crypto/bn/bn_sqrt.c

   1 /* crypto/bn/bn_mod.c */
   2 /* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
   3  * and Bodo Moeller for the OpenSSL project. */
   4 /* ====================================================================
   5  * Copyright (c) 1998-2000 The OpenSSL Project.  All rights reserved.
   6  *
   7  * Redistribution and use in source and binary forms, with or without
   8  * modification, are permitted provided that the following conditions
   9  * are met:
  10  *
  11  * 1. Redistributions of source code must retain the above copyright
  12  *    notice, this list of conditions and the following disclaimer.
  13  *
  14  * 2. Redistributions in binary form must reproduce the above copyright
  15  *    notice, this list of conditions and the following disclaimer in
  16  *    the documentation and/or other materials provided with the
  17  *    distribution.
  18  *
  19  * 3. All advertising materials mentioning features or use of this
  20  *    software must display the following acknowledgment:
  21  *    "This product includes software developed by the OpenSSL Project
  22  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
  23  *
  24  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
  25  *    endorse or promote products derived from this software without
  26  *    prior written permission. For written permission, please contact
  27  *    openssl-core@openssl.org.
  28  *
  29  * 5. Products derived from this software may not be called "OpenSSL"
  30  *    nor may "OpenSSL" appear in their names without prior written
  31  *    permission of the OpenSSL Project.
  32  *
  33  * 6. Redistributions of any form whatsoever must retain the following
  34  *    acknowledgment:
  35  *    "This product includes software developed by the OpenSSL Project
  36  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
  37  *
  38  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
  39  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  40  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
  41  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
  42  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  43  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
  44  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
  45  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  46  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
  47  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
  48  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
  49  * OF THE POSSIBILITY OF SUCH DAMAGE.
  50  * ====================================================================
  51  *
  52  * This product includes cryptographic software written by Eric Young
  53  * (eay@cryptsoft.com).  This product includes software written by Tim
  54  * Hudson (tjh@cryptsoft.com).
  55  *
  56  */
  57
  58 #include "cryptlib.h"
  59 #include "bn_lcl.h"
  60
  61
  62 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
  63 /* Returns 'ret' such that
  64  *      ret^2 == a (mod p),
  65  * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
  66  * in Algebraic Computational Number Theory", algorithm 1.5.1).
  67  * 'p' must be prime!
  68  */
  69         {
  70         BIGNUM *ret = in;
  71         int err = 1;
  72         int r;
  73         BIGNUM *b, *q, *t, *x, *y;
  74         int e, i, j;
  75
  76         if (!BN_is_odd(p) || BN_abs_is_word(p, 1))
  77                 {
  78                 if (BN_abs_is_word(p, 2))
  79                         {
  80                         if (ret == NULL)
  81                                 ret = BN_new();
  82                         if (ret == NULL)
  83                                 goto end;
  84                         if (!BN_set_word(ret, BN_is_bit_set(a, 0)))
  85                                 {
  86                                 BN_free(ret);
  87                                 return NULL;
  88                                 }
  89                         return ret;
  90                         }
  91
  92                 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
  93                 return(NULL);
  94                 }
  95
  96         if (BN_is_zero(a) || BN_is_one(a))
  97                 {
  98                 if (ret == NULL)
  99                         ret = BN_new();
 100                 if (ret == NULL)
 101                         goto end;
 102                 if (!BN_set_word(ret, BN_is_one(a)))
 103                         {
 104                         BN_free(ret);
 105                         return NULL;
 106                         }
 107                 return ret;
 108                 }
 109
 110 #if 0 /* if BN_mod_sqrt is used with correct input, this just wastes time */
 111         r = BN_kronecker(a, p, ctx);
 112         if (r < -1) return NULL;
 113         if (r == -1)
 114                 {
 115                 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 116                 return(NULL);
 117                 }
 118 #endif
 119
 120         BN_CTX_start(ctx);
 121         b = BN_CTX_get(ctx);
 122         q = BN_CTX_get(ctx);
 123         t = BN_CTX_get(ctx);
 124         x = BN_CTX_get(ctx);
 125         y = BN_CTX_get(ctx);
 126         if (y == NULL) goto end;
 127
 128         if (ret == NULL)
 129                 ret = BN_new();
 130         if (ret == NULL) goto end;
 131
 132         /* now write  |p| - 1  as  2^e*q  where  q  is odd */
 133         e = 1;
 134         while (!BN_is_bit_set(p, e))
 135                 e++;
 136         if (e > 2)
 137                 {
 138                 /* we don't need this  q  if  e = 1 or 2 */
 139                 if (!BN_rshift(q, p, e)) goto end;
 140                 q->neg = 0;
 141                 }
 142
 143         if (e == 1)
 144                 {
 145                 /* The easy case:  (p-1)/2  is odd, so 2 has an inverse
 146                  * modulo  (p-1)/2,  and square roots can be computed
 147                  * directly by modular exponentiation.
