crypto/bn/bn_sqrt.c

   1 /*
   2  * Copyright 2000-2018 The OpenSSL Project Authors. All Rights Reserved.
   3  *
   4  * Licensed under the Apache License 2.0 (the "License").  You may not use
   5  * this file except in compliance with the License.  You can obtain a copy
   6  * in the file LICENSE in the source distribution or at
   7  * https://www.openssl.org/source/license.html
   8  */
   9
  10 #include "internal/cryptlib.h"
  11 #include "bn_lcl.h"
  12
  13 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
  14 /*
  15  * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks
  16  * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number
  17  * Theory", algorithm 1.5.1). 'p' must be prime!
  18  */
  19 {
  20     BIGNUM *ret = in;
  21     int err = 1;
  22     int r;
  23     BIGNUM *A, *b, *q, *t, *x, *y;
  24     int e, i, j;
  25
  26     if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
  27         if (BN_abs_is_word(p, 2)) {
  28             if (ret == NULL)
  29                 ret = BN_new();
  30             if (ret == NULL)
  31                 goto end;
  32             if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
  33                 if (ret != in)
  34                     BN_free(ret);
  35                 return NULL;
  36             }
  37             bn_check_top(ret);
  38             return ret;
  39         }
  40
  41         BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
  42         return NULL;
  43     }
  44
  45     if (BN_is_zero(a) || BN_is_one(a)) {
  46         if (ret == NULL)
  47             ret = BN_new();
  48         if (ret == NULL)
  49             goto end;
  50         if (!BN_set_word(ret, BN_is_one(a))) {
  51             if (ret != in)
  52                 BN_free(ret);
  53             return NULL;
  54         }
  55         bn_check_top(ret);
  56         return ret;
  57     }
  58
  59     BN_CTX_start(ctx);
  60     A = BN_CTX_get(ctx);
  61     b = BN_CTX_get(ctx);
  62     q = BN_CTX_get(ctx);
  63     t = BN_CTX_get(ctx);
  64     x = BN_CTX_get(ctx);
  65     y = BN_CTX_get(ctx);
  66     if (y == NULL)
  67         goto end;
  68
  69     if (ret == NULL)
  70         ret = BN_new();
  71     if (ret == NULL)
  72         goto end;
  73
  74     /* A = a mod p */
  75     if (!BN_nnmod(A, a, p, ctx))
  76         goto end;
  77
  78     /* now write  |p| - 1  as  2^e*q  where  q  is odd */
  79     e = 1;
  80     while (!BN_is_bit_set(p, e))
  81         e++;
  82     /* we'll set  q  later (if needed) */
  83
  84     if (e == 1) {
  85         /*-
  86          * The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
  87          * modulo  (|p|-1)/2,  and square roots can be computed
  88          * directly by modular exponentiation.
  89          * We have
  90          *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
  91          * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
  92          */
  93         if (!BN_rshift(q, p, 2))
  94             goto end;
  95         q->neg = 0;
  96         if (!BN_add_word(q, 1))
  97             goto end;
  98         if (!BN_mod_exp(ret, A, q, p, ctx))
  99             goto end;
 100         err = 0;
 101         goto vrfy;
 102     }
 103
 104     if (e == 2) {
 105         /*-
 106          * |p| == 5  (mod 8)
 107          *
 108          * In this case  2  is always a non-square since
 109          * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
 110          * So if  a  really is a square, then  2*a  is a non-square.
 111          * Thus for
 112          *      b := (2*a)^((|p|-5)/8),
 113          *      i := (2*a)*b^2
 114          * we have
 115          *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
 116          *         = (2*a)^((p-1)/2)
 117          *         = -1;
 118          * so if we set
 119          *      x := a*b*(i-1),
 120          * then
 121          *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
 122          *         = a^2 * b^2 * (-2*i)
 123          *         = a*(-i)*(2*a*b^2)
 124          *         = a*(-i)*i
 125          *         = a.
 126          *
 127          * (This is due to A.O.L. Atkin,
 128          * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
 129          * November 1992.)
 130          */
 131
 132         /* t := 2*a */
 133         if (!BN_mod_lshift1_quick(t, A, p))
 134             goto end;
 135
 136         /* b := (2*a)^((|p|-5)/8) */
 137         if (!BN_rshift(q, p, 3))
 138             goto end;
 139         q->neg = 0;
 140         if (!BN_mod_exp(b, t, q, p, ctx))
 141             goto end;
 142
 143         /* y := b^2 */
 144         if (!BN_mod_sqr(y, b, p, ctx))
 145             goto end;
 146
 147         /* t := (2*a)*b^2 - 1 */
 148         if (!BN_mod_mul(t, t, y, p, ctx))
 149             goto end;
 150         if (!BN_sub_word(t, 1))
 151             goto end;
 152
 153         /* x = a*b*t */
 154         if (!BN_mod_mul(x, A, b, p, ctx))
 155             goto end;
 156         if (!BN_mod_mul(x, x, t, p, ctx))
 157             goto end;
 158
 159         if (!BN_copy(ret, x))
 160             goto end;
 161         err = 0;
 162         goto vrfy;
 163     }
 164
 165     /*
 166      * e > 2, so we really have to use the Tonelli/Shanks algorithm. First,
 167      * find some y that is not a square.
 168      */
 169     if (!BN_copy(q, p))
 170         goto end;               /* use 'q' as temp */
 171     q->neg = 0;
 172     i = 2;
 173     do {
 174         /*
 175          * For efficiency, try small numbers first; if this fails, try random
 176          * numbers.
