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) 1998-2018 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/)"
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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).
98 #ifndef OPENSSL_NO_EC2M
101 * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should
104 # define MAX_ITERATIONS 50
106 # define SQR_nibble(w) ((((w) & 8) << 3) \
112 /* Platform-specific macros to accelerate squaring. */
113 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
115 SQR_nibble((w) >> 60) << 56 | SQR_nibble((w) >> 56) << 48 | \
116 SQR_nibble((w) >> 52) << 40 | SQR_nibble((w) >> 48) << 32 | \
117 SQR_nibble((w) >> 44) << 24 | SQR_nibble((w) >> 40) << 16 | \
118 SQR_nibble((w) >> 36) << 8 | SQR_nibble((w) >> 32)
120 SQR_nibble((w) >> 28) << 56 | SQR_nibble((w) >> 24) << 48 | \
121 SQR_nibble((w) >> 20) << 40 | SQR_nibble((w) >> 16) << 32 | \
122 SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >> 8) << 16 | \
123 SQR_nibble((w) >> 4) << 8 | SQR_nibble((w) )
125 # ifdef THIRTY_TWO_BIT
127 SQR_nibble((w) >> 28) << 24 | SQR_nibble((w) >> 24) << 16 | \
128 SQR_nibble((w) >> 20) << 8 | SQR_nibble((w) >> 16)
130 SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >> 8) << 16 | \
131 SQR_nibble((w) >> 4) << 8 | SQR_nibble((w) )
134 # if !defined(OPENSSL_BN_ASM_GF2m)
136 * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is
137 * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that
138 * the variables have the right amount of space allocated.
140 # ifdef THIRTY_TWO_BIT
141 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
144 register BN_ULONG h, l, s;
145 BN_ULONG tab[8], top2b = a >> 30;
146 register BN_ULONG a1, a2, a4;
148 a1 = a & (0x3FFFFFFF);
159 tab[7] = a1 ^ a2 ^ a4;
163 s = tab[b >> 3 & 0x7];
166 s = tab[b >> 6 & 0x7];
169 s = tab[b >> 9 & 0x7];
172 s = tab[b >> 12 & 0x7];
175 s = tab[b >> 15 & 0x7];
178 s = tab[b >> 18 & 0x7];
181 s = tab[b >> 21 & 0x7];
184 s = tab[b >> 24 & 0x7];
187 s = tab[b >> 27 & 0x7];
194 /* compensate for the top two bits of a */
209 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
210 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
213 register BN_ULONG h, l, s;
214 BN_ULONG tab[16], top3b = a >> 61;
215 register BN_ULONG a1, a2, a4, a8;
217 a1 = a & (0x1FFFFFFFFFFFFFFFULL);
229 tab[7] = a1 ^ a2 ^ a4;
233 tab[11] = a1 ^ a2 ^ a8;
235 tab[13] = a1 ^ a4 ^ a8;
236 tab[14] = a2 ^ a4 ^ a8;
237 tab[15] = a1 ^ a2 ^ a4 ^ a8;
241 s = tab[b >> 4 & 0xF];
244 s = tab[b >> 8 & 0xF];
247 s = tab[b >> 12 & 0xF];
250 s = tab[b >> 16 & 0xF];
253 s = tab[b >> 20 & 0xF];
256 s = tab[b >> 24 & 0xF];
259 s = tab[b >> 28 & 0xF];
262 s = tab[b >> 32 & 0xF];
265 s = tab[b >> 36 & 0xF];
268 s = tab[b >> 40 & 0xF];
271 s = tab[b >> 44 & 0xF];
274 s = tab[b >> 48 & 0xF];
277 s = tab[b >> 52 & 0xF];
280 s = tab[b >> 56 & 0xF];
287 /* compensate for the top three bits of a */
308 * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
309 * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST
310 * ensure that the variables have the right amount of space allocated.
312 static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0,
313 const BN_ULONG b1, const BN_ULONG b0)
316 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
317 bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1);
318 bn_GF2m_mul_1x1(r + 1, r, a0, b0);
319 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
320 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
321 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
322 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
325 void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1,
330 * Add polynomials a and b and store result in r; r could be a or b, a and b
331 * could be equal; r is the bitwise XOR of a and b.
333 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
336 const BIGNUM *at, *bt;
341 if (a->top < b->top) {
349 if (bn_wexpand(r, at->top) == NULL)
352 for (i = 0; i < bt->top; i++) {
353 r->d[i] = at->d[i] ^ bt->d[i];
355 for (; i < at->top; i++) {
366 * Some functions allow for representation of the irreducible polynomials
367 * as an int[], say p. The irreducible f(t) is then of the form:
368 * t^p[0] + t^p[1] + ... + t^p[k]
369 * where m = p[0] > p[1] > ... > p[k] = 0.
