1 | /* |
2 | * Copyright 2002-2018 The OpenSSL Project Authors. All Rights Reserved. |
3 | * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved |
4 | * |
5 | * Licensed under the Apache License 2.0 (the "License"). You may not use |
6 | * this file except in compliance with the License. You can obtain a copy |
7 | * in the file LICENSE in the source distribution or at |
8 | * https://www.openssl.org/source/license.html |
9 | */ |
10 | |
11 | #include <assert.h> |
12 | #include <limits.h> |
13 | #include <stdio.h> |
14 | #include "internal/cryptlib.h" |
15 | #include "bn_local.h" |
16 | |
17 | #ifndef OPENSSL_NO_EC2M |
18 | |
19 | /* |
20 | * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should |
21 | * fail. |
22 | */ |
23 | # define MAX_ITERATIONS 50 |
24 | |
25 | # define SQR_nibble(w) ((((w) & 8) << 3) \ |
26 | | (((w) & 4) << 2) \ |
27 | | (((w) & 2) << 1) \ |
28 | | ((w) & 1)) |
29 | |
30 | |
31 | /* Platform-specific macros to accelerate squaring. */ |
32 | # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) |
33 | # define SQR1(w) \ |
34 | SQR_nibble((w) >> 60) << 56 | SQR_nibble((w) >> 56) << 48 | \ |
35 | SQR_nibble((w) >> 52) << 40 | SQR_nibble((w) >> 48) << 32 | \ |
36 | SQR_nibble((w) >> 44) << 24 | SQR_nibble((w) >> 40) << 16 | \ |
37 | SQR_nibble((w) >> 36) << 8 | SQR_nibble((w) >> 32) |
38 | # define SQR0(w) \ |
39 | SQR_nibble((w) >> 28) << 56 | SQR_nibble((w) >> 24) << 48 | \ |
40 | SQR_nibble((w) >> 20) << 40 | SQR_nibble((w) >> 16) << 32 | \ |
41 | SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >> 8) << 16 | \ |
42 | SQR_nibble((w) >> 4) << 8 | SQR_nibble((w) ) |
43 | # endif |
44 | # ifdef THIRTY_TWO_BIT |
45 | # define SQR1(w) \ |
46 | SQR_nibble((w) >> 28) << 24 | SQR_nibble((w) >> 24) << 16 | \ |
47 | SQR_nibble((w) >> 20) << 8 | SQR_nibble((w) >> 16) |
48 | # define SQR0(w) \ |
49 | SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >> 8) << 16 | \ |
50 | SQR_nibble((w) >> 4) << 8 | SQR_nibble((w) ) |
51 | # endif |
52 | |
53 | # if !defined(OPENSSL_BN_ASM_GF2m) |
54 | /* |
55 | * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is |
56 | * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that |
57 | * the variables have the right amount of space allocated. |
58 | */ |
59 | # ifdef THIRTY_TWO_BIT |
60 | static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, |
61 | const BN_ULONG b) |
62 | { |
63 | register BN_ULONG h, l, s; |
64 | BN_ULONG tab[8], top2b = a >> 30; |
65 | register BN_ULONG a1, a2, a4; |
66 | |
67 | a1 = a & (0x3FFFFFFF); |
68 | a2 = a1 << 1; |
69 | a4 = a2 << 1; |
70 | |
71 | tab[0] = 0; |
72 | tab[1] = a1; |
73 | tab[2] = a2; |
74 | tab[3] = a1 ^ a2; |
75 | tab[4] = a4; |
76 | tab[5] = a1 ^ a4; |
77 | tab[6] = a2 ^ a4; |
78 | tab[7] = a1 ^ a2 ^ a4; |
79 | |
80 | s = tab[b & 0x7]; |
81 | l = s; |
82 | s = tab[b >> 3 & 0x7]; |
83 | l ^= s << 3; |
84 | h = s >> 29; |
85 | s = tab[b >> 6 & 0x7]; |
86 | l ^= s << 6; |
87 | h ^= s >> 26; |
88 | s = tab[b >> 9 & 0x7]; |
89 | l ^= s << 9; |
90 | h ^= s >> 23; |
91 | s = tab[b >> 12 & 0x7]; |
92 | l ^= s << 12; |
93 | h ^= s >> 20; |
94 | s = tab[b >> 15 & 0x7]; |
95 | l ^= s << 15; |
96 | h ^= s >> 17; |
97 | s = tab[b >> 18 & 0x7]; |
98 | l ^= s << 18; |
99 | h ^= s >> 14; |
100 | s = tab[b >> 21 & 0x7]; |
101 | l ^= s << 21; |
102 | h ^= s >> 11; |
103 | s = tab[b >> 24 & 0x7]; |
104 | l ^= s << 24; |
105 | h ^= s >> 8; |
106 | s = tab[b >> 27 & 0x7]; |
107 | l ^= s << 27; |
108 | h ^= s >> 5; |
109 | s = tab[b >> 30]; |
110 | l ^= s << 30; |
111 | h ^= s >> 2; |
112 | |
113 | /* compensate for the top two bits of a */ |
114 | |
115 | if (top2b & 01) { |
116 | l ^= b << 30; |
117 | h ^= b >> 2; |
118 | } |
119 | if (top2b & 02) { |
120 | l ^= b << 31; |
121 | h ^= b >> 1; |
122 | } |
123 | |
124 | *r1 = h; |
125 | *r0 = l; |
126 | } |
127 | # endif |
128 | # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) |
129 | static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, |
130 | const BN_ULONG b) |
131 | { |
132 | register BN_ULONG h, l, s; |
133 | BN_ULONG tab[16], top3b = a >> 61; |
