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.
31 * NOTE: This file is licensed pursuant to the OpenSSL license below and may
32 * be modified; but after modifications, the above covenant may no longer
33 * apply! In such cases, the corresponding paragraph ["In addition, Sun
34 * covenants ... causes the infringement."] and this note can be edited out;
35 * but please keep the Sun copyright notice and attribution.
38 /* ====================================================================
39 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
41 * Redistribution and use in source and binary forms, with or without
42 * modification, are permitted provided that the following conditions
45 * 1. Redistributions of source code must retain the above copyright
46 * notice, this list of conditions and the following disclaimer.
48 * 2. Redistributions in binary form must reproduce the above copyright
49 * notice, this list of conditions and the following disclaimer in
50 * the documentation and/or other materials provided with the
53 * 3. All advertising materials mentioning features or use of this
54 * software must display the following acknowledgment:
55 * "This product includes software developed by the OpenSSL Project
56 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
58 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
59 * endorse or promote products derived from this software without
60 * prior written permission. For written permission, please contact
61 * openssl-core@openssl.org.
63 * 5. Products derived from this software may not be called "OpenSSL"
64 * nor may "OpenSSL" appear in their names without prior written
65 * permission of the OpenSSL Project.
67 * 6. Redistributions of any form whatsoever must retain the following
69 * "This product includes software developed by the OpenSSL Project
70 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
72 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
73 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
74 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
75 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
76 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
77 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
78 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
79 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
80 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
81 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
82 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
83 * OF THE POSSIBILITY OF SUCH DAMAGE.
84 * ====================================================================
86 * This product includes cryptographic software written by Eric Young
87 * (eay@cryptsoft.com). This product includes software written by Tim
88 * Hudson (tjh@cryptsoft.com).
95 #include "internal/cryptlib.h"
98 #ifndef OPENSSL_NO_EC2M
101 * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should
104 # define MAX_ITERATIONS 50
106 static const BN_ULONG SQR_tb[16] = { 0, 1, 4, 5, 16, 17, 20, 21,
107 64, 65, 68, 69, 80, 81, 84, 85
110 /* Platform-specific macros to accelerate squaring. */
111 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
113 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
114 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
115 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
116 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
118 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
119 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
120 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
121 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
123 # ifdef THIRTY_TWO_BIT
125 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
126 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
128 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
129 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
132 # if !defined(OPENSSL_BN_ASM_GF2m)
134 * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is
135 * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that
136 * the variables have the right amount of space allocated.
138 # ifdef THIRTY_TWO_BIT
139 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
142 register BN_ULONG h, l, s;
143 BN_ULONG tab[8], top2b = a >> 30;
144 register BN_ULONG a1, a2, a4;
146 a1 = a & (0x3FFFFFFF);
157 tab[7] = a1 ^ a2 ^ a4;
161 s = tab[b >> 3 & 0x7];
164 s = tab[b >> 6 & 0x7];
167 s = tab[b >> 9 & 0x7];
170 s = tab[b >> 12 & 0x7];
173 s = tab[b >> 15 & 0x7];
176 s = tab[b >> 18 & 0x7];
179 s = tab[b >> 21 & 0x7];
182 s = tab[b >> 24 & 0x7];
185 s = tab[b >> 27 & 0x7];
192 /* compensate for the top two bits of a */
207 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
208 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
211 register BN_ULONG h, l, s;
212 BN_ULONG tab[16], top3b = a >> 61;
213 register BN_ULONG a1, a2, a4, a8;
215 a1 = a & (0x1FFFFFFFFFFFFFFFULL);
227 tab[7] = a1 ^ a2 ^ a4;
231 tab[11] = a1 ^ a2 ^ a8;
233 tab[13] = a1 ^ a4 ^ a8;
234 tab[14] = a2 ^ a4 ^ a8;
235 tab[15] = a1 ^ a2 ^ a4 ^ a8;
239 s = tab[b >> 4 & 0xF];
242 s = tab[b >> 8 & 0xF];
245 s = tab[b >> 12 & 0xF];
248 s = tab[b >> 16 & 0xF];
251 s = tab[b >> 20 & 0xF];
254 s = tab[b >> 24 & 0xF];
257 s = tab[b >> 28 & 0xF];
260 s = tab[b >> 32 & 0xF];
263 s = tab[b >> 36 & 0xF];
266 s = tab[b >> 40 & 0xF];
269 s = tab[b >> 44 & 0xF];
272 s = tab[b >> 48 & 0xF];
275 s = tab[b >> 52 & 0xF];
278 s = tab[b >> 56 & 0xF];
285 /* compensate for the top three bits of a */
306 * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
307 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST
308 * ensure that the variables have the right amount of space allocated.
