1 /*
2 * Copyright 1995-2022 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
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 */
20 static ossl_inline
bn_mod_inverse_no_branch(BIGNUM * in,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx,int * pnoinv)21 BIGNUM *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 if (!BN_one(X))
51 goto err;
52 BN_zero(Y);
53 if (BN_copy(B, a) == NULL)
54 goto err;
55 if (BN_copy(A, n) == NULL)
56 goto err;
57 A->neg = 0;
58
59 if (B->neg || (BN_ucmp(B, A) >= 0)) {
60 /*
61 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
62 * BN_div_no_branch will be called eventually.
63 */
64 {
65 BIGNUM local_B;
66 bn_init(&local_B);
67 BN_with_flags(&local_B, B, BN_FLG_CONSTTIME);
68 if (!BN_nnmod(B, &local_B, A, ctx))
69 goto err;
70 /* Ensure local_B goes out of scope before any further use of B */
71 }
72 }
73 sign = -1;
74 /*-
75 * From B = a mod |n|, A = |n| it follows that
76 *
77 * 0 <= B < A,
78 * -sign*X*a == B (mod |n|),
79 * sign*Y*a == A (mod |n|).
80 */
81
82 while (!BN_is_zero(B)) {
83 BIGNUM *tmp;
84
85 /*-
86 * 0 < B < A,
87 * (*) -sign*X*a == B (mod |n|),
88 * sign*Y*a == A (mod |n|)
89 */
90
91 /*
92 * Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
93 * BN_div_no_branch will be called eventually.
94 */
95 {
96 BIGNUM local_A;
97 bn_init(&local_A);
98 BN_with_flags(&local_A, A, BN_FLG_CONSTTIME);
99
100 /* (D, M) := (A/B, A%B) ... */
101 if (!BN_div(D, M, &local_A, B, ctx))
102 goto err;
103 /* Ensure local_A goes out of scope before any further use of A */
104 }
105
106 /*-
107 * Now
108 * A = D*B + M;
109 * thus we have
110 * (**) sign*Y*a == D*B + M (mod |n|).
111 */
112
113 tmp = A; /* keep the BIGNUM object, the value does not
114 * matter */
115
116 /* (A, B) := (B, A mod B) ... */
117 A = B;
118 B = M;
119 /* ... so we have 0 <= B < A again */
120
121 /*-
122 * Since the former M is now B and the former B is now A,
123 * (**) translates into
124 * sign*Y*a == D*A + B (mod |n|),
125 * i.e.
126 * sign*Y*a - D*A == B (mod |n|).
127 * Similarly, (*) translates into
128 * -sign*X*a == A (mod |n|).
129 *
130 * Thus,
131 * sign*Y*a + D*sign*X*a == B (mod |n|),
132 * i.e.
133 * sign*(Y + D*X)*a == B (mod |n|).
134 *
135 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
136 * -sign*X*a == B (mod |n|),
137 * sign*Y*a == A (mod |n|).
138 * Note that X and Y stay non-negative all the time.
139 */
140
141 if (!BN_mul(tmp, D, X, ctx))
142 goto err;
143 if (!BN_add(tmp, tmp, Y))
144 goto err;
145
146 M = Y; /* keep the BIGNUM object, the value does not
147 * matter */
148 Y = X;
149 X = tmp;
150 sign = -sign;
151 }
152
153 /*-
154 * The while loop (Euclid's algorithm) ends when
155 * A == gcd(a,n);
156 * we have
157 * sign*Y*a == A (mod |n|),
158 * where Y is non-negative.