 148                  * We have
 149                  *     2 * (p+1)/4 == 1   (mod (p-1)/2),
 150                  * so we can use exponent  (p+1)/4,  i.e.  (p-3)/4 + 1.
 151                  */
 152                 if (!BN_rshift(q, p, 2)) goto end;
 153                 q->neg = 0;
 154                 if (!BN_add_word(q, 1)) goto end;
 155                 if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
 156                 err = 0;
 157                 goto end;
 158                 }
 159
 160         if (e == 2)
 161                 {
 162                 /* p == 5  (mod 8)
 163                  *
 164                  * In this case  2  is always a non-square since
 165                  * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
 166                  * So if  a  really is a square, then  2*a  is a non-square.
 167                  * Thus for
 168                  *      b := (2*a)^((p-5)/8),
 169                  *      i := (2*a)*b^2
 170                  * we have
 171                  *     i^2 = (2*a)^((1 + (p-5)/4)*2)
 172                  *         = (2*a)^((p-1)/2)
 173                  *         = -1;
 174                  * so if we set
 175                  *      x := a*b*(i-1),
 176                  * then
 177                  *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
 178                  *         = a^2 * b^2 * (-2*i)
 179                  *         = a*(-i)*(2*a*b^2)
 180                  *         = a*(-i)*i
 181                  *         = a.
 182                  *
 183                  * (This is due to A.O.L. Atkin,
 184                  * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
 185                  * November 1992.)
 186                  */
 187
 188                 /* make sure that  a  is reduced modulo p */
 189                 if (a->neg || BN_ucmp(a, p) >= 0)
 190                         {
 191                         if (!BN_nnmod(x, a, p, ctx)) goto end;
 192                         a = x; /* use x as temporary variable */
 193                         }
 194
 195                 /* t := 2*a */
 196                 if (!BN_mod_lshift1_quick(t, a, p)) goto end;
 197
 198                 /* b := (2*a)^((p-5)/8) */
 199                 if (!BN_rshift(q, p, 3)) goto end;
 200                 q->neg = 0;
 201                 if (!BN_mod_exp(b, t, q, p, ctx)) goto end;
 202
 203                 /* y := b^2 */
 204                 if (!BN_mod_sqr(y, b, p, ctx)) goto end;
 205
 206                 /* t := (2*a)*b^2 - 1*/
 207                 if (!BN_mod_mul(t, t, y, p, ctx)) goto end;
 208                 if (!BN_sub_word(t, 1)) goto end;
 209
 210                 /* x = a*b*t */
 211                 if (!BN_mod_mul(x, a, b, p, ctx)) goto end;
 212                 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
 213
 214                 if (!BN_copy(ret, x)) goto end;
 215                 err = 0;
 216                 goto end;
 217                 }
 218
 219         /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
 220          * First, find some  y  that is not a square. */
 221         i = 2;
 222         do
 223                 {
 224                 /* For efficiency, try small numbers first;
 225                  * if this fails, try random numbers.
 226                  */
 227                 if (i < 22)
 228                         {
 229                         if (!BN_set_word(y, i)) goto end;
 230                         }
 231                 else
 232                         {
 233                         if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
 234                         if (BN_ucmp(y, p) >= 0)
 235                                 {
 236                                 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
 237                                 }
 238                         /* now 0 <= y < |p| */
 239                         if (BN_is_zero(y))
 240                                 if (!BN_set_word(y, i)) goto end;
 241                         }
 242
 243                 r = BN_kronecker(y, p, ctx);
 244                 if (r < -1) goto end;
 245                 if (r == 0)
 246                         {
 247                         /* m divides p */
 248                         BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 249                         goto end;
 250                         }
 251                 }
 252         while (r == 1 && ++i < 82);
 253
 254         if (r != -1)
 255                 {
 256                 /* Many rounds and still no non-square -- this is more likely
 257                  * a bug than just bad luck.
