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 /* 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. */
37 /* ====================================================================
38 * Copyright (c) 1998-2002 The OpenSSL Project. All rights reserved.
40 * Redistribution and use in source and binary forms, with or without
41 * modification, are permitted provided that the following conditions
44 * 1. Redistributions of source code must retain the above copyright
45 * notice, this list of conditions and the following disclaimer.
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
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/)"
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.
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.
66 * 6. Redistributions of any form whatsoever must retain the following
68 * "This product includes software developed by the OpenSSL Project
69 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
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 * ====================================================================
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).
97 #ifndef OPENSSL_NO_EC2M
99 /* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
100 #define MAX_ITERATIONS 50
102 static const BN_ULONG SQR_tb[16] =
103 { 0, 1, 4, 5, 16, 17, 20, 21,
104 64, 65, 68, 69, 80, 81, 84, 85 };
105 /* Platform-specific macros to accelerate squaring. */
106 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
108 SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \
109 SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \
110 SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \
111 SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF]
113 SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \
114 SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \
115 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
116 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
118 #ifdef THIRTY_TWO_BIT
120 SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \
121 SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF]
123 SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \
124 SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF]
127 /* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
128 * result is a polynomial r with degree < 2 * BN_BITS - 1
129 * The caller MUST ensure that the variables have the right amount
130 * of space allocated.
132 #ifdef THIRTY_TWO_BIT
133 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
135 register BN_ULONG h, l, s;
136 BN_ULONG tab[8], top2b = a >> 30;
137 register BN_ULONG a1, a2, a4;
139 a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1;
141 tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1^a2;
142 tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4;
144 s = tab[b & 0x7]; l = s;
145 s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29;
146 s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26;
147 s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23;
148 s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20;
149 s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17;
150 s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14;
151 s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11;
152 s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8;
153 s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5;
154 s = tab[b >> 30 ]; l ^= s << 30; h ^= s >> 2;
156 /* compensate for the top two bits of a */
158 if (top2b & 01) { l ^= b << 30; h ^= b >> 2; }
159 if (top2b & 02) { l ^= b << 31; h ^= b >> 1; }
164 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
165 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
167 register BN_ULONG h, l, s;
168 BN_ULONG tab[16], top3b = a >> 61;
169 register BN_ULONG a1, a2, a4, a8;
171 a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1;
173 tab[ 0] = 0; tab[ 1] = a1; tab[ 2] = a2; tab[ 3] = a1^a2;
174 tab[ 4] = a4; tab[ 5] = a1^a4; tab[ 6] = a2^a4; tab[ 7] = a1^a2^a4;
175 tab[ 8] = a8; tab[ 9] = a1^a8; tab[10] = a2^a8; tab[11] = a1^a2^a8;
176 tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8;
178 s = tab[b & 0xF]; l = s;
179 s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60;
180 s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56;
181 s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52;
182 s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48;
183 s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44;
184 s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40;
185 s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36;
186 s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32;
187 s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28;
188 s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24;
189 s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20;
190 s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16;
191 s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12;
192 s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8;
193 s = tab[b >> 60 ]; l ^= s << 60; h ^= s >> 4;
195 /* compensate for the top three bits of a */
197 if (top3b & 01) { l ^= b << 61; h ^= b >> 3; }
198 if (top3b & 02) { l ^= b << 62; h ^= b >> 2; }
199 if (top3b & 04) { l ^= b << 63; h ^= b >> 1; }
205 /* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
206 * result is a polynomial r with degree < 4 * BN_BITS2 - 1
207 * The caller MUST ensure that the variables have the right amount
208 * of space allocated.
210 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)
213 /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
214 bn_GF2m_mul_1x1(r+3, r+2, a1, b1);
215 bn_GF2m_mul_1x1(r+1, r, a0, b0);
216 bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
217 /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
218 r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */
219 r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
223 /* Add polynomials a and b and store result in r; r could be a or b, a and b
224 * could be equal; r is the bitwise XOR of a and b.
