2 * The RSA public-key cryptosystem
4 * Copyright (C) 2006-2010, Brainspark B.V.
6 * This file is part of PolarSSL (http://www.polarssl.org)
7 * Lead Maintainer: Paul Bakker <polarssl_maintainer at polarssl.org>
11 * This program is free software; you can redistribute it and/or modify
12 * it under the terms of the GNU General Public License as published by
13 * the Free Software Foundation; either version 2 of the License, or
14 * (at your option) any later version.
16 * This program is distributed in the hope that it will be useful,
17 * but WITHOUT ANY WARRANTY; without even the implied warranty of
18 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
19 * GNU General Public License for more details.
21 * You should have received a copy of the GNU General Public License along
22 * with this program; if not, write to the Free Software Foundation, Inc.,
23 * 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
26 * RSA was designed by Ron Rivest, Adi Shamir and Len Adleman.
28 * http://theory.lcs.mit.edu/~rivest/rsapaper.pdf
29 * http://www.cacr.math.uwaterloo.ca/hac/about/chap8.pdf
39 /* *************** begin copy from x509parse.c ********************/
41 * ASN.1 DER decoding routines
43 static int asn1_get_len( unsigned char **p,
44 const unsigned char *end,
47 if( ( end - *p ) < 1 )
48 return( POLARSSL_ERR_ASN1_OUT_OF_DATA );
50 if( ( **p & 0x80 ) == 0 )
57 if( ( end - *p ) < 2 )
58 return( POLARSSL_ERR_ASN1_OUT_OF_DATA );
65 if( ( end - *p ) < 3 )
66 return( POLARSSL_ERR_ASN1_OUT_OF_DATA );
68 *len = ( (*p)[1] << 8 ) | (*p)[2];
73 return( POLARSSL_ERR_ASN1_INVALID_LENGTH );
78 if( *len > (int) ( end - *p ) )
79 return( POLARSSL_ERR_ASN1_OUT_OF_DATA );
84 static int asn1_get_tag( unsigned char **p,
85 const unsigned char *end,
88 if( ( end - *p ) < 1 )
89 return( POLARSSL_ERR_ASN1_OUT_OF_DATA );
92 return( POLARSSL_ERR_ASN1_UNEXPECTED_TAG );
96 return( asn1_get_len( p, end, len ) );
99 static int asn1_get_int( unsigned char **p,
100 const unsigned char *end,
105 if( ( ret = asn1_get_tag( p, end, &len, ASN1_INTEGER ) ) != 0 )
108 if( len > (int) sizeof( int ) || ( **p & 0x80 ) != 0 )
109 return( POLARSSL_ERR_ASN1_INVALID_LENGTH );
115 *val = ( *val << 8 ) | **p;
122 static int asn1_get_mpi( unsigned char **p,
123 const unsigned char *end,
128 if( ( ret = asn1_get_tag( p, end, &len, ASN1_INTEGER ) ) != 0 )
131 ret = mpi_read_binary( X, *p, len );
137 /* *************** end copy from x509parse.c ********************/
143 * Initialize an RSA context
145 void rsa_init( rsa_context *ctx,
149 memset( ctx, 0, sizeof( rsa_context ) );
151 ctx->padding = padding;
152 ctx->hash_id = hash_id;
155 #if defined(POLARSSL_GENPRIME)
158 * Generate an RSA keypair
160 int rsa_gen_key( rsa_context *ctx,
