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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 bn_wexpand(r, at->top);
298
299 for (i = 0; i < bt->top; i++)
300 {
301 r->d[i] = at->d[i] ^ bt->d[i];
302 }
303 for (; i < at->top; i++)
304 {
305 r->d[i] = at->d[i];
306 }
307
308 r->top = at->top;
309 bn_correct_top(r);
310
311 return 1;
312 }
313
314
315/* Some functions allow for representation of the irreducible polynomials
316 * as an int[], say p. The irreducible f(t) is then of the form:
317 * t^p[0] + t^p[1] + ... + t^p[k]
318 * where m = p[0] > p[1] > ... > p[k] = 0.
319 */
320
321
322/* Performs modular reduction of a and store result in r. r could be a. */
323int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[])
324 {
325 int j, k;
326 int n, dN, d0, d1;
327 BN_ULONG zz, *z;
328
329 bn_check_top(a);
330
331 if (!p[0])
332 {
333 /* reduction mod 1 => return 0 */
334 BN_zero(r);
335 return 1;
336 }
337
338 /* Since the algorithm does reduction in the r value, if a != r, copy
339 * the contents of a into r so we can do reduction in r.
340 */
341 if (a != r)
342 {
343 if (!bn_wexpand(r, a->top)) return 0;
344 for (j = 0; j < a->top; j++)
345 {
346 r->d[j] = a->d[j];
347 }
348 r->top = a->top;
349 }
350 z = r->d;
351
352 /* start reduction */
353 dN = p[0] / BN_BITS2;
354 for (j = r->top - 1; j > dN;)
355 {
356 zz = z[j];
357 if (z[j] == 0) { j--; continue; }
358 z[j] = 0;
359
360 for (k = 1; p[k] != 0; k++)
361 {
362 /* reducing component t^p[k] */
363 n = p[0] - p[k];
364 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0;
365 n /= BN_BITS2;
366 z[j-n] ^= (zz>>d0);
367 if (d0) z[j-n-1] ^= (zz<<d1);
368 }
369
370 /* reducing component t^0 */
371 n = dN;
372 d0 = p[0] % BN_BITS2;
373 d1 = BN_BITS2 - d0;
374 z[j-n] ^= (zz >> d0);
375 if (d0) z[j-n-1] ^= (zz << d1);
376 }
377
378 /* final round of reduction */
379 while (j == dN)
380 {
381
382 d0 = p[0] % BN_BITS2;
383 zz = z[dN] >> d0;
384 if (zz == 0) break;
385 d1 = BN_BITS2 - d0;
386
387 /* clear up the top d1 bits */
388 if (d0)
389 z[dN] = (z[dN] << d1) >> d1;
390 else
391 z[dN] = 0;
392 z[0] ^= zz; /* reduction t^0 component */
393
394 for (k = 1; p[k] != 0; k++)
395 {
396 BN_ULONG tmp_ulong;
397
398 /* reducing component t^p[k]*/
399 n = p[k] / BN_BITS2;
400 d0 = p[k] % BN_BITS2;
401 d1 = BN_BITS2 - d0;
402 z[n] ^= (zz << d0);
403 tmp_ulong = zz >> d1;
404 if (d0 && tmp_ulong)
405 z[n+1] ^= tmp_ulong;
406 }
407
408
409 }
410
411 bn_correct_top(r);
412 return 1;
413 }
414
415/* Performs modular reduction of a by p and store result in r. r could be a.
416 *
417 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
418 * function is only provided for convenience; for best performance, use the
419 * BN_GF2m_mod_arr function.
420 */
421int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
422 {
423 int ret = 0;
424 const int max = BN_num_bits(p);
425 unsigned int *arr=NULL;
426 bn_check_top(a);
427 bn_check_top(p);
428 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
429 ret = BN_GF2m_poly2arr(p, arr, max);
430 if (!ret || ret > max)
431 {
432 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
433 goto err;
434 }
435 ret = BN_GF2m_mod_arr(r, a, arr);
436 bn_check_top(r);
437err:
438 if (arr) OPENSSL_free(arr);
439 return ret;
440 }
441
442
443/* Compute the product of two polynomials a and b, reduce modulo p, and store
444 * the result in r. r could be a or b; a could be b.
