text: /* 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);
  	}
  