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      1 /* Written by Lenka Fibikova <fibikova (at) exp-math.uni-essen.de>
      2  * and Bodo Moeller for the OpenSSL project. */
      3 /* ====================================================================
      4  * Copyright (c) 1998-2000 The OpenSSL Project.  All rights reserved.
      5  *
      6  * Redistribution and use in source and binary forms, with or without
      7  * modification, are permitted provided that the following conditions
      8  * are met:
      9  *
     10  * 1. Redistributions of source code must retain the above copyright
     11  *    notice, this list of conditions and the following disclaimer.
     12  *
     13  * 2. Redistributions in binary form must reproduce the above copyright
     14  *    notice, this list of conditions and the following disclaimer in
     15  *    the documentation and/or other materials provided with the
     16  *    distribution.
     17  *
     18  * 3. All advertising materials mentioning features or use of this
     19  *    software must display the following acknowledgment:
     20  *    "This product includes software developed by the OpenSSL Project
     21  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
     22  *
     23  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
     24  *    endorse or promote products derived from this software without
     25  *    prior written permission. For written permission, please contact
     26  *    openssl-core (at) openssl.org.
     27  *
     28  * 5. Products derived from this software may not be called "OpenSSL"
     29  *    nor may "OpenSSL" appear in their names without prior written
     30  *    permission of the OpenSSL Project.
     31  *
     32  * 6. Redistributions of any form whatsoever must retain the following
     33  *    acknowledgment:
     34  *    "This product includes software developed by the OpenSSL Project
     35  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
     36  *
     37  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
     38  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
     39  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
     40  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
     41  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
     42  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
     43  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
     44  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
     45  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
     46  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
     47  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
     48  * OF THE POSSIBILITY OF SUCH DAMAGE.
     49  * ====================================================================
     50  *
     51  * This product includes cryptographic software written by Eric Young
     52  * (eay (at) cryptsoft.com).  This product includes software written by Tim
     53  * Hudson (tjh (at) cryptsoft.com). */
     54 
     55 #include <openssl/bn.h>
     56 
     57 #include <openssl/err.h>
     58 
     59 
     60 /* Returns 'ret' such that
     61  *      ret^2 == a (mod p),
     62  * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
     63  * in Algebraic Computational Number Theory", algorithm 1.5.1).
     64  * 'p' must be prime! */
     65 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) {
     66   BIGNUM *ret = in;
     67   int err = 1;
     68   int r;
     69   BIGNUM *A, *b, *q, *t, *x, *y;
     70   int e, i, j;
     71 
     72   if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) {
     73     if (BN_abs_is_word(p, 2)) {
     74       if (ret == NULL) {
     75         ret = BN_new();
     76       }
     77       if (ret == NULL) {
     78         goto end;
     79       }
     80       if (!BN_set_word(ret, BN_is_bit_set(a, 0))) {
     81         if (ret != in) {
     82           BN_free(ret);
     83         }
     84         return NULL;
     85       }
     86       return ret;
     87     }
     88 
     89     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
     90     return (NULL);
     91   }
     92 
     93   if (BN_is_zero(a) || BN_is_one(a)) {
     94     if (ret == NULL) {
     95       ret = BN_new();
     96     }
     97     if (ret == NULL) {
     98       goto end;
     99     }
    100     if (!BN_set_word(ret, BN_is_one(a))) {
    101       if (ret != in) {
    102         BN_free(ret);
    103       }
    104       return NULL;
    105     }
    106     return ret;
    107   }
    108 
    109   BN_CTX_start(ctx);
    110   A = BN_CTX_get(ctx);
    111   b = BN_CTX_get(ctx);
    112   q = BN_CTX_get(ctx);
    113   t = BN_CTX_get(ctx);
    114   x = BN_CTX_get(ctx);
    115   y = BN_CTX_get(ctx);
    116   if (y == NULL) {
    117     goto end;
    118   }
    119 
    120   if (ret == NULL) {
    121     ret = BN_new();
    122   }
    123   if (ret == NULL) {
    124     goto end;
    125   }
    126 
    127   /* A = a mod p */
    128   if (!BN_nnmod(A, a, p, ctx)) {
    129     goto end;
    130   }
    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   }
    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     }
    150     q->neg = 0;
    151     if (!BN_add_word(q, 1) ||
    152         !BN_mod_exp(ret, A, q, p, ctx)) {
    153       goto end;
    154     }
    155     err = 0;
    156     goto vrfy;
    157   }
    158 
    159   if (e == 2) {
    160     /* |p| == 5  (mod 8)
    161      *
    162      * In this case  2  is always a non-square since
    163      * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
    164      * So if  a  really is a square, then  2*a  is a non-square.
    165      * Thus for
    166      *      b := (2*a)^((|p|-5)/8),
    167      *      i := (2*a)*b^2
    168      * we have
    169      *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
    170      *         = (2*a)^((p-1)/2)
    171      *         = -1;
    172      * so if we set
    173      *      x := a*b*(i-1),
    174      * then
    175      *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
    176      *         = a^2 * b^2 * (-2*i)
    177      *         = a*(-i)*(2*a*b^2)
    178      *         = a*(-i)*i
    179      *         = a.
