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1/* $OpenBSD: bn_gf2m.c,v 1.18 2015/02/10 09:50:12 miod Exp $ */
2/* ====================================================================
3 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
4 *
5 * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
6 * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
7 * to the OpenSSL project.
8 *
9 * The ECC Code is licensed pursuant to the OpenSSL open source
10 * license provided below.
11 *
12 * In addition, Sun covenants to all licensees who provide a reciprocal
13 * covenant with respect to their own patents if any, not to sue under
14 * current and future patent claims necessarily infringed by the making,
15 * using, practicing, selling, offering for sale and/or otherwise
16 * disposing of the ECC Code as delivered hereunder (or portions thereof),
17 * provided that such covenant shall not apply:
18 * 1) for code that a licensee deletes from the ECC Code;
19 * 2) separates from the ECC Code; or
20 * 3) for infringements caused by:
21 * i) the modification of the ECC Code or
22 * ii) the combination of the ECC Code with other software or
23 * devices where such combination causes the infringement.
24 *
25 * The software is originally written by Sheueling Chang Shantz and
26 * Douglas Stebila of Sun Microsystems Laboratories.
27 *
28 */
29
30/* NOTE: This file is licensed pursuant to the OpenSSL license below
31 * and may be modified; but after modifications, the above covenant
32 * may no longer apply! In such cases, the corresponding paragraph
33 * ["In addition, Sun covenants ... causes the infringement."] and
34 * this note can be edited out; but please keep the Sun copyright
35 * notice and attribution. */
36
37/* ====================================================================
38 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
39 *
40 * Redistribution and use in source and binary forms, with or without
41 * modification, are permitted provided that the following conditions
42 * are met:
43 *
44 * 1. Redistributions of source code must retain the above copyright
45 * notice, this list of conditions and the following disclaimer.
46 *
47 * 2. Redistributions in binary form must reproduce the above copyright
48 * notice, this list of conditions and the following disclaimer in
49 * the documentation and/or other materials provided with the
50 * distribution.
51 *
52 * 3. All advertising materials mentioning features or use of this
53 * software must display the following acknowledgment:
54 * "This product includes software developed by the OpenSSL Project
55 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
56 *
57 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
58 * endorse or promote products derived from this software without
59 * prior written permission. For written permission, please contact
60 * openssl-core@openssl.org.
61 *
62 * 5. Products derived from this software may not be called "OpenSSL"
63 * nor may "OpenSSL" appear in their names without prior written
64 * permission of the OpenSSL Project.
65 *
66 * 6. Redistributions of any form whatsoever must retain the following
67 * acknowledgment:
68 * "This product includes software developed by the OpenSSL Project
69 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
70 *
71 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
72 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
73 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
74 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
75 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
76 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
77 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
78 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
79 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
80 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
81 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
82 * OF THE POSSIBILITY OF SUCH DAMAGE.
83 * ====================================================================
84 *
85 * This product includes cryptographic software written by Eric Young
86 * (eay@cryptsoft.com). This product includes software written by Tim
87 * Hudson (tjh@cryptsoft.com).
88 *
89 */
90
91#include <limits.h>
92#include <stdio.h>
93
94#include <openssl/opensslconf.h>
95
96#include <openssl/err.h>
97
98#include "bn_lcl.h"
99
100#ifndef OPENSSL_NO_EC2M
101
102/* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
103#define MAX_ITERATIONS 50
104
105static const BN_ULONG SQR_tb[16] =
106 { 0, 1, 4, 5, 16, 17, 20, 21,
10764, 65, 68, 69, 80, 81, 84, 85 };
108/* Platform-specific macros to accelerate squaring. */
109#ifdef _LP64
110#define SQR1(w) \
111 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
112 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
113 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
114 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
115#define SQR0(w) \
116 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
117 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
118 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
119 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
120#else
121#define SQR1(w) \
122 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
123 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
124#define SQR0(w) \
125 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
126 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
127#endif
128
129#if !defined(OPENSSL_BN_ASM_GF2m)
130/* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
131 * result is a polynomial r with degree < 2 * BN_BITS - 1
132 * The caller MUST ensure that the variables have the right amount
133 * of space allocated.
