crypto/bn/bn_gf2m.c

   1 /*
   2  * Copyright 2002-2018 The OpenSSL Project Authors. All Rights Reserved.
   3  * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved
   4  *
   5  * Licensed under the Apache License 2.0 (the "License").  You may not use
   6  * this file except in compliance with the License.  You can obtain a copy
   7  * in the file LICENSE in the source distribution or at
   8  * https://www.openssl.org/source/license.html
   9  */
  10
  11 #include <assert.h>
  12 #include <limits.h>
  13 #include <stdio.h>
  14 #include "internal/cryptlib.h"
  15 #include "bn_lcl.h"
  16
  17 #ifndef OPENSSL_NO_EC2M
  18
  19 /*
  20  * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should
  21  * fail.
  22  */
  23 # define MAX_ITERATIONS 50
  24
  25 # define SQR_nibble(w)   ((((w) & 8) << 3) \
  26                        |  (((w) & 4) << 2) \
  27                        |  (((w) & 2) << 1) \
  28                        |   ((w) & 1))
  29
  30
  31 /* Platform-specific macros to accelerate squaring. */
  32 # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
  33 #  define SQR1(w) \
  34     SQR_nibble((w) >> 60) << 56 | SQR_nibble((w) >> 56) << 48 | \
  35     SQR_nibble((w) >> 52) << 40 | SQR_nibble((w) >> 48) << 32 | \
  36     SQR_nibble((w) >> 44) << 24 | SQR_nibble((w) >> 40) << 16 | \
  37     SQR_nibble((w) >> 36) <<  8 | SQR_nibble((w) >> 32)
  38 #  define SQR0(w) \
  39     SQR_nibble((w) >> 28) << 56 | SQR_nibble((w) >> 24) << 48 | \
  40     SQR_nibble((w) >> 20) << 40 | SQR_nibble((w) >> 16) << 32 | \
  41     SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >>  8) << 16 | \
  42     SQR_nibble((w) >>  4) <<  8 | SQR_nibble((w)      )
  43 # endif
  44 # ifdef THIRTY_TWO_BIT
  45 #  define SQR1(w) \
  46     SQR_nibble((w) >> 28) << 24 | SQR_nibble((w) >> 24) << 16 | \
  47     SQR_nibble((w) >> 20) <<  8 | SQR_nibble((w) >> 16)
  48 #  define SQR0(w) \
  49     SQR_nibble((w) >> 12) << 24 | SQR_nibble((w) >>  8) << 16 | \
  50     SQR_nibble((w) >>  4) <<  8 | SQR_nibble((w)      )
  51 # endif
  52
  53 # if !defined(OPENSSL_BN_ASM_GF2m)
  54 /*
  55  * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is
  56  * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that
  57  * the variables have the right amount of space allocated.
  58  */
  59 #  ifdef THIRTY_TWO_BIT
  60 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
  61                             const BN_ULONG b)
  62 {
  63     register BN_ULONG h, l, s;
  64     BN_ULONG tab[8], top2b = a >> 30;
  65     register BN_ULONG a1, a2, a4;
  66
  67     a1 = a & (0x3FFFFFFF);
  68     a2 = a1 << 1;
  69     a4 = a2 << 1;
  70
  71     tab[0] = 0;
  72     tab[1] = a1;
  73     tab[2] = a2;
  74     tab[3] = a1 ^ a2;
  75     tab[4] = a4;
  76     tab[5] = a1 ^ a4;
  77     tab[6] = a2 ^ a4;
  78     tab[7] = a1 ^ a2 ^ a4;
  79
  80     s = tab[b & 0x7];
  81     l = s;
  82     s = tab[b >> 3 & 0x7];
  83     l ^= s << 3;
  84     h = s >> 29;
  85     s = tab[b >> 6 & 0x7];
  86     l ^= s << 6;
  87     h ^= s >> 26;
  88     s = tab[b >> 9 & 0x7];
  89     l ^= s << 9;
  90     h ^= s >> 23;
  91     s = tab[b >> 12 & 0x7];
  92     l ^= s << 12;
  93     h ^= s >> 20;
  94     s = tab[b >> 15 & 0x7];
  95     l ^= s << 15;
  96     h ^= s >> 17;
  97     s = tab[b >> 18 & 0x7];
  98     l ^= s << 18;
  99     h ^= s >> 14;
 100     s = tab[b >> 21 & 0x7];
 101     l ^= s << 21;
 102     h ^= s >> 11;
 103     s = tab[b >> 24 & 0x7];
 104     l ^= s << 24;
 105     h ^= s >> 8;
 106     s = tab[b >> 27 & 0x7];
 107     l ^= s << 27;
 108     h ^= s >> 5;
 109     s = tab[b >> 30];
 110     l ^= s << 30;
 111     h ^= s >> 2;
 112
 113     /* compensate for the top two bits of a */
 114
 115     if (top2b & 01) {
 116         l ^= b << 30;
