1 /* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
2 * All rights reserved.
3 *
4 * This package is an SSL implementation written
5 * by Eric Young (eay@cryptsoft.com).
6 * The implementation was written so as to conform with Netscapes SSL.
7 *
8 * This library is free for commercial and non-commercial use as long as
9 * the following conditions are aheared to. The following conditions
10 * apply to all code found in this distribution, be it the RC4, RSA,
11 * lhash, DES, etc., code; not just the SSL code. The SSL documentation
12 * included with this distribution is covered by the same copyright terms
13 * except that the holder is Tim Hudson (tjh@cryptsoft.com).
14 *
15 * Copyright remains Eric Young's, and as such any Copyright notices in
16 * the code are not to be removed.
17 * If this package is used in a product, Eric Young should be given attribution
18 * as the author of the parts of the library used.
19 * This can be in the form of a textual message at program startup or
20 * in documentation (online or textual) provided with the package.
21 *
22 * Redistribution and use in source and binary forms, with or without
23 * modification, are permitted provided that the following conditions
24 * are met:
25 * 1. Redistributions of source code must retain the copyright
26 * notice, this list of conditions and the following disclaimer.
27 * 2. Redistributions in binary form must reproduce the above copyright
28 * notice, this list of conditions and the following disclaimer in the
29 * documentation and/or other materials provided with the distribution.
30 * 3. All advertising materials mentioning features or use of this software
31 * must display the following acknowledgement:
32 * "This product includes cryptographic software written by
33 * Eric Young (eay@cryptsoft.com)"
34 * The word 'cryptographic' can be left out if the rouines from the library
35 * being used are not cryptographic related :-).
36 * 4. If you include any Windows specific code (or a derivative thereof) from
37 * the apps directory (application code) you must include an acknowledgement:
38 * "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
39 *
40 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
41 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
42 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
43 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
44 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
45 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
46 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
47 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
48 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
49 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
50 * SUCH DAMAGE.
51 *
52 * The licence and distribution terms for any publically available version or
53 * derivative of this code cannot be changed. i.e. this code cannot simply be
54 * copied and put under another distribution licence
55 * [including the GNU Public Licence.]
56 */
57 /* ====================================================================
58 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
59 *
60 * Redistribution and use in source and binary forms, with or without
61 * modification, are permitted provided that the following conditions
62 * are met:
63 *
64 * 1. Redistributions of source code must retain the above copyright
65 * notice, this list of conditions and the following disclaimer.
66 *
67 * 2. Redistributions in binary form must reproduce the above copyright
68 * notice, this list of conditions and the following disclaimer in
69 * the documentation and/or other materials provided with the
70 * distribution.
71 *
72 * 3. All advertising materials mentioning features or use of this
73 * software must display the following acknowledgment:
74 * "This product includes software developed by the OpenSSL Project
75 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
76 *
77 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
78 * endorse or promote products derived from this software without
79 * prior written permission. For written permission, please contact
80 * openssl-core@openssl.org.
81 *
82 * 5. Products derived from this software may not be called "OpenSSL"
83 * nor may "OpenSSL" appear in their names without prior written
84 * permission of the OpenSSL Project.
85 *
86 * 6. Redistributions of any form whatsoever must retain the following
87 * acknowledgment:
88 * "This product includes software developed by the OpenSSL Project
89 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
90 *
91 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
92 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
93 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
94 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
95 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
96 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
97 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
98 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
99 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
100 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
101 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
102 * OF THE POSSIBILITY OF SUCH DAMAGE.
