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1/* crypto/bn/bn_gf2m.c */
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 <assert.h>
92#include <limits.h>
93#include <stdio.h>
94#include "cryptlib.h"
95#include "bn_lcl.h"
96
97/* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
98#define MAX_ITERATIONS 50
99
100static const BN_ULONG SQR_tb[16] =
101 { 0, 1, 4, 5, 16, 17, 20, 21,
102 64, 65, 68, 69, 80, 81, 84, 85 };
103/* Platform-specific macros to accelerate squaring. */
104#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
105#define SQR1(w) \
106 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
107 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
108 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
109 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
110#define SQR0(w) \
111 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
112 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
113 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
114 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
115#endif
116#ifdef THIRTY_TWO_BIT
117#define SQR1(w) \
118 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
119 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
120#define SQR0(w) \
121 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
122 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
123#endif
124
125/* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
126 * result is a polynomial r with degree < 2 * BN_BITS - 1
127 * The caller MUST ensure that the variables have the right amount
128 * of space allocated.
129 */
130#ifdef THIRTY_TWO_BIT
131static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
132 {
133 register BN_ULONG h, l, s;
134 BN_ULONG tab[8], top2b = a >> 30;
135 register BN_ULONG a1, a2, a4;
136
137 a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1;
138
139 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
140 tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4;
141
142 s = tab[b & 0x7]; l = s;
143 s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29;
144 s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26;
145 s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23;
146 s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20;
147 s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17;
148 s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14;
149 s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11;
150 s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8;
151 s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5;
152 s = tab[b >> 30 ]; l ^= s << 30; h ^= s >> 2;
153
154 /* compensate for the top two bits of a */
155
156 if (top2b & 01) { l ^= b << 30; h ^= b >> 2; }
157 if (top2b & 02) { l ^= b << 31; h ^= b >> 1; }
158
159 *r1 = h; *r0 = l;
160 }
161#endif
162#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
163static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
164 {
165 register BN_ULONG h, l, s;
166 BN_ULONG tab[16], top3b = a >> 61;
167 register BN_ULONG a1, a2, a4, a8;
168
169 a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1;
170
171 tab[ 0] = 0; tab[ 1] = a1; tab[ 2] = a2; tab[ 3] = a1^a2;
172 tab[ 4] = a4; tab[ 5] = a1^a4; tab[ 6] = a2^a4; tab[ 7] = a1^a2^a4;
173 tab[ 8] = a8; tab[ 9] = a1^a8; tab[10] = a2^a8; tab[11] = a1^a2^a8;
174 tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8;
175
176 s = tab[b & 0xF]; l = s;
177 s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60;
178 s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56;
179 s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52;
180 s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48;
181 s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44;
182 s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40;
183 s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36;
184 s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32;
185 s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28;
186 s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24;
187 s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20;
188 s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16;
189 s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12;
190 s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8;
191 s = tab[b >> 60 ]; l ^= s << 60; h ^= s >> 4;
192
193 /* compensate for the top three bits of a */
194
195 if (top3b & 01) { l ^= b << 61; h ^= b >> 3; }
196 if (top3b & 02) { l ^= b << 62; h ^= b >> 2; }
197 if (top3b & 04) { l ^= b << 63; h ^= b >> 1; }
198
199 *r1 = h; *r0 = l;
200 }
201#endif
202
203/* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
204 * result is a polynomial r with degree < 4 * BN_BITS2 - 1
205 * The caller MUST ensure that the variables have the right amount
206 * of space allocated.
207 */
208static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0)
209 {
210 BN_ULONG m1, m0;
211 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
212 bn_GF2m_mul_1x1(r+3, r+2, a1, b1);
213 bn_GF2m_mul_1x1(r+1, r, a0, b0);
214 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
215 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
216 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
217 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
218 }
219
220
221/* Add polynomials a and b and store result in r; r could be a or b, a and b
222 * could be equal; r is the bitwise XOR of a and b.
