library/ecp.c - security/mbedtls - Rivoreo Source Code Repositories

 /*
  *  Elliptic curves over GF(p)
  *
  *  Copyright (C) 2012, Brainspark B.V.
  *
  *  This file is part of PolarSSL (http://www.polarssl.org)
  *  Lead Maintainer: Paul Bakker <polarssl_maintainer at polarssl.org>
  *
  *  All rights reserved.
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
  *  the Free Software Foundation; either version 2 of the License, or
  *  (at your option) any later version.
  *
  *  This program is distributed in the hope that it will be useful,
  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  *  GNU General Public License for more details.
  *
  *  You should have received a copy of the GNU General Public License along
  *  with this program; if not, write to the Free Software Foundation, Inc.,
  *  51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
  */

 /*
  * References:
  *
  * SEC1 http://www.secg.org/index.php?action=secg,docs_secg
  * Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
  */

 #include "polarssl/config.h"

 #if defined(POLARSSL_ECP_C)

 #include "polarssl/ecp.h"

 /*
  * Initialize (the components of) a point
  */
 void ecp_point_init( ecp_point *pt )
 {
     if( pt == NULL )
         return;

     pt->is_zero = 1;
     mpi_init( &pt->X );
     mpi_init( &pt->Y );
 }

 /*
  * Initialize (the components of) a group
  */
 void ecp_group_init( ecp_group *grp )
 {
     if( grp == NULL )
         return;

     mpi_init( &grp->P );
     mpi_init( &grp->B );
     ecp_point_init( &grp->G );
     mpi_init( &grp->N );
 }

 /*
  * Unallocate (the components of) a point
  */
 void ecp_point_free( ecp_point *pt )
 {
     if( pt == NULL )
         return;

     pt->is_zero = 1;
     mpi_free( &( pt->X ) );
     mpi_free( &( pt->Y ) );
 }

 /*
  * Unallocate (the components of) a group
  */
 void ecp_group_free( ecp_group *grp )
 {
     if( grp == NULL )
         return;

     mpi_free( &grp->P );
     mpi_free( &grp->B );
     ecp_point_free( &grp->G );
     mpi_free( &grp->N );
 }

 /*
  * Set point to zero
  */
 void ecp_set_zero( ecp_point *pt )
 {
     pt->is_zero = 1;
     mpi_free( &pt->X );
     mpi_free( &pt->Y );
 }

 /*
  * Copy the contents of Q into P
  */
 int ecp_copy( ecp_point *P, const ecp_point *Q )
 {
     int ret = 0;

     if( Q->is_zero ) {
         ecp_set_zero( P );
         return( ret );
     }

     P->is_zero = Q->is_zero;
     MPI_CHK( mpi_copy( &P->X, &Q->X ) );
     MPI_CHK( mpi_copy( &P->Y, &Q->Y ) );

 cleanup:
     return( ret );
 }

 /*
  * Import a non-zero point from ASCII strings
  */
 int ecp_point_read_string( ecp_point *P, int radix,
                            const char *x, const char *y )
 {
     int ret = 0;

     P->is_zero = 0;
     MPI_CHK( mpi_read_string( &P->X, radix, x ) );
     MPI_CHK( mpi_read_string( &P->Y, radix, y ) );

 cleanup:
     return( ret );
 }

 /*
  * Import an ECP group from ASCII strings
  */
 int ecp_group_read_string( ecp_group *grp, int radix,
                            const char *p, const char *b,
                            const char *gx, const char *gy, const char *n)
 {
     int ret = 0;

