| /* |
| * 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 ); |
| } |
| |
| /* |
| * Reduce a mpi mod p in-place, general case, to use after mpi_mul_mpi |
| */ |
| #define MOD_MUL( N ) MPI_CHK( mpi_mod_mpi( &N, &N, &grp->P ) ) |
| |
| /* |
| * Reduce a mpi mod p in-place, to use after mpi_sub_mpi |
| */ |
| #define MOD_SUB( N ) \ |
| while( mpi_cmp_int( &N, 0 ) < 0 ) \ |
| MPI_CHK( mpi_add_mpi( &N, &N, &grp->P ) ) |
| |
| /* |
| * Reduce a mpi mod p in-place, to use after mpi_add_mpi and mpi_mul_int |
| */ |
| #define MOD_ADD( N ) \ |
| while( mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \ |
| MPI_CHK( mpi_sub_mpi( &N, &N, &grp->P ) ) |
| |
| /* |
| * Internal point format used for fast addition/doubling/multiplication: |
| * Jacobian coordinates (GECC example 3.20) |
| */ |
| typedef struct |
| { |
| mpi X, Y, Z; |
| } |
| ecp_ptjac; |
| |
| /* |
| * Initialize a point in Jacobian coordinates |
| */ |
| static void ecp_ptjac_init( ecp_ptjac *P ) |
| { |
| mpi_init( &P->X ); mpi_init( &P->Y ); mpi_init( &P->Z ); |
| } |
| |
| /* |
| * Free a point in Jacobian coordinates |
| */ |
| static void ecp_ptjac_free( ecp_ptjac *P ) |
| { |
| mpi_free( &P->X ); mpi_free( &P->Y ); mpi_free( &P->Z ); |
| } |
| |
| /* |
| * Copy P to R in Jacobian coordinates |
| */ |
| static int ecp_ptjac_copy( ecp_ptjac *R, const ecp_ptjac *P ) |
| { |
| int ret = 0; |
| |
| MPI_CHK( mpi_copy( &R->X, &P->X ) ); |
| MPI_CHK( mpi_copy( &R->Y, &P->Y ) ); |
| MPI_CHK( mpi_copy( &R->Z, &P->Z ) ); |
| |
| cleanup: |
| return( ret ); |
| } |
| |
| /* |
| * Set P to zero in Jacobian coordinates |
| */ |
| static int ecp_ptjac_set_zero( ecp_ptjac *P ) |
| { |
| int ret = 0; |
| |
| MPI_CHK( mpi_lset( &P->X, 1 ) ); |
| MPI_CHK( mpi_lset( &P->Y, 1 ) ); |
| MPI_CHK( mpi_lset( &P->Z, 0 ) ); |
| |
| cleanup: |
| return( ret ); |
| } |
| |
| /* |
| * Convert from affine to Jacobian coordinates |
| */ |
| static int ecp_aff_to_jac( ecp_ptjac *jac, const ecp_point *aff ) |
| { |
| int ret = 0; |
| |
| if( aff->is_zero ) |
| return( ecp_ptjac_set_zero( jac ) ); |
| |
| 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, const 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 ) ); MOD_MUL( ZZi ); |
| MPI_CHK( mpi_mul_mpi( &aff->X, &jac->X, &ZZi ) ); MOD_MUL( aff->X ); |
| |
| /* |
| * aff.Y = jac.Y / (jac.Z)^3 mod p |
| */ |
| MPI_CHK( mpi_mul_mpi( &aff->Y, &jac->Y, &ZZi ) ); MOD_MUL( aff->Y ); |
| MPI_CHK( mpi_mul_mpi( &aff->Y, &aff->Y, &Zi ) ); MOD_MUL( aff->Y ); |
| |
| cleanup: |
| |
| mpi_free( &Zi ); mpi_free( &ZZi ); mpi_free( &T ); |
| |
| return( ret ); |
| } |
| |
| /* |
| * Point doubling R = 2 P, Jacobian coordinates (GECC 3.21) |
| */ |
| static int ecp_double_jac( const ecp_group *grp, ecp_ptjac *R, |
| const ecp_ptjac *P ) |
| { |
| int ret = 0; |
| mpi T1, T2, T3, X, Y, Z; |
| |
| if( mpi_cmp_int( &P->Z, 0 ) == 0 ) |
| return( ecp_ptjac_set_zero( R ) ); |
| |
| mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 ); |
| mpi_init( &X ); mpi_init( &Y ); mpi_init( &Z ); |
| |
| MPI_CHK( mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 ); |
| MPI_CHK( mpi_sub_mpi( &T2, &P->X, &T1 ) ); MOD_SUB( T2 ); |
| MPI_CHK( mpi_add_mpi( &T1, &P->X, &T1 ) ); MOD_ADD( T1 ); |
| MPI_CHK( mpi_mul_mpi( &T2, &T2, &T1 ) ); MOD_MUL( T2 ); |
| MPI_CHK( mpi_mul_int( &T2, &T2, 3 ) ); MOD_ADD( T2 ); |
| MPI_CHK( mpi_copy ( &Y, &P->Y ) ); |
