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1/* $OpenBSD: bn_gf2m.c,v 1.32 2023/03/27 10:25:02 tb 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_local.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
329 if (a->top < b->top) {
330 at = b;
331 bt = a;
332 } else {
333 at = a;
334 bt = b;
335 }
336
337 if (!bn_wexpand(r, at->top))
338 return 0;
339
340 for (i = 0; i < bt->top; i++) {
341 r->d[i] = at->d[i] ^ bt->d[i];
342 }
343 for (; i < at->top; i++) {
344 r->d[i] = at->d[i];
345 }
346
347 r->top = at->top;
348 bn_correct_top(r);
349
350 return 1;
351}
352
353
354/* Some functions allow for representation of the irreducible polynomials
355 * as an int[], say p. The irreducible f(t) is then of the form:
356 * t^p[0] + t^p[1] + ... + t^p[k]
357 * where m = p[0] > p[1] > ... > p[k] = 0.
358 */
359
360
361/* Performs modular reduction of a and store result in r. r could be a. */
362int
363BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
364{
365 int j, k;
366 int n, dN, d0, d1;
367 BN_ULONG zz, *z;
368
369
370 if (!p[0]) {
371 /* reduction mod 1 => return 0 */
372 BN_zero(r);
373 return 1;
374 }
375
376 /* Since the algorithm does reduction in the r value, if a != r, copy
377 * the contents of a into r so we can do reduction in r.
378 */
379 if (a != r) {
380 if (!bn_wexpand(r, a->top))
381 return 0;
382 for (j = 0; j < a->top; j++) {
383 r->d[j] = a->d[j];
384 }
385 r->top = a->top;
386 }
387 z = r->d;
388
389 /* start reduction */
390 dN = p[0] / BN_BITS2;
391 for (j = r->top - 1; j > dN; ) {
392 zz = z[j];
393 if (z[j] == 0) {
394 j--;
395 continue;
396 }
397 z[j] = 0;
398
399 for (k = 1; p[k] != 0; k++) {
400 /* reducing component t^p[k] */
401 n = p[0] - p[k];
402 d0 = n % BN_BITS2;
403 d1 = BN_BITS2 - d0;
404 n /= BN_BITS2;
405 z[j - n] ^= (zz >> d0);
406 if (d0)
407 z[j - n - 1] ^= (zz << d1);
408 }
409
410 /* reducing component t^0 */
411 n = dN;
412 d0 = p[0] % BN_BITS2;
413 d1 = BN_BITS2 - d0;
414 z[j - n] ^= (zz >> d0);
415 if (d0)
416 z[j - n - 1] ^= (zz << d1);
417 }
418
419 /* final round of reduction */
420 while (j == dN) {
421
422 d0 = p[0] % BN_BITS2;
423 zz = z[dN] >> d0;
424 if (zz == 0)
425 break;
426 d1 = BN_BITS2 - d0;
427
428 /* clear up the top d1 bits */
429 if (d0)
430 z[dN] = (z[dN] << d1) >> d1;
431 else
432 z[dN] = 0;
433 z[0] ^= zz; /* reduction t^0 component */
434
435 for (k = 1; p[k] != 0; k++) {
436 BN_ULONG tmp_ulong;
437
438 /* reducing component t^p[k]*/
439 n = p[k] / BN_BITS2;
440 d0 = p[k] % BN_BITS2;
441 d1 = BN_BITS2 - d0;
442 z[n] ^= (zz << d0);
443 if (d0 && (tmp_ulong = zz >> d1))
444 z[n + 1] ^= tmp_ulong;
445 }
446
447
448 }
449
450 bn_correct_top(r);
451 return 1;
452}
453
454/* Performs modular reduction of a by p and store result in r. r could be a.
455 *
456 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
457 * function is only provided for convenience; for best performance, use the
458 * BN_GF2m_mod_arr function.
459 */
460int
461BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
462{
463 int ret = 0;
464 const int max = BN_num_bits(p) + 1;
465 int *arr = NULL;
466
467 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
468 goto err;
469 ret = BN_GF2m_poly2arr(p, arr, max);
470 if (!ret || ret > max) {
471 BNerror(BN_R_INVALID_LENGTH);
472 goto err;
473 }
474 ret = BN_GF2m_mod_arr(r, a, arr);
475
476 err:
477 free(arr);
478 return ret;
479}
480
481
482/* Compute the product of two polynomials a and b, reduce modulo p, and store
483 * the result in r. r could be a or b; a could be b.