    180      *
    181      * (This is due to A.O.L. Atkin,
    182      * <URL:
    183      *http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
    184      * November 1992.)
    185      */
    186 
    187     /* t := 2*a */
    188     if (!BN_mod_lshift1_quick(t, A, p)) {
    189       goto end;
    190     }
    191 
    192     /* b := (2*a)^((|p|-5)/8) */
    193     if (!BN_rshift(q, p, 3)) {
    194       goto end;
    195     }
    196     q->neg = 0;
    197     if (!BN_mod_exp(b, t, q, p, ctx)) {
    198       goto end;
    199     }
    200 
    201     /* y := b^2 */
    202     if (!BN_mod_sqr(y, b, p, ctx)) {
    203       goto end;
    204     }
    205 
    206     /* t := (2*a)*b^2 - 1*/
    207     if (!BN_mod_mul(t, t, y, p, ctx) ||
    208         !BN_sub_word(t, 1)) {
    209       goto end;
    210     }
    211 
    212     /* x = a*b*t */
    213     if (!BN_mod_mul(x, A, b, p, ctx) ||
    214         !BN_mod_mul(x, x, t, p, ctx)) {
    215       goto end;
    216     }
    217 
    218     if (!BN_copy(ret, x)) {
    219       goto end;
    220     }
    221     err = 0;
    222     goto vrfy;
    223   }
    224 
    225   /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
    226    * First, find some  y  that is not a square. */
    227   if (!BN_copy(q, p)) {
    228     goto end; /* use 'q' as temp */
    229   }
    230   q->neg = 0;
    231   i = 2;
    232   do {
    233     /* For efficiency, try small numbers first;
    234      * if this fails, try random numbers.
    235      */
    236     if (i < 22) {
    237       if (!BN_set_word(y, i)) {
    238         goto end;
    239       }
    240     } else {
    241       if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) {
    242         goto end;
    243       }
    244       if (BN_ucmp(y, p) >= 0) {
    245         if (!(p->neg ? BN_add : BN_sub)(y, y, p)) {
    246           goto end;
    247         }
    248       }
    249       /* now 0 <= y < |p| */
    250       if (BN_is_zero(y)) {
    251         if (!BN_set_word(y, i)) {
    252           goto end;
    253         }
    254       }
    255     }
    256 
    257     r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
    258     if (r < -1) {
    259       goto end;
    260     }
    261     if (r == 0) {
    262       /* m divides p */
    263       OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
    264       goto end;
    265     }
    266   } while (r == 1 && ++i < 82);
    267 
    268   if (r != -1) {
    269     /* Many rounds and still no non-square -- this is more likely
    270      * a bug than just bad luck.
    271      * Even if  p  is not prime, we should have found some  y
    272      * such that r == -1.
    273      */
    274     OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS);
    275     goto end;
    276   }
    277 
    278   /* Here's our actual 'q': */
    279   if (!BN_rshift(q, q, e)) {
    280     goto end;
    281   }
    282 
    283   /* Now that we have some non-square, we can find an element
    284    * of order  2^e  by computing its q'th power. */
    285   if (!BN_mod_exp(y, y, q, p, ctx)) {
    286     goto end;
    287   }
    288   if (BN_is_one(y)) {
    289     OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME);
    290     goto end;
    291   }
    292 
    293   /* Now we know that (if  p  is indeed prime) there is an integer
    294    * k,  0 <= k < 2^e,  such that
    295    *
    296    *      a^q * y^k == 1   (mod p).
    297    *
    298    * As  a^q  is a square and  y  is not,  k  must be even.
    299    * q+1  is even, too, so there is an element
    300    *
    301    *     X := a^((q+1)/2) * y^(k/2),
    302    *
    303    * and it satisfies
    304    *
    305    *     X^2 = a^q * a     * y^k
    306    *         = a,
    307    *
    308    * so it is the square root that we are looking for.
    309    */
    310 
    311   /* t := (q-1)/2  (note that  q  is odd) */
    312   if (!BN_rshift1(t, q)) {
    313     goto end;
    314   }
    315 
    316   /* x := a^((q-1)/2) */
    317   if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
    318   {
    319     if (!BN_nnmod(t, A, p, ctx)) {
    320       goto end;
    321     }
    322     if (BN_is_zero(t)) {
    323       /* special case: a == 0  (mod p) */
    324       BN_zero(ret);
    325       err = 0;
    326       goto end;
    327     } else if (!BN_one(x)) {
    328       goto end;
    329     }
    330   } else {
    331     if (!BN_mod_exp(x, A, t, p, ctx)) {
    332       goto end;
    333     }
    334     if (BN_is_zero(x)) {
    335       /* special case: a == 0  (mod p) */
    336       BN_zero(ret);
    337       err = 0;
    338       goto end;
    339     }
    340   }
    341 
    342   /* b := a*x^2  (= a^q) */
    343   if (!BN_mod_sqr(b, x, p, ctx) ||
    344       !BN_mod_mul(b, b, A, p, ctx)) {
    345     goto end;
    346   }
    347 
    348   /* x := a*x    (= a^((q+1)/2)) */
    349   if (!BN_mod_mul(x, x, A, p, ctx)) {
    350     goto end;
    351   }
    352 
    353   while (1) {
    354     /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
    355      * where  E  refers to the original value of  e,  which we
    356      * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
    357      *
    358      * We have  a*b = x^2,
    359      *    y^2^(e-1) = -1,
    360      *    b^2^(e-1) = 1.
