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 <assert.h>
112
113 #include <openssl/err.h>
114
115 #include "internal.h"
116
euclid(BIGNUM * a,BIGNUM * b)117 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b) {
118 BIGNUM *t;
119 int shifts = 0;
120
121 /* 0 <= b <= a */
122 while (!BN_is_zero(b)) {
123 /* 0 < b <= a */
124
125 if (BN_is_odd(a)) {
126 if (BN_is_odd(b)) {
127 if (!BN_sub(a, a, b)) {
128 goto err;
129 }
130 if (!BN_rshift1(a, a)) {
131 goto err;
132 }
133 if (BN_cmp(a, b) < 0) {
134 t = a;
135 a = b;
136 b = t;
137 }
138 } else {
139 /* a odd - b even */
140 if (!BN_rshift1(b, b)) {
141 goto err;
142 }
143 if (BN_cmp(a, b) < 0) {
144 t = a;
145 a = b;
146 b = t;
147 }
148 }
149 } else {
150 /* a is even */
151 if (BN_is_odd(b)) {
152 if (!BN_rshift1(a, a)) {
153 goto err;
154 }
155 if (BN_cmp(a, b) < 0) {
156 t = a;
157 a = b;
158 b = t;
159 }
160 } else {
161 /* a even - b even */
162 if (!BN_rshift1(a, a)) {
163 goto err;
164 }
165 if (!BN_rshift1(b, b)) {
166 goto err;
167 }
168 shifts++;
169 }
170 }
171 /* 0 <= b <= a */
172 }
173
174 if (shifts) {
175 if (!BN_lshift(a, a, shifts)) {
176 goto err;
177 }
178 }
179
180 return a;
181
182 err:
183 return NULL;
184 }
185
BN_gcd(BIGNUM * r,const BIGNUM * in_a,const BIGNUM * in_b,BN_CTX * ctx)186 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx) {
187 BIGNUM *a, *b, *t;
188 int ret = 0;
189
190 BN_CTX_start(ctx);
191 a = BN_CTX_get(ctx);
192 b = BN_CTX_get(ctx);
193
194 if (a == NULL || b == NULL) {
195 goto err;
196 }
197 if (BN_copy(a, in_a) == NULL) {
198 goto err;
199 }
200 if (BN_copy(b, in_b) == NULL) {
201 goto err;
202 }
203
204 a->neg = 0;
205 b->neg = 0;
206
207 if (BN_cmp(a, b) < 0) {
208 t = a;
209 a = b;
210 b = t;
211 }
212 t = euclid(a, b);
213 if (t == NULL) {
214 goto err;
215 }
216
217 if (BN_copy(r, t) == NULL) {
218 goto err;
219 }
220 ret = 1;
221
222 err:
223 BN_CTX_end(ctx);
224 return ret;
225 }
226
227 /* solves ax == 1 (mod n) */
228 static int bn_mod_inverse_general(BIGNUM *out, int *out_no_inverse,
229 const BIGNUM *a, const BIGNUM *n,
230 BN_CTX *ctx);
231
BN_mod_inverse_odd(BIGNUM * out,int * out_no_inverse,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)232 int BN_mod_inverse_odd(BIGNUM *out, int *out_no_inverse, const BIGNUM *a,
233 const BIGNUM *n, BN_CTX *ctx) {
234 *out_no_inverse = 0;
235
236 if (!BN_is_odd(n)) {
237 OPENSSL_PUT_ERROR(BN, BN_R_CALLED_WITH_EVEN_MODULUS);
238 return 0;
239 }
240
241 if (BN_is_negative(a) || BN_cmp(a, n) >= 0) {
242 OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED);
243 return 0;
244 }
245
246 BIGNUM *A, *B, *X, *Y;
247 int ret = 0;
248 int sign;
249
250 BN_CTX_start(ctx);
251 A = BN_CTX_get(ctx);
252 B = BN_CTX_get(ctx);
253 X = BN_CTX_get(ctx);
254 Y = BN_CTX_get(ctx);
255 if (Y == NULL) {
256 goto err;
257 }
258
259 BIGNUM *R = out;
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 sign = -1;
267 /* From B = a mod |n|, A = |n| it follows that
268 *
269 * 0 <= B < A,
270 * -sign*X*a == B (mod |n|),
271 * sign*Y*a == A (mod |n|).
