blob: a5906e45392e1361d00d5e0a9cffe22872ff6113 [file] [log] [blame]
/* Copyright (c) 2014, Intel Corporation.
*
* 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. */
/* Developers and authors:
* Shay Gueron (1, 2), and Vlad Krasnov (1)
* (1) Intel Corporation, Israel Development Center
* (2) University of Haifa
* Reference:
* S.Gueron and V.Krasnov, "Fast Prime Field Elliptic Curve Cryptography with
* 256 Bit Primes" */
#include <openssl/ec.h>
#include <assert.h>
#include <stdint.h>
#include <string.h>
#include <openssl/bn.h>
#include <openssl/crypto.h>
#include <openssl/err.h>
#include "../bn/internal.h"
#include "../ec/internal.h"
#include "../internal.h"
#if !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64) && \
!defined(OPENSSL_SMALL)
#define P256_LIMBS (256 / BN_BITS2)
typedef struct {
BN_ULONG X[P256_LIMBS];
BN_ULONG Y[P256_LIMBS];
BN_ULONG Z[P256_LIMBS];
} P256_POINT;
typedef struct {
BN_ULONG X[P256_LIMBS];
BN_ULONG Y[P256_LIMBS];
} P256_POINT_AFFINE;
typedef P256_POINT_AFFINE PRECOMP256_ROW[64];
/* Arithmetic on field elements using Almost Montgomery Multiplication. The
* "almost" means, in particular, that the inputs and outputs of these
* functions are in the range [0, 2**BN_BITS2), not [0, P). Only
* |ecp_nistz256_from_mont| outputs a fully reduced value in [0, P). Almost
* Montgomery Arithmetic is described clearly in "Efficient Software
* Implementations of Modular Exponentiation" by Shay Gueron. */
/* Modular neg: res = -a mod P, where res is not fully reduced. */
void ecp_nistz256_neg(BN_ULONG res[P256_LIMBS], const BN_ULONG a[P256_LIMBS]);
/* Montgomery mul: res = a*b*2^-256 mod P, where res is not fully reduced. */
void ecp_nistz256_mul_mont(BN_ULONG res[P256_LIMBS],
const BN_ULONG a[P256_LIMBS],
const BN_ULONG b[P256_LIMBS]);
/* Montgomery sqr: res = a*a*2^-256 mod P, where res is not fully reduced. */
void ecp_nistz256_sqr_mont(BN_ULONG res[P256_LIMBS],
const BN_ULONG a[P256_LIMBS]);
/* Convert a number from Montgomery domain, by multiplying with 1, where res
* will be fully reduced mod P. */
void ecp_nistz256_from_mont(BN_ULONG res[P256_LIMBS],
const BN_ULONG in[P256_LIMBS]);
/* Functions that perform constant time access to the precomputed tables */
void ecp_nistz256_select_w5(P256_POINT *val, const P256_POINT *in_t, int index);
void ecp_nistz256_select_w7(P256_POINT_AFFINE *val,
const P256_POINT_AFFINE *in_t, int index);
/* One converted into the Montgomery domain */
static const BN_ULONG ONE[P256_LIMBS] = {
TOBN(0x00000000, 0x00000001), TOBN(0xffffffff, 0x00000000),
TOBN(0xffffffff, 0xffffffff), TOBN(0x00000000, 0xfffffffe),
};
/* Precomputed tables for the default generator */
#include "p256-x86_64-table.h"
/* Recode window to a signed digit, see ecp_nistputil.c for details */
static unsigned booth_recode_w5(unsigned in) {
unsigned s, d;
s = ~((in >> 5) - 1);
d = (1 << 6) - in - 1;
d = (d & s) | (in & ~s);
d = (d >> 1) + (d & 1);
return (d << 1) + (s & 1);