159 */
160
161 if (sign < 0) {
162 if (!BN_sub(Y, n, Y))
163 goto err;
164 }
165 /* Now Y*a == A (mod |n|). */
166
167 if (BN_is_one(A)) {
168 /* Y*a == 1 (mod |n|) */
169 if (!Y->neg && BN_ucmp(Y, n) < 0) {
170 if (!BN_copy(R, Y))
171 goto err;
172 } else {
173 if (!BN_nnmod(R, Y, n, ctx))
174 goto err;
175 }
176 } else {
177 *pnoinv = 1;
178 /* caller sets the BN_R_NO_INVERSE error */
179 goto err;
180 }
181
182 ret = R;
183 *pnoinv = 0;
184
185 err:
186 if ((ret == NULL) && (in == NULL))
187 BN_free(R);
188 BN_CTX_end(ctx);
189 bn_check_top(ret);
190 return ret;
191 }
192
193 /*
194 * This is an internal function, we assume all callers pass valid arguments:
195 * all pointers passed here are assumed non-NULL.
196 */
int_bn_mod_inverse(BIGNUM * in,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx,int * pnoinv)197 BIGNUM *int_bn_mod_inverse(BIGNUM *in,
198 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx,
199 int *pnoinv)
200 {
201 BIGNUM *A, *B, *X, *Y, *M, *D, *T, *R = NULL;
202 BIGNUM *ret = NULL;
203 int sign;
204
205 /* This is invalid input so we don't worry about constant time here */
206 if (BN_abs_is_word(n, 1) || BN_is_zero(n)) {
207 *pnoinv = 1;
208 return NULL;
209 }
210
211 *pnoinv = 0;
212
213 if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0)
214 || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0)) {
215 return bn_mod_inverse_no_branch(in, a, n, ctx, pnoinv);
216 }
217
218 bn_check_top(a);
219 bn_check_top(n);
220
221 BN_CTX_start(ctx);
222 A = BN_CTX_get(ctx);
223 B = BN_CTX_get(ctx);
224 X = BN_CTX_get(ctx);
225 D = BN_CTX_get(ctx);
226 M = BN_CTX_get(ctx);
227 Y = BN_CTX_get(ctx);
228 T = BN_CTX_get(ctx);
229 if (T == NULL)
230 goto err;
231
232 if (in == NULL)
233 R = BN_new();
234 else
235 R = in;
236 if (R == NULL)
237 goto err;
238
239 if (!BN_one(X))
240 goto err;
241 BN_zero(Y);
242 if (BN_copy(B, a) == NULL)
243 goto err;
244 if (BN_copy(A, n) == NULL)
245 goto err;
246 A->neg = 0;
247 if (B->neg || (BN_ucmp(B, A) >= 0)) {
248 if (!BN_nnmod(B, B, A, ctx))
249 goto err;
250 }
251 sign = -1;
252 /*-
253 * From B = a mod |n|, A = |n| it follows that
254 *
255 * 0 <= B < A,
256 * -sign*X*a == B (mod |n|),
257 * sign*Y*a == A (mod |n|).
258 */
259
260 if (BN_is_odd(n) && (BN_num_bits(n) <= 2048)) {
261 /*
262 * Binary inversion algorithm; requires odd modulus. This is faster
263 * than the general algorithm if the modulus is sufficiently small
264 * (about 400 .. 500 bits on 32-bit systems, but much more on 64-bit
265 * systems)
266 */
267 int shift;
268
269 while (!BN_is_zero(B)) {
270 /*-
271 * 0 < B < |n|,
272 * 0 < A <= |n|,
273 * (1) -sign*X*a == B (mod |n|),
274 * (2) sign*Y*a == A (mod |n|)
275 */
276
277 /*
278 * Now divide B by the maximum possible power of two in the
279 * integers, and divide X by the same value mod |n|. When we're
280 * done, (1) still holds.
281 */
282 shift = 0;
283 while (!BN_is_bit_set(B, shift)) { /* note that 0 < B */
284 shift++;
285
286 if (BN_is_odd(X)) {
287 if (!BN_uadd(X, X, n))
288 goto err;
289 }
290 /*
291 * now X is even, so we can easily divide it by two
292 */
293 if (!BN_rshift1(X, X))
294 goto err;
295 }
296 if (shift > 0) {
297 if (!BN_rshift(B, B, shift))
298 goto err;
299 }
300
301 /*
302 * Same for A and Y. Afterwards, (2) still holds.