272 */
273
274 /* Binary inversion algorithm; requires odd modulus. This is faster than the
275 * general algorithm if the modulus is sufficiently small (about 400 .. 500
276 * bits on 32-bit systems, but much more on 64-bit systems) */
277 int shift;
278
279 while (!BN_is_zero(B)) {
280 /* 0 < B < |n|,
281 * 0 < A <= |n|,
282 * (1) -sign*X*a == B (mod |n|),
283 * (2) sign*Y*a == A (mod |n|) */
284
285 /* Now divide B by the maximum possible power of two in the integers,
286 * and divide X by the same value mod |n|.
287 * When we're done, (1) still holds. */
288 shift = 0;
289 while (!BN_is_bit_set(B, shift)) {
290 /* note that 0 < B */
291 shift++;
292
293 if (BN_is_odd(X)) {
294 if (!BN_uadd(X, X, n)) {
295 goto err;
296 }
297 }
298 /* now X is even, so we can easily divide it by two */
299 if (!BN_rshift1(X, X)) {
300 goto err;
301 }
302 }
303 if (shift > 0) {
304 if (!BN_rshift(B, B, shift)) {
305 goto err;
306 }
307 }
308
309 /* Same for A and Y. Afterwards, (2) still holds. */
310 shift = 0;
311 while (!BN_is_bit_set(A, shift)) {
312 /* note that 0 < A */
313 shift++;
314
315 if (BN_is_odd(Y)) {
316 if (!BN_uadd(Y, Y, n)) {
317 goto err;
318 }
319 }
320 /* now Y is even */
321 if (!BN_rshift1(Y, Y)) {
322 goto err;
323 }
324 }
325 if (shift > 0) {
326 if (!BN_rshift(A, A, shift)) {
327 goto err;
328 }
329 }
330
331 /* We still have (1) and (2).
332 * Both A and B are odd.
333 * The following computations ensure that
334 *
335 * 0 <= B < |n|,
336 * 0 < A < |n|,
337 * (1) -sign*X*a == B (mod |n|),
338 * (2) sign*Y*a == A (mod |n|),
339 *
340 * and that either A or B is even in the next iteration. */
341 if (BN_ucmp(B, A) >= 0) {
342 /* -sign*(X + Y)*a == B - A (mod |n|) */
343 if (!BN_uadd(X, X, Y)) {
344 goto err;
345 }
346 /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
347 * actually makes the algorithm slower */
348 if (!BN_usub(B, B, A)) {
349 goto err;
350 }
351 } else {
352 /* sign*(X + Y)*a == A - B (mod |n|) */
353 if (!BN_uadd(Y, Y, X)) {
354 goto err;
355 }
356 /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
357 if (!BN_usub(A, A, B)) {
358 goto err;
359 }
360 }
361 }
362
363 if (!BN_is_one(A)) {
364 *out_no_inverse = 1;
365 OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
366 goto err;
367 }
368
369 /* The while loop (Euclid's algorithm) ends when
370 * A == gcd(a,n);
371 * we have
372 * sign*Y*a == A (mod |n|),
373 * where Y is non-negative. */
374
375 if (sign < 0) {
376 if (!BN_sub(Y, n, Y)) {
377 goto err;
378 }
379 }
380 /* Now Y*a == A (mod |n|). */
381
382 /* Y*a == 1 (mod |n|) */
383 if (!Y->neg && BN_ucmp(Y, n) < 0) {
384 if (!BN_copy(R, Y)) {
385 goto err;
386 }
387 } else {
388 if (!BN_nnmod(R, Y, n, ctx)) {
389 goto err;
390 }
391 }
392
393 ret = 1;
394
395 err:
396 BN_CTX_end(ctx);
397 return ret;
398 }
399
BN_mod_inverse(BIGNUM * out,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)400 BIGNUM *BN_mod_inverse(BIGNUM *out, const BIGNUM *a, const BIGNUM *n,
401 BN_CTX *ctx) {