223 */
224int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
225 {
226 int i;
227 const BIGNUM *at, *bt;
228
229 bn_check_top(a);
230 bn_check_top(b);
231
232 if (a->top < b->top) { at = b; bt = a; }
233 else { at = a; bt = b; }
234
235 if(bn_wexpand(r, at->top) == NULL)
236 return 0;
237
238 for (i = 0; i < bt->top; i++)
239 {
240 r->d[i] = at->d[i] ^ bt->d[i];
241 }
242 for (; i < at->top; i++)
243 {
244 r->d[i] = at->d[i];
245 }
246
247 r->top = at->top;
248 bn_correct_top(r);
249
250 return 1;
251 }
252
253
254/* Some functions allow for representation of the irreducible polynomials
255 * as an int[], say p. The irreducible f(t) is then of the form:
256 * t^p[0] + t^p[1] + ... + t^p[k]
257 * where m = p[0] > p[1] > ... > p[k] = 0.
258 */
259
260
261/* Performs modular reduction of a and store result in r. r could be a. */
262int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
263 {
264 int j, k;
265 int n, dN, d0, d1;
266 BN_ULONG zz, *z;
267
268 bn_check_top(a);
269
270 if (!p[0])
271 {
272 /* reduction mod 1 => return 0 */
273 BN_zero(r);
274 return 1;
275 }
276
277 /* Since the algorithm does reduction in the r value, if a != r, copy
278 * the contents of a into r so we can do reduction in r.
279 */
280 if (a != r)
281 {
282 if (!bn_wexpand(r, a->top)) return 0;
283 for (j = 0; j < a->top; j++)
284 {
285 r->d[j] = a->d[j];
286 }
287 r->top = a->top;
288 }
289 z = r->d;
290
291 /* start reduction */
292 dN = p[0] / BN_BITS2;
293 for (j = r->top - 1; j > dN;)
294 {
295 zz = z[j];
296 if (z[j] == 0) { j--; continue; }
297 z[j] = 0;
298
299 for (k = 1; p[k] != 0; k++)
300 {
301 /* reducing component t^p[k] */
302 n = p[0] - p[k];
303 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0;
304 n /= BN_BITS2;
305 z[j-n] ^= (zz>>d0);
306 if (d0) z[j-n-1] ^= (zz<<d1);
307 }
308
309 /* reducing component t^0 */
310 n = dN;
311 d0 = p[0] % BN_BITS2;
312 d1 = BN_BITS2 - d0;
313 z[j-n] ^= (zz >> d0);
314 if (d0) z[j-n-1] ^= (zz << d1);
315 }
316
317 /* final round of reduction */
318 while (j == dN)
319 {
320
321 d0 = p[0] % BN_BITS2;
322 zz = z[dN] >> d0;
323 if (zz == 0) break;
324 d1 = BN_BITS2 - d0;
325
326 /* clear up the top d1 bits */
327 if (d0)
328 z[dN] = (z[dN] << d1) >> d1;
329 else
330 z[dN] = 0;
331 z[0] ^= zz; /* reduction t^0 component */
332
333 for (k = 1; p[k] != 0; k++)
334 {
335 BN_ULONG tmp_ulong;
336
337 /* reducing component t^p[k]*/
338 n = p[k] / BN_BITS2;
339 d0 = p[k] % BN_BITS2;
340 d1 = BN_BITS2 - d0;
341 z[n] ^= (zz << d0);
342 tmp_ulong = zz >> d1;
343 if (d0 && tmp_ulong)
344 z[n+1] ^= tmp_ulong;
345 }
346
347
348 }
349
350 bn_correct_top(r);
351 return 1;
352 }
353
354/* Performs modular reduction of a by p and store result in r. r could be a.
355 *
356 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
357 * function is only provided for convenience; for best performance, use the
358 * BN_GF2m_mod_arr function.
359 */
360int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
361 {
362 int ret = 0;
363 const int max = BN_num_bits(p) + 1;
364 int *arr=NULL;
365 bn_check_top(a);
366 bn_check_top(p);
367 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
368 ret = BN_GF2m_poly2arr(p, arr, max);
369 if (!ret || ret > max)
370 {
371 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
372 goto err;
373 }
374 ret = BN_GF2m_mod_arr(r, a, arr);
375 bn_check_top(r);
376err:
377 if (arr) OPENSSL_free(arr);
378 return ret;
379 }
380
381
382/* Compute the product of two polynomials a and b, reduce modulo p, and store
383 * the result in r. r could be a or b; a could be b.
