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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#ifdef SIXTEEN_BIT
125#define SQR1(w) \
126 SQR_tb[(w) >> 12 & 0xF] << 8 | SQR_tb[(w) >> 8 & 0xF]
127#define SQR0(w) \
128 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
129#endif
130#ifdef EIGHT_BIT
131#define SQR1(w) \
132 SQR_tb[(w) >> 4 & 0xF]
133#define SQR0(w) \
134 SQR_tb[(w) & 15]
135#endif
136
137/* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
138 * result is a polynomial r with degree < 2 * BN_BITS - 1
139 * The caller MUST ensure that the variables have the right amount
140 * of space allocated.
141 */
142#ifdef EIGHT_BIT
143static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
144 {
145 register BN_ULONG h, l, s;
146 BN_ULONG tab[4], top1b = a >> 7;
147 register BN_ULONG a1, a2;
148
149 a1 = a & (0x7F); a2 = a1 << 1;
150
151 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
152
153 s = tab[b & 0x3]; l = s;
154 s = tab[b >> 2 & 0x3]; l ^= s << 2; h = s >> 6;
155 s = tab[b >> 4 & 0x3]; l ^= s << 4; h ^= s >> 4;
156 s = tab[b >> 6 ]; l ^= s << 6; h ^= s >> 2;
157
158 /* compensate for the top bit of a */
159
160 if (top1b & 01) { l ^= b << 7; h ^= b >> 1; }
161
162 *r1 = h; *r0 = l;
163 }
164#endif
165#ifdef SIXTEEN_BIT
166static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
167 {
168 register BN_ULONG h, l, s;
169 BN_ULONG tab[4], top1b = a >> 15;
170 register BN_ULONG a1, a2;
171
172 a1 = a & (0x7FFF); a2 = a1 << 1;
173
174 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
175
176 s = tab[b & 0x3]; l = s;
177 s = tab[b >> 2 & 0x3]; l ^= s << 2; h = s >> 14;
178 s = tab[b >> 4 & 0x3]; l ^= s << 4; h ^= s >> 12;
179 s = tab[b >> 6 & 0x3]; l ^= s << 6; h ^= s >> 10;
180 s = tab[b >> 8 & 0x3]; l ^= s << 8; h ^= s >> 8;
181 s = tab[b >>10 & 0x3]; l ^= s << 10; h ^= s >> 6;
182 s = tab[b >>12 & 0x3]; l ^= s << 12; h ^= s >> 4;
183 s = tab[b >>14 ]; l ^= s << 14; h ^= s >> 2;
184
185 /* compensate for the top bit of a */
186
187 if (top1b & 01) { l ^= b << 15; h ^= b >> 1; }
188
189 *r1 = h; *r0 = l;
190 }
191#endif
192#ifdef THIRTY_TWO_BIT
193static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
194 {
195 register BN_ULONG h, l, s;
196 BN_ULONG tab[8], top2b = a >> 30;
197 register BN_ULONG a1, a2, a4;
198
199 a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1;
200
201 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
202 tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4;
203
204 s = tab[b & 0x7]; l = s;
205 s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29;
206 s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26;
207 s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23;
208 s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20;
209 s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17;
210 s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14;
211 s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11;
212 s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8;
213 s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5;
214 s = tab[b >> 30 ]; l ^= s << 30; h ^= s >> 2;
215
216 /* compensate for the top two bits of a */
217
218 if (top2b & 01) { l ^= b << 30; h ^= b >> 2; }
219 if (top2b & 02) { l ^= b << 31; h ^= b >> 1; }
220
221 *r1 = h; *r0 = l;
222 }
223#endif
224#if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
225static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
226 {
227 register BN_ULONG h, l, s;
228 BN_ULONG tab[16], top3b = a >> 61;
229 register BN_ULONG a1, a2, a4, a8;
230
