1 /* crypto/bn/bn_gf2m.c */
2 /* ====================================================================
3 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
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.
9 * The ECC Code is licensed pursuant to the OpenSSL open source
10 * license provided below.
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.
25 * The software is originally written by Sheueling Chang Shantz and
26 * Douglas Stebila of Sun Microsystems Laboratories.
30 /* ====================================================================
31 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
33 * Redistribution and use in source and binary forms, with or without
34 * modification, are permitted provided that the following conditions
37 * 1. Redistributions of source code must retain the above copyright
38 * notice, this list of conditions and the following disclaimer.
40 * 2. Redistributions in binary form must reproduce the above copyright
41 * notice, this list of conditions and the following disclaimer in
42 * the documentation and/or other materials provided with the
45 * 3. All advertising materials mentioning features or use of this
46 * software must display the following acknowledgment:
47 * "This product includes software developed by the OpenSSL Project
48 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
50 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
51 * endorse or promote products derived from this software without
52 * prior written permission. For written permission, please contact
53 * openssl-core@openssl.org.
55 * 5. Products derived from this software may not be called "OpenSSL"
56 * nor may "OpenSSL" appear in their names without prior written
57 * permission of the OpenSSL Project.
59 * 6. Redistributions of any form whatsoever must retain the following
61 * "This product includes software developed by the OpenSSL Project
62 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
64 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
65 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
66 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
67 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
68 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
69 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
70 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
71 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
72 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
73 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
74 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
75 * OF THE POSSIBILITY OF SUCH DAMAGE.
76 * ====================================================================
78 * This product includes cryptographic software written by Eric Young
79 * (eay@cryptsoft.com). This product includes software written by Tim
80 * Hudson (tjh@cryptsoft.com).
90 /* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
91 #define MAX_ITERATIONS 50
93 static const BN_ULONG SQR_tb[16] =
94 { 0, 1, 4, 5, 16, 17, 20, 21,
95 64, 65, 68, 69, 80, 81, 84, 85 };
96 /* Platform-specific macros to accelerate squaring. */
97 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
99 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
100 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
101 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
102 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
104 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
105 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
106 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
107 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
109 #ifdef THIRTY_TWO_BIT
111 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
112 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
114 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
115 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
119 SQR_tb[(w) >> 12 & 0xF] << 8 | SQR_tb[(w) >> 8 & 0xF]
121 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
125 SQR_tb[(w) >> 4 & 0xF]
130 /* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
131 * result is a polynomial r with degree < 2 * BN_BITS - 1
132 * The caller MUST ensure that the variables have the right amount
133 * of space allocated.
136 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
138 register BN_ULONG h, l, s;
139 BN_ULONG tab[4], top1b = a >> 7;
140 register BN_ULONG a1, a2;
142 a1 = a & (0x7F); a2 = a1 << 1;
144 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
146 s = tab[b & 0x3]; l = s;
147 s = tab[b >> 2 & 0x3]; l ^= s << 2; h = s >> 6;
148 s = tab[b >> 4 & 0x3]; l ^= s << 4; h ^= s >> 4;
149 s = tab[b >> 6 ]; l ^= s << 6; h ^= s >> 2;
151 /* compensate for the top bit of a */
153 if (top1b & 01) { l ^= b << 7; h ^= b >> 1; }
159 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
