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[thirdparty/openssl.git] / crypto / bn / bn_gcd.c
1 /*
2 * Copyright 1995-2017 The OpenSSL Project Authors. All Rights Reserved.
3 *
4 * Licensed under the OpenSSL license (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
8 */
9
10 #include "internal/cryptlib.h"
11 #include "bn_lcl.h"
12
13 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);
14
15 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
16 {
17 BIGNUM *a, *b, *t;
18 int ret = 0;
19
20 bn_check_top(in_a);
21 bn_check_top(in_b);
22
23 BN_CTX_start(ctx);
24 a = BN_CTX_get(ctx);
25 b = BN_CTX_get(ctx);
26 if (b == NULL)
27 goto err;
28
29 if (BN_copy(a, in_a) == NULL)
30 goto err;
31 if (BN_copy(b, in_b) == NULL)
32 goto err;
33 a->neg = 0;
34 b->neg = 0;
35
36 if (BN_cmp(a, b) < 0) {
37 t = a;
38 a = b;
39 b = t;
40 }
41 t = euclid(a, b);
42 if (t == NULL)
43 goto err;
44
45 if (BN_copy(r, t) == NULL)
46 goto err;
47 ret = 1;
48 err:
49 BN_CTX_end(ctx);
50 bn_check_top(r);
51 return ret;
52 }
53
54 static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
55 {
56 BIGNUM *t;
57 int shifts = 0;
58
59 bn_check_top(a);
60 bn_check_top(b);
61
62 /* 0 <= b <= a */
63 while (!BN_is_zero(b)) {
64 /* 0 < b <= a */
65
66 if (BN_is_odd(a)) {
67 if (BN_is_odd(b)) {
68 if (!BN_sub(a, a, b))
69 goto err;
70 if (!BN_rshift1(a, a))
71 goto err;
72 if (BN_cmp(a, b) < 0) {
73 t = a;
74 a = b;
75 b = t;
76 }
77 } else { /* a odd - b even */
78
79 if (!BN_rshift1(b, b))
80 goto err;
81 if (BN_cmp(a, b) < 0) {
82 t = a;
83 a = b;
84 b = t;
85 }
86 }
87 } else { /* a is even */
88
89 if (BN_is_odd(b)) {
90 if (!BN_rshift1(a, a))
91 goto err;
92 if (BN_cmp(a, b) < 0) {
93 t = a;
94 a = b;
95 b = t;
96 }
97 } else { /* a even - b even */
98
99 if (!BN_rshift1(a, a))
100 goto err;
101 if (!BN_rshift1(b, b))
102 goto err;
103 shifts++;
104 }
105 }
106 /* 0 <= b <= a */
107 }
108
109 if (shifts) {
110 if (!BN_lshift(a, a, shifts))
111 goto err;
112 }
113 bn_check_top(a);
114 return a;
115 err:
116 return NULL;
117 }
118
119 /* solves ax == 1 (mod n) */
120 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
121 const BIGNUM *a, const BIGNUM *n,
122 BN_CTX *ctx);
123
124 BIGNUM *BN_mod_inverse(BIGNUM *in,
125 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
126 {
127 BIGNUM *rv;
128 int noinv;
129 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
130 if (noinv)
131 BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
132 return rv;
133 }
134
135 BIGNUM *int_bn_mod_inverse(BIGNUM *in,
136 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
137 int *pnoinv)
138 {
139 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
140 BIGNUM *ret = NULL;
141 int sign;
142
143 if (pnoinv)
144 *pnoinv = 0;
145
146 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
147 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
148 return BN_mod_inverse_no_branch(in, a, n, ctx);
149 }
150
151 bn_check_top(a);
152 bn_check_top(n);
153
154 BN_CTX_start(ctx);
155 A = BN_CTX_get(ctx);
156 B = BN_CTX_get(ctx);
157 X = BN_CTX_get(ctx);
158 D = BN_CTX_get(ctx);
159 M = BN_CTX_get(ctx);
160 Y = BN_CTX_get(ctx);
161 T = BN_CTX_get(ctx);
162 if (T == NULL)
163 goto err;
164
165 if (in == NULL)
166 R = BN_new();
167 else
168 R = in;
169 if (R == NULL)
170 goto err;
171
172 BN_one(X);
173 BN_zero(Y);
174 if (BN_copy(B, a) == NULL)
175 goto err;
176 if (BN_copy(A, n) == NULL)
177 goto err;
178 A->neg = 0;
179 if (B->neg || (BN_ucmp(B, A) >= 0)) {
180 if (!BN_nnmod(B, B, A, ctx))
181 goto err;
182 }
183 sign = -1;
184 /*-
185 * From B = a mod |n|, A = |n| it follows that
186 *
187 * 0 <= B < A,
188 * -sign*X*a == B (mod |n|),
189 * sign*Y*a == A (mod |n|).
