crypto/bn/bn_sqrt.c

   1 /* crypto/bn/bn_sqrt.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 *A, *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                         bn_check_top(ret);
  90                         return ret;
  91                         }
  92
  93                 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
  94                 return(NULL);
  95                 }
  96
  97         if (BN_is_zero(a) || BN_is_one(a))
  98                 {
  99                 if (ret == NULL)
 100                         ret = BN_new();
 101                 if (ret == NULL)
 102                         goto end;
 103                 if (!BN_set_word(ret, BN_is_one(a)))
 104                         {
 105                         BN_free(ret);
 106                         return NULL;
 107                         }
 108                 bn_check_top(ret);
 109                 return ret;
 110                 }
 111
 112         BN_CTX_start(ctx);
 113         A = BN_CTX_get(ctx);
 114         b = BN_CTX_get(ctx);
 115         q = BN_CTX_get(ctx);
 116         t = BN_CTX_get(ctx);
 117         x = BN_CTX_get(ctx);
 118         y = BN_CTX_get(ctx);
 119         if (y == NULL) goto end;
 120
 121         if (ret == NULL)
 122                 ret = BN_new();
 123         if (ret == NULL) goto end;
 124
 125         /* A = a mod p */
 126         if (!BN_nnmod(A, a, p, ctx)) goto end;
 127
 128         /* now write  |p| - 1  as  2^e*q  where  q  is odd */
 129         e = 1;
 130         while (!BN_is_bit_set(p, e))
 131                 e++;
 132         /* we'll set  q  later (if needed) */
 133
 134         if (e == 1)
 135                 {
 136                 /* The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
 137                  * modulo  (|p|-1)/2,  and square roots can be computed
 138                  * directly by modular exponentiation.
 139                  * We have
 140                  *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
 141                  * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
 142                  */
 143                 if (!BN_rshift(q, p, 2)) goto end;
 144                 q->neg = 0;
 145                 if (!BN_add_word(q, 1)) goto end;
 146                 if (!BN_mod_exp(ret, A, q, p, ctx)) goto end;
 147                 err = 0;
 148                 goto vrfy;
 149                 }
 150
 151         if (e == 2)
 152                 {
 153                 /* |p| == 5  (mod 8)
 154                  *
 155                  * In this case  2  is always a non-square since
 156                  * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
 157                  * So if  a  really is a square, then  2*a  is a non-square.
 158                  * Thus for
 159                  *      b := (2*a)^((|p|-5)/8),
 160                  *      i := (2*a)*b^2
 161                  * we have
 162                  *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
 163                  *         = (2*a)^((p-1)/2)
 164                  *         = -1;
 165                  * so if we set
 166                  *      x := a*b*(i-1),
 167                  * then
 168                  *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
 169                  *         = a^2 * b^2 * (-2*i)
 170                  *         = a*(-i)*(2*a*b^2)
 171                  *         = a*(-i)*i
 172                  *         = a.
 173                  *
 174                  * (This is due to A.O.L. Atkin,
 175                  * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
 176                  * November 1992.)
 177                  */
 178
 179                 /* t := 2*a */
 180                 if (!BN_mod_lshift1_quick(t, A, p)) goto end;
 181
 182                 /* b := (2*a)^((|p|-5)/8) */
 183                 if (!BN_rshift(q, p, 3)) goto end;
 184                 q->neg = 0;
 185                 if (!BN_mod_exp(b, t, q, p, ctx)) goto end;
 186
 187                 /* y := b^2 */
 188                 if (!BN_mod_sqr(y, b, p, ctx)) goto end;
 189
 190                 /* t := (2*a)*b^2 - 1*/
 191                 if (!BN_mod_mul(t, t, y, p, ctx)) goto end;
 192                 if (!BN_sub_word(t, 1)) goto end;
 193
 194                 /* x = a*b*t */
 195                 if (!BN_mod_mul(x, A, b, p, ctx)) goto end;
 196                 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
 197
 198                 if (!BN_copy(ret, x)) goto end;
 199                 err = 0;
 200                 goto vrfy;
 201                 }
 202
 203         /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
 204          * First, find some  y  that is not a square. */
 205         if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
 206         q->neg = 0;
 207         i = 2;
 208         do
 209                 {
 210                 /* For efficiency, try small numbers first;
 211                  * if this fails, try random numbers.
 212                  */
 213                 if (i < 22)
 214                         {
 215                         if (!BN_set_word(y, i)) goto end;
 216                         }
 217                 else
 218                         {
 219                         if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
 220                         if (BN_ucmp(y, p) >= 0)
 221                                 {
 222                                 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
 223                                 }
 224                         /* now 0 <= y < |p| */
 225                         if (BN_is_zero(y))
 226                                 if (!BN_set_word(y, i)) goto end;
 227                         }
 228
 229                 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
 230                 if (r < -1) goto end;
 231                 if (r == 0)
 232                         {
 233                         /* m divides p */
 234                         BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 235                         goto end;
 236                         }
 237                 }
 238         while (r == 1 && ++i < 82);
 239
 240         if (r != -1)
 241                 {
 242                 /* Many rounds and still no non-square -- this is more likely
 243                  * a bug than just bad luck.