226 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
229 const BIGNUM *at, *bt;
234 if (a->top < b->top) { at = b; bt = a; }
235 else { at = a; bt = b; }
237 if(bn_wexpand(r, at->top) == NULL)
240 for (i = 0; i < bt->top; i++)
242 r->d[i] = at->d[i] ^ bt->d[i];
244 for (; i < at->top; i++)
256 /* Some functions allow for representation of the irreducible polynomials
257 * as an int[], say p. The irreducible f(t) is then of the form:
258 * t^p[0] + t^p[1] + ... + t^p[k]
259 * where m = p[0] > p[1] > ... > p[k] = 0.
263 /* Performs modular reduction of a and store result in r. r could be a. */
264 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
274 /* reduction mod 1 => return 0 */
279 /* Since the algorithm does reduction in the r value, if a != r, copy
280 * the contents of a into r so we can do reduction in r.
284 if (!bn_wexpand(r, a->top)) return 0;
285 for (j = 0; j < a->top; j++)
293 /* start reduction */
294 dN = p[0] / BN_BITS2;
295 for (j = r->top - 1; j > dN;)
298 if (z[j] == 0) { j--; continue; }
301 for (k = 1; p[k] != 0; k++)
303 /* reducing component t^p[k] */
305 d0 = n % BN_BITS2; d1 = BN_BITS2 - d0;
308 if (d0) z[j-n-1] ^= (zz<<d1);
311 /* reducing component t^0 */
313 d0 = p[0] % BN_BITS2;
315 z[j-n] ^= (zz >> d0);
316 if (d0) z[j-n-1] ^= (zz << d1);
319 /* final round of reduction */
323 d0 = p[0] % BN_BITS2;
328 /* clear up the top d1 bits */
330 z[dN] = (z[dN] << d1) >> d1;
333 z[0] ^= zz; /* reduction t^0 component */
335 for (k = 1; p[k] != 0; k++)
339 /* reducing component t^p[k]*/
341 d0 = p[k] % BN_BITS2;
344 tmp_ulong = zz >> d1;
356 /* Performs modular reduction of a by p and store result in r. r could be a.
358 * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
359 * function is only provided for convenience; for best performance, use the
360 * BN_GF2m_mod_arr function.
362 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
365 const int max = BN_num_bits(p) + 1;
369 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
370 ret = BN_GF2m_poly2arr(p, arr, max);
371 if (!ret || ret > max)
373 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
376 ret = BN_GF2m_mod_arr(r, a, arr);
379 if (arr) OPENSSL_free(arr);
384 /* Compute the product of two polynomials a and b, reduce modulo p, and store
385 * the result in r. r could be a or b; a could be b.
387 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
389 int zlen, i, j, k, ret = 0;
391 BN_ULONG x1, x0, y1, y0, zz[4];
398 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
402 if ((s = BN_CTX_get(ctx)) == NULL) goto err;
404 zlen = a->top + b->top + 4;
405 if (!bn_wexpand(s, zlen)) goto err;
408 for (i = 0; i < zlen; i++) s->d[i] = 0;
410 for (j = 0; j < b->top; j += 2)
413 y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
414 for (i = 0; i < a->top; i += 2)
417 x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
418 bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
419 for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
424 if (BN_GF2m_mod_arr(r, s, p))
433 /* Compute the product of two polynomials a and b, reduce modulo p, and store
434 * the result in r. r could be a or b; a could equal b.
436 * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
437 * function is only provided for convenience; for best performance, use the
438 * BN_GF2m_mod_mul_arr function.
440 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
443 const int max = BN_num_bits(p) + 1;
448 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
449 ret = BN_GF2m_poly2arr(p, arr, max);
450 if (!ret || ret > max)
452 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
455 ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
458 if (arr) OPENSSL_free(arr);
463 /* Square a, reduce the result mod p, and store it in a. r could be a. */
464 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
471 if ((s = BN_CTX_get(ctx)) == NULL) return 0;
472 if (!bn_wexpand(s, 2 * a->top)) goto err;
474 for (i = a->top - 1; i >= 0; i--)
476 s->d[2*i+1] = SQR1(a->d[i]);
477 s->d[2*i ] = SQR0(a->d[i]);
482 if (!BN_GF2m_mod_arr(r, s, p)) goto err;
490 /* Square a, reduce the result mod p, and store it in a. r could be a.
492 * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
493 * function is only provided for convenience; for best performance, use the
494 * BN_GF2m_mod_sqr_arr function.
