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192.168.1.100 / R306 / e08f1058ff217e55feb109a64cd5cac3a2da9a6d / . / ap / lib / libssl / openssl-1.1.1o / crypto / bn / bn_gcd.c
blob: 0941f7b97f3f7c0a86546227c8a6c06b8f0b5a09 [file] [log] [blame]
yuezonghe824eb0c2024-06-27 02:32:26 -0700[diff] [blame]1/*
2 * Copyright 1995-2020 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_local.h"
12
13/*
14 * bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
15 * not contain branches that may leak sensitive information.
16 *
17 * This is a static function, we ensure all callers in this file pass valid
18 * arguments: all passed pointers here are non-NULL.
19 */
20static ossl_inline
21BIGNUM *bn_mod_inverse_no_branch(BIGNUM *in,
22 const BIGNUM *a, const BIGNUM *n,
23 BN_CTX *ctx, int *pnoinv)
24{
25 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
26 BIGNUM *ret = NULL;
27 int sign;
28
29 bn_check_top(a);
30 bn_check_top(n);
31
32 BN_CTX_start(ctx);
33 A = BN_CTX_get(ctx);
34 B = BN_CTX_get(ctx);
35 X = BN_CTX_get(ctx);
36 D = BN_CTX_get(ctx);
37 M = BN_CTX_get(ctx);
38 Y = BN_CTX_get(ctx);
39 T = BN_CTX_get(ctx);
40 if (T == NULL)
41 goto err;
42
43 if (in == NULL)
44 R = BN_new();
45 else
46 R = in;
47 if (R == NULL)
48 goto err;
49
50 BN_one(X);
51 BN_zero(Y);
52 if (BN_copy(B, a) == NULL)
53 goto err;
54 if (BN_copy(A, n) == NULL)
55 goto err;
56 A->neg = 0;
57
58 if (B->neg || (BN_ucmp(B, A) >= 0)) {
59 /*
60 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
61 * BN_div_no_branch will be called eventually.
62 */
63 {
64 BIGNUM local_B;
65 bn_init(&local_B);
66 BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
67 if (!BN_nnmod(B, &local_B, A, ctx))
68 goto err;
69 /* Ensure local_B goes out of scope before any further use of B */
70 }
71 }
72 sign = -1;
73 /*-
74 * From B = a mod |n|, A = |n| it follows that
75 *
76 * 0 <= B < A,
77 * -sign*X*a == B (mod |n|),
78 * sign*Y*a == A (mod |n|).
79 */
80
81 while (!BN_is_zero(B)) {
82 BIGNUM *tmp;
83
84 /*-
85 * 0 < B < A,
86 * (*) -sign*X*a == B (mod |n|),
87 * sign*Y*a == A (mod |n|)
88 */
89
90 /*
91 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
92 * BN_div_no_branch will be called eventually.
93 */
94 {
95 BIGNUM local_A;
96 bn_init(&local_A);
97 BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
98
99 /* (D, M) := (A/B, A%B) ... */
100 if (!BN_div(D, M, &local_A, B, ctx))
101 goto err;
102 /* Ensure local_A goes out of scope before any further use of A */
103 }
104
105 /*-
106 * Now
107 * A = D*B + M;
108 * thus we have
109 * (**) sign*Y*a == D*B + M (mod |n|).
110 */
111
112 tmp = A; /* keep the BIGNUM object, the value does not
113 * matter */
114
115 /* (A, B) := (B, A mod B) ... */
116 A = B;
117 B = M;
118 /* ... so we have 0 <= B < A again */
119
120 /*-
121 * Since the former M is now B and the former B is now A,
122 * (**) translates into
123 * sign*Y*a == D*A + B (mod |n|),
124 * i.e.
125 * sign*Y*a - D*A == B (mod |n|).
126 * Similarly, (*) translates into
127 * -sign*X*a == A (mod |n|).
128 *
129 * Thus,
130 * sign*Y*a + D*sign*X*a == B (mod |n|),
131 * i.e.
132 * sign*(Y + D*X)*a == B (mod |n|).
133 *
134 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
135 * -sign*X*a == B (mod |n|),
136 * sign*Y*a == A (mod |n|).
137 * Note that X and Y stay non-negative all the time.
138 */
139
140 if (!BN_mul(tmp, D, X, ctx))
141 goto err;
142 if (!BN_add(tmp, tmp, Y))
143 goto err;
144
145 M = Y; /* keep the BIGNUM object, the value does not
146 * matter */
147 Y = X;
148 X = tmp;
149 sign = -sign;
150 }
151
152 /*-
153 * The while loop (Euclid's algorithm) ends when
154 * A == gcd(a,n);
155 * we have
156 * sign*Y*a == A (mod |n|),
157 * where Y is non-negative.
