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 );
 }

 #define dbg(X)  printf(#X " = %s%lu\n", X.s < 0 ? "-" : "", X.p[0])

 /*
  * 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)

 #include "polarssl/error.h"

 /*
  * Return true iff P and Q are the same point
  */
 static int ecp_point_eq( const ecp_point *P, const ecp_point *Q )
 {
     if( P->is_zero || Q->is_zero )
         return( P->is_zero && Q->is_zero );

     return( mpi_cmp_mpi( &P->X, &Q->X ) == 0 &&
             mpi_cmp_mpi( &P->Y, &Q->Y ) == 0 );
 }

 /*
  * Print a point assuming its coordinates are small
  */
 static void ecp_point_print( const ecp_point *P )
 {
     if( P->is_zero )
         printf( "zero\n" );
     else
         printf( "(%lu, %lu)\n", P->X.p[0], P->Y.p[0] );
 }

 #define TEST_GROUP_ORDER 13

 /*
  * Checkup routine
  *
  * Data for basic tests with small values gathered from
  * http://danher6.100webspace.net/ecc/#EFp_interactivo and double-checked
  * using Pari-GP.
  */
 int ecp_self_test( int verbose )
 {
     int ret = 0;
     unsigned i;
     ecp_group grp;
     ecp_point O, A, B, C, D, E, F, G, H, TMP;
     ecp_point *add_tbl[][3] =
     {
         { &O, &O, &O }, { &O, &A, &A }, { &A, &O, &A },
         { &A, &A, &O }, { &B, &C, &O }, { &C, &B, &O },
         { &A, &D, &E }, { &D, &A, &E },
         { &B, &D, &F }, { &D, &B, &F },
         { &D, &D, &G }, { &B, &B, &H },
     };
     mpi m;
     ecp_point mul_tbl[TEST_GROUP_ORDER + 1];
     char *mul_tbl_s[TEST_GROUP_ORDER - 1][2] =
     {
         { "17", "42" },
         { "20", "01" },
         { "14", "11" },
         { "34", "33" },
         { "21", "32" },
         { "27", "30" },
         { "27", "17" },
         { "21", "15" },
         { "34", "14" },
         { "14", "36" },
         { "20", "46" },
         { "17", "05" },
     };

     ecp_group_init( &grp );

     ecp_point_init( &O ); ecp_point_init( &A ); ecp_point_init( &B );
     ecp_point_init( &C ); ecp_point_init( &D ); ecp_point_init( &E );
     ecp_point_init( &F ); ecp_point_init( &G ); ecp_point_init( &H );
     ecp_point_init( &TMP );

     mpi_init( &m );

     for( i = 0; i <= TEST_GROUP_ORDER; i++ )
         ecp_point_init( &mul_tbl[i] );

     ecp_set_zero( &O );
     MPI_CHK( ecp_group_read_string( &grp, 10, "47", "4", "17", "42", "13" ) );
     MPI_CHK( ecp_point_read_string( &A, 10, "13", "00" ) );
     MPI_CHK( ecp_point_read_string( &B, 10, "14", "11" ) );
     MPI_CHK( ecp_point_read_string( &C, 10, "14", "36" ) );
     MPI_CHK( ecp_point_read_string( &D, 10, "37", "31" ) );
     MPI_CHK( ecp_point_read_string( &E, 10, "34", "14" ) );
     MPI_CHK( ecp_point_read_string( &F, 10, "45", "07" ) );
     MPI_CHK( ecp_point_read_string( &G, 10, "21", "32" ) );
     MPI_CHK( ecp_point_read_string( &H, 10, "27", "30" ) );

     if( verbose != 0 )
         printf( "  ECP test #1 (ecp_add): " );

     for( i = 0; i < sizeof( add_tbl ) / sizeof( add_tbl[0] ); i++ )
     {
         MPI_CHK( ecp_add( &grp, &TMP, add_tbl[i][0], add_tbl[i][1] ) );
         if( ! ecp_point_eq( &TMP, add_tbl[i][2] ) )
         {
             if( verbose != 0 )
             {
                 printf( " failed (%u)\n", i );
                 printf( "        GOT: " );
                 ecp_point_print( &TMP );
                 printf( "   EXPECTED: " );
                 ecp_point_print( add_tbl[i][2] );
             }

             return( 1 );
         }
     }

     if (verbose != 0 )
         printf( "passed\n" );

