1/* crypto/bn/bn_gcd.c */ 2/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com) 3 * All rights reserved. 4 * 5 * This package is an SSL implementation written 6 * by Eric Young (eay@cryptsoft.com). 7 * The implementation was written so as to conform with Netscapes SSL. 8 * 9 * This library is free for commercial and non-commercial use as long as 10 * the following conditions are aheared to. The following conditions 11 * apply to all code found in this distribution, be it the RC4, RSA, 12 * lhash, DES, etc., code; not just the SSL code. The SSL documentation 13 * included with this distribution is covered by the same copyright terms 14 * except that the holder is Tim Hudson (tjh@cryptsoft.com). 15 * 16 * Copyright remains Eric Young's, and as such any Copyright notices in 17 * the code are not to be removed. 18 * If this package is used in a product, Eric Young should be given attribution 19 * as the author of the parts of the library used. 20 * This can be in the form of a textual message at program startup or 21 * in documentation (online or textual) provided with the package. 22 * 23 * Redistribution and use in source and binary forms, with or without 24 * modification, are permitted provided that the following conditions 25 * are met: 26 * 1. Redistributions of source code must retain the copyright 27 * notice, this list of conditions and the following disclaimer. 28 * 2. Redistributions in binary form must reproduce the above copyright 29 * notice, this list of conditions and the following disclaimer in the 30 * documentation and/or other materials provided with the distribution. 31 * 3. All advertising materials mentioning features or use of this software 32 * must display the following acknowledgement: 33 * "This product includes cryptographic software written by 34 * Eric Young (eay@cryptsoft.com)" 35 * The word 'cryptographic' can be left out if the rouines from the library 36 * being used are not cryptographic related :-). 37 * 4. If you include any Windows specific code (or a derivative thereof) from 38 * the apps directory (application code) you must include an acknowledgement: 39 * "This product includes software written by Tim Hudson (tjh@cryptsoft.com)" 40 * 41 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND 42 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 43 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 44 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 45 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 46 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 47 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 48 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 49 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 50 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 51 * SUCH DAMAGE. 52 * 53 * The licence and distribution terms for any publically available version or 54 * derivative of this code cannot be changed. i.e. this code cannot simply be 55 * copied and put under another distribution licence 56 * [including the GNU Public Licence.] 57 */ 58/* ==================================================================== 59 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved. 60 * 61 * Redistribution and use in source and binary forms, with or without 62 * modification, are permitted provided that the following conditions 63 * are met: 64 * 65 * 1. Redistributions of source code must retain the above copyright 66 * notice, this list of conditions and the following disclaimer. 67 * 68 * 2. Redistributions in binary form must reproduce the above copyright 69 * notice, this list of conditions and the following disclaimer in 70 * the documentation and/or other materials provided with the 71 * distribution. 72 * 73 * 3. All advertising materials mentioning features or use of this 74 * software must display the following acknowledgment: 75 * "This product includes software developed by the OpenSSL Project 76 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 77 * 78 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 79 * endorse or promote products derived from this software without 80 * prior written permission. For written permission, please contact 81 * openssl-core@openssl.org. 82 * 83 * 5. Products derived from this software may not be called "OpenSSL" 84 * nor may "OpenSSL" appear in their names without prior written 85 * permission of the OpenSSL Project. 