1 /*
2 * Copyright 1995-2024 The OpenSSL Project Authors. All Rights Reserved.
3 *
4 * Licensed under the Apache License 2.0 (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 #include "internal/constant_time.h"
13
14 /*
15 * bn_mod_inverse_no_branch is a special version of BN_mod_inverse. It does
16 * not contain branches that may leak sensitive information.
17 *
18 * This is a static function, we ensure all callers in this file pass valid
19 * arguments: all passed pointers here are non-NULL.
20 */
21 static ossl_inline
bn_mod_inverse_no_branch(BIGNUM * in,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx,int * pnoinv)22 BIGNUM *bn_mod_inverse_no_branch(BIGNUM *in,
23 const BIGNUM *a, const BIGNUM *n,
24 BN_CTX *ctx, int *pnoinv)
25 {
26 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
27 BIGNUM *ret = NULL;
28 int sign;
29
30 bn_check_top(a);
31 bn_check_top(n);
32
33 BN_CTX_start(ctx);
34 A = BN_CTX_get(ctx);
35 B = BN_CTX_get(ctx);
36 X = BN_CTX_get(ctx);
37 D = BN_CTX_get(ctx);
38 M = BN_CTX_get(ctx);
39 Y = BN_CTX_get(ctx);
40 T = BN_CTX_get(ctx);
41 if (T == NULL)
42 goto err;
43
44 if (in == NULL)
45 R = BN_new();
46 else
47 R = in;
48 if (R == NULL)
49 goto err;
50
51 if (!BN_one(X))
52 goto err;
53 BN_zero(Y);
54 if (BN_copy(B, a) == NULL)
55 goto err;
56 if (BN_copy(A, n) == NULL)
57 goto err;
58 A->neg = 0;
59
60 if (B->neg || (BN_ucmp(B, A) >= 0)) {
61 /*
62 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
63 * BN_div_no_branch will be called eventually.
64 */
65 {
66 BIGNUM local_B;
67 bn_init(&local_B);
68 BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
69 if (!BN_nnmod(B, &local_B, A, ctx))
70 goto err;
71 /* Ensure local_B goes out of scope before any further use of B */
72 }
73 }
74 sign = -1;
75 /*-
76 * From B = a mod |n|, A = |n| it follows that
77 *
78 * 0 <= B < A,
79 * -sign*X*a == B (mod |n|),
80 * sign*Y*a == A (mod |n|).
81 */
82
83 while (!BN_is_zero(B)) {
84 BIGNUM *tmp;
85
86 /*-
87 * 0 < B < A,
88 * (*) -sign*X*a == B (mod |n|),
89 * sign*Y*a == A (mod |n|)
90 */
91
92 /*
93 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
94 * BN_div_no_branch will be called eventually.
95 */
96 {
97 BIGNUM local_A;
98 bn_init(&local_A);
99 BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
100
101 /* (D, M) := (A/B, A%B) ... */
102 if (!BN_div(D, M, &local_A, B, ctx))
103 goto err;
104 /* Ensure local_A goes out of scope before any further use of A */
105 }
106
107 /*-
108 * Now
109 * A = D*B + M;
110 * thus we have
111 * (**) sign*Y*a == D*B + M (mod |n|).
112 */
113
114 tmp = A; /* keep the BIGNUM object, the value does not
115 * matter */
116
117 /* (A, B) := (B, A mod B) ... */
118 A = B;
119 B = M;
120 /* ... so we have 0 <= B < A again */
121
122 /*-
123 * Since the former M is now B and the former B is now A,
124 * (**) translates into
125 * sign*Y*a == D*A + B (mod |n|),
126 * i.e.
127 * sign*Y*a - D*A == B (mod |n|).
128 * Similarly, (*) translates into
129 * -sign*X*a == A (mod |n|).
130 *
131 * Thus,
132 * sign*Y*a + D*sign*X*a == B (mod |n|),
133 * i.e.
134 * sign*(Y + D*X)*a == B (mod |n|).
135 *
136 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
137 * -sign*X*a == B (mod |n|),
138 * sign*Y*a == A (mod |n|).
139 * Note that X and Y stay non-negative all the time.
