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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 if (d0) z[dN] = (z[dN] << d1) >> d1; /* clear up the top d1 bits */
388 z[0] ^= zz; /* reduction t^0 component */
389
390 for (k = 1; p[k] != 0; k++)
391 {
392 BN_ULONG tmp_ulong;
393
394 /* reducing component t^p[k]*/
395 n = p[k] / BN_BITS2;
396 d0 = p[k] % BN_BITS2;
397 d1 = BN_BITS2 - d0;
398 z[n] ^= (zz << d0);
399 tmp_ulong = zz >> d1;
400 if (d0 && tmp_ulong)
401 z[n+1] ^= tmp_ulong;
402 }
403
404
405 }
406
407 bn_correct_top(r);
408 return 1;
409 }
410
411/* Performs modular reduction of a by p and store result in r. r could be a.
412 *
413 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
414 * function is only provided for convenience; for best performance, use the
415 * BN_GF2m_mod_arr function.
416 */
417int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
418 {
419 int ret = 0;
420 const int max = BN_num_bits(p);
421 unsigned int *arr=NULL;
422 bn_check_top(a);
423 bn_check_top(p);
424 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
425 ret = BN_GF2m_poly2arr(p, arr, max);
426 if (!ret || ret > max)
427 {
428 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
429 goto err;
430 }
431 ret = BN_GF2m_mod_arr(r, a, arr);
432 bn_check_top(r);
433err:
434 if (arr) OPENSSL_free(arr);
435 return ret;
436 }
437
438
439/* Compute the product of two polynomials a and b, reduce modulo p, and store
440 * the result in r. r could be a or b; a could be b.
441 */
442int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
443 {
444 int zlen, i, j, k, ret = 0;
445 BIGNUM *s;
446 BN_ULONG x1, x0, y1, y0, zz[4];
447
448 bn_check_top(a);
449 bn_check_top(b);
450
451 if (a == b)
452 {
453 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
454 }
455
456 BN_CTX_start(ctx);
457 if ((s = BN_CTX_get(ctx)) == NULL) goto err;
458
459 zlen = a->top + b->top + 4;
460 if (!bn_wexpand(s, zlen)) goto err;
461 s->top = zlen;
462
463 for (i = 0; i < zlen; i++) s->d[i] = 0;
464
465 for (j = 0; j < b->top; j += 2)
466 {
467 y0 = b->d[j];
468 y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
469 for (i = 0; i < a->top; i += 2)
470 {
471 x0 = a->d[i];
472 x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
473 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
474 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
475 }
476 }
477
478 bn_correct_top(s);
479 if (BN_GF2m_mod_arr(r, s, p))
480 ret = 1;
481 bn_check_top(r);
482
483err:
484 BN_CTX_end(ctx);
485 return ret;
486 }
487
488/* Compute the product of two polynomials a and b, reduce modulo p, and store
489 * the result in r. r could be a or b; a could equal b.
490 *
491 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
492 * function is only provided for convenience; for best performance, use the
493 * BN_GF2m_mod_mul_arr function.
494 */
495int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
496 {
497 int ret = 0;
498 const int max = BN_num_bits(p);
499 unsigned int *arr=NULL;
500 bn_check_top(a);
501 bn_check_top(b);
502 bn_check_top(p);
503 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
504 ret = BN_GF2m_poly2arr(p, arr, max);
505 if (!ret || ret > max)
506 {
507 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
508 goto err;
509 }
510 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
511 bn_check_top(r);
512err:
513 if (arr) OPENSSL_free(arr);
514 return ret;
515 }
516
517
518/* Square a, reduce the result mod p, and store it in a. r could be a. */
519int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
520 {
521 int i, ret = 0;
522 BIGNUM *s;
523
524 bn_check_top(a);
525 BN_CTX_start(ctx);
526 if ((s = BN_CTX_get(ctx)) == NULL) return 0;
527 if (!bn_wexpand(s, 2 * a->top)) goto err;
528
529 for (i = a->top - 1; i >= 0; i--)
530 {
531 s->d[2*i+1] = SQR1(a->d[i]);
532 s->d[2*i ] = SQR0(a->d[i]);
533 }
534
535 s->top = 2 * a->top;
536 bn_correct_top(s);
537 if (!BN_GF2m_mod_arr(r, s, p)) goto err;
538 bn_check_top(r);
539 ret = 1;
540err:
541 BN_CTX_end(ctx);
542 return ret;
543 }
544
545/* Square a, reduce the result mod p, and store it in a. r could be a.
546 *
547 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
548 * function is only provided for convenience; for best performance, use the
549 * BN_GF2m_mod_sqr_arr function.