}
static unsigned booth_recode_w7(unsigned in) {
unsigned s, d;
s = ~((in >> 7) - 1);
d = (1 << 8) - in - 1;
d = (d & s) | (in & ~s);
d = (d >> 1) + (d & 1);
return (d << 1) + (s & 1);
}
static void copy_conditional(BN_ULONG dst[P256_LIMBS],
const BN_ULONG src[P256_LIMBS], BN_ULONG move) {
BN_ULONG mask1 = ((BN_ULONG)0) - move;
BN_ULONG mask2 = ~mask1;
dst[0] = (src[0] & mask1) ^ (dst[0] & mask2);
dst[1] = (src[1] & mask1) ^ (dst[1] & mask2);
dst[2] = (src[2] & mask1) ^ (dst[2] & mask2);
dst[3] = (src[3] & mask1) ^ (dst[3] & mask2);
if (P256_LIMBS == 8) {
dst[4] = (src[4] & mask1) ^ (dst[4] & mask2);
dst[5] = (src[5] & mask1) ^ (dst[5] & mask2);
dst[6] = (src[6] & mask1) ^ (dst[6] & mask2);
dst[7] = (src[7] & mask1) ^ (dst[7] & mask2);
}
}
void ecp_nistz256_point_double(P256_POINT *r, const P256_POINT *a);
void ecp_nistz256_point_add(P256_POINT *r, const P256_POINT *a,
const P256_POINT *b);
void ecp_nistz256_point_add_affine(P256_POINT *r, const P256_POINT *a,
const P256_POINT_AFFINE *b);
/* r = in^-1 mod p */
static void ecp_nistz256_mod_inverse(BN_ULONG r[P256_LIMBS],
const BN_ULONG in[P256_LIMBS]) {
/* The poly is ffffffff 00000001 00000000 00000000 00000000 ffffffff ffffffff
ffffffff
We use FLT and used poly-2 as exponent */
BN_ULONG p2[P256_LIMBS];
BN_ULONG p4[P256_LIMBS];
BN_ULONG p8[P256_LIMBS];
BN_ULONG p16[P256_LIMBS];
BN_ULONG p32[P256_LIMBS];
BN_ULONG res[P256_LIMBS];
int i;
ecp_nistz256_sqr_mont(res, in);
ecp_nistz256_mul_mont(p2, res, in); /* 3*p */
ecp_nistz256_sqr_mont(res, p2);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_mul_mont(p4, res, p2); /* f*p */
ecp_nistz256_sqr_mont(res, p4);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_mul_mont(p8, res, p4); /* ff*p */
ecp_nistz256_sqr_mont(res, p8);
for (i = 0; i < 7; i++) {
ecp_nistz256_sqr_mont(res, res);
}
ecp_nistz256_mul_mont(p16, res, p8); /* ffff*p */
ecp_nistz256_sqr_mont(res, p16);
for (i = 0; i < 15; i++) {
ecp_nistz256_sqr_mont(res, res);
}
ecp_nistz256_mul_mont(p32, res, p16); /* ffffffff*p */
ecp_nistz256_sqr_mont(res, p32);
for (i = 0; i < 31; i++) {
ecp_nistz256_sqr_mont(res, res);
}
ecp_nistz256_mul_mont(res, res, in);
for (i = 0; i < 32 * 4; i++) {
ecp_nistz256_sqr_mont(res, res);
}
ecp_nistz256_mul_mont(res, res, p32);
for (i = 0; i < 32; i++) {
ecp_nistz256_sqr_mont(res, res);
}
ecp_nistz256_mul_mont(res, res, p32);
for (i = 0; i < 16; i++) {
ecp_nistz256_sqr_mont(res, res);
}
ecp_nistz256_mul_mont(res, res, p16);
for (i = 0; i < 8; i++) {
ecp_nistz256_sqr_mont(res, res);
}
ecp_nistz256_mul_mont(res, res, p8);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_mul_mont(res, res, p4);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_mul_mont(res, res, p2);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_sqr_mont(res, res);
ecp_nistz256_mul_mont(r, res, in);
}
/* ecp_nistz256_bignum_to_field_elem copies the contents of |in| to |out| and
* returns one if it fits. Otherwise it returns zero. */