231 a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1;
232
233 tab[ 0] = 0; tab[ 1] = a1; tab[ 2] = a2; tab[ 3] = a1^a2;
234 tab[ 4] = a4; tab[ 5] = a1^a4; tab[ 6] = a2^a4; tab[ 7] = a1^a2^a4;
235 tab[ 8] = a8; tab[ 9] = a1^a8; tab[10] = a2^a8; tab[11] = a1^a2^a8;
236 tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8;
237
238 s = tab[b & 0xF]; l = s;
239 s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60;
240 s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56;
241 s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52;
242 s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48;
243 s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44;
244 s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40;
245 s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36;
246 s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32;
247 s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28;
248 s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24;
249 s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20;
250 s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16;
251 s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12;
252 s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8;
253 s = tab[b >> 60 ]; l ^= s << 60; h ^= s >> 4;
254
255 /* compensate for the top three bits of a */
256
257 if (top3b & 01) { l ^= b << 61; h ^= b >> 3; }
258 if (top3b & 02) { l ^= b << 62; h ^= b >> 2; }
259 if (top3b & 04) { l ^= b << 63; h ^= b >> 1; }
260
261 *r1 = h; *r0 = l;
262 }
263#endif
264
265/* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
266 * result is a polynomial r with degree < 4 * BN_BITS2 - 1
267 * The caller MUST ensure that the variables have the right amount
268 * of space allocated.
269 */
270static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0)
271 {
272 BN_ULONG m1, m0;
273 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
274 bn_GF2m_mul_1x1(r+3, r+2, a1, b1);
275 bn_GF2m_mul_1x1(r+1, r, a0, b0);
276 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
277 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
278 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
279 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
280 }
281
282
283/* Add polynomials a and b and store result in r; r could be a or b, a and b
284 * could be equal; r is the bitwise XOR of a and b.
285 */
286int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
287 {
288 int i;
289 const BIGNUM *at, *bt;
290
291 bn_check_top(a);
292 bn_check_top(b);
293
294 if (a->top < b->top) { at = b; bt = a; }
295 else { at = a; bt = b; }
296
297 if(bn_wexpand(r, at->top) == NULL)
298 return 0;
299
300 for (i = 0; i < bt->top; i++)
301 {
302 r->d[i] = at->d[i] ^ bt->d[i];
303 }
304 for (; i < at->top; i++)
305 {
306 r->d[i] = at->d[i];
307 }
308
309 r->top = at->top;
310 bn_correct_top(r);
311
312 return 1;
313 }
314
315
316/* Some functions allow for representation of the irreducible polynomials
317 * as an int[], say p. The irreducible f(t) is then of the form:
318 * t^p[0] + t^p[1] + ... + t^p[k]
319 * where m = p[0] > p[1] > ... > p[k] = 0.
320 */
321
322
323/* Performs modular reduction of a and store result in r. r could be a. */
324int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[])
325 {
326 int j, k;
327 int n, dN, d0, d1;
328 BN_ULONG zz, *z;
329
330 bn_check_top(a);
331
332 if (!p[0])
333 {
334 /* reduction mod 1 => return 0 */
335 BN_zero(r);
336 return 1;
337 }
338
339 /* Since the algorithm does reduction in the r value, if a != r, copy
340 * the contents of a into r so we can do reduction in r.
