1 /* crypto/bn/bn_gf2m.c */
   2 /* ====================================================================
   3  * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
   4  *
   5  * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
   6  * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
   7  * to the OpenSSL project.
   8  *
   9  * The ECC Code is licensed pursuant to the OpenSSL open source
  10  * license provided below.
  11  *
  12  * In addition, Sun covenants to all licensees who provide a reciprocal
  13  * covenant with respect to their own patents if any, not to sue under
  14  * current and future patent claims necessarily infringed by the making,
  15  * using, practicing, selling, offering for sale and/or otherwise
  16  * disposing of the ECC Code as delivered hereunder (or portions thereof),
  17  * provided that such covenant shall not apply:
  18  *  1) for code that a licensee deletes from the ECC Code;
  19  *  2) separates from the ECC Code; or
  20  *  3) for infringements caused by:
  21  *       i) the modification of the ECC Code or
  22  *      ii) the combination of the ECC Code with other software or
  23  *          devices where such combination causes the infringement.
  24  *
  25  * The software is originally written by Sheueling Chang Shantz and
  26  * Douglas Stebila of Sun Microsystems Laboratories.
  27  *
  28  */
  29 
  30 /* NOTE: This file is licensed pursuant to the OpenSSL license below
  31  * and may be modified; but after modifications, the above covenant
  32  * may no longer apply!  In such cases, the corresponding paragraph
  33  * ["In addition, Sun covenants ... causes the infringement."] and
  34  * this note can be edited out; but please keep the Sun copyright
  35  * notice and attribution. */
  36 
  37 /* ====================================================================
  38  * Copyright (c) 1998-2002 The OpenSSL Project.  All rights reserved.
  39  *
  40  * Redistribution and use in source and binary forms, with or without
  41  * modification, are permitted provided that the following conditions
  42  * are met:
  43  *
  44  * 1. Redistributions of source code must retain the above copyright
  45  *    notice, this list of conditions and the following disclaimer.
  46  *
  47  * 2. Redistributions in binary form must reproduce the above copyright
  48  *    notice, this list of conditions and the following disclaimer in
  49  *    the documentation and/or other materials provided with the
  50  *    distribution.
  51  *
  52  * 3. All advertising materials mentioning features or use of this
  53  *    software must display the following acknowledgment:
  54  *    "This product includes software developed by the OpenSSL Project
  55  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
  56  *
  57  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
  58  *    endorse or promote products derived from this software without
  59  *    prior written permission. For written permission, please contact
  60  *    openssl-core@openssl.org.
  61  *
  62  * 5. Products derived from this software may not be called "OpenSSL"
  63  *    nor may "OpenSSL" appear in their names without prior written
  64  *    permission of the OpenSSL Project.
  65  *
  66  * 6. Redistributions of any form whatsoever must retain the following
  67  *    acknowledgment:
  68  *    "This product includes software developed by the OpenSSL Project
  69  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
  70  *
  71  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
  72  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  73  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
  74  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
  75  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  76  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
  77  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
  78  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  79  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
  80  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
  81  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
  82  * OF THE POSSIBILITY OF SUCH DAMAGE.
  83  * ====================================================================
  84  *
  85  * This product includes cryptographic software written by Eric Young
  86  * (eay@cryptsoft.com).  This product includes software written by Tim
  87  * Hudson (tjh@cryptsoft.com).
  88  *
  89  */
  90 
  91 #include <assert.h>
  92 #include <limits.h>
  93 #include <stdio.h>
  94 #include "cryptlib.h"
  95 #include "bn_lcl.h"
  96 
  97 #ifndef OPENSSL_NO_EC2M
  98 
  99 /* Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should fail. */
 100 #define MAX_ITERATIONS 50
 101 
 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)
 107 #define SQR1(w) \
 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]
 112 #define SQR0(w) \
 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]
 117 #endif
 118 #ifdef THIRTY_TWO_BIT
 119 #define SQR1(w) \
 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]
 122 #define SQR0(w) \
 123     SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >>  8 & 0xF] << 16 | \
 124     SQR_tb[(w) >>  4 & 0xF] <<  8 | SQR_tb[(w)       & 0xF]
 125 #endif
 126 
 127 #if !defined(OPENSSL_BN_ASM_GF2m)
 128 /* Product of two polynomials a, b each with degree < BN_BITS2 - 1,
 129  * result is a polynomial r with degree < 2 * BN_BITS - 1
 130  * The caller MUST ensure that the variables have the right amount
 131  * of space allocated.