158 */
159
160 if (sign < 0) {
161 if (!BN_sub(Y, n, Y))
162 goto err;
163 }
164 /* Now Y*a == A (mod |n|). */
165
166 if (BN_is_one(A)) {
167 /* Y*a == 1 (mod |n|) */
168 if (!Y->neg && BN_ucmp(Y, n) < 0) {
169 if (!BN_copy(R, Y))
170 goto err;
171 } else {
172 if (!BN_nnmod(R, Y, n, ctx))
173 goto err;
174 }
175 } else {
176 *pnoinv = 1;
177 /* caller sets the BN_R_NO_INVERSE error */
178 goto err;
179 }
180
181 ret = R;
182 *pnoinv = 0;
183
184 err:
185 if ((ret == NULL) && (in == NULL))
186 BN_free(R);
187 BN_CTX_end(ctx);
188 bn_check_top(ret);
189 return ret;
190}
191
192/*
193 * This is an internal function, we assume all callers pass valid arguments:
194 * all pointers passed here are assumed non-NULL.
195 */
196BIGNUM *int_bn_mod_inverse(BIGNUM *in,
197 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
198 int *pnoinv)
199{
200 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
201 BIGNUM *ret = NULL;
202 int sign;
203
204 /* This is invalid input so we don't worry about constant time here */
205 if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
206 *pnoinv = 1;
207 return NULL;
208 }
209
210 *pnoinv = 0;
211
212 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
213 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
214 return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
215 }
216
217 bn_check_top(a);
218 bn_check_top(n);
219
220 BN_CTX_start(ctx);
221 A = BN_CTX_get(ctx);
222 B = BN_CTX_get(ctx);
223 X = BN_CTX_get(ctx);
224 D = BN_CTX_get(ctx);
225 M = BN_CTX_get(ctx);
226 Y = BN_CTX_get(ctx);
227 T = BN_CTX_get(ctx);
228 if (T == NULL)
229 goto err;
230
231 if (in == NULL)
232 R = BN_new();
233 else
234 R = in;
235 if (R == NULL)
236 goto err;
237
238 BN_one(X);
239 BN_zero(Y);
240 if (BN_copy(B, a) == NULL)
241 goto err;
242 if (BN_copy(A, n) == NULL)
243 goto err;
244 A->neg = 0;
245 if (B->neg || (BN_ucmp(B, A) >= 0)) {
246 if (!BN_nnmod(B, B, A, ctx))
247 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) <= 2048)) {
259 /*
260 * Binary inversion algorithm; requires odd modulus. This is faster
261 * than the general algorithm if the modulus is sufficiently small
262 * (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
263 * systems)
264 */
265 int shift;
266
267 while (!BN_is_zero(B)) {
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 /*
276 * Now divide B by the maximum possible power of two in the
277 * integers, and divide X by the same value mod |n|. When we're
278 * done, (1) still holds.
279 */
280 shift = 0;
281 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
282 shift++;
283
284 if (BN_is_odd(X)) {
285 if (!BN_uadd(X, X, n))
286 goto err;
287 }
288 /*
289 * now X is even, so we can easily divide it by two
290 */
291 if (!BN_rshift1(X, X))
292 goto err;
293 }
294 if (shift > 0) {
295 if (!BN_rshift(B, B, shift))
296 goto err;
297 }
298
299 /*
300 * Same for A and Y. Afterwards, (2) still holds.
301 */
302 shift = 0;
303 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
304 shift++;
305
306 if (BN_is_odd(Y)) {
307 if (!BN_uadd(Y, Y, n))
308 goto err;
309 }
310 /* now Y is even */
311 if (!BN_rshift1(Y, Y))
312 goto err;
313 }
314 if (shift > 0) {
315 if (!BN_rshift(A, A, shift))
316 goto err;
317 }
318
319 /*-
320 * We still have (1) and (2).
321 * Both A and B are odd.
322 * The following computations ensure that
323 *
324 * 0 <= B < |n|,
325 * 0 < A < |n|,
326 * (1) -sign*X*a == B (mod |n|),
327 * (2) sign*Y*a == A (mod |n|),
328 *
329 * and that either A or B is even in the next iteration.