496 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
499 const int max = BN_num_bits(p) + 1;
504 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
505 ret = BN_GF2m_poly2arr(p, arr, max);
506 if (!ret || ret > max)
508 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
511 ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
514 if (arr) OPENSSL_free(arr);
519 /* Invert a, reduce modulo p, and store the result in r. r could be a.
520 * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
521 * Hankerson, D., Hernandez, J.L., and Menezes, A. "Software Implementation
522 * of Elliptic Curve Cryptography Over Binary Fields".
524 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
526 BIGNUM *b, *c, *u, *v, *tmp;
538 if (v == NULL) goto err;
540 if (!BN_one(b)) goto err;
541 if (!BN_GF2m_mod(u, a, p)) goto err;
542 if (!BN_copy(v, p)) goto err;
544 if (BN_is_zero(u)) goto err;
548 while (!BN_is_odd(u))
550 if (BN_is_zero(u)) goto err;
551 if (!BN_rshift1(u, u)) goto err;
554 if (!BN_GF2m_add(b, b, p)) goto err;
556 if (!BN_rshift1(b, b)) goto err;
559 if (BN_abs_is_word(u, 1)) break;
561 if (BN_num_bits(u) < BN_num_bits(v))
563 tmp = u; u = v; v = tmp;
564 tmp = b; b = c; c = tmp;
567 if (!BN_GF2m_add(u, u, v)) goto err;
568 if (!BN_GF2m_add(b, b, c)) goto err;
572 if (!BN_copy(r, b)) goto err;
581 /* Invert xx, reduce modulo p, and store the result in r. r could be xx.
583 * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
584 * function is only provided for convenience; for best performance, use the
585 * BN_GF2m_mod_inv function.
587 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx)
594 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
595 if (!BN_GF2m_arr2poly(p, field)) goto err;
597 ret = BN_GF2m_mod_inv(r, xx, field, ctx);
606 #ifndef OPENSSL_SUN_GF2M_DIV
607 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
608 * or y, x could equal y.
610 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
620 xinv = BN_CTX_get(ctx);
621 if (xinv == NULL) goto err;
623 if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
624 if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
633 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
634 * or y, x could equal y.
635 * Uses algorithm Modular_Division_GF(2^m) from
636 * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to
639 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
641 BIGNUM *a, *b, *u, *v;
654 if (v == NULL) goto err;
656 /* reduce x and y mod p */
657 if (!BN_GF2m_mod(u, y, p)) goto err;
658 if (!BN_GF2m_mod(a, x, p)) goto err;
659 if (!BN_copy(b, p)) goto err;
661 while (!BN_is_odd(a))
663 if (!BN_rshift1(a, a)) goto err;
664 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
665 if (!BN_rshift1(u, u)) goto err;
670 if (BN_GF2m_cmp(b, a) > 0)
672 if (!BN_GF2m_add(b, b, a)) goto err;
673 if (!BN_GF2m_add(v, v, u)) goto err;
676 if (!BN_rshift1(b, b)) goto err;
677 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
678 if (!BN_rshift1(v, v)) goto err;
679 } while (!BN_is_odd(b));
681 else if (BN_abs_is_word(a, 1))
685 if (!BN_GF2m_add(a, a, b)) goto err;
686 if (!BN_GF2m_add(u, u, v)) goto err;
689 if (!BN_rshift1(a, a)) goto err;
690 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
691 if (!BN_rshift1(u, u)) goto err;
692 } while (!BN_is_odd(a));
696 if (!BN_copy(r, u)) goto err;
706 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
707 * or yy, xx could equal yy.
709 * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
710 * function is only provided for convenience; for best performance, use the
711 * BN_GF2m_mod_div function.
713 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx)
722 if ((field = BN_CTX_get(ctx)) == NULL) goto err;
723 if (!BN_GF2m_arr2poly(p, field)) goto err;
725 ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
734 /* Compute the bth power of a, reduce modulo p, and store
735 * the result in r. r could be a.
