1/* Originally written by Bodo Moeller for the OpenSSL project. 2 * ==================================================================== 3 * Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in 14 * the documentation and/or other materials provided with the 15 * distribution. 16 * 17 * 3. All advertising materials mentioning features or use of this 18 * software must display the following acknowledgment: 19 * "This product includes software developed by the OpenSSL Project 20 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 21 * 22 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 23 * endorse or promote products derived from this software without 24 * prior written permission. For written permission, please contact 25 * openssl-core@openssl.org. 26 * 27 * 5. Products derived from this software may not be called "OpenSSL" 28 * nor may "OpenSSL" appear in their names without prior written 29 * permission of the OpenSSL Project. 30 * 31 * 6. Redistributions of any form whatsoever must retain the following 32 * acknowledgment: 33 * "This product includes software developed by the OpenSSL Project 34 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 35 * 36 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 37 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 38 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 39 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 40 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 41 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 42 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 43 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 44 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 45 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 46 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 47 * OF THE POSSIBILITY OF SUCH DAMAGE. 48 * ==================================================================== 49 * 50 * This product includes cryptographic software written by Eric Young 51 * (eay@cryptsoft.com). This product includes software written by Tim 52 * Hudson (tjh@cryptsoft.com). 53 * 54 */ 55/* ==================================================================== 56 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. 57 * 58 * Portions of the attached software ("Contribution") are developed by 59 * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project. 60 * 61 * The Contribution is licensed pursuant to the OpenSSL open source 62 * license provided above. 63 * 64 * The elliptic curve binary polynomial software is originally written by 65 * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems 66 * Laboratories. */ 67 68#include <openssl/ec.h> 69 70#include <string.h> 71 72#include <openssl/bn.h> 73#include <openssl/err.h> 74#include <openssl/mem.h> 75#include <openssl/thread.h> 76 77#include "internal.h" 78#include "../internal.h" 79 80 81/* This file implements the wNAF-based interleaving multi-exponentation method 82 * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp>); 83 * */ 84 85/* Determine the modified width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'. 86 * This is an array r[] of values that are either zero or odd with an 87 * absolute value less than 2^w satisfying 88 * scalar = \sum_j r[j]*2^j 89 * where at most one of any w+1 consecutive digits is non-zero 90 * with the exception that the most significant digit may be only 91 * w-1 zeros away from that next non-zero digit. 92 */ 93static int8_t *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len) { 94 int window_val; 95 int ok = 0; 96 int8_t *r = NULL; 97 int sign = 1; 98 int bit, next_bit, mask; 99 size_t len = 0, j; 100 101 if (BN_is_zero(scalar)) { 102 r = OPENSSL_malloc(1); 103 if (!r) { 104 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 105 goto err; 106 } 107 r[0] = 0; 108 *ret_len = 1; 109 return r; 110 } 111 112 /* 'int8_t' can represent integers with absolute values less than 2^7. */ 113 if (w <= 0 || w > 7) { 114 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 115 goto err; 116 } 117 bit = 1 << w; /* at most 128 */ 118 next_bit = bit << 1; /* at most 256 */ 119 mask = next_bit - 1; /* at most 255 */ 120 121 if (BN_is_negative(scalar)) { 122 sign = -1; 123 } 124 125 if (scalar->d == NULL || scalar->top == 0) { 126 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 127 goto err; 128 } 129 130 len = BN_num_bits(scalar); 131 /* The modified wNAF may be one digit longer than binary representation 132 * (*ret_len will be set to the actual length, i.e. at most 133 * BN_num_bits(scalar) + 1). */ 134 r = OPENSSL_malloc(len + 1); 135 if (r == NULL) { 136 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 137 goto err; 138 } 139 window_val = scalar->d[0] & mask; 140 j = 0; 141 /* If j+w+1 >= len, window_val will not increase. */ 142 while (window_val != 0 || j + w + 1 < len) { 143 int digit = 0; 144 145 /* 0 <= window_val <= 2^(w+1) */ 146 147 if (window_val & 1) { 148 /* 0 < window_val < 2^(w+1) */ 149 150 if (window_val & bit) { 151 digit = window_val - next_bit; /* -2^w < digit < 0 */ 152 153#if 1 /* modified wNAF */ 154 if (j + w + 1 >= len) { 155 /* special case for generating modified wNAFs: 156 * no new bits will be added into window_val, 157 * so using a positive digit here will decrease 158 * the total length of the representation */ 159 160 digit = window_val & (mask >> 1); /* 0 < digit < 2^w */ 161 } 162#endif 163 } else { 164 digit = window_val; /* 0 < digit < 2^w */ 165 } 166 167 if (digit <= -bit || digit >= bit || !(digit & 1)) { 168 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 169 goto err; 170 } 171 172 window_val -= digit; 173 174 /* Now window_val is 0 or 2^(w+1) in standard wNAF generation; 175 * for modified window NAFs, it may also be 2^w. */ 176 if (window_val != 0 && window_val != next_bit && window_val != bit) { 177 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 178 goto err; 179 } 180 } 181 182 r[j++] = sign * digit; 183 184 window_val >>= 1; 185 window_val += bit * BN_is_bit_set(scalar, j + w); 186 187 if (window_val > next_bit) { 188 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 189 goto err; 190 } 191 } 192 193 if (j > len + 1) { 194 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 195 goto err; 196 } 197 len = j; 198 ok = 1; 199 200err: 201 if (!ok) { 202 OPENSSL_free(r); 203 r = NULL; 204 } 205 if (ok) { 206 *ret_len = len; 207 } 208 return r; 209} 210 211 212/* TODO: table should be optimised for the wNAF-based implementation, 213 * sometimes smaller windows will give better performance 214 * (thus the boundaries should be increased) 215 */ 216static size_t window_bits_for_scalar_size(size_t b) { 217 if (b >= 2000) { 218 return 6; 219 } 220 221 if (b >= 800) { 222 return 5; 223 } 224 225 if (b >= 300) { 226 return 4; 227 } 228 229 if (b >= 70) { 230 return 3; 231 } 232 233 if (b >= 20) { 234 return 2; 235 } 236 237 return 1; 238} 239 240int ec_wNAF_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar, 241 const EC_POINT *p, const BIGNUM *p_scalar, BN_CTX *ctx) { 242 BN_CTX *new_ctx = NULL; 243 const EC_POINT *generator = NULL; 244 EC_POINT *tmp = NULL; 245 size_t total_num = 0; 246 size_t i, j; 247 int k; 248 int r_is_inverted = 0; 249 int r_is_at_infinity = 1; 250 size_t *wsize = NULL; /* individual window sizes */ 251 int8_t **wNAF = NULL; /* individual wNAFs */ 252 size_t *wNAF_len = NULL; 253 size_t max_len = 0; 254 size_t num_val = 0; 255 EC_POINT **val = NULL; /* precomputation */ 256 EC_POINT **v; 257 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */ 258 int ret = 0; 259 260 if (ctx == NULL) { 261 ctx = new_ctx = BN_CTX_new(); 262 if (ctx == NULL) { 263 goto err; 264 } 265 } 266 267 /* TODO: This function used to take |points| and |scalars| as arrays of 268 * |num| elements. The code below should be simplified to work in terms of |p| 269 * and |p_scalar|. */ 270 size_t num = p != NULL ? 