OpenSSL  1.0.1c
 All Classes Files Functions Variables Typedefs Enumerations Enumerator Macros
bn_gcd.c
Go to the documentation of this file.
1 /* crypto/bn/bn_gcd.c */
2 /* Copyright (C) 1995-1998 Eric Young ([email protected])
3  * All rights reserved.
4  *
5  * This package is an SSL implementation written
6  * by Eric Young ([email protected]).
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 ([email protected]).
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 ([email protected])"
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 ([email protected])"
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  * [email protected].
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  * ([email protected]). This product includes software written by Tim
108  * Hudson ([email protected]).
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 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
207  const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx);
208 BIGNUM *BN_mod_inverse(BIGNUM *in,
209  const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
210  {
211  BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
212  BIGNUM *ret=NULL;
213  int sign;
214 
215  if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0))
216  {
217  return BN_mod_inverse_no_branch(in, a, n, ctx);
218  }
219 
220  bn_check_top(a);
221  bn_check_top(n);
222 
223  BN_CTX_start(ctx);
224  A = BN_CTX_get(ctx);
225  B = BN_CTX_get(ctx);
226  X = BN_CTX_get(ctx);
227  D = BN_CTX_get(ctx);
228  M = BN_CTX_get(ctx);
229  Y = BN_CTX_get(ctx);
230  T = BN_CTX_get(ctx);
231  if (T == NULL) goto err;
232 
233  if (in == NULL)
234  R=BN_new();
235  else
236  R=in;
237  if (R == NULL) goto err;
238 
239  BN_one(X);
240  BN_zero(Y);
241  if (BN_copy(B,a) == NULL) goto err;
242  if (BN_copy(A,n) == NULL) goto err;
243  A->neg = 0;
244  if (B->neg || (BN_ucmp(B, A) >= 0))
245  {
246  if (!BN_nnmod(B, B, A, ctx)) goto err;
247  }
248  sign = -1;
249  /* From B = a mod |n|, A = |n| it follows that
250  *
251  * 0 <= B < A,
252  * -sign*X*a == B (mod |n|),
253  * sign*Y*a == A (mod |n|).
254  */
255 
256  if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048)))
257  {
258  /* Binary inversion algorithm; requires odd modulus.
259  * This is faster than the general algorithm if the modulus
260  * is sufficiently small (about 400 .. 500 bits on 32-bit
261  * sytems, but much more on 64-bit systems) */
262  int shift;
263 
264  while (!BN_is_zero(B))
265  {
266  /*
267  * 0 < B < |n|,
268  * 0 < A <= |n|,
269  * (1) -sign*X*a == B (mod |n|),
270  * (2) sign*Y*a == A (mod |n|)
271  */
272 
273  /* Now divide B by the maximum possible power of two in the integers,
274  * and divide X by the same value mod |n|.
275  * When we're done, (1) still holds. */
276  shift = 0;
277  while (!BN_is_bit_set(B, shift)) /* note that 0 < B */
278  {
279  shift++;
280 
281  if (BN_is_odd(X))
282  {
283  if (!BN_uadd(X, X, n)) goto err;
284  }
285  /* now X is even, so we can easily divide it by two */
286  if (!BN_rshift1(X, X)) goto err;
287  }
288  if (shift > 0)
289  {
290  if (!BN_rshift(B, B, shift)) goto err;
291  }
292 
293 
294  /* Same for A and Y. Afterwards, (2) still holds. */
295  shift = 0;
296  while (!BN_is_bit_set(A, shift)) /* note that 0 < A */
297  {
298  shift++;
299 
300  if (BN_is_odd(Y))
301  {
302  if (!BN_uadd(Y, Y, n)) goto err;
303  }
304  /* now Y is even */
305  if (!BN_rshift1(Y, Y)) goto err;
306  }
307  if (shift > 0)
308  {
309  if (!BN_rshift(A, A, shift)) goto err;
310  }
311 
312 
313  /* We still have (1) and (2).
314  * Both A and B are odd.
315  * The following computations ensure that
316  *
317  * 0 <= B < |n|,
318  * 0 < A < |n|,
319  * (1) -sign*X*a == B (mod |n|),
320  * (2) sign*Y*a == A (mod |n|),
321  *
322  * and that either A or B is even in the next iteration.