736 * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
738 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
749 if (BN_abs_is_word(b, 1))
750 return (BN_copy(r, a) != NULL);
753 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
755 if (!BN_GF2m_mod_arr(u, a, p)) goto err;
757 n = BN_num_bits(b) - 1;
758 for (i = n - 1; i >= 0; i--)
760 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
761 if (BN_is_bit_set(b, i))
763 if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
766 if (!BN_copy(r, u)) goto err;
774 /* Compute the bth power of a, reduce modulo p, and store
775 * the result in r. r could be a.
777 * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
778 * function is only provided for convenience; for best performance, use the
779 * BN_GF2m_mod_exp_arr function.
781 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
784 const int max = BN_num_bits(p) + 1;
789 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
790 ret = BN_GF2m_poly2arr(p, arr, max);
791 if (!ret || ret > max)
793 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
796 ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
799 if (arr) OPENSSL_free(arr);
803 /* Compute the square root of a, reduce modulo p, and store
804 * the result in r. r could be a.
805 * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
807 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
816 /* reduction mod 1 => return 0 */
822 if ((u = BN_CTX_get(ctx)) == NULL) goto err;
824 if (!BN_set_bit(u, p[0] - 1)) goto err;
825 ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
833 /* Compute the square root of a, reduce modulo p, and store
834 * the result in r. r could be a.
836 * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
837 * function is only provided for convenience; for best performance, use the
838 * BN_GF2m_mod_sqrt_arr function.
840 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
843 const int max = BN_num_bits(p) + 1;
847 if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
848 ret = BN_GF2m_poly2arr(p, arr, max);
849 if (!ret || ret > max)
851 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH);
854 ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
857 if (arr) OPENSSL_free(arr);
861 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
862 * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
864 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx)
866 int ret = 0, count = 0, j;
867 BIGNUM *a, *z, *rho, *w, *w2, *tmp;
873 /* reduction mod 1 => return 0 */
882 if (w == NULL) goto err;
884 if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
893 if (p[0] & 0x1) /* m is odd */
895 /* compute half-trace of a */
896 if (!BN_copy(z, a)) goto err;
897 for (j = 1; j <= (p[0] - 1) / 2; j++)
899 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
900 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
901 if (!BN_GF2m_add(z, z, a)) goto err;
907 rho = BN_CTX_get(ctx);
908 w2 = BN_CTX_get(ctx);
909 tmp = BN_CTX_get(ctx);
910 if (tmp == NULL) goto err;
913 if (!BN_rand(rho, p[0], 0, 0)) goto err;
914 if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
916 if (!BN_copy(w, rho)) goto err;
917 for (j = 1; j <= p[0] - 1; j++)
919 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
920 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
921 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
922 if (!BN_GF2m_add(z, z, tmp)) goto err;
923 if (!BN_GF2m_add(w, w2, rho)) goto err;
926 } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
929 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
934 if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
935 if (!BN_GF2m_add(w, z, w)) goto err;
936 if (BN_GF2m_cmp(w, a))
938 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
942 if (!BN_copy(r, z)) goto err;
952 /* Find r such that r^2 + r = a mod p. r could be a. If no r exists returns 0.
954 * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
955 * function is only provided for convenience; for best performance, use the
956 * BN_GF2m_mod_solve_quad_arr function.
958 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
961 const int max = BN_num_bits(p) + 1;
965 if ((arr = (int *)OPENSSL_malloc(sizeof(int) *
966 max)) == NULL) goto err;
967 ret = BN_GF2m_poly2arr(p, arr, max);
968 if (!ret || ret > max)
970 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
973 ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
976 if (arr) OPENSSL_free(arr);
980 /* Convert the bit-string representation of a polynomial
981 * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding
982 * to the bits with non-zero coefficient. Array is terminated with -1.
983 * Up to max elements of the array will be filled. Return value is total
984 * number of array elements that would be filled if array was large enough.
986 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
994 for (i = a->top - 1; i >= 0; i--)
997 /* skip word if a->d[i] == 0 */
1000 for (j = BN_BITS2 - 1; j >= 0; j--)
1004 if (k < max) p[k] = BN_BITS2 * i + j;
1019 /* Convert the coefficient array representation of a polynomial to a
1020 * bit-string. The array must be terminated by -1.
1022 int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1028 for (i = 0; p[i] != -1; i++)
1030 if (BN_set_bit(a, p[i]) == 0)