103 * ====================================================================
104 *
105 * This product includes cryptographic software written by Eric Young
106 * (eay@cryptsoft.com). This product includes software written by Tim
107 * Hudson (tjh@cryptsoft.com). */
108
109 #include <openssl/bn.h>
110
111 #include <openssl/err.h>
112
113 #include "internal.h"
114
euclid(BIGNUM * a,BIGNUM * b)115 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) {
116 BIGNUM *t;
117 int shifts = 0;
118
119 /* 0 <= b <= a */
120 while (!BN_is_zero(b)) {
121 /* 0 < b <= a */
122
123 if (BN_is_odd(a)) {
124 if (BN_is_odd(b)) {
125 if (!BN_sub(a, a, b)) {
126 goto err;
127 }
128 if (!BN_rshift1(a, a)) {
129 goto err;
130 }
131 if (BN_cmp(a, b) < 0) {
132 t = a;
133 a = b;
134 b = t;
135 }
136 } else {
137 /* a odd - b even */
138 if (!BN_rshift1(b, b)) {
139 goto err;
140 }
141 if (BN_cmp(a, b) < 0) {
142 t = a;
143 a = b;
144 b = t;
145 }
146 }
147 } else {
148 /* a is even */
149 if (BN_is_odd(b)) {
150 if (!BN_rshift1(a, a)) {
151 goto err;
152 }
153 if (BN_cmp(a, b) < 0) {
154 t = a;
155 a = b;
156 b = t;
157 }
158 } else {
159 /* a even - b even */
160 if (!BN_rshift1(a, a)) {
161 goto err;
162 }
163 if (!BN_rshift1(b, b)) {
164 goto err;
165 }
166 shifts++;
167 }
168 }
169 /* 0 <= b <= a */
170 }
171
172 if (shifts) {
173 if (!BN_lshift(a, a, shifts)) {
174 goto err;
175 }
176 }
177
178 return a;
179
180 err:
181 return NULL;
182 }
183
BN_gcd(BIGNUM * r,const BIGNUM * in_a,const BIGNUM * in_b,BN_CTX * ctx)184 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) {
185 BIGNUM *a, *b, *t;
186 int ret = 0;
187
188 BN_CTX_start(ctx);
189 a = BN_CTX_get(ctx);
190 b = BN_CTX_get(ctx);
191
192 if (a == NULL || b == NULL) {
193 goto err;
194 }
195 if (BN_copy(a, in_a) == NULL) {
196 goto err;
197 }
198 if (BN_copy(b, in_b) == NULL) {
199 goto err;
200 }
201
202 a->neg = 0;
203 b->neg = 0;
204
205 if (BN_cmp(a, b) < 0) {
206 t = a;
207 a = b;
208 b = t;
209 }
210 t = euclid(a, b);
211 if (t == NULL) {
212 goto err;
213 }
214
215 if (BN_copy(r, t) == NULL) {
216 goto err;
217 }
218 ret = 1;
219
220 err:
221 BN_CTX_end(ctx);
222 return ret;
223 }
224
225 /* solves ax == 1 (mod n) */
226 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *out, const BIGNUM *a,
227 const BIGNUM *n, BN_CTX *ctx);
228
BN_mod_inverse(BIGNUM * out,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)229 BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n,
230 BN_CTX *ctx) {
231 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
232 BIGNUM *ret = NULL;
233 int sign;
234
235 if ((a->flags & BN_FLG_CONSTTIME) != 0 ||
236 (n->flags & BN_FLG_CONSTTIME) != 0) {
237 return BN_mod_inverse_no_branch(out, a, n, ctx);
238 }
239
240 BN_CTX_start(ctx);
241 A = BN_CTX_get(ctx);
242 B = BN_CTX_get(ctx);
243 X = BN_CTX_get(ctx);
244 D = BN_CTX_get(ctx);
245 M = BN_CTX_get(ctx);
246 Y = BN_CTX_get(ctx);
247 T = BN_CTX_get(ctx);
248 if (T == NULL) {
249 goto err;
250 }
251
252 if (out == NULL) {
253 R = BN_new();
254 } else {
255 R = out;
256 }
257 if (R == NULL) {
258 goto err;
259 }
260
261 BN_zero(Y);
262 if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
263 goto err;
264 }
265 A->neg = 0;
266 if (B->neg || (BN_ucmp(B, A) >= 0)) {
267 if (!BN_nnmod(B, B, A, ctx)) {
268 goto err;
269 }
270 }
271 sign = -1;
272 /* From B = a mod |n|, A = |n| it follows that
273 *
274 * 0 <= B < A,
275 * -sign*X*a == B (mod |n|),
276 * sign*Y*a == A (mod |n|).
277 */
278
279 if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048))) {
280 /* Binary inversion algorithm; requires odd modulus.