341 */
342 if (a != r)
343 {
344 if (!bn_wexpand(r, a->top)) return 0;
345 for (j = 0; j < a->top; j++)
346 {
347 r->d[j] = a->d[j];
348 }
349 r->top = a->top;
350 }
351 z = r->d;
352
353 /* start reduction */
354 dN = p[0] / BN_BITS2;
355 for (j = r->top - 1; j > dN;)
356 {
357 zz = z[j];
358 if (z[j] == 0) { j--; continue; }
359 z[j] = 0;
360
361 for (k = 1; p[k] != 0; k++)
362 {
363 /* reducing component t^p[k] */
364 n = p[0] - p[k];
365 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0;
366 n /= BN_BITS2;
367 z[j-n] ^= (zz>>d0);
368 if (d0) z[j-n-1] ^= (zz<<d1);
369 }
370
371 /* reducing component t^0 */
372 n = dN;
373 d0 = p[0] % BN_BITS2;
374 d1 = BN_BITS2 - d0;
375 z[j-n] ^= (zz >> d0);
376 if (d0) z[j-n-1] ^= (zz << d1);
377 }
378
379 /* final round of reduction */
380 while (j == dN)
381 {
382
383 d0 = p[0] % BN_BITS2;
384 zz = z[dN] >> d0;
385 if (zz == 0) break;
386 d1 = BN_BITS2 - d0;
387
388 /* clear up the top d1 bits */
389 if (d0)
390 z[dN] = (z[dN] << d1) >> d1;
391 else
392 z[dN] = 0;
393 z[0] ^= zz; /* reduction t^0 component */
394
395 for (k = 1; p[k] != 0; k++)
396 {
397 BN_ULONG tmp_ulong;
398
399 /* reducing component t^p[k]*/
400 n = p[k] / BN_BITS2;
401 d0 = p[k] % BN_BITS2;
402 d1 = BN_BITS2 - d0;
403 z[n] ^= (zz << d0);
404 tmp_ulong = zz >> d1;
405 if (d0 && tmp_ulong)
406 z[n+1] ^= tmp_ulong;
407 }
408
409
410 }
411
412 bn_correct_top(r);
413 return 1;
414 }
415
416/* Performs modular reduction of a by p and store result in r. r could be a.
417 *
418 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
419 * function is only provided for convenience; for best performance, use the
420 * BN_GF2m_mod_arr function.
421 */
422int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
423 {
424 int ret = 0;
425 const int max = BN_num_bits(p);
426 unsigned int *arr=NULL;
427 bn_check_top(a);
428 bn_check_top(p);
429 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
430 ret = BN_GF2m_poly2arr(p, arr, max);
431 if (!ret || ret > max)
432 {
433 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
434 goto err;
435 }
436 ret = BN_GF2m_mod_arr(r, a, arr);
437 bn_check_top(r);
438err:
439 if (arr) OPENSSL_free(arr);
440 return ret;
441 }
442
443
444/* Compute the product of two polynomials a and b, reduce modulo p, and store
445 * the result in r. r could be a or b; a could be b.
446 */
447int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
448 {
449 int zlen, i, j, k, ret = 0;
450 BIGNUM *s;
451 BN_ULONG x1, x0, y1, y0, zz[4];
452
453 bn_check_top(a);
454 bn_check_top(b);
455
456 if (a == b)
457 {
458 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
459 }
460
461 BN_CTX_start(ctx);
462 if ((s = BN_CTX_get(ctx)) == NULL) goto err;
463
464 zlen = a->top + b->top + 4;
465 if (!bn_wexpand(s, zlen)) goto err;
466 s->top = zlen;
467
468 for (i = 0; i < zlen; i++) s->d[i] = 0;
469
470 for (j = 0; j < b->top; j += 2)
471 {
472 y0 = b->d[j];
473 y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
474 for (i = 0; i < a->top; i += 2)
475 {
476 x0 = a->d[i];
477 x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
478 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
479 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
480 }
481 }
482
483 bn_correct_top(s);
484 if (BN_GF2m_mod_arr(r, s, p))
485 ret = 1;
486 bn_check_top(r);
487
488err:
489 BN_CTX_end(ctx);
490 return ret;
491 }
492
493/* Compute the product of two polynomials a and b, reduce modulo p, and store
494 * the result in r. r could be a or b; a could equal b.
495 *
496 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
497 * function is only provided for convenience; for best performance, use the
498 * BN_GF2m_mod_mul_arr function.