134 */
135static void
136bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
137{
138#ifndef _LP64
139 BN_ULONG h, l, s;
140 BN_ULONG tab[8], top2b = a >> 30;
141 BN_ULONG a1, a2, a4;
142
143 a1 = a & (0x3FFFFFFF);
144 a2 = a1 << 1;
145 a4 = a2 << 1;
146
147 tab[0] = 0;
148 tab[1] = a1;
149 tab[2] = a2;
150 tab[3] = a1 ^ a2;
151 tab[4] = a4;
152 tab[5] = a1 ^ a4;
153 tab[6] = a2 ^ a4;
154 tab[7] = a1 ^ a2 ^ a4;
155
156 s = tab[b & 0x7];
157 l = s;
158 s = tab[b >> 3 & 0x7];
159 l ^= s << 3;
160 h = s >> 29;
161 s = tab[b >> 6 & 0x7];
162 l ^= s << 6;
163 h ^= s >> 26;
164 s = tab[b >> 9 & 0x7];
165 l ^= s << 9;
166 h ^= s >> 23;
167 s = tab[b >> 12 & 0x7];
168 l ^= s << 12;
169 h ^= s >> 20;
170 s = tab[b >> 15 & 0x7];
171 l ^= s << 15;
172 h ^= s >> 17;
173 s = tab[b >> 18 & 0x7];
174 l ^= s << 18;
175 h ^= s >> 14;
176 s = tab[b >> 21 & 0x7];
177 l ^= s << 21;
178 h ^= s >> 11;
179 s = tab[b >> 24 & 0x7];
180 l ^= s << 24;
181 h ^= s >> 8;
182 s = tab[b >> 27 & 0x7];
183 l ^= s << 27;
184 h ^= s >> 5;
185 s = tab[b >> 30];
186 l ^= s << 30;
187 h ^= s >> 2;
188
189 /* compensate for the top two bits of a */
190 if (top2b & 01) {
191 l ^= b << 30;
192 h ^= b >> 2;
193 }
194 if (top2b & 02) {
195 l ^= b << 31;
196 h ^= b >> 1;
197 }
198
199 *r1 = h;
200 *r0 = l;
201#else
202 BN_ULONG h, l, s;
203 BN_ULONG tab[16], top3b = a >> 61;
204 BN_ULONG a1, a2, a4, a8;
205
206 a1 = a & (0x1FFFFFFFFFFFFFFFULL);
207 a2 = a1 << 1;
208 a4 = a2 << 1;
209 a8 = a4 << 1;
210
211 tab[0] = 0;
212 tab[1] = a1;
213 tab[2] = a2;
214 tab[3] = a1 ^ a2;
215 tab[4] = a4;
216 tab[5] = a1 ^ a4;
217 tab[6] = a2 ^ a4;
218 tab[7] = a1 ^ a2 ^ a4;
219 tab[8] = a8;
220 tab[9] = a1 ^ a8;
221 tab[10] = a2 ^ a8;
222 tab[11] = a1 ^ a2 ^ a8;
223 tab[12] = a4 ^ a8;
224 tab[13] = a1 ^ a4 ^ a8;
225 tab[14] = a2 ^ a4 ^ a8;
226 tab[15] = a1 ^ a2 ^ a4 ^ a8;
227
228 s = tab[b & 0xF];
229 l = s;
230 s = tab[b >> 4 & 0xF];
231 l ^= s << 4;
232 h = s >> 60;
233 s = tab[b >> 8 & 0xF];
234 l ^= s << 8;
235 h ^= s >> 56;
236 s = tab[b >> 12 & 0xF];
237 l ^= s << 12;
238 h ^= s >> 52;
239 s = tab[b >> 16 & 0xF];
240 l ^= s << 16;
241 h ^= s >> 48;
242 s = tab[b >> 20 & 0xF];
243 l ^= s << 20;
244 h ^= s >> 44;
245 s = tab[b >> 24 & 0xF];
246 l ^= s << 24;
247 h ^= s >> 40;
248 s = tab[b >> 28 & 0xF];
249 l ^= s << 28;
250 h ^= s >> 36;
251 s = tab[b >> 32 & 0xF];
252 l ^= s << 32;
253 h ^= s >> 32;
254 s = tab[b >> 36 & 0xF];
255 l ^= s << 36;
256 h ^= s >> 28;
257 s = tab[b >> 40 & 0xF];
258 l ^= s << 40;
259 h ^= s >> 24;
260 s = tab[b >> 44 & 0xF];
261 l ^= s << 44;
262 h ^= s >> 20;
263 s = tab[b >> 48 & 0xF];
264 l ^= s << 48;
265 h ^= s >> 16;
266 s = tab[b >> 52 & 0xF];
267 l ^= s << 52;
268 h ^= s >> 12;
269 s = tab[b >> 56 & 0xF];
270 l ^= s << 56;
271 h ^= s >> 8;
272 s = tab[b >> 60];
273 l ^= s << 60;
274 h ^= s >> 4;
275
276 /* compensate for the top three bits of a */
277 if (top3b & 01) {
278 l ^= b << 61;
279 h ^= b >> 3;
280 }
281 if (top3b & 02) {
282 l ^= b << 62;
283 h ^= b >> 2;
284 }
285 if (top3b & 04) {
286 l ^= b << 63;
287 h ^= b >> 1;
288 }
289
290 *r1 = h;
291 *r0 = l;
292#endif
293}
294
295/* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
296 * result is a polynomial r with degree < 4 * BN_BITS2 - 1
297 * The caller MUST ensure that the variables have the right amount
298 * of space allocated.
