crypto/openssl-0.9.7e/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  * If 'a' is not a square, this is not necessarily detected by
  69  * the algorithms; a bogus result must be expected in this case.
  70  */
  71         {
  72         BIGNUM *ret = in;
  73         int err = 1;
  74         int r;
  75         BIGNUM *b, *q, *t, *x, *y;
  76         int e, i, j;
  77
  78         if (!BN_is_odd(p) || BN_abs_is_word(p, 1))
  79                 {
  80                 if (BN_abs_is_word(p, 2))
  81                         {
  82                         if (ret == NULL)
  83                                 ret = BN_new();
  84                         if (ret == NULL)
  85                                 goto end;
  86                         if (!BN_set_word(ret, BN_is_bit_set(a, 0)))
  87                                 {
  88                                 BN_free(ret);
  89                                 return NULL;
  90                                 }
  91                         return ret;
  92                         }
  93
  94                 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
  95                 return(NULL);
  96                 }
  97
  98         if (BN_is_zero(a) || BN_is_one(a))
  99                 {
 100                 if (ret == NULL)
 101                         ret = BN_new();
 102                 if (ret == NULL)
 103                         goto end;
 104                 if (!BN_set_word(ret, BN_is_one(a)))
 105                         {
 106                         BN_free(ret);
 107                         return NULL;
 108                         }
 109                 return ret;
 110                 }
 111
 112 #if 0 /* if BN_mod_sqrt is used with correct input, this just wastes time */
 113         r = BN_kronecker(a, p, ctx);
 114         if (r < -1) return NULL;
 115         if (r == -1)
 116                 {
 117                 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 118                 return(NULL);
 119                 }
 120 #endif
 121
 122         BN_CTX_start(ctx);
 123         b = BN_CTX_get(ctx);
 124         q = BN_CTX_get(ctx);
 125         t = BN_CTX_get(ctx);
 126         x = BN_CTX_get(ctx);
 127         y = BN_CTX_get(ctx);
 128         if (y == NULL) goto end;
 129
 130         if (ret == NULL)
 131                 ret = BN_new();
 132         if (ret == NULL) goto end;
 133
 134         /* now write  |p| - 1  as  2^e*q  where  q  is odd */
 135         e = 1;
 136         while (!BN_is_bit_set(p, e))
 137                 e++;
 138         /* we'll set  q  later (if needed) */
 139
 140         if (e == 1)
 141                 {
 142                 /* The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
 143                  * modulo  (|p|-1)/2,  and square roots can be computed
 144                  * directly by modular exponentiation.
 145                  * We have
 146                  *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
 147                  * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
 148                  */
 149                 if (!BN_rshift(q, p, 2)) goto end;
 150                 q->neg = 0;
 151                 if (!BN_add_word(q, 1)) goto end;
 152                 if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
 153                 err = 0;
 154                 goto end;
 155                 }
 156
 157         if (e == 2)
 158                 {
 159                 /* |p| == 5  (mod 8)
 160                  *
 161                  * In this case  2  is always a non-square since
 162                  * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
 163                  * So if  a  really is a square, then  2*a  is a non-square.
 164                  * Thus for
 165                  *      b := (2*a)^((|p|-5)/8),
 166                  *      i := (2*a)*b^2
 167                  * we have
 168                  *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
 169                  *         = (2*a)^((p-1)/2)
 170                  *         = -1;
 171                  * so if we set
 172                  *      x := a*b*(i-1),
 173                  * then
 174                  *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
 175                  *         = a^2 * b^2 * (-2*i)
 176                  *         = a*(-i)*(2*a*b^2)
 177                  *         = a*(-i)*i
 178                  *         = a.
 179                  *
 180                  * (This is due to A.O.L. Atkin,
 181                  * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
 182                  * November 1992.)
 183                  */
 184
 185                 /* make sure that  a  is reduced modulo p */
 186                 if (a->neg || BN_ucmp(a, p) >= 0)
 187                         {
 188                         if (!BN_nnmod(x, a, p, ctx)) goto end;
 189                         a = x; /* use x as temporary variable */
 190                         }
 191
 192                 /* t := 2*a */
 193                 if (!BN_mod_lshift1_quick(t, a, p)) goto end;
 194
 195                 /* b := (2*a)^((|p|-5)/8) */
 196                 if (!BN_rshift(q, p, 3)) goto end;
 197                 q->neg = 0;
 198                 if (!BN_mod_exp(b, t, q, p, ctx)) goto end;
 199
 200                 /* y := b^2 */
 201                 if (!BN_mod_sqr(y, b, p, ctx)) goto end;
 202
 203                 /* t := (2*a)*b^2 - 1*/
 204                 if (!BN_mod_mul(t, t, y, p, ctx)) goto end;
 205                 if (!BN_sub_word(t, 1)) goto end;
 206
 207                 /* x = a*b*t */
 208                 if (!BN_mod_mul(x, a, b, p, ctx)) goto end;
 209                 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
 210
 211                 if (!BN_copy(ret, x)) goto end;
 212                 err = 0;
 213                 goto end;
 214                 }
 215
 216         /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
 217          * First, find some  y  that is not a square. */
 218         if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
 219         q->neg = 0;
 220         i = 2;
 221         do
 222                 {
 223                 /* For efficiency, try small numbers first;
 224                  * if this fails, try random numbers.