402 BIGNUM *new_out = NULL;
403 if (out == NULL) {
404 new_out = BN_new();
405 if (new_out == NULL) {
406 OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
407 return NULL;
408 }
409 out = new_out;
410 }
411
412 int ok = 0;
413 BIGNUM *a_reduced = NULL;
414 if (a->neg || BN_ucmp(a, n) >= 0) {
415 a_reduced = BN_dup(a);
416 if (a_reduced == NULL) {
417 goto err;
418 }
419 if (!BN_nnmod(a_reduced, a_reduced, n, ctx)) {
420 goto err;
421 }
422 a = a_reduced;
423 }
424
425 int no_inverse;
426 if (!BN_is_odd(n)) {
427 if (!bn_mod_inverse_general(out, &no_inverse, a, n, ctx)) {
428 goto err;
429 }
430 } else if (!BN_mod_inverse_odd(out, &no_inverse, a, n, ctx)) {
431 goto err;
432 }
433
434 ok = 1;
435
436 err:
437 if (!ok) {
438 BN_free(new_out);
439 out = NULL;
440 }
441 BN_free(a_reduced);
442 return out;
443 }
444
BN_mod_inverse_blinded(BIGNUM * out,int * out_no_inverse,const BIGNUM * a,const BN_MONT_CTX * mont,BN_CTX * ctx)445 int BN_mod_inverse_blinded(BIGNUM *out, int *out_no_inverse, const BIGNUM *a,
446 const BN_MONT_CTX *mont, BN_CTX *ctx) {
447 *out_no_inverse = 0;
448
449 if (BN_is_negative(a) || BN_cmp(a, &mont->N) >= 0) {
450 OPENSSL_PUT_ERROR(BN, BN_R_INPUT_NOT_REDUCED);
451 return 0;
452 }
453
454 int ret = 0;
455 BIGNUM blinding_factor;
456 BN_init(&blinding_factor);
457
458 if (!BN_rand_range_ex(&blinding_factor, 1, &mont->N) ||
459 !BN_mod_mul_montgomery(out, &blinding_factor, a, mont, ctx) ||
460 !BN_mod_inverse_odd(out, out_no_inverse, out, &mont->N, ctx) ||
461 !BN_mod_mul_montgomery(out, &blinding_factor, out, mont, ctx)) {
462 OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
463 goto err;
464 }
465
466 ret = 1;
467
468 err:
469 BN_free(&blinding_factor);
470 return ret;
471 }
472
473 /* bn_mod_inverse_general is the general inversion algorithm that works for
474 * both even and odd |n|. It was specifically designed to contain fewer
475 * branches that may leak sensitive information; see "New Branch Prediction
476 * Vulnerabilities in OpenSSL and Necessary Software Countermeasures" by
477 * Onur Acıçmez, Shay Gueron, and Jean-Pierre Seifert. */
bn_mod_inverse_general(BIGNUM * out,int * out_no_inverse,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)478 static int bn_mod_inverse_general(BIGNUM *out, int *out_no_inverse,
479 const BIGNUM *a, const BIGNUM *n,
480 BN_CTX *ctx) {
481 BIGNUM *A, *B, *X, *Y, *M, *D, *T;
482 int ret = 0;
483 int sign;
484
485 *out_no_inverse = 0;
486
487 BN_CTX_start(ctx);
488 A = BN_CTX_get(ctx);
489 B = BN_CTX_get(ctx);
490 X = BN_CTX_get(ctx);
491 D = BN_CTX_get(ctx);
492 M = BN_CTX_get(ctx);
493 Y = BN_CTX_get(ctx);
494 T = BN_CTX_get(ctx);
495 if (T == NULL) {
496 goto err;
497 }
498
499 BIGNUM *R = out;
500
501 BN_zero(Y);
502 if (!BN_one(X) || BN_copy(B, a) == NULL || BN_copy(A, n) == NULL) {
503 goto err;
504 }
505 A->neg = 0;
506
507 sign = -1;
508 /* From B = a mod |n|, A = |n| it follows that
509 *
510 * 0 <= B < A,
511 * -sign*X*a == B (mod |n|),
512 * sign*Y*a == A (mod |n|).
