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bn_gcd.c

/* crypto/bn/bn_gcd.c */
/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
 * All rights reserved.
 *
 * This package is an SSL implementation written
 * by Eric Young (eay@cryptsoft.com).
 * The implementation was written so as to conform with Netscapes SSL.
 * 
 * This library is free for commercial and non-commercial use as long as
 * the following conditions are aheared to.  The following conditions
 * apply to all code found in this distribution, be it the RC4, RSA,
 * lhash, DES, etc., code; not just the SSL code.  The SSL documentation
 * included with this distribution is covered by the same copyright terms
 * except that the holder is Tim Hudson (tjh@cryptsoft.com).
 * 
 * Copyright remains Eric Young's, and as such any Copyright notices in
 * the code are not to be removed.
 * If this package is used in a product, Eric Young should be given attribution
 * as the author of the parts of the library used.
 * This can be in the form of a textual message at program startup or
 * in documentation (online or textual) provided with the package.
 * 
 * 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 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 acknowledgement:
 *    "This product includes cryptographic software written by
 *     Eric Young (eay@cryptsoft.com)"
 *    The word 'cryptographic' can be left out if the rouines from the library
 *    being used are not cryptographic related :-).
 * 4. If you include any Windows specific code (or a derivative thereof) from 
 *    the apps directory (application code) you must include an acknowledgement:
 *    "This product includes software written by Tim Hudson (tjh@cryptsoft.com)"
 * 
 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
 * ANY EXPRESS 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 AUTHOR OR 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.
 * 
 * The licence and distribution terms for any publically available version or
 * derivative of this code cannot be changed.  i.e. this code cannot simply be
 * copied and put under another distribution licence
 * [including the GNU Public Licence.]
 */
/* ====================================================================
 * Copyright (c) 1998-2001 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"

static BIGNUM *euclid(BIGNUM *a, BIGNUM *b);

int BN_gcd(BIGNUM *r, const BIGNUM *in_a, const BIGNUM *in_b, BN_CTX *ctx)
      {
      BIGNUM *a,*b,*t;
      int ret=0;

      bn_check_top(in_a);
      bn_check_top(in_b);

      BN_CTX_start(ctx);
      a = BN_CTX_get(ctx);
      b = BN_CTX_get(ctx);
      if (a == NULL || b == NULL) goto err;

      if (BN_copy(a,in_a) == NULL) goto err;
      if (BN_copy(b,in_b) == NULL) goto err;
      a->neg = 0;
      b->neg = 0;

      if (BN_cmp(a,b) < 0) { t=a; a=b; b=t; }
      t=euclid(a,b);
      if (t == NULL) goto err;

      if (BN_copy(r,t) == NULL) goto err;
      ret=1;
err:
      BN_CTX_end(ctx);
      bn_check_top(r);
      return(ret);
      }

static BIGNUM *euclid(BIGNUM *a, BIGNUM *b)
      {
      BIGNUM *t;
      int shifts=0;

      bn_check_top(a);
      bn_check_top(b);

      /* 0 <= b <= a */
      while (!BN_is_zero(b))
            {
            /* 0 < b <= a */

            if (BN_is_odd(a))
                  {
                  if (BN_is_odd(b))
                        {
                        if (!BN_sub(a,a,b)) goto err;
                        if (!BN_rshift1(a,a)) goto err;
                        if (BN_cmp(a,b) < 0)
                              { t=a; a=b; b=t; }
                        }
                  else        /* a odd - b even */
                        {
                        if (!BN_rshift1(b,b)) goto err;
                        if (BN_cmp(a,b) < 0)
                              { t=a; a=b; b=t; }
                        }
                  }
            else              /* a is even */
                  {
                  if (BN_is_odd(b))
                        {
                        if (!BN_rshift1(a,a)) goto err;
                        if (BN_cmp(a,b) < 0)
                              { t=a; a=b; b=t; }
                        }
                  else        /* a even - b even */
                        {
                        if (!BN_rshift1(a,a)) goto err;
                        if (!BN_rshift1(b,b)) goto err;
                        shifts++;
                        }
                  }
            /* 0 <= b <= a */
            }

      if (shifts)
            {
            if (!BN_lshift(a,a,shifts)) goto err;
            }
      bn_check_top(a);
      return(a);
err:
      return(NULL);
      }


/* solves ax == 1 (mod n) */
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
        const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx);
BIGNUM *BN_mod_inverse(BIGNUM *in,
      const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
      {
      BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
      BIGNUM *ret=NULL;
      int sign;

      if ((BN_get_flags(a, BN_FLG_CONSTTIME) != 0) || (BN_get_flags(n, BN_FLG_CONSTTIME) != 0))
            {
            return BN_mod_inverse_no_branch(in, a, n, ctx);
            }

      bn_check_top(a);
      bn_check_top(n);

