1/* $OpenBSD: ec_mult.c,v 1.31 2023/06/24 17:49:44 jsing Exp $ */ 2/* 3 * Originally written by Bodo Moeller and Nils Larsch for the OpenSSL project. 4 */ 5/* ==================================================================== 6 * Copyright (c) 1998-2007 The OpenSSL Project. All rights reserved. 7 * 8 * Redistribution and use in source and binary forms, with or without 9 * modification, are permitted provided that the following conditions 10 * are met: 11 * 12 * 1. Redistributions of source code must retain the above copyright 13 * notice, this list of conditions and the following disclaimer. 14 * 15 * 2. Redistributions in binary form must reproduce the above copyright 16 * notice, this list of conditions and the following disclaimer in 17 * the documentation and/or other materials provided with the 18 * distribution. 19 * 20 * 3. All advertising materials mentioning features or use of this 21 * software must display the following acknowledgment: 22 * "This product includes software developed by the OpenSSL Project 23 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 24 * 25 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 26 * endorse or promote products derived from this software without 27 * prior written permission. For written permission, please contact 28 * openssl-core@openssl.org. 29 * 30 * 5. Products derived from this software may not be called "OpenSSL" 31 * nor may "OpenSSL" appear in their names without prior written 32 * permission of the OpenSSL Project. 33 * 34 * 6. Redistributions of any form whatsoever must retain the following 35 * acknowledgment: 36 * "This product includes software developed by the OpenSSL Project 37 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 38 * 39 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 40 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 41 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 42 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 43 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 44 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 45 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 46 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 47 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 48 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 49 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 50 * OF THE POSSIBILITY OF SUCH DAMAGE. 51 * ==================================================================== 52 * 53 * This product includes cryptographic software written by Eric Young 54 * (eay@cryptsoft.com). This product includes software written by Tim 55 * Hudson (tjh@cryptsoft.com). 56 * 57 */ 58/* ==================================================================== 59 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. 60 * Portions of this software developed by SUN MICROSYSTEMS, INC., 61 * and contributed to the OpenSSL project. 62 */ 63 64#include <string.h> 65 66#include <openssl/err.h> 67 68#include "ec_local.h" 69 70/* 71 * This file implements the wNAF-based interleaving multi-exponentation method 72 * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp>); 73 * for multiplication with precomputation, we use wNAF splitting 74 * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#fastexp>). 75 */ 76 77/* Determine the modified width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'. 78 * This is an array r[] of values that are either zero or odd with an 79 * absolute value less than 2^w satisfying 80 * scalar = \sum_j r[j]*2^j 81 * where at most one of any w+1 consecutive digits is non-zero 82 * with the exception that the most significant digit may be only 83 * w-1 zeros away from that next non-zero digit. 