| const felem in2) { |
| smallfelem small2; |
| felem_shrink(small2, in2); |
| smallfelem_mul(out, small1, small2); |
| } |
| |
| #define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4) |
| #define two100 (((limb)1) << 100) |
| #define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4) |
| |
| /* zero100 is 0 mod p */ |
| static const felem zero100 = {two100m36m4, two100, two100m36p4, two100m36p4}; |
| |
| /* Internal function for the different flavours of felem_reduce. |
| * felem_reduce_ reduces the higher coefficients in[4]-in[7]. |
| * On entry: |
| * out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7] |
| * out[1] >= in[7] + 2^32*in[4] |
| * out[2] >= in[5] + 2^32*in[5] |
| * out[3] >= in[4] + 2^32*in[5] + 2^32*in[6] |
| * On exit: |
| * out[0] <= out[0] + in[4] + 2^32*in[5] |
| * out[1] <= out[1] + in[5] + 2^33*in[6] |
| * out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7] |
| * out[3] <= out[3] + 2^32*in[4] + 3*in[7] */ |
| static void felem_reduce_(felem out, const longfelem in) { |
| int128_t c; |
| /* combine common terms from below */ |
| c = in[4] + (in[5] << 32); |
| out[0] += c; |
| out[3] -= c; |
| |
| c = in[5] - in[7]; |
| out[1] += c; |
| out[2] -= c; |
| |
| /* the remaining terms */ |
| /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */ |
| out[1] -= (in[4] << 32); |
| out[3] += (in[4] << 32); |
| |
| /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */ |
| out[2] -= (in[5] << 32); |
| |
| /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */ |
| out[0] -= in[6]; |
| out[0] -= (in[6] << 32); |
| out[1] += (in[6] << 33); |
| out[2] += (in[6] * 2); |
| out[3] -= (in[6] << 32); |
| |
| /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */ |
| out[0] -= in[7]; |
| out[0] -= (in[7] << 32); |
| out[2] += (in[7] << 33); |
| out[3] += (in[7] * 3); |
| } |
| |
| /* felem_reduce converts a longfelem into an felem. |
| * To be called directly after felem_square or felem_mul. |
| * On entry: |
| * in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64 |
| * in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64 |
| * On exit: |
| * out[i] < 2^101 */ |
| static void felem_reduce(felem out, const longfelem in) { |
| out[0] = zero100[0] + in[0]; |
| out[1] = zero100[1] + in[1]; |
| out[2] = zero100[2] + in[2]; |
| out[3] = zero100[3] + in[3]; |
| |
| felem_reduce_(out, in); |
| |
| /* out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0 |
| * out[1] > 2^100 - 2^64 - 7*2^96 > 0 |
| * out[2] > 2^100 - 2^36 + 2^4 - 5*2^64 - 5*2^96 > 0 |
| * out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96 - 3*2^96 > 0 |
| * |
| * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101 |
| * out[1] < 2^100 + 3*2^64 + 5*2^64 + 3*2^97 < 2^101 |
| * out[2] < 2^100 + 5*2^64 + 2^64 + 3*2^65 + 2^97 < 2^101 |
| * out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 < 2^101 */ |
| } |
| |
| /* felem_reduce_zero105 converts a larger longfelem into an felem. |
| * On entry: |
| * in[0] < 2^71 |
| * On exit: |
| * out[i] < 2^106 */ |
| static void felem_reduce_zero105(felem out, const longfelem in) { |
| out[0] = zero105[0] + in[0]; |
| out[1] = zero105[1] + in[1]; |
| out[2] = zero105[2] + in[2]; |
| out[3] = zero105[3] + in[3]; |
| |
| felem_reduce_(out, in); |
| |
| /* out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0 |
| * out[1] > 2^105 - 2^71 - 2^103 > 0 |
| * out[2] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 > 0 |
| * out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 - 2^103 > 0 |
| * |
| * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 |