86 * 87 * 6. Redistributions of any form whatsoever must retain the following 88 * acknowledgment: 89 * "This product includes software developed by the OpenSSL Project 90 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 91 * 92 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 93 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 94 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 95 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 96 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 97 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 98 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 99 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 100 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 101 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 102 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 103 * OF THE POSSIBILITY OF SUCH DAMAGE. 104 * ==================================================================== 105 * 106 * This product includes cryptographic software written by Eric Young 107 * (eay@cryptsoft.com). This product includes software written by Tim 108 * Hudson (tjh@cryptsoft.com). 109 * 110 */ 111 112#include "cryptlib.h" 113#include "bn_lcl.h" 114 115static BIGNUM *euclid(BIGNUM *a, BIGNUM *b); 116 117int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) 118 { 119 BIGNUM *a,*b,*t; 120 int ret=0; 121 122 bn_check_top(in_a); 123 bn_check_top(in_b); 124 125 BN_CTX_start(ctx); 126 a = BN_CTX_get(ctx); 127 b = BN_CTX_get(ctx); 128 if (a == NULL || b == NULL) goto err; 129 130 if (BN_copy(a,in_a) == NULL) goto err; 131 if (BN_copy(b,in_b) == NULL) goto err; 132 a->neg = 0; 133 b->neg = 0; 134 135 if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; } 136 t=euclid(a,b); 137 if (t == NULL) goto err; 138 139 if (BN_copy(r,t) == NULL) goto err; 140 ret=1; 141err: 142 BN_CTX_end(ctx); 143 bn_check_top(r); 144 return(ret); 145 } 146 147static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) 148 { 149 BIGNUM *t; 150 int shifts=0; 151 152 bn_check_top(a); 153 bn_check_top(b); 154 155 /* 0 <= b <= a */ 156 while (!BN_is_zero(b)) 157 { 158 /* 0 < b <= a */ 159 160 if (BN_is_odd(a)) 161 { 162 if (BN_is_odd(b)) 163 { 164 if (!BN_sub(a,a,b)) goto err; 165 if (!BN_rshift1(a,a)) goto err; 166 if (BN_cmp(a,b) < 0) 167 { t=a; a=b; b=t; } 168 } 169 else /* a odd - b even */ 170 { 171 if (!BN_rshift1(b,b)) goto err; 172 if (BN_cmp(a,b) < 0) 173 { t=a; a=b; b=t; } 174 } 175 } 176 else /* a is even */ 177 { 178 if (BN_is_odd(b)) 179 { 180 if (!BN_rshift1(a,a)) goto err; 181 if (BN_cmp(a,b) < 0) 182 { t=a; a=b; b=t; } 183 } 184 else /* a even - b even */ 185 { 186 if (!BN_rshift1(a,a)) goto err; 187 if (!BN_rshift1(b,b)) goto err; 188 shifts++; 189 } 190 } 191 /* 0 <= b <= a */ 192 } 193 194 if (shifts) 195 { 196 if (!BN_lshift(a,a,shifts)) goto err; 197 } 198 bn_check_top(a); 199 return(a); 200err: 201 return(NULL); 202 } 203 204 205/* solves ax == 1 (mod n) */ 206BIGNUM *BN_mod_inverse(BIGNUM *in, 207 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) 208 { 209 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL; 210 BIGNUM *ret=NULL; 211 int sign; 212 213 bn_check_top(a); 214 bn_check_top(n); 215 216 BN_CTX_start(ctx); 217 A = BN_CTX_get(ctx); 218 B = BN_CTX_get(ctx); 219 X = BN_CTX_get(ctx); 220 D = BN_CTX_get(ctx); 221 M = BN_CTX_get(ctx); 222 Y = BN_CTX_get(ctx); 223 T = BN_CTX_get(ctx); 224 if (T == NULL) goto err; 225 226 if (in == NULL) 227 R=BN_new(); 228 else 229 R=in; 230 if (R == NULL) goto err; 231 232 BN_one(X); 233 BN_zero(Y); 234 if (BN_copy(B,a) == NULL) goto err; 235 if (BN_copy(A,n) == NULL) goto err; 236 A->neg = 0; 237 if (B->neg || (BN_ucmp(B, A) >= 0)) 238 { 239 if (!BN_nnmod(B, B, A, ctx)) goto err; 240 } 241 sign = -1; 242 /* From B = a mod |n|, A = |n| it follows that 243 * 244 * 0 <= B < A, 245 * -sign*X*a == B (mod |n|), 246 * sign*Y*a == A (mod |n|). 247 */ 248 249 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) 250 { 251 /* Binary inversion algorithm; requires odd modulus. 252 * This is faster than the general algorithm if the modulus 253 * is sufficiently small (about 400 .. 500 bits on 32-bit 254 * sytems, but much more on 64-bit systems) */ 255 int shift; 256 257 while (!BN_is_zero(B)) 258 { 259 /* 260 * 0 < B < |n|, 261 * 0 < A <= |n|, 262 * (1) -sign*X*a == B (mod |n|), 263 * (2) sign*Y*a == A (mod |n|) 264 */ 265 266 /* Now divide B by the maximum possible power of two in the integers, 267 * and divide X by the same value mod |n|. 