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diff --git a/src/lib/libcrypto/bn/bn_bpsw.c b/src/lib/libcrypto/bn/bn_bpsw.c
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1/* $OpenBSD: bn_bpsw.c,v 1.12 2025/02/13 11:10:01 tb Exp $ */
2/*
3 * Copyright (c) 2022 Martin Grenouilloux <martin.grenouilloux@lse.epita.fr>
4 * Copyright (c) 2022 Theo Buehler <tb@openbsd.org>
5 *
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
9 *
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17 */
18
19#include <openssl/bn.h>
20
21#include "bn_local.h"
22#include "bn_prime.h"
23
24/*
25 * For an odd n compute a / 2 (mod n). If a is even, we can do a plain
26 * division, otherwise calculate (a + n) / 2. Then reduce (mod n).
27 */
28
29static int
30bn_div_by_two_mod_odd_n(BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
31{
32 if (!BN_is_odd(n))
33 return 0;
34
35 if (BN_is_odd(a)) {
36 if (!BN_add(a, a, n))
37 return 0;
38 }
39 if (!BN_rshift1(a, a))
40 return 0;
41 if (!BN_mod_ct(a, a, n, ctx))
42 return 0;
43
44 return 1;
45}
46
47/*
48 * Given the next binary digit of k and the current Lucas terms U and V, this
49 * helper computes the next terms in the Lucas sequence defined as follows:
50 *
51 * U' = U * V (mod n)
52 * V' = (V^2 + D * U^2) / 2 (mod n)
53 *
54 * If digit == 0, bn_lucas_step() returns U' and V'. If digit == 1, it returns
55 *
56 * U'' = (U' + V') / 2 (mod n)
57 * V'' = (V' + D * U') / 2 (mod n)
58 *
59 * Compare with FIPS 186-4, Appendix C.3.3, step 6.
60 */
61
62static int
63bn_lucas_step(BIGNUM *U, BIGNUM *V, int digit, const BIGNUM *D,
64 const BIGNUM *n, BN_CTX *ctx)
65{
66 BIGNUM *tmp;
67 int ret = 0;
68
69 BN_CTX_start(ctx);
70
71 if ((tmp = BN_CTX_get(ctx)) == NULL)
72 goto err;
73
74 /* Calculate D * U^2 before computing U'. */
75 if (!BN_sqr(tmp, U, ctx))
76 goto err;
77 if (!BN_mul(tmp, D, tmp, ctx))
78 goto err;
79
80 /* U' = U * V (mod n). */
81 if (!BN_mod_mul(U, U, V, n, ctx))
82 goto err;
83
84 /* V' = (V^2 + D * U^2) / 2 (mod n). */
85 if (!BN_sqr(V, V, ctx))
86 goto err;
87 if (!BN_add(V, V, tmp))
88 goto err;
89 if (!bn_div_by_two_mod_odd_n(V, n, ctx))
90 goto err;
91
92 if (digit == 1) {
93 /* Calculate D * U' before computing U''. */
94 if (!BN_mul(tmp, D, U, ctx))
95 goto err;
96
97 /* U'' = (U' + V') / 2 (mod n). */
98 if (!BN_add(U, U, V))
99 goto err;
100 if (!bn_div_by_two_mod_odd_n(U, n, ctx))
101 goto err;
102
103 /* V'' = (V' + D * U') / 2 (mod n). */
104 if (!BN_add(V, V, tmp))
105 goto err;
106 if (!bn_div_by_two_mod_odd_n(V, n, ctx))
107 goto err;
108 }
109
110 ret = 1;
111
112 err:
113 BN_CTX_end(ctx);
114
115 return ret;
116}
117
118/*
119 * Compute the Lucas terms U_k, V_k, see FIPS 186-4, Appendix C.3.3, steps 4-6.
120 */
121
122static int
123bn_lucas(BIGNUM *U, BIGNUM *V, const BIGNUM *k, const BIGNUM *D,
124 const BIGNUM *n, BN_CTX *ctx)
125{
126 int digit, i;
127 int ret = 0;
128
129 if (!BN_one(U))
130 goto err;
131 if (!BN_one(V))
132 goto err;
133
134 /*
135 * Iterate over the digits of k from MSB to LSB. Start at digit 2
136 * since the first digit is dealt with by setting U = 1 and V = 1.