 132  */
 133 #ifdef THIRTY_TWO_BIT
 134 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
 135         {
 136         register BN_ULONG h, l, s;
 137         BN_ULONG tab[8], top2b = a >> 30;
 138         register BN_ULONG a1, a2, a4;
 139 
 140         a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1;
 141 
 142         tab[0] =  0; tab[1] = a1;    tab[2] = a2;    tab[3] = a1^a2;
 143         tab[4] = a4; tab[5] = a1^a4; tab[6] = a2^a4; tab[7] = a1^a2^a4;
 144 
 145         s = tab[b       & 0x7]; l  = s;
 146         s = tab[b >>  3 & 0x7]; l ^= s <<  3; h  = s >> 29;
 147         s = tab[b >>  6 & 0x7]; l ^= s <<  6; h ^= s >> 26;
 148         s = tab[b >>  9 & 0x7]; l ^= s <<  9; h ^= s >> 23;
 149         s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20;
 150         s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17;
 151         s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14;
 152         s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11;
 153         s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >>  8;
 154         s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >>  5;
 155         s = tab[b >> 30      ]; l ^= s << 30; h ^= s >>  2;
 156 
 157         /* compensate for the top two bits of a */
 158 
 159         if (top2b & 01) { l ^= b << 30; h ^= b >> 2; }
 160         if (top2b & 02) { l ^= b << 31; h ^= b >> 1; }
 161 
 162         *r1 = h; *r0 = l;
 163         }
 164 #endif
 165 #if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
 166 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b)
 167         {
 168         register BN_ULONG h, l, s;
 169         BN_ULONG tab[16], top3b = a >> 61;
 170         register BN_ULONG a1, a2, a4, a8;
 171 
 172         a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1;
 173 
 174         tab[ 0] = 0;     tab[ 1] = a1;       tab[ 2] = a2;       tab[ 3] = a1^a2;
 175         tab[ 4] = a4;    tab[ 5] = a1^a4;    tab[ 6] = a2^a4;    tab[ 7] = a1^a2^a4;
 176         tab[ 8] = a8;    tab[ 9] = a1^a8;    tab[10] = a2^a8;    tab[11] = a1^a2^a8;
 177         tab[12] = a4^a8; tab[13] = a1^a4^a8; tab[14] = a2^a4^a8; tab[15] = a1^a2^a4^a8;
 178 
 179         s = tab[b       & 0xF]; l  = s;
 180         s = tab[b >>  4 & 0xF]; l ^= s <<  4; h  = s >> 60;
 181         s = tab[b >>  8 & 0xF]; l ^= s <<  8; h ^= s >> 56;
 182         s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52;
 183         s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48;
 184         s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44;
 185         s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40;
 186         s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36;
 187         s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32;
 188         s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28;
 189         s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24;
 190         s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20;
 191         s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16;
 192         s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12;
 193         s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >>  8;
 194         s = tab[b >> 60      ]; l ^= s << 60; h ^= s >>  4;
 195 
 196         /* compensate for the top three bits of a */
 197 
 198         if (top3b & 01) { l ^= b << 61; h ^= b >> 3; }
 199         if (top3b & 02) { l ^= b << 62; h ^= b >> 2; }
 200         if (top3b & 04) { l ^= b << 63; h ^= b >> 1; }
 201 
 202         *r1 = h; *r0 = l;
 203         }
 204 #endif
 205 
 206 /* Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
 207  * result is a polynomial r with degree < 4 * BN_BITS2 - 1
 208  * The caller MUST ensure that the variables have the right amount
 209  * of space allocated.
 210  */
 211 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)
 212         {
 213         BN_ULONG m1, m0;
 214         /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
 215         bn_GF2m_mul_1x1(r+3, r+2, a1, b1);
 216         bn_GF2m_mul_1x1(r+1, r, a0, b0);
 217         bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
 218         /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
 219         r[2] ^= m1 ^ r[1] ^ r[3];  /* h0 ^= m1 ^ l1 ^ h1; */
 220         r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0;  /* l1 ^= l0 ^ h0 ^ m0; */
 221         }
 222 #else
 223 void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1, BN_ULONG b0);
 224 #endif
 225 
 226 /* Add polynomials a and b and store result in r; r could be a or b, a and b
 227  * could be equal; r is the bitwise XOR of a and b.
 228  */
 229 int     BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
 230         {
 231         int i;
 232         const BIGNUM *at, *bt;
 233 
 234         bn_check_top(a);
 235         bn_check_top(b);
 236 
 237         if (a->top < b->top) { at = b; bt = a; }
 238         else { at = a; bt = b; }
 239 
 240         if(bn_wexpand(r, at->top) == NULL)
 241                 return 0;
 242 
 243         for (i = 0; i < bt->top; i++)
 244                 {
 245                 r->d[i] = at->d[i] ^ bt->d[i];
 246                 }
 247         for (; i < at->top; i++)
 248                 {
 249                 r->d[i] = at->d[i];
 250                 }
 251 
 252         r->top = at->top;
 253         bn_correct_top(r);
 254 
 255         return 1;
 256         }
 257 
 258 
 259 /* Some functions allow for representation of the irreducible polynomials
 260  * as an int[], say p.  The irreducible f(t) is then of the form:
 261  *     t^p[0] + t^p[1] + ... + t^p[k]
 262  * where m = p[0] > p[1] > ... > p[k] = 0.