84 */ 85static signed char * 86compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len) 87{ 88 int window_val; 89 int ok = 0; 90 signed char *r = NULL; 91 int sign = 1; 92 int bit, next_bit, mask; 93 size_t len = 0, j; 94 95 if (BN_is_zero(scalar)) { 96 r = malloc(1); 97 if (!r) { 98 ECerror(ERR_R_MALLOC_FAILURE); 99 goto err; 100 } 101 r[0] = 0; 102 *ret_len = 1; 103 return r; 104 } 105 if (w <= 0 || w > 7) { 106 /* 'signed char' can represent integers with 107 * absolute values less than 2^7 */ 108 ECerror(ERR_R_INTERNAL_ERROR); 109 goto err; 110 } 111 bit = 1 << w; /* at most 128 */ 112 next_bit = bit << 1; /* at most 256 */ 113 mask = next_bit - 1; /* at most 255 */ 114 115 if (BN_is_negative(scalar)) { 116 sign = -1; 117 } 118 if (scalar->d == NULL || scalar->top == 0) { 119 ECerror(ERR_R_INTERNAL_ERROR); 120 goto err; 121 } 122 len = BN_num_bits(scalar); 123 r = malloc(len + 1); /* modified wNAF may be one digit longer than 124 * binary representation (*ret_len will be 125 * set to the actual length, i.e. at most 126 * BN_num_bits(scalar) + 1) */ 127 if (r == NULL) { 128 ECerror(ERR_R_MALLOC_FAILURE); 129 goto err; 130 } 131 window_val = scalar->d[0] & mask; 132 j = 0; 133 while ((window_val != 0) || (j + w + 1 < len)) { 134 /* if j+w+1 >= len, window_val will not increase */ 135 int digit = 0; 136 137 /* 0 <= window_val <= 2^(w+1) */ 138 if (window_val & 1) { 139 /* 0 < window_val < 2^(w+1) */ 140 if (window_val & bit) { 141 digit = window_val - next_bit; /* -2^w < digit < 0 */ 142 143#if 1 /* modified wNAF */ 144 if (j + w + 1 >= len) { 145 /* 146 * special case for generating 147 * modified wNAFs: no new bits will 148 * be added into window_val, so using 149 * a positive digit here will 150 * decrease the total length of the 151 * representation 152 */ 153 154 digit = window_val & (mask >> 1); /* 0 < digit < 2^w */ 155 } 156#endif 157 } else { 158 digit = window_val; /* 0 < digit < 2^w */ 159 } 160 161 if (digit <= -bit || digit >= bit || !(digit & 1)) { 162 ECerror(ERR_R_INTERNAL_ERROR); 163 goto err; 164 } 165 window_val -= digit; 166 167 /* 168 * now window_val is 0 or 2^(w+1) in standard wNAF 169 * generation; for modified window NAFs, it may also 170 * be 2^w 171 */ 172 if (window_val != 0 && window_val != next_bit && window_val != bit) { 173 ECerror(ERR_R_INTERNAL_ERROR); 174 goto err; 175 } 176 } 177 r[j++] = sign * digit; 178 179 window_val >>= 1; 180 window_val += bit * BN_is_bit_set(scalar, j + w); 181 182 if (window_val > next_bit) { 183 ECerror(ERR_R_INTERNAL_ERROR); 184 goto err; 185 } 186 } 187 188 if (j > len + 1) { 189 ECerror(ERR_R_INTERNAL_ERROR); 190 goto err; 191 } 192 len = j; 193 ok = 1; 194 195 err: 196 if (!ok) { 197 free(r); 198 r = NULL; 199 } 200 if (ok) 201 *ret_len = len; 202 return r; 203} 204 205 206/* TODO: table should be optimised for the wNAF-based implementation, 207 * sometimes smaller windows will give better performance 208 * (thus the boundaries should be increased) 209 */ 210#define EC_window_bits_for_scalar_size(b) \ 211 ((size_t) \ 212 ((b) >= 2000 ? 6 : \ 213 (b) >= 800 ? 5 : \ 214 (b) >= 300 ? 4 : \ 215 (b) >= 70 ? 3 : \ 216 (b) >= 20 ? 2 : \ 217 1)) 218 219/* Compute 220 * \sum scalars[i]*points[i], 221 * also including 222 * scalar*generator 223 * in the addition if scalar != NULL 224 */ 225int 226ec_wNAF_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, 227 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) 228{ 229 const EC_POINT *generator = NULL; 230 EC_POINT *tmp = NULL; 231 size_t totalnum; 232 size_t numblocks = 0; /* for wNAF splitting */ 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 signed char *tmp_wNAF = NULL; 240 size_t *wNAF_len = NULL; 241 size_t max_len = 0; 242 size_t num_val; 243 EC_POINT **val = NULL; /* precomputation */ 244 EC_POINT **v; 245 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' or 246 * 'pre_comp->points' */ 247 int num_scalar = 0; /* flag: will be set to 1 if 'scalar' must be 248 * treated like other scalars, i.e. 249 * precomputation is not available */ 250 int ret = 0; 251 252 if (group->meth != r->meth) { 253 ECerror(EC_R_INCOMPATIBLE_OBJECTS); 254 return 0; 255 } 256 if ((scalar == NULL) && (num == 0)) { 257 return EC_POINT_set_to_infinity(group, r); 258 } 259 for (i = 0; i < num; i++) { 260 if (group->meth != points[i]->meth) { 261 ECerror(EC_R_INCOMPATIBLE_OBJECTS); 262 return 0; 263 } 264 } 265 266 if (scalar != NULL) { 267 generator = EC_GROUP_get0_generator(group); 268 if (generator == NULL) { 269 ECerror(EC_R_UNDEFINED_GENERATOR); 270 goto err; 271 } 272 273 numblocks = 1; 274 num_scalar = 