281 * This is faster than the general algorithm if the modulus
282 * is sufficiently small (about 400 .. 500 bits on 32-bit
283 * sytems, but much more on 64-bit systems) */
284 int shift;
285
286 while (!BN_is_zero(B)) {
287 /* 0 < B < |n|,
288 * 0 < A <= |n|,
289 * (1) -sign*X*a == B (mod |n|),
290 * (2) sign*Y*a == A (mod |n|) */
291
292 /* Now divide B by the maximum possible power of two in the integers,
293 * and divide X by the same value mod |n|.
294 * When we're done, (1) still holds. */
295 shift = 0;
296 while (!BN_is_bit_set(B, shift)) {
297 /* note that 0 < B */
298 shift++;
299
300 if (BN_is_odd(X)) {
301 if (!BN_uadd(X, X, n)) {
302 goto err;
303 }
304 }
305 /* now X is even, so we can easily divide it by two */
306 if (!BN_rshift1(X, X)) {
307 goto err;
308 }
309 }
310 if (shift > 0) {
311 if (!BN_rshift(B, B, shift)) {
312 goto err;
313 }
314 }
315
316 /* Same for A and Y. Afterwards, (2) still holds. */
317 shift = 0;
318 while (!BN_is_bit_set(A, shift)) {
319 /* note that 0 < A */
320 shift++;
321
322 if (BN_is_odd(Y)) {
323 if (!BN_uadd(Y, Y, n)) {
324 goto err;
325 }
326 }
327 /* now Y is even */
328 if (!BN_rshift1(Y, Y)) {
329 goto err;
330 }
331 }
332 if (shift > 0) {
333 if (!BN_rshift(A, A, shift)) {
334 goto err;
335 }
336 }
337
338 /* We still have (1) and (2).
339 * Both A and B are odd.
340 * The following computations ensure that
341 *
342 * 0 <= B < |n|,
343 * 0 < A < |n|,
344 * (1) -sign*X*a == B (mod |n|),
345 * (2) sign*Y*a == A (mod |n|),
346 *
347 * and that either A or B is even in the next iteration. */
348 if (BN_ucmp(B, A) >= 0) {
349 /* -sign*(X + Y)*a == B - A (mod |n|) */
350 if (!BN_uadd(X, X, Y)) {
351 goto err;
352 }
353 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
354 * actually makes the algorithm slower */
355 if (!BN_usub(B, B, A)) {
356 goto err;
357 }
358 } else {
359 /* sign*(X + Y)*a == A - B (mod |n|) */
360 if (!BN_uadd(Y, Y, X)) {
361 goto err;
362 }
363 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
364 if (!BN_usub(A, A, B)) {
365 goto err;
366 }
367 }
368 }
369 } else {
370 /* general inversion algorithm */
371
372 while (!BN_is_zero(B)) {
373 BIGNUM *tmp;
374
375 /*
376 * 0 < B < A,
377 * (*) -sign*X*a == B (mod |n|),
378 * sign*Y*a == A (mod |n|) */
379
380 /* (D, M) := (A/B, A%B) ... */
381 if (BN_num_bits(A) == BN_num_bits(B)) {
382 if (!BN_one(D)) {
383 goto err;
384 }
385 if (!BN_sub(M, A, B)) {
386 goto err;
387 }
388 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
389 /* A/B is 1, 2, or 3 */
390 if (!BN_lshift1(T, B)) {
391 goto err;
392 }
393 if (BN_ucmp(A, T) < 0) {
394 /* A < 2*B, so D=1 */
395 if (!BN_one(D)) {
396 goto err;
397 }
398 if (!BN_sub(M, A, B)) {
399 goto err;
400 }
401 } else {
402 /* A >= 2*B, so D=2 or D=3 */
403 if (!BN_sub(M, A, T)) {
404 goto err;
405 }
406 if (!BN_add(D, T, B)) {
407 goto err; /* use D (:= 3*B) as temp */
408 }
409 if (BN_ucmp(A, D) < 0) {
410 /* A < 3*B, so D=2 */
411 if (!BN_set_word(D, 2)) {
412 goto err;
413 }
414 /* M (= A - 2*B) already has the correct value */
415 } else {
416 /* only D=3 remains */
417 if (!BN_set_word(D, 3)) {
418 goto err;
419 }
420 /* currently M = A - 2*B, but we need M = A - 3*B */
421 if (!BN_sub(M, M, B)) {
422 goto err;
423 }
424 }
425 }
426 } else {
427 if (!BN_div(D, M, A, B, ctx)) {
428 goto err;
429 }
430 }
431
432 /* Now
433 * A = D*B + M;
434 * thus we have
435 * (**) sign*Y*a == D*B + M (mod |n|). */
436
437 tmp = A; /* keep the BIGNUM object, the value does not matter */
438
439 /* (A, B) := (B, A mod B) ... */
440 A = B;
441 B = M;
442 /* ... so we have 0 <= B < A again */
443
444 /* Since the former M is now B and the former B is now A,
445 * (**) translates into
446 * sign*Y*a == D*A + B (mod |n|),
447 * i.e.