     MPI_CHK( ecp_copy( &mul_tbl[0], &O ) );
     for( i = 1; i <= TEST_GROUP_ORDER - 1; i++ )
         MPI_CHK( ecp_point_read_string( &mul_tbl[i], 10,
                     mul_tbl_s[i-1][0], mul_tbl_s[i-1][1] ) );
     MPI_CHK( ecp_copy( &mul_tbl[TEST_GROUP_ORDER], &O ) );

     if( verbose != 0 )
         printf( "  ECP test #2 (ecp_mul): " );

     for( i = 0; i <= TEST_GROUP_ORDER; i++ )
     {
         MPI_CHK( mpi_lset( &m, i ) );
         MPI_CHK( ecp_mul( &grp, &TMP, &m, &grp.G ) );
         if( ! ecp_point_eq( &TMP, &mul_tbl[i] ) )
         {
             if( verbose != 0 )
             {
                 printf( " failed (%u)\n", i );
                 printf( "        GOT: " );
                 ecp_point_print( &TMP );
                 printf( "   EXPECTED: " );
                 ecp_point_print( &mul_tbl[i] );
             }

             return( 1 );
         }
     }

     if (verbose != 0 )
         printf( "passed\n" );

 cleanup:

     if( ret != 0 && verbose != 0 )
     {
 #if defined(POLARSSL_ERROR_C)
         char error_buf[200];
         error_strerror( ret, error_buf, 200 );
         printf( "Unexpected error: %d - %s\n\n", ret, error_buf );
 #else
         printf( "Unexpected error: %08X\n", ret );
 #endif
     }

     ecp_group_free( &grp ); ecp_point_free( &O ); ecp_point_free( &TMP );

     ecp_point_free( &A ); ecp_point_free( &B ); ecp_point_free( &C );
     ecp_point_free( &D ); ecp_point_free( &E ); ecp_point_free( &F );
     ecp_point_free( &G ); ecp_point_free( &H );

     mpi_free( &m );

     for( i = 0; i <= TEST_GROUP_ORDER; i++ )
         ecp_point_free( &mul_tbl[i] );

     if( verbose != 0 )
         printf( "\n" );

     return( ret );
 }

 #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 );
	}

	#define dbg(X) printf(#X " = %s%lu\n", X.s < 0 ? "-" : "", X.p[0])

	/*
	* 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)

	#include "polarssl/error.h"

	/*
	* Return true iff P and Q are the same point
	*/
	static int ecp_point_eq( const ecp_point P, const ecp_point Q )
	{
	if( P->is_zero \|\| Q->is_zero )
	return( P->is_zero && Q->is_zero );

	return( mpi_cmp_mpi( &P->X, &Q->X ) == 0 &&
	mpi_cmp_mpi( &P->Y, &Q->Y ) == 0 );
	}

	/*
	* Print a point assuming its coordinates are small
	*/
	static void ecp_point_print( const ecp_point *P )
	{
	if( P->is_zero )
	printf( "zero\n" );
	else
	printf( "(%lu, %lu)\n", P->X.p[0], P->Y.p[0] );
	}

	#define TEST_GROUP_ORDER 13

	/*
	* Checkup routine
	*
	* Data for basic tests with small values gathered from
	* http://danher6.100webspace.net/ecc/#EFp_interactivo and double-checked
	* using Pari-GP.
	*/
	int ecp_self_test( int verbose )
	{
	int ret = 0;
	unsigned i;
	ecp_group grp;
	ecp_point O, A, B, C, D, E, F, G, H, TMP;
	ecp_point *add_tbl[][3] =
	{
	{ &O, &O, &O }, { &O, &A, &A }, { &A, &O, &A },
	{ &A, &A, &O }, { &B, &C, &O }, { &C, &B, &O },
	{ &A, &D, &E }, { &D, &A, &E },
	{ &B, &D, &F }, { &D, &B, &F },
	{ &D, &D, &G }, { &B, &B, &H },
	};
	mpi m;
	ecp_point mul_tbl[TEST_GROUP_ORDER + 1];
	char *mul_tbl_s[TEST_GROUP_ORDER - 1][2] =
	{
	{ "17", "42" },
	{ "20", "01" },
	{ "14", "11" },
	{ "34", "33" },
	{ "21", "32" },
	{ "27", "30" },
	{ "27", "17" },
	{ "21", "15" },
	{ "34", "14" },
	{ "14", "36" },
	{ "20", "46" },
	{ "17", "05" },
	};