      BN_CTX_start(ctx);
      A = BN_CTX_get(ctx);
      B = BN_CTX_get(ctx);
      X = BN_CTX_get(ctx);
      D = BN_CTX_get(ctx);
      M = BN_CTX_get(ctx);
      Y = BN_CTX_get(ctx);
      T = BN_CTX_get(ctx);
      if (T == NULL) goto err;

      if (in == NULL)
            R=BN_new();
      else
            R=in;
      if (R == NULL) goto err;

      BN_one(X);
      BN_zero(Y);
      if (BN_copy(B,a) == NULL) goto err;
      if (BN_copy(A,n) == NULL) goto err;
      A->neg = 0;
      if (B->neg || (BN_ucmp(B, A) >= 0))
            {
            if (!BN_nnmod(B, B, A, ctx)) goto err;
            }
      sign = -1;
      /* From  B = a mod |n|,  A = |n|  it follows that
       *
       *      0 <= B < A,
       *     -sign*X*a  ==  B   (mod |n|),
       *      sign*Y*a  ==  A   (mod |n|).
       */

      if (BN_is_odd(n) && (BN_num_bits(n) <= (BN_BITS <= 32 ? 450 : 2048)))
            {
            /* Binary inversion algorithm; requires odd modulus.
             * This is faster than the general algorithm if the modulus
             * is sufficiently small (about 400 .. 500 bits on 32-bit
             * sytems, but much more on 64-bit systems) */
            int shift;
            
            while (!BN_is_zero(B))
                  {
                  /*
                   *      0 < B < |n|,
                   *      0 < A <= |n|,
                   * (1) -sign*X*a  ==  B   (mod |n|),
                   * (2)  sign*Y*a  ==  A   (mod |n|)
                   */

                  /* Now divide  B  by the maximum possible power of two in the integers,
                   * and divide  X  by the same value mod |n|.
                   * When we're done, (1) still holds. */
                  shift = 0;
                  while (!BN_is_bit_set(B, shift)) /* note that 0 < B */
                        {
                        shift++;
                        
                        if (BN_is_odd(X))
                              {
                              if (!BN_uadd(X, X, n)) goto err;
                              }
                        /* now X is even, so we can easily divide it by two */
                        if (!BN_rshift1(X, X)) goto err;
                        }
                  if (shift > 0)
                        {
                        if (!BN_rshift(B, B, shift)) goto err;
                        }


                  /* Same for  A  and  Y.  Afterwards, (2) still holds. */
                  shift = 0;
                  while (!BN_is_bit_set(A, shift)) /* note that 0 < A */
                        {
                        shift++;
                        
                        if (BN_is_odd(Y))
                              {
                              if (!BN_uadd(Y, Y, n)) goto err;
                              }
                        /* now Y is even */
                        if (!BN_rshift1(Y, Y)) goto err;
                        }
                  if (shift > 0)
                        {
                        if (!BN_rshift(A, A, shift)) goto err;
                        }

                  
                  /* We still have (1) and (2).
                   * Both  A  and  B  are odd.
                   * The following computations ensure that
                   *
                   *     0 <= B < |n|,
                   *      0 < A < |n|,
                   * (1) -sign*X*a  ==  B   (mod |n|),
                   * (2)  sign*Y*a  ==  A   (mod |n|),
                   *
                   * and that either  A  or  B  is even in the next iteration.
                   */
                  if (BN_ucmp(B, A) >= 0)
                        {
                        /* -sign*(X + Y)*a == B - A  (mod |n|) */
                        if (!BN_uadd(X, X, Y)) goto err;
                        /* NB: we could use BN_mod_add_quick(X, X, Y, n), but that
                         * actually makes the algorithm slower */
                        if (!BN_usub(B, B, A)) goto err;
                        }
                  else
                        {
                        /*  sign*(X + Y)*a == A - B  (mod |n|) */
                        if (!BN_uadd(Y, Y, X)) goto err;
                        /* as above, BN_mod_add_quick(Y, Y, X, n) would slow things down */
                        if (!BN_usub(A, A, B)) goto err;
                        }
                  }
            }
      else
            {
            /* general inversion algorithm */

            while (!BN_is_zero(B))
                  {
                  BIGNUM *tmp;
                  
                  /*
                   *      0 < B < A,
                   * (*) -sign*X*a  ==  B   (mod |n|),
                   *      sign*Y*a  ==  A   (mod |n|)
                   */
                  
