1 /*
2 * Copyright (c) 2018 Thomas Pornin <pornin@bolet.org>
3 *
4 * Permission is hereby granted, free of charge, to any person obtaining
5 * a copy of this software and associated documentation files (the
6 * "Software"), to deal in the Software without restriction, including
7 * without limitation the rights to use, copy, modify, merge, publish,
8 * distribute, sublicense, and/or sell copies of the Software, and to
9 * permit persons to whom the Software is furnished to do so, subject to
10 * the following conditions:
11 *
12 * The above copyright notice and this permission notice shall be
13 * included in all copies or substantial portions of the Software.
14 *
15 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
16 * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
17 * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
18 * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
19 * BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
20 * ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
21 * CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
22 * SOFTWARE.
23 */
25 #include "inner.h"
27 /*
28 * Make a random integer of the provided size. The size is encoded.
29 * The header word is untouched.
30 */
31 static void
32 mkrand(const br_prng_class **rng, uint32_t *x, uint32_t esize)
33 {
34 size_t u, len;
35 unsigned m;
37 len = (esize + 31) >> 5;
38 (*rng)->generate(rng, x + 1, len * sizeof(uint32_t));
39 for (u = 1; u < len; u ++) {
40 x[u] &= 0x7FFFFFFF;
41 }
42 m = esize & 31;
43 if (m == 0) {
44 x[len] &= 0x7FFFFFFF;
45 } else {
46 x[len] &= 0x7FFFFFFF >> (31 - m);
47 }
48 }
50 /*
51 * This is the big-endian unsigned representation of the product of
52 * all small primes from 13 to 1481.
53 */
54 static const unsigned char SMALL_PRIMES[] = {
55 0x2E, 0xAB, 0x92, 0xD1, 0x8B, 0x12, 0x47, 0x31, 0x54, 0x0A,
56 0x99, 0x5D, 0x25, 0x5E, 0xE2, 0x14, 0x96, 0x29, 0x1E, 0xB7,
57 0x78, 0x70, 0xCC, 0x1F, 0xA5, 0xAB, 0x8D, 0x72, 0x11, 0x37,
58 0xFB, 0xD8, 0x1E, 0x3F, 0x5B, 0x34, 0x30, 0x17, 0x8B, 0xE5,
59 0x26, 0x28, 0x23, 0xA1, 0x8A, 0xA4, 0x29, 0xEA, 0xFD, 0x9E,
60 0x39, 0x60, 0x8A, 0xF3, 0xB5, 0xA6, 0xEB, 0x3F, 0x02, 0xB6,
61 0x16, 0xC3, 0x96, 0x9D, 0x38, 0xB0, 0x7D, 0x82, 0x87, 0x0C,
62 0xF7, 0xBE, 0x24, 0xE5, 0x5F, 0x41, 0x04, 0x79, 0x76, 0x40,
63 0xE7, 0x00, 0x22, 0x7E, 0xB5, 0x85, 0x7F, 0x8D, 0x01, 0x50,
64 0xE9, 0xD3, 0x29, 0x42, 0x08, 0xB3, 0x51, 0x40, 0x7B, 0xD7,
65 0x8D, 0xCC, 0x10, 0x01, 0x64, 0x59, 0x28, 0xB6, 0x53, 0xF3,
66 0x50, 0x4E, 0xB1, 0xF2, 0x58, 0xCD, 0x6E, 0xF5, 0x56, 0x3E,
67 0x66, 0x2F, 0xD7, 0x07, 0x7F, 0x52, 0x4C, 0x13, 0x24, 0xDC,
68 0x8E, 0x8D, 0xCC, 0xED, 0x77, 0xC4, 0x21, 0xD2, 0xFD, 0x08,
69 0xEA, 0xD7, 0xC0, 0x5C, 0x13, 0x82, 0x81, 0x31, 0x2F, 0x2B,
70 0x08, 0xE4, 0x80, 0x04, 0x7A, 0x0C, 0x8A, 0x3C, 0xDC, 0x22,
71 0xE4, 0x5A, 0x7A, 0xB0, 0x12, 0x5E, 0x4A, 0x76, 0x94, 0x77,
72 0xC2, 0x0E, 0x92, 0xBA, 0x8A, 0xA0, 0x1F, 0x14, 0x51, 0x1E,