1 : 0; 271 const EC_POINT **points = p != NULL ? &p : NULL; 272 const BIGNUM **scalars = p != NULL ? &p_scalar : NULL; 273 274 total_num = num; 275 276 if (g_scalar != NULL) { 277 generator = EC_GROUP_get0_generator(group); 278 if (generator == NULL) { 279 OPENSSL_PUT_ERROR(EC, EC_R_UNDEFINED_GENERATOR); 280 goto err; 281 } 282 283 ++total_num; /* treat 'g_scalar' like 'num'-th element of 'scalars' */ 284 } 285 286 287 wsize = OPENSSL_malloc(total_num * sizeof(wsize[0])); 288 wNAF_len = OPENSSL_malloc(total_num * sizeof(wNAF_len[0])); 289 wNAF = OPENSSL_malloc(total_num * sizeof(wNAF[0])); 290 val_sub = OPENSSL_malloc(total_num * sizeof(val_sub[0])); 291 292 /* Ensure wNAF is initialised in case we end up going to err. */ 293 if (wNAF != NULL) { 294 OPENSSL_memset(wNAF, 0, total_num * sizeof(wNAF[0])); 295 } 296 297 if (!wsize || !wNAF_len || !wNAF || !val_sub) { 298 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 299 goto err; 300 } 301 302 /* num_val will be the total number of temporarily precomputed points */ 303 num_val = 0; 304 305 for (i = 0; i < total_num; i++) { 306 size_t bits; 307 308 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(g_scalar); 309 wsize[i] = window_bits_for_scalar_size(bits); 310 num_val += (size_t)1 << (wsize[i] - 1); 311 wNAF[i] = 312 compute_wNAF((i < num ? scalars[i] : g_scalar), wsize[i], &wNAF_len[i]); 313 if (wNAF[i] == NULL) { 314 goto err; 315 } 316 if (wNAF_len[i] > max_len) { 317 max_len = wNAF_len[i]; 318 } 319 } 320 321 /* All points we precompute now go into a single array 'val'. 'val_sub[i]' is 322 * a pointer to the subarray for the i-th point. */ 323 val = OPENSSL_malloc(num_val * sizeof(val[0])); 324 if (val == NULL) { 325 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 326 goto err; 327 } 328 OPENSSL_memset(val, 0, num_val * sizeof(val[0])); 329 330 /* allocate points for precomputation */ 331 v = val; 332 for (i = 0; i < total_num; i++) { 333 val_sub[i] = v; 334 for (j = 0; j < ((size_t)1 << (wsize[i] - 1)); j++) { 335 *v = EC_POINT_new(group); 336 if (*v == NULL) { 337 goto err; 338 } 339 v++; 340 } 341 } 342 if (!(v == val + num_val)) { 343 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 344 goto err; 345 } 346 347 if (!(tmp = EC_POINT_new(group))) { 348 goto err; 349 } 350 351 /* prepare precomputed values: 352 * val_sub[i][0] := points[i] 353 * val_sub[i][1] := 3 * points[i] 354 * val_sub[i][2] := 5 * points[i] 355 * ... 356 */ 357 for (i = 0; i < total_num; i++) { 358 if (i < num) { 359 if (!EC_POINT_copy(val_sub[i][0], points[i])) { 360 goto err; 361 } 362 } else if (!EC_POINT_copy(val_sub[i][0], generator)) { 363 goto err; 364 } 365 366 if (wsize[i] > 1) { 367 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) { 368 goto err; 369 } 370 for (j = 1; j < ((size_t)1 << (wsize[i] - 1)); j++) { 371 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) { 372 goto err; 373 } 374 } 375 } 376 } 377 378#if 1 /* optional; window_bits_for_scalar_size assumes we do this step */ 379 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) { 380 goto err; 381 } 382#endif 383 384 r_is_at_infinity = 1; 385 386 for (k = max_len - 1; k >= 0; k--) { 387 if (!r_is_at_infinity && !EC_POINT_dbl(group, r, r, ctx)) { 388 goto err; 389 } 390 391 for (i = 0; i < total_num; i++) { 392 if (wNAF_len[i] > (size_t)k) { 393 int digit = wNAF[i][k]; 394 int is_neg; 395 396 if (digit) { 397 is_neg = digit < 0; 398 399 if (is_neg) { 400 digit = -digit; 401 } 402 403 if (is_neg != r_is_inverted) { 404 if (!r_is_at_infinity && !EC_POINT_invert(group, r, ctx)) { 405 goto err; 406 } 407 r_is_inverted = !r_is_inverted; 408 } 409 410 /* digit > 0 */ 411 412 if (r_is_at_infinity) { 413 if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) { 414 goto err; 415 } 416 r_is_at_infinity = 0; 417 } else { 418 if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) { 419 goto err; 420 } 421 } 422 } 423 } 424 } 425 } 426 427 if (r_is_at_infinity) { 428 if (!EC_POINT_set_to_infinity(group, r)) { 429 goto err; 430 } 431 } else if (r_is_inverted && !EC_POINT_invert(group, r, ctx)) { 432 goto err; 433 } 434 435 ret = 1; 436 437err: 438 BN_CTX_free(new_ctx); 439 EC_POINT_free(tmp); 440 OPENSSL_free(wsize); 441 OPENSSL_free(wNAF_len); 442 if (wNAF != NULL) { 443 for (i = 0; i < total_num; i++) { 444 OPENSSL_free(wNAF[i]); 445 } 446 447 OPENSSL_free(wNAF); 448 } 449 if (val != NULL) { 450 for (i = 0; i < num_val; i++) { 451 EC_POINT_clear_free(val[i]); 452 } 453 454 OPENSSL_free(val); 455 } 456 OPENSSL_free(val_sub); 457 return ret; 458} 459