372 /* Performs modular reduction of a and store result in r. r could be a. */
373 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
382 /* reduction mod 1 => return 0 */
388 * Since the algorithm does reduction in the r value, if a != r, copy the
389 * contents of a into r so we can do reduction in r.
392 if (!bn_wexpand(r, a->top))
394 for (j = 0; j < a->top; j++) {
401 /* start reduction */
402 dN = p[0] / BN_BITS2;
403 for (j = r->top - 1; j > dN;) {
411 for (k = 1; p[k] != 0; k++) {
412 /* reducing component t^p[k] */
417 z[j - n] ^= (zz >> d0);
419 z[j - n - 1] ^= (zz << d1);
422 /* reducing component t^0 */
424 d0 = p[0] % BN_BITS2;
426 z[j - n] ^= (zz >> d0);
428 z[j - n - 1] ^= (zz << d1);
431 /* final round of reduction */
434 d0 = p[0] % BN_BITS2;
440 /* clear up the top d1 bits */
442 z[dN] = (z[dN] << d1) >> d1;
445 z[0] ^= zz; /* reduction t^0 component */
447 for (k = 1; p[k] != 0; k++) {
450 /* reducing component t^p[k] */
452 d0 = p[k] % BN_BITS2;
455 if (d0 && (tmp_ulong = zz >> d1))
456 z[n + 1] ^= tmp_ulong;
466 * Performs modular reduction of a by p and store result in r. r could be a.
467 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
468 * function is only provided for convenience; for best performance, use the
469 * BN_GF2m_mod_arr function.
471 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
477 ret = BN_GF2m_poly2arr(p, arr, sizeof(arr) / sizeof(arr[0]));
478 if (!ret || ret > (int)(sizeof(arr) / sizeof(arr[0]))) {
479 BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH);
482 ret = BN_GF2m_mod_arr(r, a, arr);
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 be b.
491 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
492 const int p[], BN_CTX *ctx)
494 int zlen, i, j, k, ret = 0;
496 BN_ULONG x1, x0, y1, y0, zz[4];
502 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
506 if ((s = BN_CTX_get(ctx)) == NULL)
509 zlen = a->top + b->top + 4;
510 if (!bn_wexpand(s, zlen))
514 for (i = 0; i < zlen; i++)
517 for (j = 0; j < b->top; j += 2) {
519 y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1];
520 for (i = 0; i < a->top; i += 2) {
522 x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1];
523 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
524 for (k = 0; k < 4; k++)
525 s->d[i + j + k] ^= zz[k];
530 if (BN_GF2m_mod_arr(r, s, p))
540 * Compute the product of two polynomials a and b, reduce modulo p, and store
541 * the result in r. r could be a or b; a could equal b. This function calls
542 * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is
543 * only provided for convenience; for best performance, use the
544 * BN_GF2m_mod_mul_arr function.
546 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
547 const BIGNUM *p, BN_CTX *ctx)
550 const int max = BN_num_bits(p) + 1;
555 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
557 ret = BN_GF2m_poly2arr(p, arr, max);
558 if (!ret || ret > max) {
559 BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH);
562 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
570 /* Square a, reduce the result mod p, and store it in a. r could be a. */
571 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[],
579 if ((s = BN_CTX_get(ctx)) == NULL)
581 if (!bn_wexpand(s, 2 * a->top))
584 for (i = a->top - 1; i >= 0; i--) {
585 s->d[2 * i + 1] = SQR1(a->d[i]);
586 s->d[2 * i] = SQR0(a->d[i]);
591 if (!BN_GF2m_mod_arr(r, s, p))
601 * Square a, reduce the result mod p, and store it in a. r could be a. This
602 * function calls down to the BN_GF2m_mod_sqr_arr implementation; this
603 * wrapper function is only provided for convenience; for best performance,
604 * use the BN_GF2m_mod_sqr_arr function.
606 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
609 const int max = BN_num_bits(p) + 1;
614 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
616 ret = BN_GF2m_poly2arr(p, arr, max);
617 if (!ret || ret > max) {
618 BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH);
621 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
630 * Invert a, reduce modulo p, and store the result in r. r could be a. Uses
631 * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D.,
632 * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic
633 * Curve Cryptography Over Binary Fields".