     MPI_CHK( mpi_read_string( &grp->P, radix, p ) );
     MPI_CHK( mpi_read_string( &grp->B, radix, b ) );
     MPI_CHK( ecp_point_read_string( &grp->G, radix, gx, gy ) );
     MPI_CHK( mpi_read_string( &grp->N, radix, n ) );

 cleanup:
     return( ret );
 }

 /*
  * Set a group using well-known domain parameters
  */
 int ecp_use_known_dp( ecp_group *grp, size_t index )
 {
     switch( index )
     {
         case POLARSSL_ECP_DP_SECP192R1:
             return( ecp_group_read_string( grp, 16,
                         POLARSSL_ECP_SECP192R1_P,
                         POLARSSL_ECP_SECP192R1_B,
                         POLARSSL_ECP_SECP192R1_GX,
                         POLARSSL_ECP_SECP192R1_GY,
                         POLARSSL_ECP_SECP192R1_N )
                     );
         case POLARSSL_ECP_DP_SECP224R1:
             return( ecp_group_read_string( grp, 16,
                         POLARSSL_ECP_SECP224R1_P,
                         POLARSSL_ECP_SECP224R1_B,
                         POLARSSL_ECP_SECP224R1_GX,
                         POLARSSL_ECP_SECP224R1_GY,
                         POLARSSL_ECP_SECP224R1_N )
                     );
         case POLARSSL_ECP_DP_SECP256R1:
             return( ecp_group_read_string( grp, 16,
                         POLARSSL_ECP_SECP256R1_P,
                         POLARSSL_ECP_SECP256R1_B,
                         POLARSSL_ECP_SECP256R1_GX,
                         POLARSSL_ECP_SECP256R1_GY,
                         POLARSSL_ECP_SECP256R1_N )
                     );
         case POLARSSL_ECP_DP_SECP384R1:
             return( ecp_group_read_string( grp, 16,
                         POLARSSL_ECP_SECP384R1_P,
                         POLARSSL_ECP_SECP384R1_B,
                         POLARSSL_ECP_SECP384R1_GX,
                         POLARSSL_ECP_SECP384R1_GY,
                         POLARSSL_ECP_SECP384R1_N )
                     );
         case POLARSSL_ECP_DP_SECP521R1:
             return( ecp_group_read_string( grp, 16,
                         POLARSSL_ECP_SECP521R1_P,
                         POLARSSL_ECP_SECP521R1_B,
                         POLARSSL_ECP_SECP521R1_GX,
                         POLARSSL_ECP_SECP521R1_GY,
                         POLARSSL_ECP_SECP521R1_N )
                     );
     }

     return( POLARSSL_ERR_ECP_GENERIC );
 }

 /*
  * Addition: R = P + Q, generic case (P != Q, P != 0, Q != 0, R != 0)
  * Cf SEC1 v2 p. 7, item 4
  */
 static int ecp_add_generic( const ecp_group *grp, ecp_point *R,
                             const ecp_point *P, const ecp_point *Q )
 {
     int ret = 0;
     mpi DX, DY, K, L, LL, X, Y;

     mpi_init( &DX ); mpi_init( &DY ); mpi_init( &K ); mpi_init( &L );
     mpi_init( &LL ); mpi_init( &X ); mpi_init( &Y );

     /*
      * L = (Q.Y - P.Y) / (Q.X - P.X)  mod p
      */
     MPI_CHK( mpi_sub_mpi( &DY, &Q->Y, &P->Y ) );
     MPI_CHK( mpi_sub_mpi( &DX, &Q->X, &P->X ) );
     MPI_CHK( mpi_inv_mod( &K, &DX, &grp->P ) );
     MPI_CHK( mpi_mul_mpi( &K, &K, &DY ) );
     MPI_CHK( mpi_mod_mpi( &L, &K, &grp->P ) );

     /*
      * LL = L^2  mod p
      */
     MPI_CHK( mpi_mul_mpi( &LL, &L, &L ) );
     MPI_CHK( mpi_mod_mpi( &LL, &LL, &grp->P ) );

     /*
      * X  = L^2 - P.X - Q.X
      */
     MPI_CHK( mpi_sub_mpi( &X, &LL, &P->X ) );
     MPI_CHK( mpi_sub_mpi( &X, &X,  &Q->X ) );

     /*
      * Y = L * (P.X - X) - P.Y
      */
     MPI_CHK( mpi_sub_mpi( &Y, &P->X, &X) );
     MPI_CHK( mpi_mul_mpi( &Y, &Y, &L ) );
     MPI_CHK( mpi_sub_mpi( &Y, &Y, &P->Y ) );

