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 /**
  * \file ecp.h
  *
  * \brief This file provides an API for Elliptic Curves over GF(P) (ECP).
  *
  * The use of ECP in cryptography and TLS is defined in
  * <em>Standards for Efficient Cryptography Group (SECG): SEC1
  * Elliptic Curve Cryptography</em> and
  * <em>RFC-4492: Elliptic Curve Cryptography (ECC) Cipher Suites
  * for Transport Layer Security (TLS)</em>.
  *
  * <em>RFC-2409: The Internet Key Exchange (IKE)</em> defines ECP
  * group types.
  *
  */
 
 /*
  *  Copyright The Mbed TLS Contributors
  *  SPDX-License-Identifier: Apache-2.0 OR GPL-2.0-or-later
  */
 
 #ifndef MBEDTLS_ECP_H
 #define MBEDTLS_ECP_H
 #include "mbedtls/private_access.h"
 
 #include "tf-psa-crypto/build_info.h"
 #include "mbedtls/platform_util.h"
 
 #include "mbedtls/private/bignum.h"
 
 /*
  * ECP error codes
  */
 /** Bad input parameters to function. */
 #define MBEDTLS_ERR_ECP_BAD_INPUT_DATA                    PSA_ERROR_INVALID_ARGUMENT
 /** The buffer is too small to write to. */
 #define MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL                  PSA_ERROR_BUFFER_TOO_SMALL
 /** The requested feature is not available, for example, the requested curve is not supported. */
 #define MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE               -0x4E80
 /** The signature is not valid. */
 #define MBEDTLS_ERR_ECP_VERIFY_FAILED                     PSA_ERROR_INVALID_SIGNATURE
 /** Memory allocation failed. */
 #define MBEDTLS_ERR_ECP_ALLOC_FAILED                      PSA_ERROR_INSUFFICIENT_MEMORY
 /** Generation of random value, such as ephemeral key, failed. */
 #define MBEDTLS_ERR_ECP_RANDOM_FAILED                     -0x4D00
 /** Invalid private or public key. */
 #define MBEDTLS_ERR_ECP_INVALID_KEY                       -0x4C80
 /** Operation in progress, call again with the same parameters to continue. */
 #define MBEDTLS_ERR_ECP_IN_PROGRESS                       PSA_OPERATION_INCOMPLETE
 
 #if defined(MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS)
 /* Flags indicating whether to include code that is specific to certain
  * types of curves. These flags are for internal library use only. */
 #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_BP256R1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_BP384R1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_BP512R1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED) || \
     defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
 #define MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED
 #endif
 #if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) || \
     defined(MBEDTLS_ECP_DP_CURVE448_ENABLED)
 #define MBEDTLS_ECP_MONTGOMERY_ENABLED
 #endif /* MBEDTLS_ECP_DP_CURVE25519_ENABLED OR MBEDTLS_ECP_DP_CURVE448_ENABLED */
 #endif /* MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS */
 
 #ifdef __cplusplus
 extern "C" {
 #endif
 
 /**
  * Domain-parameter identifiers: curve, subgroup, and generator.
  *
  * \note Only curves over prime fields are supported.
  *
  * \warning This library does not support validation of arbitrary domain
  * parameters. Therefore, only standardized domain parameters from trusted
  * sources should be used. See mbedtls_ecp_group_load().
  */
 /* Note: when adding a new curve:
  * - Add it at the end of this enum, otherwise you'll break the ABI by
  *   changing the numerical value for existing curves.
  * - Increment MBEDTLS_ECP_DP_MAX below if needed.
  * - Update the calculation of MBEDTLS_ECP_MAX_BITS below.
  * - Add the corresponding MBEDTLS_ECP_DP_xxx_ENABLED macro definition to
  *   mbedtls_config.h.
  * - List the curve as a dependency of MBEDTLS_ECP_C and
  *   MBEDTLS_ECDSA_C if supported in check_config.h.
  * - Add the curve to the appropriate curve type macro
  *   MBEDTLS_ECP_yyy_ENABLED above.
  * - Add the necessary definitions to ecp_curves.c.
  * - Add the curve to the ecp_supported_curves array in ecp.c.
  * - Add the curve to applicable profiles in x509_crt.c.
  * - Add the curve to applicable presets in ssl_tls.c.
  */
 typedef enum {
     MBEDTLS_ECP_DP_NONE = 0,       /*!< Curve not defined. */
     MBEDTLS_ECP_DP_SECP192R1,      /*!< Domain parameters for the 192-bit curve defined by FIPS 186-4 and SEC1. */
     MBEDTLS_ECP_DP_SECP256R1,      /*!< Domain parameters for the 256-bit curve defined by FIPS 186-4 and SEC1. */
     MBEDTLS_ECP_DP_SECP384R1,      /*!< Domain parameters for the 384-bit curve defined by FIPS 186-4 and SEC1. */
     MBEDTLS_ECP_DP_SECP521R1,      /*!< Domain parameters for the 521-bit curve defined by FIPS 186-4 and SEC1. */
     MBEDTLS_ECP_DP_BP256R1,        /*!< Domain parameters for 256-bit Brainpool curve. */
     MBEDTLS_ECP_DP_BP384R1,        /*!< Domain parameters for 384-bit Brainpool curve. */
     MBEDTLS_ECP_DP_BP512R1,        /*!< Domain parameters for 512-bit Brainpool curve. */
     MBEDTLS_ECP_DP_CURVE25519,     /*!< Domain parameters for Curve25519. */
     MBEDTLS_ECP_DP_SECP192K1,      /*!< Domain parameters for 192-bit "Koblitz" curve. */
     MBEDTLS_ECP_DP_SECP256K1,      /*!< Domain parameters for 256-bit "Koblitz" curve. */
     MBEDTLS_ECP_DP_CURVE448,       /*!< Domain parameters for Curve448. */
 } mbedtls_ecp_group_id;
 
 #if defined(MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS)
 /**
  * The number of supported curves, plus one for #MBEDTLS_ECP_DP_NONE.
  */
 #define MBEDTLS_ECP_DP_MAX     14
 
 /*
  * Curve types
  */
 typedef enum {
     MBEDTLS_ECP_TYPE_NONE = 0,
     MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS,    /* y^2 = x^3 + a x + b      */
     MBEDTLS_ECP_TYPE_MONTGOMERY,           /* y^2 = x^3 + a x^2 + x    */
 } mbedtls_ecp_curve_type;
 
 /**
  * Curve information, for use by other modules.
  *
  * The fields of this structure are part of the public API and can be
  * accessed directly by applications. Future versions of the library may
  * add extra fields or reorder existing fields.
  */
 typedef struct mbedtls_ecp_curve_info {
     mbedtls_ecp_group_id grp_id;    /*!< An internal identifier. */
     uint16_t tls_id;                /*!< The TLS NamedCurve identifier. */
     uint16_t bit_size;              /*!< The curve size in bits. */
     const char *name;               /*!< A human-friendly name. */
 } mbedtls_ecp_curve_info;
 #endif /* MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS */
 