635 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
637 BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp;
645 if ((b = BN_CTX_get(ctx)) == NULL)
647 if ((c = BN_CTX_get(ctx)) == NULL)
649 if ((u = BN_CTX_get(ctx)) == NULL)
651 if ((v = BN_CTX_get(ctx)) == NULL)
654 if (!BN_GF2m_mod(u, a, p))
666 while (!BN_is_odd(u)) {
669 if (!BN_rshift1(u, u))
672 if (!BN_GF2m_add(b, b, p))
675 if (!BN_rshift1(b, b))
679 if (BN_abs_is_word(u, 1))
682 if (BN_num_bits(u) < BN_num_bits(v)) {
691 if (!BN_GF2m_add(u, u, v))
693 if (!BN_GF2m_add(b, b, c))
699 int ubits = BN_num_bits(u);
700 int vbits = BN_num_bits(v); /* v is copy of p */
702 BN_ULONG *udp, *bdp, *vdp, *cdp;
704 if (!bn_wexpand(u, top))
707 for (i = u->top; i < top; i++)
710 if (!bn_wexpand(b, top))
714 for (i = 1; i < top; i++)
717 if (!bn_wexpand(c, top))
720 for (i = 0; i < top; i++)
723 vdp = v->d; /* It pays off to "cache" *->d pointers,
724 * because it allows optimizer to be more
725 * aggressive. But we don't have to "cache"
726 * p->d, because *p is declared 'const'... */
728 while (ubits && !(udp[0] & 1)) {
729 BN_ULONG u0, u1, b0, b1, mask;
733 mask = (BN_ULONG)0 - (b0 & 1);
734 b0 ^= p->d[0] & mask;
735 for (i = 0; i < top - 1; i++) {
737 udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2;
739 b1 = bdp[i + 1] ^ (p->d[i + 1] & mask);
740 bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2;
748 if (ubits <= BN_BITS2) {
749 if (udp[0] == 0) /* poly was reducible */
770 for (i = 0; i < top; i++) {
774 if (ubits == vbits) {
776 int utop = (ubits - 1) / BN_BITS2;
778 while ((ul = udp[utop]) == 0 && utop)
780 ubits = utop * BN_BITS2 + BN_num_bits_word(ul);
793 # ifdef BN_DEBUG /* BN_CTX_end would complain about the
804 * Invert xx, reduce modulo p, and store the result in r. r could be xx.
805 * This function calls down to the BN_GF2m_mod_inv implementation; this
806 * wrapper function is only provided for convenience; for best performance,
807 * use the BN_GF2m_mod_inv function.
809 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[],
817 if ((field = BN_CTX_get(ctx)) == NULL)
819 if (!BN_GF2m_arr2poly(p, field))
822 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
830 # ifndef OPENSSL_SUN_GF2M_DIV
832 * Divide y by x, reduce modulo p, and store the result in r. r could be x
833 * or y, x could equal y.
835 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x,
836 const BIGNUM *p, BN_CTX *ctx)
846 xinv = BN_CTX_get(ctx);
850 if (!BN_GF2m_mod_inv(xinv, x, p, ctx))
852 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx))
863 * Divide y by x, reduce modulo p, and store the result in r. r could be x
864 * or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from
865 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the
868 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x,
869 const BIGNUM *p, BN_CTX *ctx)
871 BIGNUM *a, *b, *u, *v;
887 /* reduce x and y mod p */
888 if (!BN_GF2m_mod(u, y, p))
890 if (!BN_GF2m_mod(a, x, p))
895 while (!BN_is_odd(a)) {
896 if (!BN_rshift1(a, a))
899 if (!BN_GF2m_add(u, u, p))
901 if (!BN_rshift1(u, u))
906 if (BN_GF2m_cmp(b, a) > 0) {
907 if (!BN_GF2m_add(b, b, a))
909 if (!BN_GF2m_add(v, v, u))
912 if (!BN_rshift1(b, b))
915 if (!BN_GF2m_add(v, v, p))
917 if (!BN_rshift1(v, v))
919 } while (!BN_is_odd(b));
920 } else if (BN_abs_is_word(a, 1))
923 if (!BN_GF2m_add(a, a, b))
925 if (!BN_GF2m_add(u, u, v))
928 if (!BN_rshift1(a, a))
931 if (!BN_GF2m_add(u, u, p))
933 if (!BN_rshift1(u, u))
935 } while (!BN_is_odd(a));
951 * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
952 * * or yy, xx could equal yy. This function calls down to the
953 * BN_GF2m_mod_div implementation; this wrapper function is only provided for
954 * convenience; for best performance, use the BN_GF2m_mod_div function.