     /*
      * R = (X mod p, Y mod p)
      */
     R->is_zero = 0;
     MPI_CHK( mpi_mod_mpi( &R->X, &X, &grp->P ) );
     MPI_CHK( mpi_mod_mpi( &R->Y, &Y, &grp->P ) );

 cleanup:

     mpi_free( &DX ); mpi_free( &DY ); mpi_free( &K ); mpi_free( &L );
     mpi_free( &LL ); mpi_free( &X ); mpi_free( &Y );

     return( ret );
 }

 /*
  * Doubling: R = 2 * P, generic case (P != 0, R != 0)
  * Cf SEC1 v2 p. 7, item 5
  */
 static int ecp_double_generic( const ecp_group *grp, ecp_point *R,
                                const ecp_point *P )
 {
     int ret = 0;
     mpi LN, LD, K, L, LL, X, Y;

     mpi_init( &LN ); mpi_init( &LD ); mpi_init( &K ); mpi_init( &L );
     mpi_init( &LL ); mpi_init( &X ); mpi_init( &Y );

     /*
      * L = 3 (P.X - 1) (P.X + 1) / (2 P.Y) mod p
      */
     MPI_CHK( mpi_copy( &LD, &P->Y ) );
     MPI_CHK( mpi_shift_l( &LD, 1 ) );
     MPI_CHK( mpi_inv_mod( &K, &LD, &grp->P ) );
     MPI_CHK( mpi_mul_int( &K, &K, 3 ) );
     MPI_CHK( mpi_sub_int( &LN, &P->X, 1 ) );
     MPI_CHK( mpi_mul_mpi( &K, &K, &LN ) );
     MPI_CHK( mpi_add_int( &LN, &P->X, 1 ) );
     MPI_CHK( mpi_mul_mpi( &K, &K, &LN ) );
     MPI_CHK( mpi_mod_mpi( &L, &K, &grp->P ) );

     /*
      * LL = L^2  mod p
      */
     MPI_CHK( mpi_mul_mpi( &LL, &L, &L ) );
     MPI_CHK( mpi_mod_mpi( &LL, &LL, &grp->P ) );

     /*
      * X  = L^2 - 2 * P.X
      */
     MPI_CHK( mpi_sub_mpi( &X, &LL, &P->X ) );
     MPI_CHK( mpi_sub_mpi( &X, &X,  &P->X ) );

     /*
      * Y = L * (P.X - X) - P.Y
      */
     MPI_CHK( mpi_sub_mpi( &Y, &P->X, &X) );
     MPI_CHK( mpi_mul_mpi( &Y, &Y, &L ) );
     MPI_CHK( mpi_sub_mpi( &Y, &Y, &P->Y ) );

     /*
      * R = (X mod p, Y mod p)
      */
     R->is_zero = 0;
     MPI_CHK( mpi_mod_mpi( &R->X, &X, &grp->P ) );
     MPI_CHK( mpi_mod_mpi( &R->Y, &Y, &grp->P ) );

 cleanup:

     mpi_free( &LN ); mpi_free( &LD ); mpi_free( &K ); mpi_free( &L );
     mpi_free( &LL ); mpi_free( &X ); mpi_free( &Y );

     return( ret );
 }

 /*
  * Addition: R = P + Q, cf p. 7 of SEC1 v2
  */
 int ecp_add( const ecp_group *grp, ecp_point *R,
              const ecp_point *P, const ecp_point *Q )
 {
     int ret = 0;

     if( P->is_zero )
     {
         ret = ecp_copy( R, Q );
     }
     else if( Q->is_zero )
     {
         ret = ecp_copy( R, P );
     }
     else if( mpi_cmp_mpi( &P->X, &Q->X ) != 0 )
     {
         ret = ecp_add_generic( grp, R, P, Q );
     }
     else if( mpi_cmp_int( &P->Y, 0 ) == 0 ||
              mpi_cmp_mpi( &P->Y, &Q->Y ) != 0 )
     {
         ecp_set_zero( R );
     }
     else
     {
         /*
          * P == Q
          */
         ret = ecp_double_generic( grp, R, P );
     }

