/* Originally written by Bodo Moeller for the OpenSSL project.
* ====================================================================
* Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
*
* 3. All advertising materials mentioning features or use of this
* software must display the following acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
*
* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
* endorse or promote products derived from this software without
* prior written permission. For written permission, please contact
* openssl-core@openssl.org.
*
* 5. Products derived from this software may not be called "OpenSSL"
* nor may "OpenSSL" appear in their names without prior written
* permission of the OpenSSL Project.
*
* 6. Redistributions of any form whatsoever must retain the following
* acknowledgment:
* "This product includes software developed by the OpenSSL Project
* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
*
* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
* OF THE POSSIBILITY OF SUCH DAMAGE.
* ====================================================================
*
* This product includes cryptographic software written by Eric Young
* (eay@cryptsoft.com). This product includes software written by Tim
* Hudson (tjh@cryptsoft.com).
*
*/
/* ====================================================================
* Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
*
* Portions of the attached software ("Contribution") are developed by
* SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
*
* The Contribution is licensed pursuant to the OpenSSL open source
* license provided above.
*
* The elliptic curve binary polynomial software is originally written by
* Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
* Laboratories. */
#include <openssl/ec.h>
#include <string.h>
#include <openssl/bn.h>
#include <openssl/err.h>
#include <openssl/mem.h>
#include <openssl/thread.h>
#include "internal.h"
#include "../bn/internal.h"
#include "../../internal.h"
// This file implements the wNAF-based interleaving multi-exponentiation method
// at:
// http://link.springer.com/chapter/10.1007%2F3-540-45537-X_13
// http://www.bmoeller.de/pdf/TI-01-08.multiexp.pdf
// compute_wNAF writes the modified width-(w+1) Non-Adjacent Form (wNAF) of
// |scalar| to |out| and returns one on success or zero on internal error. |out|
// must have room for |bits| + 1 elements, each of which will be either zero or
// odd with an absolute value less than 2^w satisfying
// scalar = \sum_j out[j]*2^j
// where at most one of any w+1 consecutive digits is non-zero
// with the exception that the most significant digit may be only
// w-1 zeros away from that next non-zero digit.
static int compute_wNAF(const EC_GROUP *group, int8_t *out,
const EC_SCALAR *scalar, size_t bits, int w) {
// 'int8_t' can represent integers with absolute values less than 2^7.
if (w <= 0 || w > 7 || bits == 0) {
OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
return 0;
}
int bit = 1 << w; // at most 128
int next_bit = bit << 1; // at most 256
int mask = next_bit - 1; // at most 255
int window_val = scalar->words[0] & mask;
size_t j = 0;
// If j+w+1 >= bits, window_val will not increase.
while (window_val != 0 || j + w + 1 < bits) {
int digit = 0;
// 0 <= window_val <= 2^(w+1)
if (window_val & 1) {
// 0 < window_val < 2^(w+1)
if (window_val & bit) {
digit = window_val - next_bit; // -2^w < digit < 0
#if 1 // modified wNAF
if (j + w + 1 >= bits) {
// special case for generating modified wNAFs:
// no new bits will be added into window_val,
// so using a positive digit here will decrease
// the total length of the representation
digit = window_val & (mask >> 1); // 0 < digit < 2^w
}
#endif
} else {
digit = window_val; // 0 < digit < 2^w
}
if (digit <= -bit || digit >= bit || !(digit & 1)) {
OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
return 0;
}
window_val -= digit;
// Now window_val is 0 or 2^(w+1) in standard wNAF generation;
// for modified window NAFs, it may also be 2^w.
if (window_val != 0 && window_val != next_bit && window_val != bit) {
OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
return 0;
}
}
out[j++] = digit;
window_val >>= 1;
window_val +=
bit * bn_is_bit_set_words(scalar->words, group->order.top, j + w);
if (window_val > next_bit) {
OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
return 0;
}
}
// Fill the rest of the wNAF with zeros.
if (j > bits + 1) {
OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
return 0;
}
for (size_t i = j; i < bits + 1; i++) {
out[i] = 0;
}
return 1;
}
// TODO: table should be optimised for the wNAF-based implementation,
// sometimes smaller windows will give better performance
// (thus the boundaries should be increased)
static size_t window_bits_for_scalar_size(size_t b) {
if (b >= 300) {
return 4;
}
if (b >= 70) {
return 3;
}
if (b >= 20) {
return 2;
}
return 1;
}
// EC_WNAF_MAX_WINDOW_BITS is the largest value returned by
// |window_bits_for_scalar_size|.
