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bn_sqrt.c
/* crypto/bn/bn_sqrt.c */
/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
 * and Bodo Moeller for the OpenSSL project. */
/* ====================================================================
 * Copyright (c) 1998-2000 The OpenSSL Project.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer. 
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 *
 * 3. All advertising materials mentioning features or use of this
 *    software must display the following acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
 *
 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
 *    endorse or promote products derived from this software without
 *    prior written permission. For written permission, please contact
 *    openssl-core@openssl.org.
 *
 * 5. Products derived from this software may not be called "OpenSSL"
 *    nor may "OpenSSL" appear in their names without prior written
 *    permission of the OpenSSL Project.
 *
 * 6. Redistributions of any form whatsoever must retain the following
 *    acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
 *
 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
 * OF THE POSSIBILITY OF SUCH DAMAGE.
 * ====================================================================
 *
 * This product includes cryptographic software written by Eric Young
 * (eay@cryptsoft.com).  This product includes software written by Tim
 * Hudson (tjh@cryptsoft.com).
 *
 */

#include "cryptlib.h"
#include "bn_lcl.h"


BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 
/* Returns 'ret' such that
 *      ret^2 == a (mod p),
 * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
 * in Algebraic Computational Number Theory", algorithm 1.5.1).
 * 'p' must be prime!
 */
      {
      BIGNUM *ret = in;
      int err = 1;
      int r;
      BIGNUM *A, *b, *q, *t, *x, *y;
      int e, i, j;
      
      if (!BN_is_odd(p) || BN_abs_is_word(p, 1))
            {
            if (BN_abs_is_word(p, 2))
                  {
                  if (ret == NULL)
                        ret = BN_new();
                  if (ret == NULL)
                        goto end;
                  if (!BN_set_word(ret, BN_is_bit_set(a, 0)))
                        {
                        if (ret != in)
                              BN_free(ret);
                        return NULL;
                        }
                  bn_check_top(ret);
                  return ret;
                  }

            BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
            return(NULL);
            }

      if (BN_is_zero(a) || BN_is_one(a))
            {
            if (ret == NULL)
                  ret = BN_new();
            if (ret == NULL)
                  goto end;
            if (!BN_set_word(ret, BN_is_one(a)))
                  {
                  if (ret != in)
                        BN_free(ret);
                  return NULL;
                  }
            bn_check_top(ret);
            return ret;
            }

      BN_CTX_start(ctx);
      A = BN_CTX_get(ctx);
      b = BN_CTX_get(ctx);
      q = BN_CTX_get(ctx);
      t = BN_CTX_get(ctx);
      x = BN_CTX_get(ctx);
      y = BN_CTX_get(ctx);
      if (y == NULL) goto end;
      
      if (ret == NULL)
            ret = BN_new();
      if (ret == NULL) goto end;

      /* A = a mod p */
      if (!BN_nnmod(A, a, p, ctx)) goto end;

      /* now write  |p| - 1  as  2^e*q  where  q  is odd */
      e = 1;
      while (!BN_is_bit_set(p, e))
            e++;
      /* we'll set  q  later (if needed) */

      if (e == 1)
            {
            /* The easy case:  (|p|-1)/2  is odd, so 2 has an inverse
             * modulo  (|p|-1)/2,  and square roots can be computed
             * directly by modular exponentiation.
             * We have
             *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
             * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
             */
            if (!BN_rshift(q, p, 2)) goto end;
            q->neg = 0;
            if (!BN_add_word(q, 1)) goto end;
            if (!BN_mod_exp(ret, A, q, p, ctx)) goto end;
            err = 0;
            goto vrfy;
            }
      
      if (e == 2)
            {
            /* |p| == 5  (mod 8)
             *
             * In this case  2  is always a non-square since
             * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
             * So if  a  really is a square, then  2*a  is a non-square.
             * Thus for
             *      b := (2*a)^((|p|-5)/8),
             *      i := (2*a)*b^2
             * we have
             *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
             *         = (2*a)^((p-1)/2)
             *         = -1;
             * so if we set
             *      x := a*b*(i-1),
             * then
             *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
             *         = a^2 * b^2 * (-2*i)
             *         = a*(-i)*(2*a*b^2)
             *         = a*(-i)*i
             *         = a.
             *
             * (This is due to A.O.L. Atkin, 
             * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
             * November 1992.)
             */

            /* t := 2*a */
            if (!BN_mod_lshift1_quick(t, A, p)) goto end;

            /* b := (2*a)^((|p|-5)/8) */
            if (!BN_rshift(q, p, 3)) goto end;
            q->neg = 0;
            if (!BN_mod_exp(b, t, q, p, ctx)) goto end;

            /* y := b^2 */
            if (!BN_mod_sqr(y, b, p, ctx)) goto end;

            /* t := (2*a)*b^2 - 1*/
            if (!BN_mod_mul(t, t, y, p, ctx)) goto end;
            if (!BN_sub_word(t, 1)) goto end;