134 | register BN_ULONG a1, a2, a4, a8; |
135 | |
136 | a1 = a & (0x1FFFFFFFFFFFFFFFULL); |
137 | a2 = a1 << 1; |
138 | a4 = a2 << 1; |
139 | a8 = a4 << 1; |
140 | |
141 | tab[0] = 0; |
142 | tab[1] = a1; |
143 | tab[2] = a2; |
144 | tab[3] = a1 ^ a2; |
145 | tab[4] = a4; |
146 | tab[5] = a1 ^ a4; |
147 | tab[6] = a2 ^ a4; |
148 | tab[7] = a1 ^ a2 ^ a4; |
149 | tab[8] = a8; |
150 | tab[9] = a1 ^ a8; |
151 | tab[10] = a2 ^ a8; |
152 | tab[11] = a1 ^ a2 ^ a8; |
153 | tab[12] = a4 ^ a8; |
154 | tab[13] = a1 ^ a4 ^ a8; |
155 | tab[14] = a2 ^ a4 ^ a8; |
156 | tab[15] = a1 ^ a2 ^ a4 ^ a8; |
157 | |
158 | s = tab[b & 0xF]; |
159 | l = s; |
160 | s = tab[b >> 4 & 0xF]; |
161 | l ^= s << 4; |
162 | h = s >> 60; |
163 | s = tab[b >> 8 & 0xF]; |
164 | l ^= s << 8; |
165 | h ^= s >> 56; |
166 | s = tab[b >> 12 & 0xF]; |
167 | l ^= s << 12; |
168 | h ^= s >> 52; |
169 | s = tab[b >> 16 & 0xF]; |
170 | l ^= s << 16; |
171 | h ^= s >> 48; |
172 | s = tab[b >> 20 & 0xF]; |
173 | l ^= s << 20; |
174 | h ^= s >> 44; |
175 | s = tab[b >> 24 & 0xF]; |
176 | l ^= s << 24; |
177 | h ^= s >> 40; |
178 | s = tab[b >> 28 & 0xF]; |
179 | l ^= s << 28; |
180 | h ^= s >> 36; |
181 | s = tab[b >> 32 & 0xF]; |
182 | l ^= s << 32; |
183 | h ^= s >> 32; |
184 | s = tab[b >> 36 & 0xF]; |
185 | l ^= s << 36; |
186 | h ^= s >> 28; |
187 | s = tab[b >> 40 & 0xF]; |
188 | l ^= s << 40; |
189 | h ^= s >> 24; |
190 | s = tab[b >> 44 & 0xF]; |
191 | l ^= s << 44; |
192 | h ^= s >> 20; |
193 | s = tab[b >> 48 & 0xF]; |
194 | l ^= s << 48; |
195 | h ^= s >> 16; |
196 | s = tab[b >> 52 & 0xF]; |
197 | l ^= s << 52; |
198 | h ^= s >> 12; |
199 | s = tab[b >> 56 & 0xF]; |
200 | l ^= s << 56; |
201 | h ^= s >> 8; |
202 | s = tab[b >> 60]; |
203 | l ^= s << 60; |
204 | h ^= s >> 4; |
205 | |
206 | /* compensate for the top three bits of a */ |
207 | |
208 | if (top3b & 01) { |
209 | l ^= b << 61; |
210 | h ^= b >> 3; |
211 | } |
212 | if (top3b & 02) { |
213 | l ^= b << 62; |
214 | h ^= b >> 2; |
215 | } |
216 | if (top3b & 04) { |
217 | l ^= b << 63; |
218 | h ^= b >> 1; |
219 | } |
220 | |
221 | *r1 = h; |
222 | *r0 = l; |
223 | } |
224 | # endif |
225 | |
226 | /* |
227 | * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1, |
228 | * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST |
229 | * ensure that the variables have the right amount of space allocated. |
230 | */ |
231 | static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, |
232 | const BN_ULONG b1, const BN_ULONG b0) |
233 | { |
234 | BN_ULONG m1, m0; |
235 | /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */ |
236 | bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1); |
237 | bn_GF2m_mul_1x1(r + 1, r, a0, b0); |
238 | bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1); |
239 | /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */ |
240 | r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */ |
241 | r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */ |
242 | } |
243 | # else |
244 | void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1, |
245 | BN_ULONG b0); |
246 | # endif |
247 | |
248 | /* |
249 | * Add polynomials a and b and store result in r; r could be a or b, a and b |
250 | * could be equal; r is the bitwise XOR of a and b. |
251 | */ |
252 | int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) |
253 | { |
254 | int i; |
255 | const BIGNUM *at, *bt; |
256 | |
257 | bn_check_top(a); |
258 | bn_check_top(b); |
259 | |
260 | if (a->top < b->top) { |
261 | at = b; |
262 | bt = a; |
263 | } else { |
264 | at = a; |
265 | bt = b; |
266 | } |
267 | |
268 | if (bn_wexpand(r, at->top) == NULL) |
269 | return 0; |
270 | |
271 | for (i = 0; i < bt->top; i++) { |
272 | r->d[i] = at->d[i] ^ bt->d[i]; |
273 | } |
274 | for (; i < at->top; i++) { |
275 | r->d[i] = at->d[i]; |
276 | } |
277 | |
278 | r->top = at->top; |
279 | bn_correct_top(r); |
280 | |
281 | return 1; |
282 | } |
283 | |
284 | /*- |
285 | * Some functions allow for representation of the irreducible polynomials |
286 | * as an int[], say p. The irreducible f(t) is then of the form: |