 117         h ^= b >> 2;
 118     }
 119     if (top2b & 02) {
 120         l ^= b << 31;
 121         h ^= b >> 1;
 122     }
 123
 124     *r1 = h;
 125     *r0 = l;
 126 }
 127 #  endif
 128 #  if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG)
 129 static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a,
 130                             const BN_ULONG b)
 131 {
 132     register BN_ULONG h, l, s;
 133     BN_ULONG tab[16], top3b = a >> 61;
 134     register BN_ULONG a1, a2, a4, a8;
 135
 136     a1 = a & (0x1FFFFFFFFFFFFFFFULL);
 137     a2 = a1 << 1;
 138     a4 = a2 << 1;
 139     a8 = a4 << 1;
 140
 141     tab[0] = 0;
 142     tab[1] = a1;
 143     tab[2] = a2;
 144     tab[3] = a1 ^ a2;
 145     tab[4] = a4;
 146     tab[5] = a1 ^ a4;
 147     tab[6] = a2 ^ a4;
 148     tab[7] = a1 ^ a2 ^ a4;
 149     tab[8] = a8;
 150     tab[9] = a1 ^ a8;
 151     tab[10] = a2 ^ a8;
 152     tab[11] = a1 ^ a2 ^ a8;
 153     tab[12] = a4 ^ a8;
 154     tab[13] = a1 ^ a4 ^ a8;
 155     tab[14] = a2 ^ a4 ^ a8;
 156     tab[15] = a1 ^ a2 ^ a4 ^ a8;
 157
 158     s = tab[b & 0xF];
 159     l = s;
 160     s = tab[b >> 4 & 0xF];
 161     l ^= s << 4;
 162     h = s >> 60;
 163     s = tab[b >> 8 & 0xF];
 164     l ^= s << 8;
 165     h ^= s >> 56;
 166     s = tab[b >> 12 & 0xF];
 167     l ^= s << 12;
 168     h ^= s >> 52;
 169     s = tab[b >> 16 & 0xF];
 170     l ^= s << 16;
 171     h ^= s >> 48;
 172     s = tab[b >> 20 & 0xF];
 173     l ^= s << 20;
 174     h ^= s >> 44;
 175     s = tab[b >> 24 & 0xF];
 176     l ^= s << 24;
 177     h ^= s >> 40;
 178     s = tab[b >> 28 & 0xF];
 179     l ^= s << 28;
 180     h ^= s >> 36;
 181     s = tab[b >> 32 & 0xF];
 182     l ^= s << 32;
 183     h ^= s >> 32;
 184     s = tab[b >> 36 & 0xF];
 185     l ^= s << 36;
 186     h ^= s >> 28;
 187     s = tab[b >> 40 & 0xF];
 188     l ^= s << 40;
 189     h ^= s >> 24;
 190     s = tab[b >> 44 & 0xF];
 191     l ^= s << 44;
 192     h ^= s >> 20;
 193     s = tab[b >> 48 & 0xF];
 194     l ^= s << 48;
 195     h ^= s >> 16;
 196     s = tab[b >> 52 & 0xF];
 197     l ^= s << 52;
 198     h ^= s >> 12;
 199     s = tab[b >> 56 & 0xF];
 200     l ^= s << 56;
 201     h ^= s >> 8;
 202     s = tab[b >> 60];
 203     l ^= s << 60;
 204     h ^= s >> 4;
 205
 206     /* compensate for the top three bits of a */
 207
 208     if (top3b & 01) {
 209         l ^= b << 61;
 210         h ^= b >> 3;
 211     }
 212     if (top3b & 02) {
 213         l ^= b << 62;
 214         h ^= b >> 2;
 215     }
 216     if (top3b & 04) {
 217         l ^= b << 63;
 218         h ^= b >> 1;
 219     }
 220
 221     *r1 = h;
 222     *r0 = l;
 223 }
 224 #  endif
 225
 226 /*
 227  * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1,
 228  * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST
 229  * ensure that the variables have the right amount of space allocated.
 230  */
 231 static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0,
 232                             const BN_ULONG b1, const BN_ULONG b0)
 233 {
 234     BN_ULONG m1, m0;
 235     /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */
 236     bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1);
 237     bn_GF2m_mul_1x1(r + 1, r, a0, b0);
 238     bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1);
 239     /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */
 240     r[2] ^= m1 ^ r[1] ^ r[3];   /* h0 ^= m1 ^ l1 ^ h1; */
 241     r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */
 242 }
 243 # else
 244 void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1,
 245                      BN_ULONG b0);
 246 # endif
 247
 248 /*
 249  * Add polynomials a and b and store result in r; r could be a or b, a and b
 250  * could be equal; r is the bitwise XOR of a and b.