484 */
485int
486BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[],
487 BN_CTX *ctx)
488{
489 int zlen, i, j, k, ret = 0;
490 BIGNUM *s;
491 BN_ULONG x1, x0, y1, y0, zz[4];
492
493
494 if (a == b) {
495 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
496 }
497
498 BN_CTX_start(ctx);
499 if ((s = BN_CTX_get(ctx)) == NULL)
500 goto err;
501
502 zlen = a->top + b->top + 4;
503 if (!bn_wexpand(s, zlen))
504 goto err;
505 s->top = zlen;
506
507 for (i = 0; i < zlen; i++)
508 s->d[i] = 0;
509
510 for (j = 0; j < b->top; j += 2) {
511 y0 = b->d[j];
512 y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1];
513 for (i = 0; i < a->top; i += 2) {
514 x0 = a->d[i];
515 x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1];
516 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
517 for (k = 0; k < 4; k++)
518 s->d[i + j + k] ^= zz[k];
519 }
520 }
521
522 bn_correct_top(s);
523 if (BN_GF2m_mod_arr(r, s, p))
524 ret = 1;
525
526err:
527 BN_CTX_end(ctx);
528 return ret;
529}
530
531/* Compute the product of two polynomials a and b, reduce modulo p, and store
532 * the result in r. r could be a or b; a could equal b.
533 *
534 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
535 * function is only provided for convenience; for best performance, use the
536 * BN_GF2m_mod_mul_arr function.
537 */
538int
539BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p,
540 BN_CTX *ctx)
541{
542 int ret = 0;
543 const int max = BN_num_bits(p) + 1;
544 int *arr = NULL;
545
546 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
547 goto err;
548 ret = BN_GF2m_poly2arr(p, arr, max);
549 if (!ret || ret > max) {
550 BNerror(BN_R_INVALID_LENGTH);
551 goto err;
552 }
553 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
554
555err:
556 free(arr);
557 return ret;
558}
559
560
561/* Square a, reduce the result mod p, and store it in a. r could be a. */
562int
563BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
564{
565 int i, ret = 0;
566 BIGNUM *s;
567
568 BN_CTX_start(ctx);
569 if ((s = BN_CTX_get(ctx)) == NULL)
570 goto err;
571 if (!bn_wexpand(s, 2 * a->top))
572 goto err;
573
574 for (i = a->top - 1; i >= 0; i--) {
575 s->d[2 * i + 1] = SQR1(a->d[i]);
576 s->d[2 * i] = SQR0(a->d[i]);
577 }
578
579 s->top = 2 * a->top;
580 bn_correct_top(s);
581 if (!BN_GF2m_mod_arr(r, s, p))
582 goto err;
583 ret = 1;
584
585err:
586 BN_CTX_end(ctx);
587 return ret;
588}
589
590/* Square a, reduce the result mod p, and store it in a. r could be a.
591 *
592 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
593 * function is only provided for convenience; for best performance, use the
594 * BN_GF2m_mod_sqr_arr function.
595 */
596int
597BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
598{
599 int ret = 0;
600 const int max = BN_num_bits(p) + 1;
601 int *arr = NULL;
602
603 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
604 goto err;
605 ret = BN_GF2m_poly2arr(p, arr, max);
606 if (!ret || ret > max) {
607 BNerror(BN_R_INVALID_LENGTH);
608 goto err;
609 }
610 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
611
612err:
613 free(arr);
614 return ret;
615}
616
617
618/* Invert a, reduce modulo p, and store the result in r. r could be a.
619 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
620 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
621 * of Elliptic Curve Cryptography Over Binary Fields".