 /**
  * \brief           The ECP point structure, in Jacobian coordinates.
  *
  * \note            All functions expect and return points satisfying
  *                  the following condition: <code>Z == 0</code> or
  *                  <code>Z == 1</code>. Other values of \p Z are
  *                  used only by internal functions.
  *                  The point is zero, or "at infinity", if <code>Z == 0</code>.
  *                  Otherwise, \p X and \p Y are its standard (affine)
  *                  coordinates.
  */
 typedef struct mbedtls_ecp_point {
     mbedtls_mpi MBEDTLS_PRIVATE(X);          /*!< The X coordinate of the ECP point. */
     mbedtls_mpi MBEDTLS_PRIVATE(Y);          /*!< The Y coordinate of the ECP point. */
     mbedtls_mpi MBEDTLS_PRIVATE(Z);          /*!< The Z coordinate of the ECP point. */
 }
 mbedtls_ecp_point;
 
 #define MBEDTLS_ECP_POINT_INIT { MBEDTLS_MPI_INIT, MBEDTLS_MPI_INIT, MBEDTLS_MPI_INIT }
 
 /**
  * \brief           The ECP group structure.
  *
  * We consider two types of curve equations:
  * <ul><li>Short Weierstrass: <code>y^2 = x^3 + A x + B mod P</code>
  * (SEC1 + RFC-4492)</li>
  * <li>Montgomery: <code>y^2 = x^3 + A x^2 + x mod P</code> (Curve25519,
  * Curve448)</li></ul>
  * In both cases, the generator (\p G) for a prime-order subgroup is fixed.
  *
  * For Short Weierstrass, this subgroup is the whole curve, and its
  * cardinality is denoted by \p N. Our code requires that \p N is an
  * odd prime as mbedtls_ecp_mul() requires an odd number, and
  * mbedtls_ecdsa_sign() requires that it is prime for blinding purposes.
  *
  * The default implementation only initializes \p A without setting it to the
  * authentic value for curves with <code>A = -3</code>(SECP256R1, etc), in which
  * case you need to load \p A by yourself when using domain parameters directly,
  * for example:
  * \code
  * mbedtls_mpi_init(&A);
  * mbedtls_ecp_group_init(&grp);
  * CHECK_RETURN(mbedtls_ecp_group_load(&grp, grp_id));
  * if (mbedtls_ecp_group_a_is_minus_3(&grp)) {
  *     CHECK_RETURN(mbedtls_mpi_sub_int(&A, &grp.P, 3));
  * } else {
  *     CHECK_RETURN(mbedtls_mpi_copy(&A, &grp.A));
  * }
  *
  * do_something_with_a(&A);
  *
  * cleanup:
  * mbedtls_mpi_free(&A);
  * mbedtls_ecp_group_free(&grp);
  * \endcode
  *
  * For Montgomery curves, we do not store \p A, but <code>(A + 2) / 4</code>,
  * which is the quantity used in the formulas. Additionally, \p nbits is
  * not the size of \p N but the required size for private keys.
  *
  * If \p modp is NULL, reduction modulo \p P is done using a generic algorithm.
  * Otherwise, \p modp must point to a function that takes an \p mbedtls_mpi in the
  * range of <code>0..2^(2*pbits)-1</code>, and transforms it in-place to an integer
  * which is congruent mod \p P to the given MPI, and is close enough to \p pbits
  * in size, so that it may be efficiently brought in the 0..P-1 range by a few
  * additions or subtractions. Therefore, it is only an approximate modular
  * reduction. It must return 0 on success and non-zero on failure.
  *
  * \note        Alternative implementations of the ECP module must obey the
  *              following constraints.
  *              * Group IDs must be distinct: if two group structures have
  *                the same ID, then they must be identical.
  *              * The fields \c id, \c P, \c A, \c B, \c G, \c N,
  *                \c pbits and \c nbits must have the same type and semantics
  *                as in the built-in implementation.
  *                They must be available for reading, but direct modification
  *                of these fields does not need to be supported.
  *                They do not need to be at the same offset in the structure.
  */
 typedef struct mbedtls_ecp_group {
     mbedtls_ecp_group_id id;    /*!< An internal group identifier. */
     mbedtls_mpi P;              /*!< The prime modulus of the base field. */
     mbedtls_mpi A;              /*!< For Short Weierstrass: \p A in the equation. Note that
                                      \p A is not set to the authentic value in some cases.
                                      Refer to detailed description of ::mbedtls_ecp_group if
                                      using domain parameters in the structure.
                                      For Montgomery curves: <code>(A + 2) / 4</code>. */
     mbedtls_mpi B;              /*!< For Short Weierstrass: \p B in the equation.
                                      For Montgomery curves: unused. */
     mbedtls_ecp_point G;        /*!< The generator of the subgroup used. */
     mbedtls_mpi N;              /*!< The order of \p G. */
     size_t pbits;               /*!< The number of bits in \p P.*/
     size_t nbits;               /*!< For Short Weierstrass: The number of bits in \p P.
                                      For Montgomery curves: the number of bits in the
                                      private keys. */
     /* End of public fields */
 
     unsigned int MBEDTLS_PRIVATE(h);             /*!< \internal 1 if the constants are static. */
     int(*MBEDTLS_PRIVATE(modp))(mbedtls_mpi *);  /*!< The function for fast pseudo-reduction
                                                     mod \p P (see above).*/
     int(*MBEDTLS_PRIVATE(t_pre))(mbedtls_ecp_point *, void *);   /*!< Unused. */
     int(*MBEDTLS_PRIVATE(t_post))(mbedtls_ecp_point *, void *);  /*!< Unused. */
     void *MBEDTLS_PRIVATE(t_data);               /*!< Unused. */
     mbedtls_ecp_point *MBEDTLS_PRIVATE(T);       /*!< Pre-computed points for ecp_mul_comb(). */
     size_t MBEDTLS_PRIVATE(T_size);              /*!< The number of dynamic allocated pre-computed points. */
 }
 mbedtls_ecp_group;
 
 #define MBEDTLS_ECP_GROUP_INIT { MBEDTLS_ECP_DP_NONE, MBEDTLS_MPI_INIT, MBEDTLS_MPI_INIT, \
                                  MBEDTLS_MPI_INIT, MBEDTLS_ECP_POINT_INIT, MBEDTLS_MPI_INIT, \
                                  0, 0, 0, NULL, NULL, NULL, NULL, NULL, 0 }
 
 /**
  * \name SECTION: Module settings
  *
  * The configuration options you can set for this module are in this section.
  * Either change them in mbedtls_config.h, or define them using the compiler command line.
  * \{
  */
 
 #if !defined(MBEDTLS_ECP_WINDOW_SIZE)
 /*
  * Maximum "window" size used for point multiplication.
  * Default: a point where higher memory usage yields diminishing performance
  *          returns.
  * 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    4   /**< The maximum window size used. */
 #endif /* MBEDTLS_ECP_WINDOW_SIZE */
 