73 0x66, 0x6C, 0x38, 0x03, 0x6C, 0xC7, 0x4A, 0x4B, 0x70, 0x80,
74 0xAF, 0xCA, 0x84, 0x51, 0xD8, 0xD2, 0x26, 0x49, 0xF5, 0xA8,
75 0x5E, 0x35, 0x4B, 0xAC, 0xCE, 0x29, 0x92, 0x33, 0xB7, 0xA2,
76 0x69, 0x7D, 0x0C, 0xE0, 0x9C, 0xDB, 0x04, 0xD6, 0xB4, 0xBC,
77 0x39, 0xD7, 0x7F, 0x9E, 0x9D, 0x78, 0x38, 0x7F, 0x51, 0x54,
78 0x50, 0x8B, 0x9E, 0x9C, 0x03, 0x6C, 0xF5, 0x9D, 0x2C, 0x74,
79 0x57, 0xF0, 0x27, 0x2A, 0xC3, 0x47, 0xCA, 0xB9, 0xD7, 0x5C,
80 0xFF, 0xC2, 0xAC, 0x65, 0x4E, 0xBD
81 };
83 /*
84 * We need temporary values for at least 7 integers of the same size
85 * as a factor (including header word); more space helps with performance
86 * (in modular exponentiations), but we much prefer to remain under
87 * 2 kilobytes in total, to save stack space. The macro TEMPS below
88 * exceeds 512 (which is a count in 32-bit words) when BR_MAX_RSA_SIZE
89 * is greater than 4464 (default value is 4096, so the 2-kB limit is
90 * maintained unless BR_MAX_RSA_SIZE was modified).
91 */
92 #define MAX(x, y) ((x) > (y) ? (x) : (y))
93 #define ROUND2(x) ((((x) + 1) >> 1) << 1)
95 #define TEMPS MAX(512, ROUND2(7 * ((((BR_MAX_RSA_SIZE + 1) >> 1) + 61) / 31)))
97 /*
98 * Perform trial division on a candidate prime. This computes
99 * y = SMALL_PRIMES mod x, then tries to compute y/y mod x. The
100 * br_i31_moddiv() function will report an error if y is not invertible
101 * modulo x. Returned value is 1 on success (none of the small primes
102 * divides x), 0 on error (a non-trivial GCD is obtained).
103 *
104 * This function assumes that x is odd.
105 */
106 static uint32_t
107 trial_divisions(const uint32_t *x, uint32_t *t)
108 {
109 uint32_t *y;
110 uint32_t x0i;
112 y = t;
113 t += 1 + ((x + 31) >> 5);
114 x0i = br_i31_ninv31(x);
115 br_i31_decode_reduce(y, SMALL_PRIMES, sizeof SMALL_PRIMES, x);
116 return br_i31_moddiv(y, y, x, x0i, t);
117 }
119 /*
120 * Perform n rounds of Miller-Rabin on the candidate prime x. This
121 * function assumes that x = 3 mod 4.
122 *
123 * Returned value is 1 on success (all rounds completed successfully),
124 * 0 otherwise.
125 */
126 static uint32_t
127 miller_rabin(const br_prng_class **rng, const uint32_t *x, int n,
128 uint32_t *t, size_t tlen, br_i31_modpow_opt_type mp31)
129 {
130 /*
131 * Since x = 3 mod 4, the Miller-Rabin test is simple:
132 * - get a random base a (such that 1 < a < x-1)
133 * - compute z = a^((x-1)/2) mod x
134 * - if z != 1 and z != x-1, the number x is composite
135 *
136 * We generate bases 'a' randomly with a size which is
137 * one bit less than x, which ensures that a < x-1. It
138 * is not useful to verify that a > 1 because the probability
139 * that we get a value a equal to 0 or 1 is much smaller
140 * than the probability of our Miller-Rabin tests not to
141 * detect a composite, which is already quite smaller than the
142 * probability of the hardware misbehaving and return a
143 * composite integer because of some glitch (e.g. bad RAM
144 * or ill-timed cosmic ray).