 263  */
 264 
 265 
 266 /* Performs modular reduction of a and store result in r.  r could be a. */
 267 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
 268         {
 269         int j, k;
 270         int n, dN, d0, d1;
 271         BN_ULONG zz, *z;
 272 
 273         bn_check_top(a);
 274 
 275         if (!p[0])
 276                 {
 277                 /* reduction mod 1 => return 0 */
 278                 BN_zero(r);
 279                 return 1;
 280                 }
 281 
 282         /* Since the algorithm does reduction in the r value, if a != r, copy
 283          * the contents of a into r so we can do reduction in r.
 284          */
 285         if (a != r)
 286                 {
 287                 if (!bn_wexpand(r, a->top)) return 0;
 288                 for (j = 0; j < a->top; j++)
 289                         {
 290                         r->d[j] = a->d[j];
 291                         }
 292                 r->top = a->top;
 293                 }
 294         z = r->d;
 295 
 296         /* start reduction */
 297         dN = p[0] / BN_BITS2;
 298         for (j = r->top - 1; j > dN;)
 299                 {
 300                 zz = z[j];
 301                 if (z[j] == 0) { j--; continue; }
 302                 z[j] = 0;
 303 
 304                 for (k = 1; p[k] != 0; k++)
 305                         {
 306                         /* reducing component t^p[k] */
 307                         n = p[0] - p[k];
 308                         d0 = n % BN_BITS2;  d1 = BN_BITS2 - d0;
 309                         n /= BN_BITS2;
 310                         z[j-n] ^= (zz>>d0);
 311                         if (d0) z[j-n-1] ^= (zz<<d1);
 312                         }
 313 
 314                 /* reducing component t^0 */
 315                 n = dN;
 316                 d0 = p[0] % BN_BITS2;
 317                 d1 = BN_BITS2 - d0;
 318                 z[j-n] ^= (zz >> d0);
 319                 if (d0) z[j-n-1] ^= (zz << d1);
 320                 }
 321 
 322         /* final round of reduction */
 323         while (j == dN)
 324                 {
 325 
 326                 d0 = p[0] % BN_BITS2;
 327                 zz = z[dN] >> d0;
 328                 if (zz == 0) break;
 329                 d1 = BN_BITS2 - d0;
 330 
 331                 /* clear up the top d1 bits */
 332                 if (d0)
 333                         z[dN] = (z[dN] << d1) >> d1;
 334                 else
 335                         z[dN] = 0;
 336                 z[0] ^= zz; /* reduction t^0 component */
 337 
 338                 for (k = 1; p[k] != 0; k++)
 339                         {
 340                         BN_ULONG tmp_ulong;
 341 
 342                         /* reducing component t^p[k]*/
 343                         n = p[k] / BN_BITS2;
 344                         d0 = p[k] % BN_BITS2;
 345                         d1 = BN_BITS2 - d0;
 346                         z[n] ^= (zz << d0);
 347                         tmp_ulong = zz >> d1;
 348                         if (d0 && tmp_ulong)
 349                                 z[n+1] ^= tmp_ulong;
 350                         }
 351 
 352 
 353                 }
 354 
 355         bn_correct_top(r);
 356         return 1;
 357         }
 358 
 359 /* Performs modular reduction of a by p and store result in r.  r could be a.
 360  *
 361  * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
 362  * function is only provided for convenience; for best performance, use the
 363  * BN_GF2m_mod_arr function.
 364  */
 365 int     BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
 366         {
 367         int ret = 0;
 368         int arr[6];
 369         bn_check_top(a);
 370         bn_check_top(p);
 371         ret = BN_GF2m_poly2arr(p, arr, sizeof(arr)/sizeof(arr[0]));
 372         if (!ret || ret > (int)(sizeof(arr)/sizeof(arr[0])))
 373                 {
 374                 BNerr(BN_F_BN_GF2M_MOD,BN_R_INVALID_LENGTH);
 375                 return 0;
 376                 }
 377         ret = BN_GF2m_mod_arr(r, a, arr);
 378         bn_check_top(r);
 379         return ret;
 380         }
 381 
 382 
 383 /* Compute the product of two polynomials a and b, reduce modulo p, and store
 384  * the result in r.  r could be a or b; a could be b.