287 | * t^p[0] + t^p[1] + ... + t^p[k] |
288 | * where m = p[0] > p[1] > ... > p[k] = 0. |
289 | */ |
290 | |
291 | /* Performs modular reduction of a and store result in r. r could be a. */ |
292 | int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[]) |
293 | { |
294 | int j, k; |
295 | int n, dN, d0, d1; |
296 | BN_ULONG zz, *z; |
297 | |
298 | bn_check_top(a); |
299 | |
300 | if (p[0] == 0) { |
301 | /* reduction mod 1 => return 0 */ |
302 | BN_zero(r); |
303 | return 1; |
304 | } |
305 | |
306 | /* |
307 | * Since the algorithm does reduction in the r value, if a != r, copy the |
308 | * contents of a into r so we can do reduction in r. |
309 | */ |
310 | if (a != r) { |
311 | if (!bn_wexpand(r, a->top)) |
312 | return 0; |
313 | for (j = 0; j < a->top; j++) { |
314 | r->d[j] = a->d[j]; |
315 | } |
316 | r->top = a->top; |
317 | } |
318 | z = r->d; |
319 | |
320 | /* start reduction */ |
321 | dN = p[0] / BN_BITS2; |
322 | for (j = r->top - 1; j > dN;) { |
323 | zz = z[j]; |
324 | if (z[j] == 0) { |
325 | j--; |
326 | continue; |
327 | } |
328 | z[j] = 0; |
329 | |
330 | for (k = 1; p[k] != 0; k++) { |
331 | /* reducing component t^p[k] */ |
332 | n = p[0] - p[k]; |
333 | d0 = n % BN_BITS2; |
334 | d1 = BN_BITS2 - d0; |
335 | n /= BN_BITS2; |
336 | z[j - n] ^= (zz >> d0); |
337 | if (d0) |
338 | z[j - n - 1] ^= (zz << d1); |
339 | } |
340 | |
341 | /* reducing component t^0 */ |
342 | n = dN; |
343 | d0 = p[0] % BN_BITS2; |
344 | d1 = BN_BITS2 - d0; |
345 | z[j - n] ^= (zz >> d0); |
346 | if (d0) |
347 | z[j - n - 1] ^= (zz << d1); |
348 | } |
349 | |
350 | /* final round of reduction */ |
351 | while (j == dN) { |
352 | |
353 | d0 = p[0] % BN_BITS2; |
354 | zz = z[dN] >> d0; |
355 | if (zz == 0) |
356 | break; |
357 | d1 = BN_BITS2 - d0; |
358 | |
359 | /* clear up the top d1 bits */ |
360 | if (d0) |
361 | z[dN] = (z[dN] << d1) >> d1; |
362 | else |
363 | z[dN] = 0; |
364 | z[0] ^= zz; /* reduction t^0 component */ |
365 | |
366 | for (k = 1; p[k] != 0; k++) { |
367 | BN_ULONG tmp_ulong; |
368 | |
369 | /* reducing component t^p[k] */ |
370 | n = p[k] / BN_BITS2; |
371 | d0 = p[k] % BN_BITS2; |
372 | d1 = BN_BITS2 - d0; |
373 | z[n] ^= (zz << d0); |
374 | if (d0 && (tmp_ulong = zz >> d1)) |
375 | z[n + 1] ^= tmp_ulong; |
376 | } |
377 | |
378 | } |
379 | |
380 | bn_correct_top(r); |
381 | return 1; |
382 | } |
383 | |
384 | /* |
385 | * Performs modular reduction of a by p and store result in r. r could be a. |
386 | * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper |
387 | * function is only provided for convenience; for best performance, use the |
388 | * BN_GF2m_mod_arr function. |
389 | */ |
390 | int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) |
391 | { |
392 | int ret = 0; |
393 | int arr[6]; |
394 | bn_check_top(a); |
395 | bn_check_top(p); |
396 | ret = BN_GF2m_poly2arr(p, arr, OSSL_NELEM(arr)); |
397 | if (!ret || ret > (int)OSSL_NELEM(arr)) { |
398 | BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH); |
399 | return 0; |
400 | } |
401 | ret = BN_GF2m_mod_arr(r, a, arr); |
402 | bn_check_top(r); |
403 | return ret; |
404 | } |
405 | |
406 | /* |
407 | * Compute the product of two polynomials a and b, reduce modulo p, and store |
408 | * the result in r. r could be a or b; a could be b. |
409 | */ |
410 | int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, |
411 | const int p[], BN_CTX *ctx) |
412 | { |
413 | int zlen, i, j, k, ret = 0; |
414 | BIGNUM *s; |
415 | BN_ULONG x1, x0, y1, y0, zz[4]; |
416 | |
417 | bn_check_top(a); |
418 | bn_check_top(b); |
419 | |
420 | if (a == b) { |
421 | return BN_GF2m_mod_sqr_arr(r, a, p, ctx); |
422 | } |
423 | |
424 | BN_CTX_start(ctx); |
425 | if ((s = BN_CTX_get(ctx)) == NULL) |
426 | goto err; |
427 | |
428 | zlen = a->top + b->top + 4; |
429 | if (!bn_wexpand(s, zlen)) |
430 | goto err; |
431 | s->top = zlen; |
432 | |
433 | for (i = 0; i < zlen; i++) |
434 | s->d[i] = 0; |
435 | |
436 | for (j = 0; j < b->top; j += 2) { |
437 | y0 = b->d[j]; |
438 | y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1]; |
439 | for (i = 0; i < a->top; i += 2) { |