 #if !defined(MBEDTLS_ECP_FIXED_POINT_OPTIM)
 /*
  * Trade code size 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.
  *
  * For each n-bit Short Weierstrass curve that is enabled, this adds 4n bytes
  * of code size if n < 384 and 8n otherwise.
  *
  * Change this value to 0 to reduce code size.
  */
 #define MBEDTLS_ECP_FIXED_POINT_OPTIM  1   /**< Enable fixed-point speed-up. */
 #endif /* MBEDTLS_ECP_FIXED_POINT_OPTIM */
 
 /** \} name SECTION: Module settings */
 
 /**
  * The maximum size of the groups, that is, of \c N and \c P.
  */
 #if !defined(MBEDTLS_ECP_LIGHT)
 /* Dummy definition to help code that has optional ECP support and
  * defines an MBEDTLS_ECP_MAX_BYTES-sized array unconditionally. */
 #define MBEDTLS_ECP_MAX_BITS 1
 /* Note: the curves must be listed in DECREASING size! */
 #elif defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 521
 #elif defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 512
 #elif defined(MBEDTLS_ECP_DP_CURVE448_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 448
 #elif defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 384
 #elif defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 384
 #elif defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 256
 #elif defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 256
 #elif defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 256
 #elif defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 255
 #elif defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 192
 #elif defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
 #define MBEDTLS_ECP_MAX_BITS 192
 #else /* !MBEDTLS_ECP_LIGHT */
 #error "Missing definition of MBEDTLS_ECP_MAX_BITS"
 #endif /* !MBEDTLS_ECP_LIGHT */
 
 #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_RESTARTABLE)
 
 /**
  * \brief           Internal restart context for multiplication
  *
  * \note            Opaque struct
  */
 typedef struct mbedtls_ecp_restart_mul mbedtls_ecp_restart_mul_ctx;
 
 /**
  * \brief           Internal restart context for ecp_muladd()
  *
  * \note            Opaque struct
  */
 typedef struct mbedtls_ecp_restart_muladd mbedtls_ecp_restart_muladd_ctx;
 
 /**
  * \brief           General context for resuming ECC operations
  */
 typedef struct {
     unsigned MBEDTLS_PRIVATE(ops_done);                  /*!<  current ops count             */
     unsigned MBEDTLS_PRIVATE(depth);                     /*!<  call depth (0 = top-level)    */
     mbedtls_ecp_restart_mul_ctx *MBEDTLS_PRIVATE(rsm);   /*!<  ecp_mul_comb() sub-context    */
     mbedtls_ecp_restart_muladd_ctx *MBEDTLS_PRIVATE(ma); /*!<  ecp_muladd() sub-context      */
 } mbedtls_ecp_restart_ctx;
 
 #define MBEDTLS_ECP_RESTART_INIT { 0, 0, NULL, NULL }
 
 #if defined(MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS)
 /*
  * Operation counts for restartable functions
  */
 #define MBEDTLS_ECP_OPS_CHK   3 /*!< basic ops count for ecp_check_pubkey()  */
 #define MBEDTLS_ECP_OPS_DBL   8 /*!< basic ops count for ecp_double_jac()    */
 #define MBEDTLS_ECP_OPS_ADD  11 /*!< basic ops count for see ecp_add_mixed() */
 #define MBEDTLS_ECP_OPS_INV 120 /*!< empirical equivalent for mpi_mod_inv()  */
 
 /**
  * \brief           Internal; for restartable functions in other modules.
  *                  Check and update basic ops budget.
  *
  * \param grp       Group structure
  * \param rs_ctx    Restart context
  * \param ops       Number of basic ops to do
  *
  * \return          \c 0 if doing \p ops basic ops is still allowed,
  * \return          #MBEDTLS_ERR_ECP_IN_PROGRESS otherwise.
  */
 int mbedtls_ecp_check_budget(const mbedtls_ecp_group *grp,
                              mbedtls_ecp_restart_ctx *rs_ctx,
                              unsigned ops);
 
 /* Utility macro for checking and updating ops budget */
 #define MBEDTLS_ECP_BUDGET(ops)   \
     MBEDTLS_MPI_CHK(mbedtls_ecp_check_budget(grp, rs_ctx, \
                                              (unsigned) (ops)));
 #endif /* MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS */
 
 #else /* MBEDTLS_ECP_RESTARTABLE */
 
 #if defined(MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS)
 #define MBEDTLS_ECP_BUDGET(ops)     /* no-op; for compatibility */
 #endif /* MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS */
 
 /* We want to declare restartable versions of existing functions anyway */
 typedef void mbedtls_ecp_restart_ctx;
 
 #endif /* MBEDTLS_ECP_RESTARTABLE */
 
 /**
  * \brief    The ECP key-pair structure.
  *
  * A generic key-pair that may be used for ECDSA and fixed ECDH, for example.
  *
  * \note    Members are deliberately in the same order as in the
  *          ::mbedtls_ecdsa_context structure.
  */
 typedef struct mbedtls_ecp_keypair {
     mbedtls_ecp_group MBEDTLS_PRIVATE(grp);      /*!<  Elliptic curve and base point     */
     mbedtls_mpi MBEDTLS_PRIVATE(d);              /*!<  our secret value                  */
     mbedtls_ecp_point MBEDTLS_PRIVATE(Q);        /*!<  our public value                  */
 }
 mbedtls_ecp_keypair;
 
 #define MBEDTLS_ECP_KEYPAIR_INIT { MBEDTLS_ECP_GROUP_INIT, MBEDTLS_MPI_INIT, \
                                    MBEDTLS_ECP_POINT_INIT }
 
 /**
  * The uncompressed point format for Short Weierstrass curves
  * (MBEDTLS_ECP_DP_SECP_XXX and MBEDTLS_ECP_DP_BP_XXX).
  */
 #define MBEDTLS_ECP_PF_UNCOMPRESSED    0
 /**
  * The compressed point format for Short Weierstrass curves
  * (MBEDTLS_ECP_DP_SECP_XXX and MBEDTLS_ECP_DP_BP_XXX).
  */
 #define MBEDTLS_ECP_PF_COMPRESSED      1
 
 /*
  * Some other constants from RFC 4492
  */
 #define MBEDTLS_ECP_TLS_NAMED_CURVE    3   /**< The named_curve of ECCurveType. */
 