1; /* treat 'scalar' like 'num'-th 275 * element of 'scalars' */ 276 } 277 totalnum = num + numblocks; 278 279 /* includes space for pivot */ 280 wNAF = reallocarray(NULL, (totalnum + 1), sizeof wNAF[0]); 281 if (wNAF == NULL) { 282 ECerror(ERR_R_MALLOC_FAILURE); 283 goto err; 284 } 285 286 wNAF[0] = NULL; /* preliminary pivot */ 287 288 wsize = reallocarray(NULL, totalnum, sizeof wsize[0]); 289 wNAF_len = reallocarray(NULL, totalnum, sizeof wNAF_len[0]); 290 val_sub = reallocarray(NULL, totalnum, sizeof val_sub[0]); 291 292 if (wsize == NULL || wNAF_len == NULL || val_sub == NULL) { 293 ECerror(ERR_R_MALLOC_FAILURE); 294 goto err; 295 } 296 297 /* num_val will be the total number of temporarily precomputed points */ 298 num_val = 0; 299 300 for (i = 0; i < num + num_scalar; i++) { 301 size_t bits; 302 303 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar); 304 wsize[i] = EC_window_bits_for_scalar_size(bits); 305 num_val += (size_t) 1 << (wsize[i] - 1); 306 wNAF[i + 1] = NULL; /* make sure we always have a pivot */ 307 wNAF[i] = compute_wNAF((i < num ? scalars[i] : scalar), wsize[i], &wNAF_len[i]); 308 if (wNAF[i] == NULL) 309 goto err; 310 if (wNAF_len[i] > max_len) 311 max_len = wNAF_len[i]; 312 } 313 314 if (numblocks) { 315 /* we go here iff scalar != NULL */ 316 317 if (num_scalar != 1) { 318 ECerror(ERR_R_INTERNAL_ERROR); 319 goto err; 320 } 321 } 322 /* 323 * All points we precompute now go into a single array 'val'. 324 * 'val_sub[i]' is a pointer to the subarray for the i-th point, or 325 * to a subarray of 'pre_comp->points' if we already have 326 * precomputation. 327 */ 328 val = reallocarray(NULL, (num_val + 1), sizeof val[0]); 329 if (val == NULL) { 330 ECerror(ERR_R_MALLOC_FAILURE); 331 goto err; 332 } 333 val[num_val] = NULL; /* pivot element */ 334 335 /* allocate points for precomputation */ 336 v = val; 337 for (i = 0; i < num + num_scalar; i++) { 338 val_sub[i] = v; 339 for (j = 0; j < ((size_t) 1 << (wsize[i] - 1)); j++) { 340 *v = EC_POINT_new(group); 341 if (*v == NULL) 342 goto err; 343 v++; 344 } 345 } 346 if (!(v == val + num_val)) { 347 ECerror(ERR_R_INTERNAL_ERROR); 348 goto err; 349 } 350 if (!(tmp = EC_POINT_new(group))) 351 goto err; 352 353 /* 354 * prepare precomputed values: val_sub[i][0] := points[i] 355 * val_sub[i][1] := 3 * points[i] val_sub[i][2] := 5 * points[i] ... 356 */ 357 for (i = 0; i < num + num_scalar; i++) { 358 if (i < num) { 359 if (!EC_POINT_copy(val_sub[i][0], points[i])) 360 goto err; 361 } else { 362 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 for (j = 1; j < ((size_t) 1 << (wsize[i] - 1)); j++) { 370 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) 371 goto err; 372 } 373 } 374 } 375 376 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) 377 goto err; 378 379 r_is_at_infinity = 1; 380 381 for (k = max_len - 1; k >= 0; k--) { 382 if (!r_is_at_infinity) { 383 if (!EC_POINT_dbl(group, r, r, ctx)) 384 goto err; 385 } 386 for (i = 0; i < totalnum; i++) { 387 if (wNAF_len[i] > (size_t) k) { 388 int digit = wNAF[i][k]; 389 int is_neg; 390 391 if (digit) { 392 is_neg = digit < 0; 393 394 if (is_neg) 395 digit = -digit; 396 397 if (is_neg != r_is_inverted) { 398 if (!r_is_at_infinity) { 399 if (!EC_POINT_invert(group, r, ctx)) 400 goto err; 401 } 402 r_is_inverted = !r_is_inverted; 403 } 404 /* digit > 0 */ 405 406 if (r_is_at_infinity) { 407 if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) 408 goto err; 409 r_is_at_infinity = 0; 410 } else { 411 if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) 412 goto err; 413 } 414 } 415 } 416 } 417 } 418 419 if (r_is_at_infinity) { 420 if (!EC_POINT_set_to_infinity(group, r)) 421 goto err; 422 } else { 423 if (r_is_inverted) 424 if (!EC_POINT_invert(group, r, ctx)) 425 goto err; 426 } 427 428 ret = 1; 429 430 err: 431 EC_POINT_free(tmp); 432 free(wsize); 433 free(wNAF_len); 434 free(tmp_wNAF); 435 if (wNAF != NULL) { 436 signed char **w; 437 438 for (w = wNAF; *w != NULL; w++) 439 free(*w); 440 441 free(wNAF); 442 } 443 if (val != NULL) { 444 for (v = val; *v != NULL; v++) 445 EC_POINT_free(*v); 446 free(val); 447 } 448 free(val_sub); 449 return ret; 450} 451