323  */
324  if (BN_ucmp(B, A) >= 0)
325  {
326  /* -sign*(X + Y)*a == B - A (mod |n|) */
327  if (!BN_uadd(X, X, Y)) goto err;
328  /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
329  * actually makes the algorithm slower */
330  if (!BN_usub(B, B, A)) goto err;
331  }
332  else
333  {
334  /* sign*(X + Y)*a == A - B (mod |n|) */
335  if (!BN_uadd(Y, Y, X)) goto err;
336  /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
337  if (!BN_usub(A, A, B)) goto err;
338  }
339  }
340  }
341  else
342  {
343  /* general inversion algorithm */
344 
345  while (!BN_is_zero(B))
346  {
347  BIGNUM *tmp;
348 
349  /*
350  * 0 < B < A,
351  * (*) -sign*X*a == B (mod |n|),
352  * sign*Y*a == A (mod |n|)
353  */
354 
355  /* (D, M) := (A/B, A%B) ... */
356  if (BN_num_bits(A) == BN_num_bits(B))
357  {
358  if (!BN_one(D)) goto err;
359  if (!BN_sub(M,A,B)) goto err;
360  }
361  else if (BN_num_bits(A) == BN_num_bits(B) + 1)
362  {
363  /* A/B is 1, 2, or 3 */
364  if (!BN_lshift1(T,B)) goto err;
365  if (BN_ucmp(A,T) < 0)
366  {
367  /* A < 2*B, so D=1 */
368  if (!BN_one(D)) goto err;
369  if (!BN_sub(M,A,B)) goto err;
370  }
371  else
372  {
373  /* A >= 2*B, so D=2 or D=3 */
374  if (!BN_sub(M,A,T)) goto err;
375  if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */
376  if (BN_ucmp(A,D) < 0)
377  {
378  /* A < 3*B, so D=2 */
379  if (!BN_set_word(D,2)) goto err;
380  /* M (= A - 2*B) already has the correct value */
381  }
382  else
383  {
384  /* only D=3 remains */
385  if (!BN_set_word(D,3)) goto err;
386  /* currently M = A - 2*B, but we need M = A - 3*B */
387  if (!BN_sub(M,M,B)) goto err;
388  }
389  }
390  }
391  else
392  {
393  if (!BN_div(D,M,A,B,ctx)) goto err;
394  }
395 
396  /* Now
397  * A = D*B + M;
398  * thus we have
399  * (**) sign*Y*a == D*B + M (mod |n|).
400  */
401 
402  tmp=A; /* keep the BIGNUM object, the value does not matter */
403 
404  /* (A, B) := (B, A mod B) ... */
405  A=B;
406  B=M;
407  /* ... so we have 0 <= B < A again */
408 
409  /* Since the former M is now B and the former B is now A,
410  * (**) translates into
411  * sign*Y*a == D*A + B (mod |n|),
412  * i.e.
413  * sign*Y*a - D*A == B (mod |n|).
414  * Similarly, (*) translates into
415  * -sign*X*a == A (mod |n|).
416  *
417  * Thus,
418  * sign*Y*a + D*sign*X*a == B (mod |n|),
419  * i.e.
420  * sign*(Y + D*X)*a == B (mod |n|).
421  *
422  * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
423  * -sign*X*a == B (mod |n|),
424  * sign*Y*a == A (mod |n|).
425  * Note that X and Y stay non-negative all the time.
426  */
427 
428  /* most of the time D is very small, so we can optimize tmp := D*X+Y */
429  if (BN_is_one(D))
430  {
431  if (!BN_add(tmp,X,Y)) goto err;
432  }
433  else
434  {
435  if (BN_is_word(D,2))
436  {
437  if (!BN_lshift1(tmp,X)) goto err;
438  }
439  else if (BN_is_word(D,4))
440  {
441  if (!BN_lshift(tmp,X,2)) goto err;
442  }
443  else if (D->top == 1)
444  {
445  if (!BN_copy(tmp,X)) goto err;
446  if (!BN_mul_word(tmp,D->d[0])) goto err;
447  }
448  else
449  {
450  if (!BN_mul(tmp,D,X,ctx)) goto err;
451  }
452  if (!BN_add(tmp,tmp,Y)) goto err;
453  }
454 
455  M=Y; /* keep the BIGNUM object, the value does not matter */
456  Y=X;
457  X=tmp;
458  sign = -sign;
459  }
460  }
461 
462  /*
463  * The while loop (Euclid's algorithm) ends when
464  * A == gcd(a,n);
465  * we have
466  * sign*Y*a == A (mod |n|),
467  * where Y is non-negative.
468  */
469 
470  if (sign < 0)
471  {
472  if (!BN_sub(Y,n,Y)) goto err;
473  }
474  /* Now Y*a == A (mod |n|). */
475 
476 
477  if (BN_is_one(A))
478  {
479  /* Y*a == 1 (mod |n|) */
480  if (!Y->neg && BN_ucmp(Y,n) < 0)
481  {
482  if (!BN_copy(R,Y)) goto err;
483  }
484  else
485  {
486  if (!BN_nnmod(R,Y,n,ctx)) goto err;
487  }
488  }
489  else
490  {
491  BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE);
492  goto err;
493  }
494  ret=R;
495 err:
496  if ((ret == NULL) && (in == NULL)) BN_free(R);
497  BN_CTX_end(ctx);
498  bn_check_top(ret);
499  return(ret);
500  }
501 
502 
503 /* BN_mod_inverse_no_branch is a special version of BN_mod_inverse.