 #if defined(MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS)
 #if defined(MBEDTLS_ECP_RESTARTABLE)
 /**
  * \brief           Set the maximum number of basic operations done in a row.
  *
  *                  If more operations are needed to complete a computation,
  *                  #MBEDTLS_ERR_ECP_IN_PROGRESS will be returned by the
  *                  function performing the computation. It is then the
  *                  caller's responsibility to either call again with the same
  *                  parameters until it returns 0 or an error code; or to free
  *                  the restart context if the operation is to be aborted.
  *
  *                  It is strictly required that all input parameters and the
  *                  restart context be the same on successive calls for the
  *                  same operation, but output parameters need not be the
  *                  same; they must not be used until the function finally
  *                  returns 0.
  *
  *                  This only applies to functions whose documentation
  *                  mentions they may return #MBEDTLS_ERR_ECP_IN_PROGRESS.
  *                  For functions that accept a "restart context"
  *                  argument, passing NULL disables restart and makes the
  *                  function equivalent to the function with the same name
  *                  with \c _restartable removed. For functions in the ECDH
  *                  module, restart is disabled unless the function accepts
  *                  an "ECDH context" argument and
  *                  mbedtls_ecdh_enable_restart() was previously called on
  *                  that context. For function in the SSL module, restart is
  *                  only enabled for specific sides and key exchanges
  *                  (currently only for clients and ECDHE-ECDSA).
  *
  * \warning         Using the PSA interruptible interfaces with keys in local
  *                  storage and no accelerator driver will also call this
  *                  function to set the values specified via those interfaces,
  *                  overwriting values previously set. Care should be taken if
  *                  mixing these two interfaces.
  *
  * \param max_ops   Maximum number of basic operations done in a row.
  *                  Default: 0 (unlimited).
  *                  Lower (non-zero) values mean ECC functions will block for
  *                  a lesser maximum amount of time.
  *
  * \note            A "basic operation" is defined as a rough equivalent of a
  *                  multiplication in GF(p) for the NIST P-256 curve.
  *                  As an indication, with default settings, a scalar
  *                  multiplication (full run of \c mbedtls_ecp_mul()) is:
  *                  - about 3300 basic operations for P-256
  *                  - about 9400 basic operations for P-384
  *
  * \note            Very low values are not always respected: sometimes
  *                  functions need to block for a minimum number of
  *                  operations, and will do so even if max_ops is set to a
  *                  lower value.  That minimum depends on the curve size, and
  *                  can be made lower by decreasing the value of
  *                  \c MBEDTLS_ECP_WINDOW_SIZE.  As an indication, here is the
  *                  lowest effective value for various curves and values of
  *                  that parameter (w for short):
  *                          w=6     w=5     w=4     w=3     w=2
  *                  P-256   208     208     160     136     124
  *                  P-384   682     416     320     272     248
  *                  P-521  1364     832     640     544     496
  *
  * \note            This setting is currently ignored by Curve25519.
  */
 void mbedtls_ecp_set_max_ops(unsigned max_ops);
 
 /**
  * \brief           Check if restart is enabled (max_ops != 0)
  *
  * \return          \c 0 if \c max_ops == 0 (restart disabled)
  * \return          \c 1 otherwise (restart enabled)
  */
 int mbedtls_ecp_restart_is_enabled(void);
 #endif /* MBEDTLS_ECP_RESTARTABLE */
 
 /*
  * Get the type of a curve
  */
 mbedtls_ecp_curve_type mbedtls_ecp_get_type(const mbedtls_ecp_group *grp);
 
 /**
  * \brief           This function retrieves the information defined in
  *                  mbedtls_ecp_curve_info() for all supported curves.
  *
  * \note            This function returns information about all curves
  *                  supported by the library. Some curves may not be
  *                  supported for all algorithms. Call mbedtls_ecdh_can_do()
  *                  or mbedtls_ecdsa_can_do() to check if a curve is
  *                  supported for ECDH or ECDSA.
  *
  * \return          A statically allocated array. The last entry is 0.
  */
 const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list(void);
 
 /**
  * \brief           This function retrieves the list of internal group
  *                  identifiers of all supported curves in the order of
  *                  preference.
  *
  * \note            This function returns information about all curves
  *                  supported by the library. Some curves may not be
  *                  supported for all algorithms. Call mbedtls_ecdh_can_do()
  *                  or mbedtls_ecdsa_can_do() to check if a curve is
  *                  supported for ECDH or ECDSA.
  *
  * \return          A statically allocated array,
  *                  terminated with MBEDTLS_ECP_DP_NONE.
  */
 const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list(void);
 
 /**
  * \brief           This function retrieves curve information from an internal
  *                  group identifier.
  *
  * \param grp_id    An \c MBEDTLS_ECP_DP_XXX value.
  *
  * \return          The associated curve information on success.
  * \return          NULL on failure.
  */
 const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id(mbedtls_ecp_group_id grp_id);
 
 /**
  * \brief           This function retrieves curve information from a TLS
  *                  NamedCurve value.
  *
  * \param tls_id    An \c MBEDTLS_ECP_DP_XXX value.
  *
  * \return          The associated curve information on success.
  * \return          NULL on failure.
  */
 const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id(uint16_t tls_id);
 
 /**
  * \brief           This function retrieves curve information from a
  *                  human-readable name.
  *
  * \param name      The human-readable name.
  *
  * \return          The associated curve information on success.
  * \return          NULL on failure.
  */
 const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name(const char *name);
 
 /**
  * \brief           This function initializes a point as zero.
  *
  * \param pt        The point to initialize.
  */
 void mbedtls_ecp_point_init(mbedtls_ecp_point *pt);
 
 /**
  * \brief           This function initializes an ECP group context
  *                  without loading any domain parameters.
  *
  * \note            After this function is called, domain parameters
  *                  for various ECP groups can be loaded through the
  *                  mbedtls_ecp_group_load() or mbedtls_ecp_tls_read_group()
  *                  functions.
  */
 void mbedtls_ecp_group_init(mbedtls_ecp_group *grp);
 
 /**
  * \brief           This function initializes a key pair as an invalid one.
  *
  * \param key       The key pair to initialize.
  */
 void mbedtls_ecp_keypair_init(mbedtls_ecp_keypair *key);
 
 /**
  * \brief           This function frees the components of a point.
  *
  * \param pt        The point to free.
  */
 void mbedtls_ecp_point_free(mbedtls_ecp_point *pt);
 
 /**
  * \brief           This function frees the components of an ECP group.
  *
  * \param grp       The group to free. This may be \c NULL, in which
  *                  case this function returns immediately. If it is not
  *                  \c NULL, it must point to an initialized ECP group.
  */
 void mbedtls_ecp_group_free(mbedtls_ecp_group *grp);
 
 /**
  * \brief           This function frees the components of a key pair.
  *
  * \param key       The key pair to free. This may be \c NULL, in which
  *                  case this function returns immediately. If it is not
  *                  \c NULL, it must point to an initialized ECP key pair.
  */
 void mbedtls_ecp_keypair_free(mbedtls_ecp_keypair *key);
 
 #if defined(MBEDTLS_ECP_RESTARTABLE)
 /**
  * \brief           Initialize a restart context.
  *
  * \param ctx       The restart context to initialize. This must
  *                  not be \c NULL.
  */
 void mbedtls_ecp_restart_init(mbedtls_ecp_restart_ctx *ctx);
 
 /**
  * \brief           Free the components of a restart context.
  *
  * \param ctx       The restart context to free. This may be \c NULL, in which
  *                  case this function returns immediately. If it is not
  *                  \c NULL, it must point to an initialized restart context.
  */
 void mbedtls_ecp_restart_free(mbedtls_ecp_restart_ctx *ctx);
 #endif /* MBEDTLS_ECP_RESTARTABLE */
 
