lib/libcrypto/bn/bn_sqrt.c

   1 /* $OpenBSD: bn_sqrt.c,v 1.9 2017/01/29 17:49:22 beck Exp $ */
   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 <openssl/err.h>
  59
  60 #include "bn_lcl.h"
  61
  62 BIGNUM *
  63 BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
  64 /* Returns 'ret' such that
  65  *      ret^2 == a (mod p),
  66  * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
  67  * in Algebraic Computational Number Theory", algorithm 1.5.1).
  68  * 'p' must be prime!
  69  */
  70 {
  71         BIGNUM *ret = in;
  72         int err = 1;
  73         int r;
  74         BIGNUM *A, *b, *q, *t, *x, *y;
  75         int e, i, j;
  76
  77         if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
  78                 if (BN_abs_is_word(p, 2)) {
  79                         if (ret == NULL)
  80                                 ret = BN_new();
  81                         if (ret == NULL)
  82                                 goto end;
  83                         if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
  84                                 if (ret != in)
  85                                         BN_free(ret);
  86                                 return NULL;
  87                         }
  88                         bn_check_top(ret);
  89                         return ret;
  90                 }
  91
  92                 BNerror(BN_R_P_IS_NOT_PRIME);
  93                 return (NULL);
  94         }
  95
  96         if (BN_is_zero(a) || BN_is_one(a)) {
  97                 if (ret == NULL)
  98                         ret = BN_new();
  99                 if (ret == NULL)
 100                         goto end;
 101                 if (!BN_set_word(ret, BN_is_one(a))) {
 102                         if (ret != in)
 103                                 BN_free(ret);
 104                         return NULL;
 105                 }
 106                 bn_check_top(ret);
 107                 return ret;
 108         }
 109
 110         BN_CTX_start(ctx);
 111         if ((A = BN_CTX_get(ctx)) == NULL)
 112                 goto end;
 113         if ((b = BN_CTX_get(ctx)) == NULL)
 114                 goto end;
 115         if ((q = BN_CTX_get(ctx)) == NULL)
 116                 goto end;
 117         if ((t = BN_CTX_get(ctx)) == NULL)
 118                 goto end;
 119         if ((x = BN_CTX_get(ctx)) == NULL)
 120                 goto end;
 121         if ((y = BN_CTX_get(ctx)) == NULL)
 122                 goto end;
 123
 124         if (ret == NULL)
 125                 ret = BN_new();
 126         if (ret == NULL)
 127                 goto end;
 128
 129         /* A = a mod p */
 130         if (!BN_nnmod(A, a, p, ctx))
 131                 goto end;
 132
 133         /* now write  |p| - 1  as  2^e*q  where  q  is odd */
 134         e = 1;
 135         while (!BN_is_bit_set(p, e))
 136                 e++;
 137         /* we'll set  q  later (if needed) */
 138
 139         if (e == 1) {
 140                 /* The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
 141                  * modulo  (|p|-1)/2,  and square roots can be computed
 142                  * directly by modular exponentiation.
 143                  * We have
 144                  *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
 145                  * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
 146                  */
 147                 if (!BN_rshift(q, p, 2))
 148                         goto end;
 149                 q->neg = 0;
 150                 if (!BN_add_word(q, 1))
 151                         goto end;
 152                 if (!BN_mod_exp_ct(ret, A, q, p, ctx))
 153                         goto end;
 154                 err = 0;
 155                 goto vrfy;
 156         }
 157
 158         if (e == 2) {
 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                 /* t := 2*a */
 186                 if (!BN_mod_lshift1_quick(t, A, p))
 187                         goto end;
 188
 189                 /* b := (2*a)^((|p|-5)/8) */
 190                 if (!BN_rshift(q, p, 3))
 191                         goto end;
 192                 q->neg = 0;
 193                 if (!BN_mod_exp_ct(b, t, q, p, ctx))
 194                         goto end;
 195
 196                 /* y := b^2 */
 197                 if (!BN_mod_sqr(y, b, p, ctx))
 198                         goto end;
 199
 200                 /* t := (2*a)*b^2 - 1*/
 201                 if (!BN_mod_mul(t, t, y, p, ctx))
 202                         goto end;
 203                 if (!BN_sub_word(t, 1))
 204                         goto end;
 205
 206                 /* x = a*b*t */
 207                 if (!BN_mod_mul(x, A, b, p, ctx))
 208                         goto end;
 209                 if (!BN_mod_mul(x, x, t, p, ctx))
 210                         goto end;
 211
 212                 if (!BN_copy(ret, x))
 213                         goto end;
 214                 err = 0;
 215                 goto vrfy;
 216         }
 217
 218         /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
 219          * First, find some  y  that is not a square. */
 220         if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
 221                 q->neg = 0;
 222         i = 2;
 223         do {
 224                 /* For efficiency, try small numbers first;
 225                  * if this fails, try random numbers.
 226                  */
 227                 if (i < 22) {
 228                         if (!BN_set_word(y, i))
 229                                 goto end;
 230                 } else {
 231                         if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0))
 232                                 goto end;
 233                         if (BN_ucmp(y, p) >= 0) {
 234                                 if (p->neg) {
 235                                         if (!BN_add(y, y, p))
 236                                                 goto end;
 237                                 } else {
 238                                         if (!BN_sub(y, y, p))
 239                                                 goto end;
 240                                 }
 241                         }
 242                         /* now 0 <= y < |p| */
 243                         if (BN_is_zero(y))
 244                                 if (!BN_set_word(y, i))
 245                                         goto end;
 246                 }
 247
 248                 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
 249                 if (r < -1)
 250                         goto end;
 251                 if (r == 0) {
 252                         /* m divides p */
 253                         BNerror(BN_R_P_IS_NOT_PRIME);
 254                         goto end;
 255                 }
 256         }
 257         while (r == 1 && ++i < 82);
 258
 259                 if (r != -1) {
 260                 /* Many rounds and still no non-square -- this is more likely
 261                  * a bug than just bad luck.