504  * It does not contain branches that may leak sensitive information.
505  */
506 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
507  const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
508  {
509  BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
510  BIGNUM local_A, local_B;
511  BIGNUM *pA, *pB;
512  BIGNUM *ret=NULL;
513  int sign;
514 
515  bn_check_top(a);
516  bn_check_top(n);
517 
518  BN_CTX_start(ctx);
519  A = BN_CTX_get(ctx);
520  B = BN_CTX_get(ctx);
521  X = BN_CTX_get(ctx);
522  D = BN_CTX_get(ctx);
523  M = BN_CTX_get(ctx);
524  Y = BN_CTX_get(ctx);
525  T = BN_CTX_get(ctx);
526  if (T == NULL) goto err;
527 
528  if (in == NULL)
529  R=BN_new();
530  else
531  R=in;
532  if (R == NULL) goto err;
533 
534  BN_one(X);
535  BN_zero(Y);
536  if (BN_copy(B,a) == NULL) goto err;
537  if (BN_copy(A,n) == NULL) goto err;
538  A->neg = 0;
539 
540  if (B->neg || (BN_ucmp(B, A) >= 0))
541  {
542  /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
543  * BN_div_no_branch will be called eventually.
544  */
545  pB = &local_B;
546  BN_with_flags(pB, B, BN_FLG_CONSTTIME);
547  if (!BN_nnmod(B, pB, A, ctx)) goto err;
548  }
549  sign = -1;
550  /* From B = a mod |n|, A = |n| it follows that
551  *
552  * 0 <= B < A,
553  * -sign*X*a == B (mod |n|),
554  * sign*Y*a == A (mod |n|).
555  */
556 
557  while (!BN_is_zero(B))
558  {
559  BIGNUM *tmp;
560 
561  /*
562  * 0 < B < A,
563  * (*) -sign*X*a == B (mod |n|),
564  * sign*Y*a == A (mod |n|)
565  */
566 
567  /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
568  * BN_div_no_branch will be called eventually.
569  */
570  pA = &local_A;
571  BN_with_flags(pA, A, BN_FLG_CONSTTIME);
572 
573  /* (D, M) := (A/B, A%B) ... */
574  if (!BN_div(D,M,pA,B,ctx)) goto err;
575 
576  /* Now
577  * A = D*B + M;
578  * thus we have
579  * (**) sign*Y*a == D*B + M (mod |n|).
580  */
581 
582  tmp=A; /* keep the BIGNUM object, the value does not matter */
583 
584  /* (A, B) := (B, A mod B) ... */
585  A=B;
586  B=M;
587  /* ... so we have 0 <= B < A again */
588 
589  /* Since the former M is now B and the former B is now A,
590  * (**) translates into
591  * sign*Y*a == D*A + B (mod |n|),
592  * i.e.
593  * sign*Y*a - D*A == B (mod |n|).
594  * Similarly, (*) translates into
595  * -sign*X*a == A (mod |n|).
596  *
597  * Thus,
598  * sign*Y*a + D*sign*X*a == B (mod |n|),
599  * i.e.
600  * sign*(Y + D*X)*a == B (mod |n|).
601  *
602  * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
603  * -sign*X*a == B (mod |n|),
604  * sign*Y*a == A (mod |n|).
605  * Note that X and Y stay non-negative all the time.
606  */
607 
608  if (!BN_mul(tmp,D,X,ctx)) goto err;
609  if (!BN_add(tmp,tmp,Y)) goto err;
610 
611  M=Y; /* keep the BIGNUM object, the value does not matter */
612  Y=X;
613  X=tmp;
614  sign = -sign;
615  }
616 
617  /*
618  * The while loop (Euclid's algorithm) ends when
619  * A == gcd(a,n);
620  * we have
621  * sign*Y*a == A (mod |n|),
622  * where Y is non-negative.
623  */
624 
625  if (sign < 0)
626  {
627  if (!BN_sub(Y,n,Y)) goto err;
628  }
629  /* Now Y*a == A (mod |n|). */
630 
631  if (BN_is_one(A))
632  {
633  /* Y*a == 1 (mod |n|) */
634  if (!Y->neg && BN_ucmp(Y,n) < 0)
635  {
636  if (!BN_copy(R,Y)) goto err;
637  }
638  else
639  {
640  if (!BN_nnmod(R,Y,n,ctx)) goto err;
641  }
642  }
643  else
644  {
645  BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH,BN_R_NO_INVERSE);
646  goto err;
647  }
648  ret=R;
649 err:
650  if ((ret == NULL) && (in == NULL)) BN_free(R);
651  BN_CTX_end(ctx);
652  bn_check_top(ret);
653  return(ret);
654  }
Generated on Thu Jan 10 2013 09:53:34 for OpenSSL by   doxygen 1.8.2