crypto/bn/bn_gcd.c

   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
 115 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);
 116
 117 int 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;
 141 err:
 142         BN_CTX_end(ctx);
 143         bn_check_top(r);
 144         return(ret);
 145         }
 146
 147 static 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);
 200 err:
 201         return(NULL);
 202         }
 203
 204
 205 /* solves ax == 1 (mod n) */
 206 BIGNUM *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;
 488 err:
 489         if ((ret == NULL) && (in == NULL)) BN_free(R);
 490         BN_CTX_end(ctx);
 491         bn_check_top(ret);
 492         return(ret);
 493         }