161 int (*f_rng)(void *),
163 int nbits, int exponent )
168 if( f_rng == NULL || nbits < 128 || exponent < 3 )
169 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
171 mpi_init( &P1, &Q1, &H, &G, NULL );
174 * find primes P and Q with Q < P so that:
175 * GCD( E, (P-1)*(Q-1) ) == 1
177 MPI_CHK( mpi_lset( &ctx->E, exponent ) );
181 MPI_CHK( mpi_gen_prime( &ctx->P, ( nbits + 1 ) >> 1, 0,
184 MPI_CHK( mpi_gen_prime( &ctx->Q, ( nbits + 1 ) >> 1, 0,
187 if( mpi_cmp_mpi( &ctx->P, &ctx->Q ) < 0 )
188 mpi_swap( &ctx->P, &ctx->Q );
190 if( mpi_cmp_mpi( &ctx->P, &ctx->Q ) == 0 )
193 MPI_CHK( mpi_mul_mpi( &ctx->N, &ctx->P, &ctx->Q ) );
194 if( mpi_msb( &ctx->N ) != nbits )
197 MPI_CHK( mpi_sub_int( &P1, &ctx->P, 1 ) );
198 MPI_CHK( mpi_sub_int( &Q1, &ctx->Q, 1 ) );
199 MPI_CHK( mpi_mul_mpi( &H, &P1, &Q1 ) );
200 MPI_CHK( mpi_gcd( &G, &ctx->E, &H ) );
202 while( mpi_cmp_int( &G, 1 ) != 0 );
205 * D = E^-1 mod ((P-1)*(Q-1))
210 MPI_CHK( mpi_inv_mod( &ctx->D , &ctx->E, &H ) );
211 MPI_CHK( mpi_mod_mpi( &ctx->DP, &ctx->D, &P1 ) );
212 MPI_CHK( mpi_mod_mpi( &ctx->DQ, &ctx->D, &Q1 ) );
213 MPI_CHK( mpi_inv_mod( &ctx->QP, &ctx->Q, &ctx->P ) );
215 ctx->len = ( mpi_msb( &ctx->N ) + 7 ) >> 3;
219 mpi_free( &G, &H, &Q1, &P1, NULL );
224 return( POLARSSL_ERR_RSA_KEY_GEN_FAILED | ret );
233 * Check a public RSA key
235 int rsa_check_pubkey( const rsa_context *ctx )
237 if( !ctx->N.p || !ctx->E.p )
238 return( POLARSSL_ERR_RSA_KEY_CHECK_FAILED );
240 if( ( ctx->N.p[0] & 1 ) == 0 ||
241 ( ctx->E.p[0] & 1 ) == 0 )
242 return( POLARSSL_ERR_RSA_KEY_CHECK_FAILED );
244 if( mpi_msb( &ctx->N ) < 128 ||
245 mpi_msb( &ctx->N ) > 4096 )
246 return( POLARSSL_ERR_RSA_KEY_CHECK_FAILED );
248 if( mpi_msb( &ctx->E ) < 2 ||
249 mpi_msb( &ctx->E ) > 64 )
250 return( POLARSSL_ERR_RSA_KEY_CHECK_FAILED );
256 * Check a private RSA key
258 int rsa_check_privkey( const rsa_context *ctx )
261 mpi PQ, DE, P1, Q1, H, I, G, G2, L1, L2;
263 if( ( ret = rsa_check_pubkey( ctx ) ) != 0 )
266 if( !ctx->P.p || !ctx->Q.p || !ctx->D.p )
267 return( POLARSSL_ERR_RSA_KEY_CHECK_FAILED );
269 mpi_init( &PQ, &DE, &P1, &Q1, &H, &I, &G, &G2, &L1, &L2, NULL );
271 MPI_CHK( mpi_mul_mpi( &PQ, &ctx->P, &ctx->Q ) );
272 MPI_CHK( mpi_mul_mpi( &DE, &ctx->D, &ctx->E ) );
273 MPI_CHK( mpi_sub_int( &P1, &ctx->P, 1 ) );
274 MPI_CHK( mpi_sub_int( &Q1, &ctx->Q, 1 ) );
275 MPI_CHK( mpi_mul_mpi( &H, &P1, &Q1 ) );
276 MPI_CHK( mpi_gcd( &G, &ctx->E, &H ) );
278 MPI_CHK( mpi_gcd( &G2, &P1, &Q1 ) );
279 MPI_CHK( mpi_div_mpi( &L1, &L2, &H, &G2 ) );
280 MPI_CHK( mpi_mod_mpi( &I, &DE, &L1 ) );
283 * Check for a valid PKCS1v2 private key
285 if( mpi_cmp_mpi( &PQ, &ctx->N ) == 0 &&
286 mpi_cmp_int( &L2, 0 ) == 0 &&
287 mpi_cmp_int( &I, 1 ) == 0 &&
288 mpi_cmp_int( &G, 1 ) == 0 )