	ecp_group_init( &grp );

	ecp_point_init( &O ); ecp_point_init( &A ); ecp_point_init( &B );
	ecp_point_init( &C ); ecp_point_init( &D ); ecp_point_init( &E );
	ecp_point_init( &F ); ecp_point_init( &G ); ecp_point_init( &H );
	ecp_point_init( &TMP );

	mpi_init( &m );

	for( i = 0; i <= TEST_GROUP_ORDER; i++ )
	ecp_point_init( &mul_tbl[i] );

	ecp_set_zero( &O );
	MPI_CHK( ecp_group_read_string( &grp, 10, "47", "4", "17", "42", "13" ) );
	MPI_CHK( ecp_point_read_string( &A, 10, "13", "00" ) );
	MPI_CHK( ecp_point_read_string( &B, 10, "14", "11" ) );
	MPI_CHK( ecp_point_read_string( &C, 10, "14", "36" ) );
	MPI_CHK( ecp_point_read_string( &D, 10, "37", "31" ) );
	MPI_CHK( ecp_point_read_string( &E, 10, "34", "14" ) );
	MPI_CHK( ecp_point_read_string( &F, 10, "45", "07" ) );
	MPI_CHK( ecp_point_read_string( &G, 10, "21", "32" ) );
	MPI_CHK( ecp_point_read_string( &H, 10, "27", "30" ) );

	if( verbose != 0 )
	printf( " ECP test #1 (ecp_add): " );

	for( i = 0; i < sizeof( add_tbl ) / sizeof( add_tbl[0] ); i++ )
	{
	MPI_CHK( ecp_add( &grp, &TMP, add_tbl[i][0], add_tbl[i][1] ) );
	if( ! ecp_point_eq( &TMP, add_tbl[i][2] ) )
	{
	if( verbose != 0 )
	{
	printf( " failed (%u)\n", i );
	printf( " GOT: " );
	ecp_point_print( &TMP );
	printf( " EXPECTED: " );
	ecp_point_print( add_tbl[i][2] );
	}

	return( 1 );
	}
	}

	if (verbose != 0 )
	printf( "passed\n" );

	MPI_CHK( ecp_copy( &mul_tbl[0], &O ) );
	for( i = 1; i <= TEST_GROUP_ORDER - 1; i++ )
	MPI_CHK( ecp_point_read_string( &mul_tbl[i], 10,
	mul_tbl_s[i-1][0], mul_tbl_s[i-1][1] ) );
	MPI_CHK( ecp_copy( &mul_tbl[TEST_GROUP_ORDER], &O ) );

	if( verbose != 0 )
	printf( " ECP test #2 (ecp_mul): " );

	for( i = 0; i <= TEST_GROUP_ORDER; i++ )
	{
	MPI_CHK( mpi_lset( &m, i ) );
	MPI_CHK( ecp_mul( &grp, &TMP, &m, &grp.G ) );
	if( ! ecp_point_eq( &TMP, &mul_tbl[i] ) )
	{
	if( verbose != 0 )
	{
	printf( " failed (%u)\n", i );
	printf( " GOT: " );
	ecp_point_print( &TMP );
	printf( " EXPECTED: " );
	ecp_point_print( &mul_tbl[i] );
	}

	return( 1 );
	}
	}

	if (verbose != 0 )
	printf( "passed\n" );

	cleanup:

	if( ret != 0 && verbose != 0 )
	{
	#if defined(POLARSSL_ERROR_C)
	char error_buf[200];
	error_strerror( ret, error_buf, 200 );
	printf( "Unexpected error: %d - %s\n\n", ret, error_buf );
	#else
	printf( "Unexpected error: %08X\n", ret );
	#endif
	}

	ecp_group_free( &grp ); ecp_point_free( &O ); ecp_point_free( &TMP );

	ecp_point_free( &A ); ecp_point_free( &B ); ecp_point_free( &C );
	ecp_point_free( &D ); ecp_point_free( &E ); ecp_point_free( &F );
	ecp_point_free( &G ); ecp_point_free( &H );

	mpi_free( &m );

	for( i = 0; i <= TEST_GROUP_ORDER; i++ )
	ecp_point_free( &mul_tbl[i] );

	if( verbose != 0 )
	printf( "\n" );

	return( ret );
	}

	#endif

	#endif