| * out[1] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 |
| * out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 < 2^106 |
| * out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106 */ |
| } |
| |
| /* subtract_u64 sets *result = *result - v and *carry to one if the |
| * subtraction underflowed. */ |
| static void subtract_u64(u64 *result, u64 *carry, u64 v) { |
| uint128_t r = *result; |
| r -= v; |
| *carry = (r >> 64) & 1; |
| *result = (u64)r; |
| } |
| |
| /* felem_contract converts |in| to its unique, minimal representation. On |
| * entry: in[i] < 2^109. */ |
| static void felem_contract(smallfelem out, const felem in) { |
| u64 all_equal_so_far = 0, result = 0; |
| |
| felem_shrink(out, in); |
| /* small is minimal except that the value might be > p */ |
| |
| all_equal_so_far--; |
| /* We are doing a constant time test if out >= kPrime. We need to compare |
| * each u64, from most-significant to least significant. For each one, if |
| * all words so far have been equal (m is all ones) then a non-equal |
| * result is the answer. Otherwise we continue. */ |
| size_t i; |
| for (i = 3; i < 4; i--) { |
| u64 equal; |
| uint128_t a = ((uint128_t)kPrime[i]) - out[i]; |
| /* if out[i] > kPrime[i] then a will underflow and the high 64-bits |
| * will all be set. */ |
| result |= all_equal_so_far & ((u64)(a >> 64)); |
| |
| /* if kPrime[i] == out[i] then |equal| will be all zeros and the |
| * decrement will make it all ones. */ |
| equal = kPrime[i] ^ out[i]; |
| equal--; |
| equal &= equal << 32; |
| equal &= equal << 16; |
| equal &= equal << 8; |
| equal &= equal << 4; |
| equal &= equal << 2; |
| equal &= equal << 1; |
| equal = ((s64)equal) >> 63; |
| |
| all_equal_so_far &= equal; |
| } |
| |
| /* if all_equal_so_far is still all ones then the two values are equal |
| * and so out >= kPrime is true. */ |
| result |= all_equal_so_far; |
| |
| /* if out >= kPrime then we subtract kPrime. */ |
| u64 carry; |
| subtract_u64(&out[0], &carry, result & kPrime[0]); |
| subtract_u64(&out[1], &carry, carry); |
| subtract_u64(&out[2], &carry, carry); |
| subtract_u64(&out[3], &carry, carry); |
| |
| subtract_u64(&out[1], &carry, result & kPrime[1]); |
| subtract_u64(&out[2], &carry, carry); |
| subtract_u64(&out[3], &carry, carry); |
| |
| subtract_u64(&out[2], &carry, result & kPrime[2]); |
| subtract_u64(&out[3], &carry, carry); |
| |
| subtract_u64(&out[3], &carry, result & kPrime[3]); |
| } |
| |
| /* felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0 |
| * otherwise. |
| * On entry: |
| * small[i] < 2^64 */ |
| static limb smallfelem_is_zero(const smallfelem small) { |
| limb result; |
| u64 is_p; |
| |
| u64 is_zero = small[0] | small[1] | small[2] | small[3]; |
| is_zero--; |
| is_zero &= is_zero << 32; |
| is_zero &= is_zero << 16; |
| is_zero &= is_zero << 8; |
| is_zero &= is_zero << 4; |
| is_zero &= is_zero << 2; |
| is_zero &= is_zero << 1; |
| is_zero = ((s64)is_zero) >> 63; |
| |
| is_p = (small[0] ^ kPrime[0]) | (small[1] ^ kPrime[1]) | |
| (small[2] ^ kPrime[2]) | (small[3] ^ kPrime[3]); |
| is_p--; |
| is_p &= is_p << 32; |
| is_p &= is_p << 16; |
| is_p &= is_p << 8; |
| is_p &= is_p << 4; |
| is_p &= is_p << 2; |
| is_p &= is_p << 1; |
| is_p = ((s64)is_p) >> 63; |
| |
| is_zero |= is_p; |
| |
| result = is_zero; |
| result |= ((limb)is_zero) << 64; |
| return result; |
| } |
| |
| /* 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; |
| size_t i; |
| |
| felem_square(tmp, in); |
| felem_reduce(ftmp, tmp); /* 2^1 */ |