290 mpi_free( &G, &I, &H, &Q1, &P1, &DE, &PQ, &G2, &L1, &L2, NULL );
297 mpi_free( &G, &I, &H, &Q1, &P1, &DE, &PQ, &G2, &L1, &L2, NULL );
298 return( POLARSSL_ERR_RSA_KEY_CHECK_FAILED | ret );
302 * Do an RSA public key operation
304 int rsa_public( rsa_context *ctx,
305 const unsigned char *input,
306 unsigned char *output )
311 mpi_init( &T, NULL );
313 MPI_CHK( mpi_read_binary( &T, input, ctx->len ) );
315 if( mpi_cmp_mpi( &T, &ctx->N ) >= 0 )
317 mpi_free( &T, NULL );
318 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
322 MPI_CHK( mpi_exp_mod( &T, &T, &ctx->E, &ctx->N, &ctx->RN ) );
323 MPI_CHK( mpi_write_binary( &T, output, olen ) );
327 mpi_free( &T, NULL );
330 return( POLARSSL_ERR_RSA_PUBLIC_FAILED | ret );
336 * Do an RSA private key operation
338 int rsa_private( rsa_context *ctx,
339 const unsigned char *input,
340 unsigned char *output )
345 mpi_init( &T, &T1, &T2, NULL );
347 MPI_CHK( mpi_read_binary( &T, input, ctx->len ) );
349 if( mpi_cmp_mpi( &T, &ctx->N ) >= 0 )
351 mpi_free( &T, NULL );
352 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
356 MPI_CHK( mpi_exp_mod( &T, &T, &ctx->D, &ctx->N, &ctx->RN ) );
359 * faster decryption using the CRT
361 * T1 = input ^ dP mod P
362 * T2 = input ^ dQ mod Q
364 MPI_CHK( mpi_exp_mod( &T1, &T, &ctx->DP, &ctx->P, &ctx->RP ) );
365 MPI_CHK( mpi_exp_mod( &T2, &T, &ctx->DQ, &ctx->Q, &ctx->RQ ) );
368 * T = (T1 - T2) * (Q^-1 mod P) mod P
370 MPI_CHK( mpi_sub_mpi( &T, &T1, &T2 ) );
371 MPI_CHK( mpi_mul_mpi( &T1, &T, &ctx->QP ) );
372 MPI_CHK( mpi_mod_mpi( &T, &T1, &ctx->P ) );
375 * output = T2 + T * Q
377 MPI_CHK( mpi_mul_mpi( &T1, &T, &ctx->Q ) );
378 MPI_CHK( mpi_add_mpi( &T, &T2, &T1 ) );
382 MPI_CHK( mpi_write_binary( &T, output, olen ) );
386 mpi_free( &T, &T1, &T2, NULL );
389 return( POLARSSL_ERR_RSA_PRIVATE_FAILED | ret );
395 * Add the message padding, then do an RSA operation
397 int rsa_pkcs1_encrypt( rsa_context *ctx,
398 int (*f_rng)(void *),
401 const unsigned char *input,
402 unsigned char *output )
405 unsigned char *p = output;
409 switch( ctx->padding )
413 if( ilen < 0 || olen < ilen + 11 || f_rng == NULL )
414 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
416 nb_pad = olen - 3 - ilen;
421 while( nb_pad-- > 0 )
426 *p = (unsigned char) f_rng( p_rng );
427 } while( *p == 0 && --rng_dl );
429 // Check if RNG failed to generate data
432 return POLARSSL_ERR_RSA_RNG_FAILED;
437 memcpy( p, input, ilen );
442 return( POLARSSL_ERR_RSA_INVALID_PADDING );
445 return( ( mode == RSA_PUBLIC )
446 ? rsa_public( ctx, output, output )
447 : rsa_private( ctx, output, output ) );
451 * Do an RSA operation, then remove the message padding
453 int rsa_pkcs1_decrypt( rsa_context *ctx,
455 const unsigned char *input,