445 */
446int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
447 {
448 int zlen, i, j, k, ret = 0;
449 BIGNUM *s;
450 BN_ULONG x1, x0, y1, y0, zz[4];
451
452 bn_check_top(a);
453 bn_check_top(b);
454
455 if (a == b)
456 {
457 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
458 }
459
460 BN_CTX_start(ctx);
461 if ((s = BN_CTX_get(ctx)) == NULL) goto err;
462
463 zlen = a->top + b->top + 4;
464 if (!bn_wexpand(s, zlen)) goto err;
465 s->top = zlen;
466
467 for (i = 0; i < zlen; i++) s->d[i] = 0;
468
469 for (j = 0; j < b->top; j += 2)
470 {
471 y0 = b->d[j];
472 y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
473 for (i = 0; i < a->top; i += 2)
474 {
475 x0 = a->d[i];
476 x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
477 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
478 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
479 }
480 }
481
482 bn_correct_top(s);
483 if (BN_GF2m_mod_arr(r, s, p))
484 ret = 1;
485 bn_check_top(r);
486
487err:
488 BN_CTX_end(ctx);
489 return ret;
490 }
491
492/* Compute the product of two polynomials a and b, reduce modulo p, and store
493 * the result in r. r could be a or b; a could equal b.
494 *
495 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
496 * function is only provided for convenience; for best performance, use the
497 * BN_GF2m_mod_mul_arr function.
498 */
499int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
500 {
501 int ret = 0;
502 const int max = BN_num_bits(p);
503 unsigned int *arr=NULL;
504 bn_check_top(a);
505 bn_check_top(b);
506 bn_check_top(p);
507 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
508 ret = BN_GF2m_poly2arr(p, arr, max);
509 if (!ret || ret > max)
510 {
511 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
512 goto err;
513 }
514 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
515 bn_check_top(r);
516err:
517 if (arr) OPENSSL_free(arr);
518 return ret;
519 }
520
521
522/* Square a, reduce the result mod p, and store it in a. r could be a. */
523int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
524 {
525 int i, ret = 0;
526 BIGNUM *s;
527
528 bn_check_top(a);
529 BN_CTX_start(ctx);
530 if ((s = BN_CTX_get(ctx)) == NULL) return 0;
531 if (!bn_wexpand(s, 2 * a->top)) goto err;
532
533 for (i = a->top - 1; i >= 0; i--)
534 {
535 s->d[2*i+1] = SQR1(a->d[i]);
536 s->d[2*i ] = SQR0(a->d[i]);
537 }
538
539 s->top = 2 * a->top;
540 bn_correct_top(s);
541 if (!BN_GF2m_mod_arr(r, s, p)) goto err;
542 bn_check_top(r);
543 ret = 1;
544err:
545 BN_CTX_end(ctx);
546 return ret;
547 }
548
549/* Square a, reduce the result mod p, and store it in a. r could be a.
550 *
551 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
552 * function is only provided for convenience; for best performance, use the
553 * BN_GF2m_mod_sqr_arr function.
554 */
555int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
556 {
557 int ret = 0;
558 const int max = BN_num_bits(p);
559 unsigned int *arr=NULL;
560
561 bn_check_top(a);
562 bn_check_top(p);
563 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
564 ret = BN_GF2m_poly2arr(p, arr, max);
565 if (!ret || ret > max)
566 {
567 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
568 goto err;
569 }
570 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
571 bn_check_top(r);
572err:
573 if (arr) OPENSSL_free(arr);
574 return ret;
575 }
576
577
578/* Invert a, reduce modulo p, and store the result in r. r could be a.
579 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
580 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
581 * of Elliptic Curve Cryptography Over Binary Fields".