145 */
146 unsigned char *xm1d2;
147 size_t xlen, xm1d2_len, xm1d2_len_u32, u;
148 uint32_t asize;
149 unsigned cc;
150 uint32_t x0i;
152 /*
153 * Compute (x-1)/2 (encoded).
154 */
155 xm1d2 = (unsigned char *)t;
156 xm1d2_len = ((x - (x >> 5)) + 7) >> 3;
157 br_i31_encode(xm1d2, xm1d2_len, x);
158 cc = 0;
159 for (u = 0; u < xm1d2_len; u ++) {
160 unsigned w;
162 w = xm1d2[u];
163 xm1d2[u] = (unsigned char)((w >> 1) | cc);
164 cc = w << 7;
165 }
167 /*
168 * We used some words of the provided buffer for (x-1)/2.
169 */
170 xm1d2_len_u32 = (xm1d2_len + 3) >> 2;
171 t += xm1d2_len_u32;
172 tlen -= xm1d2_len_u32;
174 xlen = (x + 31) >> 5;
175 asize = x - 1 - EQ0(x & 31);
176 x0i = br_i31_ninv31(x);
177 while (n -- > 0) {
178 uint32_t *a, *t2;
179 uint32_t eq1, eqm1;
180 size_t t2len;
182 /*
183 * Generate a random base. We don't need the base to be
184 * really uniform modulo x, so we just get a random
185 * number which is one bit shorter than x.
186 */
187 a = t;
188 a = x;
189 a[xlen] = 0;
190 mkrand(rng, a, asize);
192 /*
193 * Compute a^((x-1)/2) mod x. We assume here that the
194 * function will not fail (the temporary array is large
195 * enough).
196 */
197 t2 = t + 1 + xlen;
198 t2len = tlen - 1 - xlen;
199 if ((t2len & 1) != 0) {
200 /*
201 * Since the source array is 64-bit aligned and
202 * has an even number of elements (TEMPS), we
203 * can use the parity of the remaining length to
204 * detect and adjust alignment.
205 */
206 t2 ++;
207 t2len --;
208 }
209 mp31(a, xm1d2, xm1d2_len, x, x0i, t2, t2len);
211 /*
212 * We must obtain either 1 or x-1. Note that x is odd,
213 * hence x-1 differs from x only in its low word (no
214 * carry).
215 */
216 eq1 = a ^ 1;
217 eqm1 = a ^ (x - 1);
218 for (u = 2; u <= xlen; u ++) {
219 eq1 |= a[u];
220 eqm1 |= a[u] ^ x[u];
221 }
223 if ((EQ0(eq1) | EQ0(eqm1)) == 0) {
224 return 0;
225 }
226 }
227 return 1;
228 }
230 /*
231 * Create a random prime of the provided size. 'size' is the _encoded_
232 * bit length. The two top bits and the two bottom bits are set to 1.
233 */
234 static void
235 mkprime(const br_prng_class **rng, uint32_t *x, uint32_t esize,
236 uint32_t pubexp, uint32_t *t, size_t tlen, br_i31_modpow_opt_type mp31)
237 {
238 size_t len;
240 x = esize;
241 len = (esize + 31) >> 5;
242 for (;;) {
243 size_t u;
244 uint32_t m3, m5, m7, m11;
245 int rounds, s7, s11;
247 /*
248 * Generate random bits. We force the two top bits and the
249 * two bottom bits to 1.