| felem_mul(tmp, in, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^2 - 2^0 */ |
| felem_assign(e2, ftmp); |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^3 - 2^1 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^4 - 2^2 */ |
| felem_mul(tmp, ftmp, e2); |
| felem_reduce(ftmp, tmp); /* 2^4 - 2^0 */ |
| felem_assign(e4, ftmp); |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^5 - 2^1 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^6 - 2^2 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^7 - 2^3 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); /* 2^8 - 2^4 */ |
| felem_mul(tmp, ftmp, e4); |
| felem_reduce(ftmp, tmp); /* 2^8 - 2^0 */ |
| felem_assign(e8, ftmp); |
| for (i = 0; i < 8; i++) { |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); |
| } /* 2^16 - 2^8 */ |
| felem_mul(tmp, ftmp, e8); |
| felem_reduce(ftmp, tmp); /* 2^16 - 2^0 */ |
| felem_assign(e16, ftmp); |
| for (i = 0; i < 16; i++) { |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); |
| } /* 2^32 - 2^16 */ |
| felem_mul(tmp, ftmp, e16); |
| felem_reduce(ftmp, tmp); /* 2^32 - 2^0 */ |
| felem_assign(e32, ftmp); |
| for (i = 0; i < 32; i++) { |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); |
| } /* 2^64 - 2^32 */ |
| felem_assign(e64, ftmp); |
| felem_mul(tmp, ftmp, in); |
| felem_reduce(ftmp, tmp); /* 2^64 - 2^32 + 2^0 */ |
| for (i = 0; i < 192; i++) { |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); |
| } /* 2^256 - 2^224 + 2^192 */ |
| |
| felem_mul(tmp, e64, e32); |
| felem_reduce(ftmp2, tmp); /* 2^64 - 2^0 */ |
| for (i = 0; i < 16; i++) { |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); |
| } /* 2^80 - 2^16 */ |
| felem_mul(tmp, ftmp2, e16); |
| felem_reduce(ftmp2, tmp); /* 2^80 - 2^0 */ |
| for (i = 0; i < 8; i++) { |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); |
| } /* 2^88 - 2^8 */ |
| felem_mul(tmp, ftmp2, e8); |
| felem_reduce(ftmp2, tmp); /* 2^88 - 2^0 */ |
| for (i = 0; i < 4; i++) { |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); |
| } /* 2^92 - 2^4 */ |
| felem_mul(tmp, ftmp2, e4); |
| felem_reduce(ftmp2, tmp); /* 2^92 - 2^0 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); /* 2^93 - 2^1 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); /* 2^94 - 2^2 */ |
| felem_mul(tmp, ftmp2, e2); |
| felem_reduce(ftmp2, tmp); /* 2^94 - 2^0 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); /* 2^95 - 2^1 */ |
| felem_square(tmp, ftmp2); |
| felem_reduce(ftmp2, tmp); /* 2^96 - 2^2 */ |
| felem_mul(tmp, ftmp2, in); |
| felem_reduce(ftmp2, tmp); /* 2^96 - 3 */ |
| |
| felem_mul(tmp, ftmp2, ftmp); |
| felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */ |
| } |
| |
| /* Group operations |
| * ---------------- |
| * |
| * Building on top of the field operations we have the operations on the |
| * elliptic curve group itself. Points on the curve are represented in Jacobian |
| * coordinates. */ |
| |
| /* point_double calculates 2*(x_in, y_in, z_in) |
| * |
| * The method is taken from: |
| * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b |
| * |
| * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed. |
| * while x_out == y_in is not (maybe this works, but it's not tested). */ |
| static void point_double(felem x_out, felem y_out, felem z_out, |
| const felem x_in, const felem y_in, const felem z_in) { |
| longfelem tmp, tmp2; |
| felem delta, gamma, beta, alpha, ftmp, ftmp2; |
| smallfelem small1, small2; |
| |
| felem_assign(ftmp, x_in); |
| /* ftmp[i] < 2^106 */ |
| felem_assign(ftmp2, x_in); |
| /* ftmp2[i] < 2^106 */ |
| |
| /* delta = z^2 */ |