499 */
500int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
501 {
502 int ret = 0;
503 const int max = BN_num_bits(p);
504 unsigned int *arr=NULL;
505 bn_check_top(a);
506 bn_check_top(b);
507 bn_check_top(p);
508 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
509 ret = BN_GF2m_poly2arr(p, arr, max);
510 if (!ret || ret > max)
511 {
512 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
513 goto err;
514 }
515 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
516 bn_check_top(r);
517err:
518 if (arr) OPENSSL_free(arr);
519 return ret;
520 }
521
522
523/* Square a, reduce the result mod p, and store it in a. r could be a. */
524int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
525 {
526 int i, ret = 0;
527 BIGNUM *s;
528
529 bn_check_top(a);
530 BN_CTX_start(ctx);
531 if ((s = BN_CTX_get(ctx)) == NULL) return 0;
532 if (!bn_wexpand(s, 2 * a->top)) goto err;
533
534 for (i = a->top - 1; i >= 0; i--)
535 {
536 s->d[2*i+1] = SQR1(a->d[i]);
537 s->d[2*i ] = SQR0(a->d[i]);
538 }
539
540 s->top = 2 * a->top;
541 bn_correct_top(s);
542 if (!BN_GF2m_mod_arr(r, s, p)) goto err;
543 bn_check_top(r);
544 ret = 1;
545err:
546 BN_CTX_end(ctx);
547 return ret;
548 }
549
550/* Square a, reduce the result mod p, and store it in a. r could be a.
551 *
552 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
553 * function is only provided for convenience; for best performance, use the
554 * BN_GF2m_mod_sqr_arr function.
555 */
556int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
557 {
558 int ret = 0;
559 const int max = BN_num_bits(p);
560 unsigned int *arr=NULL;
561
562 bn_check_top(a);
563 bn_check_top(p);
564 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
565 ret = BN_GF2m_poly2arr(p, arr, max);
566 if (!ret || ret > max)
567 {
568 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
569 goto err;
570 }
571 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
572 bn_check_top(r);
573err:
574 if (arr) OPENSSL_free(arr);
575 return ret;
576 }
577
578
579/* Invert a, reduce modulo p, and store the result in r. r could be a.
580 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
581 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
582 * of Elliptic Curve Cryptography Over Binary Fields".
583 */
584int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
585 {
586 BIGNUM *b, *c, *u, *v, *tmp;
587 int ret = 0;
588
589 bn_check_top(a);
590 bn_check_top(p);
591
592 BN_CTX_start(ctx);
593
594 b = BN_CTX_get(ctx);
595 c = BN_CTX_get(ctx);
596 u = BN_CTX_get(ctx);
597 v = BN_CTX_get(ctx);
598 if (v == NULL) goto err;
599
600 if (!BN_one(b)) goto err;
601 if (!BN_GF2m_mod(u, a, p)) goto err;
602 if (!BN_copy(v, p)) goto err;
603
604 if (BN_is_zero(u)) goto err;
605
606 while (1)
607 {
608 while (!BN_is_odd(u))
609 {
610 if (!BN_rshift1(u, u)) goto err;
611 if (BN_is_odd(b))
612 {
613 if (!BN_GF2m_add(b, b, p)) goto err;
614 }
615 if (!BN_rshift1(b, b)) goto err;
616 }
617
618 if (BN_abs_is_word(u, 1)) break;
619
620 if (BN_num_bits(u) < BN_num_bits(v))
621 {
622 tmp = u; u = v; v = tmp;
623 tmp = b; b = c; c = tmp;
624 }
625
626 if (!BN_GF2m_add(u, u, v)) goto err;
627 if (!BN_GF2m_add(b, b, c)) goto err;
628 }
629
630
631 if (!BN_copy(r, b)) goto err;
632 bn_check_top(r);
633 ret = 1;
634
635err:
636 BN_CTX_end(ctx);
637 return ret;
638 }
639
640/* Invert xx, reduce modulo p, and store the result in r. r could be xx.
641 *
642 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
643 * function is only provided for convenience; for best performance, use the
644 * BN_GF2m_mod_inv function.