310 static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0,
311 const BN_ULONG b1, const BN_ULONG b0)
314 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
315 bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1);
316 bn_GF2m_mul_1x1(r + 1, r, a0, b0);
317 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
318 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
319 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
320 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
323 void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1,
328 * Add polynomials a and b and store result in r; r could be a or b, a and b
329 * could be equal; r is the bitwise XOR of a and b.
331 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
334 const BIGNUM *at, *bt;
339 if (a->top < b->top) {
347 if (bn_wexpand(r, at->top) == NULL)
350 for (i = 0; i < bt->top; i++) {
351 r->d[i] = at->d[i] ^ bt->d[i];
353 for (; i < at->top; i++) {
364 * Some functions allow for representation of the irreducible polynomials
365 * as an int[], say p. The irreducible f(t) is then of the form:
366 * t^p[0] + t^p[1] + ... + t^p[k]
367 * where m = p[0] > p[1] > ... > p[k] = 0.
370 /* Performs modular reduction of a and store result in r. r could be a. */
371 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
380 /* reduction mod 1 => return 0 */
386 * Since the algorithm does reduction in the r value, if a != r, copy the
387 * contents of a into r so we can do reduction in r.
390 if (!bn_wexpand(r, a->top))
392 for (j = 0; j < a->top; j++) {
399 /* start reduction */
400 dN = p[0] / BN_BITS2;
401 for (j = r->top - 1; j > dN;) {
409 for (k = 1; p[k] != 0; k++) {
410 /* reducing component t^p[k] */
415 z[j - n] ^= (zz >> d0);
417 z[j - n - 1] ^= (zz << d1);
420 /* reducing component t^0 */
422 d0 = p[0] % BN_BITS2;
424 z[j - n] ^= (zz >> d0);
426 z[j - n - 1] ^= (zz << d1);
429 /* final round of reduction */
432 d0 = p[0] % BN_BITS2;
438 /* clear up the top d1 bits */
440 z[dN] = (z[dN] << d1) >> d1;
443 z[0] ^= zz; /* reduction t^0 component */
445 for (k = 1; p[k] != 0; k++) {
448 /* reducing component t^p[k] */
450 d0 = p[k] % BN_BITS2;
453 if (d0 && (tmp_ulong = zz >> d1))
454 z[n + 1] ^= tmp_ulong;
464 * Performs modular reduction of a by p and store result in r. r could be a.
465 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
466 * function is only provided for convenience; for best performance, use the
467 * BN_GF2m_mod_arr function.
469 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
475 ret = BN_GF2m_poly2arr(p, arr, OSSL_NELEM(arr));
476 if (!ret || ret > (int)OSSL_NELEM(arr)) {
477 BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH);
480 ret = BN_GF2m_mod_arr(r, a, arr);
486 * Compute the product of two polynomials a and b, reduce modulo p, and store
487 * the result in r. r could be a or b; a could be b.
