539 lines
14 KiB
C
539 lines
14 KiB
C
/*
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* Elliptic curves over GF(p)
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*
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* Copyright (C) 2012, Brainspark B.V.
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*
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* This file is part of PolarSSL (http://www.polarssl.org)
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* Lead Maintainer: Paul Bakker <polarssl_maintainer at polarssl.org>
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*
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* All rights reserved.
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 2 of the License, or
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* (at your option) any later version.
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*
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* This program is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License along
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* with this program; if not, write to the Free Software Foundation, Inc.,
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* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
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*/
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/*
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* References:
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*
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* SEC1 http://www.secg.org/index.php?action=secg,docs_secg
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* GECC = Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
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*/
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#include "polarssl/config.h"
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#if defined(POLARSSL_ECP_C)
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#include "polarssl/ecp.h"
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/*
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* Initialize (the components of) a point
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*/
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void ecp_point_init( ecp_point *pt )
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{
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if( pt == NULL )
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return;
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pt->is_zero = 1;
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mpi_init( &pt->X );
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mpi_init( &pt->Y );
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}
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/*
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* Initialize (the components of) a group
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*/
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void ecp_group_init( ecp_group *grp )
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{
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if( grp == NULL )
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return;
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mpi_init( &grp->P );
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mpi_init( &grp->B );
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ecp_point_init( &grp->G );
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mpi_init( &grp->N );
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}
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/*
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* Unallocate (the components of) a point
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*/
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void ecp_point_free( ecp_point *pt )
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{
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if( pt == NULL )
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return;
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pt->is_zero = 1;
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mpi_free( &( pt->X ) );
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mpi_free( &( pt->Y ) );
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}
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/*
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* Unallocate (the components of) a group
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*/
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void ecp_group_free( ecp_group *grp )
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{
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if( grp == NULL )
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return;
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mpi_free( &grp->P );
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mpi_free( &grp->B );
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ecp_point_free( &grp->G );
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mpi_free( &grp->N );
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}
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/*
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* Set point to zero
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*/
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void ecp_set_zero( ecp_point *pt )
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{
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pt->is_zero = 1;
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mpi_free( &pt->X );
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mpi_free( &pt->Y );
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}
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/*
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* Copy the contents of Q into P
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*/
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int ecp_copy( ecp_point *P, const ecp_point *Q )
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{
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int ret = 0;
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if( Q->is_zero ) {
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ecp_set_zero( P );
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return( ret );
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}
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P->is_zero = Q->is_zero;
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MPI_CHK( mpi_copy( &P->X, &Q->X ) );
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MPI_CHK( mpi_copy( &P->Y, &Q->Y ) );
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cleanup:
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return( ret );
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}
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/*
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* Import a non-zero point from ASCII strings
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*/
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int ecp_point_read_string( ecp_point *P, int radix,
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const char *x, const char *y )
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{
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int ret = 0;
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P->is_zero = 0;
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MPI_CHK( mpi_read_string( &P->X, radix, x ) );
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MPI_CHK( mpi_read_string( &P->Y, radix, y ) );
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cleanup:
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return( ret );
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}
