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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
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 /*-
251 * From B = a mod |n|, A = |n| it follows that
252 *
253 * 0 <= B < A,
254 * -sign*X*a == B (mod |n|),
255 * sign*Y*a == A (mod |n|).
256 */
257
258 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048)))
259 {
260 /* Binary inversion algorithm; requires odd modulus.
261 * This is faster than the general algorithm if the modulus
262 * is sufficiently small (about 400 .. 500 bits on 32-bit
263 * sytems, but much more on 64-bit systems) */
264 int shift;
265
266 while (!BN_is_zero(B))
267 {
268 /*-
269 * 0 < B < |n|,
270 * 0 < A <= |n|,
271 * (1) -sign*X*a == B (mod |n|),
272 * (2) sign*Y*a == A (mod |n|)
273 */
274
275 /* Now divide B by the maximum possible power of two in the integers,
276 * and divide X by the same value mod |n|.
277 * When we're done, (1) still holds. */
278 shift = 0;
279 while (!BN_is_bit_set(B, shift)) /* note that 0 < B */
280 {
281 shift++;
282
283 if (BN_is_odd(X))
284 {
285 if (!BN_uadd(X, X, n)) goto err;
286 }
287 /* now X is even, so we can easily divide it by two */
288 if (!BN_rshift1(X, X)) goto err;
289 }
290 if (shift > 0)
291 {
292 if (!BN_rshift(B, B, shift)) goto err;
293 }
294
295
296 /* Same for A and Y. Afterwards, (2) still holds. */
297 shift = 0;
298 while (!BN_is_bit_set(A, shift)) /* note that 0 < A */
299 {
300 shift++;
301
302 if (BN_is_odd(Y))
303 {
304 if (!BN_uadd(Y, Y, n)) goto err;
305 }
306 /* now Y is even */
307 if (!BN_rshift1(Y, Y)) goto err;
308 }
309 if (shift > 0)
310 {
311 if (!BN_rshift(A, A, shift)) goto err;
312 }
313
314
315 /*-
316 * We still have (1) and (2).
317 * Both A and B are odd.
318 * The following computations ensure that
319 *
320 * 0 <= B < |n|,
321 * 0 < A < |n|,
322 * (1) -sign*X*a == B (mod |n|),
323 * (2) sign*Y*a == A (mod |n|),
324 *
325 * and that either A or B is even in the next iteration.
326 */
327 if (BN_ucmp(B, A) >= 0)
328 {
329 /* -sign*(X + Y)*a == B - A (mod |n|) */
330 if (!BN_uadd(X, X, Y)) goto err;
331 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
332 * actually makes the algorithm slower */
333 if (!BN_usub(B, B, A)) goto err;
334 }
335 else
336 {
337 /* sign*(X + Y)*a == A - B (mod |n|) */
338 if (!BN_uadd(Y, Y, X)) goto err;
339 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
340 if (!BN_usub(A, A, B)) goto err;
341 }
342 }
343 }
344 else
345 {
346 /* general inversion algorithm */
347
348 while (!BN_is_zero(B))
349 {
350 BIGNUM *tmp;
351
352 /*-
353 * 0 < B < A,
354 * (*) -sign*X*a == B (mod |n|),
355 * sign*Y*a == A (mod |n|)
356 */
357
358 /* (D, M) := (A/B, A%B) ... */
359 if (BN_num_bits(A) == BN_num_bits(B))
360 {
361 if (!BN_one(D)) goto err;
362 if (!BN_sub(M,A,B)) goto err;
363 }
364 else if (BN_num_bits(A) == BN_num_bits(B) + 1)
365 {
366 /* A/B is 1, 2, or 3 */
367 if (!BN_lshift1(T,B)) goto err;
368 if (BN_ucmp(A,T) < 0)
369 {
370 /* A < 2*B, so D=1 */
371 if (!BN_one(D)) goto err;
372 if (!BN_sub(M,A,B)) goto err;
373 }
374 else
375 {
376 /* A >= 2*B, so D=2 or D=3 */
377 if (!BN_sub(M,A,T)) goto err;
378 if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */
379 if (BN_ucmp(A,D) < 0)
380 {
381 /* A < 3*B, so D=2 */
382 if (!BN_set_word(D,2)) goto err;
383 /* M (= A - 2*B) already has the correct value */
384 }
385 else
386 {
387 /* only D=3 remains */
388 if (!BN_set_word(D,3)) goto err;
389 /* currently M = A - 2*B, but we need M = A - 3*B */
390 if (!BN_sub(M,M,B)) goto err;
391 }
392 }
393 }
394 else
395 {
396 if (!BN_div(D,M,A,B,ctx)) goto err;
397 }
398
399 /*-
400 * Now
401 * A = D*B + M;
402 * thus we have
403 * (**) sign*Y*a == D*B + M (mod |n|).