            /* x = a*b*t */
            if (!BN_mod_mul(x, A, b, p, ctx)) goto end;
            if (!BN_mod_mul(x, x, t, p, ctx)) goto end;

            if (!BN_copy(ret, x)) goto end;
            err = 0;
            goto vrfy;
            }
      
      /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
       * First, find some  y  that is not a square. */
      if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
      q->neg = 0;
      i = 2;
      do
            {
            /* For efficiency, try small numbers first;
             * if this fails, try random numbers.
             */
            if (i < 22)
                  {
                  if (!BN_set_word(y, i)) goto end;
                  }
            else
                  {
                  if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
                  if (BN_ucmp(y, p) >= 0)
                        {
                        if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
                        }
                  /* now 0 <= y < |p| */
                  if (BN_is_zero(y))
                        if (!BN_set_word(y, i)) goto end;
                  }
            
            r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
            if (r < -1) goto end;
            if (r == 0)
                  {
                  /* m divides p */
                  BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
                  goto end;
                  }
            }
      while (r == 1 && ++i < 82);
      
      if (r != -1)
            {
            /* Many rounds and still no non-square -- this is more likely
             * a bug than just bad luck.
             * Even if  p  is not prime, we should have found some  y
             * such that r == -1.
             */
            BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
            goto end;
            }

      /* Here's our actual 'q': */
      if (!BN_rshift(q, q, e)) goto end;

      /* Now that we have some non-square, we can find an element
       * of order  2^e  by computing its q'th power. */
      if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
      if (BN_is_one(y))
            {
            BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
            goto end;
            }

      /* Now we know that (if  p  is indeed prime) there is an integer
       * k,  0 <= k < 2^e,  such that
       *
       *      a^q * y^k == 1   (mod p).
       *
       * As  a^q  is a square and  y  is not,  k  must be even.
       * q+1  is even, too, so there is an element
       *
       *     X := a^((q+1)/2) * y^(k/2),
       *
       * and it satisfies
       *
       *     X^2 = a^q * a     * y^k
       *         = a,
       *
       * so it is the square root that we are looking for.
       */
      
      /* t := (q-1)/2  (note that  q  is odd) */
      if (!BN_rshift1(t, q)) goto end;
      
      /* x := a^((q-1)/2) */
      if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
            {
            if (!BN_nnmod(t, A, p, ctx)) goto end;
            if (BN_is_zero(t))
                  {
                  /* special case: a == 0  (mod p) */
                  BN_zero(ret);
                  err = 0;
                  goto end;
                  }
            else
                  if (!BN_one(x)) goto end;
            }
      else
            {
            if (!BN_mod_exp(x, A, t, p, ctx)) goto end;
            if (BN_is_zero(x))
                  {
                  /* special case: a == 0  (mod p) */
                  BN_zero(ret);
                  err = 0;
                  goto end;
                  }
            }

      /* b := a*x^2  (= a^q) */
      if (!BN_mod_sqr(b, x, p, ctx)) goto end;
      if (!BN_mod_mul(b, b, A, p, ctx)) goto end;
      
      /* x := a*x    (= a^((q+1)/2)) */
      if (!BN_mod_mul(x, x, A, p, ctx)) goto end;

      while (1)
            {
            /* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
             * where  E  refers to the original value of  e,  which we
             * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
             *
             * We have  a*b = x^2,
             *    y^2^(e-1) = -1,
             *    b^2^(e-1) = 1.
             */

            if (BN_is_one(b))
                  {
                  if (!BN_copy(ret, x)) goto end;
                  err = 0;
                  goto vrfy;
                  }


            /* find smallest  i  such that  b^(2^i) = 1 */
            i = 1;
            if (!BN_mod_sqr(t, b, p, ctx)) goto end;
            while (!BN_is_one(t))
                  {
                  i++;
                  if (i == e)
                        {
                        BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
                        goto end;
                        }
                  if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
                  }
            

            /* t := y^2^(e - i - 1) */
            if (!BN_copy(t, y)) goto end;
            for (j = e - i - 1; j > 0; j--)
                  {
                  if (!BN_mod_sqr(t, t, p, ctx)) goto end;
                  }
            if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
            if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
            if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
            e = i;
            }

 vrfy:
      if (!err)
            {
            /* verify the result -- the input might have been not a square
             * (test added in 0.9.8) */
            
            if (!BN_mod_sqr(x, ret, p, ctx))
                  err = 1;
            
            if (!err && 0 != BN_cmp(x, A))
                  {
                  BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
                  err = 1;
                  }
            }

 end:
      if (err)
            {
            if (ret != NULL && ret != in)
                  {
                  BN_clear_free(ret);
                  }
            ret = NULL;
            }
      BN_CTX_end(ctx);
      bn_check_top(ret);
      return ret;
      }

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