440 | x0 = a->d[i]; |
441 | x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1]; |
442 | bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); |
443 | for (k = 0; k < 4; k++) |
444 | s->d[i + j + k] ^= zz[k]; |
445 | } |
446 | } |
447 | |
448 | bn_correct_top(s); |
449 | if (BN_GF2m_mod_arr(r, s, p)) |
450 | ret = 1; |
451 | bn_check_top(r); |
452 | |
453 | err: |
454 | BN_CTX_end(ctx); |
455 | return ret; |
456 | } |
457 | |
458 | /* |
459 | * Compute the product of two polynomials a and b, reduce modulo p, and store |
460 | * the result in r. r could be a or b; a could equal b. This function calls |
461 | * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is |
462 | * only provided for convenience; for best performance, use the |
463 | * BN_GF2m_mod_mul_arr function. |
464 | */ |
465 | int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, |
466 | const BIGNUM *p, BN_CTX *ctx) |
467 | { |
468 | int ret = 0; |
469 | const int max = BN_num_bits(p) + 1; |
470 | int *arr = NULL; |
471 | bn_check_top(a); |
472 | bn_check_top(b); |
473 | bn_check_top(p); |
474 | if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) |
475 | goto err; |
476 | ret = BN_GF2m_poly2arr(p, arr, max); |
477 | if (!ret || ret > max) { |
478 | BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH); |
479 | goto err; |
480 | } |
481 | ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); |
482 | bn_check_top(r); |
483 | err: |
484 | OPENSSL_free(arr); |
485 | return ret; |
486 | } |
487 | |
488 | /* Square a, reduce the result mod p, and store it in a. r could be a. */ |
489 | int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], |
490 | BN_CTX *ctx) |
491 | { |
492 | int i, ret = 0; |
493 | BIGNUM *s; |
494 | |
495 | bn_check_top(a); |
496 | BN_CTX_start(ctx); |
497 | if ((s = BN_CTX_get(ctx)) == NULL) |
498 | goto err; |
499 | if (!bn_wexpand(s, 2 * a->top)) |
500 | goto err; |
501 | |
502 | for (i = a->top - 1; i >= 0; i--) { |
503 | s->d[2 * i + 1] = SQR1(a->d[i]); |
504 | s->d[2 * i] = SQR0(a->d[i]); |
505 | } |
506 | |
507 | s->top = 2 * a->top; |
508 | bn_correct_top(s); |
509 | if (!BN_GF2m_mod_arr(r, s, p)) |
510 | goto err; |
511 | bn_check_top(r); |
512 | ret = 1; |
513 | err: |
514 | BN_CTX_end(ctx); |
515 | return ret; |
516 | } |
517 | |
518 | /* |
519 | * Square a, reduce the result mod p, and store it in a. r could be a. This |
520 | * function calls down to the BN_GF2m_mod_sqr_arr implementation; this |
521 | * wrapper function is only provided for convenience; for best performance, |
522 | * use the BN_GF2m_mod_sqr_arr function. |
523 | */ |
524 | int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) |
525 | { |
526 | int ret = 0; |
527 | const int max = BN_num_bits(p) + 1; |
528 | int *arr = NULL; |
529 | |
530 | bn_check_top(a); |
531 | bn_check_top(p); |
532 | if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) |
533 | goto err; |
534 | ret = BN_GF2m_poly2arr(p, arr, max); |
535 | if (!ret || ret > max) { |
536 | BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH); |
537 | goto err; |
538 | } |
539 | ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); |
540 | bn_check_top(r); |
541 | err: |
542 | OPENSSL_free(arr); |
543 | return ret; |
544 | } |
545 | |
546 | /* |
547 | * Invert a, reduce modulo p, and store the result in r. r could be a. Uses |
548 | * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D., |
549 | * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic |
550 | * Curve Cryptography Over Binary Fields". |
551 | */ |
552 | static int BN_GF2m_mod_inv_vartime(BIGNUM *r, const BIGNUM *a, |
553 | const BIGNUM *p, BN_CTX *ctx) |
554 | { |
555 | BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp; |
556 | int ret = 0; |
557 | |
558 | bn_check_top(a); |
559 | bn_check_top(p); |
560 | |
561 | BN_CTX_start(ctx); |
562 | |
563 | b = BN_CTX_get(ctx); |
564 | c = BN_CTX_get(ctx); |
565 | u = BN_CTX_get(ctx); |
566 | v = BN_CTX_get(ctx); |
567 | if (v == NULL) |
568 | goto err; |
569 | |
570 | if (!BN_GF2m_mod(u, a, p)) |
571 | goto err; |
572 | if (BN_is_zero(u)) |
