/** * \file ecp.h * * \brief Elliptic curves over GF(p) */ /* * Copyright (C) 2006-2015, ARM Limited, All Rights Reserved * SPDX-License-Identifier: GPL-2.0 * * 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. * * This file is part of mbed TLS (https://tls.mbed.org) */ #ifndef MBEDTLS_ECP_H #define MBEDTLS_ECP_H #if !defined(MBEDTLS_CONFIG_FILE) #include "config.h" #else #include MBEDTLS_CONFIG_FILE #endif #include "bignum.h" /* * ECP error codes */ #define MBEDTLS_ERR_ECP_BAD_INPUT_DATA -0x4F80 /**< Bad input parameters to function. */ #define MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL -0x4F00 /**< The buffer is too small to write to. */ #define MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE -0x4E80 /**< Requested curve not available. */ #define MBEDTLS_ERR_ECP_VERIFY_FAILED -0x4E00 /**< The signature is not valid. */ #define MBEDTLS_ERR_ECP_ALLOC_FAILED -0x4D80 /**< Memory allocation failed. */ #define MBEDTLS_ERR_ECP_RANDOM_FAILED -0x4D00 /**< Generation of random value, such as (ephemeral) key, failed. */ #define MBEDTLS_ERR_ECP_INVALID_KEY -0x4C80 /**< Invalid private or public key. */ #define MBEDTLS_ERR_ECP_SIG_LEN_MISMATCH -0x4C00 /**< The buffer contains a valid signature followed by more data. */ #define MBEDTLS_ERR_ECP_HW_ACCEL_FAILED -0x4B80 /**< ECP hardware accelerator failed. */ #if !defined(MBEDTLS_ECP_ALT) /* * default mbed TLS elliptic curve arithmetic implementation * * (in case MBEDTLS_ECP_ALT is defined then the developer has to provide an * alternative implementation for the whole module and it will replace this * one.) */ #ifdef __cplusplus extern "C" { #endif /** * Domain parameters (curve, subgroup and generator) identifiers. * * Only curves over prime fields are supported. * * \warning This library does not support validation of arbitrary domain * parameters. Therefore, only well-known domain parameters from trusted * sources should be used. See mbedtls_ecp_group_load(). */ typedef enum { MBEDTLS_ECP_DP_NONE = 0, MBEDTLS_ECP_DP_SECP192R1, /*!< 192-bits NIST curve */ MBEDTLS_ECP_DP_SECP224R1, /*!< 224-bits NIST curve */ MBEDTLS_ECP_DP_SECP256R1, /*!< 256-bits NIST curve */ MBEDTLS_ECP_DP_SECP384R1, /*!< 384-bits NIST curve */ MBEDTLS_ECP_DP_SECP521R1, /*!< 521-bits NIST curve */ MBEDTLS_ECP_DP_BP256R1, /*!< 256-bits Brainpool curve */ MBEDTLS_ECP_DP_BP384R1, /*!< 384-bits Brainpool curve */ MBEDTLS_ECP_DP_BP512R1, /*!< 512-bits Brainpool curve */ MBEDTLS_ECP_DP_CURVE25519, /*!< Curve25519 */ MBEDTLS_ECP_DP_SECP192K1, /*!< 192-bits "Koblitz" curve */ MBEDTLS_ECP_DP_SECP224K1, /*!< 224-bits "Koblitz" curve */ MBEDTLS_ECP_DP_SECP256K1, /*!< 256-bits "Koblitz" curve */ } mbedtls_ecp_group_id; /** * Number of supported curves (plus one for NONE). * * (Montgomery curves excluded for now.) */ #define MBEDTLS_ECP_DP_MAX 12 /** * Curve information for use by other modules */ typedef struct { mbedtls_ecp_group_id grp_id; /*!< Internal identifier */ uint16_t tls_id; /*!< TLS NamedCurve identifier */ uint16_t bit_size; /*!< Curve size in bits */ const char *name; /*!< Human-friendly name */ } mbedtls_ecp_curve_info; /** * \brief ECP point structure (jacobian coordinates) * * \note All functions expect and return points satisfying * the following condition: Z == 0 or Z == 1. (Other * values of Z are used by internal functions only.) * The point is zero, or "at infinity", if Z == 0. * Otherwise, X and Y are its standard (affine) coordinates. */ typedef struct { mbedtls_mpi X; /*!< the point's X coordinate */ mbedtls_mpi Y; /*!< the point's Y coordinate */ mbedtls_mpi Z; /*!< the point's Z coordinate */ } mbedtls_ecp_point; /** * \brief ECP group structure * * We consider two types of curves equations: * 1. Short Weierstrass y^2 = x^3 + A x + B mod P (SEC1 + RFC 4492) * 2. Montgomery, y^2 = x^3 + A x^2 + x mod P (Curve25519 + draft) * In both cases, a generator G for a prime-order subgroup is fixed. In the * short weierstrass, this subgroup is actually the whole curve, and its * cardinal is denoted by N. * * In the case of Short Weierstrass curves, our code requires that N is an odd * prime. (Use odd in mbedtls_ecp_mul() and prime in mbedtls_ecdsa_sign() for blinding.) * * In the case of Montgomery curves, we don't store A but (A + 2) / 4 which is * the quantity actually used in the formulas. Also, nbits is not the size of N * but the required size for private keys. * * If modp is NULL, reduction modulo P is done using a generic algorithm. * Otherwise, it must point to a function that takes an mbedtls_mpi in the range * 0..2^(2*pbits)-1 and transforms it in-place in an integer of little more * than pbits, so that the integer may be efficiently brought in the 0..P-1 * range by a few additions or substractions. It must return 0 on success and * non-zero on failure. */ typedef struct { mbedtls_ecp_group_id id; /*!