161 register BN_ULONG h, l, s;
162 BN_ULONG tab[4], top1b = a >> 15;
163 register BN_ULONG a1, a2;
165 a1 = a & (0x7FFF); a2 = a1 << 1;
167 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
169 s = tab[b & 0x3]; l = s;
170 s = tab[b >> 2 & 0x3]; l ^= s << 2; h = s >> 14;
171 s = tab[b >> 4 & 0x3]; l ^= s << 4; h ^= s >> 12;
172 s = tab[b >> 6 & 0x3]; l ^= s << 6; h ^= s >> 10;
173 s = tab[b >> 8 & 0x3]; l ^= s << 8; h ^= s >> 8;
174 s = tab[b >>10 & 0x3]; l ^= s << 10; h ^= s >> 6;
175 s = tab[b >>12 & 0x3]; l ^= s << 12; h ^= s >> 4;
176 s = tab[b >>14 ]; l ^= s << 14; h ^= s >> 2;
178 /* compensate for the top bit of a */
180 if (top1b & 01) { l ^= b << 15; h ^= b >> 1; }
185 #ifdef THIRTY_TWO_BIT
186 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
188 register BN_ULONG h, l, s;
189 BN_ULONG tab[8], top2b = a >> 30;
190 register BN_ULONG a1, a2, a4;
192 a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1;
194 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
195 tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4;
197 s = tab[b & 0x7]; l = s;
198 s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29;
199 s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26;
200 s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23;
201 s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20;
202 s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17;
203 s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14;
204 s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11;
205 s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8;
206 s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5;
207 s = tab[b >> 30 ]; l ^= s << 30; h ^= s >> 2;
209 /* compensate for the top two bits of a */
211 if (top2b & 01) { l ^= b << 30; h ^= b >> 2; }
212 if (top2b & 02) { l ^= b << 31; h ^= b >> 1; }
217 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
218 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
220 register BN_ULONG h, l, s;
221 BN_ULONG tab[16], top3b = a >> 61;
222 register BN_ULONG a1, a2, a4, a8;
224 a1 = a & (0x1FFFFFFFFFFFFFFF); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1;
226 tab[ 0] = 0; tab[ 1] = a1; tab[ 2] = a2; tab[ 3] = a1^a2;
227 tab[ 4] = a4; tab[ 5] = a1^a4; tab[ 6] = a2^a4; tab[ 7] = a1^a2^a4;
228 tab[ 8] = a8; tab[ 9] = a1^a8; tab[10] = a2^a8; tab[11] = a1^a2^a8;
229 tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8;
231 s = tab[b & 0xF]; l = s;
232 s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60;
233 s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56;
234 s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52;
235 s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48;
236 s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44;
237 s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40;
238 s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36;
239 s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32;
240 s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28;
241 s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24;
242 s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20;
243 s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16;
244 s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12;
245 s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8;
246 s = tab[b >> 60 ]; l ^= s << 60; h ^= s >> 4;
248 /* compensate for the top three bits of a */
250 if (top3b & 01) { l ^= b << 61; h ^= b >> 3; }
251 if (top3b & 02) { l ^= b << 62; h ^= b >> 2; }
252 if (top3b & 04) { l ^= b << 63; h ^= b >> 1; }
258 /* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
259 * result is a polynomial r with degree < 4 * BN_BITS2 - 1
260 * The caller MUST ensure that the variables have the right amount
261 * of space allocated.
263 static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0)
266 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
267 bn_GF2m_mul_1x1(r+3, r+2, a1, b1);
268 bn_GF2m_mul_1x1(r+1, r, a0, b0);
269 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
270 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
271 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
272 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
276 /* Add polynomials a and b and store result in r; r could be a or b, a and b
277 * could be equal; r is the bitwise XOR of a and b.
279 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
282 const BIGNUM *at, *bt;
284 if (a->top < b->top) { at = b; bt = a; }
285 else { at = a; bt = b; }
287 bn_wexpand(r, at->top);
289 for (i = 0; i < bt->top; i++)
291 r->d[i] = at->d[i] ^ bt->d[i];
293 for (; i < at->top; i++)
305 /* Some functions allow for representation of the irreducible polynomials
306 * as an int[], say p. The irreducible f(t) is then of the form:
307 * t^p[0] + t^p[1] + ... + t^p[k]
308 * where m = p[0] > p[1] > ... > p[k] = 0.