573 | goto err; |
574 | |
575 | if (!BN_copy(v, p)) |
576 | goto err; |
577 | # if 0 |
578 | if (!BN_one(b)) |
579 | goto err; |
580 | |
581 | while (1) { |
582 | while (!BN_is_odd(u)) { |
583 | if (BN_is_zero(u)) |
584 | goto err; |
585 | if (!BN_rshift1(u, u)) |
586 | goto err; |
587 | if (BN_is_odd(b)) { |
588 | if (!BN_GF2m_add(b, b, p)) |
589 | goto err; |
590 | } |
591 | if (!BN_rshift1(b, b)) |
592 | goto err; |
593 | } |
594 | |
595 | if (BN_abs_is_word(u, 1)) |
596 | break; |
597 | |
598 | if (BN_num_bits(u) < BN_num_bits(v)) { |
599 | tmp = u; |
600 | u = v; |
601 | v = tmp; |
602 | tmp = b; |
603 | b = c; |
604 | c = tmp; |
605 | } |
606 | |
607 | if (!BN_GF2m_add(u, u, v)) |
608 | goto err; |
609 | if (!BN_GF2m_add(b, b, c)) |
610 | goto err; |
611 | } |
612 | # else |
613 | { |
614 | int i; |
615 | int ubits = BN_num_bits(u); |
616 | int vbits = BN_num_bits(v); /* v is copy of p */ |
617 | int top = p->top; |
618 | BN_ULONG *udp, *bdp, *vdp, *cdp; |
619 | |
620 | if (!bn_wexpand(u, top)) |
621 | goto err; |
622 | udp = u->d; |
623 | for (i = u->top; i < top; i++) |
624 | udp[i] = 0; |
625 | u->top = top; |
626 | if (!bn_wexpand(b, top)) |
627 | goto err; |
628 | bdp = b->d; |
629 | bdp[0] = 1; |
630 | for (i = 1; i < top; i++) |
631 | bdp[i] = 0; |
632 | b->top = top; |
633 | if (!bn_wexpand(c, top)) |
634 | goto err; |
635 | cdp = c->d; |
636 | for (i = 0; i < top; i++) |
637 | cdp[i] = 0; |
638 | c->top = top; |
639 | vdp = v->d; /* It pays off to "cache" *->d pointers, |
640 | * because it allows optimizer to be more |
641 | * aggressive. But we don't have to "cache" |
642 | * p->d, because *p is declared 'const'... */ |
643 | while (1) { |
644 | while (ubits && !(udp[0] & 1)) { |
645 | BN_ULONG u0, u1, b0, b1, mask; |
646 | |
647 | u0 = udp[0]; |
648 | b0 = bdp[0]; |
649 | mask = (BN_ULONG)0 - (b0 & 1); |
650 | b0 ^= p->d[0] & mask; |
651 | for (i = 0; i < top - 1; i++) { |
652 | u1 = udp[i + 1]; |
653 | udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2; |
654 | u0 = u1; |
655 | b1 = bdp[i + 1] ^ (p->d[i + 1] & mask); |
656 | bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2; |
657 | b0 = b1; |
658 | } |
659 | udp[i] = u0 >> 1; |
660 | bdp[i] = b0 >> 1; |
661 | ubits--; |
662 | } |
663 | |
664 | if (ubits <= BN_BITS2) { |
665 | if (udp[0] == 0) /* poly was reducible */ |
666 | goto err; |
667 | if (udp[0] == 1) |
668 | break; |
669 | } |
670 | |
671 | if (ubits < vbits) { |
672 | i = ubits; |
673 | ubits = vbits; |
674 | vbits = i; |
675 | tmp = u; |
676 | u = v; |
677 | v = tmp; |
678 | tmp = b; |
679 | b = c; |
680 | c = tmp; |
681 | udp = vdp; |
682 | vdp = v->d; |
683 | bdp = cdp; |
684 | cdp = c->d; |
685 | } |
686 | for (i = 0; i < top; i++) { |
687 | udp[i] ^= vdp[i]; |
688 | bdp[i] ^= cdp[i]; |
689 | } |
690 | if (ubits == vbits) { |
691 | BN_ULONG ul; |
692 | int utop = (ubits - 1) / BN_BITS2; |
693 | |
694 | while ((ul = udp[utop]) == 0 && utop) |
695 | utop--; |
696 | ubits = utop * BN_BITS2 + BN_num_bits_word(ul); |
697 | } |
698 | } |
699 | bn_correct_top(b); |
700 | } |
701 | # endif |
702 | |
703 | if (!BN_copy(r, b)) |
704 | goto err; |
705 | bn_check_top(r); |
706 | ret = 1; |
707 | |
708 | err: |
709 | # ifdef BN_DEBUG /* BN_CTX_end would complain about the |
710 | * expanded form */ |
711 | bn_correct_top(c); |
712 | bn_correct_top(u); |
713 | bn_correct_top(v); |
714 | # endif |
715 | BN_CTX_end(ctx); |
716 | return ret; |
717 | } |
718 | |
719 | /*- |
720 | * Wrapper for BN_GF2m_mod_inv_vartime that blinds the input before calling. |
721 | * This is not constant time. |
722 | * But it does eliminate first order deduction on the input. |
723 | */ |
724 | int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) |
725 | { |
726 | BIGNUM *b = NULL; |
727 | int ret = 0; |
728 | |
729 | BN_CTX_start(ctx); |
730 | if ((b = BN_CTX_get(ctx)) == NULL) |
731 | goto err; |
732 | |
733 | /* generate blinding value */ |
734 | do { |
735 | if (!BN_priv_rand_ex(b, BN_num_bits(p) - 1, |
736 | BN_RAND_TOP_ANY, BN_RAND_BOTTOM_ANY, ctx)) |
737 | goto err; |
738 | } while (BN_is_zero(b)); |