489 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
490 const int p[], BN_CTX *ctx)
492 int zlen, i, j, k, ret = 0;
494 BN_ULONG x1, x0, y1, y0, zz[4];
500 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
504 if ((s = BN_CTX_get(ctx)) == NULL)
507 zlen = a->top + b->top + 4;
508 if (!bn_wexpand(s, zlen))
512 for (i = 0; i < zlen; i++)
515 for (j = 0; j < b->top; j += 2) {
517 y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1];
518 for (i = 0; i < a->top; i += 2) {
520 x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1];
521 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
522 for (k = 0; k < 4; k++)
523 s->d[i + j + k] ^= zz[k];
528 if (BN_GF2m_mod_arr(r, s, p))
538 * Compute the product of two polynomials a and b, reduce modulo p, and store
539 * the result in r. r could be a or b; a could equal b. This function calls
540 * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is
541 * only provided for convenience; for best performance, use the
542 * BN_GF2m_mod_mul_arr function.
544 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
545 const BIGNUM *p, BN_CTX *ctx)
548 const int max = BN_num_bits(p) + 1;
553 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
555 ret = BN_GF2m_poly2arr(p, arr, max);
556 if (!ret || ret > max) {
557 BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH);
560 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
567 /* Square a, reduce the result mod p, and store it in a. r could be a. */
568 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[],
576 if ((s = BN_CTX_get(ctx)) == NULL)
578 if (!bn_wexpand(s, 2 * a->top))
581 for (i = a->top - 1; i >= 0; i--) {
582 s->d[2 * i + 1] = SQR1(a->d[i]);
583 s->d[2 * i] = SQR0(a->d[i]);
588 if (!BN_GF2m_mod_arr(r, s, p))
598 * Square a, reduce the result mod p, and store it in a. r could be a. This
599 * function calls down to the BN_GF2m_mod_sqr_arr implementation; this
600 * wrapper function is only provided for convenience; for best performance,
601 * use the BN_GF2m_mod_sqr_arr function.
603 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
606 const int max = BN_num_bits(p) + 1;
611 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
613 ret = BN_GF2m_poly2arr(p, arr, max);
614 if (!ret || ret > max) {
615 BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH);
618 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
626 * Invert a, reduce modulo p, and store the result in r. r could be a. Uses
627 * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D.,
628 * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic
629 * Curve Cryptography Over Binary Fields".
631 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
633 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp;
641 if ((b = BN_CTX_get(ctx)) == NULL)
643 if ((c = BN_CTX_get(ctx)) == NULL)
645 if ((u = BN_CTX_get(ctx)) == NULL)
647 if ((v = BN_CTX_get(ctx)) == NULL)
650 if (!BN_GF2m_mod(u, a, p))
662 while (!BN_is_odd(u)) {
665 if (!BN_rshift1(u, u))
668 if (!BN_GF2m_add(b, b, p))
671 if (!BN_rshift1(b, b))
675 if (BN_abs_is_word(u, 1))
678 if (BN_num_bits(u) < BN_num_bits(v)) {
687 if (!BN_GF2m_add(u, u, v))
689 if (!BN_GF2m_add(b, b, c))
695 int ubits = BN_num_bits(u);
696 int vbits = BN_num_bits(v); /* v is copy of p */
698 BN_ULONG *udp, *bdp, *vdp, *cdp;
700 if (!bn_wexpand(u, top))
703 for (i = u->top; i < top; i++)
706 if (!bn_wexpand(b, top))
710 for (i = 1; i < top; i++)
713 if (!bn_wexpand(c, top))
716 for (i = 0; i < top; i++)
719 vdp = v->d; /* It pays off to "cache" *->d pointers,
720 * because it allows optimizer to be more
721 * aggressive. But we don't have to "cache"
722 * p->d, because *p is declared 'const'... */
724 while (ubits && !(udp[0] & 1)) {
725 BN_ULONG u0, u1, b0, b1, mask;
729 mask = (BN_ULONG)0 - (b0 & 1);
730 b0 ^= p->d[0] & mask;
731 for (i = 0; i < top - 1; i++) {
733 udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2;
735 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask);
736 bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2;
744 if (ubits <= BN_BITS2) {
745 if (udp[0] == 0) /* poly was reducible */
766 for (i = 0; i < top; i++) {
770 if (ubits == vbits) {
772 int utop = (ubits - 1) / BN_BITS2;
774 while ((ul = udp[utop]) == 0 && utop)
776 ubits = utop * BN_BITS2 + BN_num_bits_word(ul);
789 # ifdef BN_DEBUG /* BN_CTX_end would complain about the
800 * Invert xx, reduce modulo p, and store the result in r. r could be xx.