| felem_square(tmp, z_in); |
| felem_reduce(delta, tmp); |
| /* delta[i] < 2^101 */ |
| |
| /* gamma = y^2 */ |
| felem_square(tmp, y_in); |
| felem_reduce(gamma, tmp); |
| /* gamma[i] < 2^101 */ |
| felem_shrink(small1, gamma); |
| |
| /* beta = x*gamma */ |
| felem_small_mul(tmp, small1, x_in); |
| felem_reduce(beta, tmp); |
| /* beta[i] < 2^101 */ |
| |
| /* alpha = 3*(x-delta)*(x+delta) */ |
| felem_diff(ftmp, delta); |
| /* ftmp[i] < 2^105 + 2^106 < 2^107 */ |
| felem_sum(ftmp2, delta); |
| /* ftmp2[i] < 2^105 + 2^106 < 2^107 */ |
| felem_scalar(ftmp2, 3); |
| /* ftmp2[i] < 3 * 2^107 < 2^109 */ |
| felem_mul(tmp, ftmp, ftmp2); |
| felem_reduce(alpha, tmp); |
| /* alpha[i] < 2^101 */ |
| felem_shrink(small2, alpha); |
| |
| /* x' = alpha^2 - 8*beta */ |
| smallfelem_square(tmp, small2); |
| felem_reduce(x_out, tmp); |
| felem_assign(ftmp, beta); |
| felem_scalar(ftmp, 8); |
| /* ftmp[i] < 8 * 2^101 = 2^104 */ |
| felem_diff(x_out, ftmp); |
| /* x_out[i] < 2^105 + 2^101 < 2^106 */ |
| |
| /* z' = (y + z)^2 - gamma - delta */ |
| felem_sum(delta, gamma); |
| /* delta[i] < 2^101 + 2^101 = 2^102 */ |
| felem_assign(ftmp, y_in); |
| felem_sum(ftmp, z_in); |
| /* ftmp[i] < 2^106 + 2^106 = 2^107 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(z_out, tmp); |
| felem_diff(z_out, delta); |
| /* z_out[i] < 2^105 + 2^101 < 2^106 */ |
| |
| /* y' = alpha*(4*beta - x') - 8*gamma^2 */ |
| felem_scalar(beta, 4); |
| /* beta[i] < 4 * 2^101 = 2^103 */ |
| felem_diff_zero107(beta, x_out); |
| /* beta[i] < 2^107 + 2^103 < 2^108 */ |
| felem_small_mul(tmp, small2, beta); |
| /* tmp[i] < 7 * 2^64 < 2^67 */ |
| smallfelem_square(tmp2, small1); |
| /* tmp2[i] < 7 * 2^64 */ |
| longfelem_scalar(tmp2, 8); |
| /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */ |
| longfelem_diff(tmp, tmp2); |
| /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */ |
| felem_reduce_zero105(y_out, tmp); |
| /* y_out[i] < 2^106 */ |
| } |
| |
| /* point_double_small is the same as point_double, except that it operates on |
| * smallfelems. */ |
| static void point_double_small(smallfelem x_out, smallfelem y_out, |
| smallfelem z_out, const smallfelem x_in, |
| const smallfelem y_in, const smallfelem z_in) { |
| felem felem_x_out, felem_y_out, felem_z_out; |
| felem felem_x_in, felem_y_in, felem_z_in; |
| |
| smallfelem_expand(felem_x_in, x_in); |
| smallfelem_expand(felem_y_in, y_in); |
| smallfelem_expand(felem_z_in, z_in); |
| point_double(felem_x_out, felem_y_out, felem_z_out, felem_x_in, felem_y_in, |
| felem_z_in); |
| felem_shrink(x_out, felem_x_out); |
| felem_shrink(y_out, felem_y_out); |
| felem_shrink(z_out, felem_z_out); |
| } |
| |
| /* copy_conditional copies in to out iff mask is all ones. */ |
| static void copy_conditional(felem out, const felem in, limb mask) { |
| size_t i; |
| for (i = 0; i < NLIMBS; ++i) { |
| const limb tmp = mask & (in[i] ^ out[i]); |
| out[i] ^= tmp; |
| } |
| } |
| |
| /* copy_small_conditional copies in to out iff mask is all ones. */ |
| static void copy_small_conditional(felem out, const smallfelem in, limb mask) { |
| size_t i; |
| const u64 mask64 = mask; |
| for (i = 0; i < NLIMBS; ++i) { |
| out[i] = ((limb)(in[i] & mask64)) | (out[i] & ~mask); |
| } |
| } |
| |
| /* point_add calcuates (x1, y1, z1) + (x2, y2, z2) |
| * |
| * The method is taken from: |
| * http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl, |
| * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity). |
| * |
| * This function includes a branch for checking whether the two input points |