140 */
141
142 if (!BN_mul(tmp, D, X, ctx))
143 goto err;
144 if (!BN_add(tmp, tmp, Y))
145 goto err;
146
147 M = Y; /* keep the BIGNUM object, the value does not
148 * matter */
149 Y = X;
150 X = tmp;
151 sign = -sign;
152 }
153
154 /*-
155 * The while loop (Euclid's algorithm) ends when
156 * A == gcd(a,n);
157 * we have
158 * sign*Y*a == A (mod |n|),
159 * where Y is non-negative.
160 */
161
162 if (sign < 0) {
163 if (!BN_sub(Y, n, Y))
164 goto err;
165 }
166 /* Now Y*a == A (mod |n|). */
167
168 if (BN_is_one(A)) {
169 /* Y*a == 1 (mod |n|) */
170 if (!Y->neg && BN_ucmp(Y, n) < 0) {
171 if (!BN_copy(R, Y))
172 goto err;
173 } else {
174 if (!BN_nnmod(R, Y, n, ctx))
175 goto err;
176 }
177 } else {
178 *pnoinv = 1;
179 /* caller sets the BN_R_NO_INVERSE error */
180 goto err;
181 }
182
183 ret = R;
184 *pnoinv = 0;
185
186 err:
187 if ((ret == NULL) && (in == NULL))
188 BN_free(R);
189 BN_CTX_end(ctx);
190 bn_check_top(ret);
191 return ret;
192 }
193
194 /*
195 * This is an internal function, we assume all callers pass valid arguments:
196 * all pointers passed here are assumed non-NULL.
197 */
int_bn_mod_inverse(BIGNUM * in,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx,int * pnoinv)198 BIGNUM *int_bn_mod_inverse(BIGNUM *in,
199 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
200 int *pnoinv)
201 {
202 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
203 BIGNUM *ret = NULL;
204 int sign;
205
206 /* This is invalid input so we don't worry about constant time here */
207 if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
208 *pnoinv = 1;
209 return NULL;
210 }
211
212 *pnoinv = 0;
213
214 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
215 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
216 return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
217 }
218
219 bn_check_top(a);
220 bn_check_top(n);
221
222 BN_CTX_start(ctx);
223 A = BN_CTX_get(ctx);
224 B = BN_CTX_get(ctx);
225 X = BN_CTX_get(ctx);
226 D = BN_CTX_get(ctx);
227 M = BN_CTX_get(ctx);
228 Y = BN_CTX_get(ctx);
229 T = BN_CTX_get(ctx);
230 if (T == NULL)
231 goto err;
232
233 if (in == NULL)
234 R = BN_new();
235 else
236 R = in;
237 if (R == NULL)
238 goto err;
239
240 if (!BN_one(X))
241 goto err;
242 BN_zero(Y);
243 if (BN_copy(B, a) == NULL)
244 goto err;
245 if (BN_copy(A, n) == NULL)
246 goto err;
247 A->neg = 0;
248 if (B->neg || (BN_ucmp(B, A) >= 0)) {
249 if (!BN_nnmod(B, B, A, ctx))
250 goto err;
251 }
252 sign = -1;
253 /*-
254 * From B = a mod |n|, A = |n| it follows that
255 *
256 * 0 <= B < A,
257 * -sign*X*a == B (mod |n|),
258 * sign*Y*a == A (mod |n|).
259 */
260
261 if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
262 /*
263 * Binary inversion algorithm; requires odd modulus. This is faster
264 * than the general algorithm if the modulus is sufficiently small
265 * (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
266 * systems)
267 */
268 int shift;
269
270 while (!BN_is_zero(B)) {
271 /*-
272 * 0 < B < |n|,
273 * 0 < A <= |n|,
274 * (1) -sign*X*a == B (mod |n|),
275 * (2) sign*Y*a == A (mod |n|)
276 */
277
278 /*
279 * Now divide B by the maximum possible power of two in the
280 * integers, and divide X by the same value mod |n|. When we're
281 * done, (1) still holds.
282 */
283 shift = 0;
284 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
285 shift++;
286
287 if (BN_is_odd(X)) {
288 if (!BN_uadd(X, X, n))
289 goto err;
290 }
291 /*
292 * now X is even, so we can easily divide it by two
293 */
294 if (!BN_rshift1(X, X))
295 goto err;
296 }
297 if (shift > 0) {
298 if (!BN_rshift(B, B, shift))
299 goto err;
300 }
301
302 /*
303 * Same for A and Y. Afterwards, (2) still holds.