 251  */
 252 int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
 253 {
 254     int i;
 255     const BIGNUM *at, *bt;
 256
 257     bn_check_top(a);
 258     bn_check_top(b);
 259
 260     if (a->top < b->top) {
 261         at = b;
 262         bt = a;
 263     } else {
 264         at = a;
 265         bt = b;
 266     }
 267
 268     if (bn_wexpand(r, at->top) == NULL)
 269         return 0;
 270
 271     for (i = 0; i < bt->top; i++) {
 272         r->d[i] = at->d[i] ^ bt->d[i];
 273     }
 274     for (; i < at->top; i++) {
 275         r->d[i] = at->d[i];
 276     }
 277
 278     r->top = at->top;
 279     bn_correct_top(r);
 280
 281     return 1;
 282 }
 283
 284 /*-
 285  * Some functions allow for representation of the irreducible polynomials
 286  * as an int[], say p.  The irreducible f(t) is then of the form:
 287  *     t^p[0] + t^p[1] + ... + t^p[k]
 288  * where m = p[0] > p[1] > ... > p[k] = 0.
 289  */
 290
 291 /* Performs modular reduction of a and store result in r.  r could be a. */
 292 int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
 293 {
 294     int j, k;
 295     int n, dN, d0, d1;
 296     BN_ULONG zz, *z;
 297
 298     bn_check_top(a);
 299
 300     if (!p[0]) {
 301         /* reduction mod 1 => return 0 */
 302         BN_zero(r);
 303         return 1;
 304     }
 305
 306     /*
 307      * Since the algorithm does reduction in the r value, if a != r, copy the
 308      * contents of a into r so we can do reduction in r.
 309      */
 310     if (a != r) {
 311         if (!bn_wexpand(r, a->top))
 312             return 0;
 313         for (j = 0; j < a->top; j++) {
 314             r->d[j] = a->d[j];
 315         }
 316         r->top = a->top;
 317     }
 318     z = r->d;
 319
 320     /* start reduction */
 321     dN = p[0] / BN_BITS2;
 322     for (j = r->top - 1; j > dN;) {
 323         zz = z[j];
 324         if (z[j] == 0) {
 325             j--;
 326             continue;
 327         }
 328         z[j] = 0;
 329
 330         for (k = 1; p[k] != 0; k++) {
 331             /* reducing component t^p[k] */
 332             n = p[0] - p[k];
 333             d0 = n % BN_BITS2;
 334             d1 = BN_BITS2 - d0;
 335             n /= BN_BITS2;
 336             z[j - n] ^= (zz >> d0);
 337             if (d0)
 338                 z[j - n - 1] ^= (zz << d1);
 339         }
 340
 341         /* reducing component t^0 */
 342         n = dN;
 343         d0 = p[0] % BN_BITS2;
 344         d1 = BN_BITS2 - d0;
 345         z[j - n] ^= (zz >> d0);
 346         if (d0)
 347             z[j - n - 1] ^= (zz << d1);
 348     }
 349
 350     /* final round of reduction */
 351     while (j == dN) {
 352
 353         d0 = p[0] % BN_BITS2;
 354         zz = z[dN] >> d0;
 355         if (zz == 0)
 356             break;
 357         d1 = BN_BITS2 - d0;
 358
 359         /* clear up the top d1 bits */
 360         if (d0)
 361             z[dN] = (z[dN] << d1) >> d1;
 362         else
 363             z[dN] = 0;
 364         z[0] ^= zz;             /* reduction t^0 component */
 365
 366         for (k = 1; p[k] != 0; k++) {
 367             BN_ULONG tmp_ulong;
 368
 369             /* reducing component t^p[k] */
 370             n = p[k] / BN_BITS2;
 371             d0 = p[k] % BN_BITS2;
 372             d1 = BN_BITS2 - d0;
 373             z[n] ^= (zz << d0);
 374             if (d0 && (tmp_ulong = zz >> d1))
 375                 z[n + 1] ^= tmp_ulong;
 376         }
 377
 378     }
 379
 380     bn_correct_top(r);
 381     return 1;
 382 }
 383
 384 /*
 385  * Performs modular reduction of a by p and store result in r.  r could be a.
 386  * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper
 387  * function is only provided for convenience; for best performance, use the
 388  * BN_GF2m_mod_arr function.
 389  */
 390 int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
 391 {
 392     int ret = 0;
 393     int arr[6];
 394     bn_check_top(a);
 395     bn_check_top(p);
 396     ret = BN_GF2m_poly2arr(p, arr, OSSL_NELEM(arr));
 397     if (!ret || ret > (int)OSSL_NELEM(arr)) {
 398         BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH);
 399         return 0;
 400     }
 401     ret = BN_GF2m_mod_arr(r, a, arr);
 402     bn_check_top(r);
 403     return ret;
 404 }
 405
 406 /*
 407  * Compute the product of two polynomials a and b, reduce modulo p, and store
 408  * the result in r.  r could be a or b; a could be b.