404 */
405
406 tmp=A; /* keep the BIGNUM object, the value does not matter */
407
408 /* (A, B) := (B, A mod B) ... */
409 A=B;
410 B=M;
411 /* ... so we have 0 <= B < A again */
412
413 /*-
414 * Since the former M is now B and the former B is now A,
415 * (**) translates into
416 * sign*Y*a == D*A + B (mod |n|),
417 * i.e.
418 * sign*Y*a - D*A == B (mod |n|).
419 * Similarly, (*) translates into
420 * -sign*X*a == A (mod |n|).
421 *
422 * Thus,
423 * sign*Y*a + D*sign*X*a == B (mod |n|),
424 * i.e.
425 * sign*(Y + D*X)*a == B (mod |n|).
426 *
427 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
428 * -sign*X*a == B (mod |n|),
429 * sign*Y*a == A (mod |n|).
430 * Note that X and Y stay non-negative all the time.
431 */
432
433 /* most of the time D is very small, so we can optimize tmp := D*X+Y */
434 if (BN_is_one(D))
435 {
436 if (!BN_add(tmp,X,Y)) goto err;
437 }
438 else
439 {
440 if (BN_is_word(D,2))
441 {
442 if (!BN_lshift1(tmp,X)) goto err;
443 }
444 else if (BN_is_word(D,4))
445 {
446 if (!BN_lshift(tmp,X,2)) goto err;
447 }
448 else if (D->top == 1)
449 {
450 if (!BN_copy(tmp,X)) goto err;
451 if (!BN_mul_word(tmp,D->d[0])) goto err;
452 }
453 else
454 {
455 if (!BN_mul(tmp,D,X,ctx)) goto err;
456 }
457 if (!BN_add(tmp,tmp,Y)) goto err;
458 }
459
460 M=Y; /* keep the BIGNUM object, the value does not matter */
461 Y=X;
462 X=tmp;
463 sign = -sign;
464 }
465 }
466
467 /*-
468 * The while loop (Euclid's algorithm) ends when
469 * A == gcd(a,n);
470 * we have
471 * sign*Y*a == A (mod |n|),
472 * where Y is non-negative.
473 */
474
475 if (sign < 0)
476 {
477 if (!BN_sub(Y,n,Y)) goto err;
478 }
479 /* Now Y*a == A (mod |n|). */
480
481
482 if (BN_is_one(A))
483 {
484 /* Y*a == 1 (mod |n|) */
485 if (!Y->neg && BN_ucmp(Y,n) < 0)
486 {
487 if (!BN_copy(R,Y)) goto err;
488 }
489 else
490 {
491 if (!BN_nnmod(R,Y,n,ctx)) goto err;
492 }
493 }
494 else
495 {
496 BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE);
497 goto err;
498 }
499 ret=R;
500 err:
501 if ((ret == NULL) && (in == NULL)) BN_free(R);
502 BN_CTX_end(ctx);
503 bn_check_top(ret);
504 return(ret);
505 }
506
507
508 /* BN_mod_inverse_no_branch is a special version of BN_mod_inverse.
509 * It does not contain branches that may leak sensitive information.