550 */
551int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
552 {
553 int ret = 0;
554 const int max = BN_num_bits(p);
555 unsigned int *arr=NULL;
556
557 bn_check_top(a);
558 bn_check_top(p);
559 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
560 ret = BN_GF2m_poly2arr(p, arr, max);
561 if (!ret || ret > max)
562 {
563 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
564 goto err;
565 }
566 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
567 bn_check_top(r);
568err:
569 if (arr) OPENSSL_free(arr);
570 return ret;
571 }
572
573
574/* Invert a, reduce modulo p, and store the result in r. r could be a.
575 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
576 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
577 * of Elliptic Curve Cryptography Over Binary Fields".
578 */
579int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
580 {
581 BIGNUM *b, *c, *u, *v, *tmp;
582 int ret = 0;
583
584 bn_check_top(a);
585 bn_check_top(p);
586
587 BN_CTX_start(ctx);
588
589 b = BN_CTX_get(ctx);
590 c = BN_CTX_get(ctx);
591 u = BN_CTX_get(ctx);
592 v = BN_CTX_get(ctx);
593 if (v == NULL) goto err;
594
595 if (!BN_one(b)) goto err;
596 if (!BN_GF2m_mod(u, a, p)) goto err;
597 if (!BN_copy(v, p)) goto err;
598
599 if (BN_is_zero(u)) goto err;
600
601 while (1)
602 {
603 while (!BN_is_odd(u))
604 {
605 if (!BN_rshift1(u, u)) goto err;
606 if (BN_is_odd(b))
607 {
608 if (!BN_GF2m_add(b, b, p)) goto err;
609 }
610 if (!BN_rshift1(b, b)) goto err;
611 }
612
613 if (BN_abs_is_word(u, 1)) break;
614
615 if (BN_num_bits(u) < BN_num_bits(v))
616 {
617 tmp = u; u = v; v = tmp;
618 tmp = b; b = c; c = tmp;
619 }
620
621 if (!BN_GF2m_add(u, u, v)) goto err;
622 if (!BN_GF2m_add(b, b, c)) goto err;
623 }
624
625
626 if (!BN_copy(r, b)) goto err;
627 bn_check_top(r);
628 ret = 1;
629
630err:
631 BN_CTX_end(ctx);
632 return ret;
633 }
634
635/* Invert xx, reduce modulo p, and store the result in r. r could be xx.
636 *
637 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
638 * function is only provided for convenience; for best performance, use the
639 * BN_GF2m_mod_inv function.
640 */
641int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
642 {
643 BIGNUM *field;
644 int ret = 0;
645
646 bn_check_top(xx);
647 BN_CTX_start(ctx);
648 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
649 if (!BN_GF2m_arr2poly(p, field)) goto err;
650
651 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
652 bn_check_top(r);
653
654err:
655 BN_CTX_end(ctx);
656 return ret;
657 }
658
659
660#ifndef OPENSSL_SUN_GF2M_DIV
661/* Divide y by x, reduce modulo p, and store the result in r. r could be x
662 * or y, x could equal y.
663 */
664int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
665 {
666 BIGNUM *xinv = NULL;
667 int ret = 0;
668
669 bn_check_top(y);
670 bn_check_top(x);
671 bn_check_top(p);
672
673 BN_CTX_start(ctx);
674 xinv = BN_CTX_get(ctx);
675 if (xinv == NULL) goto err;
676
677 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
678 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
679 bn_check_top(r);
680 ret = 1;
681
682err:
683 BN_CTX_end(ctx);
684 return ret;
685 }
686#else
687/* Divide y by x, reduce modulo p, and store the result in r. r could be x
688 * or y, x could equal y.
689 * Uses algorithm Modular_Division_GF(2^m) from
690 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
691 * the Great Divide".
692 */
693int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
694 {
695 BIGNUM *a, *b, *u, *v;
696 int ret = 0;
697
698 bn_check_top(y);
699 bn_check_top(x);
700 bn_check_top(p);
701
702 BN_CTX_start(ctx);
703
704 a = BN_CTX_get(ctx);
705 b = BN_CTX_get(ctx);
706 u = BN_CTX_get(ctx);
707 v = BN_CTX_get(ctx);
708 if (v == NULL) goto err;
709
710 /* reduce x and y mod p */
711 if (!BN_GF2m_mod(u, y, p)) goto err;
712 if (!BN_GF2m_mod(a, x, p)) goto err;
713 if (!BN_copy(b, p)) goto err;
714
715 while (!BN_is_odd(a))
716 {
717 if (!BN_rshift1(a, a)) goto err;
718 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
719 if (!BN_rshift1(u, u)) goto err;
720 }
721
722 do
723 {
724 if (BN_GF2m_cmp(b, a) > 0)
725 {
726 if (!BN_GF2m_add(b, b, a)) goto err;
727 if (!BN_GF2m_add(v, v, u)) goto err;
728 do
729 {
730 if (!BN_rshift1(b, b)) goto err;
731 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
732 if (!BN_rshift1(v, v)) goto err;
733 } while (!BN_is_odd(b));
734 }
735 else if (BN_abs_is_word(a, 1))
736 break;
737 else
738 {
739 if (!BN_GF2m_add(a, a, b)) goto err;
740 if (!BN_GF2m_add(u, u, v)) goto err;
741 do
742 {
743 if (!BN_rshift1(a, a)) goto err;
744 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
745 if (!BN_rshift1(u, u)) goto err;
746 } while (!BN_is_odd(a));
747 }
748 } while (1);
749
750 if (!BN_copy(r, u)) goto err;
751 bn_check_top(r);
752 ret = 1;
753
754err:
755 BN_CTX_end(ctx);
756 return ret;
757 }
758#endif
759
760/* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
761 * or yy, xx could equal yy.