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/*
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* Import an ECP group from ASCII strings
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*/
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int ecp_group_read_string( ecp_group *grp, int radix,
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const char *p, const char *b,
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const char *gx, const char *gy, const char *n)
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{
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int ret = 0;
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MPI_CHK( mpi_read_string( &grp->P, radix, p ) );
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MPI_CHK( mpi_read_string( &grp->B, radix, b ) );
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MPI_CHK( ecp_point_read_string( &grp->G, radix, gx, gy ) );
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MPI_CHK( mpi_read_string( &grp->N, radix, n ) );
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cleanup:
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return( ret );
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}
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/*
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* Set a group using well-known domain parameters
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*/
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int ecp_use_known_dp( ecp_group *grp, size_t index )
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{
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switch( index )
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{
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case POLARSSL_ECP_DP_SECP192R1:
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return( ecp_group_read_string( grp, 16,
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POLARSSL_ECP_SECP192R1_P,
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POLARSSL_ECP_SECP192R1_B,
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POLARSSL_ECP_SECP192R1_GX,
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POLARSSL_ECP_SECP192R1_GY,
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POLARSSL_ECP_SECP192R1_N )
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);
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case POLARSSL_ECP_DP_SECP224R1:
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return( ecp_group_read_string( grp, 16,
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POLARSSL_ECP_SECP224R1_P,
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POLARSSL_ECP_SECP224R1_B,
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POLARSSL_ECP_SECP224R1_GX,
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POLARSSL_ECP_SECP224R1_GY,
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POLARSSL_ECP_SECP224R1_N )
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);
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case POLARSSL_ECP_DP_SECP256R1:
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return( ecp_group_read_string( grp, 16,
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POLARSSL_ECP_SECP256R1_P,
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POLARSSL_ECP_SECP256R1_B,
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POLARSSL_ECP_SECP256R1_GX,
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POLARSSL_ECP_SECP256R1_GY,
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POLARSSL_ECP_SECP256R1_N )
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);
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case POLARSSL_ECP_DP_SECP384R1:
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return( ecp_group_read_string( grp, 16,
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POLARSSL_ECP_SECP384R1_P,
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POLARSSL_ECP_SECP384R1_B,
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POLARSSL_ECP_SECP384R1_GX,
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POLARSSL_ECP_SECP384R1_GY,
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POLARSSL_ECP_SECP384R1_N )
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);
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case POLARSSL_ECP_DP_SECP521R1:
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return( ecp_group_read_string( grp, 16,
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POLARSSL_ECP_SECP521R1_P,
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POLARSSL_ECP_SECP521R1_B,
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POLARSSL_ECP_SECP521R1_GX,
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POLARSSL_ECP_SECP521R1_GY,
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POLARSSL_ECP_SECP521R1_N )
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);
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}
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return( POLARSSL_ERR_ECP_GENERIC );
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}
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/*
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* Reduce a mpi mod p in-place, general case, to use after mpi_mul_mpi
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*/
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#define MOD_MUL( N ) MPI_CHK( mpi_mod_mpi( &N, &N, &grp->P ) )
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/*
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* Reduce a mpi mod p in-place, to use after mpi_sub_mpi
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*/
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#define MOD_SUB( N ) \
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while( mpi_cmp_int( &N, 0 ) < 0 ) \
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MPI_CHK( mpi_add_mpi( &N, &N, &grp->P ) )
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/*
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* Reduce a mpi mod p in-place, to use after mpi_add_mpi and mpi_mul_int
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*/
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#define MOD_ADD( N ) \
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while( mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \
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MPI_CHK( mpi_sub_mpi( &N, &N, &grp->P ) )
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/*
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* Internal point format used for fast addition/doubling/multiplication:
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* Jacobian coordinates (GECC example 3.20)
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*/
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typedef struct
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{
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mpi X, Y, Z;
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}
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ecp_ptjac;
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/*
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* Initialize a point in Jacobian coordinates
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*/
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static void ecp_ptjac_init( ecp_ptjac *P )
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{
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mpi_init( &P->X ); mpi_init( &P->Y ); mpi_init( &P->Z );
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}
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/*
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* Free a point in Jacobian coordinates
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*/
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static void ecp_ptjac_free( ecp_ptjac *P )
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{