 409  */
 410 int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
 411                         const int p[], BN_CTX *ctx)
 412 {
 413     int zlen, i, j, k, ret = 0;
 414     BIGNUM *s;
 415     BN_ULONG x1, x0, y1, y0, zz[4];
 416
 417     bn_check_top(a);
 418     bn_check_top(b);
 419
 420     if (a == b) {
 421         return BN_GF2m_mod_sqr_arr(r, a, p, ctx);
 422     }
 423
 424     BN_CTX_start(ctx);
 425     if ((s = BN_CTX_get(ctx)) == NULL)
 426         goto err;
 427
 428     zlen = a->top + b->top + 4;
 429     if (!bn_wexpand(s, zlen))
 430         goto err;
 431     s->top = zlen;
 432
 433     for (i = 0; i < zlen; i++)
 434         s->d[i] = 0;
 435
 436     for (j = 0; j < b->top; j += 2) {
 437         y0 = b->d[j];
 438         y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1];
 439         for (i = 0; i < a->top; i += 2) {
 440             x0 = a->d[i];
 441             x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1];
 442             bn_GF2m_mul_2x2(zz, x1, x0, y1, y0);
 443             for (k = 0; k < 4; k++)
 444                 s->d[i + j + k] ^= zz[k];
 445         }
 446     }
 447
 448     bn_correct_top(s);
 449     if (BN_GF2m_mod_arr(r, s, p))
 450         ret = 1;
 451     bn_check_top(r);
 452
 453  err:
 454     BN_CTX_end(ctx);
 455     return ret;
 456 }
 457
 458 /*
 459  * Compute the product of two polynomials a and b, reduce modulo p, and store
 460  * the result in r.  r could be a or b; a could equal b. This function calls
 461  * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is
 462  * only provided for convenience; for best performance, use the
 463  * BN_GF2m_mod_mul_arr function.
 464  */
 465 int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
 466                     const BIGNUM *p, BN_CTX *ctx)
 467 {
 468     int ret = 0;
 469     const int max = BN_num_bits(p) + 1;
 470     int *arr = NULL;
 471     bn_check_top(a);
 472     bn_check_top(b);
 473     bn_check_top(p);
 474     if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
 475         goto err;
 476     ret = BN_GF2m_poly2arr(p, arr, max);
 477     if (!ret || ret > max) {
 478         BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH);
 479         goto err;
 480     }
 481     ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx);
 482     bn_check_top(r);
 483  err:
 484     OPENSSL_free(arr);
 485     return ret;
 486 }
 487
 488 /* Square a, reduce the result mod p, and store it in a.  r could be a. */
 489 int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[],
 490                         BN_CTX *ctx)
 491 {
 492     int i, ret = 0;
 493     BIGNUM *s;
 494
 495     bn_check_top(a);
 496     BN_CTX_start(ctx);
 497     if ((s = BN_CTX_get(ctx)) == NULL)
 498         goto err;
 499     if (!bn_wexpand(s, 2 * a->top))
 500         goto err;
 501
 502     for (i = a->top - 1; i >= 0; i--) {
 503         s->d[2 * i + 1] = SQR1(a->d[i]);
 504         s->d[2 * i] = SQR0(a->d[i]);
 505     }
 506
 507     s->top = 2 * a->top;
 508     bn_correct_top(s);
 509     if (!BN_GF2m_mod_arr(r, s, p))
 510         goto err;
 511     bn_check_top(r);
 512     ret = 1;
 513  err:
 514     BN_CTX_end(ctx);
 515     return ret;
 516 }
 517
 518 /*
 519  * Square a, reduce the result mod p, and store it in a.  r could be a. This
 520  * function calls down to the BN_GF2m_mod_sqr_arr implementation; this
 521  * wrapper function is only provided for convenience; for best performance,
 522  * use the BN_GF2m_mod_sqr_arr function.
 523  */
 524 int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 525 {
 526     int ret = 0;
 527     const int max = BN_num_bits(p) + 1;
 528     int *arr = NULL;
 529
 530     bn_check_top(a);
 531     bn_check_top(p);
 532     if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
 533         goto err;
 534     ret = BN_GF2m_poly2arr(p, arr, max);
 535     if (!ret || ret > max) {
 536         BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH);
 537         goto err;
 538     }
 539     ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx);
 540     bn_check_top(r);
 541  err:
 542     OPENSSL_free(arr);
 543     return ret;
 544 }
 545
 546 /*
 547  * Invert a, reduce modulo p, and store the result in r. r could be a. Uses
 548  * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D.,
 549  * Hernandez, J.L., and Menezes, A.  "Software Implementation of Elliptic
 550  * Curve Cryptography Over Binary Fields".