 385  */
 386 int     BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
 387         {
 388         int zlen, i, j, k, ret = 0;
 389         BIGNUM *s;
 390         BN_ULONG x1, x0, y1, y0, zz[4];
 391 
 392         bn_check_top(a);
 393         bn_check_top(b);
 394 
 395         if (a == b)
 396                 {
 397                 return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
 398                 }
 399 
 400         BN_CTX_start(ctx);
 401         if ((s = BN_CTX_get(ctx)) == NULL) goto err;
 402 
 403         zlen = a->top + b->top + 4;
 404         if (!bn_wexpand(s, zlen)) goto err;
 405         s->top = zlen;
 406 
 407         for (i = 0; i < zlen; i++) s->d[i] = 0;
 408 
 409         for (j = 0; j < b->top; j += 2)
 410                 {
 411                 y0 = b->d[j];
 412                 y1 = ((j+1) == b->top) ? 0 : b->d[j+1];
 413                 for (i = 0; i < a->top; i += 2)
 414                         {
 415                         x0 = a->d[i];
 416                         x1 = ((i+1) == a->top) ? 0 : a->d[i+1];
 417                         bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
 418                         for (k = 0; k < 4; k++) s->d[i+j+k] ^= zz[k];
 419                         }
 420                 }
 421 
 422         bn_correct_top(s);
 423         if (BN_GF2m_mod_arr(r, s, p))
 424                 ret = 1;
 425         bn_check_top(r);
 426 
 427 err:
 428         BN_CTX_end(ctx);
 429         return ret;
 430         }
 431 
 432 /* Compute the product of two polynomials a and b, reduce modulo p, and store
 433  * the result in r.  r could be a or b; a could equal b.
 434  *
 435  * This function calls down to the BN_GF2m_mod_mul_arr implementation; this wrapper
 436  * function is only provided for convenience; for best performance, use the
 437  * BN_GF2m_mod_mul_arr function.
 438  */
 439 int     BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
 440         {
 441         int ret = 0;
 442         const int max = BN_num_bits(p) + 1;
 443         int *arr=NULL;
 444         bn_check_top(a);
 445         bn_check_top(b);
 446         bn_check_top(p);
 447         if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 448         ret = BN_GF2m_poly2arr(p, arr, max);
 449         if (!ret || ret > max)
 450                 {
 451                 BNerr(BN_F_BN_GF2M_MOD_MUL,BN_R_INVALID_LENGTH);
 452                 goto err;
 453                 }
 454         ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
 455         bn_check_top(r);
 456 err:
 457         if (arr) OPENSSL_free(arr);
 458         return ret;
 459         }
 460 
 461 
 462 /* Square a, reduce the result mod p, and store it in a.  r could be a. */
 463 int     BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
 464         {
 465         int i, ret = 0;
 466         BIGNUM *s;
 467 
 468         bn_check_top(a);
 469         BN_CTX_start(ctx);
 470         if ((s = BN_CTX_get(ctx)) == NULL) return 0;
 471         if (!bn_wexpand(s, 2 * a->top)) goto err;
 472 
 473         for (i = a->top - 1; i >= 0; i--)
 474                 {
 475                 s->d[2*i+1] = SQR1(a->d[i]);
 476                 s->d[2*i  ] = SQR0(a->d[i]);
 477                 }
 478 
 479         s->top = 2 * a->top;
 480         bn_correct_top(s);
 481         if (!BN_GF2m_mod_arr(r, s, p)) goto err;
 482         bn_check_top(r);
 483         ret = 1;
 484 err:
 485         BN_CTX_end(ctx);
 486         return ret;
 487         }
 488 
 489 /* Square a, reduce the result mod p, and store it in a.  r could be a.
 490  *
 491  * This function calls down to the BN_GF2m_mod_sqr_arr implementation; this wrapper
 492  * function is only provided for convenience; for best performance, use the
 493  * BN_GF2m_mod_sqr_arr function.
 494  */
 495 int     BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 496         {
 497         int ret = 0;
 498         const int max = BN_num_bits(p) + 1;
 499         int *arr=NULL;
 500 
 501         bn_check_top(a);
 502         bn_check_top(p);
 503         if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 504         ret = BN_GF2m_poly2arr(p, arr, max);
 505         if (!ret || ret > max)
 506                 {
 507                 BNerr(BN_F_BN_GF2M_MOD_SQR,BN_R_INVALID_LENGTH);
 508                 goto err;
 509                 }
 510         ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
 511         bn_check_top(r);
 512 err:
 513         if (arr) OPENSSL_free(arr);
 514         return ret;
 515         }
 516 
 517 
 518 /* Invert a, reduce modulo p, and store the result in r. r could be a.
 519  * Uses Modified Almost Inverse Algorithm (Algorithm 10) from
 520  *     Hankerson, D., Hernandez, J.L., and Menezes, A.  "Software Implementation
 521  *     of Elliptic Curve Cryptography Over Binary Fields".