 225                  */
 226                 if (i < 22)
 227                         {
 228                         if (!BN_set_word(y, i)) goto end;
 229                         }
 230                 else
 231                         {
 232                         if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
 233                         if (BN_ucmp(y, p) >= 0)
 234                                 {
 235                                 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
 236                                 }
 237                         /* now 0 <= y < |p| */
 238                         if (BN_is_zero(y))
 239                                 if (!BN_set_word(y, i)) goto end;
 240                         }
 241
 242                 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
 243                 if (r < -1) goto end;
 244                 if (r == 0)
 245                         {
 246                         /* m divides p */
 247                         BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 248                         goto end;
 249                         }
 250                 }
 251         while (r == 1 && ++i < 82);
 252
 253         if (r != -1)
 254                 {
 255                 /* Many rounds and still no non-square -- this is more likely
 256                  * a bug than just bad luck.
 257                  * Even if  p  is not prime, we should have found some  y
 258                  * such that r == -1.
 259                  */
 260                 BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
 261                 goto end;
 262                 }
 263
 264         /* Here's our actual 'q': */
 265         if (!BN_rshift(q, q, e)) goto end;
 266
 267         /* Now that we have some non-square, we can find an element
 268          * of order  2^e  by computing its q'th power. */
 269         if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
 270         if (BN_is_one(y))
 271                 {
 272                 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 273                 goto end;
 274                 }
 275
 276         /* Now we know that (if  p  is indeed prime) there is an integer
 277          * k,  0 <= k < 2^e,  such that
 278          *
 279          *      a^q * y^k == 1   (mod p).
 280          *
 281          * As  a^q  is a square and  y  is not,  k  must be even.
 282          * q+1  is even, too, so there is an element
 283          *
 284          *     X := a^((q+1)/2) * y^(k/2),
 285          *
 286          * and it satisfies
 287          *
 288          *     X^2 = a^q * a     * y^k
 289          *         = a,
 290          *
 291          * so it is the square root that we are looking for.
 292          */
 293
 294         /* t := (q-1)/2  (note that  q  is odd) */
 295         if (!BN_rshift1(t, q)) goto end;
 296
 297         /* x := a^((q-1)/2) */
 298         if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
 299                 {
 300                 if (!BN_nnmod(t, a, p, ctx)) goto end;
 301                 if (BN_is_zero(t))
 302                         {
 303                         /* special case: a == 0  (mod p) */
 304                         if (!BN_zero(ret)) goto end;
 305                         err = 0;
 306                         goto end;
 307                         }
 308                 else
 309                         if (!BN_one(x)) goto end;
 310                 }
 311         else
 312                 {
 313                 if (!BN_mod_exp(x, a, t, p, ctx)) goto end;
 314                 if (BN_is_zero(x))
 315                         {
 316                         /* special case: a == 0  (mod p) */
 317                         if (!BN_zero(ret)) goto end;
 318                         err = 0;
 319                         goto end;
 320                         }
 321                 }
 322
 323         /* b := a*x^2  (= a^q) */
 324         if (!BN_mod_sqr(b, x, p, ctx)) goto end;
 325         if (!BN_mod_mul(b, b, a, p, ctx)) goto end;
 326
 327         /* x := a*x    (= a^((q+1)/2)) */
 328         if (!BN_mod_mul(x, x, a, p, ctx)) goto end;
 329
 330         while (1)
 331                 {
 332                 /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
 333                  * where  E  refers to the original value of  e,  which we
 334                  * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
 335                  *
 336                  * We have  a*b = x^2,
 337                  *    y^2^(e-1) = -1,
 338                  *    b^2^(e-1) = 1.
 339                  */
 340
 341                 if (BN_is_one(b))
 342                         {
 343                         if (!BN_copy(ret, x)) goto end;
 344                         err = 0;
 345                         goto end;
 346                         }
 347
 348
 349                 /* find smallest  i  such that  b^(2^i) = 1 */
 350                 i = 1;
 351                 if (!BN_mod_sqr(t, b, p, ctx)) goto end;
 352                 while (!BN_is_one(t))
 353                         {
 354                         i++;
 355                         if (i == e)
 356                                 {
 357                                 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 358                                 goto end;
 359                                 }
 360                         if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
 361                         }
 362
 363
 364                 /* t := y^2^(e - i - 1) */
 365                 if (!BN_copy(t, y)) goto end;
 366                 for (j = e - i - 1; j > 0; j--)
 367                         {
 368                         if (!BN_mod_sqr(t, t, p, ctx)) goto end;
 369                         }
 370                 if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
 371                 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
 372                 if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
 373                 e = i;
 374                 }
 375
 376  end:
 377         if (err)
 378                 {
 379                 if (ret != NULL && ret != in)
 380                         {
 381                         BN_clear_free(ret);
 382                         }
 383                 ret = NULL;
 384                 }
 385         BN_CTX_end(ctx);
 386         return ret;
 387         }