250 */
251 mkrand(rng, x, esize);
252 if ((esize & 31) == 0) {
253 x[len] |= 0x60000000;
254 } else if ((esize & 31) == 1) {
255 x[len] |= 0x00000001;
256 x[len - 1] |= 0x40000000;
257 } else {
258 x[len] |= 0x00000003 << ((esize & 31) - 2);
259 }
260 x |= 0x00000003;
262 /*
263 * Trial division with low primes (3, 5, 7 and 11). We
264 * use the following properties:
265 *
266 * 2^2 = 1 mod 3
267 * 2^4 = 1 mod 5
268 * 2^3 = 1 mod 7
269 * 2^10 = 1 mod 11
270 */
271 m3 = 0;
272 m5 = 0;
273 m7 = 0;
274 m11 = 0;
275 s7 = 0;
276 s11 = 0;
277 for (u = 0; u < len; u ++) {
278 uint32_t w, w3, w5, w7, w11;
280 w = x[1 + u];
281 w3 = (w & 0xFFFF) + (w >> 16); /* max: 98302 */
282 w5 = (w & 0xFFFF) + (w >> 16); /* max: 98302 */
283 w7 = (w & 0x7FFF) + (w >> 15); /* max: 98302 */
284 w11 = (w & 0xFFFFF) + (w >> 20); /* max: 1050622 */
286 m3 += w3 << (u & 1);
287 m3 = (m3 & 0xFF) + (m3 >> 8); /* max: 1025 */
289 m5 += w5 << ((4 - u) & 3);
290 m5 = (m5 & 0xFFF) + (m5 >> 12); /* max: 4479 */
292 m7 += w7 << s7;
293 m7 = (m7 & 0x1FF) + (m7 >> 9); /* max: 1280 */
294 if (++ s7 == 3) {
295 s7 = 0;
296 }
298 m11 += w11 << s11;
299 if (++ s11 == 10) {
300 s11 = 0;
301 }
302 m11 = (m11 & 0x3FF) + (m11 >> 10); /* max: 526847 */
303 }
305 m3 = (m3 & 0x3F) + (m3 >> 6); /* max: 78 */
306 m3 = (m3 & 0x0F) + (m3 >> 4); /* max: 18 */
307 m3 = ((m3 * 43) >> 5) & 3;
309 m5 = (m5 & 0xFF) + (m5 >> 8); /* max: 271 */
310 m5 = (m5 & 0x0F) + (m5 >> 4); /* max: 31 */
311 m5 -= 20 & -GT(m5, 19);
312 m5 -= 10 & -GT(m5, 9);
313 m5 -= 5 & -GT(m5, 4);
315 m7 = (m7 & 0x3F) + (m7 >> 6); /* max: 82 */
316 m7 = (m7 & 0x07) + (m7 >> 3); /* max: 16 */
317 m7 = ((m7 * 147) >> 7) & 7;
319 /*
320 * 2^5 = 32 = -1 mod 11.
321 */
322 m11 = (m11 & 0x3FF) + (m11 >> 10); /* max: 1536 */
323 m11 = (m11 & 0x3FF) + (m11 >> 10); /* max: 1023 */
324 m11 = (m11 & 0x1F) + 33 - (m11 >> 5); /* max: 64 */
325 m11 -= 44 & -GT(m11, 43);
326 m11 -= 22 & -GT(m11, 21);
327 m11 -= 11 & -GT(m11, 10);
329 /*
330 * If any of these modulo is 0, then the candidate is
331 * not prime. Also, if pubexp is 3, 5, 7 or 11, and the
332 * corresponding modulus is 1, then the candidate must
333 * be rejected, because we need e to be invertible
334 * modulo p-1. We can use simple comparisons here
335 * because they won't leak information on a candidate
336 * that we keep, only on one that we reject (and is thus
337 * not secret).
338 */
339 if (m3 == 0 || m5 == 0 || m7 == 0 || m11 == 0) {
340 continue;
341 }
342 if ((pubexp == 3 && m3 == 1)
343 || (pubexp == 5 && m5 == 1)
344 || (pubexp == 7 && m7 == 1)
345 || (pubexp == 11 && m11 == 1))
346 {
347 continue;
348 }
350 /*
351 * More trial divisions.
352 */
353 if (!trial_divisions(x, t)) {
354 continue;
355 }
357 /*
358 * Miller-Rabin algorithm. Since we selected a random
359 * integer, not a maliciously crafted integer, we can use
360 * relatively few rounds to lower the risk of a false
361 * positive (i.e. declaring prime a non-prime) under
362 * 2^(-80). It is not useful to lower the probability much
363 * below that, since that would be substantially below
364 * the probability of the hardware misbehaving. Sufficient
365 * numbers of rounds are extracted from the Handbook of
366 * Applied Cryptography, note 4.49 (page 149).
367 *
368 * Since we work on the encoded size (esize), we need to
369 * compare with encoded thresholds.