 522  */
 523 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 524         {
 525         BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp;
 526         int ret = 0;
 527 
 528         bn_check_top(a);
 529         bn_check_top(p);
 530 
 531         BN_CTX_start(ctx);
 532 
 533         if ((b = BN_CTX_get(ctx))==NULL) goto err;
 534         if ((c = BN_CTX_get(ctx))==NULL) goto err;
 535         if ((u = BN_CTX_get(ctx))==NULL) goto err;
 536         if ((v = BN_CTX_get(ctx))==NULL) goto err;
 537 
 538         if (!BN_GF2m_mod(u, a, p)) goto err;
 539         if (BN_is_zero(u)) goto err;
 540 
 541         if (!BN_copy(v, p)) goto err;
 542 #if 0
 543         if (!BN_one(b)) goto err;
 544 
 545         while (1)
 546                 {
 547                 while (!BN_is_odd(u))
 548                         {
 549                         if (BN_is_zero(u)) goto err;
 550                         if (!BN_rshift1(u, u)) goto err;
 551                         if (BN_is_odd(b))
 552                                 {
 553                                 if (!BN_GF2m_add(b, b, p)) goto err;
 554                                 }
 555                         if (!BN_rshift1(b, b)) goto err;
 556                         }
 557 
 558                 if (BN_abs_is_word(u, 1)) break;
 559 
 560                 if (BN_num_bits(u) < BN_num_bits(v))
 561                         {
 562                         tmp = u; u = v; v = tmp;
 563                         tmp = b; b = c; c = tmp;
 564                         }
 565 
 566                 if (!BN_GF2m_add(u, u, v)) goto err;
 567                 if (!BN_GF2m_add(b, b, c)) goto err;
 568                 }
 569 #else
 570         {
 571         int i,  ubits = BN_num_bits(u),
 572                 vbits = BN_num_bits(v), /* v is copy of p */
 573                 top = p->top;
 574         BN_ULONG *udp,*bdp,*vdp,*cdp;
 575 
 576         bn_wexpand(u,top);      udp = u->d;
 577                                 for (i=u->top;i<top;i++) udp[i] = 0;
 578                                 u->top = top;
 579         bn_wexpand(b,top);      bdp = b->d;
 580                                 bdp[0] = 1;
 581                                 for (i=1;i<top;i++) bdp[i] = 0;
 582                                 b->top = top;
 583         bn_wexpand(c,top);      cdp = c->d;
 584                                 for (i=0;i<top;i++) cdp[i] = 0;
 585                                 c->top = top;
 586         vdp = v->d;  /* It pays off to "cache" *->d pointers, because
 587                          * it allows optimizer to be more aggressive.
 588                          * But we don't have to "cache" p->d, because *p
 589                          * is declared 'const'... */
 590         while (1)
 591                 {
 592                 while (ubits && !(udp[0]&1))
 593                         {
 594                         BN_ULONG u0,u1,b0,b1,mask;
 595 
 596                         u0   = udp[0];
 597                         b0   = bdp[0];
 598                         mask = (BN_ULONG)0-(b0&1);
 599                         b0  ^= p->d[0]&mask;
 600                         for (i=0;i<top-1;i++)
 601                                 {
 602                                 u1 = udp[i+1];
 603                                 udp[i] = ((u0>>1)|(u1<<(BN_BITS2-1)))&BN_MASK2;
 604                                 u0 = u1;
 605                                 b1 = bdp[i+1]^(p->d[i+1]&mask);
 606                                 bdp[i] = ((b0>>1)|(b1<<(BN_BITS2-1)))&BN_MASK2;
 607                                 b0 = b1;
 608                                 }
 609                         udp[i] = u0>>1;
 610                         bdp[i] = b0>>1;
 611                         ubits--;
 612                         }
 613 
 614                 if (ubits<=BN_BITS2 && udp[0]==1) break;
 615 
 616                 if (ubits<vbits)
 617                         {
 618                         i = ubits; ubits = vbits; vbits = i;
 619                         tmp = u; u = v; v = tmp;
 620                         tmp = b; b = c; c = tmp;
 621                         udp = vdp; vdp = v->d;
 622                         bdp = cdp; cdp = c->d;
 623                         }
 624                 for(i=0;i<top;i++)
 625                         {
 626                         udp[i] ^= vdp[i];
 627                         bdp[i] ^= cdp[i];
 628                         }
 629                 if (ubits==vbits)
 630                         {
 631                         BN_ULONG ul;
 632                         int utop = (ubits-1)/BN_BITS2;
 633 
 634                         while ((ul=udp[utop])==0 && utop) utop--;
 635                         ubits = utop*BN_BITS2 + BN_num_bits_word(ul);
 636                         }
 637                 }
 638         bn_correct_top(b);
 639         }
 640 #endif
 641 
 642         if (!BN_copy(r, b)) goto err;
 643         bn_check_top(r);
 644         ret = 1;
 645 
 646 err:
 647 #ifdef BN_DEBUG /* BN_CTX_end would complain about the expanded form */
 648         bn_correct_top(c);
 649         bn_correct_top(u);
 650         bn_correct_top(v);
 651 #endif
 652         BN_CTX_end(ctx);
 653         return ret;
 654         }
 655 
 656 /* Invert xx, reduce modulo p, and store the result in r. r could be xx.