384 */
385int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
386 {
387 int zlen, i, j, k, ret = 0;
388 BIGNUM *s;
389 BN_ULONG x1, x0, y1, y0, zz[4];
390
391 bn_check_top(a);
392 bn_check_top(b);
393
394 if (a == b)
395 {
396 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
397 }
398
399 BN_CTX_start(ctx);
400 if ((s = BN_CTX_get(ctx)) == NULL) goto err;
401
402 zlen = a->top + b->top + 4;
403 if (!bn_wexpand(s, zlen)) goto err;
404 s->top = zlen;
405
406 for (i = 0; i < zlen; i++) s->d[i] = 0;
407
408 for (j = 0; j < b->top; j += 2)
409 {
410 y0 = b->d[j];
411 y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
412 for (i = 0; i < a->top; i += 2)
413 {
414 x0 = a->d[i];
415 x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
416 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
417 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
418 }
419 }
420
421 bn_correct_top(s);
422 if (BN_GF2m_mod_arr(r, s, p))
423 ret = 1;
424 bn_check_top(r);
425
426err:
427 BN_CTX_end(ctx);
428 return ret;
429 }
430
431/* Compute the product of two polynomials a and b, reduce modulo p, and store
432 * the result in r. r could be a or b; a could equal b.
433 *
434 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
435 * function is only provided for convenience; for best performance, use the
436 * BN_GF2m_mod_mul_arr function.
437 */
438int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
439 {
440 int ret = 0;
441 const int max = BN_num_bits(p) + 1;
442 int *arr=NULL;
443 bn_check_top(a);
444 bn_check_top(b);
445 bn_check_top(p);
446 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
447 ret = BN_GF2m_poly2arr(p, arr, max);
448 if (!ret || ret > max)
449 {
450 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
451 goto err;
452 }
453 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
454 bn_check_top(r);
455err:
456 if (arr) OPENSSL_free(arr);
457 return ret;
458 }
459
460
461/* Square a, reduce the result mod p, and store it in a. r could be a. */
462int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
463 {
464 int i, ret = 0;
465 BIGNUM *s;
466
467 bn_check_top(a);
468 BN_CTX_start(ctx);
469 if ((s = BN_CTX_get(ctx)) == NULL) return 0;
470 if (!bn_wexpand(s, 2 * a->top)) goto err;
471
472 for (i = a->top - 1; i >= 0; i--)
473 {
474 s->d[2*i+1] = SQR1(a->d[i]);
475 s->d[2*i ] = SQR0(a->d[i]);
476 }
477
478 s->top = 2 * a->top;
479 bn_correct_top(s);
480 if (!BN_GF2m_mod_arr(r, s, p)) goto err;
481 bn_check_top(r);
482 ret = 1;
483err:
484 BN_CTX_end(ctx);
485 return ret;
486 }
487
488/* Square a, reduce the result mod p, and store it in a. r could be a.
489 *
490 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
491 * function is only provided for convenience; for best performance, use the
492 * BN_GF2m_mod_sqr_arr function.
493 */
494int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
495 {
496 int ret = 0;
497 const int max = BN_num_bits(p) + 1;
498 int *arr=NULL;
499
500 bn_check_top(a);
501 bn_check_top(p);
502 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
503 ret = BN_GF2m_poly2arr(p, arr, max);
504 if (!ret || ret > max)
505 {
506 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
507 goto err;
508 }
509 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
510 bn_check_top(r);
511err:
512 if (arr) OPENSSL_free(arr);
513 return ret;
514 }
515
516
517/* Invert a, reduce modulo p, and store the result in r. r could be a.
518 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
519 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
520 * of Elliptic Curve Cryptography Over Binary Fields".