 /**
  * \brief           This function copies the contents of point \p Q into
  *                  point \p P.
  *
  * \param P         The destination point. This must be initialized.
  * \param Q         The source point. This must be initialized.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          Another negative error code for other kinds of failure.
  */
 int mbedtls_ecp_copy(mbedtls_ecp_point *P, const mbedtls_ecp_point *Q);
 
 /**
  * \brief           This function copies the contents of group \p src into
  *                  group \p dst.
  *
  * \param dst       The destination group. This must be initialized.
  * \param src       The source group. This must be initialized.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_group_copy(mbedtls_ecp_group *dst,
                            const mbedtls_ecp_group *src);
 
 /**
  * \brief           This function sets a point to the point at infinity.
  *
  * \param pt        The point to set. This must be initialized.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_set_zero(mbedtls_ecp_point *pt);
 
 /**
  * \brief           This function checks if a point is the point at infinity.
  *
  * \param pt        The point to test. This must be initialized.
  *
  * \return          \c 1 if the point is zero.
  * \return          \c 0 if the point is non-zero.
  * \return          A negative error code on failure.
  */
 int mbedtls_ecp_is_zero(mbedtls_ecp_point *pt);
 
 /**
  * \brief           This function compares two points.
  *
  * \note            This assumes that the points are normalized. Otherwise,
  *                  they may compare as "not equal" even if they are.
  *
  * \param P         The first point to compare. This must be initialized.
  * \param Q         The second point to compare. This must be initialized.
  *
  * \return          \c 0 if the points are equal.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if the points are not equal.
  */
 int mbedtls_ecp_point_cmp(const mbedtls_ecp_point *P,
                           const mbedtls_ecp_point *Q);
 
 /**
  * \brief           This function imports a non-zero point from two ASCII
  *                  strings.
  *
  * \param P         The destination point. This must be initialized.
  * \param radix     The numeric base of the input.
  * \param x         The first affine coordinate, as a null-terminated string.
  * \param y         The second affine coordinate, as a null-terminated string.
  *
  * \return          \c 0 on success.
  * \return          An \c MBEDTLS_ERR_MPI_XXX error code on failure.
  */
 int mbedtls_ecp_point_read_string(mbedtls_ecp_point *P, int radix,
                                   const char *x, const char *y);
 
 /**
  * \brief           This function exports a point into unsigned binary data.
  *
  * \param grp       The group to which the point should belong.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param P         The point to export. This must be initialized.
  * \param format    The point format. This must be either
  *                  #MBEDTLS_ECP_PF_COMPRESSED or #MBEDTLS_ECP_PF_UNCOMPRESSED.
  *                  (For groups without these formats, this parameter is
  *                  ignored. But it still has to be either of the above
  *                  values.)
  * \param olen      The address at which to store the length of
  *                  the output in Bytes. This must not be \c NULL.
  * \param buf       The output buffer. This must be a writable buffer
  *                  of length \p buflen Bytes.
  * \param buflen    The length of the output buffer \p buf in Bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL if the output buffer
  *                  is too small to hold the point.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the point format
  *                  or the export for the given group is not implemented.
  * \return          Another negative error code on other kinds of failure.
  */
 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           This function imports a point from unsigned binary data.
  *
  * \note            This function does not check that the point actually
  *                  belongs to the given group, see mbedtls_ecp_check_pubkey()
  *                  for that.
  *
  * \note            For compressed points, see #MBEDTLS_ECP_PF_COMPRESSED for
  *                  limitations.
  *
  * \param grp       The group to which the point should belong.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param P         The destination context to import the point to.
  *                  This must be initialized.
  * \param buf       The input buffer. This must be a readable buffer
  *                  of length \p ilen Bytes.
  * \param ilen      The length of the input buffer \p buf in Bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if the input is invalid.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the import for the
  *                  given group is not implemented.
  */
 int mbedtls_ecp_point_read_binary(const mbedtls_ecp_group *grp,
                                   mbedtls_ecp_point *P,
                                   const unsigned char *buf, size_t ilen);
 
 /**
  * \brief           This function imports a point from a TLS ECPoint record.
  *
  * \note            On function return, \p *buf is updated to point immediately
  *                  after the ECPoint record.
  *
  * \param grp       The ECP group to use.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param pt        The destination point.
  * \param buf       The address of the pointer to the start of the input buffer.
  * \param len       The length of the buffer.
  *
  * \return          \c 0 on success.
  * \return          An \c MBEDTLS_ERR_MPI_XXX error code on initialization
  *                  failure.
  * \return          #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           This function exports a point as a TLS ECPoint record
  *                  defined in RFC 4492, Section 5.4.
  *
  * \param grp       The ECP group to use.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param pt        The point to be exported. This must be initialized.
  * \param format    The point format to use. This must be either
  *                  #MBEDTLS_ECP_PF_COMPRESSED or #MBEDTLS_ECP_PF_UNCOMPRESSED.
  * \param olen      The address at which to store the length in Bytes
  *                  of the data written.
  * \param buf       The target buffer. This must be a writable buffer of
  *                  length \p blen Bytes.
  * \param blen      The length of the target buffer \p buf in Bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if the input is invalid.
  * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL if the target buffer
  *                  is too small to hold the exported point.
  * \return          Another negative error code on other kinds of failure.
  */
 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           This function sets up an ECP group context
  *                  from a standardized set of domain parameters.
  *
  * \note            The index should be a value of the NamedCurve enum,
  *                  as defined in <em>RFC-4492: Elliptic Curve Cryptography
  *                  (ECC) Cipher Suites for Transport Layer Security (TLS)</em>,
  *                  usually in the form of an \c MBEDTLS_ECP_DP_XXX macro.
  *
  * \param grp       The group context to setup. This must be initialized.
  * \param id        The identifier of the domain parameter set to load.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if \p id doesn't
  *                  correspond to a known group.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_group_load(mbedtls_ecp_group *grp, mbedtls_ecp_group_id id);
 
 /**
  * \brief           This function sets up an ECP group context from a TLS
  *                  ECParameters record as defined in RFC 4492, Section 5.4.
  *
  * \note            The read pointer \p buf is updated to point right after
  *                  the ECParameters record on exit.
  *
  * \param grp       The group context to setup. This must be initialized.
  * \param buf       The address of the pointer to the start of the input buffer.
  * \param len       The length of the input buffer \c *buf in Bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the group is not
  *                  recognized.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_tls_read_group(mbedtls_ecp_group *grp,
                                const unsigned char **buf, size_t len);
 
 /**
  * \brief           This function extracts an elliptic curve group ID from a
  *                  TLS ECParameters record as defined in RFC 4492, Section 5.4.
  *
  * \note            The read pointer \p buf is updated to point right after
  *                  the ECParameters record on exit.
  *
  * \param grp       The address at which to store the group id.
  *                  This must not be \c NULL.
  * \param buf       The address of the pointer to the start of the input buffer.
  * \param len       The length of the input buffer \c *buf in Bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the group is not
  *                  recognized.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_tls_read_group_id(mbedtls_ecp_group_id *grp,
                                   const unsigned char **buf,
                                   size_t len);
 /**
  * \brief           This function exports an elliptic curve as a TLS
  *                  ECParameters record as defined in RFC 4492, Section 5.4.
  *
  * \param grp       The ECP group to be exported.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param olen      The address at which to store the number of Bytes written.
  *                  This must not be \c NULL.
  * \param buf       The buffer to write to. This must be a writable buffer
  *                  of length \p blen Bytes.
  * \param blen      The length of the output buffer \p buf in Bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL if the output
  *                  buffer is too small to hold the exported group.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_tls_write_group(const mbedtls_ecp_group *grp,
                                 size_t *olen,
                                 unsigned char *buf, size_t blen);
 
