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d070f51224
mbedtls/library/ecp.c
Manuel Pégourié-Gonnard d070f51224 Started introducting Jacobian coordinates
2013-01-16 16:31:51 +01:00

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/*
* 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
* GECC = 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 );
}
/*
* Internal point format used for fast addition/doubling/multiplication:
* Jacobian coordinates (GECC example 3.20)
*/
typedef struct
{
mpi X, Y, Z;
}
ecp_ptjac;
/*
* Convert from affine to Jacobian coordinates
*/
static int ecp_aff_to_jac( ecp_ptjac *jac, ecp_point *aff )
{
int ret = 0;
if( aff->is_zero )
{
MPI_CHK( mpi_lset( &jac->X, 1 ) );
MPI_CHK( mpi_lset( &jac->Y, 1 ) );
MPI_CHK( mpi_lset( &jac->Z, 0 ) );
}
else
{
MPI_CHK( mpi_copy( &jac->X, &aff->X ) );
MPI_CHK( mpi_copy( &jac->Y, &aff->Y ) );
MPI_CHK( mpi_lset( &jac->Z, 1 ) );
}
cleanup:
return( ret );
}
/*
* Convert from Jacobian to affine coordinates
*/
static int ecp_jac_to_aff( const ecp_group *grp,
ecp_point *aff, ecp_ptjac *jac )
{
int ret = 0;
mpi Zi, ZZi, T;
if( mpi_cmp_int( &jac->Z, 0 ) == 0 ) {
ecp_set_zero( aff );
return( 0 );
}
mpi_init( &Zi ); mpi_init( &ZZi ); mpi_init( &T );
aff->is_zero = 0;
/*
* aff.X = jac.X / (jac.Z)^2 mod p
*/
MPI_CHK( mpi_inv_mod( &Zi, &jac->Z, &grp->P ) );
MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) );
MPI_CHK( mpi_mul_mpi( &T, &jac->X, &ZZi ) );
MPI_CHK( mpi_mod_mpi( &aff->X, &T, &grp->P ) );
/*
* aff.Y = jac.Y / (jac.Z)^3 mod p
*/
MPI_CHK( mpi_mul_mpi( &T, &jac->Y, &ZZi ) );
MPI_CHK( mpi_mul_mpi( &T, &T, &Zi ) );
MPI_CHK( mpi_mod_mpi( &aff->Y, &T, &grp->P ) );
cleanup:
mpi_free( &Zi ); mpi_free( &ZZi ); mpi_free( &T );
return( ret );
}
/*
* 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
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