956 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx,
957 const int p[], BN_CTX *ctx)
966 if ((field = BN_CTX_get(ctx)) == NULL)
968 if (!BN_GF2m_arr2poly(p, field))
971 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
980 * Compute the bth power of a, reduce modulo p, and store the result in r. r
981 * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE
984 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
985 const int p[], BN_CTX *ctx)
996 if (BN_abs_is_word(b, 1))
997 return (BN_copy(r, a) != NULL);
1000 if ((u = BN_CTX_get(ctx)) == NULL)
1003 if (!BN_GF2m_mod_arr(u, a, p))
1006 n = BN_num_bits(b) - 1;
1007 for (i = n - 1; i >= 0; i--) {
1008 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx))
1010 if (BN_is_bit_set(b, i)) {
1011 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx))
1025 * Compute the bth power of a, reduce modulo p, and store the result in r. r
1026 * could be a. This function calls down to the BN_GF2m_mod_exp_arr
1027 * implementation; this wrapper function is only provided for convenience;
1028 * for best performance, use the BN_GF2m_mod_exp_arr function.
1030 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
1031 const BIGNUM *p, BN_CTX *ctx)
1034 const int max = BN_num_bits(p) + 1;
1039 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
1041 ret = BN_GF2m_poly2arr(p, arr, max);
1042 if (!ret || ret > max) {
1043 BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH);
1046 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
1055 * Compute the square root of a, reduce modulo p, and store the result in r.
1056 * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
1058 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[],
1067 /* reduction mod 1 => return 0 */
1073 if ((u = BN_CTX_get(ctx)) == NULL)
1076 if (!BN_set_bit(u, p[0] - 1))
1078 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
1087 * Compute the square root of a, reduce modulo p, and store the result in r.
1088 * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr
1089 * implementation; this wrapper function is only provided for convenience;
1090 * for best performance, use the BN_GF2m_mod_sqrt_arr function.
1092 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1095 const int max = BN_num_bits(p) + 1;
1099 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
1101 ret = BN_GF2m_poly2arr(p, arr, max);
1102 if (!ret || ret > max) {
1103 BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH);
1106 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
1115 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
1116 * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
1118 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[],
1121 int ret = 0, count = 0, j;
1122 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
1127 /* reduction mod 1 => return 0 */
1133 a = BN_CTX_get(ctx);
1134 z = BN_CTX_get(ctx);
1135 w = BN_CTX_get(ctx);
1139 if (!BN_GF2m_mod_arr(a, a_, p))
1142 if (BN_is_zero(a)) {
1148 if (p[0] & 0x1) { /* m is odd */
1149 /* compute half-trace of a */
1152 for (j = 1; j <= (p[0] - 1) / 2; j++) {
1153 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1155 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1157 if (!BN_GF2m_add(z, z, a))
1161 } else { /* m is even */
1163 rho = BN_CTX_get(ctx);
1164 w2 = BN_CTX_get(ctx);
1165 tmp = BN_CTX_get(ctx);
1169 if (!BN_rand(rho, p[0], 0, 0))
1171 if (!BN_GF2m_mod_arr(rho, rho, p))
1174 if (!BN_copy(w, rho))
1176 for (j = 1; j <= p[0] - 1; j++) {
1177 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1179 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx))
1181 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx))
1183 if (!BN_GF2m_add(z, z, tmp))
1185 if (!BN_GF2m_add(w, w2, rho))
1189 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
1190 if (BN_is_zero(w)) {
1191 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS);
1196 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx))
1198 if (!BN_GF2m_add(w, z, w))
1200 if (BN_GF2m_cmp(w, a)) {
1201 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
1217 * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns
1218 * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr
1219 * implementation; this wrapper function is only provided for convenience;
1220 * for best performance, use the BN_GF2m_mod_solve_quad_arr function.
1222 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
1226 const int max = BN_num_bits(p) + 1;
1230 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL)
1232 ret = BN_GF2m_poly2arr(p, arr, max);
1233 if (!ret || ret > max) {
1234 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH);
1237 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1246 * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i *
1247 * x^i) into an array of integers corresponding to the bits with non-zero
1248 * coefficient. Array is terminated with -1. Up to max elements of the array
1249 * will be filled. Return value is total number of array elements that would
1250 * be filled if array was large enough.
1252 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
1260 for (i = a->top - 1; i >= 0; i--) {
1262 /* skip word if a->d[i] == 0 */
1265 for (j = BN_BITS2 - 1; j >= 0; j--) {
1266 if (a->d[i] & mask) {
1268 p[k] = BN_BITS2 * i + j;
1284 * Convert the coefficient array representation of a polynomial to a
1285 * bit-string. The array must be terminated by -1.
1287 int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1293 for (i = 0; p[i] != -1; i++) {
1294 if (BN_set_bit(a, p[i]) == 0)