645 */
646int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
647 {
648 BIGNUM *field;
649 int ret = 0;
650
651 bn_check_top(xx);
652 BN_CTX_start(ctx);
653 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
654 if (!BN_GF2m_arr2poly(p, field)) goto err;
655
656 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
657 bn_check_top(r);
658
659err:
660 BN_CTX_end(ctx);
661 return ret;
662 }
663
664
665#ifndef OPENSSL_SUN_GF2M_DIV
666/* Divide y by x, reduce modulo p, and store the result in r. r could be x
667 * or y, x could equal y.
668 */
669int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
670 {
671 BIGNUM *xinv = NULL;
672 int ret = 0;
673
674 bn_check_top(y);
675 bn_check_top(x);
676 bn_check_top(p);
677
678 BN_CTX_start(ctx);
679 xinv = BN_CTX_get(ctx);
680 if (xinv == NULL) goto err;
681
682 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
683 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
684 bn_check_top(r);
685 ret = 1;
686
687err:
688 BN_CTX_end(ctx);
689 return ret;
690 }
691#else
692/* Divide y by x, reduce modulo p, and store the result in r. r could be x
693 * or y, x could equal y.
694 * Uses algorithm Modular_Division_GF(2^m) from
695 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
696 * the Great Divide".
697 */
698int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
699 {
700 BIGNUM *a, *b, *u, *v;
701 int ret = 0;
702
703 bn_check_top(y);
704 bn_check_top(x);
705 bn_check_top(p);
706
707 BN_CTX_start(ctx);
708
709 a = BN_CTX_get(ctx);
710 b = BN_CTX_get(ctx);
711 u = BN_CTX_get(ctx);
712 v = BN_CTX_get(ctx);
713 if (v == NULL) goto err;
714
715 /* reduce x and y mod p */
716 if (!BN_GF2m_mod(u, y, p)) goto err;
717 if (!BN_GF2m_mod(a, x, p)) goto err;
718 if (!BN_copy(b, p)) goto err;
719
720 while (!BN_is_odd(a))
721 {
722 if (!BN_rshift1(a, a)) goto err;
723 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
724 if (!BN_rshift1(u, u)) goto err;
725 }
726
727 do
728 {
729 if (BN_GF2m_cmp(b, a) > 0)
730 {
731 if (!BN_GF2m_add(b, b, a)) goto err;
732 if (!BN_GF2m_add(v, v, u)) goto err;
733 do
734 {
735 if (!BN_rshift1(b, b)) goto err;
736 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
737 if (!BN_rshift1(v, v)) goto err;
738 } while (!BN_is_odd(b));
739 }
740 else if (BN_abs_is_word(a, 1))
741 break;
742 else
743 {
744 if (!BN_GF2m_add(a, a, b)) goto err;
745 if (!BN_GF2m_add(u, u, v)) goto err;
746 do
747 {
748 if (!BN_rshift1(a, a)) goto err;
749 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
750 if (!BN_rshift1(u, u)) goto err;
751 } while (!BN_is_odd(a));
752 }
753 } while (1);
754
755 if (!BN_copy(r, u)) goto err;
756 bn_check_top(r);
757 ret = 1;
758
759err:
760 BN_CTX_end(ctx);
761 return ret;
762 }
763#endif
764
765/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
766 * or yy, xx could equal yy.
767 *
768 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
769 * function is only provided for convenience; for best performance, use the
770 * BN_GF2m_mod_div function.
771 */
772int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
773 {
774 BIGNUM *field;
775 int ret = 0;
776
777 bn_check_top(yy);
778 bn_check_top(xx);
779
780 BN_CTX_start(ctx);
781 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
782 if (!BN_GF2m_arr2poly(p, field)) goto err;
783
784 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
785 bn_check_top(r);
786
787err:
788 BN_CTX_end(ctx);
789 return ret;
790 }
791
792
793/* Compute the bth power of a, reduce modulo p, and store
794 * the result in r. r could be a.