304 */
305 shift = 0;
306 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
307 shift++;
308
309 if (BN_is_odd(Y)) {
310 if (!BN_uadd(Y, Y, n))
311 goto err;
312 }
313 /* now Y is even */
314 if (!BN_rshift1(Y, Y))
315 goto err;
316 }
317 if (shift > 0) {
318 if (!BN_rshift(A, A, shift))
319 goto err;
320 }
321
322 /*-
323 * We still have (1) and (2).
324 * Both A and B are odd.
325 * The following computations ensure that
326 *
327 * 0 <= B < |n|,
328 * 0 < A < |n|,
329 * (1) -sign*X*a == B (mod |n|),
330 * (2) sign*Y*a == A (mod |n|),
331 *
332 * and that either A or B is even in the next iteration.
333 */
334 if (BN_ucmp(B, A) >= 0) {
335 /* -sign*(X + Y)*a == B - A (mod |n|) */
336 if (!BN_uadd(X, X, Y))
337 goto err;
338 /*
339 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
340 * actually makes the algorithm slower
341 */
342 if (!BN_usub(B, B, A))
343 goto err;
344 } else {
345 /* sign*(X + Y)*a == A - B (mod |n|) */
346 if (!BN_uadd(Y, Y, X))
347 goto err;
348 /*
349 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
350 */
351 if (!BN_usub(A, A, B))
352 goto err;
353 }
354 }
355 } else {
356 /* general inversion algorithm */
357
358 while (!BN_is_zero(B)) {
359 BIGNUM *tmp;
360
361 /*-
362 * 0 < B < A,
363 * (*) -sign*X*a == B (mod |n|),
364 * sign*Y*a == A (mod |n|)
365 */
366
367 /* (D, M) := (A/B, A%B) ... */
368 if (BN_num_bits(A) == BN_num_bits(B)) {
369 if (!BN_one(D))
370 goto err;
371 if (!BN_sub(M, A, B))
372 goto err;
373 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
374 /* A/B is 1, 2, or 3 */
375 if (!BN_lshift1(T, B))
376 goto err;
377 if (BN_ucmp(A, T) < 0) {
378 /* A < 2*B, so D=1 */
379 if (!BN_one(D))
380 goto err;
381 if (!BN_sub(M, A, B))
382 goto err;
383 } else {
384 /* A >= 2*B, so D=2 or D=3 */
385 if (!BN_sub(M, A, T))
386 goto err;
387 if (!BN_add(D, T, B))
388 goto err; /* use D (:= 3*B) as temp */
389 if (BN_ucmp(A, D) < 0) {
390 /* A < 3*B, so D=2 */
391 if (!BN_set_word(D, 2))
392 goto err;
393 /*
394 * M (= A - 2*B) already has the correct value
395 */
396 } else {
397 /* only D=3 remains */
398 if (!BN_set_word(D, 3))
399 goto err;
400 /*
401 * currently M = A - 2*B, but we need M = A - 3*B
402 */
403 if (!BN_sub(M, M, B))
404 goto err;
405 }
406 }
407 } else {
408 if (!BN_div(D, M, A, B, ctx))
409 goto err;
410 }
411
412 /*-
413 * Now
414 * A = D*B + M;
415 * thus we have
416 * (**) sign*Y*a == D*B + M (mod |n|).
417 */
418
419 tmp = A; /* keep the BIGNUM object, the value does not matter */
420
421 /* (A, B) := (B, A mod B) ... */
422 A = B;
423 B = M;
424 /* ... so we have 0 <= B < A again */
425
426 /*-
427 * Since the former M is now B and the former B is now A,
428 * (**) translates into
429 * sign*Y*a == D*A + B (mod |n|),
430 * i.e.
431 * sign*Y*a - D*A == B (mod |n|).
432 * Similarly, (*) translates into
433 * -sign*X*a == A (mod |n|).
434 *
435 * Thus,
436 * sign*Y*a + D*sign*X*a == B (mod |n|),
437 * i.e.
438 * sign*(Y + D*X)*a == B (mod |n|).
439 *
440 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
441 * -sign*X*a == B (mod |n|),
442 * sign*Y*a == A (mod |n|).
443 * Note that X and Y stay non-negative all the time.