static int ecp_nistz256_bignum_to_field_elem(BN_ULONG out[P256_LIMBS],
const BIGNUM *in) {
if (in->top > P256_LIMBS) {
return 0;
}
memset(out, 0, sizeof(BN_ULONG) * P256_LIMBS);
memcpy(out, in->d, sizeof(BN_ULONG) * in->top);
return 1;
}
/* r = p * p_scalar */
static int ecp_nistz256_windowed_mul(const EC_GROUP *group, P256_POINT *r,
const EC_POINT *p, const BIGNUM *p_scalar,
BN_CTX *ctx) {
assert(p != NULL);
assert(p_scalar != NULL);
static const unsigned kWindowSize = 5;
static const unsigned kMask = (1 << (5 /* kWindowSize */ + 1)) - 1;
/* A |P256_POINT| is (3 * 32) = 96 bytes, and the 64-byte alignment should
* add no more than 63 bytes of overhead. Thus, |table| should require
* ~1599 ((96 * 16) + 63) bytes of stack space. */
alignas(64) P256_POINT table[16];
uint8_t p_str[33];
int ret = 0;
BN_CTX *new_ctx = NULL;
int ctx_started = 0;
if (BN_num_bits(p_scalar) > 256 || BN_is_negative(p_scalar)) {
if (ctx == NULL) {
new_ctx = BN_CTX_new();
if (new_ctx == NULL) {
OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
goto err;
}
ctx = new_ctx;
}
BN_CTX_start(ctx);
ctx_started = 1;
BIGNUM *mod = BN_CTX_get(ctx);
if (mod == NULL) {
OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
goto err;
}
if (!BN_nnmod(mod, p_scalar, &group->order, ctx)) {
OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
goto err;
}
p_scalar = mod;
}
int j;
for (j = 0; j < p_scalar->top * BN_BYTES; j += BN_BYTES) {
BN_ULONG d = p_scalar->d[j / BN_BYTES];
p_str[j + 0] = d & 0xff;
p_str[j + 1] = (d >> 8) & 0xff;
p_str[j + 2] = (d >> 16) & 0xff;
p_str[j + 3] = (d >>= 24) & 0xff;
if (BN_BYTES == 8) {
d >>= 8;
p_str[j + 4] = d & 0xff;
p_str[j + 5] = (d >> 8) & 0xff;
p_str[j + 6] = (d >> 16) & 0xff;
p_str[j + 7] = (d >> 24) & 0xff;
}
}
for (; j < 33; j++) {
p_str[j] = 0;
}
/* table[0] is implicitly (0,0,0) (the point at infinity), therefore it is
* not stored. All other values are actually stored with an offset of -1 in
* table. */
P256_POINT *row = table;
if (!ecp_nistz256_bignum_to_field_elem(row[1 - 1].X, &p->X) ||
!ecp_nistz256_bignum_to_field_elem(row[1 - 1].Y, &p->Y) ||
!ecp_nistz256_bignum_to_field_elem(row[1 - 1].Z, &p->Z)) {
OPENSSL_PUT_ERROR(EC, EC_R_COORDINATES_OUT_OF_RANGE);
goto err;
}
ecp_nistz256_point_double(&row[2 - 1], &row[1 - 1]);
ecp_nistz256_point_add(&row[3 - 1], &row[2 - 1], &row[1 - 1]);
ecp_nistz256_point_double(&row[4 - 1], &row[2 - 1]);
ecp_nistz256_point_double(&row[6 - 1], &row[3 - 1]);
ecp_nistz256_point_double(&row[8 - 1], &row[4 - 1]);
ecp_nistz256_point_double(&row[12 - 1], &row[6 - 1]);
ecp_nistz256_point_add(&row[5 - 1], &row[4 - 1], &row[1 - 1]);
ecp_nistz256_point_add(&row[7 - 1], &row[6 - 1], &row[1 - 1]);
ecp_nistz256_point_add(&row[9 - 1], &row[8 - 1], &row[1 - 1]);
ecp_nistz256_point_add(&row[13 - 1], &row[12 - 1], &row[1 - 1]);
ecp_nistz256_point_double(&row[14 - 1], &row[7 - 1]);
ecp_nistz256_point_double(&row[10 - 1], &row[5 - 1]);