                  /* (D, M) := (A/B, A%B) ... */
                  if (BN_num_bits(A) == BN_num_bits(B))
                        {
                        if (!BN_one(D)) goto err;
                        if (!BN_sub(M,A,B)) goto err;
                        }
                  else if (BN_num_bits(A) == BN_num_bits(B) + 1)
                        {
                        /* A/B is 1, 2, or 3 */
                        if (!BN_lshift1(T,B)) goto err;
                        if (BN_ucmp(A,T) < 0)
                              {
                              /* A < 2*B, so D=1 */
                              if (!BN_one(D)) goto err;
                              if (!BN_sub(M,A,B)) goto err;
                              }
                        else
                              {
                              /* A >= 2*B, so D=2 or D=3 */
                              if (!BN_sub(M,A,T)) goto err;
                              if (!BN_add(D,T,B)) goto err; /* use D (:= 3*B) as temp */
                              if (BN_ucmp(A,D) < 0)
                                    {
                                    /* A < 3*B, so D=2 */
                                    if (!BN_set_word(D,2)) goto err;
                                    /* M (= A - 2*B) already has the correct value */
                                    }
                              else
                                    {
                                    /* only D=3 remains */
                                    if (!BN_set_word(D,3)) goto err;
                                    /* currently  M = A - 2*B,  but we need  M = A - 3*B */
                                    if (!BN_sub(M,M,B)) goto err;
                                    }
                              }
                        }
                  else
                        {
                        if (!BN_div(D,M,A,B,ctx)) goto err;
                        }
                  
                  /* Now
                   *      A = D*B + M;
                   * thus we have
                   * (**)  sign*Y*a  ==  D*B + M   (mod |n|).
                   */
                  
                  tmp=A; /* keep the BIGNUM object, the value does not matter */
                  
                  /* (A, B) := (B, A mod B) ... */
                  A=B;
                  B=M;
                  /* ... so we have  0 <= B < A  again */
                  
                  /* Since the former  M  is now  B  and the former  B  is now  A,
                   * (**) translates into
                   *       sign*Y*a  ==  D*A + B    (mod |n|),
                   * i.e.
                   *       sign*Y*a - D*A  ==  B    (mod |n|).
                   * Similarly, (*) translates into
                   *      -sign*X*a  ==  A          (mod |n|).
                   *
                   * Thus,
                   *   sign*Y*a + D*sign*X*a  ==  B  (mod |n|),
                   * i.e.
                   *        sign*(Y + D*X)*a  ==  B  (mod |n|).
                   *
                   * So if we set  (X, Y, sign) := (Y + D*X, X, -sign),  we arrive back at
                   *      -sign*X*a  ==  B   (mod |n|),
                   *       sign*Y*a  ==  A   (mod |n|).
                   * Note that  X  and  Y  stay non-negative all the time.
                   */
                  
                  /* most of the time D is very small, so we can optimize tmp := D*X+Y */
                  if (BN_is_one(D))
                        {
                        if (!BN_add(tmp,X,Y)) goto err;
                        }
                  else
                        {
                        if (BN_is_word(D,2))
                              {
                              if (!BN_lshift1(tmp,X)) goto err;
                              }
                        else if (BN_is_word(D,4))
                              {
                              if (!BN_lshift(tmp,X,2)) goto err;
                              }
                        else if (D->top == 1)
                              {
                              if (!BN_copy(tmp,X)) goto err;
                              if (!BN_mul_word(tmp,D->d[0])) goto err;
                              }
                        else
                              {
                              if (!BN_mul(tmp,D,X,ctx)) goto err;
                              }
                        if (!BN_add(tmp,tmp,Y)) goto err;
                        }
                  
                  M=Y; /* keep the BIGNUM object, the value does not matter */
                  Y=X;
                  X=tmp;
                  sign = -sign;
                  }
            }
            
      /*
       * The while loop (Euclid's algorithm) ends when
       *      A == gcd(a,n);
       * we have
       *       sign*Y*a  ==  A  (mod |n|),
       * where  Y  is non-negative.
       */

      if (sign < 0)
            {
            if (!BN_sub(Y,n,Y)) goto err;
            }
      /* Now  Y*a  ==  A  (mod |n|).  */
      

      if (BN_is_one(A))
            {
            /* Y*a == 1  (mod |n|) */
            if (!Y->neg && BN_ucmp(Y,n) < 0)
                  {
                  if (!BN_copy(R,Y)) goto err;
                  }
            else
                  {
                  if (!BN_nnmod(R,Y,n,ctx)) goto err;
                  }
            }
      else
            {
            BNerr(BN_F_BN_MOD_INVERSE,BN_R_NO_INVERSE);
            goto err;
            }
      ret=R;
err:
      if ((ret == NULL) && (in == NULL)) BN_free(R);
      BN_CTX_end(ctx);
      bn_check_top(ret);
      return(ret);
      }