312 /* Performs modular reduction of a and store result in r. r could be a. */
313 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[])
319 /* Since the algorithm does reduction in the r value, if a != r, copy the
320 * contents of a into r so we can do reduction in r.
324 if (!bn_wexpand(r, a->top)) return 0;
325 for (j = 0; j < a->top; j++)
333 /* start reduction */
334 dN = p[0] / BN_BITS2;
335 for (j = r->top - 1; j > dN;)
338 if (z[j] == 0) { j--; continue; }
341 for (k = 1; p[k] > 0; k++)
343 /* reducing component t^p[k] */
345 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0;
348 if (d0) z[j-n-1] ^= (zz<<d1);
351 /* reducing component t^0 */
353 d0 = p[0] % BN_BITS2;
355 z[j-n] ^= (zz >> d0);
356 if (d0) z[j-n-1] ^= (zz << d1);
359 /* final round of reduction */
363 d0 = p[0] % BN_BITS2;
368 if (d0) z[dN] = (z[dN] << d1) >> d1; /* clear up the top d1 bits */
369 z[0] ^= zz; /* reduction t^0 component */
371 for (k = 1; p[k] > 0; k++)
373 /* reducing component t^p[k]*/
375 d0 = p[k] % BN_BITS2;
378 if (d0) z[n+1] ^= (zz >> d1);
389 /* Performs modular reduction of a by p and store result in r. r could be a.
391 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
392 * function is only provided for convenience; for best performance, use the
393 * BN_GF2m_mod_arr function.
395 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
397 const int max = BN_num_bits(p);
398 unsigned int *arr=NULL, ret = 0;
399 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
400 if (BN_GF2m_poly2arr(p, arr, max) > max)
402 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
405 ret = BN_GF2m_mod_arr(r, a, arr);
407 if (arr) OPENSSL_free(arr);
412 /* Compute the product of two polynomials a and b, reduce modulo p, and store
413 * the result in r. r could be a or b; a could be b.
415 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
417 int zlen, i, j, k, ret = 0;
419 BN_ULONG x1, x0, y1, y0, zz[4];
423 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
428 if ((s = BN_CTX_get(ctx)) == NULL) goto err;
430 zlen = a->top + b->top + 4;
431 if (!bn_wexpand(s, zlen)) goto err;
434 for (i = 0; i < zlen; i++) s->d[i] = 0;
436 for (j = 0; j < b->top; j += 2)
439 y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
440 for (i = 0; i < a->top; i += 2)
443 x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
444 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
445 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
450 BN_GF2m_mod_arr(r, s, p);
459 /* Compute the product of two polynomials a and b, reduce modulo p, and store
460 * the result in r. r could be a or b; a could equal b.
462 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
463 * function is only provided for convenience; for best performance, use the
464 * BN_GF2m_mod_mul_arr function.
466 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
468 const int max = BN_num_bits(p);
469 unsigned int *arr=NULL, ret = 0;
470 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
471 if (BN_GF2m_poly2arr(p, arr, max) > max)
473 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
476 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
478 if (arr) OPENSSL_free(arr);
483 /* Square a, reduce the result mod p, and store it in a. r could be a. */
484 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
490 if ((s = BN_CTX_get(ctx)) == NULL) return 0;
491 if (!bn_wexpand(s, 2 * a->top)) goto err;
493 for (i = a->top - 1; i >= 0; i--)
495 s->d[2*i+1] = SQR1(a->d[i]);
496 s->d[2*i ] = SQR0(a->d[i]);
501 if (!BN_GF2m_mod_arr(r, s, p)) goto err;
508 /* Square a, reduce the result mod p, and store it in a. r could be a.
510 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
511 * function is only provided for convenience; for best performance, use the
512 * BN_GF2m_mod_sqr_arr function.
514 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
516 const int max = BN_num_bits(p);
517 unsigned int *arr=NULL, ret = 0;
518 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
519 if (BN_GF2m_poly2arr(p, arr, max) > max)
521 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
524 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
526 if (arr) OPENSSL_free(arr);
531 /* Invert a, reduce modulo p, and store the result in r. r could be a.