| * are equal, (while not equal to the point at infinity). This case never |
| * happens during single point multiplication, so there is no timing leak for |
| * ECDH or ECDSA signing. */ |
| static void point_add(felem x3, felem y3, felem z3, const felem x1, |
| const felem y1, const felem z1, const int mixed, |
| const smallfelem x2, const smallfelem y2, |
| const smallfelem z2) { |
| felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out; |
| longfelem tmp, tmp2; |
| smallfelem small1, small2, small3, small4, small5; |
| limb x_equal, y_equal, z1_is_zero, z2_is_zero; |
| |
| felem_shrink(small3, z1); |
| |
| z1_is_zero = smallfelem_is_zero(small3); |
| z2_is_zero = smallfelem_is_zero(z2); |
| |
| /* ftmp = z1z1 = z1**2 */ |
| smallfelem_square(tmp, small3); |
| felem_reduce(ftmp, tmp); |
| /* ftmp[i] < 2^101 */ |
| felem_shrink(small1, ftmp); |
| |
| if (!mixed) { |
| /* ftmp2 = z2z2 = z2**2 */ |
| smallfelem_square(tmp, z2); |
| felem_reduce(ftmp2, tmp); |
| /* ftmp2[i] < 2^101 */ |
| felem_shrink(small2, ftmp2); |
| |
| felem_shrink(small5, x1); |
| |
| /* u1 = ftmp3 = x1*z2z2 */ |
| smallfelem_mul(tmp, small5, small2); |
| felem_reduce(ftmp3, tmp); |
| /* ftmp3[i] < 2^101 */ |
| |
| /* ftmp5 = z1 + z2 */ |
| felem_assign(ftmp5, z1); |
| felem_small_sum(ftmp5, z2); |
| /* ftmp5[i] < 2^107 */ |
| |
| /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */ |
| felem_square(tmp, ftmp5); |
| felem_reduce(ftmp5, tmp); |
| /* ftmp2 = z2z2 + z1z1 */ |
| felem_sum(ftmp2, ftmp); |
| /* ftmp2[i] < 2^101 + 2^101 = 2^102 */ |
| felem_diff(ftmp5, ftmp2); |
| /* ftmp5[i] < 2^105 + 2^101 < 2^106 */ |
| |
| /* ftmp2 = z2 * z2z2 */ |
| smallfelem_mul(tmp, small2, z2); |
| felem_reduce(ftmp2, tmp); |
| |
| /* s1 = ftmp2 = y1 * z2**3 */ |
| felem_mul(tmp, y1, ftmp2); |
| felem_reduce(ftmp6, tmp); |
| /* ftmp6[i] < 2^101 */ |
| } else { |
| /* We'll assume z2 = 1 (special case z2 = 0 is handled later). */ |
| |
| /* u1 = ftmp3 = x1*z2z2 */ |
| felem_assign(ftmp3, x1); |
| /* ftmp3[i] < 2^106 */ |
| |
| /* ftmp5 = 2z1z2 */ |
| felem_assign(ftmp5, z1); |
| felem_scalar(ftmp5, 2); |
| /* ftmp5[i] < 2*2^106 = 2^107 */ |
| |
| /* s1 = ftmp2 = y1 * z2**3 */ |
| felem_assign(ftmp6, y1); |
| /* ftmp6[i] < 2^106 */ |
| } |
| |
| /* u2 = x2*z1z1 */ |
| smallfelem_mul(tmp, x2, small1); |
| felem_reduce(ftmp4, tmp); |
| |
| /* h = ftmp4 = u2 - u1 */ |
| felem_diff_zero107(ftmp4, ftmp3); |
| /* ftmp4[i] < 2^107 + 2^101 < 2^108 */ |
| felem_shrink(small4, ftmp4); |
| |
| x_equal = smallfelem_is_zero(small4); |
| |
| /* z_out = ftmp5 * h */ |
| felem_small_mul(tmp, small4, ftmp5); |
| felem_reduce(z_out, tmp); |
| /* z_out[i] < 2^101 */ |
| |
| /* ftmp = z1 * z1z1 */ |
| smallfelem_mul(tmp, small1, small3); |
| felem_reduce(ftmp, tmp); |
| |
| /* s2 = tmp = y2 * z1**3 */ |
| felem_small_mul(tmp, y2, ftmp); |
| felem_reduce(ftmp5, tmp); |
| |
| /* r = ftmp5 = (s2 - s1)*2 */ |
| felem_diff_zero107(ftmp5, ftmp6); |
| /* ftmp5[i] < 2^107 + 2^107 = 2^108 */ |
| felem_scalar(ftmp5, 2); |
| /* ftmp5[i] < 2^109 */ |
| felem_shrink(small1, ftmp5); |
| y_equal = smallfelem_is_zero(small1); |
| |
| if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) { |
| point_double(x3, y3, z3, x1, y1, z1); |
| return; |
| } |
| |
| /* I = ftmp = (2h)**2 */ |
| felem_assign(ftmp, ftmp4); |
| felem_scalar(ftmp, 2); |
| /* ftmp[i] < 2*2^108 = 2^109 */ |
| felem_square(tmp, ftmp); |
| felem_reduce(ftmp, tmp); |
| |
| /* J = ftmp2 = h * I */ |
| felem_mul(tmp, ftmp4, ftmp); |
| felem_reduce(ftmp2, tmp); |
| |
| /* V = ftmp4 = U1 * I */ |
| felem_mul(tmp, ftmp3, ftmp); |