 /**
  * \brief           This function performs a scalar multiplication of a point
  *                  by an integer: \p R = \p m * \p P.
  *
  *                  It is not thread-safe to use same group in multiple threads.
  *
  * \note            To prevent timing attacks, this function
  *                  executes the exact same sequence of base-field
  *                  operations for any valid \p m. It avoids any if-branch or
  *                  array index depending on the value of \p m. It also uses
  *                  \p f_rng to randomize some intermediate results.
  *
  * \param grp       The ECP group to use.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param R         The point in which to store the result of the calculation.
  *                  This must be initialized.
  * \param m         The integer by which to multiply. This must be initialized.
  * \param P         The point to multiply. This must be initialized.
  * \param f_rng     The RNG function. This must not be \c NULL.
  * \param p_rng     The RNG context to be passed to \p f_rng. This may be \c
  *                  NULL if \p f_rng doesn't need a context.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_INVALID_KEY if \p m is not a valid private
  *                  key, or \p P is not a valid public key.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          Another negative error code on other kinds of failure.
  */
 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           This function performs multiplication of a point by
  *                  an integer: \p R = \p m * \p P in a restartable way.
  *
  * \see             mbedtls_ecp_mul()
  *
  * \note            This function does the same as \c mbedtls_ecp_mul(), but
  *                  it can return early and restart according to the limit set
  *                  with \c mbedtls_ecp_set_max_ops() to reduce blocking.
  *
  * \param grp       The ECP group to use.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param R         The point in which to store the result of the calculation.
  *                  This must be initialized.
  * \param m         The integer by which to multiply. This must be initialized.
  * \param P         The point to multiply. This must be initialized.
  * \param f_rng     The RNG function. This must not be \c NULL.
  * \param p_rng     The RNG context to be passed to \p f_rng. This may be \c
  *                  NULL if \p f_rng doesn't need a context.
  * \param rs_ctx    The restart context (NULL disables restart).
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_INVALID_KEY if \p m is not a valid private
  *                  key, or \p P is not a valid public key.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          #MBEDTLS_ERR_ECP_IN_PROGRESS if maximum number of
  *                  operations was reached: see \c mbedtls_ecp_set_max_ops().
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_mul_restartable(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,
                                 mbedtls_ecp_restart_ctx *rs_ctx);
 
 #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED)
 /**
  * \brief           This function checks if domain parameter A of the curve is
  *                  \c -3.
  *
  * \note            This function is only defined for short Weierstrass curves.
  *                  It may not be included in builds without any short
  *                  Weierstrass curve.
  *
  * \param grp       The ECP group to use.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  *
  * \return          \c 1 if <code>A = -3</code>.
  * \return          \c 0 Otherwise.
  */
 static inline int mbedtls_ecp_group_a_is_minus_3(const mbedtls_ecp_group *grp)
 {
     return grp->A.MBEDTLS_PRIVATE(p) == NULL;
 }
 
 /**
  * \brief           This function performs multiplication and addition of two
  *                  points by integers: \p R = \p m * \p P + \p n * \p Q
  *
  *                  It is 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.
  *
  * \note            This function is only defined for short Weierstrass curves.
  *                  It may not be included in builds without any short
  *                  Weierstrass curve.
  *
  * \param grp       The ECP group to use.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param R         The point in which to store the result of the calculation.
  *                  This must be initialized.
  * \param m         The integer by which to multiply \p P.
  *                  This must be initialized.
  * \param P         The point to multiply by \p m. This must be initialized.
  * \param n         The integer by which to multiply \p Q.
  *                  This must be initialized.
  * \param Q         The point to be multiplied by \p n.
  *                  This must be initialized.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_INVALID_KEY if \p m or \p n are not
  *                  valid private keys, or \p P or \p Q are not valid public
  *                  keys.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if \p grp does not
  *                  designate a short Weierstrass curve.
  * \return          Another negative error code on other kinds of failure.
  */
 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           This function performs multiplication and addition of two
  *                  points by integers: \p R = \p m * \p P + \p n * \p Q in a
  *                  restartable way.
  *
  * \see             \c mbedtls_ecp_muladd()
  *
  * \note            This function works the same as \c mbedtls_ecp_muladd(),
  *                  but it can return early and restart according to the limit
  *                  set with \c mbedtls_ecp_set_max_ops() to reduce blocking.
  *
  * \note            This function is only defined for short Weierstrass curves.
  *                  It may not be included in builds without any short
  *                  Weierstrass curve.
  *
  * \param grp       The ECP group to use.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param R         The point in which to store the result of the calculation.
  *                  This must be initialized.
  * \param m         The integer by which to multiply \p P.
  *                  This must be initialized.
  * \param P         The point to multiply by \p m. This must be initialized.
  * \param n         The integer by which to multiply \p Q.
  *                  This must be initialized.
  * \param Q         The point to be multiplied by \p n.
  *                  This must be initialized.
  * \param rs_ctx    The restart context (NULL disables restart).
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_INVALID_KEY if \p m or \p n are not
  *                  valid private keys, or \p P or \p Q are not valid public
  *                  keys.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if \p grp does not
  *                  designate a short Weierstrass curve.
  * \return          #MBEDTLS_ERR_ECP_IN_PROGRESS if maximum number of
  *                  operations was reached: see \c mbedtls_ecp_set_max_ops().
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_muladd_restartable(
     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,
     mbedtls_ecp_restart_ctx *rs_ctx);
 #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */
 
 /**
  * \brief           This function checks that a point is a valid public key
  *                  on this curve.
  *
  *                  It only checks that the point is non-zero, has
  *                  valid coordinates and lies on the curve. It does not verify
  *                  that it is indeed a multiple of \c G. This additional
  *                  check is computationally more expensive, is not required
  *                  by standards, and should not 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            This function uses bare components rather than an
  *                  ::mbedtls_ecp_keypair structure, to ease use with other
  *                  structures, such as ::mbedtls_ecdh_context or
  *                  ::mbedtls_ecdsa_context.
  *
  * \param grp       The ECP group the point should belong to.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param pt        The point to check. This must be initialized.
  *
  * \return          \c 0 if the point is a valid public key.
  * \return          #MBEDTLS_ERR_ECP_INVALID_KEY if the point is not
  *                  a valid public key for the given curve.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_check_pubkey(const mbedtls_ecp_group *grp,
                              const mbedtls_ecp_point *pt);
 