 657  *
 658  * This function calls down to the BN_GF2m_mod_inv implementation; this wrapper
 659  * function is only provided for convenience; for best performance, use the
 660  * BN_GF2m_mod_inv function.
 661  */
 662 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx)
 663         {
 664         BIGNUM *field;
 665         int ret = 0;
 666 
 667         bn_check_top(xx);
 668         BN_CTX_start(ctx);
 669         if ((field = BN_CTX_get(ctx)) == NULL) goto err;
 670         if (!BN_GF2m_arr2poly(p, field)) goto err;
 671 
 672         ret = BN_GF2m_mod_inv(r, xx, field, ctx);
 673         bn_check_top(r);
 674 
 675 err:
 676         BN_CTX_end(ctx);
 677         return ret;
 678         }
 679 
 680 
 681 #ifndef OPENSSL_SUN_GF2M_DIV
 682 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
 683  * or y, x could equal y.
 684  */
 685 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
 686         {
 687         BIGNUM *xinv = NULL;
 688         int ret = 0;
 689 
 690         bn_check_top(y);
 691         bn_check_top(x);
 692         bn_check_top(p);
 693 
 694         BN_CTX_start(ctx);
 695         xinv = BN_CTX_get(ctx);
 696         if (xinv == NULL) goto err;
 697 
 698         if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err;
 699         if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err;
 700         bn_check_top(r);
 701         ret = 1;
 702 
 703 err:
 704         BN_CTX_end(ctx);
 705         return ret;
 706         }
 707 #else
 708 /* Divide y by x, reduce modulo p, and store the result in r. r could be x
 709  * or y, x could equal y.
 710  * Uses algorithm Modular_Division_GF(2^m) from
 711  *     Chang-Shantz, S.  "From Euclid's GCD to Montgomery Multiplication to
 712  *     the Great Divide".
 713  */
 714 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx)
 715         {
 716         BIGNUM *a, *b, *u, *v;
 717         int ret = 0;
 718 
 719         bn_check_top(y);
 720         bn_check_top(x);
 721         bn_check_top(p);
 722 
 723         BN_CTX_start(ctx);
 724 
 725         a = BN_CTX_get(ctx);
 726         b = BN_CTX_get(ctx);
 727         u = BN_CTX_get(ctx);
 728         v = BN_CTX_get(ctx);
 729         if (v == NULL) goto err;
 730 
 731         /* reduce x and y mod p */
 732         if (!BN_GF2m_mod(u, y, p)) goto err;
 733         if (!BN_GF2m_mod(a, x, p)) goto err;
 734         if (!BN_copy(b, p)) goto err;
 735 
 736         while (!BN_is_odd(a))
 737                 {
 738                 if (!BN_rshift1(a, a)) goto err;
 739                 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
 740                 if (!BN_rshift1(u, u)) goto err;
 741                 }
 742 
 743         do
 744                 {
 745                 if (BN_GF2m_cmp(b, a) > 0)
 746                         {
 747                         if (!BN_GF2m_add(b, b, a)) goto err;
 748                         if (!BN_GF2m_add(v, v, u)) goto err;
 749                         do
 750                                 {
 751                                 if (!BN_rshift1(b, b)) goto err;
 752                                 if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err;
 753                                 if (!BN_rshift1(v, v)) goto err;
 754                                 } while (!BN_is_odd(b));
 755                         }
 756                 else if (BN_abs_is_word(a, 1))
 757                         break;
 758                 else
 759                         {
 760                         if (!BN_GF2m_add(a, a, b)) goto err;
 761                         if (!BN_GF2m_add(u, u, v)) goto err;
 762                         do
 763                                 {
 764                                 if (!BN_rshift1(a, a)) goto err;
 765                                 if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err;
 766                                 if (!BN_rshift1(u, u)) goto err;
 767                                 } while (!BN_is_odd(a));
 768                         }
 769                 } while (1);
 770 
 771         if (!BN_copy(r, u)) goto err;
 772         bn_check_top(r);
 773         ret = 1;
 774 
 775 err:
 776         BN_CTX_end(ctx);
 777         return ret;
 778         }
 779 #endif
 780 
 781 /* Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
 782  * or yy, xx could equal yy.
 783  *
 784  * This function calls down to the BN_GF2m_mod_div implementation; this wrapper
 785  * function is only provided for convenience; for best performance, use the
 786  * BN_GF2m_mod_div function.
 787  */
 788 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx)
 789         {
 790         BIGNUM *field;
 791         int ret = 0;
 792 
 793         bn_check_top(yy);
 794         bn_check_top(xx);
 795 
 796         BN_CTX_start(ctx);
 797         if ((field = BN_CTX_get(ctx)) == NULL) goto err;
 798         if (!BN_GF2m_arr2poly(p, field)) goto err;
 799 
 800         ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
 801         bn_check_top(r);
 802 
 803 err:
 804         BN_CTX_end(ctx);
 805         return ret;
 806         }
 807 
 808 
 809 /* Compute the bth power of a, reduce modulo p, and store
 810  * the result in r.  r could be a.