| felem_reduce(ftmp4, tmp); |
| |
| /* x_out = r**2 - J - 2V */ |
| smallfelem_square(tmp, small1); |
| felem_reduce(x_out, tmp); |
| felem_assign(ftmp3, ftmp4); |
| felem_scalar(ftmp4, 2); |
| felem_sum(ftmp4, ftmp2); |
| /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */ |
| felem_diff(x_out, ftmp4); |
| /* x_out[i] < 2^105 + 2^101 */ |
| |
| /* y_out = r(V-x_out) - 2 * s1 * J */ |
| felem_diff_zero107(ftmp3, x_out); |
| /* ftmp3[i] < 2^107 + 2^101 < 2^108 */ |
| felem_small_mul(tmp, small1, ftmp3); |
| felem_mul(tmp2, ftmp6, ftmp2); |
| longfelem_scalar(tmp2, 2); |
| /* tmp2[i] < 2*2^67 = 2^68 */ |
| longfelem_diff(tmp, tmp2); |
| /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */ |
| felem_reduce_zero105(y_out, tmp); |
| /* y_out[i] < 2^106 */ |
| |
| copy_small_conditional(x_out, x2, z1_is_zero); |
| copy_conditional(x_out, x1, z2_is_zero); |
| copy_small_conditional(y_out, y2, z1_is_zero); |
| copy_conditional(y_out, y1, z2_is_zero); |
| copy_small_conditional(z_out, z2, z1_is_zero); |
| copy_conditional(z_out, z1, z2_is_zero); |
| felem_assign(x3, x_out); |
| felem_assign(y3, y_out); |
| felem_assign(z3, z_out); |
| } |
| |
| /* point_add_small is the same as point_add, except that it operates on |
| * smallfelems. */ |
| static void point_add_small(smallfelem x3, smallfelem y3, smallfelem z3, |
| smallfelem x1, smallfelem y1, smallfelem z1, |
| smallfelem x2, smallfelem y2, smallfelem z2) { |
| felem felem_x3, felem_y3, felem_z3; |
| felem felem_x1, felem_y1, felem_z1; |
| smallfelem_expand(felem_x1, x1); |
| smallfelem_expand(felem_y1, y1); |
| smallfelem_expand(felem_z1, z1); |
| point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0, x2, |
| y2, z2); |
| felem_shrink(x3, felem_x3); |
| felem_shrink(y3, felem_y3); |
| felem_shrink(z3, felem_z3); |
| } |
| |
| /* Base point pre computation |
| * -------------------------- |
| * |
| * Two different sorts of precomputed tables are used in the following code. |
| * Each contain various points on the curve, where each point is three field |
| * elements (x, y, z). |
| * |
| * For the base point table, z is usually 1 (0 for the point at infinity). |
| * This table has 2 * 16 elements, starting with the following: |
| * index | bits | point |
| * ------+---------+------------------------------ |
| * 0 | 0 0 0 0 | 0G |
| * 1 | 0 0 0 1 | 1G |
| * 2 | 0 0 1 0 | 2^64G |
| * 3 | 0 0 1 1 | (2^64 + 1)G |
| * 4 | 0 1 0 0 | 2^128G |
| * 5 | 0 1 0 1 | (2^128 + 1)G |
| * 6 | 0 1 1 0 | (2^128 + 2^64)G |
| * 7 | 0 1 1 1 | (2^128 + 2^64 + 1)G |
| * 8 | 1 0 0 0 | 2^192G |
| * 9 | 1 0 0 1 | (2^192 + 1)G |
| * 10 | 1 0 1 0 | (2^192 + 2^64)G |
| * 11 | 1 0 1 1 | (2^192 + 2^64 + 1)G |
| * 12 | 1 1 0 0 | (2^192 + 2^128)G |
| * 13 | 1 1 0 1 | (2^192 + 2^128 + 1)G |
| * 14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G |
| * 15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G |
| * followed by a copy of this with each element multiplied by 2^32. |
| * |
| * The reason for this is so that we can clock bits into four different |
| * locations when doing simple scalar multiplies against the base point, |
| * and then another four locations using the second 16 elements. |
| * |
| * Tables for other points have table[i] = iG for i in 0 .. 16. */ |
| |
| /* 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 u64 idx, size_t size, |
| const smallfelem pre_comp[/*size*/][3], |
| smallfelem out[3]) { |
| u64 *outlimbs = &out[0][0]; |
| memset(outlimbs, 0, 3 * sizeof(smallfelem)); |
| |
| size_t i; |
| for (i = 0; i < size; i++) { |
