bn_gcd.c revision 109998
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 return(ret); 144 } 145 146static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) 147 { 148 BIGNUM *t; 149 int shifts=0; 150 151 bn_check_top(a); 152 bn_check_top(b); 153 154 /* 0 <= b <= a */ 155 while (!BN_is_zero(b)) 156 { 157 /* 0 < b <= a */ 158 159 if (BN_is_odd(a)) 160 { 161 if (BN_is_odd(b)) 162 { 163 if (!BN_sub(a,a,b)) goto err; 164 if (!BN_rshift1(a,a)) goto err; 165 if (BN_cmp(a,b) < 0) 166 { t=a; a=b; b=t; } 167 } 168 else /* a odd - b even */ 169 { 170 if (!BN_rshift1(b,b)) goto err; 171 if (BN_cmp(a,b) < 0) 172 { t=a; a=b; b=t; } 173 } 174 } 175 else /* a is even */ 176 { 177 if (BN_is_odd(b)) 178 { 179 if (!BN_rshift1(a,a)) goto err; 180 if (BN_cmp(a,b) < 0) 181 { t=a; a=b; b=t; } 182 } 183 else /* a even - b even */ 184 { 185 if (!BN_rshift1(a,a)) goto err; 186 if (!BN_rshift1(b,b)) goto err; 187 shifts++; 188 } 189 } 190 /* 0 <= b <= a */ 191 } 192 193 if (shifts) 194 { 195 if (!BN_lshift(a,a,shifts)) goto err; 196 } 197 return(a); 198err: 199 return(NULL); 200 } 201 202 203/* solves ax == 1 (mod n) */ 204BIGNUM *BN_mod_inverse(BIGNUM *in, 205 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx) 206 { 207 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL; 208 BIGNUM *ret=NULL; 209 int sign; 210 211 bn_check_top(a); 212 bn_check_top(n); 213 214 BN_CTX_start(ctx); 215 A = BN_CTX_get(ctx); 216 B = BN_CTX_get(ctx); 217 X = BN_CTX_get(ctx); 218 D = BN_CTX_get(ctx); 219 M = BN_CTX_get(ctx); 220 Y = BN_CTX_get(ctx); 221 T = BN_CTX_get(ctx); 222 if (T == NULL) goto err; 223 224 if (in == NULL) 225 R=BN_new(); 226 else 227 R=in; 228 if (R == NULL) goto err; 229 230 BN_one(X); 231 BN_zero(Y); 232 if (BN_copy(B,a) == NULL) goto err; 233 if (BN_copy(A,n) == NULL) goto err; 234 A->neg = 0; 235 if (B->neg || (BN_ucmp(B, A) >= 0)) 236 { 237 if (!BN_nnmod(B, B, A, ctx)) goto err; 238 } 239 sign = -1; 240 /* From B = a mod |n|, A = |n| it follows that 241 * 242 * 0 <= B < A, 243 * -sign*X*a == B (mod |n|), 244 * sign*Y*a == A (mod |n|). 245 */ 246 247 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) 248 { 249 /* Binary inversion algorithm; requires odd modulus. 250 * This is faster than the general algorithm if the modulus 251 * is sufficiently small (about 400 .. 500 bits on 32-bit 252 * sytems, but much more on 64-bit systems) */ 253 int shift; 254 255 while (!BN_is_zero(B)) 256 { 257 /* 258 * 0 < B < |n|, 259 * 0 < A <= |n|, 260 * (1) -sign*X*a == B (mod |n|), 261 * (2) sign*Y*a == A (mod |n|) 262 */ 263 264 /* Now divide B by the maximum possible power of two in the integers, 265 * and divide X by the same value mod |n|. 266 * When we're done, (1) still holds. */ 267 shift = 0; 268 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */ 269 { 270 shift++; 271 272 if (BN_is_odd(X)) 273 { 274 if (!BN_uadd(X, X, n)) goto err; 275 } 276 /* now X is even, so we can easily divide it by two */ 277 if (!BN_rshift1(X, X)) goto err; 278 } 279 if (shift > 0) 280 { 281 if (!BN_rshift(B, B, shift)) goto err; 282 } 283 284 285 /* Same for A and Y. Afterwards, (2) still holds. */ 286 shift = 0; 287 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */ 288 { 289 shift++; 290 291 if (BN_is_odd(Y)) 292 { 293 if (!BN_uadd(Y, Y, n)) goto err; 294 } 295 /* now Y is even */ 296 if (!BN_rshift1(Y, Y)) goto err; 297 } 298 if (shift > 0) 299 { 300 if (!BN_rshift(A, A, shift)) goto err; 301 } 302 303 304 /* We still have (1) and (2). 305 * Both A and B are odd. 306 * The following computations ensure that 307 * 308 * 0 <= B < |n|, 309 * 0 < A < |n|, 310 * (1) -sign*X*a == B (mod |n|), 311 * (2) sign*Y*a == A (mod |n|), 312 * 313 * and that either A or B is even in the next iteration. 