513 */
514
515 while (!BN_is_zero(B)) {
516 BIGNUM *tmp;
517
518 /*
519 * 0 < B < A,
520 * (*) -sign*X*a == B (mod |n|),
521 * sign*Y*a == A (mod |n|)
522 */
523
524 /* (D, M) := (A/B, A%B) ... */
525 if (!BN_div(D, M, A, B, ctx)) {
526 goto err;
527 }
528
529 /* Now
530 * A = D*B + M;
531 * thus we have
532 * (**) sign*Y*a == D*B + M (mod |n|).
533 */
534
535 tmp = A; /* keep the BIGNUM object, the value does not matter */
536
537 /* (A, B) := (B, A mod B) ... */
538 A = B;
539 B = M;
540 /* ... so we have 0 <= B < A again */
541
542 /* Since the former M is now B and the former B is now A,
543 * (**) translates into
544 * sign*Y*a == D*A + B (mod |n|),
545 * i.e.
546 * sign*Y*a - D*A == B (mod |n|).
547 * Similarly, (*) translates into
548 * -sign*X*a == A (mod |n|).
549 *
550 * Thus,
551 * sign*Y*a + D*sign*X*a == B (mod |n|),
552 * i.e.
553 * sign*(Y + D*X)*a == B (mod |n|).
554 *
555 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
556 * -sign*X*a == B (mod |n|),
557 * sign*Y*a == A (mod |n|).
558 * Note that X and Y stay non-negative all the time.
559 */
560
561 if (!BN_mul(tmp, D, X, ctx)) {
562 goto err;
563 }
564 if (!BN_add(tmp, tmp, Y)) {
565 goto err;
566 }
567
568 M = Y; /* keep the BIGNUM object, the value does not matter */
569 Y = X;
570 X = tmp;
571 sign = -sign;
572 }
573
574 if (!BN_is_one(A)) {
575 *out_no_inverse = 1;
576 OPENSSL_PUT_ERROR(BN, BN_R_NO_INVERSE);
577 goto err;
578 }
579
580 /*
581 * The while loop (Euclid's algorithm) ends when
582 * A == gcd(a,n);
583 * we have
584 * sign*Y*a == A (mod |n|),
585 * where Y is non-negative.
586 */
587
588 if (sign < 0) {
589 if (!BN_sub(Y, n, Y)) {
590 goto err;
591 }
592 }
593 /* Now Y*a == A (mod |n|). */
594
595 /* Y*a == 1 (mod |n|) */
596 if (!Y->neg && BN_ucmp(Y, n) < 0) {
597 if (!BN_copy(R, Y)) {
598 goto err;
599 }
600 } else {
601 if (!BN_nnmod(R, Y, n, ctx)) {
602 goto err;
603 }
604 }
605
606 ret = 1;
607
608 err:
609 BN_CTX_end(ctx);
610 return ret;
611 }
612
bn_mod_inverse_prime(BIGNUM * out,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx,const BN_MONT_CTX * mont_p)613 int bn_mod_inverse_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
614 BN_CTX *ctx, const BN_MONT_CTX *mont_p) {
615 BN_CTX_start(ctx);
616 BIGNUM *p_minus_2 = BN_CTX_get(ctx);
617 int ok = p_minus_2 != NULL &&
618 BN_copy(p_minus_2, p) &&
619 BN_sub_word(p_minus_2, 2) &&
620 BN_mod_exp_mont(out, a, p_minus_2, p, ctx, mont_p);
621 BN_CTX_end(ctx);
622 return ok;
623 }
624
bn_mod_inverse_secret_prime(BIGNUM * out,const BIGNUM * a,const BIGNUM * p,BN_CTX * ctx,const BN_MONT_CTX * mont_p)625 int bn_mod_inverse_secret_prime(BIGNUM *out, const BIGNUM *a, const BIGNUM *p,
626 BN_CTX *ctx, const BN_MONT_CTX *mont_p) {
627 BN_CTX_start(ctx);
628 BIGNUM *p_minus_2 = BN_CTX_get(ctx);
629 int ok = p_minus_2 != NULL &&
630 BN_copy(p_minus_2, p) &&
631 BN_sub_word(p_minus_2, 2) &&
632 BN_mod_exp_mont_consttime(out, a, p_minus_2, p, ctx, mont_p);
633 BN_CTX_end(ctx);
634 return ok;
635 }
636