/* * Elliptic curves over GF(p): generic functions * * Copyright (C) 2006-2014, Brainspark B.V. * * This file is part of PolarSSL (http://www.polarssl.org) * Lead Maintainer: Paul Bakker * * 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 * FIPS 186-3 http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf * RFC 4492 for the related TLS structures and constants * * [M255] http://cr.yp.to/ecdh/curve25519-20060209.pdf * * [2] CORON, Jean-Sébastien. Resistance against differential power analysis * for elliptic curve cryptosystems. In : Cryptographic Hardware and * Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302. * * * [3] HEDABOU, Mustapha, PINEL, Pierre, et BÉNÉTEAU, Lucien. A comb method to * render ECC resistant against Side Channel Attacks. IACR Cryptology * ePrint Archive, 2004, vol. 2004, p. 342. * */ #if !defined(POLARSSL_CONFIG_FILE) #include "polarssl/config.h" #else #include POLARSSL_CONFIG_FILE #endif #if defined(POLARSSL_ECP_C) #include "polarssl/ecp.h" #if defined(POLARSSL_PLATFORM_C) #include "polarssl/platform.h" #else #define polarssl_printf printf #define polarssl_malloc malloc #define polarssl_free free #endif #include #if defined(_MSC_VER) && !defined strcasecmp && !defined(EFIX64) && \ !defined(EFI32) #define strcasecmp _stricmp #endif #if defined(_MSC_VER) && !defined(inline) #define inline _inline #else #if defined(__ARMCC_VERSION) && !defined(inline) #define inline __inline #endif /* __ARMCC_VERSION */ #endif /*_MSC_VER */ /* Implementation that should never be optimized out by the compiler */ static void polarssl_zeroize( void *v, size_t n ) { volatile unsigned char *p = v; while( n-- ) *p++ = 0; } #if defined(POLARSSL_SELF_TEST) /* * Counts of point addition and doubling, and field multiplications. * Used to test resistance of point multiplication to simple timing attacks. */ static unsigned long add_count, dbl_count, mul_count; #endif #if defined(POLARSSL_ECP_DP_SECP192R1_ENABLED) || \ defined(POLARSSL_ECP_DP_SECP224R1_ENABLED) || \ defined(POLARSSL_ECP_DP_SECP256R1_ENABLED) || \ defined(POLARSSL_ECP_DP_SECP384R1_ENABLED) || \ defined(POLARSSL_ECP_DP_SECP521R1_ENABLED) || \ defined(POLARSSL_ECP_DP_BP256R1_ENABLED) || \ defined(POLARSSL_ECP_DP_BP384R1_ENABLED) || \ defined(POLARSSL_ECP_DP_BP512R1_ENABLED) || \ defined(POLARSSL_ECP_DP_SECP192K1_ENABLED) || \ defined(POLARSSL_ECP_DP_SECP224K1_ENABLED) || \ defined(POLARSSL_ECP_DP_SECP256K1_ENABLED) #define POLARSSL_ECP_SHORT_WEIERSTRASS #endif #if defined(POLARSSL_ECP_DP_M221_ENABLED) || \ defined(POLARSSL_ECP_DP_M255_ENABLED) || \ defined(POLARSSL_ECP_DP_M383_ENABLED) || \ defined(POLARSSL_ECP_DP_M511_ENABLED) #define POLARSSL_ECP_MONTGOMERY #endif /* * Curve types: internal for now, might be exposed later */ typedef enum { POLARSSL_ECP_TYPE_NONE = 0, POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS, /* y^2 = x^3 + a x + b */ POLARSSL_ECP_TYPE_MONTGOMERY, /* y^2 = x^3 + a x^2 + x */ } ecp_curve_type; /* * List of supported curves: * - internal ID * - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2) * - size in bits * - readable name * * Curves are listed in order: largest curves first, and for a given size, * fastest curves first. This provides the default order for the SSL module. */ static const ecp_curve_info ecp_supported_curves[] = { #if defined(POLARSSL_ECP_DP_SECP521R1_ENABLED) { POLARSSL_ECP_DP_SECP521R1, 25, 521, "secp521r1" }, #endif #if defined(POLARSSL_ECP_DP_BP512R1_ENABLED) { POLARSSL_ECP_DP_BP512R1, 28, 512, "brainpoolP512r1" }, #endif #if defined(POLARSSL_ECP_DP_SECP384R1_ENABLED) { POLARSSL_ECP_DP_SECP384R1, 24, 384, "secp384r1" }, #endif #if defined(POLARSSL_ECP_DP_BP384R1_ENABLED) { POLARSSL_ECP_DP_BP384R1, 27, 384, "brainpoolP384r1" }, #endif #if defined(POLARSSL_ECP_DP_SECP256R1_ENABLED) { POLARSSL_ECP_DP_SECP256R1, 23, 256, "secp256r1" }, #endif #if defined(POLARSSL_ECP_DP_SECP256K1_ENABLED) { POLARSSL_ECP_DP_SECP256K1, 22, 256, "secp256k1" }, #endif #if defined(POLARSSL_ECP_DP_BP256R1_ENABLED) { POLARSSL_ECP_DP_BP256R1, 26, 256, "brainpoolP256r1" }, #endif #if defined(POLARSSL_ECP_DP_SECP224R1_ENABLED) { POLARSSL_ECP_DP_SECP224R1, 21, 224, "secp224r1" }, #endif #if defined(POLARSSL_ECP_DP_SECP224K1_ENABLED) { POLARSSL_ECP_DP_SECP224K1, 20, 224, "secp224k1" }, #endif #if defined(POLARSSL_ECP_DP_SECP192R1_ENABLED) { POLARSSL_ECP_DP_SECP192R1, 19, 192, "secp192r1" }, #endif #if defined(POLARSSL_ECP_DP_SECP192K1_ENABLED) { POLARSSL_ECP_DP_SECP192K1, 18, 192, "secp192k1" }, #endif { POLARSSL_ECP_DP_NONE, 0, 0, NULL }, }; #define ECP_NB_CURVES sizeof( ecp_supported_curves ) / \ sizeof( ecp_supported_curves[0] ) static ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES]; /* * List of supported curves and associated info */ const ecp_curve_info *ecp_curve_list( void ) { return( ecp_supported_curves ); } /* * List of supported curves, group ID only */ const ecp_group_id *ecp_grp_id_list( void ) { static int init_done = 0; if( ! init_done ) { size_t i = 0; const ecp_curve_info *curve_info; for( curve_info = ecp_curve_list(); curve_info->grp_id != POLARSSL_ECP_DP_NONE; curve_info++ ) { ecp_supported_grp_id[i++] = curve_info->grp_id; } ecp_supported_grp_id[i] = POLARSSL_ECP_DP_NONE; init_done = 1; } return( ecp_supported_grp_id ); } /* * Get the curve info for the internal identifier */ const ecp_curve_info *ecp_curve_info_from_grp_id( ecp_group_id grp_id ) { const ecp_curve_info *curve_info; for( curve_info = ecp_curve_list(); curve_info->grp_id != POLARSSL_ECP_DP_NONE; curve_info++ ) { if( curve_info->grp_id == grp_id ) return( curve_info ); } return( NULL ); } /* * Get the curve info from the TLS identifier */ const ecp_curve_info *ecp_curve_info_from_tls_id( uint16_t tls_id ) { const ecp_curve_info *curve_info; for( curve_info = ecp_curve_list(); curve_info->grp_id != POLARSSL_ECP_DP_NONE; curve_info++ ) { if( curve_info->tls_id == tls_id ) return( curve_info ); } return( NULL ); } /* * Get the curve info from the name */ const ecp_curve_info *ecp_curve_info_from_name( const char *name ) { const ecp_curve_info *curve_info; for( curve_info = ecp_curve_list(); curve_info->grp_id != POLARSSL_ECP_DP_NONE; curve_info++ ) { if( strcasecmp( curve_info->name, name ) == 0 ) return( curve_info ); } return( NULL ); } /* * Get the type of a curve */ static inline ecp_curve_type ecp_get_type( const ecp_group *grp ) { if( grp->G.X.p == NULL ) return( POLARSSL_ECP_TYPE_NONE ); if( grp->G.Y.p == NULL ) return( POLARSSL_ECP_TYPE_MONTGOMERY ); else return( POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS ); } /* * Initialize (the components of) a point */ void ecp_point_init( ecp_point *pt ) { if( pt == NULL ) return; mpi_init( &pt->X ); mpi_init( &pt->Y ); mpi_init( &pt->Z ); } /* * Initialize (the components of) a group */ void ecp_group_init( ecp_group *grp ) { if( grp == NULL ) return; memset( grp, 0, sizeof( ecp_group ) ); } /* * Initialize (the components of) a key pair */ void ecp_keypair_init( ecp_keypair *key ) { if( key == NULL ) return; ecp_group_init( &key->grp ); mpi_init( &key->d ); ecp_point_init( &key->Q ); } /* * Unallocate (the components of) a point */ void ecp_point_free( ecp_point *pt ) { if( pt == NULL ) return; mpi_free( &( pt->X ) ); mpi_free( &( pt->Y ) ); mpi_free( &( pt->Z ) ); } /* * Unallocate (the components of) a group */ void ecp_group_free( ecp_group *grp ) { size_t i; if( grp == NULL ) return; if( grp->h != 1 ) { mpi_free( &grp->P ); mpi_free( &grp->A ); mpi_free( &grp->B ); ecp_point_free( &grp->G ); mpi_free( &grp->N ); } if( grp->T != NULL ) { for( i = 0; i < grp->T_size; i++ ) ecp_point_free( &grp->T[i] ); polarssl_free( grp->T ); } polarssl_zeroize( grp, sizeof( ecp_group ) ); } /* * Unallocate (the components of) a key pair */ void ecp_keypair_free( ecp_keypair *key ) { if( key == NULL ) return; ecp_group_free( &key->grp ); mpi_free( &key->d ); ecp_point_free( &key->Q ); } /* * Copy the contents of a point */ int ecp_copy( ecp_point *P, const ecp_point *Q ) { int ret; MPI_CHK( mpi_copy( &P->X, &Q->X ) ); MPI_CHK( mpi_copy( &P->Y, &Q->Y ) ); MPI_CHK( mpi_copy( &P->Z, &Q->Z ) ); cleanup: return( ret ); } /* * Copy the contents of a group object */ int ecp_group_copy( ecp_group *dst, const ecp_group *src ) { return ecp_use_known_dp( dst, src->id ); } /* * Set point to zero */ int ecp_set_zero( ecp_point *pt ) { int ret; MPI_CHK( mpi_lset( &pt->X , 1 ) ); MPI_CHK( mpi_lset( &pt->Y , 1 ) ); MPI_CHK( mpi_lset( &pt->Z , 0 ) ); cleanup: return( ret ); } /* * Tell if a point is zero */ int ecp_is_zero( ecp_point *pt ) { return( mpi_cmp_int( &pt->Z, 0 ) == 0 ); } /* * 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; MPI_CHK( mpi_read_string( &P->X, radix, x ) ); MPI_CHK( mpi_read_string( &P->Y, radix, y ) ); MPI_CHK( mpi_lset( &P->Z, 1 ) ); cleanup: return( ret ); } /* * Export a point into unsigned binary data (SEC1 2.3.3) */ int ecp_point_write_binary( const ecp_group *grp, const ecp_point *P, int format, size_t *olen, unsigned char *buf, size_t buflen ) { int ret = 0; size_t plen; if( format != POLARSSL_ECP_PF_UNCOMPRESSED && format != POLARSSL_ECP_PF_COMPRESSED ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); /* * Common case: P == 0 */ if( mpi_cmp_int( &P->Z, 0 ) == 0 ) { if( buflen < 1 ) return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL ); buf[0] = 0x00; *olen = 1; return( 0 ); } plen = mpi_size( &grp->P ); if( format == POLARSSL_ECP_PF_UNCOMPRESSED ) { *olen = 2 * plen + 1; if( buflen < *olen ) return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL ); buf[0] = 0x04; MPI_CHK( mpi_write_binary( &P->X, buf + 1, plen ) ); MPI_CHK( mpi_write_binary( &P->Y, buf + 1 + plen, plen ) ); } else if( format == POLARSSL_ECP_PF_COMPRESSED ) { *olen = plen + 1; if( buflen < *olen ) return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL ); buf[0] = 0x02 + mpi_get_bit( &P->Y, 0 ); MPI_CHK( mpi_write_binary( &P->X, buf + 1, plen ) ); } cleanup: return( ret ); } /* * Import a point from unsigned binary data (SEC1 2.3.4) */ int ecp_point_read_binary( const ecp_group *grp, ecp_point *pt, const unsigned char *buf, size_t ilen ) { int ret; size_t plen; if( ilen < 1 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); if( buf[0] == 0x00 ) { if( ilen == 1 ) return( ecp_set_zero( pt ) ); else return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); } plen = mpi_size( &grp->P ); if( buf[0] != 0x04 ) return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE ); if( ilen != 2 * plen + 1 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); MPI_CHK( mpi_read_binary( &pt->X, buf + 1, plen ) ); MPI_CHK( mpi_read_binary( &pt->Y, buf + 1 + plen, plen ) ); MPI_CHK( mpi_lset( &pt->Z, 1 ) ); cleanup: return( ret ); } /* * Import a point from a TLS ECPoint record (RFC 4492) * struct { * opaque point <1..2^8-1>; * } ECPoint; */ int ecp_tls_read_point( const ecp_group *grp, ecp_point *pt, const unsigned char **buf, size_t buf_len ) { unsigned char data_len; const unsigned char *buf_start; /* * We must have at least two bytes (1 for length, at least one for data) */ if( buf_len < 2 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); data_len = *(*buf)++; if( data_len < 1 || data_len > buf_len - 1 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); /* * Save buffer start for read_binary and update buf */ buf_start = *buf; *buf += data_len; return ecp_point_read_binary( grp, pt, buf_start, data_len ); } /* * Export a point as a TLS ECPoint record (RFC 4492) * struct { * opaque point <1..2^8-1>; * } ECPoint; */ int ecp_tls_write_point( const ecp_group *grp, const ecp_point *pt, int format, size_t *olen, unsigned char *buf, size_t blen ) { int ret; /* * buffer length must be at least one, for our length byte */ if( blen < 1 