762 *
763 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
764 * function is only provided for convenience; for best performance, use the
765 * BN_GF2m_mod_div function.
766 */
767int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
768 {
769 BIGNUM *field;
770 int ret = 0;
771
772 bn_check_top(yy);
773 bn_check_top(xx);
774
775 BN_CTX_start(ctx);
776 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
777 if (!BN_GF2m_arr2poly(p, field)) goto err;
778
779 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
780 bn_check_top(r);
781
782err:
783 BN_CTX_end(ctx);
784 return ret;
785 }
786
787
788/* Compute the bth power of a, reduce modulo p, and store
789 * the result in r. r could be a.
790 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
791 */
792int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
793 {
794 int ret = 0, i, n;
795 BIGNUM *u;
796
797 bn_check_top(a);
798 bn_check_top(b);
799
800 if (BN_is_zero(b))
801 return(BN_one(r));
802
803 if (BN_abs_is_word(b, 1))
804 return (BN_copy(r, a) != NULL);
805
806 BN_CTX_start(ctx);
807 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
808
809 if (!BN_GF2m_mod_arr(u, a, p)) goto err;
810
811 n = BN_num_bits(b) - 1;
812 for (i = n - 1; i >= 0; i--)
813 {
814 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
815 if (BN_is_bit_set(b, i))
816 {
817 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
818 }
819 }
820 if (!BN_copy(r, u)) goto err;
821 bn_check_top(r);
822 ret = 1;
823err:
824 BN_CTX_end(ctx);
825 return ret;
826 }
827
828/* Compute the bth power of a, reduce modulo p, and store
829 * the result in r. r could be a.
830 *
831 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
832 * function is only provided for convenience; for best performance, use the
833 * BN_GF2m_mod_exp_arr function.
834 */
835int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
836 {
837 int ret = 0;
838 const int max = BN_num_bits(p);
839 unsigned int *arr=NULL;
840 bn_check_top(a);
841 bn_check_top(b);
842 bn_check_top(p);
843 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
844 ret = BN_GF2m_poly2arr(p, arr, max);
845 if (!ret || ret > max)
846 {
847 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
848 goto err;
849 }
850 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
851 bn_check_top(r);
852err:
853 if (arr) OPENSSL_free(arr);
854 return ret;
855 }
856
857/* Compute the square root of a, reduce modulo p, and store
858 * the result in r. r could be a.
859 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
860 */
861int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
862 {
863 int ret = 0;
864 BIGNUM *u;
865
866 bn_check_top(a);
867
868 if (!p[0])
869 {
870 /* reduction mod 1 => return 0 */
871 BN_zero(r);
872 return 1;
873 }
874
875 BN_CTX_start(ctx);
876 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
877
878 if (!BN_set_bit(u, p[0] - 1)) goto err;
879 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
880 bn_check_top(r);
881
882err:
883 BN_CTX_end(ctx);
884 return ret;
885 }
886
887/* Compute the square root of a, reduce modulo p, and store
888 * the result in r. r could be a.
889 *
890 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
891 * function is only provided for convenience; for best performance, use the
892 * BN_GF2m_mod_sqrt_arr function.