 177          */
 178         if (i < 22) {
 179             if (!BN_set_word(y, i))
 180                 goto end;
 181         } else {
 182             if (!BN_priv_rand(y, BN_num_bits(p), 0, 0))
 183                 goto end;
 184             if (BN_ucmp(y, p) >= 0) {
 185                 if (!(p->neg ? BN_add : BN_sub) (y, y, p))
 186                     goto end;
 187             }
 188             /* now 0 <= y < |p| */
 189             if (BN_is_zero(y))
 190                 if (!BN_set_word(y, i))
 191                     goto end;
 192         }
 193
 194         r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
 195         if (r < -1)
 196             goto end;
 197         if (r == 0) {
 198             /* m divides p */
 199             BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 200             goto end;
 201         }
 202     }
 203     while (r == 1 && ++i < 82);
 204
 205     if (r != -1) {
 206         /*
 207          * Many rounds and still no non-square -- this is more likely a bug
 208          * than just bad luck. Even if p is not prime, we should have found
 209          * some y such that r == -1.
 210          */
 211         BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
 212         goto end;
 213     }
 214
 215     /* Here's our actual 'q': */
 216     if (!BN_rshift(q, q, e))
 217         goto end;
 218
 219     /*
 220      * Now that we have some non-square, we can find an element of order 2^e
 221      * by computing its q'th power.
 222      */
 223     if (!BN_mod_exp(y, y, q, p, ctx))
 224         goto end;
 225     if (BN_is_one(y)) {
 226         BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
 227         goto end;
 228     }
 229
 230     /*-
 231      * Now we know that (if  p  is indeed prime) there is an integer
 232      * k,  0 <= k < 2^e,  such that
 233      *
 234      *      a^q * y^k == 1   (mod p).
 235      *
 236      * As  a^q  is a square and  y  is not,  k  must be even.
 237      * q+1  is even, too, so there is an element
 238      *
 239      *     X := a^((q+1)/2) * y^(k/2),
 240      *
 241      * and it satisfies
 242      *
 243      *     X^2 = a^q * a     * y^k
 244      *         = a,
 245      *
 246      * so it is the square root that we are looking for.
 247      */
 248
 249     /* t := (q-1)/2  (note that  q  is odd) */
 250     if (!BN_rshift1(t, q))
 251         goto end;
 252
 253     /* x := a^((q-1)/2) */
 254     if (BN_is_zero(t)) {        /* special case: p = 2^e + 1 */
 255         if (!BN_nnmod(t, A, p, ctx))
 256             goto end;
 257         if (BN_is_zero(t)) {
 258             /* special case: a == 0  (mod p) */
 259             BN_zero(ret);
 260             err = 0;
 261             goto end;
 262         } else if (!BN_one(x))
 263             goto end;
 264     } else {
 265         if (!BN_mod_exp(x, A, t, p, ctx))
 266             goto end;
 267         if (BN_is_zero(x)) {
 268             /* special case: a == 0  (mod p) */
 269             BN_zero(ret);
 270             err = 0;
 271             goto end;
 272         }
 273     }
 274
 275     /* b := a*x^2  (= a^q) */
 276     if (!BN_mod_sqr(b, x, p, ctx))
 277         goto end;
 278     if (!BN_mod_mul(b, b, A, p, ctx))
 279         goto end;
 280
 281     /* x := a*x    (= a^((q+1)/2)) */
 282     if (!BN_mod_mul(x, x, A, p, ctx))
 283         goto end;
 284
 285     while (1) {
 286         /*-
 287          * Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
 288          * where  E  refers to the original value of  e,  which we
 289          * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
 290          *
 291          * We have  a*b = x^2,
 292          *    y^2^(e-1) = -1,
 293          *    b^2^(e-1) = 1.
 294          */
 295
 296         if (BN_is_one(b)) {
 297             if (!BN_copy(ret, x))
 298                 goto end;
 299             err = 0;
 300             goto vrfy;
 301         }
 302
 303         /* find smallest  i  such that  b^(2^i) = 1 */
 304         i = 1;
 305         if (!BN_mod_sqr(t, b, p, ctx))
 306             goto end;
 307         while (!BN_is_one(t)) {
 308             i++;
 309             if (i == e) {
 310                 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 311                 goto end;
 312             }
 313             if (!BN_mod_mul(t, t, t, p, ctx))
 314                 goto end;
 315         }
 316
 317         /* t := y^2^(e - i - 1) */
 318         if (!BN_copy(t, y))
 319             goto end;
 320         for (j = e - i - 1; j > 0; j--) {
 321             if (!BN_mod_sqr(t, t, p, ctx))
 322                 goto end;
 323         }
 324         if (!BN_mod_mul(y, t, t, p, ctx))
 325             goto end;
 326         if (!BN_mod_mul(x, x, t, p, ctx))
 327             goto end;
 328         if (!BN_mod_mul(b, b, y, p, ctx))
 329             goto end;
 330         e = i;
 331     }
 332
 333  vrfy:
 334     if (!err) {
 335         /*
 336          * verify the result -- the input might have been not a square (test
 337          * added in 0.9.8)
 338          */
 339
 340         if (!BN_mod_sqr(x, ret, p, ctx))
 341             err = 1;
 342
 343         if (!err && 0 != BN_cmp(x, A)) {
 344             BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
 345             err = 1;
 346         }
 347     }
 348
 349  end:
 350     if (err) {
 351         if (ret != in)
 352             BN_clear_free(ret);
 353         ret = NULL;
 354     }
 355     BN_CTX_end(ctx);
 356     bn_check_top(ret);
 357     return ret;
 358 }