ecp_nistz256_point_add(&row[15 - 1], &row[14 - 1], &row[1 - 1]);
ecp_nistz256_point_add(&row[11 - 1], &row[10 - 1], &row[1 - 1]);
ecp_nistz256_point_double(&row[16 - 1], &row[8 - 1]);
BN_ULONG tmp[P256_LIMBS];
alignas(32) P256_POINT h;
unsigned index = 255;
unsigned wvalue = p_str[(index - 1) / 8];
wvalue = (wvalue >> ((index - 1) % 8)) & kMask;
ecp_nistz256_select_w5(r, table, booth_recode_w5(wvalue) >> 1);
while (index >= 5) {
if (index != 255) {
unsigned off = (index - 1) / 8;
wvalue = p_str[off] | p_str[off + 1] << 8;
wvalue = (wvalue >> ((index - 1) % 8)) & kMask;
wvalue = booth_recode_w5(wvalue);
ecp_nistz256_select_w5(&h, table, wvalue >> 1);
ecp_nistz256_neg(tmp, h.Y);
copy_conditional(h.Y, tmp, (wvalue & 1));
ecp_nistz256_point_add(r, r, &h);
}
index -= kWindowSize;
ecp_nistz256_point_double(r, r);
ecp_nistz256_point_double(r, r);
ecp_nistz256_point_double(r, r);
ecp_nistz256_point_double(r, r);
ecp_nistz256_point_double(r, r);
}
/* Final window */
wvalue = p_str[0];
wvalue = (wvalue << 1) & kMask;
wvalue = booth_recode_w5(wvalue);
ecp_nistz256_select_w5(&h, table, wvalue >> 1);
ecp_nistz256_neg(tmp, h.Y);
copy_conditional(h.Y, tmp, wvalue & 1);
ecp_nistz256_point_add(r, r, &h);
ret = 1;
err:
if (ctx_started) {
BN_CTX_end(ctx);
}
BN_CTX_free(new_ctx);
return ret;
}
static int ecp_nistz256_points_mul(
const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar,
const EC_POINT *p_, const BIGNUM *p_scalar, BN_CTX *ctx) {
assert((p_ != NULL) == (p_scalar != NULL));
static const unsigned kWindowSize = 7;
static const unsigned kMask = (1 << (7 /* kWindowSize */ + 1)) - 1;
alignas(32) union {
P256_POINT p;
P256_POINT_AFFINE a;
} t, p;
int ret = 0;
BN_CTX *new_ctx = NULL;
int ctx_started = 0;
if (g_scalar != NULL) {
if (BN_num_bits(g_scalar) > 256 || BN_is_negative(g_scalar)) {
if (ctx == NULL) {
new_ctx = BN_CTX_new();
if (new_ctx == NULL) {
goto err;
}
ctx = new_ctx;
}
BN_CTX_start(ctx);
ctx_started = 1;
BIGNUM *tmp_scalar = BN_CTX_get(ctx);
if (tmp_scalar == NULL) {
goto err;
}
if (!BN_nnmod(tmp_scalar, g_scalar, &group->order, ctx)) {
OPENSSL_PUT_ERROR(EC, ERR_R_BN_LIB);
goto err;
}
g_scalar = tmp_scalar;
}
uint8_t p_str[33] = {0};
int i;
for (i = 0; i < g_scalar->top * BN_BYTES; i += BN_BYTES) {
BN_ULONG d = g_scalar->d[i / BN_BYTES];
p_str[i + 0] = d & 0xff;
p_str[i + 1] = (d >> 8) & 0xff;
p_str[i + 2] = (d >> 16) & 0xff;
p_str[i + 3] = (d >>= 24) & 0xff;
if (BN_BYTES == 8) {
d >>= 8;
p_str[i + 4] = d & 0xff;
p_str[i + 5] = (d >> 8) & 0xff;
p_str[i + 6] = (d >> 16) & 0xff;
p_str[i + 7] = (d >> 24) & 0xff;
}
}
for (; i < (int) sizeof(p_str); i++) {
p_str[i] = 0;
}
/* First window */
unsigned wvalue = (p_str[0] << 1) & kMask;
unsigned index = kWindowSize;
wvalue = booth_recode_w7(wvalue);
const PRECOMP256_ROW *const precomputed_table =
(const PRECOMP256_ROW *)ecp_nistz256_precomputed;
ecp_nistz256_select_w7(&p.a, precomputed_table[0], wvalue >> 1);
ecp_nistz256_neg(p.p.Z, p.p.Y);