299 */
300static void
301bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0,
302 const BN_ULONG b1, const BN_ULONG b0)
303{
304 BN_ULONG m1, m0;
305
306 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
307 bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1);
308 bn_GF2m_mul_1x1(r + 1, r, a0, b0);
309 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
310 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
311 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
312 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
313}
314#else
315void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1,
316 BN_ULONG b0);
317#endif
318
319/* Add polynomials a and b and store result in r; r could be a or b, a and b
320 * could be equal; r is the bitwise XOR of a and b.
321 */
322int
323BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
324{
325 int i;
326 const BIGNUM *at, *bt;
327
328 bn_check_top(a);
329 bn_check_top(b);
330
331 if (a->top < b->top) {
332 at = b;
333 bt = a;
334 } else {
335 at = a;
336 bt = b;
337 }
338
339 if (bn_wexpand(r, at->top) == NULL)
340 return 0;
341
342 for (i = 0; i < bt->top; i++) {
343 r->d[i] = at->d[i] ^ bt->d[i];
344 }
345 for (; i < at->top; i++) {
346 r->d[i] = at->d[i];
347 }
348
349 r->top = at->top;
350 bn_correct_top(r);
351
352 return 1;
353}
354
355
356/* Some functions allow for representation of the irreducible polynomials
357 * as an int[], say p. The irreducible f(t) is then of the form:
358 * t^p[0] + t^p[1] + ... + t^p[k]
359 * where m = p[0] > p[1] > ... > p[k] = 0.
360 */
361
362
363/* Performs modular reduction of a and store result in r. r could be a. */
364int
365BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
366{
367 int j, k;
368 int n, dN, d0, d1;
369 BN_ULONG zz, *z;
370
371 bn_check_top(a);
372
373 if (!p[0]) {
374 /* reduction mod 1 => return 0 */
375 BN_zero(r);
376 return 1;
377 }
378
379 /* Since the algorithm does reduction in the r value, if a != r, copy
380 * the contents of a into r so we can do reduction in r.
381 */
382 if (a != r) {
383 if (!bn_wexpand(r, a->top))
384 return 0;
385 for (j = 0; j < a->top; j++) {
386 r->d[j] = a->d[j];
387 }
388 r->top = a->top;
389 }
390 z = r->d;
391
392 /* start reduction */
393 dN = p[0] / BN_BITS2;
394 for (j = r->top - 1; j > dN; ) {
395 zz = z[j];
396 if (z[j] == 0) {
397 j--;
398 continue;
399 }
400 z[j] = 0;
401
402 for (k = 1; p[k] != 0; k++) {
403 /* reducing component t^p[k] */
404 n = p[0] - p[k];
405 d0 = n % BN_BITS2;
406 d1 = BN_BITS2 - d0;
407 n /= BN_BITS2;
408 z[j - n] ^= (zz >> d0);
409 if (d0)
410 z[j - n - 1] ^= (zz << d1);
411 }
412
413 /* reducing component t^0 */
414 n = dN;
415 d0 = p[0] % BN_BITS2;
416 d1 = BN_BITS2 - d0;
417 z[j - n] ^= (zz >> d0);
418 if (d0)
419 z[j - n - 1] ^= (zz << d1);
420 }
421
422 /* final round of reduction */
423 while (j == dN) {
424
425 d0 = p[0] % BN_BITS2;
426 zz = z[dN] >> d0;
427 if (zz == 0)
428 break;
429 d1 = BN_BITS2 - d0;
430
431 /* clear up the top d1 bits */
432 if (d0)
433 z[dN] = (z[dN] << d1) >> d1;
434 else
435 z[dN] = 0;
436 z[0] ^= zz; /* reduction t^0 component */
437
438 for (k = 1; p[k] != 0; k++) {
439 BN_ULONG tmp_ulong;
440
441 /* reducing component t^p[k]*/
442 n = p[k] / BN_BITS2;
443 d0 = p[k] % BN_BITS2;
444 d1 = BN_BITS2 - d0;
445 z[n] ^= (zz << d0);
446 tmp_ulong = zz >> d1;
447 if (d0 && tmp_ulong)
448 z[n + 1] ^= tmp_ulong;
449 }
450
451
452 }
453
454 bn_correct_top(r);
455 return 1;
456}
457
458/* Performs modular reduction of a by p and store result in r. r could be a.