444 */
445
446 /*
447 * most of the time D is very small, so we can optimize tmp := D*X+Y
448 */
449 if (BN_is_one(D)) {
450 if (!BN_add(tmp, X, Y))
451 goto err;
452 } else {
453 if (BN_is_word(D, 2)) {
454 if (!BN_lshift1(tmp, X))
455 goto err;
456 } else if (BN_is_word(D, 4)) {
457 if (!BN_lshift(tmp, X, 2))
458 goto err;
459 } else if (D->top == 1) {
460 if (!BN_copy(tmp, X))
461 goto err;
462 if (!BN_mul_word(tmp, D->d[0]))
463 goto err;
464 } else {
465 if (!BN_mul(tmp, D, X, ctx))
466 goto err;
467 }
468 if (!BN_add(tmp, tmp, Y))
469 goto err;
470 }
471
472 M = Y; /* keep the BIGNUM object, the value does not matter */
473 Y = X;
474 X = tmp;
475 sign = -sign;
476 }
477 }
478
479 /*-
480 * The while loop (Euclid's algorithm) ends when
481 * A == gcd(a,n);
482 * we have
483 * sign*Y*a == A (mod |n|),
484 * where Y is non-negative.
485 */
486
487 if (sign < 0) {
488 if (!BN_sub(Y, n, Y))
489 goto err;
490 }
491 /* Now Y*a == A (mod |n|). */
492
493 if (BN_is_one(A)) {
494 /* Y*a == 1 (mod |n|) */
495 if (!Y->neg && BN_ucmp(Y, n) < 0) {
496 if (!BN_copy(R, Y))
497 goto err;
498 } else {
499 if (!BN_nnmod(R, Y, n, ctx))
500 goto err;
501 }
502 } else {
503 *pnoinv = 1;
504 goto err;
505 }
506 ret = R;
507 err:
508 if ((ret == NULL) && (in == NULL))
509 BN_free(R);
510 BN_CTX_end(ctx);
511 bn_check_top(ret);
512 return ret;
513 }
514
515 /* solves ax == 1 (mod n) */
BN_mod_inverse(BIGNUM * in,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)516 BIGNUM *BN_mod_inverse(BIGNUM *in,
517 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
518 {
519 BN_CTX *new_ctx = NULL;
520 BIGNUM *rv;
521 int noinv = 0;
522
523 if (ctx == NULL) {
524 ctx = new_ctx = BN_CTX_new_ex(NULL);
525 if (ctx == NULL) {
526 ERR_raise(ERR_LIB_BN, ERR_R_BN_LIB);
527 return NULL;
528 }
529 }
530
531 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
532 if (noinv)
533 ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE);
534 BN_CTX_free(new_ctx);
535 return rv;
536 }
537
538 /*
539 * The numbers a and b are coprime if the only positive integer that is a
540 * divisor of both of them is 1.
541 * i.e. gcd(a,b) = 1.
542 *
543 * Coprimes have the property: b has a multiplicative inverse modulo a
544 * i.e there is some value x such that bx = 1 (mod a).
545 *
546 * Testing the modulo inverse is currently much faster than the constant
547 * time version of BN_gcd().
548 */
BN_are_coprime(BIGNUM * a,const BIGNUM * b,BN_CTX * ctx)549 int BN_are_coprime(BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
550 {
551 int ret = 0;
552 BIGNUM *tmp;
553
554 BN_CTX_start(ctx);
555 tmp = BN_CTX_get(ctx);
556 if (tmp == NULL)
557 goto end;
558
559 ERR_set_mark();
560 BN_set_flags(a, BN_FLG_CONSTTIME);
561 ret = (BN_mod_inverse(tmp, a, b, ctx) != NULL);
562 /* Clear any errors (an error is returned if there is no inverse) */
563 ERR_pop_to_mark();
564 end:
565 BN_CTX_end(ctx);
566 return ret;
567 }
568
569 /*-
570 * This function is based on the constant-time GCD work by Bernstein and Yang:
571 * https://eprint.iacr.org/2019/266
572 * Generalized fast GCD function to allow even inputs.
573 * The algorithm first finds the shared powers of 2 between
574 * the inputs, and removes them, reducing at least one of the
575 * inputs to an odd value. Then it proceeds to calculate the GCD.
576 * Before returning the resulting GCD, we take care of adding
577 * back the powers of two removed at the beginning.
578 * Note 1: we assume the bit length of both inputs is public information,
579 * since access to top potentially leaks this information.