448 * sign*Y*a - D*A == B (mod |n|).
449 * Similarly, (*) translates into
450 * -sign*X*a == A (mod |n|).
451 *
452 * Thus,
453 * sign*Y*a + D*sign*X*a == B (mod |n|),
454 * i.e.
455 * sign*(Y + D*X)*a == B (mod |n|).
456 *
457 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
458 * -sign*X*a == B (mod |n|),
459 * sign*Y*a == A (mod |n|).
460 * Note that X and Y stay non-negative all the time. */
461
462 /* most of the time D is very small, so we can optimize tmp := D*X+Y */
463 if (BN_is_one(D)) {
464 if (!BN_add(tmp, X, Y)) {
465 goto err;
466 }
467 } else {
468 if (BN_is_word(D, 2)) {
469 if (!BN_lshift1(tmp, X)) {
470 goto err;
471 }
472 } else if (BN_is_word(D, 4)) {
473 if (!BN_lshift(tmp, X, 2)) {
474 goto err;
475 }
476 } else if (D->top == 1) {
477 if (!BN_copy(tmp, X)) {
478 goto err;
479 }
480 if (!BN_mul_word(tmp, D->d[0])) {
481 goto err;
482 }
483 } else {
484 if (!BN_mul(tmp, D, X, ctx)) {
485 goto err;
486 }
487 }
488 if (!BN_add(tmp, tmp, Y)) {
489 goto err;
490 }
491 }
492
493 M = Y; /* keep the BIGNUM object, the value does not matter */
494 Y = X;
495 X = tmp;
496 sign = -sign;
497 }
498 }
499
500 /* The while loop (Euclid's algorithm) ends when
501 * A == gcd(a,n);
502 * we have
503 * sign*Y*a == A (mod |n|),
504 * where Y is non-negative. */
505
506 if (sign < 0) {
507 if (!BN_sub(Y, n, Y)) {
508 goto err;
509 }
510 }
511 /* Now Y*a == A (mod |n|). */
512
513 if (BN_is_one(A)) {
514 /* Y*a == 1 (mod |n|) */
515 if (!Y->neg && BN_ucmp(Y, n) < 0) {
516 if (!BN_copy(R, Y)) {
517 goto err;
518 }
519 } else {
520 if (!BN_nnmod(R, Y, n, ctx)) {
521 goto err;
522 }
523 }
524 } else {
525 OPENSSL_PUT_ERROR(BN, BN_mod_inverse, BN_R_NO_INVERSE);
526 goto err;
527 }
528 ret = R;
529
530 err:
531 if (ret == NULL && out == NULL) {
532 BN_free(R);
533 }
534 BN_CTX_end(ctx);
535 return ret;
536 }
537
538 /* BN_mod_inverse_no_branch is a special version of BN_mod_inverse.