459 *
460 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
461 * function is only provided for convenience; for best performance, use the
462 * BN_GF2m_mod_arr function.
463 */
464int
465BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
466{
467 int ret = 0;
468 int arr[6];
469
470 bn_check_top(a);
471 bn_check_top(p);
472 ret = BN_GF2m_poly2arr(p, arr, sizeof(arr) / sizeof(arr[0]));
473 if (!ret || ret > (int)(sizeof(arr) / sizeof(arr[0]))) {
474 BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH);
475 return 0;
476 }
477 ret = BN_GF2m_mod_arr(r, a, arr);
478 bn_check_top(r);
479 return ret;
480}
481
482
483/* Compute the product of two polynomials a and b, reduce modulo p, and store
484 * the result in r. r could be a or b; a could be b.
485 */
486int
487BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[],
488 BN_CTX *ctx)
489{
490 int zlen, i, j, k, ret = 0;
491 BIGNUM *s;
492 BN_ULONG x1, x0, y1, y0, zz[4];
493
494 bn_check_top(a);
495 bn_check_top(b);
496
497 if (a == b) {
498 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
499 }
500
501 BN_CTX_start(ctx);
502 if ((s = BN_CTX_get(ctx)) == NULL)
503 goto err;
504
505 zlen = a->top + b->top + 4;
506 if (!bn_wexpand(s, zlen))
507 goto err;
508 s->top = zlen;
509
510 for (i = 0; i < zlen; i++)
511 s->d[i] = 0;
512
513 for (j = 0; j < b->top; j += 2) {
514 y0 = b->d[j];
515 y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1];
516 for (i = 0; i < a->top; i += 2) {
517 x0 = a->d[i];
518 x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1];
519 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
520 for (k = 0; k < 4; k++)
521 s->d[i + j + k] ^= zz[k];
522 }
523 }
524
525 bn_correct_top(s);
526 if (BN_GF2m_mod_arr(r, s, p))
527 ret = 1;
528 bn_check_top(r);
529
530err:
531 BN_CTX_end(ctx);
532 return ret;
533}
534
535/* Compute the product of two polynomials a and b, reduce modulo p, and store
536 * the result in r. r could be a or b; a could equal b.
537 *
538 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
539 * function is only provided for convenience; for best performance, use the
540 * BN_GF2m_mod_mul_arr function.
541 */
542int
543BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p,
544 BN_CTX *ctx)
545{
546 int ret = 0;
547 const int max = BN_num_bits(p) + 1;
548 int *arr = NULL;
549
550 bn_check_top(a);
551 bn_check_top(b);
552 bn_check_top(p);
553 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
554 goto err;
555 ret = BN_GF2m_poly2arr(p, arr, max);
556 if (!ret || ret > max) {
557 BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH);
558 goto err;
559 }
560 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
561 bn_check_top(r);
562
563err:
564 free(arr);
565 return ret;
566}
567
568
569/* Square a, reduce the result mod p, and store it in a. r could be a. */
570int
571BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
572{
573 int i, ret = 0;
574 BIGNUM *s;
575
576 bn_check_top(a);
577 BN_CTX_start(ctx);
578 if ((s = BN_CTX_get(ctx)) == NULL)
579 return 0;
580 if (!bn_wexpand(s, 2 * a->top))
581 goto err;
582
583 for (i = a->top - 1; i >= 0; i--) {
584 s->d[2 * i + 1] = SQR1(a->d[i]);
585 s->d[2 * i] = SQR0(a->d[i]);
586 }
587
588 s->top = 2 * a->top;
589 bn_correct_top(s);
590 if (!BN_GF2m_mod_arr(r, s, p))
591 goto err;
592 bn_check_top(r);
593 ret = 1;
594
595err:
596 BN_CTX_end(ctx);
597 return ret;
598}
599
600/* Square a, reduce the result mod p, and store it in a. r could be a.
601 *
602 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
603 * function is only provided for convenience; for best performance, use the
604 * BN_GF2m_mod_sqr_arr function.
605 */
606int
607BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
608{
609 int ret = 0;
610 const int max = BN_num_bits(p) + 1;
611 int *arr = NULL;
612
613 bn_check_top(a);
614 bn_check_top(p);
615 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
616 goto err;
617 ret = BN_GF2m_poly2arr(p, arr, max);
618 if (!ret || ret > max) {
619 BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH);
620 goto err;
621 }
622 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
623 bn_check_top(r);
624
625err:
626 free(arr);
627 return ret;
628}
629
630
631/* Invert a, reduce modulo p, and store the result in r. r could be a.
632 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
633 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
634 * of Elliptic Curve Cryptography Over Binary Fields".