#define EC_WNAF_MAX_WINDOW_BITS 4
// compute_precomp sets |out[i]| to a newly-allocated |EC_POINT| containing
// (2*i+1)*p, for i from 0 to |len|. It returns one on success and
// zero on error.
static int compute_precomp(const EC_GROUP *group, EC_POINT **out,
const EC_POINT *p, size_t len, BN_CTX *ctx) {
out[0] = EC_POINT_new(group);
if (out[0] == NULL ||
!EC_POINT_copy(out[0], p)) {
return 0;
}
int ret = 0;
EC_POINT *two_p = EC_POINT_new(group);
if (two_p == NULL ||
!EC_POINT_dbl(group, two_p, p, ctx)) {
goto err;
}
for (size_t i = 1; i < len; i++) {
out[i] = EC_POINT_new(group);
if (out[i] == NULL ||
!EC_POINT_add(group, out[i], out[i - 1], two_p, ctx)) {
goto err;
}
}
ret = 1;
err:
EC_POINT_free(two_p);
return ret;
}
static int lookup_precomp(const EC_GROUP *group, EC_POINT *out,
EC_POINT *const *precomp, int digit, BN_CTX *ctx) {
if (digit < 0) {
digit = -digit;
return EC_POINT_copy(out, precomp[digit >> 1]) &&
EC_POINT_invert(group, out, ctx);
}
return EC_POINT_copy(out, precomp[digit >> 1]);
}
int ec_wNAF_mul(const EC_GROUP *group, EC_POINT *r, const EC_SCALAR *g_scalar,
const EC_POINT *p, const EC_SCALAR *p_scalar, BN_CTX *ctx) {
BN_CTX *new_ctx = NULL;
EC_POINT *precomp_storage[2 * (1 << (EC_WNAF_MAX_WINDOW_BITS - 1))] = {NULL};
EC_POINT **g_precomp = NULL, **p_precomp = NULL;
int8_t g_wNAF[EC_MAX_SCALAR_BYTES * 8 + 1];
int8_t p_wNAF[EC_MAX_SCALAR_BYTES * 8 + 1];
EC_POINT *tmp = NULL;
int ret = 0;
if (ctx == NULL) {
ctx = new_ctx = BN_CTX_new();
if (ctx == NULL) {
goto err;
}
}
size_t bits = BN_num_bits(&group->order);
size_t wsize = window_bits_for_scalar_size(bits);
size_t wNAF_len = bits + 1;
size_t precomp_len = (size_t)1 << (wsize - 1);
if (wNAF_len > OPENSSL_ARRAY_SIZE(g_wNAF) ||
wNAF_len > OPENSSL_ARRAY_SIZE(p_wNAF) ||
2 * precomp_len > OPENSSL_ARRAY_SIZE(precomp_storage)) {
OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR);
goto err;
}
// TODO(davidben): |mul_public| is for ECDSA verification which can assume
// non-NULL inputs, but this code is also used for |mul| which cannot. It's
// not constant-time, so replace the generic |mul| and remove the NULL checks.
size_t total_precomp = 0;
if (g_scalar != NULL) {
const EC_POINT *g = EC_GROUP_get0_generator(group);
if (g == NULL) {
OPENSSL_PUT_ERROR(EC, EC_R_UNDEFINED_GENERATOR);
goto err;
}
g_precomp = precomp_storage + total_precomp;
total_precomp += precomp_len;
if (!compute_wNAF(group, g_wNAF, g_scalar, bits, wsize) ||
!compute_precomp(group, g_precomp, g, precomp_len, ctx)) {
goto err;
}
}
if (p_scalar != NULL) {
p_precomp = precomp_storage + total_precomp;
total_precomp += precomp_len;
if (!compute_wNAF(group, p_wNAF, p_scalar, bits, wsize) ||
!compute_precomp(group, p_precomp, p, precomp_len, ctx)) {
goto err;
}
}
tmp = EC_POINT_new(group);
if (tmp == NULL ||
// |window_bits_for_scalar_size| assumes we do this step.
!EC_POINTs_make_affine(group, total_precomp, precomp_storage, ctx)) {
goto err;
}
int r_is_at_infinity = 1;
for (size_t k = wNAF_len - 1; k < wNAF_len; k--) {
if (!r_is_at_infinity && !EC_POINT_dbl(group, r, r, ctx)) {
goto err;
}
if (g_scalar != NULL) {
if (g_wNAF[k] != 0) {
if (!lookup_precomp(group, tmp, g_precomp, g_wNAF[k], ctx)) {
goto err;
}
if (r_is_at_infinity) {
if (!EC_POINT_copy(r, tmp)) {
goto err;
}
r_is_at_infinity = 0;
} else if (!EC_POINT_add(group, r, r, tmp, ctx)) {
goto err;
}
}
}
if (p_scalar != NULL) {
if (p_wNAF[k] != 0) {
if (!lookup_precomp(group, tmp, p_precomp, p_wNAF[k], ctx)) {
goto err;
}
if (r_is_at_infinity) {
if (!EC_POINT_copy(r, tmp)) {
goto err;
}
r_is_at_infinity = 0;
} else if (!EC_POINT_add(group, r, r, tmp, ctx)) {
goto err;
}
}
}
}
if (r_is_at_infinity &&
!EC_POINT_set_to_infinity(group, r)) {
goto err;
}
ret = 1;
err:
BN_CTX_free(new_ctx);
EC_POINT_free(tmp);
OPENSSL_cleanse(&g_wNAF, sizeof(g_wNAF));
OPENSSL_cleanse(&p_wNAF, sizeof(p_wNAF));
for (size_t i = 0; i < OPENSSL_ARRAY_SIZE(precomp_storage); i++) {
EC_POINT_free(precomp_storage[i]);
}
return ret;
}