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 (at) 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 (at) cryptsoft.com). This product includes software written by Tim 52 * Hudson (tjh (at) 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-exponentiation method 82 * at: 83 * http://link.springer.com/chapter/10.1007%2F3-540-45537-X_13 84 * http://www.bmoeller.de/pdf/TI-01-08.multiexp.pdf */ 85 86 /* Determine the modified width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'. 87 * This is an array r[] of values that are either zero or odd with an 88 * absolute value less than 2^w satisfying 89 * scalar = \sum_j r[j]*2^j 90 * where at most one of any w+1 consecutive digits is non-zero 91 * with the exception that the most significant digit may be only 92 * w-1 zeros away from that next non-zero digit. 93 */ 94 static int8_t *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len) { 95 int window_val; 96 int ok = 0; 97 int8_t *r = NULL; 98 int sign = 1; 99 int bit, next_bit, mask; 100 size_t len = 0, j; 101 102 if (BN_is_zero(scalar)) { 103 r = OPENSSL_malloc(1); 104 if (!r) { 105 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 106 goto err; 107 } 108 r[0] = 0; 109 *ret_len = 1; 110 return r; 111 } 112 113 /* 'int8_t' can represent integers with absolute values less than 2^7. */ 114 if (w <= 0 || w > 7) { 115 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 116 goto err; 117 } 118 bit = 1 << w; /* at most 128 */ 119 next_bit = bit << 1; /* at most 256 */ 120 mask = next_bit - 1; /* at most 255 */ 121 122 if (BN_is_negative(scalar)) { 123 sign = -1; 124 } 125 126 if (scalar->d == NULL || scalar->top == 0) { 127 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 128 goto err; 129 } 130 131 len = BN_num_bits(scalar); 132 /* The modified wNAF may be one digit longer than binary representation 133 * (*ret_len will be set to the actual length, i.e. at most 134 * BN_num_bits(scalar) + 1). */ 135 r = OPENSSL_malloc(len + 1); 136 if (r == NULL) { 137 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 138 goto err; 139 } 140 window_val = scalar->d[0] & mask; 141 j = 0; 142 /* If j+w+1 >= len, window_val will not increase. */ 143 while (window_val != 0 || j + w + 1 < len) { 144 int digit = 0; 145 146 /* 0 <= window_val <= 2^(w+1) */ 147 148 if (window_val & 1) { 149 /* 0 < window_val < 2^(w+1) */ 150 151 if (window_val & bit) { 152 digit = window_val - next_bit; /* -2^w < digit < 0 */ 153 154 #if 1 /* modified wNAF */ 155 if (j + w + 1 >= len) { 156 /* special case for generating modified wNAFs: 157 * no new bits will be added into window_val, 158 * so using a positive digit here will decrease 159 * the total length of the representation */ 160 161 digit = window_val & (mask >> 1); /* 0 < digit < 2^w */ 162 } 163 #endif 164 } else { 165 digit = window_val; /* 0 < digit < 2^w */ 166 } 167 168 if (digit <= -bit || digit >= bit || !