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); if( ( ret = ecp_point_write_binary( grp, pt, format, olen, buf + 1, blen - 1) ) != 0 ) return( ret ); /* * write length to the first byte and update total length */ buf[0] = (unsigned char) *olen; ++*olen; return( 0 ); } /* * Import an ECP group from ASCII strings, case A == -3 */ 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; 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 ) ); grp->pbits = mpi_msb( &grp->P ); grp->nbits = mpi_msb( &grp->N ); cleanup: if( ret != 0 ) ecp_group_free( grp ); return( ret ); } /* * Set a group from an ECParameters record (RFC 4492) */ int ecp_tls_read_group( ecp_group *grp, const unsigned char **buf, size_t len ) { uint16_t tls_id; const ecp_curve_info *curve_info; /* * We expect at least three bytes (see below) */ if( len < 3 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); /* * First byte is curve_type; only named_curve is handled */ if( *(*buf)++ != POLARSSL_ECP_TLS_NAMED_CURVE ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); /* * Next two bytes are the namedcurve value */ tls_id = *(*buf)++; tls_id <<= 8; tls_id |= *(*buf)++; if( ( curve_info = ecp_curve_info_from_tls_id( tls_id ) ) == NULL ) return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE ); return ecp_use_known_dp( grp, curve_info->grp_id ); } /* * Write the ECParameters record corresponding to a group (RFC 4492) */ int ecp_tls_write_group( const ecp_group *grp, size_t *olen, unsigned char *buf, size_t blen ) { const ecp_curve_info *curve_info; if( ( curve_info = ecp_curve_info_from_grp_id( grp->id ) ) == NULL ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); /* * We are going to write 3 bytes (see below) */ *olen = 3; if( blen < *olen ) return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL ); /* * First byte is curve_type, always named_curve */ *buf++ = POLARSSL_ECP_TLS_NAMED_CURVE; /* * Next two bytes are the namedcurve value */ buf[0] = curve_info->tls_id >> 8; buf[1] = curve_info->tls_id & 0xFF; return( 0 ); } /* * Wrapper around fast quasi-modp functions, with fall-back to mpi_mod_mpi. * See the documentation of struct ecp_group. * * This function is in the critial loop for ecp_mul, so pay attention to perf. */ static int ecp_modp( mpi *N, const ecp_group *grp ) { int ret; if( grp->modp == NULL ) return( mpi_mod_mpi( N, N, &grp->P ) ); /* N->s < 0 is a much faster test, which fails only if N is 0 */ if( ( N->s < 0 && mpi_cmp_int( N, 0 ) != 0 ) || mpi_msb( N ) > 2 * grp->pbits ) { return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); } MPI_CHK( grp->modp( N ) ); /* N->s < 0 is a much faster test, which fails only if N is 0 */ while( N->s < 0 && mpi_cmp_int( N, 0 ) != 0 ) MPI_CHK( mpi_add_mpi( N, N, &grp->P ) ); while( mpi_cmp_mpi( N, &grp->P ) >= 0 ) /* we known P, N and the result are positive */ MPI_CHK( mpi_sub_abs( N, N, &grp->P ) ); cleanup: return( ret ); } /* * Fast mod-p functions expect their argument to be in the 0..p^2 range. * * In order to guarantee that, we need to ensure that operands of * mpi_mul_mpi are in the 0..p range. So, after each operation we will * bring the result back to this range. * * The following macros are shortcuts for doing that. */ /* * Reduce a mpi mod p in-place, general case, to use after mpi_mul_mpi */ #if defined(POLARSSL_SELF_TEST) #define INC_MUL_COUNT mul_count++; #else #define INC_MUL_COUNT #endif #define MOD_MUL( N ) do { MPI_CHK( ecp_modp( &N, grp ) ); INC_MUL_COUNT } \ while( 0 ) /* * Reduce a mpi mod p in-place, to use after mpi_sub_mpi * N->s < 0 is a very fast test, which fails only if N is 0 */ #define MOD_SUB( N ) \ while( N.s < 0 && 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. * We known P, N and the result are positive, so sub_abs is correct, and * a bit faster. */ #define MOD_ADD( N ) \ while( mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \ MPI_CHK( mpi_sub_abs( &N, &N, &grp->P ) ) #if defined(POLARSSL_ECP_SHORT_WEIERSTRASS) /* * For curves in short Weierstrass form, we do all the internal operations in * Jacobian coordinates. * * For multiplication, we'll use a comb method with coutermeasueres against * SPA, hence timing attacks. */ /* * Normalize jacobian coordinates so that Z == 0 || Z == 1 (GECC 3.2.1) * Cost: 1N := 1I + 3M + 1S */ static int ecp_normalize_jac( const ecp_group *grp, ecp_point *pt ) { int ret; mpi Zi, ZZi; if( mpi_cmp_int( &pt->Z, 0 ) == 0 ) return( 0 ); mpi_init( &Zi ); mpi_init( &ZZi ); /* * X = X / Z^2 mod p */ MPI_CHK( mpi_inv_mod( &Zi, &pt->Z, &grp->P ) ); MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi ); MPI_CHK( mpi_mul_mpi( &pt->X, &pt->X, &ZZi ) ); MOD_MUL( pt->X ); /* * Y = Y / Z^3 mod p */ MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &ZZi ) ); MOD_MUL( pt->Y ); MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &Zi ) ); MOD_MUL( pt->Y ); /* * Z = 1 */ MPI_CHK( mpi_lset( &pt->Z, 1 ) ); cleanup: mpi_free( &Zi ); mpi_free( &ZZi ); return( ret ); } /* * Normalize jacobian coordinates of an array of (pointers to) points, * using Montgomery's trick to perform only one inversion mod P. * (See for example Cohen's "A Course in Computational Algebraic Number * Theory", Algorithm 10.3.4.) * * Warning: fails (returning an error) if one of the points is zero! * This should never happen, see choice of w in ecp_mul_comb(). * * Cost: 1N(t) := 1I + (6t - 3)M + 1S */ static int ecp_normalize_jac_many( const ecp_group *grp, ecp_point *T[], size_t t_len ) { int ret; size_t i; mpi *c, u, Zi, ZZi; if( t_len < 2 ) return( ecp_normalize_jac( grp, *T ) ); if( ( c = (mpi *) polarssl_malloc( t_len * sizeof( mpi ) ) ) == NULL ) return( POLARSSL_ERR_ECP_MALLOC_FAILED ); mpi_init( &u ); mpi_init( &Zi ); mpi_init( &ZZi ); for( i = 0; i < t_len; i++ ) mpi_init( &c[i] ); /* * c[i] = Z_0 * ... * Z_i */ MPI_CHK( mpi_copy( &c[0], &T[0]->Z ) ); for( i = 1; i < t_len; i++ ) { MPI_CHK( mpi_mul_mpi( &c[i], &c[i-1], &T[i]->Z ) ); MOD_MUL( c[i] ); } /* * u = 1 / (Z_0 * ... * Z_n) mod P */ MPI_CHK( mpi_inv_mod( &u, &c[t_len-1], &grp->P ) ); for( i = t_len - 1; ; i-- ) { /* * Zi = 1 / Z_i mod p * u = 1 / (Z_0 * ... * Z_i) mod P */ if( i == 0 ) { MPI_CHK( mpi_copy( &Zi, &u ) ); } else { MPI_CHK( mpi_mul_mpi( &Zi, &u, &c[i-1] ) ); MOD_MUL( Zi ); MPI_CHK( mpi_mul_mpi( &u, &u, &T[i]->Z ) ); MOD_MUL( u ); } /* * proceed as in normalize() */ MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi ); MPI_CHK( mpi_mul_mpi( &T[i]->X, &T[i]->X, &ZZi ) ); MOD_MUL( T[i]->X ); MPI_CHK( mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &ZZi ) ); MOD_MUL( T[i]->Y ); MPI_CHK( mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &Zi ) ); MOD_MUL( T[i]->Y ); /* * Post-precessing: reclaim some memory by shrinking coordinates * - not storing Z (always 1) * - shrinking other coordinates, but still keeping the same number of * limbs as P, as otherwise it will too likely be regrown too fast. */ MPI_CHK( mpi_shrink( &T[i]->X, grp->P.n ) ); MPI_CHK( mpi_shrink( &T[i]->Y, grp->P.n ) ); mpi_free( &T[i]->Z ); if( i == 0 ) break; } cleanup: mpi_free( &u ); mpi_free( &Zi ); mpi_free( &ZZi ); for( i = 0; i < t_len; i++ ) mpi_free( &c[i] ); polarssl_free( c ); return( ret ); } /* * Conditional point inversion: Q -> -Q = (Q.X, -Q.Y, Q.Z) without leak. * "inv" must be 0 (don't invert) or 1 (invert) or the result will be invalid */ static int ecp_safe_invert_jac( const ecp_group *grp, ecp_point *Q, unsigned char inv ) { int ret; unsigned char nonzero; mpi mQY; mpi_init( &mQY ); /* Use the fact that -Q.Y mod P = P - Q.Y unless Q.Y == 0 */ MPI_CHK( mpi_sub_mpi( &mQY, &grp->P, &Q->Y ) ); nonzero = mpi_cmp_int( &Q->Y, 0 ) != 0; MPI_CHK( mpi_safe_cond_assign( &Q->Y, &mQY, inv & nonzero ) ); cleanup: mpi_free( &mQY ); return( ret ); } /* * Point doubling R = 2 P, Jacobian coordinates * * http://www.hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian/doubling/dbl-2007-bl.op3 * with heavy variable renaming, some reordering and one minor modification * (a = 2 * b, c = d - 2a replaced with c = d, c = c - b, c = c - b) * in order to use a lot less intermediate variables (6 vs 25). * * Cost: 1D := 2M + 8S */ static int ecp_double_jac( const ecp_group *grp, ecp_point *R, const ecp_point *P ) { int ret; mpi T1, T2, T3, X3, Y3, Z3; #if defined(POLARSSL_SELF_TEST) dbl_count++; #endif mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 ); mpi_init( &X3 ); mpi_init( &Y3 ); mpi_init( &Z3 ); MPI_CHK( mpi_mul_mpi( &T3, &P->X, &P->X ) ); MOD_MUL( T3 ); MPI_CHK( mpi_mul_mpi( &T2, &P->Y, &P->Y ) ); MOD_MUL( T2 ); MPI_CHK( mpi_mul_mpi( &Y3, &T2, &T2 ) ); MOD_MUL( Y3 ); MPI_CHK( mpi_add_mpi( &X3, &P->X, &T2 ) ); MOD_ADD( X3 ); MPI_CHK( mpi_mul_mpi( &X3, &X3, &X3 ) ); MOD_MUL( X3 ); MPI_CHK( mpi_sub_mpi( &X3, &X3, &Y3 ) ); MOD_SUB( X3 ); MPI_CHK( mpi_sub_mpi( &X3, &X3, &T3 ) ); MOD_SUB( X3 ); MPI_CHK( mpi_mul_int( &T1, &X3, 2 ) ); MOD_ADD( T1 ); MPI_CHK( mpi_mul_mpi( &Z3, &P->Z, &P->Z ) ); MOD_MUL( Z3 ); MPI_CHK( mpi_mul_mpi( &X3, &Z3, &Z3 ) ); MOD_MUL( X3 ); MPI_CHK( mpi_mul_int( &T3, &T3, 3 ) ); MOD_ADD( T3 ); /* Special case for A = -3 */ if( grp->A.p == NULL ) { MPI_CHK( mpi_mul_int( &X3, &X3, 3 ) ); X3.s = -1; /* mpi_mul_int doesn't handle negative numbers */ MOD_SUB( X3 ); } else { MPI_CHK( mpi_mul_mpi( &X3, &X3, &grp->A ) ); MOD_MUL( X3 ); } MPI_CHK( mpi_add_mpi( &T3, &T3, &X3 ) ); MOD_ADD( T3 ); MPI_CHK( mpi_mul_mpi( &X3, &T3, &T3 ) ); MOD_MUL( X3 ); MPI_CHK( mpi_sub_mpi( &X3, &X3, &T1 ) ); MOD_SUB( X3 ); MPI_CHK( mpi_sub_mpi( &X3, &X3, &T1 ) ); MOD_SUB( X3 ); MPI_CHK( mpi_sub_mpi( &T1, &T1, &X3 ) ); MOD_SUB( T1 ); MPI_CHK( mpi_mul_mpi( &T1, &T3, &T1 ) ); MOD_MUL( T1 ); MPI_CHK( mpi_mul_int( &T3, &Y3, 8 ) ); MOD_ADD( T3 ); MPI_CHK( mpi_sub_mpi( &Y3, &T1, &T3 ) ); MOD_SUB( Y3 ); MPI_CHK( mpi_add_mpi( &T1, &P->Y, &P->Z ) ); MOD_ADD( T1 ); MPI_CHK( mpi_mul_mpi( &T1, &T1, &T1 ) ); MOD_MUL( T1 ); MPI_CHK( mpi_sub_mpi( &T1, &T1, &T2 ) ); MOD_SUB( T1 ); MPI_CHK( mpi_sub_mpi( &Z3, &T1, &Z3 ) ); MOD_SUB( Z3 ); MPI_CHK( mpi_copy( &R->X, &X3 ) ); MPI_CHK( mpi_copy( &R->Y, &Y3 ) ); MPI_CHK( mpi_copy( &R->Z, &Z3 ) ); cleanup: mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 ); mpi_free( &X3 ); mpi_free( &Y3 ); mpi_free( &Z3 ); return( ret ); } /* * Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22) * * The coordinates of Q must be normalized (= affine), * but those of P don't need to. R is not normalized. * * Special cases: (1) P or Q is zero, (2) R is zero, (3) P == Q. * None of these cases can happen as intermediate step in ecp_mul_comb(): * - at each step, P, Q and R are multiples of the base point, the factor * being less than its order, so none of them is zero; * - Q is an odd multiple of the base point, P an even multiple, * due to the choice of precomputed points in the modified comb method. * So branches for these cases do not leak secret information. * * We accept Q->Z being unset (saving memory in tables) as meaning 1. * * Cost: 1A := 8M + 3S */ static int ecp_add_mixed( const ecp_group *grp, ecp_point *R, const ecp_point *P, const ecp_point *Q ) { int ret; mpi T1, T2, T3, T4, X, Y, Z; #if defined(POLARSSL_SELF_TEST) add_count++; #endif /* * Trivial cases: P == 0 or Q == 0 (case 1) */ if( mpi_cmp_int( &P->Z, 0 ) == 0 ) return( ecp_copy( R, Q ) ); if( Q->Z.p != NULL && mpi_cmp_int( &Q->Z, 0 ) == 0 ) return( ecp_copy( R, P ) ); /* * Make sure Q coordinates are normalized */ if( Q->Z.p != NULL && mpi_cmp_int( &Q->Z, 1 ) != 0 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); 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 ); /* Special cases (2) and (3) */ 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_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, result's coordinates normalized */ int ecp_add( const ecp_group *grp, ecp_point *R, const ecp_point *P, const ecp_point *Q ) { int ret; if( ecp_get_type( grp ) != POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS ) return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE ); MPI_CHK( ecp_add_mixed( grp, R, P, Q ) ); MPI_CHK( ecp_normalize_jac( grp, R ) ); cleanup: return( ret ); } /* * Subtraction: R = P - Q, result's coordinates normalized */ int ecp_sub( const ecp_group *grp, ecp_point *R, const ecp_point *P, const ecp_point *Q ) { int ret; ecp_point mQ; ecp_point_init( &mQ ); if( ecp_get_type( grp ) != POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS ) return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE ); /* mQ = - Q */ MPI_CHK( ecp_copy( &mQ, Q ) ); if( mpi_cmp_int( &mQ.Y, 0 ) != 0 ) MPI_CHK( mpi_sub_mpi( &mQ.Y, &grp->P, &mQ.Y ) ); MPI_CHK( ecp_add_mixed( grp, R, P, &mQ ) ); MPI_CHK( ecp_normalize_jac( grp, R ) ); cleanup: ecp_point_free( &mQ ); return( ret ); } /* * Randomize jacobian coordinates: * (X, Y, Z) -> (l^2 X, l^3 Y, l Z) for random l * This is sort of the reverse operation of ecp_normalize_jac(). * * This countermeasure was first suggested in [2]. */ static int ecp_randomize_jac( const ecp_group *grp, ecp_point *pt, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; mpi l, ll; size_t p_size = ( grp->pbits + 7 ) / 8; int count = 0; mpi_init( &l ); mpi_init( &ll ); /* Generate l such that 1 < l < p */ do { mpi_fill_random( &l, p_size, f_rng, p_rng ); while( mpi_cmp_mpi( &l, &grp->P ) >= 0 ) MPI_CHK( mpi_shift_r( &l, 1 ) ); if( count++ > 10 ) return( POLARSSL_ERR_ECP_RANDOM_FAILED ); } while( mpi_cmp_int( &l, 1 ) <= 0 ); /* Z = l * Z */ MPI_CHK( mpi_mul_mpi( &pt->Z, &pt->Z, &l ) ); MOD_MUL( pt->Z ); /* X = l^2 * X */ MPI_CHK( mpi_mul_mpi( &ll, &l, &l ) ); MOD_MUL( ll ); MPI_CHK( mpi_mul_mpi( &pt->X, &pt->X, &ll ) ); MOD_MUL( pt->X ); /* Y = l^3 * Y */ MPI_CHK( mpi_mul_mpi( &ll, &ll, &l ) ); MOD_MUL( ll ); MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &ll ) ); MOD_MUL( pt->Y ); cleanup: mpi_free( &l ); mpi_free( &ll ); return( ret ); } /* * Check and define parameters used by the comb method (see below for details) */ #if POLARSSL_ECP_WINDOW_SIZE < 2 || POLARSSL_ECP_WINDOW_SIZE > 7 #error "POLARSSL_ECP_WINDOW_SIZE out of bounds" #endif /* d = ceil( n / w ) */ #define COMB_MAX_D ( POLARSSL_ECP_MAX_BITS + 1 ) / 2 /* number of precomputed points */ #define COMB_MAX_PRE ( 1 << ( POLARSSL_ECP_WINDOW_SIZE - 1 ) ) /* * Compute the representation of m that will be used with our comb method. * * The basic comb method is described in GECC 3.44 for example. We use a * modified version that provides resistance to SPA by avoiding zero * digits in the representation as in [3]. We modify the method further by * requiring that all K_i be odd, which has the small cost that our * representation uses one more K_i, due to carries. * * Also, for the sake of compactness, only the seven low-order bits of x[i] * are used to represent K_i, and the msb of x[i] encodes the the sign (s_i in * the paper): it is set if and only if if s_i == -1; * * Calling conventions: * - x is an array of size d + 1 * - w is the size, ie number of teeth, of the comb, and must be between * 2 and 7 (in practice, between 2 and POLARSSL_ECP_WINDOW_SIZE) * - m is the MPI, expected to be odd and such that bitlength(m) <= w * d * (the result will be incorrect if these assumptions are not satisfied) */ static void ecp_comb_fixed( unsigned char x[], size_t d, unsigned char w, const mpi *m ) { size_t i, j; unsigned char c, cc, adjust; memset( x, 0, d+1 ); /* First get the classical comb values (except for x_d = 0) */ for( i = 0; i < d; i++ ) for( j = 0; j < w; j++ ) x[i] |= mpi_get_bit( m, i + d * j ) << j; /* Now make sure x_1 .. x_d are odd */ c = 0; for( i = 1; i <= d; i++ ) { /* Add carry and update it */ cc = x[i] & c; x[i] = x[i] ^ c; c = cc; /* Adjust if needed, avoiding branches */ adjust = 1 - ( x[i] & 0x01 ); c |= x[i] & ( x[i-1] * adjust ); x[i] = x[i] ^ ( x[i-1] * adjust ); x[i-1] |= adjust << 7; } } /* * Precompute points for the comb method * * If i = i_{w-1} ... i_1 is the binary representation of i, then * T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P * * T must be able to hold 2^{w - 1} elements * * Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1) */ static int ecp_precompute_comb( const ecp_group *grp, ecp_point T[], const ecp_point *P, unsigned char w, size_t d ) { int ret; unsigned char i, k; size_t j; ecp_point *cur, *TT[COMB_MAX_PRE - 1]; /* * Set T[0] = P and * T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value) */ MPI_CHK( ecp_copy( &T[0], P ) ); k = 0; for( i = 1; i < ( 1U << ( w - 1 ) ); i <<= 1 ) { cur = T + i; MPI_CHK( ecp_copy( cur, T + ( i >> 1 ) ) ); for( j = 0; j < d; j++ ) MPI_CHK( ecp_double_jac( grp, cur, cur ) ); TT[k++] = cur; } MPI_CHK( ecp_normalize_jac_many( grp, TT, k ) ); /* * Compute the remaining ones using the minimal number of additions * Be careful to update T[2^l] only after using it! */ k = 0; for( i = 1; i < ( 1U << ( w - 1 ) ); i <<= 1 ) { j = i; while( j-- ) { MPI_CHK( ecp_add_mixed( grp, &T[i + j], &T[j], &T[i] ) ); TT[k++] = &T[i + j]; } } MPI_CHK( ecp_normalize_jac_many( grp, TT, k ) ); cleanup: return( ret ); } /* * Select precomputed point: R = sign(i) * T[ abs(i) / 2 ] */ static int ecp_select_comb( const ecp_group *grp, ecp_point *R, const ecp_point T[], unsigned char t_len, unsigned char i ) { int ret; unsigned char ii, j; /* Ignore the "sign" bit and scale down */ ii = ( i & 0x7Fu ) >> 1; /* Read the whole table to thwart cache-based timing attacks */ for( j = 0; j < t_len; j++ ) { MPI_CHK( mpi_safe_cond_assign( &R->X, &T[j].X, j == ii ) ); MPI_CHK( mpi_safe_cond_assign( &R->Y, &T[j].Y, j == ii ) ); } /* Safely invert result if i is "negative" */ MPI_CHK( ecp_safe_invert_jac( grp, R, i >> 7 ) ); cleanup: return( ret ); } /* * Core multiplication algorithm for the (modified) comb method. * This part