190 */
191
192 if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
193 /*
194 * Binary inversion algorithm; requires odd modulus. This is faster
195 * than the general algorithm if the modulus is sufficiently small
196 * (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
197 * systems)
198 */
199 int shift;
200
201 while (!BN_is_zero(B)) {
202 /*-
203 * 0 < B < |n|,
204 * 0 < A <= |n|,
205 * (1) -sign*X*a == B (mod |n|),
206 * (2) sign*Y*a == A (mod |n|)
207 */
208
209 /*
210 * Now divide B by the maximum possible power of two in the
211 * integers, and divide X by the same value mod |n|. When we're
212 * done, (1) still holds.
213 */
214 shift = 0;
215 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
216 shift++;
217
218 if (BN_is_odd(X)) {
219 if (!BN_uadd(X, X, n))
220 goto err;
221 }
222 /*
223 * now X is even, so we can easily divide it by two
224 */
225 if (!BN_rshift1(X, X))
226 goto err;
227 }
228 if (shift > 0) {
229 if (!BN_rshift(B, B, shift))
230 goto err;
231 }
232
233 /*
234 * Same for A and Y. Afterwards, (2) still holds.
235 */
236 shift = 0;
237 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
238 shift++;
239
240 if (BN_is_odd(Y)) {
241 if (!BN_uadd(Y, Y, n))
242 goto err;
243 }
244 /* now Y is even */
245 if (!BN_rshift1(Y, Y))
246 goto err;
247 }
248 if (shift > 0) {
249 if (!BN_rshift(A, A, shift))
250 goto err;
251 }
252
253 /*-
254 * We still have (1) and (2).
255 * Both A and B are odd.
256 * The following computations ensure that
257 *
258 * 0 <= B < |n|,
259 * 0 < A < |n|,
260 * (1) -sign*X*a == B (mod |n|),
261 * (2) sign*Y*a == A (mod |n|),
262 *
263 * and that either A or B is even in the next iteration.
264 */
265 if (BN_ucmp(B, A) >= 0) {
266 /* -sign*(X + Y)*a == B - A (mod |n|) */
267 if (!BN_uadd(X, X, Y))
268 goto err;
269 /*
270 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
271 * actually makes the algorithm slower
272 */
273 if (!BN_usub(B, B, A))
274 goto err;
275 } else {
276 /* sign*(X + Y)*a == A - B (mod |n|) */
277 if (!BN_uadd(Y, Y, X))
278 goto err;
279 /*
280 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
281 */
282 if (!BN_usub(A, A, B))
283 goto err;
284 }
285 }
286 } else {
287 /* general inversion algorithm */
288
289 while (!BN_is_zero(B)) {
290 BIGNUM *tmp;
291
292 /*-
293 * 0 < B < A,
294 * (*) -sign*X*a == B (mod |n|),
295 * sign*Y*a == A (mod |n|)
296 */
297
298 /* (D, M) := (A/B, A%B) ... */
299 if (BN_num_bits(A) == BN_num_bits(B)) {
300 if (!BN_one(D))
301 goto err;
302 if (!BN_sub(M, A, B))
303 goto err;
304 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
305 /* A/B is 1, 2, or 3 */
306 if (!BN_lshift1(T, B))
307 goto err;
308 if (BN_ucmp(A, T) < 0) {
309 /* A < 2*B, so D=1 */
310 if (!BN_one(D))
311 goto err;
312 if (!BN_sub(M, A, B))
313 goto err;
314 } else {
315 /* A >= 2*B, so D=2 or D=3 */
316 if (!BN_sub(M, A, T))
317 goto err;
318 if (!BN_add(D, T, B))
319 goto err; /* use D (:= 3*B) as temp */
320 if (BN_ucmp(A, D) < 0) {
321 /* A < 3*B, so D=2 */
322 if (!BN_set_word(D, 2))
323 goto err;
324 /*
325 * M (= A - 2*B) already has the correct value
326 */
327 } else {
328 /* only D=3 remains */
329 if (!BN_set_word(D, 3))
330 goto err;
331 /*
332 * currently M = A - 2*B, but we need M = A - 3*B
333 */
334 if (!BN_sub(M, M, B))
335 goto err;
336 }
337 }
338 } else {
339 if (!BN_div(D, M, A, B, ctx))
340 goto err;
341 }
342
343 /*-
344 * Now
345 * A = D*B + M;
346 * thus we have
347 * (**) sign*Y*a == D*B + M (mod |n|).