739 | |
740 | /* r := a * b */ |
741 | if (!BN_GF2m_mod_mul(r, a, b, p, ctx)) |
742 | goto err; |
743 | |
744 | /* r := 1/(a * b) */ |
745 | if (!BN_GF2m_mod_inv_vartime(r, r, p, ctx)) |
746 | goto err; |
747 | |
748 | /* r := b/(a * b) = 1/a */ |
749 | if (!BN_GF2m_mod_mul(r, r, b, p, ctx)) |
750 | goto err; |
751 | |
752 | ret = 1; |
753 | |
754 | err: |
755 | BN_CTX_end(ctx); |
756 | return ret; |
757 | } |
758 | |
759 | /* |
760 | * Invert xx, reduce modulo p, and store the result in r. r could be xx. |
761 | * This function calls down to the BN_GF2m_mod_inv implementation; this |
762 | * wrapper function is only provided for convenience; for best performance, |
763 | * use the BN_GF2m_mod_inv function. |
764 | */ |
765 | int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], |
766 | BN_CTX *ctx) |
767 | { |
768 | BIGNUM *field; |
769 | int ret = 0; |
770 | |
771 | bn_check_top(xx); |
772 | BN_CTX_start(ctx); |
773 | if ((field = BN_CTX_get(ctx)) == NULL) |
774 | goto err; |
775 | if (!BN_GF2m_arr2poly(p, field)) |
776 | goto err; |
777 | |
778 | ret = BN_GF2m_mod_inv(r, xx, field, ctx); |
779 | bn_check_top(r); |
780 | |
781 | err: |
782 | BN_CTX_end(ctx); |
783 | return ret; |
784 | } |
785 | |
786 | /* |
787 | * Divide y by x, reduce modulo p, and store the result in r. r could be x |
788 | * or y, x could equal y. |
789 | */ |
790 | int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, |
791 | const BIGNUM *p, BN_CTX *ctx) |
792 | { |
793 | BIGNUM *xinv = NULL; |
794 | int ret = 0; |
795 | |
796 | bn_check_top(y); |
797 | bn_check_top(x); |
798 | bn_check_top(p); |
799 | |
800 | BN_CTX_start(ctx); |
801 | xinv = BN_CTX_get(ctx); |
802 | if (xinv == NULL) |
803 | goto err; |
804 | |
805 | if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) |
806 | goto err; |
807 | if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) |
808 | goto err; |
809 | bn_check_top(r); |
810 | ret = 1; |
811 | |
812 | err: |
813 | BN_CTX_end(ctx); |
814 | return ret; |
815 | } |
816 | |
817 | /* |
818 | * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx |
819 | * * or yy, xx could equal yy. This function calls down to the |
820 | * BN_GF2m_mod_div implementation; this wrapper function is only provided for |
821 | * convenience; for best performance, use the BN_GF2m_mod_div function. |
822 | */ |
823 | int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, |
824 | const int p[], BN_CTX *ctx) |
825 | { |
826 | BIGNUM *field; |
827 | int ret = 0; |
828 | |
829 | bn_check_top(yy); |
830 | bn_check_top(xx); |
831 | |
832 | BN_CTX_start(ctx); |
833 | if ((field = BN_CTX_get(ctx)) == NULL) |
834 | goto err; |
835 | if (!BN_GF2m_arr2poly(p, field)) |
836 | goto err; |
837 | |
838 | ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); |
839 | bn_check_top(r); |
840 | |
841 | err: |
842 | BN_CTX_end(ctx); |
843 | return ret; |
844 | } |
845 | |
846 | /* |
847 | * Compute the bth power of a, reduce modulo p, and store the result in r. r |
848 | * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE |
849 | * P1363. |
850 | */ |
851 | int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, |
852 | const int p[], BN_CTX *ctx) |
853 | { |
854 | int ret = 0, i, n; |
855 | BIGNUM *u; |
856 | |
857 | bn_check_top(a); |
858 | bn_check_top(b); |
859 | |
860 | if (BN_is_zero(b)) |
861 | return BN_one(r); |
862 | |
863 | if (BN_abs_is_word(b, 1)) |
864 | return (BN_copy(r, a) != NULL); |
865 | |
866 | BN_CTX_start(ctx); |
867 | if ((u = BN_CTX_get(ctx)) == NULL) |
868 | goto err; |
869 | |
870 | if (!BN_GF2m_mod_arr(u, a, p)) |
871 | goto err; |
872 | |
873 | n = BN_num_bits(b) - 1; |
874 | for (i = n - 1; i >= 0; i--) { |
875 | if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) |
876 | goto err; |
877 | if (BN_is_bit_set(b, i)) { |
878 | if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) |
879 | goto err; |
880 | } |
881 | } |
882 | if (!BN_copy(r, u)) |
883 | goto err; |
884 | bn_check_top(r); |
885 | ret = 1; |
886 | err: |
887 | BN_CTX_end(ctx); |
888 | return ret; |
889 | } |
890 | |
891 | /* |