 551  */
 552 static int BN_GF2m_mod_inv_vartime(BIGNUM *r, const BIGNUM *a,
 553                                    const BIGNUM *p, BN_CTX *ctx)
 554 {
 555     BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp;
 556     int ret = 0;
 557
 558     bn_check_top(a);
 559     bn_check_top(p);
 560
 561     BN_CTX_start(ctx);
 562
 563     b = BN_CTX_get(ctx);
 564     c = BN_CTX_get(ctx);
 565     u = BN_CTX_get(ctx);
 566     v = BN_CTX_get(ctx);
 567     if (v == NULL)
 568         goto err;
 569
 570     if (!BN_GF2m_mod(u, a, p))
 571         goto err;
 572     if (BN_is_zero(u))
 573         goto err;
 574
 575     if (!BN_copy(v, p))
 576         goto err;
 577 # if 0
 578     if (!BN_one(b))
 579         goto err;
 580
 581     while (1) {
 582         while (!BN_is_odd(u)) {
 583             if (BN_is_zero(u))
 584                 goto err;
 585             if (!BN_rshift1(u, u))
 586                 goto err;
 587             if (BN_is_odd(b)) {
 588                 if (!BN_GF2m_add(b, b, p))
 589                     goto err;
 590             }
 591             if (!BN_rshift1(b, b))
 592                 goto err;
 593         }
 594
 595         if (BN_abs_is_word(u, 1))
 596             break;
 597
 598         if (BN_num_bits(u) < BN_num_bits(v)) {
 599             tmp = u;
 600             u = v;
 601             v = tmp;
 602             tmp = b;
 603             b = c;
 604             c = tmp;
 605         }
 606
 607         if (!BN_GF2m_add(u, u, v))
 608             goto err;
 609         if (!BN_GF2m_add(b, b, c))
 610             goto err;
 611     }
 612 # else
 613     {
 614         int i;
 615         int ubits = BN_num_bits(u);
 616         int vbits = BN_num_bits(v); /* v is copy of p */
 617         int top = p->top;
 618         BN_ULONG *udp, *bdp, *vdp, *cdp;
 619
 620         if (!bn_wexpand(u, top))
 621             goto err;
 622         udp = u->d;
 623         for (i = u->top; i < top; i++)
 624             udp[i] = 0;
 625         u->top = top;
 626         if (!bn_wexpand(b, top))
 627           goto err;
 628         bdp = b->d;
 629         bdp[0] = 1;
 630         for (i = 1; i < top; i++)
 631             bdp[i] = 0;
 632         b->top = top;
 633         if (!bn_wexpand(c, top))
 634           goto err;
 635         cdp = c->d;
 636         for (i = 0; i < top; i++)
 637             cdp[i] = 0;
 638         c->top = top;
 639         vdp = v->d;             /* It pays off to "cache" *->d pointers,
 640                                  * because it allows optimizer to be more
 641                                  * aggressive. But we don't have to "cache"
 642                                  * p->d, because *p is declared 'const'... */
 643         while (1) {
 644             while (ubits && !(udp[0] & 1)) {
 645                 BN_ULONG u0, u1, b0, b1, mask;
 646
 647                 u0 = udp[0];
 648                 b0 = bdp[0];
 649                 mask = (BN_ULONG)0 - (b0 & 1);
 650                 b0 ^= p->d[0] & mask;
 651                 for (i = 0; i < top - 1; i++) {
 652                     u1 = udp[i + 1];
 653                     udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2;
 654                     u0 = u1;
 655                     b1 = bdp[i + 1] ^ (p->d[i + 1] & mask);
 656                     bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2;
 657                     b0 = b1;
 658                 }
 659                 udp[i] = u0 >> 1;
 660                 bdp[i] = b0 >> 1;
 661                 ubits--;
 662             }
 663
 664             if (ubits <= BN_BITS2) {
 665                 if (udp[0] == 0) /* poly was reducible */
 666                     goto err;
 667                 if (udp[0] == 1)
 668                     break;
 669             }
 670
 671             if (ubits < vbits) {
 672                 i = ubits;
 673                 ubits = vbits;
 674                 vbits = i;
 675                 tmp = u;
 676                 u = v;
 677                 v = tmp;
 678                 tmp = b;
 679                 b = c;
 680                 c = tmp;
 681                 udp = vdp;
 682                 vdp = v->d;
 683                 bdp = cdp;
 684                 cdp = c->d;
 685             }
 686             for (i = 0; i < top; i++) {
 687                 udp[i] ^= vdp[i];
 688                 bdp[i] ^= cdp[i];
 689             }
 690             if (ubits == vbits) {
 691                 BN_ULONG ul;
 692                 int utop = (ubits - 1) / BN_BITS2;
 693
 694                 while ((ul = udp[utop]) == 0 && utop)
 695                     utop--;
 696                 ubits = utop * BN_BITS2 + BN_num_bits_word(ul);
 697             }
 698         }
 699         bn_correct_top(b);
 700     }
 701 # endif
 702
 703     if (!BN_copy(r, b))
 704         goto err;
 705     bn_check_top(r);
 706     ret = 1;
 707
 708  err:
 709 # ifdef BN_DEBUG                /* BN_CTX_end would complain about the
 710                                  * expanded form */
 711     bn_correct_top(c);
 712     bn_correct_top(u);
 713     bn_correct_top(v);
 714 # endif
 715     BN_CTX_end(ctx);
 716     return ret;
 717 }
 718
 719 /*-
 720  * Wrapper for BN_GF2m_mod_inv_vartime that blinds the input before calling.