copy_conditional(p.p.Y, p.p.Z, wvalue & 1);
memcpy(p.p.Z, ONE, sizeof(ONE));
for (i = 1; i < 37; i++) {
unsigned off = (index - 1) / 8;
wvalue = p_str[off] | p_str[off + 1] << 8;
wvalue = (wvalue >> ((index - 1) % 8)) & kMask;
index += kWindowSize;
wvalue = booth_recode_w7(wvalue);
ecp_nistz256_select_w7(&t.a, precomputed_table[i], wvalue >> 1);
ecp_nistz256_neg(t.p.Z, t.a.Y);
copy_conditional(t.a.Y, t.p.Z, wvalue & 1);
ecp_nistz256_point_add_affine(&p.p, &p.p, &t.a);
}
}
const int p_is_infinity = g_scalar == NULL;
if (p_scalar != NULL) {
P256_POINT *out = &t.p;
if (p_is_infinity) {
out = &p.p;
}
if (!ecp_nistz256_windowed_mul(group, out, p_, p_scalar, ctx)) {
goto err;
}
if (!p_is_infinity) {
ecp_nistz256_point_add(&p.p, &p.p, out);
}
}
/* Not constant-time, but we're only operating on the public output. */
if (!bn_set_words(&r->X, p.p.X, P256_LIMBS) ||
!bn_set_words(&r->Y, p.p.Y, P256_LIMBS) ||
!bn_set_words(&r->Z, p.p.Z, P256_LIMBS)) {
return 0;
}
ret = 1;
err:
if (ctx_started) {
BN_CTX_end(ctx);
}
BN_CTX_free(new_ctx);
return ret;
}
static int ecp_nistz256_get_affine(const EC_GROUP *group, const EC_POINT *point,
BIGNUM *x, BIGNUM *y, BN_CTX *ctx) {
BN_ULONG z_inv2[P256_LIMBS];
BN_ULONG z_inv3[P256_LIMBS];
BN_ULONG point_x[P256_LIMBS], point_y[P256_LIMBS], point_z[P256_LIMBS];
if (EC_POINT_is_at_infinity(group, point)) {
OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY);
return 0;
}
if (!ecp_nistz256_bignum_to_field_elem(point_x, &point->X) ||
!ecp_nistz256_bignum_to_field_elem(point_y, &point->Y) ||
!ecp_nistz256_bignum_to_field_elem(point_z, &point->Z)) {
OPENSSL_PUT_ERROR(EC, EC_R_COORDINATES_OUT_OF_RANGE);
return 0;
}
ecp_nistz256_mod_inverse(z_inv3, point_z);
ecp_nistz256_sqr_mont(z_inv2, z_inv3);
/* Unlike the |BN_mod_mul_montgomery|-based implementation, we cannot factor
* out the two calls to |ecp_nistz256_from_mont| into one call, because
* |ecp_nistz256_from_mont| must be the last operation to ensure that the
* result is fully reduced mod P. */
if (x != NULL) {
BN_ULONG x_aff[P256_LIMBS];
ecp_nistz256_mul_mont(x_aff, z_inv2, point_x);
ecp_nistz256_from_mont(x_aff, x_aff);
if (!bn_set_words(x, x_aff, P256_LIMBS)) {
OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
return 0;
}
}
if (y != NULL) {
BN_ULONG y_aff[P256_LIMBS];
ecp_nistz256_mul_mont(z_inv3, z_inv3, z_inv2);
ecp_nistz256_mul_mont(y_aff, z_inv3, point_y);
ecp_nistz256_from_mont(y_aff, y_aff);
if (!bn_set_words(y, y_aff, P256_LIMBS)) {
OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE);
return 0;
}
}
return 1;
}
const EC_METHOD EC_GFp_nistz256_method = {
ec_GFp_mont_group_init,
ec_GFp_mont_group_finish,
ec_GFp_mont_group_copy,
ec_GFp_mont_group_set_curve,
ecp_nistz256_get_affine,
ecp_nistz256_points_mul,
ec_GFp_mont_field_mul,
ec_GFp_mont_field_sqr,
ec_GFp_mont_field_encode,
ec_GFp_mont_field_decode,
};
#endif /* !defined(OPENSSL_NO_ASM) && defined(OPENSSL_X86_64) && \
!defined(OPENSSL_SMALL) */