303 */
304 shift = 0;
305 while (!BN_is_bit_set(A, shift)) { /* note that 0 < A */
306 shift++;
307
308 if (BN_is_odd(Y)) {
309 if (!BN_uadd(Y, Y, n))
310 goto err;
311 }
312 /* now Y is even */
313 if (!BN_rshift1(Y, Y))
314 goto err;
315 }
316 if (shift > 0) {
317 if (!BN_rshift(A, A, shift))
318 goto err;
319 }
320
321 /*-
322 * We still have (1) and (2).
323 * Both A and B are odd.
324 * The following computations ensure that
325 *
326 * 0 <= B < |n|,
327 * 0 < A < |n|,
328 * (1) -sign*X*a == B (mod |n|),
329 * (2) sign*Y*a == A (mod |n|),
330 *
331 * and that either A or B is even in the next iteration.
332 */
333 if (BN_ucmp(B, A) >= 0) {
334 /* -sign*(X + Y)*a == B - A (mod |n|) */
335 if (!BN_uadd(X, X, Y))
336 goto err;
337 /*
338 * NB: we could use BN_mod_add_quick(X, X, Y, n), but that
339 * actually makes the algorithm slower
340 */
341 if (!BN_usub(B, B, A))
342 goto err;
343 } else {
344 /* sign*(X + Y)*a == A - B (mod |n|) */
345 if (!BN_uadd(Y, Y, X))
346 goto err;
347 /*
348 * as above, BN_mod_add_quick(Y, Y, X, n) would slow things down
349 */
350 if (!BN_usub(A, A, B))
351 goto err;
352 }
353 }
354 } else {
355 /* general inversion algorithm */
356
357 while (!BN_is_zero(B)) {
358 BIGNUM *tmp;
359
360 /*-
361 * 0 < B < A,
362 * (*) -sign*X*a == B (mod |n|),
363 * sign*Y*a == A (mod |n|)
364 */
365
366 /* (D, M) := (A/B, A%B) ... */
367 if (BN_num_bits(A) == BN_num_bits(B)) {
368 if (!BN_one(D))
369 goto err;
370 if (!BN_sub(M, A, B))
371 goto err;
372 } else if (BN_num_bits(A) == BN_num_bits(B) + 1) {
373 /* A/B is 1, 2, or 3 */
374 if (!BN_lshift1(T, B))
375 goto err;
376 if (BN_ucmp(A, T) < 0) {
377 /* A < 2*B, so D=1 */
378 if (!BN_one(D))
379 goto err;
380 if (!BN_sub(M, A, B))
381 goto err;
382 } else {
383 /* A >= 2*B, so D=2 or D=3 */
384 if (!BN_sub(M, A, T))
385 goto err;
386 if (!BN_add(D, T, B))
387 goto err; /* use D (:= 3*B) as temp */
388 if (BN_ucmp(A, D) < 0) {
389 /* A < 3*B, so D=2 */
390 if (!BN_set_word(D, 2))
391 goto err;
392 /*
393 * M (= A - 2*B) already has the correct value
394 */
395 } else {
396 /* only D=3 remains */
397 if (!BN_set_word(D, 3))
398 goto err;
399 /*
400 * currently M = A - 2*B, but we need M = A - 3*B
401 */
402 if (!BN_sub(M, M, B))
403 goto err;
404 }
405 }
406 } else {
407 if (!BN_div(D, M, A, B, ctx))
408 goto err;
409 }
410
411 /*-
412 * Now
413 * A = D*B + M;
414 * thus we have
415 * (**) sign*Y*a == D*B + M (mod |n|).
416 */
417
418 tmp = A; /* keep the BIGNUM object, the value does not matter */
419
420 /* (A, B) := (B, A mod B) ... */
421 A = B;
422 B = M;
423 /* ... so we have 0 <= B < A again */
424
425 /*-
426 * Since the former M is now B and the former B is now A,
427 * (**) translates into
428 * sign*Y*a == D*A + B (mod |n|),
429 * i.e.