 721  * This is not constant time.
 722  * But it does eliminate first order deduction on the input.
 723  */
 724 int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 725 {
 726     BIGNUM *b = NULL;
 727     int ret = 0;
 728
 729     BN_CTX_start(ctx);
 730     if ((b = BN_CTX_get(ctx)) == NULL)
 731         goto err;
 732
 733     /* generate blinding value */
 734     do {
 735         if (!BN_priv_rand_ex(b, BN_num_bits(p) - 1,
 736                              BN_RAND_TOP_ANY, BN_RAND_BOTTOM_ANY, ctx))
 737             goto err;
 738     } while (BN_is_zero(b));
 739
 740     /* r := a * b */
 741     if (!BN_GF2m_mod_mul(r, a, b, p, ctx))
 742         goto err;
 743
 744     /* r := 1/(a * b) */
 745     if (!BN_GF2m_mod_inv_vartime(r, r, p, ctx))
 746         goto err;
 747
 748     /* r := b/(a * b) = 1/a */
 749     if (!BN_GF2m_mod_mul(r, r, b, p, ctx))
 750         goto err;
 751
 752     ret = 1;
 753
 754  err:
 755     BN_CTX_end(ctx);
 756     return ret;
 757 }
 758
 759 /*
 760  * Invert xx, reduce modulo p, and store the result in r. r could be xx.
 761  * This function calls down to the BN_GF2m_mod_inv implementation; this
 762  * wrapper function is only provided for convenience; for best performance,
 763  * use the BN_GF2m_mod_inv function.
 764  */
 765 int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[],
 766                         BN_CTX *ctx)
 767 {
 768     BIGNUM *field;
 769     int ret = 0;
 770
 771     bn_check_top(xx);
 772     BN_CTX_start(ctx);
 773     if ((field = BN_CTX_get(ctx)) == NULL)
 774         goto err;
 775     if (!BN_GF2m_arr2poly(p, field))
 776         goto err;
 777
 778     ret = BN_GF2m_mod_inv(r, xx, field, ctx);
 779     bn_check_top(r);
 780
 781  err:
 782     BN_CTX_end(ctx);
 783     return ret;
 784 }
 785
 786 /*
 787  * Divide y by x, reduce modulo p, and store the result in r. r could be x
 788  * or y, x could equal y.
 789  */
 790 int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x,
 791                     const BIGNUM *p, BN_CTX *ctx)
 792 {
 793     BIGNUM *xinv = NULL;
 794     int ret = 0;
 795
 796     bn_check_top(y);
 797     bn_check_top(x);
 798     bn_check_top(p);
 799
 800     BN_CTX_start(ctx);
 801     xinv = BN_CTX_get(ctx);
 802     if (xinv == NULL)
 803         goto err;
 804
 805     if (!BN_GF2m_mod_inv(xinv, x, p, ctx))
 806         goto err;
 807     if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx))
 808         goto err;
 809     bn_check_top(r);
 810     ret = 1;
 811
 812  err:
 813     BN_CTX_end(ctx);
 814     return ret;
 815 }
 816
 817 /*
 818  * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx
 819  * * or yy, xx could equal yy. This function calls down to the
 820  * BN_GF2m_mod_div implementation; this wrapper function is only provided for
 821  * convenience; for best performance, use the BN_GF2m_mod_div function.
 822  */
 823 int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx,
 824                         const int p[], BN_CTX *ctx)
 825 {
 826     BIGNUM *field;
 827     int ret = 0;
 828
 829     bn_check_top(yy);
 830     bn_check_top(xx);
 831
 832     BN_CTX_start(ctx);
 833     if ((field = BN_CTX_get(ctx)) == NULL)
 834         goto err;
 835     if (!BN_GF2m_arr2poly(p, field))
 836         goto err;
 837
 838     ret = BN_GF2m_mod_div(r, yy, xx, field, ctx);
 839     bn_check_top(r);
 840
 841  err:
 842     BN_CTX_end(ctx);
 843     return ret;
 844 }
 845
 846 /*
 847  * Compute the bth power of a, reduce modulo p, and store the result in r.  r
 848  * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE
 849  * P1363.
