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 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
207 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx);
208
209 BIGNUM *BN_mod_inverse(BIGNUM *in,
210 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
211 {
212 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
213 BIGNUM *ret=NULL;
214 int sign;
215
216 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0))
217 {
218 return BN_mod_inverse_no_branch(in, a, n, ctx);
219 }
220
221 bn_check_top(a);
222 bn_check_top(n);
223
224 BN_CTX_start(ctx);
225 A = BN_CTX_get(ctx);
226 B = BN_CTX_get(ctx);
227 X = BN_CTX_get(ctx);
228 D = BN_CTX_get(ctx);
229 M = BN_CTX_get(ctx);
230 Y = BN_CTX_get(ctx);
231 T = BN_CTX_get(ctx);
232 if (T == NULL) goto err;
233
234 if (in == NULL)
235 R=BN_new();
236 else
237 R=in;
238 if (R == NULL) goto err;
239
240 BN_one(X);
241 BN_zero(Y);
242 if (BN_copy(B,a) == NULL) goto err;
243 if (BN_copy(A,n) == NULL) goto err;
244 A->neg = 0;
245 if (B->neg || (BN_ucmp(B, A) >= 0))
246 {
247 if (!BN_nnmod(B, B, A, ctx)) goto err;
248 }
249 sign = -1;
250 /* From B = a mod |n|, A = |n| it follows that
251 *
252 * 0 <= B < A,
253 * -sign*X*a == B (mod |n|),
254 * sign*Y*a == A (mod |n|).
255 */
256
257 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048)))
258 {
259 /* Binary inversion algorithm; requires odd modulus.
260 * This is faster than the general algorithm if the modulus
261 * is sufficiently small (about 400 .. 500 bits on 32-bit
262 * sytems, but much more on 64-bit systems) */
263 int shift;
264
265 while (!BN_is_zero(B))
266 {
267 /*
268 * 0 < B < |n|,
269 * 0 < A <= |n|,
270 * (1) -sign*X*a == B (mod |n|),
271 * (2) sign*Y*a == A (mod |n|)
272 */
273
274 /* Now divide B by the maximum possible power of two in the integers,
275 * and divide X by the same value mod |n|.
276 * When we're done, (1) still holds. */
277 shift = 0;
278 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */
279 {
280 shift++;
281
282 if (BN_is_odd(X))
283 {
284 if (!BN_uadd(X, X, n)) goto err;
285 }
286 /* now X is even, so we can easily divide it by two */
287 if (!BN_rshift1(X, X)) goto err;
288 }
289 if (shift > 0)
290 {
291 if (!BN_rshift(B, B, shift)) goto err;
292 }
293
294
295 /* Same for A and Y. Afterwards, (2) still holds. */
296 shift = 0;
297 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */
298 {
299 shift++;
300
301 if (BN_is_odd(Y))
302 {
303 if (!BN_uadd(Y, Y, n)) goto err;
304 }
305 /* now Y is even */
306 if (!BN_rshift1(Y, Y)) goto err;
307 }
308 if (shift > 0)
309 {
310 if (!BN_rshift(A, A, shift)) goto err;
311 }
312
313
314 /* We still have (1) and (2).
315 * Both A and B are odd.
316 * The following computations ensure that
317 *
318 * 0 <= B < |n|,
319 * 0 < A < |n|,
320 * (1) -sign*X*a == B (mod |n|),
321 * (2) sign*Y*a == A (mod |n|),
322 *
323 * and that either A or B is even in the next iteration.