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mpi_free( &P->X ); mpi_free( &P->Y ); mpi_free( &P->Z );
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}
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/*
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* Copy P to R in Jacobian coordinates
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*/
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static int ecp_ptjac_copy( ecp_ptjac *R, const ecp_ptjac *P )
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{
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int ret = 0;
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MPI_CHK( mpi_copy( &R->X, &P->X ) );
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MPI_CHK( mpi_copy( &R->Y, &P->Y ) );
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MPI_CHK( mpi_copy( &R->Z, &P->Z ) );
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cleanup:
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return( ret );
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}
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/*
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* Set P to zero in Jacobian coordinates
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*/
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static int ecp_ptjac_set_zero( ecp_ptjac *P )
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{
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int ret = 0;
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MPI_CHK( mpi_lset( &P->X, 1 ) );
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MPI_CHK( mpi_lset( &P->Y, 1 ) );
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MPI_CHK( mpi_lset( &P->Z, 0 ) );
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cleanup:
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return( ret );
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}
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/*
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* Convert from affine to Jacobian coordinates
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*/
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static int ecp_aff_to_jac( ecp_ptjac *jac, const ecp_point *aff )
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{
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int ret = 0;
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if( aff->is_zero )
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return( ecp_ptjac_set_zero( jac ) );
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MPI_CHK( mpi_copy( &jac->X, &aff->X ) );
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MPI_CHK( mpi_copy( &jac->Y, &aff->Y ) );
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MPI_CHK( mpi_lset( &jac->Z, 1 ) );
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cleanup:
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return( ret );
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}
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/*
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* Convert from Jacobian to affine coordinates
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*/
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static int ecp_jac_to_aff( const ecp_group *grp,
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ecp_point *aff, const ecp_ptjac *jac )
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{
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int ret = 0;
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mpi Zi, ZZi, T;
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if( mpi_cmp_int( &jac->Z, 0 ) == 0 ) {
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ecp_set_zero( aff );
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return( 0 );
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}
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mpi_init( &Zi ); mpi_init( &ZZi ); mpi_init( &T );
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aff->is_zero = 0;
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/*
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* aff.X = jac.X / (jac.Z)^2 mod p
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*/
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MPI_CHK( mpi_inv_mod( &Zi, &jac->Z, &grp->P ) );
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MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
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MPI_CHK( mpi_mul_mpi( &aff->X, &jac->X, &ZZi ) ); MOD_MUL( aff->X );
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/*
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* aff.Y = jac.Y / (jac.Z)^3 mod p
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*/
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MPI_CHK( mpi_mul_mpi( &aff->Y, &jac->Y, &ZZi ) ); MOD_MUL( aff->Y );
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MPI_CHK( mpi_mul_mpi( &aff->Y, &aff->Y, &Zi ) ); MOD_MUL( aff->Y );
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cleanup:
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mpi_free( &Zi ); mpi_free( &ZZi ); mpi_free( &T );
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return( ret );
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}
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/*
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* Point doubling R = 2 P, Jacobian coordinates (GECC 3.21)
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*/
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static int ecp_double_jac( const ecp_group *grp, ecp_ptjac *R,
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const ecp_ptjac *P )
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{
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int ret = 0;
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mpi T1, T2, T3, X, Y, Z;
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if( mpi_cmp_int( &P->Z, 0 ) == 0 )
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return( ecp_ptjac_set_zero( R ) );
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mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 );
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mpi_init( &X ); mpi_init( &Y ); mpi_init( &Z );
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MPI_CHK( mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 );
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MPI_CHK( mpi_sub_mpi( &T2, &P->X, &T1 ) ); MOD_SUB( T2 );
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MPI_CHK( mpi_add_mpi( &T1, &P->X, &T1 ) ); MOD_ADD( T1 );
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MPI_CHK( mpi_mul_mpi( &T2, &T2, &T1 ) ); MOD_MUL( T2 );
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MPI_CHK( mpi_mul_int( &T2, &T2, 3 ) ); MOD_ADD( T2 );
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MPI_CHK( mpi_copy ( &Y, &P->Y ) );
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MPI_CHK( mpi_shift_l( &Y, 1 ) ); MOD_ADD( Y );
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MPI_CHK( mpi_mul_mpi( &Z, &Y, &P->Z ) ); MOD_MUL( Z );
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MPI_CHK( mpi_mul_mpi( &Y, &Y, &Y ) ); MOD_MUL( Y );
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MPI_CHK( mpi_mul_mpi( &T3, &Y, &P->X ) ); MOD_MUL( T3 );
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MPI_CHK( mpi_mul_mpi( &Y, &Y, &Y ) ); MOD_MUL( Y );
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/*
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* For Y = Y / 2 mod p, we must make sure that Y is even before
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* using right-shift. No need to reduce mod p afterwards.
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*/
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if( mpi_get_bit( &Y, 0 ) == 1 )