< internal group identifier */ mbedtls_mpi P; /*!< prime modulus of the base field */ mbedtls_mpi A; /*!< 1. A in the equation, or 2. (A + 2) / 4 */ mbedtls_mpi B; /*!< 1. B in the equation, or 2. unused */ mbedtls_ecp_point G; /*!< generator of the (sub)group used */ mbedtls_mpi N; /*!< 1. the order of G, or 2. unused */ size_t pbits; /*!< number of bits in P */ size_t nbits; /*!< number of bits in 1. P, or 2. private keys */ unsigned int h; /*!< internal: 1 if the constants are static */ int (*modp)(mbedtls_mpi *); /*!< function for fast reduction mod P */ int (*t_pre)(mbedtls_ecp_point *, void *); /*!< unused */ int (*t_post)(mbedtls_ecp_point *, void *); /*!< unused */ void *t_data; /*!< unused */ mbedtls_ecp_point *T; /*!< pre-computed points for ecp_mul_comb() */ size_t T_size; /*!< number for pre-computed points */ } mbedtls_ecp_group; /** * \brief ECP key pair structure * * A generic key pair that could be used for ECDSA, fixed ECDH, etc. * * \note Members purposefully in the same order as struc mbedtls_ecdsa_context. */ typedef struct { mbedtls_ecp_group grp; /*!< Elliptic curve and base point */ mbedtls_mpi d; /*!< our secret value */ mbedtls_ecp_point Q; /*!< our public value */ } mbedtls_ecp_keypair; /** * \name SECTION: Module settings * * The configuration options you can set for this module are in this section. * Either change them in config.h or define them on the compiler command line. * \{ */ #if !defined(MBEDTLS_ECP_MAX_BITS) /** * Maximum size of the groups (that is, of N and P) */ #define MBEDTLS_ECP_MAX_BITS 521 /**< Maximum bit size of groups */ #endif #define MBEDTLS_ECP_MAX_BYTES ( ( MBEDTLS_ECP_MAX_BITS + 7 ) / 8 ) #define MBEDTLS_ECP_MAX_PT_LEN ( 2 * MBEDTLS_ECP_MAX_BYTES + 1 ) #if !defined(MBEDTLS_ECP_WINDOW_SIZE) /* * Maximum "window" size used for point multiplication. * Default: 6. * Minimum value: 2. Maximum value: 7. * * Result is an array of at most ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) ) * points used for point multiplication. This value is directly tied to EC * peak memory usage, so decreasing it by one should roughly cut memory usage * by two (if large curves are in use). * * Reduction in size may reduce speed, but larger curves are impacted first. * Sample performances (in ECDHE handshakes/s, with FIXED_POINT_OPTIM = 1): * w-size: 6 5 4 3 2 * 521 145 141 135 120 97 * 384 214 209 198 177 146 * 256 320 320 303 262 226 * 224 475 475 453 398 342 * 192 640 640 633 587 476 */ #define MBEDTLS_ECP_WINDOW_SIZE 6 /**< Maximum window size used */ #endif /* MBEDTLS_ECP_WINDOW_SIZE */ #if !defined(MBEDTLS_ECP_FIXED_POINT_OPTIM) /* * Trade memory for speed on fixed-point multiplication. * * This speeds up repeated multiplication of the generator (that is, the * multiplication in ECDSA signatures, and half of the multiplications in * ECDSA verification and ECDHE) by a factor roughly 3 to 4. * * The cost is increasing EC peak memory usage by a factor roughly 2. * * Change this value to 0 to reduce peak memory usage. */ #define MBEDTLS_ECP_FIXED_POINT_OPTIM 1 /**< Enable fixed-point speed-up */ #endif /* MBEDTLS_ECP_FIXED_POINT_OPTIM */ /* \} name SECTION: Module settings */ /* * Point formats, from RFC 4492's enum ECPointFormat */ #define MBEDTLS_ECP_PF_UNCOMPRESSED 0 /**< Uncompressed point format */ #define MBEDTLS_ECP_PF_COMPRESSED 1 /**< Compressed point format */ /* * Some other constants from RFC 4492 */ #define MBEDTLS_ECP_TLS_NAMED_CURVE 3 /**< ECCurveType's named_curve */ /** * \brief Get the list of supported curves in order of preferrence * (full information) * * \return A statically allocated array, the last entry is 0. */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list( void ); /** * \brief Get the list of supported curves in order of preferrence * (grp_id