521 */
522int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
523 {
524 BIGNUM *b, *c, *u, *v, *tmp;
525 int ret = 0;
526
527 bn_check_top(a);
528 bn_check_top(p);
529
530 BN_CTX_start(ctx);
531
532 b = BN_CTX_get(ctx);
533 c = BN_CTX_get(ctx);
534 u = BN_CTX_get(ctx);
535 v = BN_CTX_get(ctx);
536 if (v == NULL) goto err;
537
538 if (!BN_one(b)) goto err;
539 if (!BN_GF2m_mod(u, a, p)) goto err;
540 if (!BN_copy(v, p)) goto err;
541
542 if (BN_is_zero(u)) goto err;
543
544 while (1)
545 {
546 while (!BN_is_odd(u))
547 {
548 if (BN_is_zero(u)) goto err;
549 if (!BN_rshift1(u, u)) goto err;
550 if (BN_is_odd(b))
551 {
552 if (!BN_GF2m_add(b, b, p)) goto err;
553 }
554 if (!BN_rshift1(b, b)) goto err;
555 }
556
557 if (BN_abs_is_word(u, 1)) break;
558
559 if (BN_num_bits(u) < BN_num_bits(v))
560 {
561 tmp = u; u = v; v = tmp;
562 tmp = b; b = c; c = tmp;
563 }
564
565 if (!BN_GF2m_add(u, u, v)) goto err;
566 if (!BN_GF2m_add(b, b, c)) goto err;
567 }
568
569
570 if (!BN_copy(r, b)) goto err;
571 bn_check_top(r);
572 ret = 1;
573
574err:
575 BN_CTX_end(ctx);
576 return ret;
577 }
578
579/* Invert xx, reduce modulo p, and store the result in r. r could be xx.
580 *
581 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
582 * function is only provided for convenience; for best performance, use the
583 * BN_GF2m_mod_inv function.
584 */
585int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx)
586 {
587 BIGNUM *field;
588 int ret = 0;
589
590 bn_check_top(xx);
591 BN_CTX_start(ctx);
592 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
593 if (!BN_GF2m_arr2poly(p, field)) goto err;
594
595 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
596 bn_check_top(r);
597
598err:
599 BN_CTX_end(ctx);
600 return ret;
601 }
602
603
604#ifndef OPENSSL_SUN_GF2M_DIV
605/* Divide y by x, reduce modulo p, and store the result in r. r could be x
606 * or y, x could equal y.
607 */
608int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
609 {
610 BIGNUM *xinv = NULL;
611 int ret = 0;
612
613 bn_check_top(y);
614 bn_check_top(x);
615 bn_check_top(p);
616
617 BN_CTX_start(ctx);
618 xinv = BN_CTX_get(ctx);
619 if (xinv == NULL) goto err;
620
621 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
622 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
623 bn_check_top(r);
624 ret = 1;
625
626err:
627 BN_CTX_end(ctx);
628 return ret;
629 }
630#else
631/* Divide y by x, reduce modulo p, and store the result in r. r could be x
632 * or y, x could equal y.
633 * Uses algorithm Modular_Division_GF(2^m) from
634 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
635 * the Great Divide".
636 */
637int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
638 {
639 BIGNUM *a, *b, *u, *v;
640 int ret = 0;
641
642 bn_check_top(y);
643 bn_check_top(x);
644 bn_check_top(p);
645
646 BN_CTX_start(ctx);
647
648 a = BN_CTX_get(ctx);
649 b = BN_CTX_get(ctx);
650 u = BN_CTX_get(ctx);
651 v = BN_CTX_get(ctx);
652 if (v == NULL) goto err;
653
654 /* reduce x and y mod p */
655 if (!BN_GF2m_mod(u, y, p)) goto err;
656 if (!BN_GF2m_mod(a, x, p)) goto err;
657 if (!BN_copy(b, p)) goto err;
658
659 while (!BN_is_odd(a))
660 {
661 if (!BN_rshift1(a, a)) goto err;
662 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
663 if (!BN_rshift1(u, u)) goto err;
664 }
665
666 do
667 {
668 if (BN_GF2m_cmp(b, a) > 0)
669 {
670 if (!BN_GF2m_add(b, b, a)) goto err;
671 if (!BN_GF2m_add(v, v, u)) goto err;
672 do
673 {
674 if (!BN_rshift1(b, b)) goto err;
675 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
676 if (!BN_rshift1(v, v)) goto err;
677 } while (!BN_is_odd(b));
678 }
679 else if (BN_abs_is_word(a, 1))
680 break;
681 else
682 {
683 if (!BN_GF2m_add(a, a, b)) goto err;
684 if (!BN_GF2m_add(u, u, v)) goto err;
685 do
686 {
687 if (!BN_rshift1(a, a)) goto err;
688 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
689 if (!BN_rshift1(u, u)) goto err;
690 } while (!BN_is_odd(a));
691 }
692 } while (1);
693
694 if (!BN_copy(r, u)) goto err;
695 bn_check_top(r);
696 ret = 1;
697
698err:
699 BN_CTX_end(ctx);
700 return ret;
701 }
702#endif
703
704/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
705 * or yy, xx could equal yy.