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MPI_CHK( mpi_add_mpi( &Y, &Y, &grp->P ) );
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MPI_CHK( mpi_shift_r( &Y, 1 ) );
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MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X );
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MPI_CHK( mpi_copy ( &T1, &T3 ) );
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MPI_CHK( mpi_shift_l( &T1, 1 ) ); MOD_ADD( T1 );
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MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X );
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MPI_CHK( mpi_sub_mpi( &T1, &T3, &X ) ); MOD_SUB( T1 );
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MPI_CHK( mpi_mul_mpi( &T1, &T1, &T2 ) ); MOD_MUL( T1 );
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MPI_CHK( mpi_sub_mpi( &Y, &T1, &Y ) ); MOD_SUB( Y );
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MPI_CHK( mpi_copy( &R->X, &X ) );
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MPI_CHK( mpi_copy( &R->Y, &Y ) );
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MPI_CHK( mpi_copy( &R->Z, &Z ) );
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cleanup:
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mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 );
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mpi_free( &X ); mpi_free( &Y ); mpi_free( &Z );
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return( ret );
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}
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/*
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* Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22)
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*/
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static int ecp_add_mixed( const ecp_group *grp, ecp_ptjac *R,
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const ecp_ptjac *P, const ecp_point *Q )
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{
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int ret = 0;
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mpi T1, T2, T3, T4, X, Y, Z;
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/*
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* Trivial cases: P == 0 or Q == 0
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*/
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if( mpi_cmp_int( &P->Z, 0 ) == 0 )
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return( ecp_aff_to_jac( R, Q ) );
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if( Q->is_zero )
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return( ecp_ptjac_copy( R, P ) );
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mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 ); mpi_init( &T4 );
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mpi_init( &X ); mpi_init( &Y ); mpi_init( &Z );
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MPI_CHK( mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 );
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MPI_CHK( mpi_mul_mpi( &T2, &T1, &P->Z ) ); MOD_MUL( T2 );
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MPI_CHK( mpi_mul_mpi( &T1, &T1, &Q->X ) ); MOD_MUL( T1 );
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MPI_CHK( mpi_mul_mpi( &T2, &T2, &Q->Y ) ); MOD_MUL( T2 );
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MPI_CHK( mpi_sub_mpi( &T1, &T1, &P->X ) ); MOD_SUB( T1 );
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MPI_CHK( mpi_sub_mpi( &T2, &T2, &P->Y ) ); MOD_SUB( T2 );
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if( mpi_cmp_int( &T1, 0 ) == 0 )
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{
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if( mpi_cmp_int( &T2, 0 ) == 0 )
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{
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ret = ecp_double_jac( grp, R, P );
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goto cleanup;
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}
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else
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{
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ret = ecp_ptjac_set_zero( R );
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goto cleanup;
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}
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}
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MPI_CHK( mpi_mul_mpi( &Z, &P->Z, &T1 ) ); MOD_MUL( Z );
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MPI_CHK( mpi_mul_mpi( &T3, &T1, &T1 ) ); MOD_MUL( T3 );
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MPI_CHK( mpi_mul_mpi( &T4, &T3, &T1 ) ); MOD_MUL( T4 );
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MPI_CHK( mpi_mul_mpi( &T3, &T3, &P->X ) ); MOD_MUL( T3 );
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MPI_CHK( mpi_mul_int( &T1, &T3, 2 ) ); MOD_ADD( T1 );
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MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X );
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MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X );
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MPI_CHK( mpi_sub_mpi( &X, &X, &T4 ) ); MOD_SUB( X );
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MPI_CHK( mpi_sub_mpi( &T3, &T3, &X ) ); MOD_SUB( T3 );
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MPI_CHK( mpi_mul_mpi( &T3, &T3, &T2 ) ); MOD_MUL( T3 );
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MPI_CHK( mpi_mul_mpi( &T4, &T4, &P->Y ) ); MOD_MUL( T4 );
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MPI_CHK( mpi_sub_mpi( &Y, &T3, &T4 ) ); MOD_SUB( Y );
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MPI_CHK( mpi_copy( &R->X, &X ) );
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MPI_CHK( mpi_copy( &R->Y, &Y ) );
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MPI_CHK( mpi_copy( &R->Z, &Z ) );
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cleanup:
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|
|
mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 ); mpi_free( &T4 );
|
|
mpi_free( &X ); mpi_free( &Y ); mpi_free( &Z );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Addition: R = P + Q, affine wrapper
|
|
*/
|
|
int ecp_add( const ecp_group *grp, ecp_point *R,
|
|
const ecp_point *P, const ecp_point *Q )
|
|
{
|
|
int ret = 0;
|
|
ecp_ptjac J;
|
|
|
|
ecp_ptjac_init( &J );
|
|
|
|
MPI_CHK( ecp_aff_to_jac( &J, P ) );
|
|
MPI_CHK( ecp_add_mixed( grp, &J, &J, Q ) );
|
|
MPI_CHK( ecp_jac_to_aff( grp, R, &J ) );
|
|
|
|
cleanup:
|
|
|
|
ecp_ptjac_free( &J );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Integer multiplication: R = m * P (GECC 5.7, SPA-resistant variant)
|
|
*/
|
|
int ecp_mul( const ecp_group *grp, ecp_point *R,
|
|
const mpi *m, const ecp_point *P )
|
|
{
|
|
int ret = 0;
|
|
size_t pos;
|
|
ecp_ptjac Q[2];
|
|
|
|
ecp_ptjac_init( &Q[0] ); ecp_ptjac_init( &Q[1] );
|
|
|
|
/*
|
|
* The general method works only for m >= 1
|
|
*/
|
|
if( mpi_cmp_int( m, 0 ) == 0 ) {
|
|
ecp_set_zero( R );
|
|
goto cleanup;
|
|
}
|
|
|
|
ecp_ptjac_set_zero( &Q[0] );
|
|
|
|
for( pos = mpi_msb( m ) - 1 ; ; pos-- )
|
|
{
|
|
MPI_CHK( ecp_double_jac( grp, &Q[0], &Q[0] ) );
|
|
MPI_CHK( ecp_add_mixed( grp, &Q[1], &Q[0], P ) );
|
|
MPI_CHK( ecp_ptjac_copy( &Q[0], &Q[ mpi_get_bit( m, pos ) ] ) );
|
|
|
|
if( pos == 0 )
|
|
break;
|
|
}
|
|
|
|
MPI_CHK( ecp_jac_to_aff( grp, R, &Q[0] ) );
|
|
|
|
cleanup:
|
|
|
|
ecp_ptjac_free( &Q[0] ); ecp_ptjac_free( &Q[1] );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
|
|
#if defined(POLARSSL_SELF_TEST)
|
|
|
|
/*
|
|
* Checkup routine
|
|
*/
|
|
int ecp_self_test( int verbose )
|
|
{
|
|
return( verbose++ );
|
|
}
|
|
|
|
#endif
|
|
|
|
#endif
|