324 */
325 if (BN_ucmp(B, A) >= 0)
326 {
327 /* -sign*(X + Y)*a == B - A (mod |n|) */
328 if (!BN_uadd(X, X, Y)) goto err;
329 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
330 * actually makes the algorithm slower */
331 if (!BN_usub(B, B, A)) goto err;
332 }
333 else
334 {
335 /* sign*(X + Y)*a == A - B (mod |n|) */
336 if (!BN_uadd(Y, Y, X)) goto err;
337 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
338 if (!BN_usub(A, A, B)) goto err;
339 }
340 }
341 }
342 else
343 {
344 /* general inversion algorithm */
345
346 while (!BN_is_zero(B))
347 {
348 BIGNUM *tmp;
349
350 /*
351 * 0 < B < A,
352 * (*) -sign*X*a == B (mod |n|),
353 * sign*Y*a == A (mod |n|)
354 */
355
356 /* (D, M) := (A/B, A%B) ... */
357 if (BN_num_bits(A) == BN_num_bits(B))
358 {
359 if (!BN_one(D)) goto err;
360 if (!BN_sub(M,A,B)) goto err;
361 }
362 else if (BN_num_bits(A) == BN_num_bits(B) + 1)
363 {
364 /* A/B is 1, 2, or 3 */
365 if (!BN_lshift1(T,B)) goto err;
366 if (BN_ucmp(A,T) < 0)
367 {
368 /* A < 2*B, so D=1 */
369 if (!BN_one(D)) goto err;
370 if (!BN_sub(M,A,B)) goto err;
371 }
372 else
373 {
374 /* A >= 2*B, so D=2 or D=3 */
375 if (!BN_sub(M,A,T)) goto err;
376 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */
377 if (BN_ucmp(A,D) < 0)
378 {
379 /* A < 3*B, so D=2 */
380 if (!BN_set_word(D,2)) goto err;
381 /* M (= A - 2*B) already has the correct value */
382 }
383 else
384 {
385 /* only D=3 remains */
386 if (!BN_set_word(D,3)) goto err;
387 /* currently M = A - 2*B, but we need M = A - 3*B */
388 if (!BN_sub(M,M,B)) goto err;
389 }
390 }
391 }
392 else
393 {
394 if (!BN_div(D,M,A,B,ctx)) goto err;
395 }
396
397 /* Now
398 * A = D*B + M;
399 * thus we have
400 * (**) sign*Y*a == D*B + M (mod |n|).
401 */
402
403 tmp=A; /* keep the BIGNUM object, the value does not matter */
404
405 /* (A, B) := (B, A mod B) ... */
406 A=B;
407 B=M;
408 /* ... so we have 0 <= B < A again */
409
410 /* Since the former M is now B and the former B is now A,
411 * (**) translates into
412 * sign*Y*a == D*A + B (mod |n|),
413 * i.e.
414 * sign*Y*a - D*A == B (mod |n|).
415 * Similarly, (*) translates into
416 * -sign*X*a == A (mod |n|).
417 *
418 * Thus,
419 * sign*Y*a + D*sign*X*a == B (mod |n|),
420 * i.e.
421 * sign*(Y + D*X)*a == B (mod |n|).
422 *
423 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
424 * -sign*X*a == B (mod |n|),
425 * sign*Y*a == A (mod |n|).
426 * Note that X and Y stay non-negative all the time.
427 */
428
429 /* most of the time D is very small, so we can optimize tmp := D*X+Y */
430 if (BN_is_one(D))
431 {
432 if (!BN_add(tmp,X,Y)) goto err;
433 }
434 else
435 {
436 if (BN_is_word(D,2))
437 {
438 if (!BN_lshift1(tmp,X)) goto err;
439 }
440 else if (BN_is_word(D,4))
441 {
442 if (!BN_lshift(tmp,X,2)) goto err;
443 }
444 else if (D->top == 1)
445 {
446 if (!BN_copy(tmp,X)) goto err;
447 if (!BN_mul_word(tmp,D->d[0])) goto err;
448 }
449 else
450 {
451 if (!BN_mul(tmp,D,X,ctx)) goto err;
452 }
453 if (!BN_add(tmp,tmp,Y)) goto err;
454 }
455
456 M=Y; /* keep the BIGNUM object, the value does not matter */
457 Y=X;
458 X=tmp;
459 sign = -sign;
460 }
461 }
462
463 /*
464 * The while loop (Euclid's algorithm) ends when
465 * A == gcd(a,n);
466 * we have
467 * sign*Y*a == A (mod |n|),
468 * where Y is non-negative.
469 */
470
471 if (sign < 0)
472 {
473 if (!BN_sub(Y,n,Y)) goto err;
474 }
475 /* Now Y*a == A (mod |n|). */
476
477
478 if (BN_is_one(A))
479 {
480 /* Y*a == 1 (mod |n|) */
481 if (!Y->neg && BN_ucmp(Y,n) < 0)
482 {
483 if (!BN_copy(R,Y)) goto err;
484 }
485 else
486 {
487 if (!BN_nnmod(R,Y,n,ctx)) goto err;
488 }
489 }
490 else
491 {
492 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE);
493 goto err;
494 }
495 ret=R;
496 err:
497 if ((ret == NULL) && (in == NULL)) BN_free(R);
498 BN_CTX_end(ctx);
499 bn_check_top(ret);
500 return(ret);
501 }
502
503
504 /* BN_mod_inverse_no_branch is a special version of BN_mod_inverse.
505 * It does not contain branches that may leak sensitive information.