     return ret;
 }

 /*
  * Integer multiplication: R = m * P
  * Using Montgomery's Ladder to avoid leaking information about m
  */
 int ecp_mul( const ecp_group *grp, ecp_point *R,
              const mpi *m, const ecp_point *P )
 {
     int ret = 0;
     size_t pos;
     ecp_point A, B;

     ecp_point_init( &A ); ecp_point_init( &B );

     /*
      * The general method works only for m >= 2
      */
     if( mpi_cmp_int( m, 0 ) == 0 ) {
         ecp_set_zero( R );
         goto cleanup;
     }

     if( mpi_cmp_int( m, 1 ) == 0 ) {
         MPI_CHK( ecp_copy( R, P ) );
         goto cleanup;
     }

     MPI_CHK( ecp_copy( &A, P ) );
     MPI_CHK( ecp_add( grp, &B, P, P ) );

     for( pos = mpi_msb( m ) - 2; ; pos-- )
     {
         if( mpi_get_bit( m, pos ) == 0 )
         {
             MPI_CHK( ecp_add( grp, &B, &A, &B ) );
             MPI_CHK( ecp_add( grp, &A, &A, &A ) ) ;
         }
         else
         {
             MPI_CHK( ecp_add( grp, &A, &A, &B ) );
             MPI_CHK( ecp_add( grp, &B, &B, &B ) ) ;
         }

         if( pos == 0 )
             break;
     }

     MPI_CHK( ecp_copy( R, &A ) );

 cleanup:

     ecp_point_free( &A ); ecp_point_free( &B );

     return( ret );
 }


 #if defined(POLARSSL_SELF_TEST)

 /*
  * Checkup routine
  */
 int ecp_self_test( int verbose )
 {
     return( verbose++ );
 }

 #endif

 #endif
	/*
	* Elliptic curves over GF(p)
	*
	* Copyright (C) 2012, Brainspark B.V.
	*
	* This file is part of PolarSSL (http://www.polarssl.org)
	* Lead Maintainer: Paul Bakker <polarssl_maintainer at polarssl.org>
	*
	* All rights reserved.
	*
	* This program is free software; you can redistribute it and/or modify
	* it under the terms of the GNU General Public License as published by
	* the Free Software Foundation; either version 2 of the License, or
	* (at your option) any later version.
	*
	* This program is distributed in the hope that it will be useful,
	* but WITHOUT ANY WARRANTY; without even the implied warranty of
	* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
	* GNU General Public License for more details.
	*
	* You should have received a copy of the GNU General Public License along
	* with this program; if not, write to the Free Software Foundation, Inc.,
	* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
	*/

	/*
	* References:
	*
	* SEC1 http://www.secg.org/index.php?action=secg,docs_secg
	* Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
	*/

	#include "polarssl/config.h"

	#if defined(POLARSSL_ECP_C)

	#include "polarssl/ecp.h"

	/*
	* Initialize (the components of) a point
	*/
	void ecp_point_init( ecp_point *pt )
	{
	if( pt == NULL )
	return;

	pt->is_zero = 1;
	mpi_init( &pt->X );
	mpi_init( &pt->Y );
	}

	/*
	* Initialize (the components of) a group
	*/
	void ecp_group_init( ecp_group *grp )
	{
	if( grp == NULL )
	return;

	mpi_init( &grp->P );
	mpi_init( &grp->B );
	ecp_point_init( &grp->G );
	mpi_init( &grp->N );
	}

	/*
	* Unallocate (the components of) a point
	*/
	void ecp_point_free( ecp_point *pt )
	{
	if( pt == NULL )
	return;

	pt->is_zero = 1;
	mpi_free( &( pt->X ) );
	mpi_free( &( pt->Y ) );
	}

	/*
	* Unallocate (the components of) a group
	*/
	void ecp_group_free( ecp_group *grp )
	{
	if( grp == NULL )
	return;

	mpi_free( &grp->P );
	mpi_free( &grp->B );
	ecp_point_free( &grp->G );
	mpi_free( &grp->N );
	}