137 */
138
139 for (i = BN_num_bits(k) - 2; i >= 0; i--) {
140 digit = BN_is_bit_set(k, i);
141
142 if (!bn_lucas_step(U, V, digit, D, n, ctx))
143 goto err;
144 }
145
146 ret = 1;
147
148 err:
149 return ret;
150}
151
152/*
153 * This is a stronger variant of the Lucas test in FIPS 186-4, Appendix C.3.3.
154 * Every strong Lucas pseudoprime n is also a Lucas pseudoprime since
155 * U_{n+1} == 0 follows from U_k == 0 or V_{k * 2^r} == 0 for 0 <= r < s.
156 */
157
158static int
159bn_strong_lucas_test(int *is_pseudoprime, const BIGNUM *n, const BIGNUM *D,
160 BN_CTX *ctx)
161{
162 BIGNUM *k, *U, *V;
163 int r, s;
164 int ret = 0;
165
166 BN_CTX_start(ctx);
167
168 if ((k = BN_CTX_get(ctx)) == NULL)
169 goto err;
170 if ((U = BN_CTX_get(ctx)) == NULL)
171 goto err;
172 if ((V = BN_CTX_get(ctx)) == NULL)
173 goto err;
174
175 /*
176 * Factorize n + 1 = k * 2^s with odd k: shift away the s trailing ones
177 * of n and set the lowest bit of the resulting number k.
178 */
179
180 s = 0;
181 while (BN_is_bit_set(n, s))
182 s++;
183 if (!BN_rshift(k, n, s))
184 goto err;
185 if (!BN_set_bit(k, 0))
186 goto err;
187
188 /*
189 * Calculate the Lucas terms U_k and V_k. If either of them is zero,
190 * then n is a strong Lucas pseudoprime.
191 */
192
193 if (!bn_lucas(U, V, k, D, n, ctx))
194 goto err;
195
196 if (BN_is_zero(U) || BN_is_zero(V)) {
197 *is_pseudoprime = 1;
198 goto done;
199 }
200
201 /*
202 * Calculate the Lucas terms U_{k * 2^r}, V_{k * 2^r} for 1 <= r < s.
203 * If any V_{k * 2^r} is zero then n is a strong Lucas pseudoprime.
204 */
205
206 for (r = 1; r < s; r++) {
207 if (!bn_lucas_step(U, V, 0, D, n, ctx))
208 goto err;
209
210 if (BN_is_zero(V)) {
211 *is_pseudoprime = 1;
212 goto done;
213 }
214 }
215
216 /*
217 * If we got here, n is definitely composite.
218 */
219
220 *is_pseudoprime = 0;
221
222 done:
223 ret = 1;
224
225 err:
226 BN_CTX_end(ctx);
227
228 return ret;
229}
230
231/*
232 * Test n for primality using the strong Lucas test with Selfridge's Method A.
233 * Returns 1 if n is prime or a strong Lucas-Selfridge pseudoprime.
234 * If it returns 0 then n is definitely composite.
235 */
236
237static int
238bn_strong_lucas_selfridge(int *is_pseudoprime, const BIGNUM *n, BN_CTX *ctx)
239{
240 BIGNUM *D, *two;
241 int is_perfect_square, jacobi_symbol, sign;
242 int ret = 0;
243
244 BN_CTX_start(ctx);
245
246 /* If n is a perfect square, it is composite. */
247 if (!bn_is_perfect_square(&is_perfect_square, n, ctx))
248 goto err;
249 if (is_perfect_square) {
250 *is_pseudoprime = 0;
251 goto done;
252 }
253
254 /*
255 * Find the first D in the Selfridge sequence 5, -7, 9, -11, 13, ...
256 * such that the Jacobi symbol (D/n) is -1.