539 * It does not contain branches that may leak sensitive information. */
BN_mod_inverse_no_branch(BIGNUM * out,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)540 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *out, const BIGNUM *a,
541 const BIGNUM *n, BN_CTX *ctx) {
542 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
543 BIGNUM local_A, local_B;
544 BIGNUM *pA, *pB;
545 BIGNUM *ret = NULL;
546 int sign;
547
548 BN_CTX_start(ctx);
549 A = BN_CTX_get(ctx);
550 B = BN_CTX_get(ctx);
551 X = BN_CTX_get(ctx);
552 D = BN_CTX_get(ctx);
553 M = BN_CTX_get(ctx);
554 Y = BN_CTX_get(ctx);
555 T = BN_CTX_get(ctx);
556 if (T == NULL) {
557 goto err;
558 }
559
560 if (out == NULL) {
561 R = BN_new();
562 } else {
563 R = out;
564 }
565 if (R == NULL) {
566 goto err;
567 }
568
569 BN_zero(Y);
570 if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
571 goto err;
572 }
573 A->neg = 0;
574
575 if (B->neg || (BN_ucmp(B, A) >= 0)) {
576 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
577 * BN_div_no_branch will be called eventually.
578 */
579 pB = &local_B;
580 BN_with_flags(pB, B, BN_FLG_CONSTTIME);
581 if (!BN_nnmod(B, pB, A, ctx)) {
582 goto err;
583 }
584 }
585 sign = -1;
586 /* From B = a mod |n|, A = |n| it follows that
587 *
588 * 0 <= B < A,
589 * -sign*X*a == B (mod |n|),
590 * sign*Y*a == A (mod |n|).
591 */
592
593 while (!BN_is_zero(B)) {
594 BIGNUM *tmp;
595
596 /*
597 * 0 < B < A,
598 * (*) -sign*X*a == B (mod |n|),
599 * sign*Y*a == A (mod |n|)
600 */
601
602 /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
603 * BN_div_no_branch will be called eventually.
604 */
605 pA = &local_A;
606 BN_with_flags(pA, A, BN_FLG_CONSTTIME);
607
608 /* (D, M) := (A/B, A%B) ... */
609 if (!BN_div(D, M, pA, B, ctx)) {
610 goto err;
611 }
612
613 /* Now
614 * A = D*B + M;
615 * thus we have
616 * (**) sign*Y*a == D*B + M (mod |n|).
617 */
618
619 tmp = A; /* keep the BIGNUM object, the value does not matter */
620
621 /* (A, B) := (B, A mod B) ... */
622 A = B;
623 B = M;
624 /* ... so we have 0 <= B < A again */
625
626 /* Since the former M is now B and the former B is now A,
627 * (**) translates into
628 * sign*Y*a == D*A + B (mod |n|),
629 * i.e.
630 * sign*Y*a - D*A == B (mod |n|).
631 * Similarly, (*) translates into
632 * -sign*X*a == A (mod |n|).
633 *
634 * Thus,
635 * sign*Y*a + D*sign*X*a == B (mod |n|),
636 * i.e.
637 * sign*(Y + D*X)*a == B (mod |n|).
638 *
639 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
640 * -sign*X*a == B (mod |n|),
641 * sign*Y*a == A (mod |n|).
642 * Note that X and Y stay non-negative all the time.
643 */
644
645 if (!BN_mul(tmp, D, X, ctx)) {
646 goto err;
647 }
648 if (!BN_add(tmp, tmp, Y)) {
649 goto err;
650 }
651
652 M = Y; /* keep the BIGNUM object, the value does not matter */
653 Y = X;
654 X = tmp;
655 sign = -sign;
656 }
657
658 /*
659 * The while loop (Euclid's algorithm) ends when
660 * A == gcd(a,n);
661 * we have
662 * sign*Y*a == A (mod |n|),
663 * where Y is non-negative.
664 */
665
666 if (sign < 0) {
667 if (!BN_sub(Y, n, Y)) {
668 goto err;
669 }
670 }
671 /* Now Y*a == A (mod |n|). */
672
673 if (BN_is_one(A)) {
674 /* Y*a == 1 (mod |n|) */
675 if (!Y->neg && BN_ucmp(Y, n) < 0) {
676 if (!BN_copy(R, Y)) {
677 goto err;
678 }
679 } else {
680 if (!BN_nnmod(R, Y, n, ctx)) {
681 goto err;
682 }
683 }
684 } else {
685 OPENSSL_PUT_ERROR(BN, BN_mod_inverse_no_branch, BN_R_NO_INVERSE);
686 goto err;
687 }
688 ret = R;
689
690 err:
691 if (ret == NULL && out == NULL) {
692 BN_free(R);
693 }
694
695 BN_CTX_end(ctx);
696 return ret;
697 }
698