	/*
	* Set point to zero
	*/
	void ecp_set_zero( ecp_point *pt )
	{
	pt->is_zero = 1;
	mpi_free( &pt->X );
	mpi_free( &pt->Y );
	}

	/*
	* Copy the contents of Q into P
	*/
	int ecp_copy( ecp_point P, const ecp_point Q )
	{
	int ret = 0;

	if( Q->is_zero ) {
	ecp_set_zero( P );
	return( ret );
	}

	P->is_zero = Q->is_zero;
	MPI_CHK( mpi_copy( &P->X, &Q->X ) );
	MPI_CHK( mpi_copy( &P->Y, &Q->Y ) );

	cleanup:
	return( ret );
	}

	/*
	* Import a non-zero point from ASCII strings
	*/
	int ecp_point_read_string( ecp_point *P, int radix,
	const char x, const char y )
	{
	int ret = 0;

	P->is_zero = 0;
	MPI_CHK( mpi_read_string( &P->X, radix, x ) );
	MPI_CHK( mpi_read_string( &P->Y, radix, y ) );

	cleanup:
	return( ret );
	}

	/*
	* Import an ECP group from ASCII strings
	*/
	int ecp_group_read_string( ecp_group *grp, int radix,
	const char p, const char b,
	const char gx, const char gy, const char *n)
	{
	int ret = 0;

	MPI_CHK( mpi_read_string( &grp->P, radix, p ) );
	MPI_CHK( mpi_read_string( &grp->B, radix, b ) );
	MPI_CHK( ecp_point_read_string( &grp->G, radix, gx, gy ) );
	MPI_CHK( mpi_read_string( &grp->N, radix, n ) );

	cleanup:
	return( ret );
	}

	/*
	* Set a group using well-known domain parameters
	*/
	int ecp_use_known_dp( ecp_group *grp, size_t index )
	{
	switch( index )
	{
	case POLARSSL_ECP_DP_SECP192R1:
	return( ecp_group_read_string( grp, 16,
	POLARSSL_ECP_SECP192R1_P,
	POLARSSL_ECP_SECP192R1_B,
	POLARSSL_ECP_SECP192R1_GX,
	POLARSSL_ECP_SECP192R1_GY,
	POLARSSL_ECP_SECP192R1_N )
	);
	case POLARSSL_ECP_DP_SECP224R1:
	return( ecp_group_read_string( grp, 16,
	POLARSSL_ECP_SECP224R1_P,
	POLARSSL_ECP_SECP224R1_B,
	POLARSSL_ECP_SECP224R1_GX,
	POLARSSL_ECP_SECP224R1_GY,
	POLARSSL_ECP_SECP224R1_N )
	);
	case POLARSSL_ECP_DP_SECP256R1:
	return( ecp_group_read_string( grp, 16,
	POLARSSL_ECP_SECP256R1_P,
	POLARSSL_ECP_SECP256R1_B,
	POLARSSL_ECP_SECP256R1_GX,
	POLARSSL_ECP_SECP256R1_GY,
	POLARSSL_ECP_SECP256R1_N )
	);
	case POLARSSL_ECP_DP_SECP384R1:
	return( ecp_group_read_string( grp, 16,
	POLARSSL_ECP_SECP384R1_P,
	POLARSSL_ECP_SECP384R1_B,
	POLARSSL_ECP_SECP384R1_GX,
	POLARSSL_ECP_SECP384R1_GY,
	POLARSSL_ECP_SECP384R1_N )
	);
	case POLARSSL_ECP_DP_SECP521R1:
	return( ecp_group_read_string( grp, 16,
	POLARSSL_ECP_SECP521R1_P,
	POLARSSL_ECP_SECP521R1_B,
	POLARSSL_ECP_SECP521R1_GX,
	POLARSSL_ECP_SECP521R1_GY,
	POLARSSL_ECP_SECP521R1_N )
	);
	}

