/* 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 #include #include #include #include #include #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.width, 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; }