635 */
636int
637BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
638{
639 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp;
640 int ret = 0;
641
642 bn_check_top(a);
643 bn_check_top(p);
644
645 BN_CTX_start(ctx);
646
647 if ((b = BN_CTX_get(ctx)) == NULL)
648 goto err;
649 if ((c = BN_CTX_get(ctx)) == NULL)
650 goto err;
651 if ((u = BN_CTX_get(ctx)) == NULL)
652 goto err;
653 if ((v = BN_CTX_get(ctx)) == NULL)
654 goto err;
655
656 if (!BN_GF2m_mod(u, a, p))
657 goto err;
658 if (BN_is_zero(u))
659 goto err;
660
661 if (!BN_copy(v, p))
662 goto err;
663#if 0
664 if (!BN_one(b))
665 goto err;
666
667 while (1) {
668 while (!BN_is_odd(u)) {
669 if (BN_is_zero(u))
670 goto err;
671 if (!BN_rshift1(u, u))
672 goto err;
673 if (BN_is_odd(b)) {
674 if (!BN_GF2m_add(b, b, p))
675 goto err;
676 }
677 if (!BN_rshift1(b, b))
678 goto err;
679 }
680
681 if (BN_abs_is_word(u, 1))
682 break;
683
684 if (BN_num_bits(u) < BN_num_bits(v)) {
685 tmp = u;
686 u = v;
687 v = tmp;
688 tmp = b;
689 b = c;
690 c = tmp;
691 }
692
693 if (!BN_GF2m_add(u, u, v))
694 goto err;
695 if (!BN_GF2m_add(b, b, c))
696 goto err;
697 }
698#else
699 {
700 int i, ubits = BN_num_bits(u),
701 vbits = BN_num_bits(v), /* v is copy of p */
702 top = p->top;
703 BN_ULONG *udp, *bdp, *vdp, *cdp;
704
705 bn_wexpand(u, top);
706 udp = u->d;
707 for (i = u->top; i < top; i++)
708 udp[i] = 0;
709 u->top = top;
710 bn_wexpand(b, top);
711 bdp = b->d;
712 bdp[0] = 1;
713 for (i = 1; i < top; i++)
714 bdp[i] = 0;
715 b->top = top;
716 bn_wexpand(c, top);
717 cdp = c->d;
718 for (i = 0; i < top; i++)
719 cdp[i] = 0;
720 c->top = top;
721 vdp = v->d; /* It pays off to "cache" *->d pointers, because
722 * it allows optimizer to be more aggressive.
723 * But we don't have to "cache" p->d, because *p
724 * is declared 'const'... */
725 while (1) {
726 while (ubits && !(udp[0]&1)) {
727 BN_ULONG u0, u1, b0, b1, mask;
728
729 u0 = udp[0];
730 b0 = bdp[0];
731 mask = (BN_ULONG)0 - (b0 & 1);
732 b0 ^= p->d[0] & mask;
733 for (i = 0; i < top - 1; i++) {
734 u1 = udp[i + 1];
735 udp[i] = ((u0 >> 1) |
736 (u1 << (BN_BITS2 - 1))) & BN_MASK2;
737 u0 = u1;
738 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask);
739 bdp[i] = ((b0 >> 1) |
740 (b1 << (BN_BITS2 - 1))) & BN_MASK2;
741 b0 = b1;
742 }
743 udp[i] = u0 >> 1;
744 bdp[i] = b0 >> 1;
745 ubits--;
746 }
747
748 if (ubits <= BN_BITS2 && udp[0] == 1)
749 break;
750
751 if (ubits < vbits) {
752 i = ubits;
753 ubits = vbits;
754 vbits = i;
755 tmp = u;
756 u = v;
757 v = tmp;
758 tmp = b;
759 b = c;
760 c = tmp;
761 udp = vdp;
762 vdp = v->d;
763 bdp = cdp;
764 cdp = c->d;
765 }
766 for (i = 0; i < top; i++) {
767 udp[i] ^= vdp[i];
768 bdp[i] ^= cdp[i];
769 }
770 if (ubits == vbits) {
771 BN_ULONG ul;
772 int utop = (ubits - 1) / BN_BITS2;
773
774 while ((ul = udp[utop]) == 0 && utop)
775 utop--;
776 ubits = utop*BN_BITS2 + BN_num_bits_word(ul);
777 }
778 }
779 bn_correct_top(b);
780 }
781#endif
782
783 if (!BN_copy(r, b))
784 goto err;
785 bn_check_top(r);
786 ret = 1;
787
788err:
789#ifdef BN_DEBUG /* BN_CTX_end would complain about the expanded form */
790 bn_correct_top(c);
791 bn_correct_top(u);
792 bn_correct_top(v);
793#endif
794 BN_CTX_end(ctx);
795 return ret;
796}
797
798/* Invert xx, reduce modulo p, and store the result in r. r could be xx.