622 */
623int
624BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
625{
626 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp;
627 int ret = 0;
628
629
630 BN_CTX_start(ctx);
631
632 if ((b = BN_CTX_get(ctx)) == NULL)
633 goto err;
634 if ((c = BN_CTX_get(ctx)) == NULL)
635 goto err;
636 if ((u = BN_CTX_get(ctx)) == NULL)
637 goto err;
638 if ((v = BN_CTX_get(ctx)) == NULL)
639 goto err;
640
641 if (!BN_GF2m_mod(u, a, p))
642 goto err;
643 if (BN_is_zero(u))
644 goto err;
645
646 if (!bn_copy(v, p))
647 goto err;
648#if 0
649 if (!BN_one(b))
650 goto err;
651
652 while (1) {
653 while (!BN_is_odd(u)) {
654 if (BN_is_zero(u))
655 goto err;
656 if (!BN_rshift1(u, u))
657 goto err;
658 if (BN_is_odd(b)) {
659 if (!BN_GF2m_add(b, b, p))
660 goto err;
661 }
662 if (!BN_rshift1(b, b))
663 goto err;
664 }
665
666 if (BN_abs_is_word(u, 1))
667 break;
668
669 if (BN_num_bits(u) < BN_num_bits(v)) {
670 tmp = u;
671 u = v;
672 v = tmp;
673 tmp = b;
674 b = c;
675 c = tmp;
676 }
677
678 if (!BN_GF2m_add(u, u, v))
679 goto err;
680 if (!BN_GF2m_add(b, b, c))
681 goto err;
682 }
683#else
684 {
685 int i, ubits = BN_num_bits(u),
686 vbits = BN_num_bits(v), /* v is copy of p */
687 top = p->top;
688 BN_ULONG *udp, *bdp, *vdp, *cdp;
689
690 if (!bn_wexpand(u, top))
691 goto err;
692 udp = u->d;
693 for (i = u->top; i < top; i++)
694 udp[i] = 0;
695 u->top = top;
696 if (!bn_wexpand(b, top))
697 goto err;
698 bdp = b->d;
699 bdp[0] = 1;
700 for (i = 1; i < top; i++)
701 bdp[i] = 0;
702 b->top = top;
703 if (!bn_wexpand(c, top))
704 goto err;
705 cdp = c->d;
706 for (i = 0; i < top; i++)
707 cdp[i] = 0;
708 c->top = top;
709 vdp = v->d; /* It pays off to "cache" *->d pointers, because
710 * it allows optimizer to be more aggressive.
711 * But we don't have to "cache" p->d, because *p
712 * is declared 'const'... */
713 while (1) {
714 while (ubits && !(udp[0]&1)) {
715 BN_ULONG u0, u1, b0, b1, mask;
716
717 u0 = udp[0];
718 b0 = bdp[0];
719 mask = (BN_ULONG)0 - (b0 & 1);
720 b0 ^= p->d[0] & mask;
721 for (i = 0; i < top - 1; i++) {
722 u1 = udp[i + 1];
723 udp[i] = ((u0 >> 1) |
724 (u1 << (BN_BITS2 - 1))) & BN_MASK2;
725 u0 = u1;
726 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask);
727 bdp[i] = ((b0 >> 1) |
728 (b1 << (BN_BITS2 - 1))) & BN_MASK2;
729 b0 = b1;
730 }
731 udp[i] = u0 >> 1;
732 bdp[i] = b0 >> 1;
733 ubits--;
734 }
735
736 if (ubits <= BN_BITS2) {
737 /* See if poly was reducible. */
738 if (udp[0] == 0)
739 goto err;
740 if (udp[0] == 1)
741 break;
742 }
743
744 if (ubits < vbits) {
745 i = ubits;
746 ubits = vbits;
747 vbits = i;
748 tmp = u;
749 u = v;
750 v = tmp;
751 tmp = b;
752 b = c;
753 c = tmp;
754 udp = vdp;
755 vdp = v->d;
756 bdp = cdp;
757 cdp = c->d;
758 }
759 for (i = 0; i < top; i++) {
760 udp[i] ^= vdp[i];
761 bdp[i] ^= cdp[i];
762 }
763 if (ubits == vbits) {
764 BN_ULONG ul;
765 int utop = (ubits - 1) / BN_BITS2;
766
767 while ((ul = udp[utop]) == 0 && utop)
768 utop--;
769 ubits = utop*BN_BITS2 + BN_num_bits_word(ul);
770 }
771 }
772 bn_correct_top(b);
773 }
774#endif
775
776 if (!bn_copy(r, b))
777 goto err;
778 ret = 1;
779
780err:
781 BN_CTX_end(ctx);
782 return ret;
783}
784
785/* Invert xx, reduce modulo p, and store the result in r. r could be xx.