330 */
331 if (BN_ucmp(B, A) >= 0) {
332 /* -sign*(X + Y)*a == B - A (mod |n|) */
333 if (!BN_uadd(X, X, Y))
334 goto err;
335 /*
336 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
337 * actually makes the algorithm slower
338 */
339 if (!BN_usub(B, B, A))
340 goto err;
341 } else {
342 /* sign*(X + Y)*a == A - B (mod |n|) */
343 if (!BN_uadd(Y, Y, X))
344 goto err;
345 /*
346 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
347 */
348 if (!BN_usub(A, A, B))
349 goto err;
350 }
351 }
352 } else {
353 /* general inversion algorithm */
354
355 while (!BN_is_zero(B)) {
356 BIGNUM *tmp;
357
358 /*-
359 * 0 < B < A,
360 * (*) -sign*X*a == B (mod |n|),
361 * sign*Y*a == A (mod |n|)
362 */
363
364 /* (D, M) := (A/B, A%B) ... */
365 if (BN_num_bits(A) == BN_num_bits(B)) {
366 if (!BN_one(D))
367 goto err;
368 if (!BN_sub(M, A, B))
369 goto err;
370 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
371 /* A/B is 1, 2, or 3 */
372 if (!BN_lshift1(T, B))
373 goto err;
374 if (BN_ucmp(A, T) < 0) {
375 /* A < 2*B, so D=1 */
376 if (!BN_one(D))
377 goto err;
378 if (!BN_sub(M, A, B))
379 goto err;
380 } else {
381 /* A >= 2*B, so D=2 or D=3 */
382 if (!BN_sub(M, A, T))
383 goto err;
384 if (!BN_add(D, T, B))
385 goto err; /* use D (:= 3*B) as temp */
386 if (BN_ucmp(A, D) < 0) {
387 /* A < 3*B, so D=2 */
388 if (!BN_set_word(D, 2))
389 goto err;
390 /*
391 * M (= A - 2*B) already has the correct value
392 */
393 } else {
394 /* only D=3 remains */
395 if (!BN_set_word(D, 3))
396 goto err;
397 /*
398 * currently M = A - 2*B, but we need M = A - 3*B
399 */
400 if (!BN_sub(M, M, B))
401 goto err;
402 }
403 }
404 } else {
405 if (!BN_div(D, M, A, B, ctx))
406 goto err;
407 }
408
409 /*-
410 * Now
411 * A = D*B + M;
412 * thus we have
413 * (**) sign*Y*a == D*B + M (mod |n|).
414 */
415
416 tmp = A; /* keep the BIGNUM object, the value does not matter */
417
418 /* (A, B) := (B, A mod B) ... */
419 A = B;
420 B = M;
421 /* ... so we have 0 <= B < A again */
422
423 /*-
424 * Since the former M is now B and the former B is now A,
425 * (**) translates into
426 * sign*Y*a == D*A + B (mod |n|),
427 * i.e.
428 * sign*Y*a - D*A == B (mod |n|).
429 * Similarly, (*) translates into
430 * -sign*X*a == A (mod |n|).
431 *
432 * Thus,
433 * sign*Y*a + D*sign*X*a == B (mod |n|),
434 * i.e.
435 * sign*(Y + D*X)*a == B (mod |n|).
436 *
437 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
438 * -sign*X*a == B (mod |n|),
439 * sign*Y*a == A (mod |n|).
440 * Note that X and Y stay non-negative all the time.
441 */
442
443 /*
444 * most of the time D is very small, so we can optimize tmp := D*X+Y
445 */
446 if (BN_is_one(D)) {
447 if (!BN_add(tmp, X, Y))
448 goto err;
449 } else {
450 if (BN_is_word(D, 2)) {
451 if (!BN_lshift1(tmp, X))
452 goto err;
453 } else if (BN_is_word(D, 4)) {
454 if (!BN_lshift(tmp, X, 2))
455 goto err;
456 } else if (D->top == 1) {
457 if (!BN_copy(tmp, X))
458 goto err;
459 if (!BN_mul_word(tmp, D->d[0]))
460 goto err;
461 } else {
462 if (!BN_mul(tmp, D, X, ctx))
463 goto err;
464 }
465 if (!BN_add(tmp, tmp, Y))
466 goto err;
467 }
468
469 M = Y; /* keep the BIGNUM object, the value does not matter */
470 Y = X;
471 X = tmp;
472 sign = -sign;
473 }
474 }
475
476 /*-
477 * The while loop (Euclid's algorithm) ends when
478 * A == gcd(a,n);
479 * we have
480 * sign*Y*a == A (mod |n|),
481 * where Y is non-negative.