	return( POLARSSL_ERR_ECP_GENERIC );
	}

	/*
	* Addition: R = P + Q, generic case (P != Q, P != 0, Q != 0, R != 0)
	* Cf SEC1 v2 p. 7, item 4
	*/
	static int ecp_add_generic( const ecp_group grp, ecp_point R,
	const ecp_point P, const ecp_point Q )
	{
	int ret = 0;
	mpi DX, DY, K, L, LL, X, Y;

	mpi_init( &DX ); mpi_init( &DY ); mpi_init( &K ); mpi_init( &L );
	mpi_init( &LL ); mpi_init( &X ); mpi_init( &Y );

	/*
	* L = (Q.Y - P.Y) / (Q.X - P.X) mod p
	*/
	MPI_CHK( mpi_sub_mpi( &DY, &Q->Y, &P->Y ) );
	MPI_CHK( mpi_sub_mpi( &DX, &Q->X, &P->X ) );
	MPI_CHK( mpi_inv_mod( &K, &DX, &grp->P ) );
	MPI_CHK( mpi_mul_mpi( &K, &K, &DY ) );
	MPI_CHK( mpi_mod_mpi( &L, &K, &grp->P ) );

	/*
	* LL = L^2 mod p
	*/
	MPI_CHK( mpi_mul_mpi( &LL, &L, &L ) );
	MPI_CHK( mpi_mod_mpi( &LL, &LL, &grp->P ) );

	/*
	* X = L^2 - P.X - Q.X
	*/
	MPI_CHK( mpi_sub_mpi( &X, &LL, &P->X ) );
	MPI_CHK( mpi_sub_mpi( &X, &X, &Q->X ) );

	/*
	* Y = L * (P.X - X) - P.Y
	*/
	MPI_CHK( mpi_sub_mpi( &Y, &P->X, &X) );
	MPI_CHK( mpi_mul_mpi( &Y, &Y, &L ) );
	MPI_CHK( mpi_sub_mpi( &Y, &Y, &P->Y ) );

	/*
	* R = (X mod p, Y mod p)
	*/
	R->is_zero = 0;
	MPI_CHK( mpi_mod_mpi( &R->X, &X, &grp->P ) );
	MPI_CHK( mpi_mod_mpi( &R->Y, &Y, &grp->P ) );

	cleanup:

	mpi_free( &DX ); mpi_free( &DY ); mpi_free( &K ); mpi_free( &L );
	mpi_free( &LL ); mpi_free( &X ); mpi_free( &Y );

	return( ret );
	}

	/*
	* Doubling: R = 2 * P, generic case (P != 0, R != 0)
	* Cf SEC1 v2 p. 7, item 5
	*/
	static int ecp_double_generic( const ecp_group grp, ecp_point R,
	const ecp_point *P )
	{
	int ret = 0;
	mpi LN, LD, K, L, LL, X, Y;

	mpi_init( &LN ); mpi_init( &LD ); mpi_init( &K ); mpi_init( &L );
	mpi_init( &LL ); mpi_init( &X ); mpi_init( &Y );

	/*
	* L = 3 (P.X - 1) (P.X + 1) / (2 P.Y) mod p
	*/
	MPI_CHK( mpi_copy( &LD, &P->Y ) );
	MPI_CHK( mpi_shift_l( &LD, 1 ) );
	MPI_CHK( mpi_inv_mod( &K, &LD, &grp->P ) );
	MPI_CHK( mpi_mul_int( &K, &K, 3 ) );
	MPI_CHK( mpi_sub_int( &LN, &P->X, 1 ) );
	MPI_CHK( mpi_mul_mpi( &K, &K, &LN ) );
	MPI_CHK( mpi_add_int( &LN, &P->X, 1 ) );
	MPI_CHK( mpi_mul_mpi( &K, &K, &LN ) );
	MPI_CHK( mpi_mod_mpi( &L, &K, &grp->P ) );