370 */
371 if (esize < 309) {
372 rounds = 12;
373 } else if (esize < 464) {
374 rounds = 9;
375 } else if (esize < 670) {
376 rounds = 6;
377 } else if (esize < 877) {
378 rounds = 4;
379 } else if (esize < 1341) {
380 rounds = 3;
381 } else {
382 rounds = 2;
383 }
385 if (miller_rabin(rng, x, rounds, t, tlen, mp31)) {
386 return;
387 }
388 }
389 }
391 /*
392 * Let p be a prime (p > 2^33, p = 3 mod 4). Let m = (p-1)/2, provided
393 * as parameter (with announced bit length equal to that of p). This
394 * function computes d = 1/e mod p-1 (for an odd integer e). Returned
395 * value is 1 on success, 0 on error (an error is reported if e is not
396 * invertible modulo p-1).
397 *
398 * The temporary buffer (t) must have room for at least 4 integers of
399 * the size of p.
400 */
401 static uint32_t
402 invert_pubexp(uint32_t *d, const uint32_t *m, uint32_t e, uint32_t *t)
403 {
404 uint32_t *f;
405 uint32_t r;
407 f = t;
408 t += 1 + ((m + 31) >> 5);
410 /*
411 * Compute d = 1/e mod m. Since p = 3 mod 4, m is odd.
412 */
413 br_i31_zero(d, m);
414 d = 1;
415 br_i31_zero(f, m);
416 f = e & 0x7FFFFFFF;
417 f = e >> 31;
418 r = br_i31_moddiv(d, f, m, br_i31_ninv31(m), t);
420 /*
421 * We really want d = 1/e mod p-1, with p = 2m. By the CRT,
422 * the result is either the d we got, or d + m.
423 *
424 * Let's write e*d = 1 + k*m, for some integer k. Integers e
425 * and m are odd. If d is odd, then e*d is odd, which implies
426 * that k must be even; in that case, e*d = 1 + (k/2)*2m, and
427 * thus d is already fine. Conversely, if d is even, then k
428 * is odd, and we must add m to d in order to get the correct
429 * result.
430 */
431 br_i31_add(d, m, (uint32_t)(1 - (d & 1)));
433 return r;
434 }
436 /*
437 * Swap two buffers in RAM. They must be disjoint.
438 */
439 static void
440 bufswap(void *b1, void *b2, size_t len)
441 {
442 size_t u;
443 unsigned char *buf1, *buf2;
445 buf1 = b1;
446 buf2 = b2;
447 for (u = 0; u < len; u ++) {
448 unsigned w;
450 w = buf1[u];
451 buf1[u] = buf2[u];
452 buf2[u] = w;
453 }
454 }
456 /* see inner.h */
457 uint32_t
458 br_rsa_i31_keygen_inner(const br_prng_class **rng,
459 br_rsa_private_key *sk, void *kbuf_priv,
460 br_rsa_public_key *pk, void *kbuf_pub,
461 unsigned size, uint32_t pubexp, br_i31_modpow_opt_type mp31)
462 {
463 uint32_t esize_p, esize_q;
464 size_t plen, qlen, tlen;
465 uint32_t *p, *q, *t;
466 union {
467 uint32_t t32[TEMPS];
468 uint64_t t64[TEMPS >> 1]; /* for 64-bit alignment */
469 } tmp;
470 uint32_t r;
472 if (size < BR_MIN_RSA_SIZE || size > BR_MAX_RSA_SIZE) {
473 return 0;
474 }
475 if (pubexp == 0) {
476 pubexp = 3;
477 } else if (pubexp == 1 || (pubexp & 1) == 0) {
478 return 0;
479 }
481 esize_p = (size + 1) >> 1;
482 esize_q = size - esize_p;
483 sk->n_bitlen = size;
484 sk->p = kbuf_priv;
485 sk->plen = (esize_p + 7) >> 3;
486 sk->q = sk->p + sk->plen;
487 sk->qlen = (esize_q + 7) >> 3;
488 sk->dp = sk->q + sk->qlen;
489 sk->dplen = sk->plen;
490 sk->dq = sk->dp + sk->dplen;
491 sk->dqlen = sk->qlen;
492 sk->iq = sk->dq + sk->dqlen;
493 sk->iqlen = sk->plen;
495 if (pk != NULL) {
496 pk->n = kbuf_pub;
497 pk->nlen = (size + 7) >> 3;
498 pk->e = pk->n + pk->nlen;
499 pk->elen = 4;
500 br_enc32be(pk->e, pubexp);
501 while (*pk->e == 0) {
502 pk->e ++;
503 pk->elen --;
504 }
505 }
507 /*
508 * We now switch to encoded sizes.