 262                  * Even if  p  is not prime, we should have found some  y
 263                  * such that r == -1.
 264                  */
 265                 BNerror(BN_R_TOO_MANY_ITERATIONS);
 266                 goto end;
 267         }
 268
 269         /* Here's our actual 'q': */
 270         if (!BN_rshift(q, q, e))
 271                 goto end;
 272
 273         /* Now that we have some non-square, we can find an element
 274          * of order  2^e  by computing its q'th power. */
 275         if (!BN_mod_exp_ct(y, y, q, p, ctx))
 276                 goto end;
 277         if (BN_is_one(y)) {
 278                 BNerror(BN_R_P_IS_NOT_PRIME);
 279                 goto end;
 280         }
 281
 282         /* Now we know that (if  p  is indeed prime) there is an integer
 283          * k,  0 <= k < 2^e,  such that
 284          *
 285          *      a^q * y^k == 1   (mod p).
 286          *
 287          * As  a^q  is a square and  y  is not,  k  must be even.
 288          * q+1  is even, too, so there is an element
 289          *
 290          *     X := a^((q+1)/2) * y^(k/2),
 291          *
 292          * and it satisfies
 293          *
 294          *     X^2 = a^q * a     * y^k
 295          *         = a,
 296          *
 297          * so it is the square root that we are looking for.
 298          */
 299
 300         /* t := (q-1)/2  (note that  q  is odd) */
 301         if (!BN_rshift1(t, q))
 302                 goto end;
 303
 304         /* x := a^((q-1)/2) */
 305         if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
 306         {
 307                 if (!BN_nnmod(t, A, p, ctx))
 308                         goto end;
 309                 if (BN_is_zero(t)) {
 310                         /* special case: a == 0  (mod p) */
 311                         BN_zero(ret);
 312                         err = 0;
 313                         goto end;
 314                 } else if (!BN_one(x))
 315                         goto end;
 316         } else {
 317                 if (!BN_mod_exp_ct(x, A, t, p, ctx))
 318                         goto end;
 319                 if (BN_is_zero(x)) {
 320                         /* special case: a == 0  (mod p) */
 321                         BN_zero(ret);
 322                         err = 0;
 323                         goto end;
 324                 }
 325         }
 326
 327         /* b := a*x^2  (= a^q) */
 328         if (!BN_mod_sqr(b, x, p, ctx))
 329                 goto end;
 330         if (!BN_mod_mul(b, b, A, p, ctx))
 331                 goto end;
 332
 333         /* x := a*x    (= a^((q+1)/2)) */
 334         if (!BN_mod_mul(x, x, A, p, ctx))
 335                 goto end;
 336
 337         while (1) {
 338                 /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
 339                  * where  E  refers to the original value of  e,  which we
 340                  * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
 341                  *
 342                  * We have  a*b = x^2,
 343                  *    y^2^(e-1) = -1,
 344                  *    b^2^(e-1) = 1.
 345                  */
 346
 347                 if (BN_is_one(b)) {
 348                         if (!BN_copy(ret, x))
 349                                 goto end;
 350                         err = 0;
 351                         goto vrfy;
 352                 }
 353
 354
 355                 /* find smallest  i  such that  b^(2^i) = 1 */
 356                 i = 1;
 357                 if (!BN_mod_sqr(t, b, p, ctx))
 358                         goto end;
 359                 while (!BN_is_one(t)) {
 360                         i++;
 361                         if (i == e) {
 362                                 BNerror(BN_R_NOT_A_SQUARE);
 363                                 goto end;
 364                         }
 365                         if (!BN_mod_mul(t, t, t, p, ctx))
 366                                 goto end;
 367                 }
 368
 369
 370                 /* t := y^2^(e - i - 1) */
 371                 if (!BN_copy(t, y))
 372                         goto end;
 373                 for (j = e - i - 1; j > 0; j--) {
 374                         if (!BN_mod_sqr(t, t, p, ctx))
 375                                 goto end;
 376                 }
 377                 if (!BN_mod_mul(y, t, t, p, ctx))
 378                         goto end;
 379                 if (!BN_mod_mul(x, x, t, p, ctx))
 380                         goto end;
 381                 if (!BN_mod_mul(b, b, y, p, ctx))
 382                         goto end;
 383                 e = i;
 384         }
 385
 386 vrfy:
 387         if (!err) {
 388                 /* verify the result -- the input might have been not a square
 389                  * (test added in 0.9.8) */
 390
 391                 if (!BN_mod_sqr(x, ret, p, ctx))
 392                         err = 1;
 393
 394                 if (!err && 0 != BN_cmp(x, A)) {
 395                         BNerror(BN_R_NOT_A_SQUARE);
 396                         err = 1;
 397                 }
 398         }
 399
 400 end:
 401         if (err) {
 402                 if (ret != NULL && ret != in) {
 403                         BN_clear_free(ret);
 404                 }
 405                 ret = NULL;
 406         }
 407         BN_CTX_end(ctx);
 408         bn_check_top(ret);
 409         return ret;
 410 }