786 *
787 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
788 * function is only provided for convenience; for best performance, use the
789 * BN_GF2m_mod_inv function.
790 */
791int
792BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx)
793{
794 BIGNUM *field;
795 int ret = 0;
796
797 BN_CTX_start(ctx);
798 if ((field = BN_CTX_get(ctx)) == NULL)
799 goto err;
800 if (!BN_GF2m_arr2poly(p, field))
801 goto err;
802
803 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
804
805err:
806 BN_CTX_end(ctx);
807 return ret;
808}
809
810
811#ifndef OPENSSL_SUN_GF2M_DIV
812/* Divide y by x, reduce modulo p, and store the result in r. r could be x
813 * or y, x could equal y.
814 */
815int
816BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p,
817 BN_CTX *ctx)
818{
819 BIGNUM *xinv = NULL;
820 int ret = 0;
821
822
823 BN_CTX_start(ctx);
824 if ((xinv = BN_CTX_get(ctx)) == NULL)
825 goto err;
826
827 if (!BN_GF2m_mod_inv(xinv, x, p, ctx))
828 goto err;
829 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx))
830 goto err;
831 ret = 1;
832
833err:
834 BN_CTX_end(ctx);
835 return ret;
836}
837#else
838/* Divide y by x, reduce modulo p, and store the result in r. r could be x
839 * or y, x could equal y.
840 * Uses algorithm Modular_Division_GF(2^m) from
841 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
842 * the Great Divide".
843 */
844int
845BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p,
846 BN_CTX *ctx)
847{
848 BIGNUM *a, *b, *u, *v;
849 int ret = 0;
850
851
852 BN_CTX_start(ctx);
853
854 if ((a = BN_CTX_get(ctx)) == NULL)
855 goto err;
856 if ((b = BN_CTX_get(ctx)) == NULL)
857 goto err;
858 if ((u = BN_CTX_get(ctx)) == NULL)
859 goto err;
860 if ((v = BN_CTX_get(ctx)) == NULL)
861 goto err;
862
863 /* reduce x and y mod p */
864 if (!BN_GF2m_mod(u, y, p))
865 goto err;
866 if (!BN_GF2m_mod(a, x, p))
867 goto err;
868 if (!bn_copy(b, p))
869 goto err;
870
871 while (!BN_is_odd(a)) {
872 if (!BN_rshift1(a, a))
873 goto err;
874 if (BN_is_odd(u))
875 if (!BN_GF2m_add(u, u, p))
876 goto err;
877 if (!BN_rshift1(u, u))
878 goto err;
879 }
880
881 do {
882 if (BN_GF2m_cmp(b, a) > 0) {
883 if (!BN_GF2m_add(b, b, a))
884 goto err;
885 if (!BN_GF2m_add(v, v, u))
886 goto err;
887 do {
888 if (!BN_rshift1(b, b))
889 goto err;
890 if (BN_is_odd(v))
891 if (!BN_GF2m_add(v, v, p))
892 goto err;
893 if (!BN_rshift1(v, v))
894 goto err;
895 } while (!BN_is_odd(b));
896 } else if (BN_abs_is_word(a, 1))
897 break;
898 else {
899 if (!BN_GF2m_add(a, a, b))
900 goto err;
901 if (!BN_GF2m_add(u, u, v))
902 goto err;
903 do {
904 if (!BN_rshift1(a, a))
905 goto err;
906 if (BN_is_odd(u))
907 if (!BN_GF2m_add(u, u, p))
908 goto err;
909 if (!BN_rshift1(u, u))
910 goto err;
911 } while (!BN_is_odd(a));
912 }
913 } while (1);
914
915 if (!bn_copy(r, u))
916 goto err;
917 ret = 1;
918
919err:
920 BN_CTX_end(ctx);
921 return ret;
922}
923#endif
924
925/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
926 * or yy, xx could equal yy.
927 *
928 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
929 * function is only provided for convenience; for best performance, use the
930 * BN_GF2m_mod_div function.
931 */
932int
933BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx,
934 const int p[], BN_CTX *ctx)
935{
936 BIGNUM *field;
937 int ret = 0;
938
939
940 BN_CTX_start(ctx);
941 if ((field = BN_CTX_get(ctx)) == NULL)
942 goto err;
943 if (!BN_GF2m_arr2poly(p, field))
944 goto err;
945
946 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
947
948err:
949 BN_CTX_end(ctx);
950 return ret;
951}
952
953
954/* Compute the bth power of a, reduce modulo p, and store
955 * the result in r. r could be a.