 850  */
 851 int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
 852                         const int p[], BN_CTX *ctx)
 853 {
 854     int ret = 0, i, n;
 855     BIGNUM *u;
 856
 857     bn_check_top(a);
 858     bn_check_top(b);
 859
 860     if (BN_is_zero(b))
 861         return BN_one(r);
 862
 863     if (BN_abs_is_word(b, 1))
 864         return (BN_copy(r, a) != NULL);
 865
 866     BN_CTX_start(ctx);
 867     if ((u = BN_CTX_get(ctx)) == NULL)
 868         goto err;
 869
 870     if (!BN_GF2m_mod_arr(u, a, p))
 871         goto err;
 872
 873     n = BN_num_bits(b) - 1;
 874     for (i = n - 1; i >= 0; i--) {
 875         if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx))
 876             goto err;
 877         if (BN_is_bit_set(b, i)) {
 878             if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx))
 879                 goto err;
 880         }
 881     }
 882     if (!BN_copy(r, u))
 883         goto err;
 884     bn_check_top(r);
 885     ret = 1;
 886  err:
 887     BN_CTX_end(ctx);
 888     return ret;
 889 }
 890
 891 /*
 892  * Compute the bth power of a, reduce modulo p, and store the result in r.  r
 893  * could be a. This function calls down to the BN_GF2m_mod_exp_arr
 894  * implementation; this wrapper function is only provided for convenience;
 895  * for best performance, use the BN_GF2m_mod_exp_arr function.
 896  */
 897 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
 898                     const BIGNUM *p, BN_CTX *ctx)
 899 {
 900     int ret = 0;
 901     const int max = BN_num_bits(p) + 1;
 902     int *arr = NULL;
 903     bn_check_top(a);
 904     bn_check_top(b);
 905     bn_check_top(p);
 906     if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
 907         goto err;
 908     ret = BN_GF2m_poly2arr(p, arr, max);
 909     if (!ret || ret > max) {
 910         BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH);
 911         goto err;
 912     }
 913     ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx);
 914     bn_check_top(r);
 915  err:
 916     OPENSSL_free(arr);
 917     return ret;
 918 }
 919
 920 /*
 921  * Compute the square root of a, reduce modulo p, and store the result in r.
 922  * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
 923  */
 924 int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[],
 925                          BN_CTX *ctx)
 926 {
 927     int ret = 0;
 928     BIGNUM *u;
 929
 930     bn_check_top(a);
 931
 932     if (!p[0]) {
 933         /* reduction mod 1 => return 0 */
 934         BN_zero(r);
 935         return 1;
 936     }
 937
 938     BN_CTX_start(ctx);
 939     if ((u = BN_CTX_get(ctx)) == NULL)
 940         goto err;
 941
 942     if (!BN_set_bit(u, p[0] - 1))
 943         goto err;
 944     ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx);
 945     bn_check_top(r);
 946
 947  err:
 948     BN_CTX_end(ctx);
 949     return ret;
 950 }
 951
 952 /*
 953  * Compute the square root of a, reduce modulo p, and store the result in r.
 954  * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr
 955  * implementation; this wrapper function is only provided for convenience;
 956  * for best performance, use the BN_GF2m_mod_sqrt_arr function.