| MPI_CHK( mpi_shift_l( &Y, 1 ) ); MOD_ADD( Y ); |
| MPI_CHK( mpi_mul_mpi( &Z, &Y, &P->Z ) ); MOD_MUL( Z ); |
| MPI_CHK( mpi_mul_mpi( &Y, &Y, &Y ) ); MOD_MUL( Y ); |
| MPI_CHK( mpi_mul_mpi( &T3, &Y, &P->X ) ); MOD_MUL( T3 ); |
| MPI_CHK( mpi_mul_mpi( &Y, &Y, &Y ) ); MOD_MUL( Y ); |
| |
| /* |
| * For Y = Y / 2 mod p, we must make sure that Y is even before |
| * using right-shift. No need to reduce mod p afterwards. |
| */ |
| if( mpi_get_bit( &Y, 0 ) == 1 ) |
| MPI_CHK( mpi_add_mpi( &Y, &Y, &grp->P ) ); |
| MPI_CHK( mpi_shift_r( &Y, 1 ) ); |
| |
| MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X ); |
| MPI_CHK( mpi_copy ( &T1, &T3 ) ); |
| MPI_CHK( mpi_shift_l( &T1, 1 ) ); MOD_ADD( T1 ); |
| MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X ); |
| MPI_CHK( mpi_sub_mpi( &T1, &T3, &X ) ); MOD_SUB( T1 ); |
| MPI_CHK( mpi_mul_mpi( &T1, &T1, &T2 ) ); MOD_MUL( T1 ); |
| MPI_CHK( mpi_sub_mpi( &Y, &T1, &Y ) ); MOD_SUB( Y ); |
| |
| MPI_CHK( mpi_copy( &R->X, &X ) ); |
| MPI_CHK( mpi_copy( &R->Y, &Y ) ); |
| MPI_CHK( mpi_copy( &R->Z, &Z ) ); |
| |
| cleanup: |
| |
| mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 ); |
| mpi_free( &X ); mpi_free( &Y ); mpi_free( &Z ); |
| |
| return( ret ); |
| } |
| |
| /* |
| * Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22) |
| */ |
| static int ecp_add_mixed( const ecp_group *grp, ecp_ptjac *R, |
| const ecp_ptjac *P, const ecp_point *Q ) |
| { |
| int ret = 0; |
| mpi T1, T2, T3, T4, X, Y, Z; |
| |
| /* |
| * Trivial cases: P == 0 or Q == 0 |
| */ |
| if( mpi_cmp_int( &P->Z, 0 ) == 0 ) |
| return( ecp_aff_to_jac( R, Q ) ); |
| |
| if( Q->is_zero ) |
| return( ecp_ptjac_copy( R, P ) ); |
| |
| mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 ); mpi_init( &T4 ); |
| mpi_init( &X ); mpi_init( &Y ); mpi_init( &Z ); |
| |
| MPI_CHK( mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 ); |
| MPI_CHK( mpi_mul_mpi( &T2, &T1, &P->Z ) ); MOD_MUL( T2 ); |
| MPI_CHK( mpi_mul_mpi( &T1, &T1, &Q->X ) ); MOD_MUL( T1 ); |
| MPI_CHK( mpi_mul_mpi( &T2, &T2, &Q->Y ) ); MOD_MUL( T2 ); |
| MPI_CHK( mpi_sub_mpi( &T1, &T1, &P->X ) ); MOD_SUB( T1 ); |
| MPI_CHK( mpi_sub_mpi( &T2, &T2, &P->Y ) ); MOD_SUB( T2 ); |
| |
| if( mpi_cmp_int( &T1, 0 ) == 0 ) |
| { |
| if( mpi_cmp_int( &T2, 0 ) == 0 ) |
| { |
| ret = ecp_double_jac( grp, R, P ); |
| goto cleanup; |
| } |
| else |
| { |
| ret = ecp_ptjac_set_zero( R ); |
| goto cleanup; |
| } |
| } |
| |
| MPI_CHK( mpi_mul_mpi( &Z, &P->Z, &T1 ) ); MOD_MUL( Z ); |
| MPI_CHK( mpi_mul_mpi( &T3, &T1, &T1 ) ); MOD_MUL( T3 ); |
| MPI_CHK( mpi_mul_mpi( &T4, &T3, &T1 ) ); MOD_MUL( T4 ); |
| MPI_CHK( mpi_mul_mpi( &T3, &T3, &P->X ) ); MOD_MUL( T3 ); |
| MPI_CHK( mpi_mul_int( &T1, &T3, 2 ) ); MOD_ADD( T1 ); |
| MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X ); |
| MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X ); |
| MPI_CHK( mpi_sub_mpi( &X, &X, &T4 ) ); MOD_SUB( X ); |
| MPI_CHK( mpi_sub_mpi( &T3, &T3, &X ) ); MOD_SUB( T3 ); |
| MPI_CHK( mpi_mul_mpi( &T3, &T3, &T2 ) ); MOD_MUL( T3 ); |
| MPI_CHK( mpi_mul_mpi( &T4, &T4, &P->Y ) ); MOD_MUL( T4 ); |
| MPI_CHK( mpi_sub_mpi( &Y, &T3, &T4 ) ); MOD_SUB( Y ); |
| |
| MPI_CHK( mpi_copy( &R->X, &X ) ); |
| MPI_CHK( mpi_copy( &R->Y, &Y ) ); |
| MPI_CHK( mpi_copy( &R->Z, &Z ) ); |
| |
| cleanup: |
| |
| 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 |