430 * sign*Y*a - D*A == B (mod |n|).
431 * Similarly, (*) translates into
432 * -sign*X*a == A (mod |n|).
433 *
434 * Thus,
435 * sign*Y*a + D*sign*X*a == B (mod |n|),
436 * i.e.
437 * sign*(Y + D*X)*a == B (mod |n|).
438 *
439 * So if we set (X, Y, sign) := (Y + D*X, X, -sign), we arrive back at
440 * -sign*X*a == B (mod |n|),
441 * sign*Y*a == A (mod |n|).
442 * Note that X and Y stay non-negative all the time.
443 */
444
445 /*
446 * most of the time D is very small, so we can optimize tmp := D*X+Y
447 */
448 if (BN_is_one(D)) {
449 if (!BN_add(tmp, X, Y))
450 goto err;
451 } else {
452 if (BN_is_word(D, 2)) {
453 if (!BN_lshift1(tmp, X))
454 goto err;
455 } else if (BN_is_word(D, 4)) {
456 if (!BN_lshift(tmp, X, 2))
457 goto err;
458 } else if (D->top == 1) {
459 if (!BN_copy(tmp, X))
460 goto err;
461 if (!BN_mul_word(tmp, D->d[0]))
462 goto err;
463 } else {
464 if (!BN_mul(tmp, D, X, ctx))
465 goto err;
466 }
467 if (!BN_add(tmp, tmp, Y))
468 goto err;
469 }
470
471 M = Y; /* keep the BIGNUM object, the value does not matter */
472 Y = X;
473 X = tmp;
474 sign = -sign;
475 }
476 }
477
478 /*-
479 * The while loop (Euclid's algorithm) ends when
480 * A == gcd(a,n);
481 * we have
482 * sign*Y*a == A (mod |n|),
483 * where Y is non-negative.
484 */
485
486 if (sign < 0) {
487 if (!BN_sub(Y, n, Y))
488 goto err;
489 }
490 /* Now Y*a == A (mod |n|). */
491
492 if (BN_is_one(A)) {
493 /* Y*a == 1 (mod |n|) */
494 if (!Y->neg && BN_ucmp(Y, n) < 0) {
495 if (!BN_copy(R, Y))
496 goto err;
497 } else {
498 if (!BN_nnmod(R, Y, n, ctx))
499 goto err;
500 }
501 } else {
502 *pnoinv = 1;
503 goto err;
504 }
505 ret = R;
506 err:
507 if ((ret == NULL) && (in == NULL))
508 BN_free(R);
509 BN_CTX_end(ctx);
510 bn_check_top(ret);
511 return ret;
512 }
513
514 /* solves ax == 1 (mod n) */
BN_mod_inverse(BIGNUM * in,const BIGNUM * a,const BIGNUM * n,BN_CTX * ctx)515 BIGNUM *BN_mod_inverse(BIGNUM *in,
516 const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
517 {
518 BN_CTX *new_ctx = NULL;
519 BIGNUM *rv;
520 int noinv = 0;
521
522 if (ctx == NULL) {
523 ctx = new_ctx = BN_CTX_new_ex(NULL);
524 if (ctx == NULL) {
525 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE);
526 return NULL;
527 }
528 }
529
530 rv = int_bn_mod_inverse(in, a, n, ctx, &noinv);
531 if (noinv)
532 ERR_raise(ERR_LIB_BN, BN_R_NO_INVERSE);
533 BN_CTX_free(new_ctx);
534 return rv;
535 }
536
537 /*-
538 * This function is based on the constant-time GCD work by Bernstein and Yang:
539 * https://eprint.iacr.org/2019/266
540 * Generalized fast GCD function to allow even inputs.
541 * The algorithm first finds the shared powers of 2 between
542 * the inputs, and removes them, reducing at least one of the
543 * inputs to an odd value. Then it proceeds to calculate the GCD.