956 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
957 */
958int
959BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[],
960 BN_CTX *ctx)
961{
962 int ret = 0, i, n;
963 BIGNUM *u;
964
965
966 if (BN_is_zero(b))
967 return BN_one(r);
968
969 if (BN_abs_is_word(b, 1))
970 return bn_copy(r, a);
971
972 BN_CTX_start(ctx);
973 if ((u = BN_CTX_get(ctx)) == NULL)
974 goto err;
975
976 if (!BN_GF2m_mod_arr(u, a, p))
977 goto err;
978
979 n = BN_num_bits(b) - 1;
980 for (i = n - 1; i >= 0; i--) {
981 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx))
982 goto err;
983 if (BN_is_bit_set(b, i)) {
984 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx))
985 goto err;
986 }
987 }
988 if (!bn_copy(r, u))
989 goto err;
990 ret = 1;
991
992err:
993 BN_CTX_end(ctx);
994 return ret;
995}
996
997/* Compute the bth power of a, reduce modulo p, and store
998 * the result in r. r could be a.
999 *
1000 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
1001 * function is only provided for convenience; for best performance, use the
1002 * BN_GF2m_mod_exp_arr function.
1003 */
1004int
1005BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p,
1006 BN_CTX *ctx)
1007{
1008 int ret = 0;
1009 const int max = BN_num_bits(p) + 1;
1010 int *arr = NULL;
1011
1012 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
1013 goto err;
1014 ret = BN_GF2m_poly2arr(p, arr, max);
1015 if (!ret || ret > max) {
1016 BNerror(BN_R_INVALID_LENGTH);
1017 goto err;
1018 }
1019 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
1020
1021err:
1022 free(arr);
1023 return ret;
1024}
1025
1026/* Compute the square root of a, reduce modulo p, and store
1027 * the result in r. r could be a.
1028 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
1029 */
1030int
1031BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
1032{
1033 int ret = 0;
1034 BIGNUM *u;
1035
1036
1037 if (!p[0]) {
1038 /* reduction mod 1 => return 0 */
1039 BN_zero(r);
1040 return 1;
1041 }
1042
1043 BN_CTX_start(ctx);
1044 if ((u = BN_CTX_get(ctx)) == NULL)
1045 goto err;
1046
1047 if (!BN_set_bit(u, p[0] - 1))
1048 goto err;
1049 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
1050
1051err:
1052 BN_CTX_end(ctx);
1053 return ret;
1054}
1055
1056/* Compute the square root of a, reduce modulo p, and store
1057 * the result in r. r could be a.
1058 *
1059 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
1060 * function is only provided for convenience; for best performance, use the
1061 * BN_GF2m_mod_sqrt_arr function.
1062 */
1063int
1064BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1065{
1066 int ret = 0;
1067 const int max = BN_num_bits(p) + 1;
1068 int *arr = NULL;
1069 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
1070 goto err;
1071 ret = BN_GF2m_poly2arr(p, arr, max);
1072 if (!ret || ret > max) {
1073 BNerror(BN_R_INVALID_LENGTH);
1074 goto err;
1075 }
1076 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
1077
1078err:
1079 free(arr);
1080 return ret;
1081}
1082
1083/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
1084 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
1085 */
1086int
1087BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[],
1088 BN_CTX *ctx)
1089{
1090 int ret = 0, count = 0, j;
1091 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
1092
1093
1094 if (!p[0]) {
1095 /* reduction mod 1 => return 0 */
1096 BN_zero(r);
1097 return 1;
1098 }
1099
1100 BN_CTX_start(ctx);
1101 if ((a = BN_CTX_get(ctx)) == NULL)
1102 goto err;
1103 if ((z = BN_CTX_get(ctx)) == NULL)
1104 goto err;
1105 if ((w = BN_CTX_get(ctx)) == NULL)
1106 goto err;
1107
1108 if (!BN_GF2m_mod_arr(a, a_, p))
1109 goto err;
1110
1111 if (BN_is_zero(a)) {
1112 BN_zero(r);
1113 ret = 1;
1114 goto err;