    361      */
    362 
    363     if (BN_is_one(b)) {
    364       if (!BN_copy(ret, x)) {
    365         goto end;
    366       }
    367       err = 0;
    368       goto vrfy;
    369     }
    370 
    371 
    372     /* find smallest  i  such that  b^(2^i) = 1 */
    373     i = 1;
    374     if (!BN_mod_sqr(t, b, p, ctx)) {
    375       goto end;
    376     }
    377     while (!BN_is_one(t)) {
    378       i++;
    379       if (i == e) {
    380         OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
    381         goto end;
    382       }
    383       if (!BN_mod_mul(t, t, t, p, ctx)) {
    384         goto end;
    385       }
    386     }
    387 
    388 
    389     /* t := y^2^(e - i - 1) */
    390     if (!BN_copy(t, y)) {
    391       goto end;
    392     }
    393     for (j = e - i - 1; j > 0; j--) {
    394       if (!BN_mod_sqr(t, t, p, ctx)) {
    395         goto end;
    396       }
    397     }
    398     if (!BN_mod_mul(y, t, t, p, ctx) ||
    399         !BN_mod_mul(x, x, t, p, ctx) ||
    400         !BN_mod_mul(b, b, y, p, ctx)) {
    401       goto end;
    402     }
    403     e = i;
    404   }
    405 
    406 vrfy:
    407   if (!err) {
    408     /* verify the result -- the input might have been not a square
    409      * (test added in 0.9.8) */
    410 
    411     if (!BN_mod_sqr(x, ret, p, ctx)) {
    412       err = 1;
    413     }
    414 
    415     if (!err && 0 != BN_cmp(x, A)) {
    416       OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
    417       err = 1;
    418     }
    419   }
    420 
    421 end:
    422   if (err) {
    423     if (ret != in) {
    424       BN_clear_free(ret);
    425     }
    426     ret = NULL;
    427   }
    428   BN_CTX_end(ctx);
    429   return ret;
    430 }
    431 
    432 int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) {
    433   BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2;
    434   int ok = 0, last_delta_valid = 0;
    435 
    436   if (in->neg) {
    437     OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER);
    438     return 0;
    439   }
    440   if (BN_is_zero(in)) {
    441     BN_zero(out_sqrt);
    442     return 1;
    443   }
    444 
    445   BN_CTX_start(ctx);
    446   if (out_sqrt == in) {
    447     estimate = BN_CTX_get(ctx);
    448   } else {
    449     estimate = out_sqrt;
    450   }
    451   tmp = BN_CTX_get(ctx);
    452   last_delta = BN_CTX_get(ctx);
    453   delta = BN_CTX_get(ctx);
    454   if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) {
    455     OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE);
    456     goto err;
    457   }
    458 
    459   /* We estimate that the square root of an n-bit number is 2^{n/2}. */
    460   BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2);
    461 
    462   /* This is Newton's method for finding a root of the equation |estimate|^2 -
    463    * |in| = 0. */
    464   for (;;) {
    465     /* |estimate| = 1/2 * (|estimate| + |in|/|estimate|) */
    466     if (!BN_div(tmp, NULL, in, estimate, ctx) ||
    467         !BN_add(tmp, tmp, estimate) ||
    468         !BN_rshift1(estimate, tmp) ||
    469         /* |tmp| = |estimate|^2 */
    470         !BN_sqr(tmp, estimate, ctx) ||
    471         /* |delta| = |in| - |tmp| */
    472         !BN_sub(delta, in, tmp)) {
    473       OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB);
    474       goto err;
    475     }
    476 
    477     delta->neg = 0;
    478     /* The difference between |in| and |estimate| squared is required to always
    479      * decrease. This ensures that the loop always terminates, but I don't have
    480      * a proof that it always finds the square root for a given square. */
    481     if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) {
    482       break;
    483     }
    484 
    485     last_delta_valid = 1;
    486 
    487     tmp2 = last_delta;
    488     last_delta = delta;
    489     delta = tmp2;
    490   }
    491 
    492   if (BN_cmp(tmp, in) != 0) {
    493     OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE);
    494     goto err;
    495   }
    496 
    497   ok = 1;
    498 
    499 err:
    500   if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) {
    501     ok = 0;
    502   }
    503   BN_CTX_end(ctx);
    504   return ok;
    505 }
    506