582 */
583int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
584 {
585 BIGNUM *b, *c, *u, *v, *tmp;
586 int ret = 0;
587
588 bn_check_top(a);
589 bn_check_top(p);
590
591 BN_CTX_start(ctx);
592
593 b = BN_CTX_get(ctx);
594 c = BN_CTX_get(ctx);
595 u = BN_CTX_get(ctx);
596 v = BN_CTX_get(ctx);
597 if (v == NULL) goto err;
598
599 if (!BN_one(b)) goto err;
600 if (!BN_GF2m_mod(u, a, p)) goto err;
601 if (!BN_copy(v, p)) goto err;
602
603 if (BN_is_zero(u)) goto err;
604
605 while (1)
606 {
607 while (!BN_is_odd(u))
608 {
609 if (!BN_rshift1(u, u)) goto err;
610 if (BN_is_odd(b))
611 {
612 if (!BN_GF2m_add(b, b, p)) goto err;
613 }
614 if (!BN_rshift1(b, b)) goto err;
615 }
616
617 if (BN_abs_is_word(u, 1)) break;
618
619 if (BN_num_bits(u) < BN_num_bits(v))
620 {
621 tmp = u; u = v; v = tmp;
622 tmp = b; b = c; c = tmp;
623 }
624
625 if (!BN_GF2m_add(u, u, v)) goto err;
626 if (!BN_GF2m_add(b, b, c)) goto err;
627 }
628
629
630 if (!BN_copy(r, b)) goto err;
631 bn_check_top(r);
632 ret = 1;
633
634err:
635 BN_CTX_end(ctx);
636 return ret;
637 }
638
639/* Invert xx, reduce modulo p, and store the result in r. r could be xx.
640 *
641 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
642 * function is only provided for convenience; for best performance, use the
643 * BN_GF2m_mod_inv function.
644 */
645int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
646 {
647 BIGNUM *field;
648 int ret = 0;
649
650 bn_check_top(xx);
651 BN_CTX_start(ctx);
652 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
653 if (!BN_GF2m_arr2poly(p, field)) goto err;
654
655 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
656 bn_check_top(r);
657
658err:
659 BN_CTX_end(ctx);
660 return ret;
661 }
662
663
664#ifndef OPENSSL_SUN_GF2M_DIV
665/* Divide y by x, reduce modulo p, and store the result in r. r could be x
666 * or y, x could equal y.
667 */
668int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
669 {
670 BIGNUM *xinv = NULL;
671 int ret = 0;
672
673 bn_check_top(y);
674 bn_check_top(x);
675 bn_check_top(p);
676
677 BN_CTX_start(ctx);
678 xinv = BN_CTX_get(ctx);
679 if (xinv == NULL) goto err;
680
681 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
682 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
683 bn_check_top(r);
684 ret = 1;
685
686err:
687 BN_CTX_end(ctx);
688 return ret;
689 }
690#else
691/* Divide y by x, reduce modulo p, and store the result in r. r could be x
692 * or y, x could equal y.
693 * Uses algorithm Modular_Division_GF(2^m) from
694 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
695 * the Great Divide".
696 */
697int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
698 {
699 BIGNUM *a, *b, *u, *v;
700 int ret = 0;
701
702 bn_check_top(y);
703 bn_check_top(x);
704 bn_check_top(p);
705
706 BN_CTX_start(ctx);
707
708 a = BN_CTX_get(ctx);
709 b = BN_CTX_get(ctx);
710 u = BN_CTX_get(ctx);
711 v = BN_CTX_get(ctx);
712 if (v == NULL) goto err;
713
714 /* reduce x and y mod p */
715 if (!BN_GF2m_mod(u, y, p)) goto err;
716 if (!BN_GF2m_mod(a, x, p)) goto err;
717 if (!BN_copy(b, p)) goto err;
718
719 while (!BN_is_odd(a))
720 {
721 if (!BN_rshift1(a, a)) goto err;
722 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
723 if (!BN_rshift1(u, u)) goto err;
724 }
725
726 do
727 {
728 if (BN_GF2m_cmp(b, a) > 0)
729 {
730 if (!BN_GF2m_add(b, b, a)) goto err;
731 if (!BN_GF2m_add(v, v, u)) goto err;
732 do
733 {
734 if (!BN_rshift1(b, b)) goto err;
735 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
736 if (!BN_rshift1(v, v)) goto err;
737 } while (!BN_is_odd(b));
738 }
739 else if (BN_abs_is_word(a, 1))
740 break;
741 else
742 {
743 if (!BN_GF2m_add(a, a, b)) goto err;
744 if (!BN_GF2m_add(u, u, v)) goto err;
745 do
746 {
747 if (!BN_rshift1(a, a)) goto err;
748 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
749 if (!BN_rshift1(u, u)) goto err;
750 } while (!BN_is_odd(a));
751 }
752 } while (1);
753
754 if (!BN_copy(r, u)) goto err;
755 bn_check_top(r);
756 ret = 1;
757
758err:
759 BN_CTX_end(ctx);
760 return ret;
761 }
762#endif
763
764/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
765 * or yy, xx could equal yy.