893 */
894int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
895 {
896 int ret = 0;
897 const int max = BN_num_bits(p);
898 unsigned int *arr=NULL;
899 bn_check_top(a);
900 bn_check_top(p);
901 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
902 ret = BN_GF2m_poly2arr(p, arr, max);
903 if (!ret || ret > max)
904 {
905 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH);
906 goto err;
907 }
908 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
909 bn_check_top(r);
910err:
911 if (arr) OPENSSL_free(arr);
912 return ret;
913 }
914
915/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
916 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
917 */
918int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx)
919 {
920 int ret = 0, count = 0;
921 unsigned int j;
922 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
923
924 bn_check_top(a_);
925
926 if (!p[0])
927 {
928 /* reduction mod 1 => return 0 */
929 BN_zero(r);
930 return 1;
931 }
932
933 BN_CTX_start(ctx);
934 a = BN_CTX_get(ctx);
935 z = BN_CTX_get(ctx);
936 w = BN_CTX_get(ctx);
937 if (w == NULL) goto err;
938
939 if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
940
941 if (BN_is_zero(a))
942 {
943 BN_zero(r);
944 ret = 1;
945 goto err;
946 }
947
948 if (p[0] & 0x1) /* m is odd */
949 {
950 /* compute half-trace of a */
951 if (!BN_copy(z, a)) goto err;
952 for (j = 1; j <= (p[0] - 1) / 2; j++)
953 {
954 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
955 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
956 if (!BN_GF2m_add(z, z, a)) goto err;
957 }
958
959 }
960 else /* m is even */
961 {
962 rho = BN_CTX_get(ctx);
963 w2 = BN_CTX_get(ctx);
964 tmp = BN_CTX_get(ctx);
965 if (tmp == NULL) goto err;
966 do
967 {
968 if (!BN_rand(rho, p[0], 0, 0)) goto err;
969 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
970 BN_zero(z);
971 if (!BN_copy(w, rho)) goto err;
972 for (j = 1; j <= p[0] - 1; j++)
973 {
974 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
975 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
976 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
977 if (!BN_GF2m_add(z, z, tmp)) goto err;
978 if (!BN_GF2m_add(w, w2, rho)) goto err;
979 }
980 count++;
981 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
982 if (BN_is_zero(w))
983 {
984 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
985 goto err;
986 }
987 }
988
989 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
990 if (!BN_GF2m_add(w, z, w)) goto err;
991 if (BN_GF2m_cmp(w, a))
992 {
993 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
994 goto err;
995 }
996
997 if (!BN_copy(r, z)) goto err;
998 bn_check_top(r);
999
1000 ret = 1;
1001
1002err:
1003 BN_CTX_end(ctx);
1004 return ret;
1005 }
1006
1007/* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
1008 *
1009 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
1010 * function is only provided for convenience; for best performance, use the
1011 * BN_GF2m_mod_solve_quad_arr function.
1012 */
1013int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1014 {
1015 int ret = 0;
1016 const int max = BN_num_bits(p);
1017 unsigned int *arr=NULL;
1018 bn_check_top(a);
1019 bn_check_top(p);
1020 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) *
1021 max)) == NULL) goto err;
1022 ret = BN_GF2m_poly2arr(p, arr, max);
1023 if (!ret || ret > max)
1024 {
1025 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
1026 goto err;
1027 }
1028 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1029 bn_check_top(r);
1030err:
1031 if (arr) OPENSSL_free(arr);
1032 return ret;
1033 }
1034
1035/* Convert the bit-string representation of a polynomial
1036 * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array
1037 * of integers corresponding to the bits with non-zero coefficient.
1038 * Up to max elements of the array will be filled. Return value is total
1039 * number of coefficients that would be extracted if array was large enough.
1040 */
1041int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max)
1042 {
1043 int i, j, k = 0;
1044 BN_ULONG mask;
1045
1046 if (BN_is_zero(a) || !BN_is_bit_set(a, 0))
1047 /* a_0 == 0 => return error (the unsigned int array
1048 * must be terminated by 0)
1049 */
1050 return 0;
1051
1052 for (i = a->top - 1; i >= 0; i--)
1053 {
1054 if (!a->d[i])
1055 /* skip word if a->d[i] == 0 */
1056 continue;
1057 mask = BN_TBIT;
1058 for (j = BN_BITS2 - 1; j >= 0; j--)
1059 {
1060 if (a->d[i] & mask)
1061 {
1062 if (k < max) p[k] = BN_BITS2 * i + j;
1063 k++;
1064 }
1065 mask >>= 1;
1066 }
1067 }
1068
1069 return k;
1070 }
1071
1072/* Convert the coefficient array representation of a polynomial to a
1073 * bit-string. The array must be terminated by 0.
1074 */
1075int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a)
1076 {
1077 int i;
1078
1079 bn_check_top(a);
1080 BN_zero(a);
1081 for (i = 0; p[i] != 0; i++)
1082 {
1083 if (BN_set_bit(a, p[i]) == 0)
1084 return 0;
1085 }
1086 BN_set_bit(a, 0);
1087 bn_check_top(a);
1088
1089 return 1;
1090 }
1091
generated by cgit v1.2.3-55-g6feb (git 2.39.1) at 2025-08-04 22:29:45 +0000