only) * * \return A statically allocated array, * terminated with MBEDTLS_ECP_DP_NONE. */ const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list( void ); /** * \brief Get curve information from an internal group identifier * * \param grp_id A MBEDTLS_ECP_DP_XXX value * * \return The associated curve information or NULL */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id( mbedtls_ecp_group_id grp_id ); /** * \brief Get curve information from a TLS NamedCurve value * * \param tls_id A MBEDTLS_ECP_DP_XXX value * * \return The associated curve information or NULL */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id( uint16_t tls_id ); /** * \brief Get curve information from a human-readable name * * \param name The name * * \return The associated curve information or NULL */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name( const char *name ); /** * \brief Initialize a point (as zero) */ void mbedtls_ecp_point_init( mbedtls_ecp_point *pt ); /** * \brief Initialize a group (to something meaningless) */ void mbedtls_ecp_group_init( mbedtls_ecp_group *grp ); /** * \brief Initialize a key pair (as an invalid one) */ void mbedtls_ecp_keypair_init( mbedtls_ecp_keypair *key ); /** * \brief Free the components of a point */ void mbedtls_ecp_point_free( mbedtls_ecp_point *pt ); /** * \brief Free the components of an ECP group */ void mbedtls_ecp_group_free( mbedtls_ecp_group *grp ); /** * \brief Free the components of a key pair */ void mbedtls_ecp_keypair_free( mbedtls_ecp_keypair *key ); /** * \brief Copy the contents of point Q into P * * \param P Destination point * \param Q Source point * * \return 0 if successful, * MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed */ int mbedtls_ecp_copy( mbedtls_ecp_point *P, const mbedtls_ecp_point *Q ); /** * \brief Copy the contents of a group object * * \param dst Destination group * \param src Source group * * \return 0 if successful, * MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed */ int mbedtls_ecp_group_copy( mbedtls_ecp_group *dst, const mbedtls_ecp_group *src ); /** * \brief Set a point to zero * * \param pt Destination point * * \return 0 if successful, * MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed */ int mbedtls_ecp_set_zero( mbedtls_ecp_point *pt ); /** * \brief Tell if a point is zero * * \param pt Point to test * * \return 1 if point is zero, 0 otherwise */ int mbedtls_ecp_is_zero( mbedtls_ecp_point *pt ); /** * \brief Compare two points * * \note This assumes the points are normalized. Otherwise, * they may compare as "not equal" even if they are. * * \param P First point to compare * \param Q Second point to compare * * \return 0 if the points are equal, * MBEDTLS_ERR_ECP_BAD_INPUT_DATA otherwise */ int mbedtls_ecp_point_cmp( const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q ); /** * \brief Import a non-zero point from two ASCII strings * * \param P Destination point * \param radix Input numeric base * \param x First affine coordinate as a null-terminated string * \param y Second affine coordinate as a null-terminated string * * \return 0 if successful, or a MBEDTLS_ERR_MPI_XXX error code */ int mbedtls_ecp_point_read_string( mbedtls_ecp_point *P, int radix, const char *x, const char *y ); /** * \brief Export a point into unsigned binary data * * \param grp Group to which the point should belong * \param P Point to export * \param format Point format, should be a MBEDTLS_ECP_PF_XXX macro * \param olen Length of the actual output * \param buf Output buffer * \param buflen Length of the output buffer * * \return 0 if successful, * or MBEDTLS_ERR_ECP_BAD_INPUT_DATA * or MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL */ int mbedtls_ecp_point_write_binary( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *P, int format, size_t *olen, unsigned char *buf, size_t buflen ); /** * \brief Import a point from unsigned binary data * * \param grp Group to which the point should belong * \param P Point to import * \param buf Input buffer * \param ilen Actual length of input * * \return 0 if successful, * MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid, * MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed, * MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the point format * is not implemented. * * \note This function