 /**
  * \brief           This function checks that an \c mbedtls_mpi is a
  *                  valid private key for this curve.
  *
  * \note            This function uses bare components rather than an
  *                  ::mbedtls_ecp_keypair structure to ease use with other
  *                  structures, such as ::mbedtls_ecdh_context or
  *                  ::mbedtls_ecdsa_context.
  *
  * \param grp       The ECP group the private key should belong to.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param d         The integer to check. This must be initialized.
  *
  * \return          \c 0 if the point is a valid private key.
  * \return          #MBEDTLS_ERR_ECP_INVALID_KEY if the point is not a valid
  *                  private key for the given curve.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_check_privkey(const mbedtls_ecp_group *grp,
                               const mbedtls_mpi *d);
 
 /**
  * \brief           This function generates a private key.
  *
  * \param grp       The ECP group to generate a private key for.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param d         The destination MPI (secret part). This must be initialized.
  * \param f_rng     The RNG function. This must not be \c NULL.
  * \param p_rng     The RNG parameter to be passed to \p f_rng. This may be
  *                  \c NULL if \p f_rng doesn't need a context argument.
  *
  * \return          \c 0 on success.
  * \return          An \c MBEDTLS_ERR_ECP_XXX or \c MBEDTLS_MPI_XXX error code
  *                  on failure.
  */
 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           This function generates a keypair with a configurable base
  *                  point.
  *
  * \note            This function uses bare components rather than an
  *                  ::mbedtls_ecp_keypair structure to ease use with other
  *                  structures, such as ::mbedtls_ecdh_context or
  *                  ::mbedtls_ecdsa_context.
  *
  * \param grp       The ECP group to generate a key pair for.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param G         The base point to use. This must be initialized
  *                  and belong to \p grp. It replaces the default base
  *                  point \c grp->G used by mbedtls_ecp_gen_keypair().
  * \param d         The destination MPI (secret part).
  *                  This must be initialized.
  * \param Q         The destination point (public part).
  *                  This must be initialized.
  * \param f_rng     The RNG function. This must not be \c NULL.
  * \param p_rng     The RNG context to be passed to \p f_rng. This may
  *                  be \c NULL if \p f_rng doesn't need a context argument.
  *
  * \return          \c 0 on success.
  * \return          An \c MBEDTLS_ERR_ECP_XXX or \c MBEDTLS_MPI_XXX error code
  *                  on failure.
  */
 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           This function generates an ECP keypair.
  *
  * \note            This function uses bare components rather than an
  *                  ::mbedtls_ecp_keypair structure to ease use with other
  *                  structures, such as ::mbedtls_ecdh_context or
  *                  ::mbedtls_ecdsa_context.
  *
  * \param grp       The ECP group to generate a key pair for.
  *                  This must be initialized and have group parameters
  *                  set, for example through mbedtls_ecp_group_load().
  * \param d         The destination MPI (secret part).
  *                  This must be initialized.
  * \param Q         The destination point (public part).
  *                  This must be initialized.
  * \param f_rng     The RNG function. This must not be \c NULL.
  * \param p_rng     The RNG context to be passed to \p f_rng. This may
  *                  be \c NULL if \p f_rng doesn't need a context argument.
  *
  * \return          \c 0 on success.
  * \return          An \c MBEDTLS_ERR_ECP_XXX or \c MBEDTLS_MPI_XXX error code
  *                  on failure.
  */
 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           This function generates an ECP key.
  *
  * \param grp_id    The ECP group identifier.
  * \param key       The destination key. This must be initialized.
  * \param f_rng     The RNG function to use. This must not be \c NULL.
  * \param p_rng     The RNG context to be passed to \p f_rng. This may
  *                  be \c NULL if \p f_rng doesn't need a context argument.
  *
  * \return          \c 0 on success.
  * \return          An \c MBEDTLS_ERR_ECP_XXX or \c MBEDTLS_MPI_XXX error code
  *                  on failure.
  */
 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          Set the public key in a key pair object.
  *
  * \note            This function does not check that the point actually
  *                  belongs to the given group. Call mbedtls_ecp_check_pubkey()
  *                  on \p Q before calling this function to check that.
  *
  * \note            This function does not check that the public key matches
  *                  the private key that is already in \p key, if any.
  *                  To check the consistency of the resulting key pair object,
  *                  call mbedtls_ecp_check_pub_priv() after setting both
  *                  the public key and the private key.
  *
  * \param grp_id    The ECP group identifier.
  * \param key       The key pair object. It must be initialized.
  *                  If its group has already been set, it must match \p grp_id.
  *                  If its group has not been set, it will be set to \p grp_id.
  *                  If the public key has already been set, it is overwritten.
  * \param Q         The public key to copy. This must be a point on the
  *                  curve indicated by \p grp_id.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if \p key does not
  *                  match \p grp_id.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the operation for
  *                  the group is not implemented.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_set_public_key(mbedtls_ecp_group_id grp_id,
                                mbedtls_ecp_keypair *key,
                                const mbedtls_ecp_point *Q);
 
 /**
  * \brief           This function reads an elliptic curve private key.
  *
  * \note            This function does not set the public key in the
  *                  key pair object. Without a public key, the key pair object
  *                  cannot be used with operations that require the public key.
  *                  Call mbedtls_ecp_keypair_calc_public() to set the public
  *                  key from the private key. Alternatively, you can call
  *                  mbedtls_ecp_set_public_key() to set the public key part,
  *                  and then optionally mbedtls_ecp_check_pub_priv() to check
  *                  that the private and public parts are consistent.
  *
  * \note            If a public key has already been set in the key pair
  *                  object, this function does not check that it is consistent
  *                  with the private key. Call mbedtls_ecp_check_pub_priv()
  *                  after setting both the public key and the private key
  *                  to make that check.
  *
  * \param grp_id    The ECP group identifier.
  * \param key       The destination key.
  * \param buf       The buffer containing the binary representation of the
  *                  key. (Big endian integer for Weierstrass curves, byte
  *                  string for Montgomery curves.)
  * \param buflen    The length of the buffer in bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_INVALID_KEY error if the key is
  *                  invalid.
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the operation for
  *                  the group is not implemented.
  * \return          Another negative error code on different kinds of failure.
  */
 int mbedtls_ecp_read_key(mbedtls_ecp_group_id grp_id, mbedtls_ecp_keypair *key,
                          const unsigned char *buf, size_t buflen);
 