532 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
533 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
534 * of Elliptic Curve Cryptography Over Binary Fields".
536 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
538 BIGNUM *b, *c, *u, *v, *tmp;
547 if (v == NULL) goto err;
549 if (!BN_one(b)) goto err;
550 if (!BN_zero(c)) goto err;
551 if (!BN_GF2m_mod(u, a, p)) goto err;
552 if (!BN_copy(v, p)) goto err;
554 u->neg = 0; /* Need to set u->neg = 0 because BN_is_one(u) checks
555 * the neg flag of the bignum.
558 if (BN_is_zero(u)) goto err;
562 while (!BN_is_odd(u))
564 if (!BN_rshift1(u, u)) goto err;
567 if (!BN_GF2m_add(b, b, p)) goto err;
569 if (!BN_rshift1(b, b)) goto err;
572 if (BN_is_one(u)) break;
574 if (BN_num_bits(u) < BN_num_bits(v))
576 tmp = u; u = v; v = tmp;
577 tmp = b; b = c; c = tmp;
580 if (!BN_GF2m_add(u, u, v)) goto err;
581 if (!BN_GF2m_add(b, b, c)) goto err;
585 if (!BN_copy(r, b)) goto err;
593 /* Invert xx, reduce modulo p, and store the result in r. r could be xx.
595 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
596 * function is only provided for convenience; for best performance, use the
597 * BN_GF2m_mod_inv function.
599 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
605 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
606 if (!BN_GF2m_arr2poly(p, field)) goto err;
608 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
616 #ifndef OPENSSL_SUN_GF2M_DIV
617 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
618 * or y, x could equal y.
620 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
626 xinv = BN_CTX_get(ctx);
627 if (xinv == NULL) goto err;
629 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
630 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
638 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
639 * or y, x could equal y.
640 * Uses algorithm Modular_Division_GF(2^m) from
641 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
644 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
646 BIGNUM *a, *b, *u, *v;
655 if (v == NULL) goto err;
657 /* reduce x and y mod p */
658 if (!BN_GF2m_mod(u, y, p)) goto err;
659 if (!BN_GF2m_mod(a, x, p)) goto err;
660 if (!BN_copy(b, p)) goto err;
661 if (!BN_zero(v)) goto err;
663 a->neg = 0; /* Need to set a->neg = 0 because BN_is_one(a) checks
664 * the neg flag of the bignum.
667 while (!BN_is_odd(a))
669 if (!BN_rshift1(a, a)) goto err;
670 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
671 if (!BN_rshift1(u, u)) goto err;
676 if (BN_GF2m_cmp(b, a) > 0)
678 if (!BN_GF2m_add(b, b, a)) goto err;
679 if (!BN_GF2m_add(v, v, u)) goto err;
682 if (!BN_rshift1(b, b)) goto err;
683 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
684 if (!BN_rshift1(v, v)) goto err;
685 } while (!BN_is_odd(b));
687 else if (BN_is_one(a))
691 if (!BN_GF2m_add(a, a, b)) goto err;
692 if (!BN_GF2m_add(u, u, v)) goto err;
695 if (!BN_rshift1(a, a)) goto err;
696 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
697 if (!BN_rshift1(u, u)) goto err;
698 } while (!BN_is_odd(a));
702 if (!BN_copy(r, u)) goto err;
711 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
712 * or yy, xx could equal yy.
714 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
715 * function is only provided for convenience; for best performance, use the
716 * BN_GF2m_mod_div function.
718 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
724 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
725 if (!BN_GF2m_arr2poly(p, field)) goto err;
727 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
735 /* Compute the bth power of a, reduce modulo p, and store
736 * the result in r. r could be a.