 244                  * Even if  p  is not prime, we should have found some  y
 245                  * such that r == -1.
 246                  */
 247                 BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
 248                 goto end;
 249                 }
 250
 251         /* Here's our actual 'q': */
 252         if (!BN_rshift(q, q, e)) goto end;
 253
 254         /* Now that we have some non-square, we can find an element
 255          * of order  2^e  by computing its q'th power. */
 256         if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
 257         if (BN_is_one(y))
 258                 {
 259                 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 260                 goto end;
 261                 }
 262
 263         /* Now we know that (if  p  is indeed prime) there is an integer
 264          * k,  0 <= k < 2^e,  such that
 265          *
 266          *      a^q * y^k == 1   (mod p).
 267          *
 268          * As  a^q  is a square and  y  is not,  k  must be even.
 269          * q+1  is even, too, so there is an element
 270          *
 271          *     X := a^((q+1)/2) * y^(k/2),
 272          *
 273          * and it satisfies
 274          *
 275          *     X^2 = a^q * a     * y^k
 276          *         = a,
 277          *
 278          * so it is the square root that we are looking for.
 279          */
 280
 281         /* t := (q-1)/2  (note that  q  is odd) */
 282         if (!BN_rshift1(t, q)) goto end;
 283
 284         /* x := a^((q-1)/2) */
 285         if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
 286                 {
 287                 if (!BN_nnmod(t, A, p, ctx)) goto end;
 288                 if (BN_is_zero(t))
 289                         {
 290                         /* special case: a == 0  (mod p) */
 291                         BN_zero(ret);
 292                         err = 0;
 293                         goto end;
 294                         }
 295                 else
 296                         if (!BN_one(x)) goto end;
 297                 }
 298         else
 299                 {
 300                 if (!BN_mod_exp(x, A, t, p, ctx)) goto end;
 301                 if (BN_is_zero(x))
 302                         {
 303                         /* special case: a == 0  (mod p) */
 304                         BN_zero(ret);
 305                         err = 0;
 306                         goto end;
 307                         }
 308                 }
 309
 310         /* b := a*x^2  (= a^q) */
 311         if (!BN_mod_sqr(b, x, p, ctx)) goto end;
 312         if (!BN_mod_mul(b, b, A, p, ctx)) goto end;
 313
 314         /* x := a*x    (= a^((q+1)/2)) */
 315         if (!BN_mod_mul(x, x, A, p, ctx)) goto end;
 316
 317         while (1)
 318                 {
 319                 /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
 320                  * where  E  refers to the original value of  e,  which we
 321                  * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
 322                  *
 323                  * We have  a*b = x^2,
 324                  *    y^2^(e-1) = -1,
 325                  *    b^2^(e-1) = 1.
 326                  */
 327
 328                 if (BN_is_one(b))
 329                         {
 330                         if (!BN_copy(ret, x)) goto end;
 331                         err = 0;
 332                         goto vrfy;
 333                         }
 334
 335
 336                 /* find smallest  i  such that  b^(2^i) = 1 */
 337                 i = 1;
 338                 if (!BN_mod_sqr(t, b, p, ctx)) goto end;
 339                 while (!BN_is_one(t))
 340                         {
 341                         i++;
 342                         if (i == e)
 343                                 {
 344                                 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 345                                 goto end;
 346                                 }
 347                         if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
 348                         }
 349
 350
 351                 /* t := y^2^(e - i - 1) */
 352                 if (!BN_copy(t, y)) goto end;
 353                 for (j = e - i - 1; j > 0; j--)
 354                         {
 355                         if (!BN_mod_sqr(t, t, p, ctx)) goto end;
 356                         }
 357                 if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
 358                 if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
 359                 if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
 360                 e = i;
 361                 }
 362
 363  vrfy:
 364         if (!err)
 365                 {
 366                 /* verify the result -- the input might have been not a square
 367                  * (test added in 0.9.8) */
 368
 369                 if (!BN_mod_sqr(x, ret, p, ctx))
 370                         err = 1;
 371
 372                 if (!err && 0 != BN_cmp(x, A))
 373                         {
 374                         BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 375                         err = 1;
 376                         }
 377                 }
 378
 379  end:
 380         if (err)
 381                 {
 382                 if (ret != NULL && ret != in)
 383                         {
 384                         BN_clear_free(ret);
 385                         }
 386                 ret = NULL;
 387                 }
 388         BN_CTX_end(ctx);
 389         bn_check_top(ret);
 390         return ret;
 391         }