795 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
796 */
797int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
798 {
799 int ret = 0, i, n;
800 BIGNUM *u;
801
802 bn_check_top(a);
803 bn_check_top(b);
804
805 if (BN_is_zero(b))
806 return(BN_one(r));
807
808 if (BN_abs_is_word(b, 1))
809 return (BN_copy(r, a) != NULL);
810
811 BN_CTX_start(ctx);
812 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
813
814 if (!BN_GF2m_mod_arr(u, a, p)) goto err;
815
816 n = BN_num_bits(b) - 1;
817 for (i = n - 1; i >= 0; i--)
818 {
819 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
820 if (BN_is_bit_set(b, i))
821 {
822 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
823 }
824 }
825 if (!BN_copy(r, u)) goto err;
826 bn_check_top(r);
827 ret = 1;
828err:
829 BN_CTX_end(ctx);
830 return ret;
831 }
832
833/* Compute the bth power of a, reduce modulo p, and store
834 * the result in r. r could be a.
835 *
836 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
837 * function is only provided for convenience; for best performance, use the
838 * BN_GF2m_mod_exp_arr function.
839 */
840int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
841 {
842 int ret = 0;
843 const int max = BN_num_bits(p);
844 unsigned int *arr=NULL;
845 bn_check_top(a);
846 bn_check_top(b);
847 bn_check_top(p);
848 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
849 ret = BN_GF2m_poly2arr(p, arr, max);
850 if (!ret || ret > max)
851 {
852 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
853 goto err;
854 }
855 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
856 bn_check_top(r);
857err:
858 if (arr) OPENSSL_free(arr);
859 return ret;
860 }
861
862/* Compute the square root of a, reduce modulo p, and store
863 * the result in r. r could be a.
864 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
865 */
866int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
867 {
868 int ret = 0;
869 BIGNUM *u;
870
871 bn_check_top(a);
872
873 if (!p[0])
874 {
875 /* reduction mod 1 => return 0 */
876 BN_zero(r);
877 return 1;
878 }
879
880 BN_CTX_start(ctx);
881 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
882
883 if (!BN_set_bit(u, p[0] - 1)) goto err;
884 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
885 bn_check_top(r);
886
887err:
888 BN_CTX_end(ctx);
889 return ret;
890 }
891
892/* Compute the square root of a, reduce modulo p, and store
893 * the result in r. r could be a.
894 *
895 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
896 * function is only provided for convenience; for best performance, use the
897 * BN_GF2m_mod_sqrt_arr function.
898 */
899int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
900 {
901 int ret = 0;
902 const int max = BN_num_bits(p);
903 unsigned int *arr=NULL;
904 bn_check_top(a);
905 bn_check_top(p);
906 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
907 ret = BN_GF2m_poly2arr(p, arr, max);
908 if (!ret || ret > max)
909 {
910 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH);
911 goto err;
912 }
913 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
914 bn_check_top(r);
915err:
916 if (arr) OPENSSL_free(arr);
917 return ret;
918 }
919
920/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
921 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
922 */
923int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx)
924 {
925 int ret = 0, count = 0;
926 unsigned int j;
927 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
928
929 bn_check_top(a_);
930
931 if (!p[0])
932 {
933 /* reduction mod 1 => return 0 */
934 BN_zero(r);
935 return 1;
936 }
937
938 BN_CTX_start(ctx);
939 a = BN_CTX_get(ctx);
940 z = BN_CTX_get(ctx);
941 w = BN_CTX_get(ctx);
942 if (w == NULL) goto err;
943
944 if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
945
946 if (BN_is_zero(a))
947 {
948 BN_zero(r);
949 ret = 1;
950 goto err;