| const u64 *inlimbs = (const u64 *)&pre_comp[i][0][0]; |
| u64 mask = i ^ idx; |
| mask |= mask >> 4; |
| mask |= mask >> 2; |
| mask |= mask >> 1; |
| mask &= 1; |
| mask--; |
| size_t j; |
| for (j = 0; j < NLIMBS * 3; j++) { |
| outlimbs[j] |= inlimbs[j] & mask; |
| } |
| } |
| } |
| |
| /* get_bit returns the |i|th bit in |in| */ |
| static char get_bit(const felem_bytearray in, int i) { |
| if (i < 0 || i >= 256) { |
| return 0; |
| } |
| return (in[i >> 3] >> (i & 7)) & 1; |
| } |
| |
| /* Interleaved point multiplication using precomputed point multiples: The |
| * small point multiples 0*P, 1*P, ..., 17*P are in pre_comp[], the scalars |
| * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the |
| * generator, using certain (large) precomputed multiples in g_pre_comp. |
| * Output point (X, Y, Z) is stored in x_out, y_out, z_out. */ |
| static void batch_mul(felem x_out, felem y_out, felem z_out, |
| const felem_bytearray scalars[], |
| const size_t num_points, const u8 *g_scalar, |
| const smallfelem pre_comp[][17][3]) { |
| felem nq[3], ftmp; |
| smallfelem tmp[3]; |
| u64 bits; |
| u8 sign, digit; |
| |
| /* set nq to the point at infinity */ |
| memset(nq, 0, 3 * sizeof(felem)); |
| |
| /* Loop over all scalars msb-to-lsb, interleaving additions of multiples |
| * of the generator (two in each of the last 32 rounds) and additions of |
| * other points multiples (every 5th round). */ |
| |
| int skip = 1; /* save two point operations in the first round */ |
| size_t i = num_points != 0 ? 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 (num_points != 0 && i % 5 == 0) { |
| /* loop over all scalars */ |
| size_t num; |
| for (num = 0; num < num_points; ++num) { |
| bits = get_bit(scalars[num], i + 4) << 5; |
| bits |= get_bit(scalars[num], i + 3) << 4; |
| bits |= get_bit(scalars[num], i + 2) << 3; |
| bits |= get_bit(scalars[num], i + 1) << 2; |
| bits |= get_bit(scalars[num], i) << 1; |
| bits |= get_bit(scalars[num], i - 1); |
| ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); |
| |
| /* select the point to add or subtract, in constant time. */ |
| select_point(digit, 17, pre_comp[num], tmp); |
| smallfelem_neg(ftmp, tmp[1]); /* (X, -Y, Z) is the negative |
| * point */ |
| copy_small_conditional(ftmp, tmp[1], (((limb)sign) - 1)); |
| felem_contract(tmp[1], ftmp); |
| |
| if (!skip) { |
| point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 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 BIGNUM *g_scalar, |
| const EC_POINT *p_, |
| const BIGNUM *p_scalar_, |
| BN_CTX *ctx) { |
| /* TODO: This function used to take |points| and |scalars| as arrays of |
| * |num| elements. The code below should be simplified to work in terms of |p| |
| * and |p_scalar|. */ |
| size_t num = p_ != NULL ? 1 : 0; |
| const EC_POINT **points = p_ != NULL ? &p_ : NULL; |
| BIGNUM const *const *scalars = p_ != NULL ? &p_scalar_ : NULL; |
| |
| int ret = 0; |
| BN_CTX *new_ctx = NULL; |
| BIGNUM *x, *y, *z, *tmp_scalar; |
| felem_bytearray g_secret; |
| felem_bytearray *secrets = NULL; |
| smallfelem(*pre_comp)[17][3] = NULL; |
| felem_bytearray tmp; |
| size_t num_points = num; |
| smallfelem x_in, y_in, z_in; |
| felem x_out, y_out, z_out; |
| const EC_POINT *p = NULL; |
| const BIGNUM *p_scalar = NULL; |
| |
| if (ctx == NULL) { |
| ctx = new_ctx = BN_CTX_new(); |
| if (ctx == NULL) { |
| return 0; |
| } |
| } |
| |
| BN_CTX_start(ctx); |
| if ((x = BN_CTX_get(ctx)) == NULL || |
| (y = BN_CTX_get(ctx)) == NULL || |
| (z = BN_CTX_get(ctx)) == NULL || |
| (tmp_scalar = BN_CTX_get(ctx)) == NULL) { |
| goto err; |