 811  * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
 812  */
 813 int     BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
 814         {
 815         int ret = 0, i, n;
 816         BIGNUM *u;
 817 
 818         bn_check_top(a);
 819         bn_check_top(b);
 820 
 821         if (BN_is_zero(b))
 822                 return(BN_one(r));
 823 
 824         if (BN_abs_is_word(b, 1))
 825                 return (BN_copy(r, a) != NULL);
 826 
 827         BN_CTX_start(ctx);
 828         if ((u = BN_CTX_get(ctx)) == NULL) goto err;
 829 
 830         if (!BN_GF2m_mod_arr(u, a, p)) goto err;
 831 
 832         n = BN_num_bits(b) - 1;
 833         for (i = n - 1; i >= 0; i--)
 834                 {
 835                 if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err;
 836                 if (BN_is_bit_set(b, i))
 837                         {
 838                         if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err;
 839                         }
 840                 }
 841         if (!BN_copy(r, u)) goto err;
 842         bn_check_top(r);
 843         ret = 1;
 844 err:
 845         BN_CTX_end(ctx);
 846         return ret;
 847         }
 848 
 849 /* Compute the bth power of a, reduce modulo p, and store
 850  * the result in r.  r could be a.
 851  *
 852  * This function calls down to the BN_GF2m_mod_exp_arr implementation; this wrapper
 853  * function is only provided for convenience; for best performance, use the
 854  * BN_GF2m_mod_exp_arr function.
 855  */
 856 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
 857         {
 858         int ret = 0;
 859         const int max = BN_num_bits(p) + 1;
 860         int *arr=NULL;
 861         bn_check_top(a);
 862         bn_check_top(b);
 863         bn_check_top(p);
 864         if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 865         ret = BN_GF2m_poly2arr(p, arr, max);
 866         if (!ret || ret > max)
 867                 {
 868                 BNerr(BN_F_BN_GF2M_MOD_EXP,BN_R_INVALID_LENGTH);
 869                 goto err;
 870                 }
 871         ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
 872         bn_check_top(r);
 873 err:
 874         if (arr) OPENSSL_free(arr);
 875         return ret;
 876         }
 877 
 878 /* Compute the square root of a, reduce modulo p, and store
 879  * the result in r.  r could be a.
 880  * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
 881  */
 882 int     BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
 883         {
 884         int ret = 0;
 885         BIGNUM *u;
 886 
 887         bn_check_top(a);
 888 
 889         if (!p[0])
 890                 {
 891                 /* reduction mod 1 => return 0 */
 892                 BN_zero(r);
 893                 return 1;
 894                 }
 895 
 896         BN_CTX_start(ctx);
 897         if ((u = BN_CTX_get(ctx)) == NULL) goto err;
 898 
 899         if (!BN_set_bit(u, p[0] - 1)) goto err;
 900         ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
 901         bn_check_top(r);
 902 
 903 err:
 904         BN_CTX_end(ctx);
 905         return ret;
 906         }
 907 
 908 /* Compute the square root of a, reduce modulo p, and store
 909  * the result in r.  r could be a.
 910  *
 911  * This function calls down to the BN_GF2m_mod_sqrt_arr implementation; this wrapper
 912  * function is only provided for convenience; for best performance, use the
 913  * BN_GF2m_mod_sqrt_arr function.
 914  */
 915 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 916         {
 917         int ret = 0;
 918         const int max = BN_num_bits(p) + 1;
 919         int *arr=NULL;
 920         bn_check_top(a);
 921         bn_check_top(p);
 922         if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 923         ret = BN_GF2m_poly2arr(p, arr, max);
 924         if (!ret || ret > max)
 925                 {
 926                 BNerr(BN_F_BN_GF2M_MOD_SQRT,BN_R_INVALID_LENGTH);
 927                 goto err;
 928                 }
 929         ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
 930         bn_check_top(r);
 931 err:
 932         if (arr) OPENSSL_free(arr);
 933         return ret;
 934         }
 935 
 936 /* Find r such that r^2 + r = a mod p.  r could be a. If no r exists returns 0.