is actually common with the basic comb method (GECC 3.44) * * Cost: d A + d D + 1 R */ static int ecp_mul_comb_core( const ecp_group *grp, ecp_point *R, const ecp_point T[], unsigned char t_len, const unsigned char x[], size_t d, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; ecp_point Txi; size_t i; ecp_point_init( &Txi ); /* Start with a non-zero point and randomize its coordinates */ i = d; MPI_CHK( ecp_select_comb( grp, R, T, t_len, x[i] ) ); MPI_CHK( mpi_lset( &R->Z, 1 ) ); if( f_rng != 0 ) MPI_CHK( ecp_randomize_jac( grp, R, f_rng, p_rng ) ); while( i-- != 0 ) { MPI_CHK( ecp_double_jac( grp, R, R ) ); MPI_CHK( ecp_select_comb( grp, &Txi, T, t_len, x[i] ) ); MPI_CHK( ecp_add_mixed( grp, R, R, &Txi ) ); } cleanup: ecp_point_free( &Txi ); return( ret ); } /* * Multiplication using the comb method, * for curves in short Weierstrass form */ static int ecp_mul_comb( ecp_group *grp, ecp_point *R, const mpi *m, const ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; unsigned char w, m_is_odd, p_eq_g, pre_len, i; size_t d; unsigned char k[COMB_MAX_D + 1]; ecp_point *T; mpi M, mm; mpi_init( &M ); mpi_init( &mm ); /* we need N to be odd to trnaform m in an odd number, check now */ if( mpi_get_bit( &grp->N, 0 ) != 1 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); /* * Minimize the number of multiplications, that is minimize * 10 * d * w + 18 * 2^(w-1) + 11 * d + 7 * w, with d = ceil( nbits / w ) * (see costs of the various parts, with 1S = 1M) */ w = grp->nbits >= 384 ? 5 : 4; /* * If P == G, pre-compute a bit more, since this may be re-used later. * Just adding one avoids upping the cost of the first mul too much, * and the memory cost too. */ #if POLARSSL_ECP_FIXED_POINT_OPTIM == 1 p_eq_g = ( mpi_cmp_mpi( &P->Y, &grp->G.Y ) == 0 && mpi_cmp_mpi( &P->X, &grp->G.X ) == 0 ); if( p_eq_g ) w++; #else p_eq_g = 0; #endif /* * Make sure w is within bounds. * (The last test is useful only for very small curves in the test suite.) */ if( w > POLARSSL_ECP_WINDOW_SIZE ) w = POLARSSL_ECP_WINDOW_SIZE; if( w >= grp->nbits ) w = 2; /* Other sizes that depend on w */ pre_len = 1U << ( w - 1 ); d = ( grp->nbits + w - 1 ) / w; /* * Prepare precomputed points: if P == G we want to * use grp->T if already initialized, or initialize it. */ T = p_eq_g ? grp->T : NULL; if( T == NULL ) { T = (ecp_point *) polarssl_malloc( pre_len * sizeof( ecp_point ) ); if( T == NULL ) { ret = POLARSSL_ERR_ECP_MALLOC_FAILED; goto cleanup; } for( i = 0; i < pre_len; i++ ) ecp_point_init( &T[i] ); MPI_CHK( ecp_precompute_comb( grp, T, P, w, d ) ); if( p_eq_g ) { grp->T = T; grp->T_size = pre_len; } } /* * Make sure M is odd (M = m or M = N - m, since N is odd) * using the fact that m * P = - (N - m) * P */ m_is_odd = ( mpi_get_bit( m, 0 ) == 1 ); MPI_CHK( mpi_copy( &M, m ) ); MPI_CHK( mpi_sub_mpi( &mm, &grp->N, m ) ); MPI_CHK( mpi_safe_cond_assign( &M, &mm, ! m_is_odd ) ); /* * Go for comb multiplication, R = M * P */ ecp_comb_fixed( k, d, w, &M ); MPI_CHK( ecp_mul_comb_core( grp, R, T, pre_len, k, d, f_rng, p_rng ) ); /* * Now get m * P from M * P and normalize it */ MPI_CHK( ecp_safe_invert_jac( grp, R, ! m_is_odd ) ); MPI_CHK( ecp_normalize_jac( grp, R ) ); cleanup: if( T != NULL && ! p_eq_g ) { for( i = 0; i < pre_len; i++ ) ecp_point_free( &T[i] ); polarssl_free( T ); } mpi_free( &M ); mpi_free( &mm ); if( ret != 0 ) ecp_point_free( R ); return( ret ); } #endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */ #if defined(POLARSSL_ECP_MONTGOMERY) /* * For Montgomery curves, we do all the internal arithmetic in projective * coordinates. Import/export of points uses only the x coordinates, which is * internaly represented as X / Z. * * For scalar multiplication, we'll use a Montgomery ladder. */ /* * Normalize Montgomery x/z coordinates: X = X/Z, Z = 1 * Cost: 1M + 1I */ static int ecp_normalize_mxz( const ecp_group *grp, ecp_point *P ) { int ret; MPI_CHK( mpi_inv_mod( &P->Z, &P->Z, &grp->P ) ); MPI_CHK( mpi_mul_mpi( &P->X, &P->X, &P->Z ) ); MOD_MUL( P->X ); MPI_CHK( mpi_lset( &P->Z, 1 ) ); cleanup: return( ret ); } /* * Randomize projective x/z coordinates: * (X, Z) -> (l X, l Z) for random l * This is sort of the reverse operation of ecp_normalize_mxz(). * * This countermeasure was first suggested in [2]. * Cost: 2M */ static int ecp_randomize_mxz( const ecp_group *grp, ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; mpi l; size_t p_size = ( grp->pbits + 7 ) / 8; int count = 0; mpi_init( &l ); /* Generate l such that 1 < l < p */ do { mpi_fill_random( &l, p_size, f_rng, p_rng ); while( mpi_cmp_mpi( &l, &grp->P ) >= 0 ) MPI_CHK( mpi_shift_r( &l, 1 ) ); if( count++ > 10 ) return( POLARSSL_ERR_ECP_RANDOM_FAILED ); } while( mpi_cmp_int( &l, 1 ) <= 0 ); MPI_CHK( mpi_mul_mpi( &P->X, &P->X, &l ) ); MOD_MUL( P->X ); MPI_CHK( mpi_mul_mpi( &P->Z, &P->Z, &l ) ); MOD_MUL( P->Z ); cleanup: mpi_free( &l ); return( ret ); } /* * Double-and-add: R = 2P, S = P + Q, with d = X(P - Q), * for Montgomery curves in x/z coordinates. * * http://www.hyperelliptic.org/EFD/g1p/auto-code/montgom/xz/ladder/mladd-1987-m.op3 * with * d = X1 * P = (X2, Z2) * Q = (X3, Z3) * R = (X4, Z4) * S = (X5, Z5) * and eliminating temporary variables tO, ..., t4. * * Cost: 5M + 4S */ static int ecp_double_add_mxz( const ecp_group *grp, ecp_point *R, ecp_point *S, const ecp_point *P, const ecp_point *Q, const mpi *d ) { int ret; mpi A, AA, B, BB, E, C, D, DA, CB; mpi_init( &A ); mpi_init( &AA ); mpi_init( &B ); mpi_init( &BB ); mpi_init( &E ); mpi_init( &C ); mpi_init( &D ); mpi_init( &DA ); mpi_init( &CB ); MPI_CHK( mpi_add_mpi( &A, &P->X, &P->Z ) ); MOD_ADD( A ); MPI_CHK( mpi_mul_mpi( &AA, &A, &A ) ); MOD_MUL( AA ); MPI_CHK( mpi_sub_mpi( &B, &P->X, &P->Z ) ); MOD_SUB( B ); MPI_CHK( mpi_mul_mpi( &BB, &B, &B ) ); MOD_MUL( BB ); MPI_CHK( mpi_sub_mpi( &E, &AA, &BB ) ); MOD_SUB( E ); MPI_CHK( mpi_add_mpi( &C, &Q->X, &Q->Z ) ); MOD_ADD( C ); MPI_CHK( mpi_sub_mpi( &D, &Q->X, &Q->Z ) ); MOD_SUB( D ); MPI_CHK( mpi_mul_mpi( &DA, &D, &A ) ); MOD_MUL( DA ); MPI_CHK( mpi_mul_mpi( &CB, &C, &B ) ); MOD_MUL( CB ); MPI_CHK( mpi_add_mpi( &S->X, &DA, &CB ) ); MOD_MUL( S->X ); MPI_CHK( mpi_mul_mpi( &S->X, &S->X, &S->X ) ); MOD_MUL( S->X ); MPI_CHK( mpi_sub_mpi( &S->Z, &DA, &CB ) ); MOD_SUB( S->Z ); MPI_CHK( mpi_mul_mpi( &S->Z, &S->Z, &S->Z ) ); MOD_MUL( S->Z ); MPI_CHK( mpi_mul_mpi( &S->Z, d, &S->Z ) ); MOD_MUL( S->Z ); MPI_CHK( mpi_mul_mpi( &R->X, &AA, &BB ) ); MOD_MUL( R->X ); MPI_CHK( mpi_mul_mpi( &R->Z, &grp->A, &E ) ); MOD_MUL( R->Z ); MPI_CHK( mpi_add_mpi( &R->Z, &BB, &R->Z ) ); MOD_ADD( R->Z ); MPI_CHK( mpi_mul_mpi( &R->Z, &E, &R->Z ) ); MOD_MUL( R->Z ); cleanup: mpi_free( &A ); mpi_free( &AA ); mpi_free( &B ); mpi_free( &BB ); mpi_free( &E ); mpi_free( &C ); mpi_free( &D ); mpi_free( &DA ); mpi_free( &CB ); return( ret ); } /* * Multiplication with Montgomery ladder in x/z coordinates, * for curves in Montgomery form */ static int ecp_mul_mxz( ecp_group *grp, ecp_point *R, const mpi *m, const ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; size_t i; unsigned char b; ecp_point RP; mpi PX; ecp_point_init( &RP ); mpi_init( &PX ); /* Save PX and read from P before writing to R, in case P == R */ MPI_CHK( mpi_copy( &PX, &P->X ) ); MPI_CHK( ecp_copy( &RP, P ) ); /* Set R to zero in modified x/z coordinates */ MPI_CHK( mpi_lset( &R->X, 1 ) ); MPI_CHK( mpi_lset( &R->Z, 0 ) ); mpi_free( &R->Y ); /* RP.X might be sligtly larger than P, so reduce it */ MOD_ADD( RP.X ); /* Randomize coordinates of the starting point */ if( f_rng != NULL ) MPI_CHK( ecp_randomize_mxz( grp, &RP, f_rng, p_rng ) ); /* Loop invariant: R = result so far, RP = R + P */ i = mpi_msb( m ); /* one past the (zero-based) most significant bit */ while( i-- > 0 ) { b = mpi_get_bit( m, i ); /* * if (b) R = 2R + P else R = 2R, * which is: * if (b) double_add( RP, R, RP, R ) * else double_add( R, RP, R, RP ) * but using safe conditional swaps to avoid leaks */ MPI_CHK( mpi_safe_cond_swap( &R->X, &RP.X, b ) ); MPI_CHK( mpi_safe_cond_swap( &R->Z, &RP.Z, b ) ); MPI_CHK( ecp_double_add_mxz( grp, R, &RP, R, &RP, &PX ) ); MPI_CHK( mpi_safe_cond_swap( &R->X, &RP.X, b ) ); MPI_CHK( mpi_safe_cond_swap( &R->Z, &RP.Z, b ) ); } MPI_CHK( ecp_normalize_mxz( grp, R ) ); cleanup: ecp_point_free( &RP ); mpi_free( &PX ); return( ret ); } #endif /* POLARSSL_ECP_MONTGOMERY */ /* * Multiplication R = m * P */ int ecp_mul( ecp_group *grp, ecp_point *R, const mpi *m, const ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; /* Common sanity checks */ if( mpi_cmp_int( &P->Z, 1 ) != 0 ) return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); if( ( ret = ecp_check_privkey( grp, m ) ) != 0 || ( ret = ecp_check_pubkey( grp, P ) ) != 0 ) return( ret ); #if defined(POLARSSL_ECP_MONTGOMERY) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY ) return( ecp_mul_mxz( grp, R, m, P, f_rng, p_rng ) ); #endif #if defined(POLARSSL_ECP_SHORT_WEIERSTRASS) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS ) return( ecp_mul_comb( grp, R, m, P, f_rng, p_rng ) ); #endif return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); } #if defined(POLARSSL_ECP_SHORT_WEIERSTRASS) /* * Check that an affine point is valid as a public key, * short weierstrass curves (SEC1 3.2.3.1) */ static int ecp_check_pubkey_sw( const ecp_group *grp, const ecp_point *pt ) { int ret; mpi YY, RHS; /* pt coordinates must be normalized for our checks */ if( mpi_cmp_int( &pt->X, 0 ) < 0 || mpi_cmp_int( &pt->Y, 0 ) < 0 || mpi_cmp_mpi( &pt->X, &grp->P ) >= 0 || mpi_cmp_mpi( &pt->Y, &grp->P ) >= 0 ) return( POLARSSL_ERR_ECP_INVALID_KEY ); mpi_init( &YY ); mpi_init( &RHS ); /* * YY = Y^2 * RHS = X (X^2 + A) + B = X^3 + A X + B */ MPI_CHK( mpi_mul_mpi( &YY, &pt->Y, &pt->Y ) ); MOD_MUL( YY ); MPI_CHK( mpi_mul_mpi( &RHS, &pt->X, &pt->X ) ); MOD_MUL( RHS ); /* Special case for A = -3 */ if( grp->A.p == NULL ) { MPI_CHK( mpi_sub_int( &RHS, &RHS, 3 ) ); MOD_SUB( RHS ); } else { MPI_CHK( mpi_add_mpi( &RHS, &RHS, &grp->A ) ); MOD_ADD( RHS ); } MPI_CHK( mpi_mul_mpi( &RHS, &RHS, &pt->X ) ); MOD_MUL( RHS ); MPI_CHK( mpi_add_mpi( &RHS, &RHS, &grp->B ) ); MOD_ADD( RHS ); if( mpi_cmp_mpi( &YY, &RHS ) != 0 ) ret = POLARSSL_ERR_ECP_INVALID_KEY; cleanup: mpi_free( &YY ); mpi_free( &RHS ); return( ret ); } #endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */ #if defined(POLARSSL_ECP_MONTGOMERY) /* * Check validity of a public key for Montgomery curves with x-only schemes */ static int ecp_check_pubkey_mx( const ecp_group *grp, const ecp_point *pt ) { /* [M255 p. 5] Just check X is the correct number of bytes */ if( mpi_size( &pt->X ) > ( grp->nbits + 7 ) / 8 ) return( POLARSSL_ERR_ECP_INVALID_KEY ); return( 0 ); } #endif /* POLARSSL_ECP_MONTGOMERY */ /* * Check that a point is valid as a public key */ int ecp_check_pubkey( const ecp_group *grp, const ecp_point *pt ) { /* Must use affine coordinates */ if( mpi_cmp_int( &pt->Z, 1 ) != 0 ) return( POLARSSL_ERR_ECP_INVALID_KEY ); #if defined(POLARSSL_ECP_MONTGOMERY) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY ) return( ecp_check_pubkey_mx( grp, pt ) ); #endif #if defined(POLARSSL_ECP_SHORT_WEIERSTRASS) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS ) return( ecp_check_pubkey_sw( grp, pt ) ); #endif return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); } /* * Check that an mpi is valid as a private key */ int ecp_check_privkey( const ecp_group *grp, const mpi *d ) { #if defined(POLARSSL_ECP_MONTGOMERY) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY ) { /* see [M255] page 5 */ if( mpi_get_bit( d, 0 ) != 0 || mpi_get_bit( d, 1 ) != 0 || mpi_get_bit( d, 2 ) != 0 || mpi_msb( d ) - 1 != grp->nbits ) /* mpi_msb is one-based! */ return( POLARSSL_ERR_ECP_INVALID_KEY ); else return( 0 ); } #endif /* POLARSSL_ECP_MONTGOMERY */ #if defined(POLARSSL_ECP_SHORT_WEIERSTRASS) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS ) { /* see SEC1 3.2 */ if( mpi_cmp_int( d, 1 ) < 0 || mpi_cmp_mpi( d, &grp->N ) >= 0 ) return( POLARSSL_ERR_ECP_INVALID_KEY ); else return( 0 ); } #endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */ return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); } /* * Generate a keypair */ int ecp_gen_keypair( ecp_group *grp, mpi *d, ecp_point *Q, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; size_t n_size = ( grp->nbits + 7 ) / 8; #if defined(POLARSSL_ECP_MONTGOMERY) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY ) { /* [M225] page 5 */ size_t b; MPI_CHK( mpi_fill_random( d, n_size, f_rng, p_rng ) ); /* Make sure the most significant bit is nbits */ b = mpi_msb( d ) - 1; /* mpi_msb is one-based */ if( b > grp->nbits ) MPI_CHK( mpi_shift_r( d, b - grp->nbits ) ); else MPI_CHK( mpi_set_bit( d, grp->nbits, 1 ) ); /* Make sure the last three bits are unset */ MPI_CHK( mpi_set_bit( d, 0, 0 ) ); MPI_CHK( mpi_set_bit( d, 1, 0 ) ); MPI_CHK( mpi_set_bit( d, 2, 0 ) ); } else #endif /* POLARSSL_ECP_MONTGOMERY */ #if defined(POLARSSL_ECP_SHORT_WEIERSTRASS) if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS ) { /* SEC1 3.2.1: Generate d such that 1 <= n < N */ int count = 0; unsigned char rnd[POLARSSL_ECP_MAX_BYTES]; /* * Match the procedure given in RFC 6979 (deterministic ECDSA): * - use the same byte ordering; * - keep the leftmost nbits bits of the generated octet string; * - try until result is in the desired range. * This also avoids any biais, which is especially important for ECDSA. */ do { MPI_CHK( f_rng( p_rng, rnd, n_size ) ); MPI_CHK( mpi_read_binary( d, rnd, n_size ) ); MPI_CHK( mpi_shift_r( d, 8 * n_size - grp->nbits ) ); /* * Each try has at worst a probability 1/2 of failing (the msb has * a probability 1/2 of being 0, and then the result will be < N), * so after 30 tries failure probability is a most 2**(-30). * * For most curves, 1 try is enough with overwhelming probability, * since N starts with a lot of 1s in binary, but some curves * such as secp224k1 are actually very close to the worst case. */ if( ++count > 30 ) return( POLARSSL_ERR_ECP_RANDOM_FAILED ); } while( mpi_cmp_int( d, 1 ) < 0 || mpi_cmp_mpi( d, &grp->N ) >= 0 ); } else #endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */ return( POLARSSL_ERR_ECP_BAD_INPUT_DATA ); cleanup: if( ret != 0 ) return( ret ); return( ecp_mul( grp, Q, d, &grp->G, f_rng, p_rng ) ); } /* * Generate a keypair, prettier wrapper */ int ecp_gen_key( ecp_group_id grp_id, ecp_keypair *key, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret; if( ( ret = ecp_use_known_dp( &key->grp, grp_id ) ) != 0 ) return( ret ); return( ecp_gen_keypair( &key->grp, &key->d, &key->Q, f_rng, p_rng ) ); } #if defined(POLARSSL_SELF_TEST) /* * Checkup routine */ int ecp_self_test( int verbose ) { int ret; size_t i; ecp_group grp; ecp_point R, P; mpi m; unsigned long add_c_prev, dbl_c_prev, mul_c_prev; /* exponents especially adapted for secp192r1 */ const char *exponents[] = { "000000000000000000000000000000000000000000000001", /* one */ "FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22830", /* N - 1 */ "5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */ "400000000000000000000000000000000000000000000000", /* one and zeros */ "7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF", /* all ones */ "555555555555555555555555555555555555555555555555", /* 101010... */ }; ecp_group_init( &grp ); ecp_point_init( &R ); ecp_point_init( &P ); mpi_init( &m ); /* Use secp192r1 if available, or any available curve */ #if defined(POLARSSL_ECP_DP_SECP192R1_ENABLED) MPI_CHK( ecp_use_known_dp( &grp, POLARSSL_ECP_DP_SECP192R1 ) ); #else MPI_CHK( ecp_use_known_dp( &grp, ecp_curve_list()->grp_id ) ); #endif if( verbose != 0 ) polarssl_printf( " ECP test #1 (constant op_count, base point G): " ); /* Do a dummy multiplication first to trigger precomputation */ MPI_CHK( mpi_lset( &m, 2 ) ); MPI_CHK( ecp_mul( &grp, &P, &m, &grp.G, NULL, NULL ) ); add_count = 0; dbl_count = 0; mul_count = 0; MPI_CHK( mpi_read_string( &m, 16, exponents[0] ) ); MPI_CHK( ecp_mul( &grp, &R, &m, &grp.G, NULL, NULL ) ); for( i = 1; i < sizeof( exponents ) / sizeof( exponents[0] ); i++ ) { add_c_prev = add_count; dbl_c_prev = dbl_count; mul_c_prev = mul_count; add_count = 0; dbl_count = 0; mul_count = 0; MPI_CHK( mpi_read_string( &m, 16, exponents[i] ) ); MPI_CHK( ecp_mul( &grp, &R, &m, &grp.G, NULL, NULL ) ); if( add_count != add_c_prev || dbl_count != dbl_c_prev || mul_count != mul_c_prev ) { if( verbose != 0 ) polarssl_printf( "failed (%u)\n", (unsigned int) i ); ret = 1; goto cleanup; } } if( verbose != 0 ) polarssl_printf( "passed\n" ); if( verbose != 0 ) polarssl_printf( " ECP test #2 (constant op_count, other point): " ); /* We computed P = 2G last time, use it */ add_count = 0; dbl_count = 0; mul_count = 0; MPI_CHK( mpi_read_string( &m, 16, exponents[0] ) ); MPI_CHK( ecp_mul( &grp, &R, &m, &P, NULL, NULL ) ); for( i = 1; i < sizeof( exponents ) / sizeof( exponents[0] ); i++ ) { add_c_prev = add_count; dbl_c_prev = dbl_count; mul_c_prev = mul_count; add_count = 0; dbl_count = 0; mul_count = 0; MPI_CHK( mpi_read_string( &m, 16, exponents[i] ) ); MPI_CHK( ecp_mul( &grp, &R, &m, &P, NULL, NULL ) ); if( add_count != add_c_prev || dbl_count != dbl_c_prev || mul_count != mul_c_prev ) { if( verbose != 0 ) polarssl_printf( "failed (%u)\n", (unsigned int) i ); ret = 1; goto cleanup; } } if( verbose != 0 ) polarssl_printf( "passed\n" ); cleanup: if( ret < 0 && verbose != 0 ) polarssl_printf( "Unexpected error, return code = %08X\n", ret ); ecp_group_free( &grp ); ecp_point_free( &R ); ecp_point_free( &P ); mpi_free( &m ); if( verbose != 0 ) polarssl_printf( "\n" ); return( ret ); } #endif /* POLARSSL_SELF_TEST */ #endif /* POLARSSL_ECP_C */