257 */
258
259 if ((D = BN_CTX_get(ctx)) == NULL)
260 goto err;
261 if ((two = BN_CTX_get(ctx)) == NULL)
262 goto err;
263
264 sign = 1;
265 if (!BN_set_word(D, 5))
266 goto err;
267 if (!BN_set_word(two, 2))
268 goto err;
269
270 while (1) {
271 /* For odd n the Kronecker symbol computes the Jacobi symbol. */
272 if ((jacobi_symbol = BN_kronecker(D, n, ctx)) == -2)
273 goto err;
274
275 /* We found the value for D. */
276 if (jacobi_symbol == -1)
277 break;
278
279 /* n and D have prime factors in common. */
280 if (jacobi_symbol == 0) {
281 *is_pseudoprime = 0;
282 goto done;
283 }
284
285 sign = -sign;
286 if (!BN_uadd(D, D, two))
287 goto err;
288 BN_set_negative(D, sign == -1);
289 }
290
291 if (!bn_strong_lucas_test(is_pseudoprime, n, D, ctx))
292 goto err;
293
294 done:
295 ret = 1;
296
297 err:
298 BN_CTX_end(ctx);
299
300 return ret;
301}
302
303/*
304 * Fermat criterion in Miller-Rabin test.
305 *
306 * Check whether 1 < base < n - 1 witnesses that n is composite. For prime n:
307 *
308 * * Fermat's little theorem: base^(n-1) = 1 (mod n).
309 * * The only square roots of 1 (mod n) are 1 and -1.
310 *
311 * Calculate base^((n-1)/2) by writing n - 1 = k * 2^s with odd k. Iteratively
312 * compute power = (base^k)^(2^(s-1)) by successive squaring of base^k.
313 *
314 * If power ever reaches -1, base^(n-1) is equal to 1 and n is a pseudoprime
315 * for base. If power reaches 1 before -1 during successive squaring, we have
316 * an unexpected square root of 1 and n is composite. Otherwise base^(n-1) != 1,
317 * and n is composite.
318 */
319
320static int
321bn_fermat(int *is_pseudoprime, const BIGNUM *n, const BIGNUM *n_minus_one,
322 const BIGNUM *k, int s, const BIGNUM *base, BN_CTX *ctx, BN_MONT_CTX *mctx)
323{
324 BIGNUM *power;
325 int ret = 0;
326 int i;
327
328 BN_CTX_start(ctx);
329
330 if ((power = BN_CTX_get(ctx)) == NULL)
331 goto err;
332
333 /* Sanity check: ensure that 1 < base < n - 1. */
334 if (BN_cmp(base, BN_value_one()) <= 0 || BN_cmp(base, n_minus_one) >= 0)
335 goto err;
336
337 if (!BN_mod_exp_mont_ct(power, base, k, n, ctx, mctx))
338 goto err;
339
340 if (BN_is_one(power) || BN_cmp(power, n_minus_one) == 0) {
341 *is_pseudoprime = 1;
342 goto done;
343 }
344
345 /* Loop invariant: power is neither 1 nor -1 (mod n). */
346 for (i = 1; i < s; i++) {
347 if (!BN_mod_sqr(power, power, n, ctx))
348 goto err;
349
350 /* n is a pseudoprime for base. */
351 if (BN_cmp(power, n_minus_one) == 0) {
352 *is_pseudoprime = 1;
353 goto done;
354 }
355
356 /* n is composite: there's a square root of unity != 1 or -1. */
357 if (BN_is_one(power)) {
358 *is_pseudoprime = 0;
359 goto done;
360 }
361 }
362
363 /*
364 * If we get here, n is definitely composite: base^(n-1) != 1.
365 */
366
367 *is_pseudoprime = 0;
368
369 done:
370 ret = 1;
371
372 err:
373 BN_CTX_end(ctx);
374
375 return ret;
376}
377
378/*
379 * Miller-Rabin primality test for base 2 and for |rounds| of random bases.
380 * On success: is_pseudoprime == 0 implies that n is composite.