706 *
707 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
708 * function is only provided for convenience; for best performance, use the
709 * BN_GF2m_mod_div function.
710 */
711int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx)
712 {
713 BIGNUM *field;
714 int ret = 0;
715
716 bn_check_top(yy);
717 bn_check_top(xx);
718
719 BN_CTX_start(ctx);
720 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
721 if (!BN_GF2m_arr2poly(p, field)) goto err;
722
723 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
724 bn_check_top(r);
725
726err:
727 BN_CTX_end(ctx);
728 return ret;
729 }
730
731
732/* Compute the bth power of a, reduce modulo p, and store
733 * the result in r. r could be a.
734 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
735 */
736int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
737 {
738 int ret = 0, i, n;
739 BIGNUM *u;
740
741 bn_check_top(a);
742 bn_check_top(b);
743
744 if (BN_is_zero(b))
745 return(BN_one(r));
746
747 if (BN_abs_is_word(b, 1))
748 return (BN_copy(r, a) != NULL);
749
750 BN_CTX_start(ctx);
751 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
752
753 if (!BN_GF2m_mod_arr(u, a, p)) goto err;
754
755 n = BN_num_bits(b) - 1;
756 for (i = n - 1; i >= 0; i--)
757 {
758 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
759 if (BN_is_bit_set(b, i))
760 {
761 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
762 }
763 }
764 if (!BN_copy(r, u)) goto err;
765 bn_check_top(r);
766 ret = 1;
767err:
768 BN_CTX_end(ctx);
769 return ret;
770 }
771
772/* Compute the bth power of a, reduce modulo p, and store
773 * the result in r. r could be a.
774 *
775 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
776 * function is only provided for convenience; for best performance, use the
777 * BN_GF2m_mod_exp_arr function.
778 */
779int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
780 {
781 int ret = 0;
782 const int max = BN_num_bits(p) + 1;
783 int *arr=NULL;
784 bn_check_top(a);
785 bn_check_top(b);
786 bn_check_top(p);
787 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
788 ret = BN_GF2m_poly2arr(p, arr, max);
789 if (!ret || ret > max)
790 {
791 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
792 goto err;
793 }
794 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
795 bn_check_top(r);
796err:
797 if (arr) OPENSSL_free(arr);
798 return ret;
799 }
800
801/* Compute the square root of a, reduce modulo p, and store
802 * the result in r. r could be a.
803 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
804 */
805int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
806 {
807 int ret = 0;
808 BIGNUM *u;
809
810 bn_check_top(a);
811
812 if (!p[0])
813 {
814 /* reduction mod 1 => return 0 */
815 BN_zero(r);
816 return 1;
817 }
818
819 BN_CTX_start(ctx);
820 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
821
822 if (!BN_set_bit(u, p[0] - 1)) goto err;
823 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
824 bn_check_top(r);
825
826err:
827 BN_CTX_end(ctx);
828 return ret;
829 }
830
831/* Compute the square root of a, reduce modulo p, and store
832 * the result in r. r could be a.
833 *
834 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
835 * function is only provided for convenience; for best performance, use the
836 * BN_GF2m_mod_sqrt_arr function.