510 */
511 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
512 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
513 {
514 BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
515 BIGNUM local_A, local_B;
516 BIGNUM *pA, *pB;
517 BIGNUM *ret=NULL;
518 int sign;
519
520 bn_check_top(a);
521 bn_check_top(n);
522
523 BN_CTX_start(ctx);
524 A = BN_CTX_get(ctx);
525 B = BN_CTX_get(ctx);
526 X = BN_CTX_get(ctx);
527 D = BN_CTX_get(ctx);
528 M = BN_CTX_get(ctx);
529 Y = BN_CTX_get(ctx);
530 T = BN_CTX_get(ctx);
531 if (T == NULL) goto err;
532
533 if (in == NULL)
534 R=BN_new();
535 else
536 R=in;
537 if (R == NULL) goto err;
538
539 BN_one(X);
540 BN_zero(Y);
541 if (BN_copy(B,a) == NULL) goto err;
542 if (BN_copy(A,n) == NULL) goto err;
543 A->neg = 0;
544
545 if (B->neg || (BN_ucmp(B, A) >= 0))
546 {
547 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
548 * BN_div_no_branch will be called eventually.
549 */
550 pB = &local_B;
551 BN_with_flags(pB, B, BN_FLG_CONSTTIME);
552 if (!BN_nnmod(B, pB, A, ctx)) goto err;
553 }
554 sign = -1;
555 /*-
556 * From B = a mod |n|, A = |n| it follows that
557 *
558 * 0 <= B < A,
559 * -sign*X*a == B (mod |n|),
560 * sign*Y*a == A (mod |n|).
561 */
562
563 while (!BN_is_zero(B))
564 {
565 BIGNUM *tmp;
566
567 /*-
568 * 0 < B < A,
569 * (*) -sign*X*a == B (mod |n|),
570 * sign*Y*a == A (mod |n|)
571 */
572
573 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
574 * BN_div_no_branch will be called eventually.
575 */
576 pA = &local_A;
577 BN_with_flags(pA, A, BN_FLG_CONSTTIME);
578
579 /* (D, M) := (A/B, A%B) ... */
580 if (!BN_div(D,M,pA,B,ctx)) goto err;
581
582 /*-
583 * Now
584 * A = D*B + M;
585 * thus we have
586 * (**) sign*Y*a == D*B + M (mod |n|).
587 */
588
589 tmp=A; /* keep the BIGNUM object, the value does not matter */
590
591 /* (A, B) := (B, A mod B) ... */
592 A=B;
593 B=M;
594 /* ... so we have 0 <= B < A again */
595
596 /*-
597 * Since the former M is now B and the former B is now A,
598 * (**) translates into
599 * sign*Y*a == D*A + B (mod |n|),
600 * i.e.
601 * sign*Y*a - D*A == B (mod |n|).
602 * Similarly, (*) translates into
603 * -sign*X*a == A (mod |n|).
604 *
605 * Thus,
606 * sign*Y*a + D*sign*X*a == B (mod |n|),
607 * i.e.
608 * sign*(Y + D*X)*a == B (mod |n|).
609 *
610 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
611 * -sign*X*a == B (mod |n|),
612 * sign*Y*a == A (mod |n|).
613 * Note that X and Y stay non-negative all the time.
614 */
615
616 if (!BN_mul(tmp,D,X,ctx)) goto err;
617 if (!BN_add(tmp,tmp,Y)) goto err;
618
619 M=Y; /* keep the BIGNUM object, the value does not matter */
620 Y=X;
621 X=tmp;
622 sign = -sign;
623 }
624
625 /*-
626 * The while loop (Euclid's algorithm) ends when
627 * A == gcd(a,n);
628 * we have
629 * sign*Y*a == A (mod |n|),
630 * where Y is non-negative.
631 */
632
633 if (sign < 0)
634 {
635 if (!BN_sub(Y,n,Y)) goto err;
636 }
637 /* Now Y*a == A (mod |n|). */
638
639 if (BN_is_one(A))
640 {
641 /* Y*a == 1 (mod |n|) */
642 if (!Y->neg && BN_ucmp(Y,n) < 0)
643 {
644 if (!BN_copy(R,Y)) goto err;
645 }
646 else
647 {
648 if (!BN_nnmod(R,Y,n,ctx)) goto err;
649 }
650 }
651 else
652 {
653 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH,BN_R_NO_INVERSE);
654 goto err;
655 }
656 ret=R;
657 err:
658 if ((ret == NULL) && (in == NULL)) BN_free(R);
659 BN_CTX_end(ctx);
660 bn_check_top(ret);
661 return(ret);
662 }
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