456 unsigned char *output,
461 unsigned char buf[1024];
465 if( ilen < 16 || ilen > (int) sizeof( buf ) )
466 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
468 ret = ( mode == RSA_PUBLIC )
469 ? rsa_public( ctx, input, buf )
470 : rsa_private( ctx, input, buf );
477 switch( ctx->padding )
481 if( *p++ != 0 || *p++ != RSA_CRYPT )
482 return( POLARSSL_ERR_RSA_INVALID_PADDING );
486 if( p >= buf + ilen - 1 )
487 return( POLARSSL_ERR_RSA_INVALID_PADDING );
495 return( POLARSSL_ERR_RSA_INVALID_PADDING );
498 if (ilen - (int)(p - buf) > output_max_len)
499 return( POLARSSL_ERR_RSA_OUTPUT_TOO_LARGE );
501 *olen = ilen - (int)(p - buf);
502 memcpy( output, p, *olen );
508 * Do an RSA operation to sign the message digest
510 int rsa_pkcs1_sign( rsa_context *ctx,
514 const unsigned char *hash,
518 unsigned char *p = sig;
522 switch( ctx->padding )
529 nb_pad = olen - 3 - hashlen;
535 nb_pad = olen - 3 - 34;
539 nb_pad = olen - 3 - 35;
543 nb_pad = olen - 3 - 47;
547 nb_pad = olen - 3 - 51;
551 nb_pad = olen - 3 - 67;
555 nb_pad = olen - 3 - 83;
560 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
564 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
568 memset( p, 0xFF, nb_pad );
575 return( POLARSSL_ERR_RSA_INVALID_PADDING );
581 memcpy( p, hash, hashlen );
585 memcpy( p, ASN1_HASH_MDX, 18 );
586 memcpy( p + 18, hash, 16 );
590 memcpy( p, ASN1_HASH_MDX, 18 );
591 memcpy( p + 18, hash, 16 );
595 memcpy( p, ASN1_HASH_MDX, 18 );
596 memcpy( p + 18, hash, 16 );
600 memcpy( p, ASN1_HASH_SHA1, 15 );
601 memcpy( p + 15, hash, 20 );
605 memcpy( p, ASN1_HASH_SHA2X, 19 );
606 memcpy( p + 19, hash, 28 );
607 p[1] += 28; p[14] = 4; p[18] += 28; break;
610 memcpy( p, ASN1_HASH_SHA2X, 19 );
611 memcpy( p + 19, hash, 32 );
612 p[1] += 32; p[14] = 1; p[18] += 32; break;
615 memcpy( p, ASN1_HASH_SHA2X, 19 );
616 memcpy( p + 19, hash, 48 );
617 p[1] += 48; p[14] = 2; p[18] += 48; break;
620 memcpy( p, ASN1_HASH_SHA2X, 19 );
621 memcpy( p + 19, hash, 64 );
622 p[1] += 64; p[14] = 3; p[18] += 64; break;
625 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
628 return( ( mode == RSA_PUBLIC )
629 ? rsa_public( ctx, sig, sig )
630 : rsa_private( ctx, sig, sig ) );
634 * Do an RSA operation and check the message digest
636 int rsa_pkcs1_verify( rsa_context *ctx,
640 const unsigned char *hash,
643 int ret, len, siglen;
645 unsigned char buf[1024];
649 if( siglen < 16 || siglen > (int) sizeof( buf ) )
650 return( POLARSSL_ERR_RSA_BAD_INPUT_DATA );
652 ret = ( mode == RSA_PUBLIC )
653 ? rsa_public( ctx, sig, buf )
654 : rsa_private( ctx, sig, buf );
661 switch( ctx->padding )
665 if( *p++ != 0 || *p++ != RSA_SIGN )
666 return( POLARSSL_ERR_RSA_INVALID_PADDING );
670 if( p >= buf + siglen - 1 || *p != 0xFF )