892 | * Compute the bth power of a, reduce modulo p, and store the result in r. r |
893 | * could be a. This function calls down to the BN_GF2m_mod_exp_arr |
894 | * implementation; this wrapper function is only provided for convenience; |
895 | * for best performance, use the BN_GF2m_mod_exp_arr function. |
896 | */ |
897 | int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, |
898 | const BIGNUM *p, BN_CTX *ctx) |
899 | { |
900 | int ret = 0; |
901 | const int max = BN_num_bits(p) + 1; |
902 | int *arr = NULL; |
903 | bn_check_top(a); |
904 | bn_check_top(b); |
905 | bn_check_top(p); |
906 | if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) |
907 | goto err; |
908 | ret = BN_GF2m_poly2arr(p, arr, max); |
909 | if (!ret || ret > max) { |
910 | BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH); |
911 | goto err; |
912 | } |
913 | ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); |
914 | bn_check_top(r); |
915 | err: |
916 | OPENSSL_free(arr); |
917 | return ret; |
918 | } |
919 | |
920 | /* |
921 | * Compute the square root of a, reduce modulo p, and store the result in r. |
922 | * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363. |
923 | */ |
924 | int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], |
925 | BN_CTX *ctx) |
926 | { |
927 | int ret = 0; |
928 | BIGNUM *u; |
929 | |
930 | bn_check_top(a); |
931 | |
932 | if (p[0] == 0) { |
933 | /* reduction mod 1 => return 0 */ |
934 | BN_zero(r); |
935 | return 1; |
936 | } |
937 | |
938 | BN_CTX_start(ctx); |
939 | if ((u = BN_CTX_get(ctx)) == NULL) |
940 | goto err; |
941 | |
942 | if (!BN_set_bit(u, p[0] - 1)) |
943 | goto err; |
944 | ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); |
945 | bn_check_top(r); |
946 | |
947 | err: |
948 | BN_CTX_end(ctx); |
949 | return ret; |
950 | } |
951 | |
952 | /* |
953 | * Compute the square root of a, reduce modulo p, and store the result in r. |
954 | * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr |
955 | * implementation; this wrapper function is only provided for convenience; |
956 | * for best performance, use the BN_GF2m_mod_sqrt_arr function. |
957 | */ |
958 | int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) |
959 | { |
960 | int ret = 0; |
961 | const int max = BN_num_bits(p) + 1; |
962 | int *arr = NULL; |
963 | bn_check_top(a); |
964 | bn_check_top(p); |
965 | if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) |
966 | goto err; |
967 | ret = BN_GF2m_poly2arr(p, arr, max); |
968 | if (!ret || ret > max) { |
969 | BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH); |
970 | goto err; |
971 | } |
972 | ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); |
973 | bn_check_top(r); |
974 | err: |
975 | OPENSSL_free(arr); |
976 | return ret; |
977 | } |
978 | |
979 | /* |
980 | * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns |
981 | * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363. |
982 | */ |
983 | int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], |
984 | BN_CTX *ctx) |
985 | { |
986 | int ret = 0, count = 0, j; |
987 | BIGNUM *a, *z, *rho, *w, *w2, *tmp; |
988 | |
989 | bn_check_top(a_); |
990 | |
991 | if (p[0] == 0) { |
992 | /* reduction mod 1 => return 0 */ |
993 | BN_zero(r); |
994 | return 1; |
995 | } |
996 | |
997 | BN_CTX_start(ctx); |
998 | a = BN_CTX_get(ctx); |
999 | z = BN_CTX_get(ctx); |
1000 | w = BN_CTX_get(ctx); |
1001 | if (w == NULL) |
1002 | goto err; |
1003 | |
1004 | if (!BN_GF2m_mod_arr(a, a_, p)) |
1005 | goto err; |
1006 | |
1007 | if (BN_is_zero(a)) { |
1008 | BN_zero(r); |
1009 | ret = 1; |
1010 | goto err; |
1011 | } |
1012 | |
1013 | if (p[0] & 0x1) { /* m is odd */ |
1014 | /* compute half-trace of a */ |
1015 | if (!BN_copy(z, a)) |
1016 | goto err; |
1017 | for (j = 1; j <= (p[0] - 1) / 2; j++) { |
1018 | if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) |
1019 | goto err; |
1020 | if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) |
1021 | goto err; |
1022 | if (!BN_GF2m_add(z, z, a)) |
1023 | goto err; |
1024 | } |
1025 | |
1026 | } else { /* m is even */ |
1027 | |
1028 | rho = BN_CTX_get(ctx); |