799 *
800 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
801 * function is only provided for convenience; for best performance, use the
802 * BN_GF2m_mod_inv function.
803 */
804int
805BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx)
806{
807 BIGNUM *field;
808 int ret = 0;
809
810 bn_check_top(xx);
811 BN_CTX_start(ctx);
812 if ((field = BN_CTX_get(ctx)) == NULL)
813 goto err;
814 if (!BN_GF2m_arr2poly(p, field))
815 goto err;
816
817 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
818 bn_check_top(r);
819
820err:
821 BN_CTX_end(ctx);
822 return ret;
823}
824
825
826#ifndef OPENSSL_SUN_GF2M_DIV
827/* Divide y by x, reduce modulo p, and store the result in r. r could be x
828 * or y, x could equal y.
829 */
830int
831BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p,
832 BN_CTX *ctx)
833{
834 BIGNUM *xinv = NULL;
835 int ret = 0;
836
837 bn_check_top(y);
838 bn_check_top(x);
839 bn_check_top(p);
840
841 BN_CTX_start(ctx);
842 if ((xinv = BN_CTX_get(ctx)) == NULL)
843 goto err;
844
845 if (!BN_GF2m_mod_inv(xinv, x, p, ctx))
846 goto err;
847 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx))
848 goto err;
849 bn_check_top(r);
850 ret = 1;
851
852err:
853 BN_CTX_end(ctx);
854 return ret;
855}
856#else
857/* Divide y by x, reduce modulo p, and store the result in r. r could be x
858 * or y, x could equal y.
859 * Uses algorithm Modular_Division_GF(2^m) from
860 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
861 * the Great Divide".
862 */
863int
864BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p,
865 BN_CTX *ctx)
866{
867 BIGNUM *a, *b, *u, *v;
868 int ret = 0;
869
870 bn_check_top(y);
871 bn_check_top(x);
872 bn_check_top(p);
873
874 BN_CTX_start(ctx);
875
876 if ((a = BN_CTX_get(ctx)) == NULL)
877 goto err;
878 if ((b = BN_CTX_get(ctx)) == NULL)
879 goto err;
880 if ((u = BN_CTX_get(ctx)) == NULL)
881 goto err;
882 if ((v = BN_CTX_get(ctx)) == NULL)
883 goto err;
884
885 /* reduce x and y mod p */
886 if (!BN_GF2m_mod(u, y, p))
887 goto err;
888 if (!BN_GF2m_mod(a, x, p))
889 goto err;
890 if (!BN_copy(b, p))
891 goto err;
892
893 while (!BN_is_odd(a)) {
894 if (!BN_rshift1(a, a))
895 goto err;
896 if (BN_is_odd(u))
897 if (!BN_GF2m_add(u, u, p))
898 goto err;
899 if (!BN_rshift1(u, u))
900 goto err;
901 }
902
903 do {
904 if (BN_GF2m_cmp(b, a) > 0) {
905 if (!BN_GF2m_add(b, b, a))
906 goto err;
907 if (!BN_GF2m_add(v, v, u))
908 goto err;
909 do {
910 if (!BN_rshift1(b, b))
911 goto err;
912 if (BN_is_odd(v))
913 if (!BN_GF2m_add(v, v, p))
914 goto err;
915 if (!BN_rshift1(v, v))
916 goto err;
917 } while (!BN_is_odd(b));
918 } else if (BN_abs_is_word(a, 1))
919 break;
920 else {
921 if (!BN_GF2m_add(a, a, b))
922 goto err;
923 if (!BN_GF2m_add(u, u, v))
924 goto err;
925 do {
926 if (!BN_rshift1(a, a))
927 goto err;
928 if (BN_is_odd(u))
929 if (!BN_GF2m_add(u, u, p))
930 goto err;
931 if (!BN_rshift1(u, u))
932 goto err;
933 } while (!BN_is_odd(a));
934 }
935 } while (1);
936
937 if (!BN_copy(r, u))
938 goto err;
939 bn_check_top(r);
940 ret = 1;
941
942err:
943 BN_CTX_end(ctx);
944 return ret;
945}
946#endif
947
948/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
949 * or yy, xx could equal yy.
950 *
951 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
952 * function is only provided for convenience; for best performance, use the
953 * BN_GF2m_mod_div function.