348 */
349
350 tmp = A; /* keep the BIGNUM object, the value does not matter */
351
352 /* (A, B) := (B, A mod B) ... */
353 A = B;
354 B = M;
355 /* ... so we have 0 <= B < A again */
356
357 /*-
358 * Since the former M is now B and the former B is now A,
359 * (**) translates into
360 * sign*Y*a == D*A + B (mod |n|),
361 * i.e.
362 * sign*Y*a - D*A == B (mod |n|).
363 * Similarly, (*) translates into
364 * -sign*X*a == A (mod |n|).
365 *
366 * Thus,
367 * sign*Y*a + D*sign*X*a == B (mod |n|),
368 * i.e.
369 * sign*(Y + D*X)*a == B (mod |n|).
370 *
371 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
372 * -sign*X*a == B (mod |n|),
373 * sign*Y*a == A (mod |n|).
374 * Note that X and Y stay non-negative all the time.
375 */
376
377 /*
378 * most of the time D is very small, so we can optimize tmp := D*X+Y
379 */
380 if (BN_is_one(D)) {
381 if (!BN_add(tmp, X, Y))
382 goto err;
383 } else {
384 if (BN_is_word(D, 2)) {
385 if (!BN_lshift1(tmp, X))
386 goto err;
387 } else if (BN_is_word(D, 4)) {
388 if (!BN_lshift(tmp, X, 2))
389 goto err;
390 } else if (D->top == 1) {
391 if (!BN_copy(tmp, X))
392 goto err;
393 if (!BN_mul_word(tmp, D->d[0]))
394 goto err;
395 } else {
396 if (!BN_mul(tmp, D, X, ctx))
397 goto err;
398 }
399 if (!BN_add(tmp, tmp, Y))
400 goto err;
401 }
402
403 M = Y; /* keep the BIGNUM object, the value does not matter */
404 Y = X;
405 X = tmp;
406 sign = -sign;
407 }
408 }
409
410 /*-
411 * The while loop (Euclid's algorithm) ends when
412 * A == gcd(a,n);
413 * we have
414 * sign*Y*a == A (mod |n|),
415 * where Y is non-negative.
416 */
417
418 if (sign < 0) {
419 if (!BN_sub(Y, n, Y))
420 goto err;
421 }
422 /* Now Y*a == A (mod |n|). */
423
424 if (BN_is_one(A)) {
425 /* Y*a == 1 (mod |n|) */
426 if (!Y->neg && BN_ucmp(Y, n) < 0) {
427 if (!BN_copy(R, Y))
428 goto err;
429 } else {
430 if (!BN_nnmod(R, Y, n, ctx))
431 goto err;
432 }
433 } else {
434 if (pnoinv)
435 *pnoinv = 1;
436 goto err;
437 }
438 ret = R;
439 err:
440 if ((ret == NULL) && (in == NULL))
441 BN_free(R);
442 BN_CTX_end(ctx);
443 bn_check_top(ret);
444 return ret;
445 }
446
447 /*
448 * BN_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
449 * not contain branches that may leak sensitive information.