1029 | w2 = BN_CTX_get(ctx); |
1030 | tmp = BN_CTX_get(ctx); |
1031 | if (tmp == NULL) |
1032 | goto err; |
1033 | do { |
1034 | if (!BN_priv_rand_ex(rho, p[0], BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ANY, |
1035 | ctx)) |
1036 | goto err; |
1037 | if (!BN_GF2m_mod_arr(rho, rho, p)) |
1038 | goto err; |
1039 | BN_zero(z); |
1040 | if (!BN_copy(w, rho)) |
1041 | goto err; |
1042 | for (j = 1; j <= p[0] - 1; j++) { |
1043 | if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) |
1044 | goto err; |
1045 | if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) |
1046 | goto err; |
1047 | if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) |
1048 | goto err; |
1049 | if (!BN_GF2m_add(z, z, tmp)) |
1050 | goto err; |
1051 | if (!BN_GF2m_add(w, w2, rho)) |
1052 | goto err; |
1053 | } |
1054 | count++; |
1055 | } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); |
1056 | if (BN_is_zero(w)) { |
1057 | BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS); |
1058 | goto err; |
1059 | } |
1060 | } |
1061 | |
1062 | if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) |
1063 | goto err; |
1064 | if (!BN_GF2m_add(w, z, w)) |
1065 | goto err; |
1066 | if (BN_GF2m_cmp(w, a)) { |
1067 | BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); |
1068 | goto err; |
1069 | } |
1070 | |
1071 | if (!BN_copy(r, z)) |
1072 | goto err; |
1073 | bn_check_top(r); |
1074 | |
1075 | ret = 1; |
1076 | |
1077 | err: |
1078 | BN_CTX_end(ctx); |
1079 | return ret; |
1080 | } |
1081 | |
1082 | /* |
1083 | * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns |
1084 | * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr |
1085 | * implementation; this wrapper function is only provided for convenience; |
1086 | * for best performance, use the BN_GF2m_mod_solve_quad_arr function. |
1087 | */ |
1088 | int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, |
1089 | BN_CTX *ctx) |
1090 | { |
1091 | int ret = 0; |
1092 | const int max = BN_num_bits(p) + 1; |
1093 | int *arr = NULL; |
1094 | bn_check_top(a); |
1095 | bn_check_top(p); |
1096 | if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) |
1097 | goto err; |
1098 | ret = BN_GF2m_poly2arr(p, arr, max); |
1099 | if (!ret || ret > max) { |
1100 | BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH); |
1101 | goto err; |
1102 | } |
1103 | ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); |
1104 | bn_check_top(r); |
1105 | err: |
1106 | OPENSSL_free(arr); |
1107 | return ret; |
1108 | } |
1109 | |
1110 | /* |
1111 | * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i * |
1112 | * x^i) into an array of integers corresponding to the bits with non-zero |
1113 | * coefficient. Array is terminated with -1. Up to max elements of the array |
1114 | * will be filled. Return value is total number of array elements that would |
1115 | * be filled if array was large enough. |
1116 | */ |
1117 | int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) |
1118 | { |
1119 | int i, j, k = 0; |
1120 | BN_ULONG mask; |
1121 | |
1122 | if (BN_is_zero(a)) |
1123 | return 0; |
1124 | |
1125 | for (i = a->top - 1; i >= 0; i--) { |
1126 | if (!a->d[i]) |
1127 | /* skip word if a->d[i] == 0 */ |
1128 | continue; |
1129 | mask = BN_TBIT; |
1130 | for (j = BN_BITS2 - 1; j >= 0; j--) { |
1131 | if (a->d[i] & mask) { |
1132 | if (k < max) |
1133 | p[k] = BN_BITS2 * i + j; |
1134 | k++; |
1135 | } |
1136 | mask >>= 1; |
1137 | } |
1138 | } |
1139 | |
1140 | if (k < max) { |
1141 | p[k] = -1; |
1142 | k++; |
1143 | } |
1144 | |
1145 | return k; |
1146 | } |
1147 | |
1148 | /* |
1149 | * Convert the coefficient array representation of a polynomial to a |
1150 | * bit-string. The array must be terminated by -1. |
1151 | */ |
1152 | int BN_GF2m_arr2poly(const int p[], BIGNUM *a) |
1153 | { |
1154 | int i; |
1155 | |
1156 | bn_check_top(a); |
1157 | BN_zero(a); |
1158 | for (i = 0; p[i] != -1; i++) { |
1159 | if (BN_set_bit(a, p[i]) == 0) |
1160 | return 0; |
1161 | } |
1162 | bn_check_top(a); |
1163 | |
1164 | return 1; |
1165 | } |
1166 | |
1167 | #endif |
1168 | |