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boringssl / boringssl / f0322b2abce91458f62db44dc6b777fc5e0323d9 / . / crypto / ec / p256-64.c
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/* 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,
0 /* check_pub_key_order */,
ec_GFp_simple_field_mul, ec_GFp_simple_field_sqr,
0 /* field_encode */, 0 /* field_decode */,
};
return &ret;
}
#endif /* 64_BIT && !WINDOWS */