954 */
955int
956BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx,
957 const int p[], BN_CTX *ctx)
958{
959 BIGNUM *field;
960 int ret = 0;
961
962 bn_check_top(yy);
963 bn_check_top(xx);
964
965 BN_CTX_start(ctx);
966 if ((field = BN_CTX_get(ctx)) == NULL)
967 goto err;
968 if (!BN_GF2m_arr2poly(p, field))
969 goto err;
970
971 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
972 bn_check_top(r);
973
974err:
975 BN_CTX_end(ctx);
976 return ret;
977}
978
979
980/* Compute the bth power of a, reduce modulo p, and store
981 * the result in r. r could be a.
982 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
983 */
984int
985BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[],
986 BN_CTX *ctx)
987{
988 int ret = 0, i, n;
989 BIGNUM *u;
990
991 bn_check_top(a);
992 bn_check_top(b);
993
994 if (BN_is_zero(b))
995 return (BN_one(r));
996
997 if (BN_abs_is_word(b, 1))
998 return (BN_copy(r, a) != NULL);
999
1000 BN_CTX_start(ctx);
1001 if ((u = BN_CTX_get(ctx)) == NULL)
1002 goto err;
1003
1004 if (!BN_GF2m_mod_arr(u, a, p))
1005 goto err;
1006
1007 n = BN_num_bits(b) - 1;
1008 for (i = n - 1; i >= 0; i--) {
1009 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx))
1010 goto err;
1011 if (BN_is_bit_set(b, i)) {
1012 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx))
1013 goto err;
1014 }
1015 }
1016 if (!BN_copy(r, u))
1017 goto err;
1018 bn_check_top(r);
1019 ret = 1;
1020
1021err:
1022 BN_CTX_end(ctx);
1023 return ret;
1024}
1025
1026/* Compute the bth power of a, reduce modulo p, and store
1027 * the result in r. r could be a.
1028 *
1029 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
1030 * function is only provided for convenience; for best performance, use the
1031 * BN_GF2m_mod_exp_arr function.
1032 */
1033int
1034BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p,
1035 BN_CTX *ctx)
1036{
1037 int ret = 0;
1038 const int max = BN_num_bits(p) + 1;
1039 int *arr = NULL;
1040
1041 bn_check_top(a);
1042 bn_check_top(b);
1043 bn_check_top(p);
1044 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
1045 goto err;
1046 ret = BN_GF2m_poly2arr(p, arr, max);
1047 if (!ret || ret > max) {
1048 BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH);
1049 goto err;
1050 }
1051 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
1052 bn_check_top(r);
1053
1054err:
1055 free(arr);
1056 return ret;
1057}
1058
1059/* Compute the square root of a, reduce modulo p, and store
1060 * the result in r. r could be a.
1061 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
1062 */
1063int
1064BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
1065{
1066 int ret = 0;
1067 BIGNUM *u;
1068
1069 bn_check_top(a);
1070
1071 if (!p[0]) {
1072 /* reduction mod 1 => return 0 */
1073 BN_zero(r);
1074 return 1;
1075 }
1076
1077 BN_CTX_start(ctx);
1078 if ((u = BN_CTX_get(ctx)) == NULL)
1079 goto err;
1080
1081 if (!BN_set_bit(u, p[0] - 1))
1082 goto err;
1083 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
1084 bn_check_top(r);
1085
1086err:
1087 BN_CTX_end(ctx);
1088 return ret;
1089}
1090
1091/* Compute the square root of a, reduce modulo p, and store
1092 * the result in r. r could be a.
1093 *
1094 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
1095 * function is only provided for convenience; for best performance, use the
1096 * BN_GF2m_mod_sqrt_arr function.