509 *
510 * floor((x * 16913) / (2^19)) is equal to floor(x/31) for all
511 * integers x from 0 to 34966; the intermediate product fits on
512 * 30 bits, thus we can use MUL31().
513 */
514 esize_p += MUL31(esize_p, 16913) >> 19;
515 esize_q += MUL31(esize_q, 16913) >> 19;
516 plen = (esize_p + 31) >> 5;
517 qlen = (esize_q + 31) >> 5;
518 p = tmp.t32;
519 q = p + 1 + plen;
520 t = q + 1 + qlen;
521 tlen = ((sizeof tmp.t32) / sizeof(uint32_t)) - (2 + plen + qlen);
523 /*
524 * When looking for primes p and q, we temporarily divide
525 * candidates by 2, in order to compute the inverse of the
526 * public exponent.
527 */
529 for (;;) {
530 mkprime(rng, p, esize_p, pubexp, t, tlen, mp31);
531 br_i31_rshift(p, 1);
532 if (invert_pubexp(t, p, pubexp, t + 1 + plen)) {
534 p |= 1;
535 br_i31_encode(sk->p, sk->plen, p);
536 br_i31_encode(sk->dp, sk->dplen, t);
537 break;
538 }
539 }
541 for (;;) {
542 mkprime(rng, q, esize_q, pubexp, t, tlen, mp31);
543 br_i31_rshift(q, 1);
544 if (invert_pubexp(t, q, pubexp, t + 1 + qlen)) {
546 q |= 1;
547 br_i31_encode(sk->q, sk->qlen, q);
548 br_i31_encode(sk->dq, sk->dqlen, t);
549 break;
550 }
551 }
553 /*
554 * If p and q have the same size, then it is possible that q > p
555 * (when the target modulus size is odd, we generate p with a
556 * greater bit length than q). If q > p, we want to swap p and q
557 * (and also dp and dq) for two reasons:
558 * - The final step below (inversion of q modulo p) is easier if
559 * p > q.
560 * - While BearSSL's RSA code is perfectly happy with RSA keys such
561 * that p < q, some other implementations have restrictions and
562 * require p > q.
563 *
564 * Note that we can do a simple non-constant-time swap here,
565 * because the only information we leak here is that we insist on
566 * returning p and q such that p > q, which is not a secret.
567 */
568 if (esize_p == esize_q && br_i31_sub(p, q, 0) == 1) {
569 bufswap(p, q, (1 + plen) * sizeof *p);
570 bufswap(sk->p, sk->q, sk->plen);
571 bufswap(sk->dp, sk->dq, sk->dplen);
572 }
574 /*
575 * We have produced p, q, dp and dq. We can now compute iq = 1/d mod p.
576 *
577 * We ensured that p >= q, so this is just a matter of updating the
578 * header word for q (and possibly adding an extra word).
579 *
580 * Theoretically, the call below may fail, in case we were
581 * extraordinarily unlucky, and p = q. Another failure case is if
582 * Miller-Rabin failed us _twice_, and p and q are non-prime and
583 * have a factor is common. We report the error mostly because it
584 * is cheap and we can, but in practice this never happens (or, at
585 * least, it happens way less often than hardware glitches).
586 */
587 q = p;
588 if (plen > qlen) {
589 q[plen] = 0;
590 t ++;
591 tlen --;
592 }
593 br_i31_zero(t, p);
594 t = 1;
595 r = br_i31_moddiv(t, q, p, br_i31_ninv31(p), t + 1 + plen);
596 br_i31_encode(sk->iq, sk->iqlen, t);
598 /*
599 * Compute the public modulus too, if required.
600 */
601 if (pk != NULL) {
602 br_i31_zero(t, p);
603 br_i31_mulacc(t, p, q);
604 br_i31_encode(pk->n, pk->nlen, t);
605 }
607 return r;
608 }
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