506 */
507 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
508 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
509 {
510 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
511 BIGNUM local_A, local_B;
512 BIGNUM *pA, *pB;
513 BIGNUM *ret=NULL;
514 int sign;
515
516 bn_check_top(a);
517 bn_check_top(n);
518
519 BN_CTX_start(ctx);
520 A = BN_CTX_get(ctx);
521 B = BN_CTX_get(ctx);
522 X = BN_CTX_get(ctx);
523 D = BN_CTX_get(ctx);
524 M = BN_CTX_get(ctx);
525 Y = BN_CTX_get(ctx);
526 T = BN_CTX_get(ctx);
527 if (T == NULL) goto err;
528
529 if (in == NULL)
530 R=BN_new();
531 else
532 R=in;
533 if (R == NULL) goto err;
534
535 BN_one(X);
536 BN_zero(Y);
537 if (BN_copy(B,a) == NULL) goto err;
538 if (BN_copy(A,n) == NULL) goto err;
539 A->neg = 0;
540
541 if (B->neg || (BN_ucmp(B, A) >= 0))
542 {
543 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
544 * BN_div_no_branch will be called eventually.
545 */
546 pB = &local_B;
547 BN_with_flags(pB, B, BN_FLG_CONSTTIME);
548 if (!BN_nnmod(B, pB, A, ctx)) goto err;
549 }
550 sign = -1;
551 /* From B = a mod |n|, A = |n| it follows that
552 *
553 * 0 <= B < A,
554 * -sign*X*a == B (mod |n|),
555 * sign*Y*a == A (mod |n|).
556 */
557
558 while (!BN_is_zero(B))
559 {
560 BIGNUM *tmp;
561
562 /*
563 * 0 < B < A,
564 * (*) -sign*X*a == B (mod |n|),
565 * sign*Y*a == A (mod |n|)
566 */
567
568 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
569 * BN_div_no_branch will be called eventually.
570 */
571 pA = &local_A;
572 BN_with_flags(pA, A, BN_FLG_CONSTTIME);
573
574 /* (D, M) := (A/B, A%B) ... */
575 if (!BN_div(D,M,pA,B,ctx)) goto err;
576
577 /* Now
578 * A = D*B + M;
579 * thus we have
580 * (**) sign*Y*a == D*B + M (mod |n|).
581 */
582
583 tmp=A; /* keep the BIGNUM object, the value does not matter */
584
585 /* (A, B) := (B, A mod B) ... */
586 A=B;
587 B=M;
588 /* ... so we have 0 <= B < A again */
589
590 /* Since the former M is now B and the former B is now A,
591 * (**) translates into
592 * sign*Y*a == D*A + B (mod |n|),
593 * i.e.
594 * sign*Y*a - D*A == B (mod |n|).
595 * Similarly, (*) translates into
596 * -sign*X*a == A (mod |n|).
597 *
598 * Thus,
599 * sign*Y*a + D*sign*X*a == B (mod |n|),
600 * i.e.
601 * sign*(Y + D*X)*a == B (mod |n|).
602 *
603 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
604 * -sign*X*a == B (mod |n|),
605 * sign*Y*a == A (mod |n|).
606 * Note that X and Y stay non-negative all the time.
607 */
608
609 if (!BN_mul(tmp,D,X,ctx)) goto err;
610 if (!BN_add(tmp,tmp,Y)) goto err;
611
612 M=Y; /* keep the BIGNUM object, the value does not matter */
613 Y=X;
614 X=tmp;
615 sign = -sign;
616 }
617
618 /*
619 * The while loop (Euclid's algorithm) ends when
620 * A == gcd(a,n);
621 * we have
622 * sign*Y*a == A (mod |n|),
623 * where Y is non-negative.
624 */
625
626 if (sign < 0)
627 {
628 if (!BN_sub(Y,n,Y)) goto err;
629 }
630 /* Now Y*a == A (mod |n|). */
631
632 if (BN_is_one(A))
633 {
634 /* Y*a == 1 (mod |n|) */
635 if (!Y->neg && BN_ucmp(Y,n) < 0)
636 {
637 if (!BN_copy(R,Y)) goto err;
638 }
639 else
640 {
641 if (!BN_nnmod(R,Y,n,ctx)) goto err;
642 }
643 }
644 else
645 {
646 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH,BN_R_NO_INVERSE);
647 goto err;
648 }
649 ret=R;
650 err:
651 if ((ret == NULL) && (in == NULL)) BN_free(R);
652 BN_CTX_end(ctx);
653 bn_check_top(ret);
654 return(ret);
655 }