801 * This function calls down to the BN_GF2m_mod_inv implementation; this
802 * wrapper function is only provided for convenience; for best performance,
803 * use the BN_GF2m_mod_inv function.
805 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[],
813 if ((field = BN_CTX_get(ctx)) == NULL)
815 if (!BN_GF2m_arr2poly(p, field))
818 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
826 # ifndef OPENSSL_SUN_GF2M_DIV
828 * Divide y by x, reduce modulo p, and store the result in r. r could be x
829 * or y, x could equal y.
831 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x,
832 const BIGNUM *p, BN_CTX *ctx)
842 xinv = BN_CTX_get(ctx);
846 if (!BN_GF2m_mod_inv(xinv, x, p, ctx))
848 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx))
859 * Divide y by x, reduce modulo p, and store the result in r. r could be x
860 * or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from
861 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the
864 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x,
865 const BIGNUM *p, BN_CTX *ctx)
867 BIGNUM *a, *b, *u, *v;
883 /* reduce x and y mod p */
884 if (!BN_GF2m_mod(u, y, p))
886 if (!BN_GF2m_mod(a, x, p))
891 while (!BN_is_odd(a)) {
892 if (!BN_rshift1(a, a))
895 if (!BN_GF2m_add(u, u, p))
897 if (!BN_rshift1(u, u))
902 if (BN_GF2m_cmp(b, a) > 0) {
903 if (!BN_GF2m_add(b, b, a))
905 if (!BN_GF2m_add(v, v, u))
908 if (!BN_rshift1(b, b))
911 if (!BN_GF2m_add(v, v, p))
913 if (!BN_rshift1(v, v))
915 } while (!BN_is_odd(b));
916 } else if (BN_abs_is_word(a, 1))
919 if (!BN_GF2m_add(a, a, b))
921 if (!BN_GF2m_add(u, u, v))
924 if (!BN_rshift1(a, a))
927 if (!BN_GF2m_add(u, u, p))
929 if (!BN_rshift1(u, u))
931 } while (!BN_is_odd(a));
947 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
948 * * or yy, xx could equal yy. This function calls down to the
949 * BN_GF2m_mod_div implementation; this wrapper function is only provided for
950 * convenience; for best performance, use the BN_GF2m_mod_div function.
952 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx,
953 const int p[], BN_CTX *ctx)
962 if ((field = BN_CTX_get(ctx)) == NULL)
964 if (!BN_GF2m_arr2poly(p, field))
967 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
976 * Compute the bth power of a, reduce modulo p, and store the result in r. r
977 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE
980 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
981 const int p[], BN_CTX *ctx)
992 if (BN_abs_is_word(b, 1))
993 return (BN_copy(r, a) != NULL);
996 if ((u = BN_CTX_get(ctx)) == NULL)
999 if (!BN_GF2m_mod_arr(u, a, p))
1002 n = BN_num_bits(b) - 1;
1003 for (i = n - 1; i >= 0; i--) {
1004 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx))
1006 if (BN_is_bit_set(b, i)) {
1007 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx))
1021 * Compute the bth power of a, reduce modulo p, and store the result in r. r
1022 * could be a. This function calls down to the BN_GF2m_mod_exp_arr
1023 * implementation; this wrapper function is only provided for convenience;
1024 * for best performance, use the BN_GF2m_mod_exp_arr function.