671 return( POLARSSL_ERR_RSA_INVALID_PADDING );
679 return( POLARSSL_ERR_RSA_INVALID_PADDING );
682 len = siglen - (int)( p - buf );
689 if( memcmp( p, ASN1_HASH_MDX, 18 ) != 0 )
690 return( POLARSSL_ERR_RSA_VERIFY_FAILED );
692 if( ( c == 2 && hash_id == SIG_RSA_MD2 ) ||
693 ( c == 4 && hash_id == SIG_RSA_MD4 ) ||
694 ( c == 5 && hash_id == SIG_RSA_MD5 ) )
696 if( memcmp( p + 18, hash, 16 ) == 0 )
699 return( POLARSSL_ERR_RSA_VERIFY_FAILED );
703 if( len == 35 && hash_id == SIG_RSA_SHA1 )
705 if( memcmp( p, ASN1_HASH_SHA1, 15 ) == 0 &&
706 memcmp( p + 15, hash, 20 ) == 0 )
709 return( POLARSSL_ERR_RSA_VERIFY_FAILED );
711 if( ( len == 19 + 28 && p[14] == 4 && hash_id == SIG_RSA_SHA224 ) ||
712 ( len == 19 + 32 && p[14] == 1 && hash_id == SIG_RSA_SHA256 ) ||
713 ( len == 19 + 48 && p[14] == 2 && hash_id == SIG_RSA_SHA384 ) ||
714 ( len == 19 + 64 && p[14] == 3 && hash_id == SIG_RSA_SHA512 ) )
721 memcmp( p, ASN1_HASH_SHA2X, 18 ) == 0 &&
722 memcmp( p + 19, hash, c ) == 0 )
725 return( POLARSSL_ERR_RSA_VERIFY_FAILED );
728 if( len == hashlen && hash_id == SIG_RSA_RAW )
730 if( memcmp( p, hash, hashlen ) == 0 )
733 return( POLARSSL_ERR_RSA_VERIFY_FAILED );
736 return( POLARSSL_ERR_RSA_INVALID_PADDING );
740 * Free the components of an RSA key
742 void rsa_free( rsa_context *ctx )
744 mpi_free( &ctx->RQ, &ctx->RP, &ctx->RN,
745 &ctx->QP, &ctx->DQ, &ctx->DP,
746 &ctx->Q, &ctx->P, &ctx->D,
747 &ctx->E, &ctx->N, NULL );
751 /* PDKIM code (not copied from polarssl) */
753 * Parse a public RSA key
755 OpenSSL RSA public key ASN1 container
756 0:d=0 hl=3 l= 159 cons: SEQUENCE
757 3:d=1 hl=2 l= 13 cons: SEQUENCE
758 5:d=2 hl=2 l= 9 prim: OBJECT:rsaEncryption
759 16:d=2 hl=2 l= 0 prim: NULL
760 18:d=1 hl=3 l= 141 prim: BIT STRING:RSAPublicKey (below)
762 RSAPublicKey ASN1 container
763 0:d=0 hl=3 l= 137 cons: SEQUENCE
764 3:d=1 hl=3 l= 129 prim: INTEGER:Public modulus
765 135:d=1 hl=2 l= 3 prim: INTEGER:Public exponent
768 int rsa_parse_public_key( rsa_context *rsa, unsigned char *buf, int buflen )
770 unsigned char *p, *end;
776 if( ( ret = asn1_get_tag( &p, end, &len,
777 ASN1_CONSTRUCTED | ASN1_SEQUENCE ) ) != 0 ) {
778 return( POLARSSL_ERR_X509_KEY_INVALID_FORMAT | ret );
781 if( ( ret = asn1_get_tag( &p, end, &len,
782 ASN1_CONSTRUCTED | ASN1_SEQUENCE ) ) == 0 ) {
783 /* Skip over embedded rsaEncryption Object */
786 /* The RSAPublicKey ASN1 container is wrapped in a BIT STRING */
787 if( ( ret = asn1_get_tag( &p, end, &len,
788 ASN1_BIT_STRING ) ) != 0 ) {
789 return( POLARSSL_ERR_X509_KEY_INVALID_FORMAT | ret );
792 /* Limit range to that BIT STRING */
796 if( ( ret = asn1_get_tag( &p, end, &len,
797 ASN1_CONSTRUCTED | ASN1_SEQUENCE ) ) != 0 ) {
798 return( POLARSSL_ERR_X509_KEY_INVALID_FORMAT | ret );