544 * Before returning the resulting GCD, we take care of adding
545 * back the powers of two removed at the beginning.
546 * Note 1: we assume the bit length of both inputs is public information,
547 * since access to top potentially leaks this information.
548 */
BN_gcd(BIGNUM * r,const BIGNUM * in_a,const BIGNUM * in_b,BN_CTX * ctx)549 int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
550 {
551 BIGNUM *g, *temp = NULL;
552 BN_ULONG mask = 0;
553 int i, j, top, rlen, glen, m, bit = 1, delta = 1, cond = 0, shifts = 0, ret = 0;
554
555 /* Note 2: zero input corner cases are not constant-time since they are
556 * handled immediately. An attacker can run an attack under this
557 * assumption without the need of side-channel information. */
558 if (BN_is_zero(in_b)) {
559 ret = BN_copy(r, in_a) != NULL;
560 r->neg = 0;
561 return ret;
562 }
563 if (BN_is_zero(in_a)) {
564 ret = BN_copy(r, in_b) != NULL;
565 r->neg = 0;
566 return ret;
567 }
568
569 bn_check_top(in_a);
570 bn_check_top(in_b);
571
572 BN_CTX_start(ctx);
573 temp = BN_CTX_get(ctx);
574 g = BN_CTX_get(ctx);
575
576 /* make r != 0, g != 0 even, so BN_rshift is not a potential nop */
577 if (g == NULL
578 || !BN_lshift1(g, in_b)
579 || !BN_lshift1(r, in_a))
580 goto err;
581
582 /* find shared powers of two, i.e. "shifts" >= 1 */
583 for (i = 0; i < r->dmax && i < g->dmax; i++) {
584 mask = ~(r->d[i] | g->d[i]);
585 for (j = 0; j < BN_BITS2; j++) {
586 bit &= mask;
587 shifts += bit;
588 mask >>= 1;
589 }
590 }
591
592 /* subtract shared powers of two; shifts >= 1 */
593 if (!BN_rshift(r, r, shifts)
594 || !BN_rshift(g, g, shifts))
595 goto err;
596
597 /* expand to biggest nword, with room for a possible extra word */
598 top = 1 + ((r->top >= g->top) ? r->top : g->top);
599 if (bn_wexpand(r, top) == NULL
600 || bn_wexpand(g, top) == NULL
601 || bn_wexpand(temp, top) == NULL)
602 goto err;
603
604 /* re arrange inputs s.t. r is odd */
605 BN_consttime_swap((~r->d[0]) & 1, r, g, top);
606
607 /* compute the number of iterations */
608 rlen = BN_num_bits(r);
609 glen = BN_num_bits(g);
610 m = 4 + 3 * ((rlen >= glen) ? rlen : glen);
611
612 for (i = 0; i < m; i++) {
613 /* conditionally flip signs if delta is positive and g is odd */
614 cond = (-delta >> (8 * sizeof(delta) - 1)) & g->d[0] & 1
615 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
616 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1)));
617 delta = (-cond & -delta) | ((cond - 1) & delta);
618 r->neg ^= cond;
619 /* swap */
620 BN_consttime_swap(cond, r, g, top);
621
622 /* elimination step */
623 delta++;
624 if (!BN_add(temp, g, r))
625 goto err;
626 BN_consttime_swap(g->d[0] & 1 /* g is odd */
627 /* make sure g->top > 0 (i.e. if top == 0 then g == 0 always) */
628 & (~((g->top - 1) >> (sizeof(g->top) * 8 - 1))),
629 g, temp, top);
630 if (!BN_rshift1(g, g))
631 goto err;
632 }
633
634 /* remove possible negative sign */
635 r->neg = 0;
636 /* add powers of 2 removed, then correct the artificial shift */
637 if (!BN_lshift(r, r, shifts)
638 || !BN_rshift1(r, r))
639 goto err;
640
641 ret = 1;
642
643 err:
644 BN_CTX_end(ctx);
645 bn_check_top(r);
646 return ret;
647 }
648