314 */ 315 if (BN_ucmp(B, A) >= 0) 316 { 317 /* -sign*(X + Y)*a == B - A (mod |n|) */ 318 if (!BN_uadd(X, X, Y)) goto err; 319 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that 320 * actually makes the algorithm slower */ 321 if (!BN_usub(B, B, A)) goto err; 322 } 323 else 324 { 325 /* sign*(X + Y)*a == A - B (mod |n|) */ 326 if (!BN_uadd(Y, Y, X)) goto err; 327 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */ 328 if (!BN_usub(A, A, B)) goto err; 329 } 330 } 331 } 332 else 333 { 334 /* general inversion algorithm */ 335 336 while (!BN_is_zero(B)) 337 { 338 BIGNUM *tmp; 339 340 /* 341 * 0 < B < A, 342 * (*) -sign*X*a == B (mod |n|), 343 * sign*Y*a == A (mod |n|) 344 */ 345 346 /* (D, M) := (A/B, A%B) ... */ 347 if (BN_num_bits(A) == BN_num_bits(B)) 348 { 349 if (!BN_one(D)) goto err; 350 if (!BN_sub(M,A,B)) goto err; 351 } 352 else if (BN_num_bits(A) == BN_num_bits(B) + 1) 353 { 354 /* A/B is 1, 2, or 3 */ 355 if (!BN_lshift1(T,B)) goto err; 356 if (BN_ucmp(A,T) < 0) 357 { 358 /* A < 2*B, so D=1 */ 359 if (!BN_one(D)) goto err; 360 if (!BN_sub(M,A,B)) goto err; 361 } 362 else 363 { 364 /* A >= 2*B, so D=2 or D=3 */ 365 if (!BN_sub(M,A,T)) goto err; 366 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */ 367 if (BN_ucmp(A,D) < 0) 368 { 369 /* A < 3*B, so D=2 */ 370 if (!BN_set_word(D,2)) goto err; 371 /* M (= A - 2*B) already has the correct value */ 372 } 373 else 374 { 375 /* only D=3 remains */ 376 if (!BN_set_word(D,3)) goto err; 377 /* currently M = A - 2*B, but we need M = A - 3*B */ 378 if (!BN_sub(M,M,B)) goto err; 379 } 380 } 381 } 382 else 383 { 384 if (!BN_div(D,M,A,B,ctx)) goto err; 385 } 386 387 /* Now 388 * A = D*B + M; 389 * thus we have 390 * (**) sign*Y*a == D*B + M (mod |n|). 391 */ 392 393 tmp=A; /* keep the BIGNUM object, the value does not matter */ 394 395 /* (A, B) := (B, A mod B) ... */ 396 A=B; 397 B=M; 398 /* ... so we have 0 <= B < A again */ 399 400 /* Since the former M is now B and the former B is now A, 401 * (**) translates into 402 * sign*Y*a == D*A + B (mod |n|), 403 * i.e. 404 * sign*Y*a - D*A == B (mod |n|). 405 * Similarly, (*) translates into 406 * -sign*X*a == A (mod |n|). 407 * 408 * Thus, 409 * sign*Y*a + D*sign*X*a == B (mod |n|), 410 * i.e. 411 * sign*(Y + D*X)*a == B (mod |n|). 412 * 413 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at 414 * -sign*X*a == B (mod |n|), 415 * sign*Y*a == A (mod |n|). 416 * Note that X and Y stay non-negative all the time. 417 */ 418 419 /* most of the time D is very small, so we can optimize tmp := D*X+Y */ 420 if (BN_is_one(D)) 421 { 422 if (!BN_add(tmp,X,Y)) goto err; 423 } 424 else 425 { 426 if (BN_is_word(D,2)) 427 { 428 if (!BN_lshift1(tmp,X)) goto err; 429 } 430 else if (BN_is_word(D,4)) 431 { 432 if (!BN_lshift(tmp,X,2)) goto err; 433 } 434 else if (D->top == 1) 435 { 436 if (!BN_copy(tmp,X)) goto err; 437 if (!BN_mul_word(tmp,D->d[0])) goto err; 438 } 439 else 440 { 441 if (!BN_mul(tmp,D,X,ctx)) goto err; 442 } 443 if (!BN_add(tmp,tmp,Y)) goto err; 444 } 445 446 M=Y; /* keep the BIGNUM object, the value does not matter */ 447 Y=X; 448 X=tmp; 449 sign = -sign; 450 } 451 } 452 453 /* 454 * The while loop (Euclid's algorithm) ends when 455 * A == gcd(a,n); 456 * we have 457 * sign*Y*a == A (mod |n|), 458 * where Y is non-negative. 459 */ 460 461 if (sign < 0) 462 { 463 if (!BN_sub(Y,n,Y)) goto err; 464 } 465 /* Now Y*a == A (mod |n|). */ 466 467 468 if (BN_is_one(A)) 469 { 470 /* Y*a == 1 (mod |n|) */ 471 if (!Y->neg && BN_ucmp(Y,n) < 0) 472 { 473 if (!BN_copy(R,Y)) goto err; 474 } 475 else 476 { 477 if (!BN_nnmod(R,Y,n,ctx)) goto err; 478 } 479 } 480 else 481 { 482 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE); 483 goto err; 484 } 485 ret=R; 486err: 487 if ((ret == NULL) && (in == NULL)) BN_free(R); 488 BN_CTX_end(ctx); 489 return(ret); 490 } 491