 258                  * Even if  p  is not prime, we should have found some  y
 259                  * such that r == -1.
 260                  */
 261                 BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
 262                 goto end;
 263                 }
 264
 265
 266         /* Now that we have some non-square, we can find an element
 267          * of order  2^e  by computing its q'th power. */
 268         if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
 269         if (BN_is_one(y))
 270                 {
 271                 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 272                 goto end;
 273                 }
 274
 275         /* Now we know that (if  p  is indeed prime) there is an integer
 276          * k,  0 <= k < 2^e,  such that
 277          *
 278          *      a^q * y^k == 1   (mod p).
 279          *
 280          * As  a^q  is a square and  y  is not,  k  must be even.
 281          * q+1  is even, too, so there is an element
 282          *
 283          *     X := a^((q+1)/2) * y^(k/2),
 284          *
 285          * and it satisfies
 286          *
 287          *     X^2 = a^q * a     * y^k
 288          *         = a,
 289          *
 290          * so it is the square root that we are looking for.
 291          */
 292
 293         /* t := (q-1)/2  (note that  q  is odd) */
 294         if (!BN_rshift1(t, q)) goto end;
 295
 296         /* x := a^((q-1)/2) */
 297         if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
 298                 {
 299                 if (!BN_nnmod(t, a, p, ctx)) goto end;
 300                 if (BN_is_zero(t))
 301                         {
 302                         /* special case: a == 0  (mod p) */
 303                         if (!BN_zero(ret)) goto end;
 304                         err = 0;
 305                         goto end;
 306                         }
 307                 else
 308                         if (!BN_one(x)) goto end;
 309                 }
 310         else
 311                 {
 312                 if (!BN_mod_exp(x, a, t, p, ctx)) goto end;
 313                 if (BN_is_zero(x))
 314                         {
 315                         /* special case: a == 0  (mod p) */
 316                         if (!BN_zero(ret)) goto end;
 317                         err = 0;
 318                         goto end;
 319                         }
 320                 }
 321
 322         /* b := a*x^2  (= a^q) */
 323         if (!BN_mod_sqr(b, x, p, ctx)) goto end;
 324         if (!BN_mod_mul(b, b, a, p, ctx)) goto end;
 325
 326         /* x := a*x    (= a^((q+1)/2)) */
 327         if (!BN_mod_mul(x, x, a, p, ctx)) goto end;
 328
 329         while (1)
 330                 {
 331                 /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
 332                  * where  E  refers to the original value of  e,  which we
 333                  * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
 334                  *
 335                  * We have  a*b = x^2,
 336                  *    y^2^(e-1) = -1,
 337                  *    b^2^(e-1) = 1.
 338                  */
 339
 340                 if (BN_is_one(b))
 341                         {
 342                         if (!BN_copy(ret, x)) goto end;
 343                         err = 0;
 344                         goto end;
 345                         }
 346
 347
 348                 /* find smallest  i  such that  b^(2^i) = 1 */
 349                 i = 1;
 350                 if (!BN_mod_sqr(t, b, p, ctx)) goto end;
 351                 while (!BN_is_one(t))
 352                         {
 353                         i++;
 354                         if (i == e)
 355                                 {
 356                                 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 357                                 goto end;
 358                                 }
 359                         if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
 360                         }
 361
 362
 363                 /* t := y^2^(e - i - 1) */
 364                 if (!BN_copy(t, y)) goto end;
 365                 for (j = e - i - 1; j > 0; j--)
 366                         {
 367                         if (!BN_mod_sqr(t, t, p, ctx)) goto end;
 368                         }
 369                 if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
 370                 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
 371                 if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
 372                 e = i;
 373                 }
 374
 375  end:
 376         if (err)
 377                 {
 378                 if (ret != NULL && ret != in)
 379                         {
 380                         BN_clear_free(ret);
 381                         }
 382                 ret = NULL;
 383                 }
 384         BN_CTX_end(ctx);
 385         return ret;
 386         }