1097 */
1098int
1099BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1100{
1101 int ret = 0;
1102 const int max = BN_num_bits(p) + 1;
1103 int *arr = NULL;
1104 bn_check_top(a);
1105 bn_check_top(p);
1106 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
1107 goto err;
1108 ret = BN_GF2m_poly2arr(p, arr, max);
1109 if (!ret || ret > max) {
1110 BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH);
1111 goto err;
1112 }
1113 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
1114 bn_check_top(r);
1115
1116err:
1117 free(arr);
1118 return ret;
1119}
1120
1121/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
1122 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
1123 */
1124int
1125BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[],
1126 BN_CTX *ctx)
1127{
1128 int ret = 0, count = 0, j;
1129 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
1130
1131 bn_check_top(a_);
1132
1133 if (!p[0]) {
1134 /* reduction mod 1 => return 0 */
1135 BN_zero(r);
1136 return 1;
1137 }
1138
1139 BN_CTX_start(ctx);
1140 if ((a = BN_CTX_get(ctx)) == NULL)
1141 goto err;
1142 if ((z = BN_CTX_get(ctx)) == NULL)
1143 goto err;
1144 if ((w = BN_CTX_get(ctx)) == NULL)
1145 goto err;
1146
1147 if (!BN_GF2m_mod_arr(a, a_, p))
1148 goto err;
1149
1150 if (BN_is_zero(a)) {
1151 BN_zero(r);
1152 ret = 1;
1153 goto err;
1154 }
1155
1156 if (p[0] & 0x1) /* m is odd */
1157 {
1158 /* compute half-trace of a */
1159 if (!BN_copy(z, a))
1160 goto err;
1161 for (j = 1; j <= (p[0] - 1) / 2; j++) {
1162 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1163 goto err;
1164 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1165 goto err;
1166 if (!BN_GF2m_add(z, z, a))
1167 goto err;
1168 }
1169
1170 }
1171 else /* m is even */
1172 {
1173 if ((rho = BN_CTX_get(ctx)) == NULL)
1174 goto err;
1175 if ((w2 = BN_CTX_get(ctx)) == NULL)
1176 goto err;
1177 if ((tmp = BN_CTX_get(ctx)) == NULL)
1178 goto err;
1179 do {
1180 if (!BN_rand(rho, p[0], 0, 0))
1181 goto err;
1182 if (!BN_GF2m_mod_arr(rho, rho, p))
1183 goto err;
1184 BN_zero(z);
1185 if (!BN_copy(w, rho))
1186 goto err;
1187 for (j = 1; j <= p[0] - 1; j++) {
1188 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1189 goto err;
1190 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx))
1191 goto err;
1192 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx))
1193 goto err;
1194 if (!BN_GF2m_add(z, z, tmp))
1195 goto err;
1196 if (!BN_GF2m_add(w, w2, rho))
1197 goto err;
1198 }
1199 count++;
1200 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
1201 if (BN_is_zero(w)) {
1202 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,
1203 BN_R_TOO_MANY_ITERATIONS);
1204 goto err;
1205 }
1206 }
1207
1208 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx))
1209 goto err;
1210 if (!BN_GF2m_add(w, z, w))
1211 goto err;
1212 if (BN_GF2m_cmp(w, a)) {
1213 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
1214 goto err;
1215 }
1216
1217 if (!BN_copy(r, z))
1218 goto err;
1219 bn_check_top(r);
1220
1221 ret = 1;
1222
1223err:
1224 BN_CTX_end(ctx);
1225 return ret;
1226}
1227
1228/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
1229 *
1230 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
1231 * function is only provided for convenience; for best performance, use the
1232 * BN_GF2m_mod_solve_quad_arr function.
1233 */
1234int
1235BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1236{
1237 int ret = 0;
1238 const int max = BN_num_bits(p) + 1;
1239 int *arr = NULL;
1240
1241 bn_check_top(a);
1242 bn_check_top(p);
1243 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
1244 goto err;
1245 ret = BN_GF2m_poly2arr(p, arr, max);
1246 if (!ret || ret > max) {
1247 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH);
1248 goto err;
1249 }
1250 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1251 bn_check_top(r);
1252
1253err:
1254 free(arr);
1255 return ret;
1256}
1257
1258/* Convert the bit-string representation of a polynomial
1259 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding
1260 * to the bits with non-zero coefficient. Array is terminated with -1.
1261 * Up to max elements of the array will be filled. Return value is total
1262 * number of array elements that would be filled if array was large enough.
1263 */
1264int
1265BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
1266{
1267 int i, j, k = 0;
1268 BN_ULONG mask;
1269
1270 if (BN_is_zero(a))
1271 return 0;
1272
1273 for (i = a->top - 1; i >= 0; i--) {
1274 if (!a->d[i])
1275 /* skip word if a->d[i] == 0 */
1276 continue;
1277 mask = BN_TBIT;
1278 for (j = BN_BITS2 - 1; j >= 0; j--) {
1279 if (a->d[i] & mask) {
1280 if (k < max)
1281 p[k] = BN_BITS2 * i + j;
1282 k++;
1283 }
1284 mask >>= 1;
1285 }
1286 }
1287
1288 if (k < max) {
1289 p[k] = -1;
1290 k++;
1291 }
1292
1293 return k;
1294}
1295
1296/* Convert the coefficient array representation of a polynomial to a
1297 * bit-string. The array must be terminated by -1.
1298 */
1299int
1300BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1301{
1302 int i;
1303
1304 bn_check_top(a);
1305 BN_zero(a);
1306 for (i = 0; p[i] != -1; i++) {
1307 if (BN_set_bit(a, p[i]) == 0)
1308 return 0;
1309 }
1310 bn_check_top(a);
1311
1312 return 1;
1313}
1314
1315#endif
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-08-19 07:06:26 +0000