| /* |
| * 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) |
| |
| #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" ); |
| |
| if( verbose != 0 ) |
| printf( " ECP test #3 (use_known_dp): " ); |
| |
| for( i = 0; i <= POLARSSL_ECP_DP_SECP521R1; i++ ) |
| { |
| MPI_CHK( ecp_use_known_dp( &grp, i ) ); |
| } |
| |
| 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 |