802 if ( ( ( ret = asn1_get_mpi( &p, end, &(rsa->N) ) ) == 0 ) &&
803 ( ( ret = asn1_get_mpi( &p, end, &(rsa->E) ) ) == 0 ) ) {
804 rsa->len = mpi_size( &rsa->N );
808 return( POLARSSL_ERR_X509_KEY_INVALID_FORMAT | ret );
812 * Parse a private RSA key
814 int rsa_parse_key( rsa_context *rsa, unsigned char *buf, int buflen,
815 unsigned char *pwd, int pwdlen )
818 unsigned char *s1, *s2;
819 unsigned char *p, *end;
821 s1 = (unsigned char *) strstr( (char *) buf,
822 "-----BEGIN RSA PRIVATE KEY-----" );
826 s2 = (unsigned char *) strstr( (char *) buf,
827 "-----END RSA PRIVATE KEY-----" );
829 if( s2 == NULL || s2 <= s1 )
830 return( POLARSSL_ERR_X509_KEY_INVALID_PEM );
833 if( *s1 == '\r' ) s1++;
834 if( *s1 == '\n' ) s1++;
835 else return( POLARSSL_ERR_X509_KEY_INVALID_PEM );
839 if( memcmp( s1, "Proc-Type: 4,ENCRYPTED", 22 ) == 0 )
841 return( POLARSSL_ERR_X509_FEATURE_UNAVAILABLE );
845 ret = base64_decode( NULL, &len, s1, s2 - s1 );
847 if( ret == POLARSSL_ERR_BASE64_INVALID_CHARACTER )
848 return( ret | POLARSSL_ERR_X509_KEY_INVALID_PEM );
850 if( ( buf = (unsigned char *) malloc( len ) ) == NULL )
853 if( ( ret = base64_decode( buf, &len, s1, s2 - s1 ) ) != 0 )
856 return( ret | POLARSSL_ERR_X509_KEY_INVALID_PEM );
863 return( POLARSSL_ERR_X509_FEATURE_UNAVAILABLE );
867 memset( rsa, 0, sizeof( rsa_context ) );
873 * RSAPrivateKey ::= SEQUENCE {
875 * modulus INTEGER, -- n
876 * publicExponent INTEGER, -- e
877 * privateExponent INTEGER, -- d
878 * prime1 INTEGER, -- p
879 * prime2 INTEGER, -- q
880 * exponent1 INTEGER, -- d mod (p-1)
881 * exponent2 INTEGER, -- d mod (q-1)
882 * coefficient INTEGER, -- (inverse of q) mod p
883 * otherPrimeInfos OtherPrimeInfos OPTIONAL
886 if( ( ret = asn1_get_tag( &p, end, &len,
887 ASN1_CONSTRUCTED | ASN1_SEQUENCE ) ) != 0 )
893 return( POLARSSL_ERR_X509_KEY_INVALID_FORMAT | ret );
898 if( ( ret = asn1_get_int( &p, end, &rsa->ver ) ) != 0 )
904 return( POLARSSL_ERR_X509_KEY_INVALID_FORMAT | ret );
913 return( ret | POLARSSL_ERR_X509_KEY_INVALID_VERSION );
916 if( ( ret = asn1_get_mpi( &p, end, &rsa->N ) ) != 0 ||
917 ( ret = asn1_get_mpi( &p, end, &rsa->E ) ) != 0 ||
918 ( ret = asn1_get_mpi( &p, end, &rsa->D ) ) != 0 ||
919 ( ret = asn1_get_mpi( &p, end, &rsa->P ) ) != 0 ||
920 ( ret = asn1_get_mpi( &p, end, &rsa->Q ) ) != 0 ||
921 ( ret = asn1_get_mpi( &p, end, &rsa->DP ) ) != 0 ||
922 ( ret = asn1_get_mpi( &p, end, &rsa->DQ ) ) != 0 ||
923 ( ret = asn1_get_mpi( &p, end, &rsa->QP ) ) != 0 )
929 return( ret | POLARSSL_ERR_X509_KEY_INVALID_FORMAT );
932 rsa->len = mpi_size( &rsa->N );
940 return( POLARSSL_ERR_X509_KEY_INVALID_FORMAT |
941 POLARSSL_ERR_ASN1_LENGTH_MISMATCH );
944 if( ( ret = rsa_check_privkey( rsa ) ) != 0 )