| } |
| |
| if (num_points > 0) { |
| secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray)); |
| pre_comp = OPENSSL_malloc(num_points * sizeof(smallfelem[17][3])); |
| if (secrets == NULL || pre_comp == NULL) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); |
| goto err; |
| } |
| |
| /* we treat NULL scalars as 0, and NULL points as points at infinity, |
| * i.e., they contribute nothing to the linear combination. */ |
| memset(secrets, 0, num_points * sizeof(felem_bytearray)); |
| memset(pre_comp, 0, num_points * 17 * 3 * sizeof(smallfelem)); |
| size_t i; |
| for (i = 0; i < num_points; ++i) { |
| if (i == num) { |
| /* we didn't have a valid precomputation, so we pick the generator. */ |
| p = EC_GROUP_get0_generator(group); |
| p_scalar = g_scalar; |
| } else { |
| /* the i^th point */ |
| p = points[i]; |
| p_scalar = scalars[i]; |
| } |
| if (p_scalar != NULL && p != NULL) { |
| size_t num_bytes; |
| /* reduce g_scalar to 0 <= g_scalar < 2^256 */ |
| if (BN_num_bits(p_scalar) > 256 || BN_is_negative(p_scalar)) { |
| /* this is an unusual input, and we don't guarantee |
| * constant-timeness. */ |
| if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx)) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); |
| goto err; |
| } |
| num_bytes = BN_bn2bin(tmp_scalar, tmp); |
| } else { |
| num_bytes = BN_bn2bin(p_scalar, tmp); |
| } |
| flip_endian(secrets[i], tmp, num_bytes); |
| /* precompute multiples */ |
| if (!BN_to_felem(x_out, &p->X) || |
| !BN_to_felem(y_out, &p->Y) || |
| !BN_to_felem(z_out, &p->Z)) { |
| goto err; |
| } |
| felem_shrink(pre_comp[i][1][0], x_out); |
| felem_shrink(pre_comp[i][1][1], y_out); |
| felem_shrink(pre_comp[i][1][2], z_out); |
| size_t j; |
| for (j = 2; j <= 16; ++j) { |
| if (j & 1) { |
| point_add_small(pre_comp[i][j][0], pre_comp[i][j][1], |
| pre_comp[i][j][2], pre_comp[i][1][0], |
| pre_comp[i][1][1], pre_comp[i][1][2], |
| pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], |
| pre_comp[i][j - 1][2]); |
| } else { |
| point_double_small(pre_comp[i][j][0], pre_comp[i][j][1], |
| pre_comp[i][j][2], pre_comp[i][j / 2][0], |
| pre_comp[i][j / 2][1], pre_comp[i][j / 2][2]); |
| } |
| } |
| } |
| } |
| } |
| |
| if (g_scalar != NULL) { |
| size_t num_bytes; |
| |
| memset(g_secret, 0, sizeof(g_secret)); |
| /* reduce g_scalar to 0 <= g_scalar < 2^256 */ |
| if (BN_num_bits(g_scalar) > 256 || BN_is_negative(g_scalar)) { |
| /* this is an unusual input, and we don't guarantee |
| * constant-timeness. */ |
| if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) { |
| OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB); |
| goto err; |
| } |
| num_bytes = BN_bn2bin(tmp_scalar, tmp); |
| } else { |
| num_bytes = BN_bn2bin(g_scalar, tmp); |
| } |
| flip_endian(g_secret, tmp, num_bytes); |
| } |
| batch_mul(x_out, y_out, z_out, (const felem_bytearray(*))secrets, |
| num_points, g_scalar != NULL ? g_secret : NULL, |
| (const smallfelem(*)[17][3])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); |
| OPENSSL_free(secrets); |
| OPENSSL_free(pre_comp); |
| return ret; |
| } |
| |
| const EC_METHOD *EC_GFp_nistp256_method(void) { |
| static const EC_METHOD ret = { |
| ec_GFp_simple_group_init, |
| ec_GFp_simple_group_finish, |
| ec_GFp_simple_group_copy, |
| ec_GFp_simple_group_set_curve, |
| ec_GFp_nistp256_point_get_affine_coordinates, |
| ec_GFp_nistp256_points_mul, |
| ec_GFp_simple_field_mul, ec_GFp_simple_field_sqr, |
| 0 /* field_encode */, 0 /* field_decode */, |
| }; |
| |
| return &ret; |
| } |
| |
| #endif /* 64_BIT && !WINDOWS */ |