1026 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
1027 const BIGNUM *p, BN_CTX *ctx)
1030 const int max = BN_num_bits(p) + 1;
1035 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
1037 ret = BN_GF2m_poly2arr(p, arr, max);
1038 if (!ret || ret > max) {
1039 BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH);
1042 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
1050 * Compute the square root of a, reduce modulo p, and store the result in r.
1051 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
1053 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[],
1062 /* reduction mod 1 => return 0 */
1068 if ((u = BN_CTX_get(ctx)) == NULL)
1071 if (!BN_set_bit(u, p[0] - 1))
1073 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
1082 * Compute the square root of a, reduce modulo p, and store the result in r.
1083 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr
1084 * implementation; this wrapper function is only provided for convenience;
1085 * for best performance, use the BN_GF2m_mod_sqrt_arr function.
1087 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1090 const int max = BN_num_bits(p) + 1;
1094 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
1096 ret = BN_GF2m_poly2arr(p, arr, max);
1097 if (!ret || ret > max) {
1098 BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH);
1101 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
1109 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
1110 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
1112 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[],
1115 int ret = 0, count = 0, j;
1116 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
1121 /* reduction mod 1 => return 0 */
1127 a = BN_CTX_get(ctx);
1128 z = BN_CTX_get(ctx);
1129 w = BN_CTX_get(ctx);
1133 if (!BN_GF2m_mod_arr(a, a_, p))
1136 if (BN_is_zero(a)) {
1142 if (p[0] & 0x1) { /* m is odd */
1143 /* compute half-trace of a */
1146 for (j = 1; j <= (p[0] - 1) / 2; j++) {
1147 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1149 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1151 if (!BN_GF2m_add(z, z, a))
1155 } else { /* m is even */
1157 rho = BN_CTX_get(ctx);
1158 w2 = BN_CTX_get(ctx);
1159 tmp = BN_CTX_get(ctx);
1163 if (!BN_rand(rho, p[0], 0, 0))
1165 if (!BN_GF2m_mod_arr(rho, rho, p))
1168 if (!BN_copy(w, rho))
1170 for (j = 1; j <= p[0] - 1; j++) {
1171 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1173 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx))
1175 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx))
1177 if (!BN_GF2m_add(z, z, tmp))
1179 if (!BN_GF2m_add(w, w2, rho))
1183 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
1184 if (BN_is_zero(w)) {
1185 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS);
1190 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx))
1192 if (!BN_GF2m_add(w, z, w))
1194 if (BN_GF2m_cmp(w, a)) {
1195 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
1211 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
1212 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr
1213 * implementation; this wrapper function is only provided for convenience;
1214 * for best performance, use the BN_GF2m_mod_solve_quad_arr function.
1216 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
1220 const int max = BN_num_bits(p) + 1;
1224 if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
1226 ret = BN_GF2m_poly2arr(p, arr, max);
1227 if (!ret || ret > max) {
1228 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH);
1231 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1239 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i *
1240 * x^i) into an array of integers corresponding to the bits with non-zero
1241 * coefficient. Array is terminated with -1. Up to max elements of the array
1242 * will be filled. Return value is total number of array elements that would
1243 * be filled if array was large enough.
1245 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
1253 for (i = a->top - 1; i >= 0; i--) {
1255 /* skip word if a->d[i] == 0 */
1258 for (j = BN_BITS2 - 1; j >= 0; j--) {
1259 if (a->d[i] & mask) {
1261 p[k] = BN_BITS2 * i + j;
1277 * Convert the coefficient array representation of a polynomial to a
1278 * bit-string. The array must be terminated by -1.
1280 int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1286 for (i = 0; p[i] != -1; i++) {
1287 if (BN_set_bit(a, p[i]) == 0)