 957  */
 958 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 959 {
 960     int ret = 0;
 961     const int max = BN_num_bits(p) + 1;
 962     int *arr = NULL;
 963     bn_check_top(a);
 964     bn_check_top(p);
 965     if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
 966         goto err;
 967     ret = BN_GF2m_poly2arr(p, arr, max);
 968     if (!ret || ret > max) {
 969         BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH);
 970         goto err;
 971     }
 972     ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx);
 973     bn_check_top(r);
 974  err:
 975     OPENSSL_free(arr);
 976     return ret;
 977 }
 978
 979 /*
 980  * Find r such that r^2 + r = a mod p.  r could be a. If no r exists returns
 981  * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
 982  */
 983 int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[],
 984                                BN_CTX *ctx)
 985 {
 986     int ret = 0, count = 0, j;
 987     BIGNUM *a, *z, *rho, *w, *w2, *tmp;
 988
 989     bn_check_top(a_);
 990
 991     if (!p[0]) {
 992         /* reduction mod 1 => return 0 */
 993         BN_zero(r);
 994         return 1;
 995     }
 996
 997     BN_CTX_start(ctx);
 998     a = BN_CTX_get(ctx);
 999     z = BN_CTX_get(ctx);
1000     w = BN_CTX_get(ctx);
1001     if (w == NULL)
1002         goto err;
1003
1004     if (!BN_GF2m_mod_arr(a, a_, p))
1005         goto err;
1006
1007     if (BN_is_zero(a)) {
1008         BN_zero(r);
1009         ret = 1;
1010         goto err;
1011     }
1012
1013     if (p[0] & 0x1) {           /* m is odd */
1014         /* compute half-trace of a */
1015         if (!BN_copy(z, a))
1016             goto err;
1017         for (j = 1; j <= (p[0] - 1) / 2; j++) {
1018             if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1019                 goto err;
1020             if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1021                 goto err;
1022             if (!BN_GF2m_add(z, z, a))
1023                 goto err;
1024         }
1025
1026     } else {                    /* m is even */
1027
1028         rho = BN_CTX_get(ctx);
1029         w2 = BN_CTX_get(ctx);
1030         tmp = BN_CTX_get(ctx);
1031         if (tmp == NULL)
1032             goto err;
1033         do {
1034             if (!BN_priv_rand_ex(rho, p[0], BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ANY,
1035                                  ctx))
1036                 goto err;
1037             if (!BN_GF2m_mod_arr(rho, rho, p))
1038                 goto err;
1039             BN_zero(z);
1040             if (!BN_copy(w, rho))
1041                 goto err;
1042             for (j = 1; j <= p[0] - 1; j++) {
1043                 if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx))
1044                     goto err;
1045                 if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx))
1046                     goto err;
1047                 if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx))
1048                     goto err;
1049                 if (!BN_GF2m_add(z, z, tmp))
1050                     goto err;
1051                 if (!BN_GF2m_add(w, w2, rho))
1052                     goto err;
1053             }
1054             count++;
1055         } while (BN_is_zero(w) && (count < MAX_ITERATIONS));
1056         if (BN_is_zero(w)) {
1057             BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS);
1058             goto err;
1059         }
1060     }
1061
1062     if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx))
1063         goto err;
1064     if (!BN_GF2m_add(w, z, w))
1065         goto err;
1066     if (BN_GF2m_cmp(w, a)) {
1067         BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION);
1068         goto err;
1069     }
1070
1071     if (!BN_copy(r, z))
1072         goto err;
1073     bn_check_top(r);
1074
1075     ret = 1;
1076
1077  err:
1078     BN_CTX_end(ctx);
1079     return ret;
1080 }
1081
1082 /*
1083  * Find r such that r^2 + r = a mod p.  r could be a. If no r exists returns
1084  * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr
1085  * implementation; this wrapper function is only provided for convenience;
1086  * for best performance, use the BN_GF2m_mod_solve_quad_arr function.
1087  */
1088 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
1089                            BN_CTX *ctx)
1090 {
1091     int ret = 0;
1092     const int max = BN_num_bits(p) + 1;
1093     int *arr = NULL;
1094     bn_check_top(a);
1095     bn_check_top(p);
1096     if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL)
1097         goto err;
1098     ret = BN_GF2m_poly2arr(p, arr, max);
1099     if (!ret || ret > max) {
1100         BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH);
1101         goto err;
1102     }
1103     ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx);
1104     bn_check_top(r);
1105  err:
1106     OPENSSL_free(arr);
1107     return ret;
1108 }
1109
1110 /*
1111  * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i *
1112  * x^i) into an array of integers corresponding to the bits with non-zero
1113  * coefficient.  Array is terminated with -1. Up to max elements of the array
1114  * will be filled.  Return value is total number of array elements that would
1115  * be filled if array was large enough.
1116  */
1117 int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
1118 {
1119     int i, j, k = 0;
1120     BN_ULONG mask;
1121
1122     if (BN_is_zero(a))
1123         return 0;
1124
1125     for (i = a->top - 1; i >= 0; i--) {
1126         if (!a->d[i])
1127             /* skip word if a->d[i] == 0 */
1128             continue;
1129         mask = BN_TBIT;
1130         for (j = BN_BITS2 - 1; j >= 0; j--) {
1131             if (a->d[i] & mask) {
1132                 if (k < max)
1133                     p[k] = BN_BITS2 * i + j;
1134                 k++;
1135             }
1136             mask >>= 1;
1137         }
1138     }
1139
1140     if (k < max) {
1141         p[k] = -1;
1142         k++;
1143     }
1144
1145     return k;
1146 }
1147
1148 /*
1149  * Convert the coefficient array representation of a polynomial to a
1150  * bit-string.  The array must be terminated by -1.
1151  */
1152 int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
1153 {
1154     int i;
1155
1156     bn_check_top(a);
1157     BN_zero(a);
1158     for (i = 0; p[i] != -1; i++) {
1159         if (BN_set_bit(a, p[i]) == 0)
1160             return 0;
1161     }
1162     bn_check_top(a);
1163
1164     return 1;
1165 }
1166
1167 #endif