482 */
483
484 if (sign < 0) {
485 if (!BN_sub(Y, n, Y))
486 goto err;
487 }
488 /* Now Y*a == A (mod |n|). */
489
490 if (BN_is_one(A)) {
491 /* Y*a == 1 (mod |n|) */
492 if (!Y->neg && BN_ucmp(Y, n) < 0) {
493 if (!BN_copy(R, Y))
494 goto err;
495 } else {
496 if (!BN_nnmod(R, Y, n, ctx))
497 goto err;
498 }
499 } else {
500 *pnoinv = 1;
501 goto err;
502 }
503 ret = R;
504 err:
505 if ((ret == NULL) && (in == NULL))
506 BN_free(R);
507 BN_CTX_end(ctx);
508 bn_check_top(ret);
509 return ret;
510}
511
512/* solves ax == 1 (mod n) */
513BIGNUM *BN_mod_inverse(BIGNUM *in,
514 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
515{
516 BN_CTX *new_ctx = NULL;
517 BIGNUM *rv;
518 int noinv = 0;
519
520 if (ctx == NULL) {
521 ctx = new_ctx = BN_CTX_new();
522 if (ctx == NULL) {
523 BNerr(BN_F_BN_MOD_INVERSE, ERR_R_MALLOC_FAILURE);
524 return NULL;
525 }
526 }
527
528 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
529 if (noinv)
530 BNerr(BN_F_BN_MOD_INVERSE, BN_R_NO_INVERSE);
531 BN_CTX_free(new_ctx);
532 return rv;
533}
534
535/*-
536 * This function is based on the constant-time GCD work by Bernstein and Yang:
537 * https://eprint.iacr.org/2019/266
538 * Generalized fast GCD function to allow even inputs.
539 * The algorithm first finds the shared powers of 2 between
540 * the inputs, and removes them, reducing at least one of the
541 * inputs to an odd value. Then it proceeds to calculate the GCD.
542 * Before returning the resulting GCD, we take care of adding
543 * back the powers of two removed at the beginning.
544 * Note 1: we assume the bit length of both inputs is public information,
545 * since access to top potentially leaks this information.
546 */
547int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
548{
549 BIGNUM *g, *temp = NULL;
550 BN_ULONG mask = 0;
551 int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;
552
553 /* Note 2: zero input corner cases are not constant-time since they are
554 * handled immediately. An attacker can run an attack under this
555 * assumption without the need of side-channel information. */
556 if (BN_is_zero(in_b)) {
557 ret = BN_copy(r, in_a) != NULL;
558 r->neg = 0;
559 return ret;
560 }
561 if (BN_is_zero(in_a)) {
562 ret = BN_copy(r, in_b) != NULL;
563 r->neg = 0;
564 return ret;
565 }
566
567 bn_check_top(in_a);
568 bn_check_top(in_b);
569
570 BN_CTX_start(ctx);
571 temp = BN_CTX_get(ctx);
572 g = BN_CTX_get(ctx);
573
574 /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
575 if (g == NULL
576 || !BN_lshift1(g, in_b)
577 || !BN_lshift1(r, in_a))
578 goto err;
579
580 /* find shared powers of two, i.e. "shifts" >= 1 */
581 for (i = 0; i < r->dmax && i < g->dmax; i++) {
582 mask = ~(r->d[i] | g->d[i]);
583 for (j = 0; j < BN_BITS2; j++) {
584 bit &= mask;
585 shifts += bit;
586 mask >>= 1;
587 }
588 }
589
590 /* subtract shared powers of two; shifts >= 1 */
591 if (!BN_rshift(r, r, shifts)
592 || !BN_rshift(g, g, shifts))
593 goto err;
594
595 /* expand to biggest nword, with room for a possible extra word */
596 top = 1 + ((r->top >= g->top) ? r->top : g->top);
597 if (bn_wexpand(r, top) == NULL
598 || bn_wexpand(g, top) == NULL
599 || bn_wexpand(temp, top) == NULL)
600 goto err;
601
602 /* re arrange inputs s.t. r is odd */
603 BN_consttime_swap((~r->d[0]) & 1, r, g, top);
604
605 /* compute the number of iterations */
606 rlen = BN_num_bits(r);
607 glen = BN_num_bits(g);
608 m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
609
610 for (i = 0; i < m; i++) {
611 /* conditionally flip signs if delta is positive and g is odd */
612 cond = (-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
613 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
614 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1)));
615 delta = (-cond & -delta) | ((cond - 1) & delta);
616 r->neg ^= cond;
617 /* swap */
618 BN_consttime_swap(cond, r, g, top);
619
620 /* elimination step */
621 delta++;
622 if (!BN_add(temp, g, r))
623 goto err;
624 BN_consttime_swap(g->d[0] & 1 /* g is odd */
625 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
626 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1))),
627 g, temp, top);
628 if (!BN_rshift1(g, g))
629 goto err;
630 }
631
632 /* remove possible negative sign */
633 r->neg = 0;
634 /* add powers of 2 removed, then correct the artificial shift */
635 if (!BN_lshift(r, r, shifts)
636 || !BN_rshift1(r, r))
637 goto err;
638
639 ret = 1;
640
641 err:
642 BN_CTX_end(ctx);
643 bn_check_top(r);
644 return ret;
645}