951 }
952
953 if (p[0] & 0x1) /* m is odd */
954 {
955 /* compute half-trace of a */
956 if (!BN_copy(z, a)) goto err;
957 for (j = 1; j <= (p[0] - 1) / 2; j++)
958 {
959 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
960 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
961 if (!BN_GF2m_add(z, z, a)) goto err;
962 }
963
964 }
965 else /* m is even */
966 {
967 rho = BN_CTX_get(ctx);
968 w2 = BN_CTX_get(ctx);
969 tmp = BN_CTX_get(ctx);
970 if (tmp == NULL) goto err;
971 do
972 {
973 if (!BN_rand(rho, p[0], 0, 0)) goto err;
974 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
975 BN_zero(z);
976 if (!BN_copy(w, rho)) goto err;
977 for (j = 1; j <= p[0] - 1; j++)
978 {
979 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
980 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
981 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
982 if (!BN_GF2m_add(z, z, tmp)) goto err;
983 if (!BN_GF2m_add(w, w2, rho)) goto err;
984 }
985 count++;
986 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
987 if (BN_is_zero(w))
988 {
989 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
990 goto err;
991 }
992 }
993
994 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
995 if (!BN_GF2m_add(w, z, w)) goto err;
996 if (BN_GF2m_cmp(w, a))
997 {
998 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
999 goto err;
1000 }
1001
1002 if (!BN_copy(r, z)) goto err;
1003 bn_check_top(r);
1004
1005 ret = 1;
1006
1007err:
1008 BN_CTX_end(ctx);
1009 return ret;
1010 }
1011
1012/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
1013 *
1014 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
1015 * function is only provided for convenience; for best performance, use the
1016 * BN_GF2m_mod_solve_quad_arr function.
1017 */
1018int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1019 {
1020 int ret = 0;
1021 const int max = BN_num_bits(p);
1022 unsigned int *arr=NULL;
1023 bn_check_top(a);
1024 bn_check_top(p);
1025 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) *
1026 max)) == NULL) goto err;
1027 ret = BN_GF2m_poly2arr(p, arr, max);
1028 if (!ret || ret > max)
1029 {
1030 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
1031 goto err;
1032 }
1033 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1034 bn_check_top(r);
1035err:
1036 if (arr) OPENSSL_free(arr);
1037 return ret;
1038 }
1039
1040/* Convert the bit-string representation of a polynomial
1041 * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array
1042 * of integers corresponding to the bits with non-zero coefficient.
1043 * Up to max elements of the array will be filled. Return value is total
1044 * number of coefficients that would be extracted if array was large enough.
1045 */
1046int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max)
1047 {
1048 int i, j, k = 0;
1049 BN_ULONG mask;
1050
1051 if (BN_is_zero(a) || !BN_is_bit_set(a, 0))
1052 /* a_0 == 0 => return error (the unsigned int array
1053 * must be terminated by 0)
1054 */
1055 return 0;
1056
1057 for (i = a->top - 1; i >= 0; i--)
1058 {
1059 if (!a->d[i])
1060 /* skip word if a->d[i] == 0 */
1061 continue;
1062 mask = BN_TBIT;
1063 for (j = BN_BITS2 - 1; j >= 0; j--)
1064 {
1065 if (a->d[i] & mask)
1066 {
1067 if (k < max) p[k] = BN_BITS2 * i + j;
1068 k++;
1069 }
1070 mask >>= 1;
1071 }
1072 }
1073
1074 return k;
1075 }
1076
1077/* Convert the coefficient array representation of a polynomial to a
1078 * bit-string. The array must be terminated by 0.
1079 */
1080int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a)
1081 {
1082 int i;
1083
1084 bn_check_top(a);
1085 BN_zero(a);
1086 for (i = 0; p[i] != 0; i++)
1087 {
1088 if (BN_set_bit(a, p[i]) == 0)
1089 return 0;
1090 }
1091 BN_set_bit(a, 0);
1092 bn_check_top(a);
1093
1094 return 1;
1095 }
1096
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-12-25 07:15:12 +0000