(digit & 1)) { 169 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 170 goto err; 171 } 172 173 window_val -= digit; 174 175 /* Now window_val is 0 or 2^(w+1) in standard wNAF generation; 176 * for modified window NAFs, it may also be 2^w. */ 177 if (window_val != 0 && window_val != next_bit && window_val != bit) { 178 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 179 goto err; 180 } 181 } 182 183 r[j++] = sign * digit; 184 185 window_val >>= 1; 186 window_val += bit * BN_is_bit_set(scalar, j + w); 187 188 if (window_val > next_bit) { 189 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 190 goto err; 191 } 192 } 193 194 if (j > len + 1) { 195 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 196 goto err; 197 } 198 len = j; 199 ok = 1; 200 201 err: 202 if (!ok) { 203 OPENSSL_free(r); 204 r = NULL; 205 } 206 if (ok) { 207 *ret_len = len; 208 } 209 return r; 210 } 211 212 213 /* TODO: table should be optimised for the wNAF-based implementation, 214 * sometimes smaller windows will give better performance 215 * (thus the boundaries should be increased) 216 */ 217 static size_t window_bits_for_scalar_size(size_t b) { 218 if (b >= 2000) { 219 return 6; 220 } 221 222 if (b >= 800) { 223 return 5; 224 } 225 226 if (b >= 300) { 227 return 4; 228 } 229 230 if (b >= 70) { 231 return 3; 232 } 233 234 if (b >= 20) { 235 return 2; 236 } 237 238 return 1; 239 } 240 241 int ec_wNAF_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar, 242 const EC_POINT *p, const BIGNUM *p_scalar, BN_CTX *ctx) { 243 BN_CTX *new_ctx = NULL; 244 const EC_POINT *generator = NULL; 245 EC_POINT *tmp = NULL; 246 size_t total_num = 0; 247 size_t i, j; 248 int k; 249 int r_is_inverted = 0; 250 int r_is_at_infinity = 1; 251 size_t *wsize = NULL; /* individual window sizes */ 252 int8_t **wNAF = NULL; /* individual wNAFs */ 253 size_t *wNAF_len = NULL; 254 size_t max_len = 0; 255 size_t num_val = 0; 256 EC_POINT **val = NULL; /* precomputation */ 257 EC_POINT **v; 258 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */ 259 int ret = 0; 260 261 if (ctx == NULL) { 262 ctx = new_ctx = BN_CTX_new(); 263 if (ctx == NULL) { 264 goto err; 265 } 266 } 267 268 /* TODO: This function used to take |points| and |scalars| as arrays of 269 * |num| elements. The code below should be simplified to work in terms of |p| 270 * and |p_scalar|. */ 271 size_t num = p != NULL ? 1 : 0; 272 const EC_POINT **points = p != NULL ? &p : NULL; 273 const BIGNUM **scalars = p != NULL ? &p_scalar : NULL; 274 275 total_num = num; 276 277 if (g_scalar != NULL) { 278 generator = EC_GROUP_get0_generator(group); 279 if (generator == NULL) { 280 OPENSSL_PUT_ERROR(EC, EC_R_UNDEFINED_GENERATOR); 281 goto err; 282 } 283 284 ++total_num; /* treat 'g_scalar' like 'num'-th element of 'scalars' */ 285 } 286 287 288 wsize = OPENSSL_malloc(total_num * sizeof(wsize[0])); 289 wNAF_len = OPENSSL_malloc(total_num * sizeof(wNAF_len[0])); 290 wNAF = OPENSSL_malloc(total_num * sizeof(wNAF[0])); 291 val_sub = OPENSSL_malloc(total_num * sizeof(val_sub[0])); 292 293 /* Ensure wNAF is initialised in case we end up going to err. */ 294 if (wNAF != NULL) { 295 OPENSSL_memset(wNAF, 0, total_num * sizeof(wNAF[0])); 296 } 297 298 if (!wsize || !wNAF_len || !wNAF || !val_sub) { 299 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 300 goto err; 301 } 302 303 /* num_val will be the total number of temporarily precomputed points */ 304 num_val = 0; 305 306 for (i = 0; i < total_num; i++) { 307 size_t bits; 308 309 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(g_scalar); 310 wsize[i] = window_bits_for_scalar_size(bits); 311 num_val += (size_t)1 << (wsize[i] - 1); 312 wNAF[i] = 313 compute_wNAF((i < num ? scalars[i] : g_scalar), wsize[i], &wNAF_len[i]); 314 if (wNAF[i] == NULL) { 315 goto err; 316 } 317 if (wNAF_len[i] > max_len) { 318 max_len = wNAF_len[i]; 319 } 320 } 321 322 /* All points we precompute now go into a single array 'val'. 