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