766 *
767 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
768 * function is only provided for convenience; for best performance, use the
769 * BN_GF2m_mod_div function.
770 */
771int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
772 {
773 BIGNUM *field;
774 int ret = 0;
775
776 bn_check_top(yy);
777 bn_check_top(xx);
778
779 BN_CTX_start(ctx);
780 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
781 if (!BN_GF2m_arr2poly(p, field)) goto err;
782
783 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
784 bn_check_top(r);
785
786err:
787 BN_CTX_end(ctx);
788 return ret;
789 }
790
791
792/* Compute the bth power of a, reduce modulo p, and store
793 * the result in r. r could be a.
794 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
795 */
796int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
797 {
798 int ret = 0, i, n;
799 BIGNUM *u;
800
801 bn_check_top(a);
802 bn_check_top(b);
803
804 if (BN_is_zero(b))
805 return(BN_one(r));
806
807 if (BN_abs_is_word(b, 1))
808 return (BN_copy(r, a) != NULL);
809
810 BN_CTX_start(ctx);
811 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
812
813 if (!BN_GF2m_mod_arr(u, a, p)) goto err;
814
815 n = BN_num_bits(b) - 1;
816 for (i = n - 1; i >= 0; i--)
817 {
818 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
819 if (BN_is_bit_set(b, i))
820 {
821 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
822 }
823 }
824 if (!BN_copy(r, u)) goto err;
825 bn_check_top(r);
826 ret = 1;
827err:
828 BN_CTX_end(ctx);
829 return ret;
830 }
831
832/* Compute the bth power of a, reduce modulo p, and store
833 * the result in r. r could be a.
834 *
835 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
836 * function is only provided for convenience; for best performance, use the
837 * BN_GF2m_mod_exp_arr function.
838 */
839int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
840 {
841 int ret = 0;
842 const int max = BN_num_bits(p);
843 unsigned int *arr=NULL;
844 bn_check_top(a);
845 bn_check_top(b);
846 bn_check_top(p);
847 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
848 ret = BN_GF2m_poly2arr(p, arr, max);
849 if (!ret || ret > max)
850 {
851 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
852 goto err;
853 }
854 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
855 bn_check_top(r);
856err:
857 if (arr) OPENSSL_free(arr);
858 return ret;
859 }
860
861/* Compute the square root of a, reduce modulo p, and store
862 * the result in r. r could be a.
863 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
864 */
865int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
866 {
867 int ret = 0;
868 BIGNUM *u;
869
870 bn_check_top(a);
871
872 if (!p[0])
873 {
874 /* reduction mod 1 => return 0 */
875 BN_zero(r);
876 return 1;
877 }
878
879 BN_CTX_start(ctx);
880 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
881
882 if (!BN_set_bit(u, p[0] - 1)) goto err;
883 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
884 bn_check_top(r);
885
886err:
887 BN_CTX_end(ctx);
888 return ret;
889 }
890
891/* Compute the square root of a, reduce modulo p, and store
892 * the result in r. r could be a.
893 *
894 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
895 * function is only provided for convenience; for best performance, use the
896 * BN_GF2m_mod_sqrt_arr function.