837 */
838int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
839 {
840 int ret = 0;
841 const int max = BN_num_bits(p) + 1;
842 int *arr=NULL;
843 bn_check_top(a);
844 bn_check_top(p);
845 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
846 ret = BN_GF2m_poly2arr(p, arr, max);
847 if (!ret || ret > max)
848 {
849 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH);
850 goto err;
851 }
852 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
853 bn_check_top(r);
854err:
855 if (arr) OPENSSL_free(arr);
856 return ret;
857 }
858
859/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
860 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
861 */
862int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx)
863 {
864 int ret = 0, count = 0, j;
865 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
866
867 bn_check_top(a_);
868
869 if (!p[0])
870 {
871 /* reduction mod 1 => return 0 */
872 BN_zero(r);
873 return 1;
874 }
875
876 BN_CTX_start(ctx);
877 a = BN_CTX_get(ctx);
878 z = BN_CTX_get(ctx);
879 w = BN_CTX_get(ctx);
880 if (w == NULL) goto err;
881
882 if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
883
884 if (BN_is_zero(a))
885 {
886 BN_zero(r);
887 ret = 1;
888 goto err;
889 }
890
891 if (p[0] & 0x1) /* m is odd */
892 {
893 /* compute half-trace of a */
894 if (!BN_copy(z, a)) goto err;
895 for (j = 1; j <= (p[0] - 1) / 2; j++)
896 {
897 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
898 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
899 if (!BN_GF2m_add(z, z, a)) goto err;
900 }
901
902 }
903 else /* m is even */
904 {
905 rho = BN_CTX_get(ctx);
906 w2 = BN_CTX_get(ctx);
907 tmp = BN_CTX_get(ctx);
908 if (tmp == NULL) goto err;
909 do
910 {
911 if (!BN_rand(rho, p[0], 0, 0)) goto err;
912 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
913 BN_zero(z);
914 if (!BN_copy(w, rho)) goto err;
915 for (j = 1; j <= p[0] - 1; j++)
916 {
917 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
918 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
919 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
920 if (!BN_GF2m_add(z, z, tmp)) goto err;
921 if (!BN_GF2m_add(w, w2, rho)) goto err;
922 }
923 count++;
924 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
925 if (BN_is_zero(w))
926 {
927 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
928 goto err;
929 }
930 }
931
932 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
933 if (!BN_GF2m_add(w, z, w)) goto err;
934 if (BN_GF2m_cmp(w, a))
935 {
936 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
937 goto err;
938 }
939
940 if (!BN_copy(r, z)) goto err;
941 bn_check_top(r);
942
943 ret = 1;
944
945err:
946 BN_CTX_end(ctx);
947 return ret;
948 }
949
950/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
951 *
952 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
953 * function is only provided for convenience; for best performance, use the
954 * BN_GF2m_mod_solve_quad_arr function.
955 */
956int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
957 {
958 int ret = 0;
959 const int max = BN_num_bits(p) + 1;
960 int *arr=NULL;
961 bn_check_top(a);
962 bn_check_top(p);
963 if ((arr = (int *)OPENSSL_malloc(sizeof(int) *
964 max)) == NULL) goto err;
965 ret = BN_GF2m_poly2arr(p, arr, max);
966 if (!ret || ret > max)
967 {
968 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
969 goto err;
970 }
971 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
972 bn_check_top(r);
973err:
974 if (arr) OPENSSL_free(arr);
975 return ret;
976 }
977
978/* Convert the bit-string representation of a polynomial
979 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding
980 * to the bits with non-zero coefficient. Array is terminated with -1.
981 * Up to max elements of the array will be filled. Return value is total
982 * number of array elements that would be filled if array was large enough.
983 */
984int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
985 {
986 int i, j, k = 0;
987 BN_ULONG mask;
988
989 if (BN_is_zero(a))
990 return 0;
991
992 for (i = a->top - 1; i >= 0; i--)
993 {
994 if (!a->d[i])
995 /* skip word if a->d[i] == 0 */
996 continue;
997 mask = BN_TBIT;
998 for (j = BN_BITS2 - 1; j >= 0; j--)
999 {
1000 if (a->d[i] & mask)
1001 {
1002 if (k < max) p[k] = BN_BITS2 * i + j;
1003 k++;
1004 }
1005 mask >>= 1;
1006 }
1007 }
1008
1009 if (k < max) {
1010 p[k] = -1;
1011 k++;
1012 }
1013
1014 return k;
1015 }
1016
1017/* Convert the coefficient array representation of a polynomial to a
1018 * bit-string. The array must be terminated by -1.
1019 */
1020int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1021 {
1022 int i;
1023
1024 bn_check_top(a);
1025 BN_zero(a);
1026 for (i = 0; p[i] != -1; i++)
1027 {
1028 if (BN_set_bit(a, p[i]) == 0)
1029 return 0;
1030 }
1031 bn_check_top(a);
1032
1033 return 1;
1034 }
1035
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-03-21 08:43:44 +0000