/* BN_mod_inverse_no_branch is a special version of BN_mod_inverse. 
 * It does not contain branches that may leak sensitive information.
 */
static BIGNUM *BN_mod_inverse_no_branch(BIGNUM *in,
      const BIGNUM *a, const BIGNUM *n, BN_CTX *ctx)
      {
      BIGNUM *A,*B,*X,*Y,*M,*D,*T,*R=NULL;
      BIGNUM local_A, local_B;
      BIGNUM *pA, *pB;
      BIGNUM *ret=NULL;
      int sign;

      bn_check_top(a);
      bn_check_top(n);

      BN_CTX_start(ctx);
      A = BN_CTX_get(ctx);
      B = BN_CTX_get(ctx);
      X = BN_CTX_get(ctx);
      D = BN_CTX_get(ctx);
      M = BN_CTX_get(ctx);
      Y = BN_CTX_get(ctx);
      T = BN_CTX_get(ctx);
      if (T == NULL) goto err;

      if (in == NULL)
            R=BN_new();
      else
            R=in;
      if (R == NULL) goto err;

      BN_one(X);
      BN_zero(Y);
      if (BN_copy(B,a) == NULL) goto err;
      if (BN_copy(A,n) == NULL) goto err;
      A->neg = 0;

      if (B->neg || (BN_ucmp(B, A) >= 0))
            {
            /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
             * BN_div_no_branch will be called eventually.
             */
            pB = &local_B;
            BN_with_flags(pB, B, BN_FLG_CONSTTIME);   
            if (!BN_nnmod(B, pB, A, ctx)) goto err;
            }
      sign = -1;
      /* From  B = a mod |n|,  A = |n|  it follows that
       *
       *      0 <= B < A,
       *     -sign*X*a  ==  B   (mod |n|),
       *      sign*Y*a  ==  A   (mod |n|).
       */

      while (!BN_is_zero(B))
            {
            BIGNUM *tmp;
            
            /*
             *      0 < B < A,
             * (*) -sign*X*a  ==  B   (mod |n|),
             *      sign*Y*a  ==  A   (mod |n|)
             */

            /* Turn BN_FLG_CONSTTIME flag on, so that when BN_div is invoked,
             * BN_div_no_branch will be called eventually.
             */
            pA = &local_A;
            BN_with_flags(pA, A, BN_FLG_CONSTTIME);   
            
            /* (D, M) := (A/B, A%B) ... */            
            if (!BN_div(D,M,pA,B,ctx)) goto err;
            
            /* Now
             *      A = D*B + M;
             * thus we have
             * (**)  sign*Y*a  ==  D*B + M   (mod |n|).
             */
            
            tmp=A; /* keep the BIGNUM object, the value does not matter */
            
            /* (A, B) := (B, A mod B) ... */
            A=B;
            B=M;
            /* ... so we have  0 <= B < A  again */
            
            /* Since the former  M  is now  B  and the former  B  is now  A,
             * (**) translates into
             *       sign*Y*a  ==  D*A + B    (mod |n|),
             * i.e.
             *       sign*Y*a - D*A  ==  B    (mod |n|).
             * Similarly, (*) translates into
             *      -sign*X*a  ==  A          (mod |n|).
             *
             * Thus,
             *   sign*Y*a + D*sign*X*a  ==  B  (mod |n|),
             * i.e.
             *        sign*(Y + D*X)*a  ==  B  (mod |n|).
             *
             * So if we set  (X, Y, sign) := (Y + D*X, X, -sign),  we arrive back at
             *      -sign*X*a  ==  B   (mod |n|),
             *       sign*Y*a  ==  A   (mod |n|).
             * Note that  X  and  Y  stay non-negative all the time.
             */
                  
            if (!BN_mul(tmp,D,X,ctx)) goto err;
            if (!BN_add(tmp,tmp,Y)) goto err;

            M=Y; /* keep the BIGNUM object, the value does not matter */
            Y=X;
            X=tmp;
            sign = -sign;
            }
            
      /*
       * The while loop (Euclid's algorithm) ends when
       *      A == gcd(a,n);
       * we have
       *       sign*Y*a  ==  A  (mod |n|),
       * where  Y  is non-negative.
       */

      if (sign < 0)
            {
            if (!BN_sub(Y,n,Y)) goto err;
            }
      /* Now  Y*a  ==  A  (mod |n|).  */

      if (BN_is_one(A))
            {
            /* Y*a == 1  (mod |n|) */
            if (!Y->neg && BN_ucmp(Y,n) < 0)
                  {
                  if (!BN_copy(R,Y)) goto err;
                  }
            else
                  {
                  if (!BN_nnmod(R,Y,n,ctx)) goto err;
                  }
            }
      else
            {
            BNerr(BN_F_BN_MOD_INVERSE_NO_BRANCH,BN_R_NO_INVERSE);
            goto err;
            }
      ret=R;
err:
      if ((ret == NULL) && (in == NULL)) BN_free(R);
      BN_CTX_end(ctx);
      bn_check_top(ret);
      return(ret);
      }

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