	/*
	* LL = L^2 mod p
	*/
	MPI_CHK( mpi_mul_mpi( &LL, &L, &L ) );
	MPI_CHK( mpi_mod_mpi( &LL, &LL, &grp->P ) );

	/*
	* X = L^2 - 2 * P.X
	*/
	MPI_CHK( mpi_sub_mpi( &X, &LL, &P->X ) );
	MPI_CHK( mpi_sub_mpi( &X, &X, &P->X ) );

	/*
	* Y = L * (P.X - X) - P.Y
	*/
	MPI_CHK( mpi_sub_mpi( &Y, &P->X, &X) );
	MPI_CHK( mpi_mul_mpi( &Y, &Y, &L ) );
	MPI_CHK( mpi_sub_mpi( &Y, &Y, &P->Y ) );

	/*
	* R = (X mod p, Y mod p)
	*/
	R->is_zero = 0;
	MPI_CHK( mpi_mod_mpi( &R->X, &X, &grp->P ) );
	MPI_CHK( mpi_mod_mpi( &R->Y, &Y, &grp->P ) );

	cleanup:

	mpi_free( &LN ); mpi_free( &LD ); mpi_free( &K ); mpi_free( &L );
	mpi_free( &LL ); mpi_free( &X ); mpi_free( &Y );

	return( ret );
	}

	/*
	* Addition: R = P + Q, cf p. 7 of SEC1 v2
	*/
	int ecp_add( const ecp_group grp, ecp_point R,
	const ecp_point P, const ecp_point Q )
	{
	int ret = 0;

	if( P->is_zero )
	{
	ret = ecp_copy( R, Q );
	}
	else if( Q->is_zero )
	{
	ret = ecp_copy( R, P );
	}
	else if( mpi_cmp_mpi( &P->X, &Q->X ) != 0 )
	{
	ret = ecp_add_generic( grp, R, P, Q );
	}
	else if( mpi_cmp_int( &P->Y, 0 ) == 0 \|\|
	mpi_cmp_mpi( &P->Y, &Q->Y ) != 0 )
	{
	ecp_set_zero( R );
	}
	else
	{
	/*
	* P == Q
	*/
	ret = ecp_double_generic( grp, R, P );
	}

	return ret;
	}

	/*
	* Integer multiplication: R = m * P
	* Using Montgomery's Ladder to avoid leaking information about m
	*/
	int ecp_mul( const ecp_group grp, ecp_point R,
	const mpi m, const ecp_point P )
	{
	int ret = 0;
	size_t pos;
	ecp_point A, B;

	ecp_point_init( &A ); ecp_point_init( &B );

	/*
	* The general method works only for m >= 2
	*/
	if( mpi_cmp_int( m, 0 ) == 0 ) {
	ecp_set_zero( R );
	goto cleanup;
	}

	if( mpi_cmp_int( m, 1 ) == 0 ) {
	MPI_CHK( ecp_copy( R, P ) );
	goto cleanup;
	}

	MPI_CHK( ecp_copy( &A, P ) );
	MPI_CHK( ecp_add( grp, &B, P, P ) );

	for( pos = mpi_msb( m ) - 2; ; pos-- )
	{
	if( mpi_get_bit( m, pos ) == 0 )
	{
	MPI_CHK( ecp_add( grp, &B, &A, &B ) );
	MPI_CHK( ecp_add( grp, &A, &A, &A ) ) ;
	}
	else
	{
	MPI_CHK( ecp_add( grp, &A, &A, &B ) );
	MPI_CHK( ecp_add( grp, &B, &B, &B ) ) ;
	}

	if( pos == 0 )
	break;
	}

	MPI_CHK( ecp_copy( R, &A ) );

	cleanup:

	ecp_point_free( &A ); ecp_point_free( &B );

	return( ret );
	}


	#if defined(POLARSSL_SELF_TEST)

	/*
	* Checkup routine
	*/
	int ecp_self_test( int verbose )
	{
	return( verbose++ );
	}

	#endif

	#endif