897 */
898int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
899 {
900 int ret = 0;
901 const int max = BN_num_bits(p);
902 unsigned int *arr=NULL;
903 bn_check_top(a);
904 bn_check_top(p);
905 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
906 ret = BN_GF2m_poly2arr(p, arr, max);
907 if (!ret || ret > max)
908 {
909 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH);
910 goto err;
911 }
912 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
913 bn_check_top(r);
914err:
915 if (arr) OPENSSL_free(arr);
916 return ret;
917 }
918
919/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
920 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
921 */
922int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx)
923 {
924 int ret = 0, count = 0;
925 unsigned int j;
926 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
927
928 bn_check_top(a_);
929
930 if (!p[0])
931 {
932 /* reduction mod 1 => return 0 */
933 BN_zero(r);
934 return 1;
935 }
936
937 BN_CTX_start(ctx);
938 a = BN_CTX_get(ctx);
939 z = BN_CTX_get(ctx);
940 w = BN_CTX_get(ctx);
941 if (w == NULL) goto err;
942
943 if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
944
945 if (BN_is_zero(a))
946 {
947 BN_zero(r);
948 ret = 1;
949 goto err;
950 }
951
952 if (p[0] & 0x1) /* m is odd */
953 {
954 /* compute half-trace of a */
955 if (!BN_copy(z, a)) goto err;
956 for (j = 1; j <= (p[0] - 1) / 2; j++)
957 {
958 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
959 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
960 if (!BN_GF2m_add(z, z, a)) goto err;
961 }
962
963 }
964 else /* m is even */
965 {
966 rho = BN_CTX_get(ctx);
967 w2 = BN_CTX_get(ctx);
968 tmp = BN_CTX_get(ctx);
969 if (tmp == NULL) goto err;
970 do
971 {
972 if (!BN_rand(rho, p[0], 0, 0)) goto err;
973 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
974 BN_zero(z);
975 if (!BN_copy(w, rho)) goto err;
976 for (j = 1; j <= p[0] - 1; j++)
977 {
978 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
979 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
980 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
981 if (!BN_GF2m_add(z, z, tmp)) goto err;
982 if (!BN_GF2m_add(w, w2, rho)) goto err;
983 }
984 count++;
985 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
986 if (BN_is_zero(w))
987 {
988 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
989 goto err;
990 }
991 }
992
993 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
994 if (!BN_GF2m_add(w, z, w)) goto err;
995 if (BN_GF2m_cmp(w, a))
996 {
997 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
998 goto err;
999 }
1000
1001 if (!BN_copy(r, z)) goto err;
1002 bn_check_top(r);
1003
1004 ret = 1;
1005
1006err:
1007 BN_CTX_end(ctx);
1008 return ret;
1009 }
1010
1011/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
1012 *
1013 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
1014 * function is only provided for convenience; for best performance, use the
1015 * BN_GF2m_mod_solve_quad_arr function.
1016 */
1017int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1018 {
1019 int ret = 0;
1020 const int max = BN_num_bits(p);
1021 unsigned int *arr=NULL;
1022 bn_check_top(a);
1023 bn_check_top(p);
1024 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) *
1025 max)) == NULL) goto err;
1026 ret = BN_GF2m_poly2arr(p, arr, max);
1027 if (!ret || ret > max)
1028 {
1029 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
1030 goto err;
1031 }
1032 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1033 bn_check_top(r);
1034err:
1035 if (arr) OPENSSL_free(arr);
1036 return ret;
1037 }
1038
1039/* Convert the bit-string representation of a polynomial
1040 * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array
1041 * of integers corresponding to the bits with non-zero coefficient.
1042 * Up to max elements of the array will be filled. Return value is total
1043 * number of coefficients that would be extracted if array was large enough.
1044 */
1045int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max)
1046 {
1047 int i, j, k = 0;
1048 BN_ULONG mask;
1049
1050 if (BN_is_zero(a) || !BN_is_bit_set(a, 0))
1051 /* a_0 == 0 => return error (the unsigned int array
1052 * must be terminated by 0)
1053 */
1054 return 0;
1055
1056 for (i = a->top - 1; i >= 0; i--)
1057 {
1058 if (!a->d[i])
1059 /* skip word if a->d[i] == 0 */
1060 continue;
1061 mask = BN_TBIT;
1062 for (j = BN_BITS2 - 1; j >= 0; j--)
1063 {
1064 if (a->d[i] & mask)
1065 {
1066 if (k < max) p[k] = BN_BITS2 * i + j;
1067 k++;
1068 }
1069 mask >>= 1;
1070 }
1071 }
1072
1073 return k;
1074 }
1075
1076/* Convert the coefficient array representation of a polynomial to a
1077 * bit-string. The array must be terminated by 0.
1078 */
1079int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a)
1080 {
1081 int i;
1082
1083 bn_check_top(a);
1084 BN_zero(a);
1085 for (i = 0; p[i] != 0; i++)
1086 {
1087 if (BN_set_bit(a, p[i]) == 0)
1088 return 0;
1089 }
1090 BN_set_bit(a, 0);
1091 bn_check_top(a);
1092
1093 return 1;
1094 }
1095
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-04-03 03:48:20 +0000