381 */
382
383static int
384bn_miller_rabin(int *is_pseudoprime, const BIGNUM *n, BN_CTX *ctx,
385 size_t rounds)
386{
387 BN_MONT_CTX *mctx = NULL;
388 BIGNUM *base, *k, *n_minus_one;
389 size_t i;
390 int s;
391 int ret = 0;
392
393 BN_CTX_start(ctx);
394
395 if ((base = BN_CTX_get(ctx)) == NULL)
396 goto err;
397 if ((k = BN_CTX_get(ctx)) == NULL)
398 goto err;
399 if ((n_minus_one = BN_CTX_get(ctx)) == NULL)
400 goto err;
401
402 if (BN_is_word(n, 2) || BN_is_word(n, 3)) {
403 *is_pseudoprime = 1;
404 goto done;
405 }
406
407 if (BN_cmp(n, BN_value_one()) <= 0 || !BN_is_odd(n)) {
408 *is_pseudoprime = 0;
409 goto done;
410 }
411
412 if (!BN_sub(n_minus_one, n, BN_value_one()))
413 goto err;
414
415 /*
416 * Factorize n - 1 = k * 2^s.
417 */
418
419 s = 0;
420 while (!BN_is_bit_set(n_minus_one, s))
421 s++;
422 if (!BN_rshift(k, n_minus_one, s))
423 goto err;
424
425 /*
426 * Montgomery setup for n.
427 */
428
429 if ((mctx = BN_MONT_CTX_create(n, ctx)) == NULL)
430 goto err;
431
432 /*
433 * Perform a Miller-Rabin test for base 2 as required by BPSW.
434 */
435
436 if (!BN_set_word(base, 2))
437 goto err;
438
439 if (!bn_fermat(is_pseudoprime, n, n_minus_one, k, s, base, ctx, mctx))
440 goto err;
441 if (!*is_pseudoprime)
442 goto done;
443
444 /*
445 * Perform Miller-Rabin tests with random 3 <= base < n - 1 to reduce
446 * risk of false positives in BPSW.
447 */
448
449 for (i = 0; i < rounds; i++) {
450 if (!bn_rand_interval(base, 3, n_minus_one))
451 goto err;
452
453 if (!bn_fermat(is_pseudoprime, n, n_minus_one, k, s, base, ctx,
454 mctx))
455 goto err;
456 if (!*is_pseudoprime)
457 goto done;
458 }
459
460 /*
461 * If we got here, we have a Miller-Rabin pseudoprime.
462 */
463
464 *is_pseudoprime = 1;
465
466 done:
467 ret = 1;
468
469 err:
470 BN_MONT_CTX_free(mctx);
471 BN_CTX_end(ctx);
472
473 return ret;
474}
475
476/*
477 * The Baillie-Pomerance-Selfridge-Wagstaff algorithm combines a Miller-Rabin
478 * test for base 2 with a Strong Lucas pseudoprime test.
479 */
480
481int
482bn_is_prime_bpsw(int *is_pseudoprime, const BIGNUM *n, BN_CTX *in_ctx,
483 size_t rounds)
484{
485 BN_CTX *ctx = NULL;
486 BN_ULONG mod;
487 int i;
488 int ret = 0;
489
490 if (BN_is_word(n, 2)) {
491 *is_pseudoprime = 1;
492 goto done;
493 }
494
495 if (BN_cmp(n, BN_value_one()) <= 0 || !BN_is_odd(n)) {
496 *is_pseudoprime = 0;
497 goto done;
498 }
499
500 /* Trial divisions with the first 2048 primes. */
501 for (i = 0; i < NUMPRIMES; i++) {
502 if ((mod = BN_mod_word(n, primes[i])) == (BN_ULONG)-1)
503 goto err;
504 if (mod == 0) {
505 *is_pseudoprime = BN_is_word(n, primes[i]);
506 goto done;
507 }
508 }
509
510 if ((ctx = in_ctx) == NULL)
511 ctx = BN_CTX_new();
512 if (ctx == NULL)
513 goto err;
514
515 if (!bn_miller_rabin(is_pseudoprime, n, ctx, rounds))
516 goto err;
517 if (!*is_pseudoprime)
518 goto done;
519
520 if (!bn_strong_lucas_selfridge(is_pseudoprime, n, ctx))
521 goto err;
522
523 done:
524 ret = 1;
525
526 err:
527 if (ctx != in_ctx)
528 BN_CTX_free(ctx);
529
530 return ret;
531}
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-05-06 18:02:23 +0000