580 */
BN_gcd(BIGNUM * r,const BIGNUM * in_a,const BIGNUM * in_b,BN_CTX * ctx)581 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
582 {
583 BIGNUM *g, *temp = NULL;
584 BN_ULONG pow2_numbits, pow2_numbits_temp, pow2_condition_mask, pow2_flag;
585 int i, j, top, rlen, glen, m, delta = 1, cond = 0, pow2_shifts, ret = 0;
586
587 /* Note 2: zero input corner cases are not constant-time since they are
588 * handled immediately. An attacker can run an attack under this
589 * assumption without the need of side-channel information. */
590 if (BN_is_zero(in_b)) {
591 ret = BN_copy(r, in_a) != NULL;
592 r->neg = 0;
593 return ret;
594 }
595 if (BN_is_zero(in_a)) {
596 ret = BN_copy(r, in_b) != NULL;
597 r->neg = 0;
598 return ret;
599 }
600
601 bn_check_top(in_a);
602 bn_check_top(in_b);
603
604 BN_CTX_start(ctx);
605 temp = BN_CTX_get(ctx);
606 g = BN_CTX_get(ctx);
607
608 /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
609 if (g == NULL
610 || !BN_lshift1(g, in_b)
611 || !BN_lshift1(r, in_a))
612 goto err;
613
614 /* find shared powers of two, i.e. "shifts" >= 1 */
615 pow2_flag = 1;
616 pow2_shifts = 0;
617 pow2_numbits = 0;
618 for (i = 0; i < r->dmax && i < g->dmax; i++) {
619 pow2_numbits_temp = r->d[i] | g->d[i];
620 pow2_condition_mask = constant_time_is_zero_bn(pow2_flag);
621 pow2_flag &= constant_time_is_zero_bn(pow2_numbits_temp);
622 pow2_shifts += pow2_flag;
623 pow2_numbits = constant_time_select_bn(pow2_condition_mask,
624 pow2_numbits, pow2_numbits_temp);
625 }
626 pow2_numbits = ~pow2_numbits;
627 pow2_shifts *= BN_BITS2;
628 pow2_flag = 1;
629 for (j = 0; j < BN_BITS2; j++) {
630 pow2_flag &= pow2_numbits;
631 pow2_shifts += pow2_flag;
632 pow2_numbits >>= 1;
633 }
634
635 /* subtract shared powers of two; shifts >= 1 */
636 if (!BN_rshift(r, r, pow2_shifts)
637 || !BN_rshift(g, g, pow2_shifts))
638 goto err;
639
640 /* expand to biggest nword, with room for a possible extra word */
641 top = 1 + ((r->top >= g->top) ? r->top : g->top);
642 if (bn_wexpand(r, top) == NULL
643 || bn_wexpand(g, top) == NULL
644 || bn_wexpand(temp, top) == NULL)
645 goto err;
646
647 /* re arrange inputs s.t. r is odd */
648 BN_consttime_swap((~r->d[0]) & 1, r, g, top);
649
650 /* compute the number of iterations */
651 rlen = BN_num_bits(r);
652 glen = BN_num_bits(g);
653 m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
654
655 for (i = 0; i < m; i++) {
656 /* conditionally flip signs if delta is positive and g is odd */
657 cond = ((unsigned int)-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
658 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
659 & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1)));
660 delta = (-cond & -delta) | ((cond - 1) & delta);
661 r->neg ^= cond;
662 /* swap */
663 BN_consttime_swap(cond, r, g, top);
664
665 /* elimination step */
666 delta++;
667 if (!BN_add(temp, g, r))
668 goto err;
669 BN_consttime_swap(g->d[0] & 1 /* g is odd */
670 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
671 & (~((unsigned int)(g->top - 1) >> (sizeof(g->top) * 8 - 1))),
672 g, temp, top);
673 if (!BN_rshift1(g, g))
674 goto err;
675 }
676
677 /* remove possible negative sign */
678 r->neg = 0;
679 /* add powers of 2 removed, then correct the artificial shift */
680 if (!BN_lshift(r, r, pow2_shifts)
681 || !BN_rshift1(r, r))
682 goto err;
683
684 ret = 1;
685
686 err:
687 BN_CTX_end(ctx);
688 bn_check_top(r);
689 return ret;
690 }
691