737 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
739 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
751 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
753 if (!BN_GF2m_mod_arr(u, a, p)) goto err;
755 n = BN_num_bits(b) - 1;
756 for (i = n - 1; i >= 0; i--)
758 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
759 if (BN_is_bit_set(b, i))
761 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
764 if (!BN_copy(r, u)) goto err;
773 /* Compute the bth power of a, reduce modulo p, and store
774 * the result in r. r could be a.
776 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
777 * function is only provided for convenience; for best performance, use the
778 * BN_GF2m_mod_exp_arr function.
780 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
782 const int max = BN_num_bits(p);
783 unsigned int *arr=NULL, ret = 0;
784 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
785 if (BN_GF2m_poly2arr(p, arr, max) > max)
787 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
790 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
792 if (arr) OPENSSL_free(arr);
796 /* Compute the square root of a, reduce modulo p, and store
797 * the result in r. r could be a.
798 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
800 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
806 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
808 if (!BN_zero(u)) goto err;
809 if (!BN_set_bit(u, p[0] - 1)) goto err;
810 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
817 /* Compute the square root of a, reduce modulo p, and store
818 * the result in r. r could be a.
820 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
821 * function is only provided for convenience; for best performance, use the
822 * BN_GF2m_mod_sqrt_arr function.
824 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
826 const int max = BN_num_bits(p);
827 unsigned int *arr=NULL, ret = 0;
828 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
829 if (BN_GF2m_poly2arr(p, arr, max) > max)
831 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
834 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
836 if (arr) OPENSSL_free(arr);
840 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
841 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
843 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx)
845 int ret = 0, i, count = 0;
846 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
852 if (w == NULL) goto err;
854 if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
862 if (p[0] & 0x1) /* m is odd */
864 /* compute half-trace of a */
865 if (!BN_copy(z, a)) goto err;
866 for (i = 1; i <= (p[0] - 1) / 2; i++)
868 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
869 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
870 if (!BN_GF2m_add(z, z, a)) goto err;
876 rho = BN_CTX_get(ctx);
877 w2 = BN_CTX_get(ctx);
878 tmp = BN_CTX_get(ctx);
879 if (tmp == NULL) goto err;
882 if (!BN_rand(rho, p[0], 0, 0)) goto err;
883 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
884 if (!BN_zero(z)) goto err;
885 if (!BN_copy(w, rho)) goto err;
886 for (i = 1; i <= p[0] - 1; i++)
888 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
889 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
890 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
891 if (!BN_GF2m_add(z, z, tmp)) goto err;
892 if (!BN_GF2m_add(w, w2, rho)) goto err;
895 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
898 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
903 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
904 if (!BN_GF2m_add(w, z, w)) goto err;
905 if (BN_GF2m_cmp(w, a)) goto err;
907 if (!BN_copy(r, z)) goto err;
916 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
918 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
919 * function is only provided for convenience; for best performance, use the
920 * BN_GF2m_mod_solve_quad_arr function.
922 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
924 const int max = BN_num_bits(p);
925 unsigned int *arr=NULL, ret = 0;
926 if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
927 if (BN_GF2m_poly2arr(p, arr, max) > max)
929 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
932 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
934 if (arr) OPENSSL_free(arr);
938 /* Convert the bit-string representation of a polynomial a into an array
939 * of integers corresponding to the bits with non-zero coefficient.
940 * Up to max elements of the array will be filled. Return value is total
941 * number of coefficients that would be extracted if array was large enough.
943 int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max)
948 for (k = 0; k < max; k++) p[k] = 0;
951 for (i = a->top - 1; i >= 0; i--)
954 for (j = BN_BITS2 - 1; j >= 0; j--)
958 if (k < max) p[k] = BN_BITS2 * i + j;
968 /* Convert the coefficient array representation of a polynomial to a
969 * bit-string. The array must be terminated by 0.
971 int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a)
976 for (i = 0; p[i] > 0; i++)