does NOT check that the point actually * belongs to the given group, see mbedtls_ecp_check_pubkey() for * that. */ int mbedtls_ecp_point_read_binary( const mbedtls_ecp_group *grp, mbedtls_ecp_point *P, const unsigned char *buf, size_t ilen ); /** * \brief Import a point from a TLS ECPoint record * * \param grp ECP group used * \param pt Destination point * \param buf $(Start of input buffer) * \param len Buffer length * * \note buf is updated to point right after the ECPoint on exit * * \return 0 if successful, * MBEDTLS_ERR_MPI_XXX if initialization failed * MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid */ int mbedtls_ecp_tls_read_point( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt, const unsigned char **buf, size_t len ); /** * \brief Export a point as a TLS ECPoint record * * \param grp ECP group used * \param pt Point to export * \param format Export format * \param olen length of data written * \param buf Buffer to write to * \param blen Buffer length * * \return 0 if successful, * or MBEDTLS_ERR_ECP_BAD_INPUT_DATA * or MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL */ int mbedtls_ecp_tls_write_point( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt, int format, size_t *olen, unsigned char *buf, size_t blen ); /** * \brief Set a group using well-known domain parameters * * \param grp Destination group * \param id Index in the list of well-known domain parameters * * \return 0 if successful, * MBEDTLS_ERR_MPI_XXX if initialization failed * MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE for unkownn groups * * \note Index should be a value of RFC 4492's enum NamedCurve, * usually in the form of a MBEDTLS_ECP_DP_XXX macro. */ int mbedtls_ecp_group_load( mbedtls_ecp_group *grp, mbedtls_ecp_group_id id ); /** * \brief Set a group from a TLS ECParameters record * * \param grp Destination group * \param buf &(Start of input buffer) * \param len Buffer length * * \note buf is updated to point right after ECParameters on exit * * \return 0 if successful, * MBEDTLS_ERR_MPI_XXX if initialization failed * MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid */ int mbedtls_ecp_tls_read_group( mbedtls_ecp_group *grp, const unsigned char **buf, size_t len ); /** * \brief Write the TLS ECParameters record for a group * * \param grp ECP group used * \param olen Number of bytes actually written * \param buf Buffer to write to * \param blen Buffer length * * \return 0 if successful, * or MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL */ int mbedtls_ecp_tls_write_group( const mbedtls_ecp_group *grp, size_t *olen, unsigned char *buf, size_t blen ); /** * \brief Multiplication by an integer: R = m * P * (Not thread-safe to use same group in multiple threads) * * \note In order to prevent timing attacks, this function * executes the exact same sequence of (base field) * operations for any valid m. It avoids any if-branch or * array index depending on the value of m. * * \note If f_rng is not NULL, it is used to randomize intermediate * results in order to prevent potential timing attacks * targeting these results. It is recommended to always * provide a non-NULL f_rng (the overhead is negligible). * * \param grp ECP group * \param R Destination point * \param m Integer by which to multiply * \param P Point to multiply * \param f_rng RNG function (see notes) * \param p_rng RNG parameter * * \return 0 if successful, * MBEDTLS_ERR_ECP_INVALID_KEY if m is not a valid privkey * or P is not a valid pubkey, * MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed */ int mbedtls_ecp_mul( mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ); /** * \brief Multiplication and addition of two points by integers: * R = m * P + n * Q * (Not thread-safe to use same group in multiple threads) * * \note In contrast to mbedtls_ecp_mul(), this function does not guarantee * a constant execution flow and timing. * * \param grp ECP group * \param R Destination point * \param m Integer by which to multiply P * \param P Point to multiply by m * \param n Integer by which to multiply Q * \param Q Point to be multiplied by n * * \return 0 if successful, * MBEDTLS_ERR_ECP_INVALID_KEY if m or n is not a valid privkey * or P or Q is not a valid pubkey, * MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed */ int mbedtls_ecp_muladd( mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, const mbedtls_mpi *n, const mbedtls_ecp_point *Q ); /** * \brief Check that a point is a valid public key on this curve * * \param grp Curve/group the point should belong to * \param pt Point to check * * \return 0 if point is a valid public key, * MBEDTLS_ERR_ECP_INVALID_KEY otherwise. * * \note This function only checks the point is non-zero, has valid * coordinates and lies on the curve, but not that it is * indeed a multiple of G. This is additional check is more * expensive, isn't required by standards, and shouldn't be * necessary if the group used has a small cofactor. In * particular, it is useless for the NIST groups which all * have a cofactor of 1. * * \note Uses bare components rather than an mbedtls_ecp_keypair structure * in order to ease use with other structures such as * mbedtls_ecdh_context of mbedtls_ecdsa_context. */ int mbedtls_ecp_check_pubkey( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt ); /** * \brief Check that an mbedtls_mpi is a valid private key for this curve * * \param grp Group used * \param d Integer to check * * \return 0 if point is a valid private key, * MBEDTLS_ERR_ECP_INVALID_KEY otherwise. * * \note Uses bare components rather than an mbedtls_ecp_keypair structure * in order to ease use with other structures such as * mbedtls_ecdh_context of mbedtls_ecdsa_context. */ int mbedtls_ecp_check_privkey( const mbedtls_ecp_group *grp, const mbedtls_mpi *d ); /** * \brief Generate a private key * * \param grp ECP group * \param d Destination MPI (secret part) * \param f_rng RNG function * \param p_rng RNG parameter * * \return 0 if successful, * or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code */ int mbedtls_ecp_gen_privkey( const mbedtls_ecp_group *grp, mbedtls_mpi *d, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ); /** * \brief Generate a keypair with configurable base point * * \param grp ECP group * \param G Chosen base point * \param d Destination MPI (secret part) * \param Q Destination point (public part) * \param f_rng RNG function * \param p_rng RNG parameter * * \return 0 if successful, * or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code * * \note Uses bare components rather than an mbedtls_ecp_keypair structure * in order to ease use with other structures such as * mbedtls_ecdh_context of mbedtls_ecdsa_context. */ int mbedtls_ecp_gen_keypair_base( mbedtls_ecp_group *grp, const mbedtls_ecp_point *G, mbedtls_mpi *d, mbedtls_ecp_point *Q, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ); /** * \brief Generate a keypair * * \param grp ECP group * \param d Destination MPI (secret part) * \param Q Destination point (public part) * \param f_rng RNG function * \param p_rng RNG parameter * * \return 0 if successful, * or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code * * \note Uses bare components rather than an mbedtls_ecp_keypair structure * in order to ease use with other structures such as * mbedtls_ecdh_context of mbedtls_ecdsa_context. */ int mbedtls_ecp_gen_keypair( mbedtls_ecp_group *grp, mbedtls_mpi *d, mbedtls_ecp_point *Q, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ); /** * \brief Generate a keypair * * \param grp_id ECP group identifier * \param key Destination keypair * \param f_rng RNG function * \param p_rng RNG parameter * * \return 0 if successful, * or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code */ int mbedtls_ecp_gen_key( mbedtls_ecp_group_id grp_id, mbedtls_ecp_keypair *key, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ); /** * \brief Check a public-private key pair * * \param pub Keypair structure holding a public key * \param prv Keypair structure holding a private (plus public) key * * \return 0 if successful (keys are valid and match), or * MBEDTLS_ERR_ECP_BAD_INPUT_DATA, or * a MBEDTLS_ERR_ECP_XXX or MBEDTLS_ERR_MPI_XXX code. */ int mbedtls_ecp_check_pub_priv( const mbedtls_ecp_keypair *pub, const mbedtls_ecp_keypair *prv ); #if defined(MBEDTLS_SELF_TEST) /** * \brief Checkup routine * * \return 0 if successful, or 1 if a test failed */ int mbedtls_ecp_self_test( int verbose ); #endif /* MBEDTLS_SELF_TEST */ #ifdef __cplusplus } #endif #else /* MBEDTLS_ECP_ALT */ #include "ecp_alt.h" #endif /* MBEDTLS_ECP_ALT */ #endif /* ecp.h */