268 * When we're done, (1) still holds. */ 269 shift = 0; 270 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */ 271 { 272 shift++; 273 274 if (BN_is_odd(X)) 275 { 276 if (!BN_uadd(X, X, n)) goto err; 277 } 278 /* now X is even, so we can easily divide it by two */ 279 if (!BN_rshift1(X, X)) goto err; 280 } 281 if (shift > 0) 282 { 283 if (!BN_rshift(B, B, shift)) goto err; 284 } 285 286 287 /* Same for A and Y. Afterwards, (2) still holds. */ 288 shift = 0; 289 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */ 290 { 291 shift++; 292 293 if (BN_is_odd(Y)) 294 { 295 if (!BN_uadd(Y, Y, n)) goto err; 296 } 297 /* now Y is even */ 298 if (!BN_rshift1(Y, Y)) goto err; 299 } 300 if (shift > 0) 301 { 302 if (!BN_rshift(A, A, shift)) goto err; 303 } 304 305 306 /* We still have (1) and (2). 307 * Both A and B are odd. 308 * The following computations ensure that 309 * 310 * 0 <= B < |n|, 311 * 0 < A < |n|, 312 * (1) -sign*X*a == B (mod |n|), 313 * (2) sign*Y*a == A (mod |n|), 314 * 315 * and that either A or B is even in the next iteration. 316 */ 317 if (BN_ucmp(B, A) >= 0) 318 { 319 /* -sign*(X + Y)*a == B - A (mod |n|) */ 320 if (!BN_uadd(X, X, Y)) goto err; 321 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that 322 * actually makes the algorithm slower */ 323 if (!BN_usub(B, B, A)) goto err; 324 } 325 else 326 { 327 /* sign*(X + Y)*a == A - B (mod |n|) */ 328 if (!BN_uadd(Y, Y, X)) goto err; 329 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */ 330 if (!BN_usub(A, A, B)) goto err; 331 } 332 } 333 } 334 else 335 { 336 /* general inversion algorithm */ 337 338 while (!BN_is_zero(B)) 339 { 340 BIGNUM *tmp; 341 342 /* 343 * 0 < B < A, 344 * (*) -sign*X*a == B (mod |n|), 345 * sign*Y*a == A (mod |n|) 346 */ 347 348 /* (D, M) := (A/B, A%B) ... */ 349 if (BN_num_bits(A) == BN_num_bits(B)) 350 { 351 if (!BN_one(D)) goto err; 352 if (!BN_sub(M,A,B)) goto err; 353 } 354 else if (BN_num_bits(A) == BN_num_bits(B) + 1) 355 { 356 /* A/B is 1, 2, or 3 */ 357 if (!BN_lshift1(T,B)) goto err; 358 if (BN_ucmp(A,T) < 0) 359 { 360 /* A < 2*B, so D=1 */ 361 if (!BN_one(D)) goto err; 362 if (!BN_sub(M,A,B)) goto err; 363 } 364 else 365 { 366 /* A >= 2*B, so D=2 or D=3 */ 367 if (!BN_sub(M,A,T)) goto err; 368 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */ 369 if (BN_ucmp(A,D) < 0) 370 { 371 /* A < 3*B, so D=2 */ 372 if (!BN_set_word(D,2)) goto err; 373 /* M (= A - 2*B) already has the correct value */ 374 } 375 else 376 { 377 /* only D=3 remains */ 378 if (!BN_set_word(D,3)) goto err; 379 /* currently M = A - 2*B, but we need M = A - 3*B */ 380 if (!BN_sub(M,M,B)) goto err; 381 } 382 } 383 } 384 else 385 { 386 if (!BN_div(D,M,A,B,ctx)) goto err; 387 } 388 389 /* Now 390 * A = D*B + M; 391 * thus we have 392 * (**) sign*Y*a == D*B + M (mod |n|). 393 */ 394 395 tmp=A; /* keep the BIGNUM object, the value does not matter */ 396 397 /* (A, B) := (B, A mod B) ... */ 398 A=B; 399 B=M; 400 /* ... so we have 0 <= B < A again */ 401 402 /* Since the former M is now B and the former B is now A, 403 * (**) translates into 404 * sign*Y*a == D*A + B (mod |n|), 405 * i.e. 406 * sign*Y*a - D*A == B (mod |n|). 407 * Similarly, (*) translates into 408 * -sign*X*a == A (mod |n|). 409 * 410 * Thus, 411 * sign*Y*a + D*sign*X*a == B (mod |n|), 412 * i.e. 413 * sign*(Y + D*X)*a == B (mod |n|). 414 * 415 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 416 * -sign*X*a == B (mod |n|), 417 * sign*Y*a == A (mod |n|). 418 * Note that X and Y stay non-negative all the time. 419 */ 420 421 /* most of the time D is very small, so we can optimize tmp := D*X+Y */ 422 if (BN_is_one(D)) 423 { 424 if (!BN_add(tmp,X,Y)) goto err; 425 } 426 else 427 { 428 if (BN_is_word(D,2)) 429 { 430 if (!BN_lshift1(tmp,X)) goto err; 431 } 432 else if (BN_is_word(D,4)) 433 { 434 if (!BN_lshift(tmp,X,2)) goto err; 435 } 436 else if (D->top == 1) 437 { 438 if (!BN_copy(tmp,X)) goto err; 439 if (!BN_mul_word(tmp,D->d[0])) goto err; 440 } 441 else 442 { 443 if (!BN_mul(tmp,D,X,ctx)) goto err; 444 } 445 if (!BN_add(tmp,tmp,Y)) goto err; 446 } 447 448 M=Y; /* keep the BIGNUM object, the value does not matter */ 449 Y=X; 450 X=tmp; 451 sign = -sign; 452 } 453 } 454 455 /* 456 * The while loop (Euclid's algorithm) ends when 457 * A == gcd(a,n); 458 * we have 459 * sign*Y*a == A (mod |n|), 460 * where Y is non-negative. 461 */ 462 463 if (sign < 0) 464 { 465 if (!BN_sub(Y,n,Y)) goto err; 466 } 467 /* Now Y*a == A (mod |n|). */ 468 469 470 if (BN_is_one(A)) 471 { 472 /* Y*a == 1 (mod |n|) */ 473 if (!Y->neg && BN_ucmp(Y,n) < 0) 474 { 475 if (!BN_copy(R,Y)) goto err; 476 } 477 else 478 { 479 if (!BN_nnmod(R,Y,n,ctx)) goto err; 480 } 481 } 482 else 483 { 484 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE); 485 goto err; 486 } 487 ret=R; 488err: 489 if ((ret == NULL) && (in == NULL)) BN_free(R); 490 BN_CTX_end(ctx); 491 bn_check_top(ret); 492 return(ret); 493 } 494