450 */
451 static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
452 const BIGNUM *a, const BIGNUM *n,
453 BN_CTX *ctx)
454 {
455 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
456 BIGNUM *ret = NULL;
457 int sign;
458
459 bn_check_top(a);
460 bn_check_top(n);
461
462 BN_CTX_start(ctx);
463 A = BN_CTX_get(ctx);
464 B = BN_CTX_get(ctx);
465 X = BN_CTX_get(ctx);
466 D = BN_CTX_get(ctx);
467 M = BN_CTX_get(ctx);
468 Y = BN_CTX_get(ctx);
469 T = BN_CTX_get(ctx);
470 if (T == NULL)
471 goto err;
472
473 if (in == NULL)
474 R = BN_new();
475 else
476 R = in;
477 if (R == NULL)
478 goto err;
479
480 BN_one(X);
481 BN_zero(Y);
482 if (BN_copy(B, a) == NULL)
483 goto err;
484 if (BN_copy(A, n) == NULL)
485 goto err;
486 A->neg = 0;
487
488 if (B->neg || (BN_ucmp(B, A) >= 0)) {
489 /*
490 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
491 * BN_div_no_branch will be called eventually.
492 */
493 {
494 BIGNUM local_B;
495 bn_init(&local_B);
496 BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
497 if (!BN_nnmod(B, &local_B, A, ctx))
498 goto err;
499 /* Ensure local_B goes out of scope before any further use of B */
500 }
501 }
502 sign = -1;
503 /*-
504 * From B = a mod |n|, A = |n| it follows that
505 *
506 * 0 <= B < A,
507 * -sign*X*a == B (mod |n|),
508 * sign*Y*a == A (mod |n|).
509 */
510
511 while (!BN_is_zero(B)) {
512 BIGNUM *tmp;
513
514 /*-
515 * 0 < B < A,
516 * (*) -sign*X*a == B (mod |n|),
517 * sign*Y*a == A (mod |n|)
518 */
519
520 /*
521 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
522 * BN_div_no_branch will be called eventually.
523 */
524 {
525 BIGNUM local_A;
526 bn_init(&local_A);
527 BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
528
529 /* (D, M) := (A/B, A%B) ... */
530 if (!BN_div(D, M, &local_A, B, ctx))
531 goto err;
532 /* Ensure local_A goes out of scope before any further use of A */
533 }
534
535 /*-
536 * Now
537 * A = D*B + M;
538 * thus we have
539 * (**) sign*Y*a == D*B + M (mod |n|).
540 */
541
542 tmp = A; /* keep the BIGNUM object, the value does not
543 * matter */
544
545 /* (A, B) := (B, A mod B) ... */
546 A = B;
547 B = M;
548 /* ... so we have 0 <= B < A again */
549
550 /*-
551 * Since the former M is now B and the former B is now A,
552 * (**) translates into
553 * sign*Y*a == D*A + B (mod |n|),
554 * i.e.
555 * sign*Y*a - D*A == B (mod |n|).
556 * Similarly, (*) translates into
557 * -sign*X*a == A (mod |n|).
558 *
559 * Thus,
560 * sign*Y*a + D*sign*X*a == B (mod |n|),
561 * i.e.
562 * sign*(Y + D*X)*a == B (mod |n|).
563 *
564 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
565 * -sign*X*a == B (mod |n|),
566 * sign*Y*a == A (mod |n|).
567 * Note that X and Y stay non-negative all the time.
568 */
569
570 if (!BN_mul(tmp, D, X, ctx))
571 goto err;
572 if (!BN_add(tmp, tmp, Y))
573 goto err;
574
575 M = Y; /* keep the BIGNUM object, the value does not
576 * matter */
577 Y = X;
578 X = tmp;
579 sign = -sign;
580 }
581
582 /*-
583 * The while loop (Euclid's algorithm) ends when
584 * A == gcd(a,n);
585 * we have
586 * sign*Y*a == A (mod |n|),
587 * where Y is non-negative.
588 */
589
590 if (sign < 0) {
591 if (!BN_sub(Y, n, Y))
592 goto err;
593 }
594 /* Now Y*a == A (mod |n|). */
595
596 if (BN_is_one(A)) {
597 /* Y*a == 1 (mod |n|) */
598 if (!Y->neg && BN_ucmp(Y, n) < 0) {
599 if (!BN_copy(R, Y))
600 goto err;
601 } else {
602 if (!BN_nnmod(R, Y, n, ctx))
603 goto err;
604 }
605 } else {
606 BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH, BN_R_NO_INVERSE);
607 goto err;
608 }
609 ret = R;
610 err:
611 if ((ret == NULL) && (in == NULL))
612 BN_free(R);
613 BN_CTX_end(ctx);
614 bn_check_top(ret);
615 return ret;
616 }
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