'val_sub[i]' is 323 * a pointer to the subarray for the i-th point. */ 324 val = OPENSSL_malloc(num_val * sizeof(val[0])); 325 if (val == NULL) { 326 OPENSSL_PUT_ERROR(EC, ERR_R_MALLOC_FAILURE); 327 goto err; 328 } 329 OPENSSL_memset(val, 0, num_val * sizeof(val[0])); 330 331 /* allocate points for precomputation */ 332 v = val; 333 for (i = 0; i < total_num; i++) { 334 val_sub[i] = v; 335 for (j = 0; j < ((size_t)1 << (wsize[i] - 1)); j++) { 336 *v = EC_POINT_new(group); 337 if (*v == NULL) { 338 goto err; 339 } 340 v++; 341 } 342 } 343 if (!(v == val + num_val)) { 344 OPENSSL_PUT_ERROR(EC, ERR_R_INTERNAL_ERROR); 345 goto err; 346 } 347 348 if (!(tmp = EC_POINT_new(group))) { 349 goto err; 350 } 351 352 /* prepare precomputed values: 353 * val_sub[i][0] := points[i] 354 * val_sub[i][1] := 3 * points[i] 355 * val_sub[i][2] := 5 * points[i] 356 * ... 357 */ 358 for (i = 0; i < total_num; i++) { 359 if (i < num) { 360 if (!EC_POINT_copy(val_sub[i][0], points[i])) { 361 goto err; 362 } 363 } else if (!EC_POINT_copy(val_sub[i][0], generator)) { 364 goto err; 365 } 366 367 if (wsize[i] > 1) { 368 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) { 369 goto err; 370 } 371 for (j = 1; j < ((size_t)1 << (wsize[i] - 1)); j++) { 372 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) { 373 goto err; 374 } 375 } 376 } 377 } 378 379 #if 1 /* optional; window_bits_for_scalar_size assumes we do this step */ 380 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) { 381 goto err; 382 } 383 #endif 384 385 r_is_at_infinity = 1; 386 387 for (k = max_len - 1; k >= 0; k--) { 388 if (!r_is_at_infinity && !EC_POINT_dbl(group, r, r, ctx)) { 389 goto err; 390 } 391 392 for (i = 0; i < total_num; i++) { 393 if (wNAF_len[i] > (size_t)k) { 394 int digit = wNAF[i][k]; 395 int is_neg; 396 397 if (digit) { 398 is_neg = digit < 0; 399 400 if (is_neg) { 401 digit = -digit; 402 } 403 404 if (is_neg != r_is_inverted) { 405 if (!r_is_at_infinity && !EC_POINT_invert(group, r, ctx)) { 406 goto err; 407 } 408 r_is_inverted = !r_is_inverted; 409 } 410 411 /* digit > 0 */ 412 413 if (r_is_at_infinity) { 414 if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) { 415 goto err; 416 } 417 r_is_at_infinity = 0; 418 } else { 419 if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) { 420 goto err; 421 } 422 } 423 } 424 } 425 } 426 } 427 428 if (r_is_at_infinity) { 429 if (!EC_POINT_set_to_infinity(group, r)) { 430 goto err; 431 } 432 } else if (r_is_inverted && !EC_POINT_invert(group, r, ctx)) { 433 goto err; 434 } 435 436 ret = 1; 437 438 err: 439 BN_CTX_free(new_ctx); 440 EC_POINT_free(tmp); 441 OPENSSL_free(wsize); 442 OPENSSL_free(wNAF_len); 443 if (wNAF != NULL) { 444 for (i = 0; i < total_num; i++) { 445 OPENSSL_free(wNAF[i]); 446 } 447 448 OPENSSL_free(wNAF); 449 } 450 if (val != NULL) { 451 for (i = 0; i < num_val; i++) { 452 EC_POINT_clear_free(val[i]); 453 } 454 455 OPENSSL_free(val); 456 } 457 OPENSSL_free(val_sub); 458 return ret; 459 } 460