 #if !defined(MBEDTLS_DEPRECATED_REMOVED)
 /**
  * \brief           This function exports an elliptic curve private key.
  *
  * \deprecated      Note that although this function accepts an output
  *                  buffer that is smaller or larger than the key, most key
  *                  import interfaces require the output to have exactly
  *                  key's nominal length. It is generally simplest to
  *                  pass the key's nominal length as \c buflen, after
  *                  checking that the output buffer is large enough.
  *                  See the description of the \p buflen parameter for
  *                  how to calculate the nominal length.
  *                  To avoid this difficulty, use mbedtls_ecp_write_key_ext()
  *                  instead.
  *                  mbedtls_ecp_write_key() is deprecated and will be
  *                  removed in a future version of the library.
  *
  * \note            If the private key was not set in \p key,
  *                  the output is unspecified. Future versions
  *                  may return an error in that case.
  *
  * \param key       The private key.
  * \param buf       The output buffer for containing the binary representation
  *                  of the key.
  *                  For Weierstrass curves, this is the big-endian
  *                  representation, padded with null bytes at the beginning
  *                  to reach \p buflen bytes.
  *                  For Montgomery curves, this is the standard byte string
  *                  representation (which is little-endian), padded with
  *                  null bytes at the end to reach \p buflen bytes.
  * \param buflen    The total length of the buffer in bytes.
  *                  The length of the output is
  *                  (`grp->nbits` + 7) / 8 bytes
  *                  where `grp->nbits` is the private key size in bits.
  *                  For Weierstrass keys, if the output buffer is smaller,
  *                  leading zeros are trimmed to fit if possible. For
  *                  Montgomery keys, the output buffer must always be large
  *                  enough for the nominal length.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL or
  *                  #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if the \p key
  *                  representation is larger than the available space in \p buf.
  * \return          Another negative error code on different kinds of failure.
  */
 int MBEDTLS_DEPRECATED mbedtls_ecp_write_key(mbedtls_ecp_keypair *key,
                                              unsigned char *buf, size_t buflen);
 #endif /* MBEDTLS_DEPRECATED_REMOVED */
 
 /**
  * \brief           This function exports an elliptic curve private key.
  *
  * \param key       The private key.
  * \param olen      On success, the length of the private key.
  *                  This is always (`grp->nbits` + 7) / 8 bytes
  *                  where `grp->nbits` is the private key size in bits.
  * \param buf       The output buffer for containing the binary representation
  *                  of the key.
  * \param buflen    The total length of the buffer in bytes.
  *                  #MBEDTLS_ECP_MAX_BYTES is always sufficient.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL if the \p key
  *                  representation is larger than the available space in \p buf.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if no private key is
  *                  set in \p key.
  * \return          Another negative error code on different kinds of failure.
  */
 int mbedtls_ecp_write_key_ext(const mbedtls_ecp_keypair *key,
                               size_t *olen, unsigned char *buf, size_t buflen);
 
 /**
  * \brief           This function exports an elliptic curve public key.
  *
  * \note            If the public key was not set in \p key,
  *                  the output is unspecified. Future versions
  *                  may return an error in that case.
  *
  * \param key       The public key.
  * \param format    The point format. This must be either
  *                  #MBEDTLS_ECP_PF_COMPRESSED or #MBEDTLS_ECP_PF_UNCOMPRESSED.
  *                  (For groups without these formats, this parameter is
  *                  ignored. But it still has to be either of the above
  *                  values.)
  * \param olen      The address at which to store the length of
  *                  the output in Bytes. This must not be \c NULL.
  * \param buf       The output buffer. This must be a writable buffer
  *                  of length \p buflen Bytes.
  * \param buflen    The length of the output buffer \p buf in Bytes.
  *
  * \return          \c 0 on success.
  * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL if the output buffer
  *                  is too small to hold the point.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if the point format
  *                  or the export for the given group is not implemented.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_write_public_key(const mbedtls_ecp_keypair *key,
                                  int format, size_t *olen,
                                  unsigned char *buf, size_t buflen);
 
 /**
  * \brief           This function checks that the keypair objects
  *                  \p pub and \p prv have the same group and the
  *                  same public point, and that the private key in
  *                  \p prv is consistent with the public key.
  *
  * \param pub       The keypair structure holding the public key. This
  *                  must be initialized. If it contains a private key, that
  *                  part is ignored.
  * \param prv       The keypair structure holding the full keypair.
  *                  This must be initialized.
  * \param f_rng     The RNG function. This must not be \c NULL.
  * \param p_rng     The RNG context to be passed to \p f_rng. This may be \c
  *                  NULL if \p f_rng doesn't need a context.
  *
  * \return          \c 0 on success, meaning that the keys are valid and match.
  * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if the keys are invalid or do not match.
  * \return          An \c MBEDTLS_ERR_ECP_XXX or an \c MBEDTLS_ERR_MPI_XXX
  *                  error code on calculation failure.
  */
 int mbedtls_ecp_check_pub_priv(
     const mbedtls_ecp_keypair *pub, const mbedtls_ecp_keypair *prv,
     int (*f_rng)(void *, unsigned char *, size_t), void *p_rng);
 
 /** \brief          Calculate the public key from a private key in a key pair.
  *
  * \param key       A keypair structure. It must have a private key set.
  *                  If the public key is set, it will be overwritten.
  * \param f_rng     The RNG function. This must not be \c NULL.
  * \param p_rng     The RNG context to be passed to \p f_rng. This may be \c
  *                  NULL if \p f_rng doesn't need a context.
  *
  * \return          \c 0 on success. The key pair object can be used for
  *                  operations that require the public key.
  * \return          An \c MBEDTLS_ERR_ECP_XXX or an \c MBEDTLS_ERR_MPI_XXX
  *                  error code on calculation failure.
  */
 int mbedtls_ecp_keypair_calc_public(
     mbedtls_ecp_keypair *key,
     int (*f_rng)(void *, unsigned char *, size_t), void *p_rng);
 
 /** \brief          Query the group that a key pair belongs to.
  *
  * \param key       The key pair to query.
  *
  * \return          The group ID for the group registered in the key pair
  *                  object.
  *                  This is \c MBEDTLS_ECP_DP_NONE if no group has been set
  *                  in the key pair object.
  */
 mbedtls_ecp_group_id mbedtls_ecp_keypair_get_group_id(
     const mbedtls_ecp_keypair *key);
 
 /**
  * \brief           This function exports generic key-pair parameters.
  *
  *                  Each of the output parameters can be a null pointer
  *                  if you do not need that parameter.
  *
  * \note            If the private key or the public key was not set in \p key,
  *                  the corresponding output is unspecified. Future versions
  *                  may return an error in that case.
  *
  * \param key       The key pair to export from.
  * \param grp       Slot for exported ECP group.
  *                  It must either be null or point to an initialized ECP group.
  * \param d         Slot for the exported secret value.
  *                  It must either be null or point to an initialized mpi.
  * \param Q         Slot for the exported public value.
  *                  It must either be null or point to an initialized ECP point.
  *
  * \return          \c 0 on success,
  * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE if key id doesn't
  *                  correspond to a known group.
  * \return          Another negative error code on other kinds of failure.
  */
 int mbedtls_ecp_export(const mbedtls_ecp_keypair *key, mbedtls_ecp_group *grp,
                        mbedtls_mpi *d, mbedtls_ecp_point *Q);
 
 #if defined(MBEDTLS_SELF_TEST)
 
 /**
  * \brief          The ECP checkup routine.
  *
  * \return         \c 0 on success.
  * \return         \c 1 on failure.
  */
 int mbedtls_ecp_self_test(int verbose);
 
 #endif /* MBEDTLS_SELF_TEST */
 
 #endif /* MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS */
 
 #ifdef __cplusplus
 }
 #endif
 
 #endif /* ecp.h */

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