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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) */
206static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
207 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx);
208BIGNUM *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;
495err:
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 */
506static 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;
649err:
650 if ((ret == NULL) && (in == NULL)) BN_free(R);
651 BN_CTX_end(ctx);
652 bn_check_top(ret);
653 return(ret);
654 }
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