 937  * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
 938  */
 939 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx)
 940         {
 941         int ret = 0, count = 0, j;
 942         BIGNUM *a, *z, *rho, *w, *w2, *tmp;
 943 
 944         bn_check_top(a_);
 945 
 946         if (!p[0])
 947                 {
 948                 /* reduction mod 1 => return 0 */
 949                 BN_zero(r);
 950                 return 1;
 951                 }
 952 
 953         BN_CTX_start(ctx);
 954         a = BN_CTX_get(ctx);
 955         z = BN_CTX_get(ctx);
 956         w = BN_CTX_get(ctx);
 957         if (w == NULL) goto err;
 958 
 959         if (!BN_GF2m_mod_arr(a, a_, p)) goto err;
 960 
 961         if (BN_is_zero(a))
 962                 {
 963                 BN_zero(r);
 964                 ret = 1;
 965                 goto err;
 966                 }
 967 
 968         if (p[0] & 0x1) /* m is odd */
 969                 {
 970                 /* compute half-trace of a */
 971                 if (!BN_copy(z, a)) goto err;
 972                 for (j = 1; j <= (p[0] - 1) / 2; j++)
 973                         {
 974                         if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
 975                         if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
 976                         if (!BN_GF2m_add(z, z, a)) goto err;
 977                         }
 978 
 979                 }
 980         else /* m is even */
 981                 {
 982                 rho = BN_CTX_get(ctx);
 983                 w2 = BN_CTX_get(ctx);
 984                 tmp = BN_CTX_get(ctx);
 985                 if (tmp == NULL) goto err;
 986                 do
 987                         {
 988                         if (!BN_rand(rho, p[0], 0, 0)) goto err;
 989                         if (!BN_GF2m_mod_arr(rho, rho, p)) goto err;
 990                         BN_zero(z);
 991                         if (!BN_copy(w, rho)) goto err;
 992                         for (j = 1; j <= p[0] - 1; j++)
 993                                 {
 994                                 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err;
 995                                 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err;
 996                                 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err;
 997                                 if (!BN_GF2m_add(z, z, tmp)) goto err;
 998                                 if (!BN_GF2m_add(w, w2, rho)) goto err;
 999                                 }
1000                         count++;
1001                         } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
1002                 if (BN_is_zero(w))
1003                         {
1004                         BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR,BN_R_TOO_MANY_ITERATIONS);
1005                         goto err;
1006                         }
1007                 }
1008 
1009         if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err;
1010         if (!BN_GF2m_add(w, z, w)) goto err;
1011         if (BN_GF2m_cmp(w, a))
1012                 {
1013                 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
1014                 goto err;
1015                 }
1016 
1017         if (!BN_copy(r, z)) goto err;
1018         bn_check_top(r);
1019 
1020         ret = 1;
1021 
1022 err:
1023         BN_CTX_end(ctx);
1024         return ret;
1025         }
1026 
1027 /* Find r such that r^2 + r = a mod p.  r could be a. If no r exists returns 0.
1028  *
1029  * This function calls down to the BN_GF2m_mod_solve_quad_arr implementation; this wrapper
1030  * function is only provided for convenience; for best performance, use the
1031  * BN_GF2m_mod_solve_quad_arr function.
1032  */
1033 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
1034         {
1035         int ret = 0;
1036         const int max = BN_num_bits(p) + 1;
1037         int *arr=NULL;
1038         bn_check_top(a);
1039         bn_check_top(p);
1040         if ((arr = (int *)OPENSSL_malloc(sizeof(int) *
1041                                                 max)) == NULL) goto err;
1042         ret = BN_GF2m_poly2arr(p, arr, max);
1043         if (!ret || ret > max)
1044                 {
1045                 BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD,BN_R_INVALID_LENGTH);
1046                 goto err;
1047                 }
1048         ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1049         bn_check_top(r);
1050 err:
1051         if (arr) OPENSSL_free(arr);
1052         return ret;
1053         }
1054 
1055 /* Convert the bit-string representation of a polynomial
1056  * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding
1057  * to the bits with non-zero coefficient.  Array is terminated with -1.
1058  * Up to max elements of the array will be filled.  Return value is total
1059  * number of array elements that would be filled if array was large enough.
1060  */
1061 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
1062         {
1063         int i, j, k = 0;
1064         BN_ULONG mask;
1065 
1066         if (BN_is_zero(a))
1067                 return 0;
1068 
1069         for (i = a->top - 1; i >= 0; i--)
1070                 {
1071                 if (!a->d[i])
1072                         /* skip word if a->d[i] == 0 */
1073                         continue;
1074                 mask = BN_TBIT;
1075                 for (j = BN_BITS2 - 1; j >= 0; j--)
1076                         {
1077                         if (a->d[i] & mask)
1078                                 {
1079                                 if (k < max) p[k] = BN_BITS2 * i + j;
1080                                 k++;
1081                                 }
1082                         mask >>= 1;
1083                         }
1084                 }
1085 
1086         if (k < max) {
1087                 p[k] = -1;
1088                 k++;
1089         }
1090 
1091         return k;
1092         }
1093 
1094 /* Convert the coefficient array representation of a polynomial to a
1095  * bit-string.  The array must be terminated by -1.
1096  */
1097 int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1098         {
1099         int i;
1100 
1101         bn_check_top(a);
1102         BN_zero(a);
1103         for (i = 0; p[i] != -1; i++)
1104                 {
1105                 if (BN_set_bit(a, p[i]) == 0)
1106                         return 0;
1107                 }
1108         bn_check_top(a);
1109 
1110         return 1;
1111         }
1112 
1113 #endif