1115 }
1116
1117 if (p[0] & 0x1) /* m is odd */
1118 {
1119 /* compute half-trace of a */
1120 if (!bn_copy(z, a))
1121 goto err;
1122 for (j = 1; j <= (p[0] - 1) / 2; j++) {
1123 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1124 goto err;
1125 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1126 goto err;
1127 if (!BN_GF2m_add(z, z, a))
1128 goto err;
1129 }
1130
1131 }
1132 else /* m is even */
1133 {
1134 if ((rho = BN_CTX_get(ctx)) == NULL)
1135 goto err;
1136 if ((w2 = BN_CTX_get(ctx)) == NULL)
1137 goto err;
1138 if ((tmp = BN_CTX_get(ctx)) == NULL)
1139 goto err;
1140 do {
1141 if (!BN_rand(rho, p[0], 0, 0))
1142 goto err;
1143 if (!BN_GF2m_mod_arr(rho, rho, p))
1144 goto err;
1145 BN_zero(z);
1146 if (!bn_copy(w, rho))
1147 goto err;
1148 for (j = 1; j <= p[0] - 1; j++) {
1149 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1150 goto err;
1151 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx))
1152 goto err;
1153 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx))
1154 goto err;
1155 if (!BN_GF2m_add(z, z, tmp))
1156 goto err;
1157 if (!BN_GF2m_add(w, w2, rho))
1158 goto err;
1159 }
1160 count++;
1161 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
1162 if (BN_is_zero(w)) {
1163 BNerror(BN_R_TOO_MANY_ITERATIONS);
1164 goto err;
1165 }
1166 }
1167
1168 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx))
1169 goto err;
1170 if (!BN_GF2m_add(w, z, w))
1171 goto err;
1172 if (BN_GF2m_cmp(w, a)) {
1173 BNerror(BN_R_NO_SOLUTION);
1174 goto err;
1175 }
1176
1177 if (!bn_copy(r, z))
1178 goto err;
1179
1180 ret = 1;
1181
1182err:
1183 BN_CTX_end(ctx);
1184 return ret;
1185}
1186
1187/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
1188 *
1189 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
1190 * function is only provided for convenience; for best performance, use the
1191 * BN_GF2m_mod_solve_quad_arr function.
1192 */
1193int
1194BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1195{
1196 int ret = 0;
1197 const int max = BN_num_bits(p) + 1;
1198 int *arr = NULL;
1199
1200 if ((arr = reallocarray(NULL, max, sizeof(int))) == NULL)
1201 goto err;
1202 ret = BN_GF2m_poly2arr(p, arr, max);
1203 if (!ret || ret > max) {
1204 BNerror(BN_R_INVALID_LENGTH);
1205 goto err;
1206 }
1207 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1208
1209err:
1210 free(arr);
1211 return ret;
1212}
1213
1214/* Convert the bit-string representation of a polynomial
1215 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding
1216 * to the bits with non-zero coefficient. Array is terminated with -1.
1217 * Up to max elements of the array will be filled. Return value is total
1218 * number of array elements that would be filled if array was large enough.
1219 */
1220int
1221BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
1222{
1223 int i, j, k = 0;
1224 BN_ULONG mask;
1225
1226 if (BN_is_zero(a))
1227 return 0;
1228
1229 for (i = a->top - 1; i >= 0; i--) {
1230 if (!a->d[i])
1231 /* skip word if a->d[i] == 0 */
1232 continue;
1233 mask = BN_TBIT;
1234 for (j = BN_BITS2 - 1; j >= 0; j--) {
1235 if (a->d[i] & mask) {
1236 if (k < max)
1237 p[k] = BN_BITS2 * i + j;
1238 k++;
1239 }
1240 mask >>= 1;
1241 }
1242 }
1243
1244 if (k < max)
1245 p[k] = -1;
1246 k++;
1247
1248 return k;
1249}
1250
1251/* Convert the coefficient array representation of a polynomial to a
1252 * bit-string. The array must be terminated by -1.
1253 */
1254int
1255BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1256{
1257 int i;
1258
1259 BN_zero(a);
1260 for (i = 0; p[i] != -1; i++) {
1261 if (BN_set_bit(a, p[i]) == 0)
1262 return 0;
1263 }
1264
1265 return 1;
1266}
1267
1268#endif
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-08-04 15:43:54 +0000