diff --git a/include/mbedtls/ecp.h b/include/mbedtls/ecp.h
index a30e876..465d750 100644
--- a/include/mbedtls/ecp.h
+++ b/include/mbedtls/ecp.h
@@ -1,9 +1,21 @@
 /**
  * \file ecp.h
  *
- * \brief Elliptic curves over GF(p)
+ * \brief This file provides an API for Elliptic Curves over GF(P) (ECP).
  *
- *  Copyright (C) 2006-2015, ARM Limited, All Rights Reserved
+ * 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 (C) 2006-2018, Arm Limited (or its affiliates), All Rights Reserved
  *  SPDX-License-Identifier: Apache-2.0
  *
  *  Licensed under the Apache License, Version 2.0 (the "License"); you may
@@ -18,8 +30,9 @@
  *  See the License for the specific language governing permissions and
  *  limitations under the License.
  *
- *  This file is part of mbed TLS (https://tls.mbed.org)
+ *  This file is part of Mbed TLS (https://tls.mbed.org)
  */
+
 #ifndef MBEDTLS_ECP_H
 #define MBEDTLS_ECP_H
 
@@ -30,13 +43,82 @@
  */
 #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_FEATURE_UNAVAILABLE               -0x4E80  /**< The requested feature is not available, for example, the requested curve is not supported. */
 #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_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  /**< Signature is valid but shorter than the user-supplied length. */
-#define MBEDTLS_ERR_ECP_IN_PROGRESS                       -0x4B80  /**< Operation in progress, try again with the same parameters. */
+#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  /**< The ECP hardware accelerator failed. */
+#define MBEDTLS_ERR_ECP_IN_PROGRESS                       -0x4B00  /**< Operation in progress, try again with the same parameters. */
+
+#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().
+ */
+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_SECP224R1,      /*!< Domain parameters for the 224-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_SECP224K1,      /*!< Domain parameters for 224-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;
+
+/**
+ * The number of supported curves, plus one for #MBEDTLS_ECP_DP_NONE.
+ *
+ * \note Montgomery curves are currently excluded.
+ */
+#define MBEDTLS_ECP_DP_MAX     12
+
+/**
+ * Curve information, for use by other modules.
+ */
+typedef struct
+{
+    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;
+
+/**
+ * \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_mpi X;          /*!< The X coordinate of the ECP point. */
+    mbedtls_mpi Y;          /*!< The Y coordinate of the ECP point. */
+    mbedtls_mpi Z;          /*!< The Z coordinate of the ECP point. */
+}
+mbedtls_ecp_point;
 
 #if !defined(MBEDTLS_ECP_ALT)
 /*
@@ -47,130 +129,59 @@
  * one.)
  */
 
-#ifdef __cplusplus
-extern "C" {
-#endif
-
 /**
- * Domain parameters (curve, subgroup and generator) identifiers.
+ * \brief           The ECP group structure.
  *
- * Only curves over prime fields are supported.
+ * 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.
  *
- * \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).
+ * 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.
  *
- * (Montgomery curves excluded for now.)
- */
-#define MBEDTLS_ECP_DP_MAX     12
-
-/**
- * Curve information for use by other modules
+ * 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 approximative modular
+ * reduction. It must return 0 on success and non-zero on failure.
+ *
  */
 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 case, 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_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. 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. */
+    unsigned int h;             /*!< \internal 1 if the constants are static. */
+    int (*modp)(mbedtls_mpi *); /*!< The function for fast pseudo-reduction
+                                     mod \p P (see above).*/
+    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;              /*!< The number of 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;
-
 #if defined(MBEDTLS_ECP_RESTARTABLE)
 
 /**
@@ -237,15 +248,15 @@
  * \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.
+ * Either change them in config.h, or define them using the compiler command line.
  * \{
  */
 
 #if !defined(MBEDTLS_ECP_MAX_BITS)
 /**
- * Maximum size of the groups (that is, of N and P)
+ * The maximum size of the groups, that is, of \c N and \c P.
  */
-#define MBEDTLS_ECP_MAX_BITS     521   /**< Maximum bit size of groups */
+#define MBEDTLS_ECP_MAX_BITS     521   /**< The maximum size of groups, in bits. */
 #endif
 
 #define MBEDTLS_ECP_MAX_BYTES    ( ( MBEDTLS_ECP_MAX_BITS + 7 ) / 8 )
@@ -271,7 +282,7 @@
  *      224       475     475     453     398     342
  *      192       640     640     633     587     476
  */
-#define MBEDTLS_ECP_WINDOW_SIZE    6   /**< Maximum window size used */
+#define MBEDTLS_ECP_WINDOW_SIZE    6   /**< The maximum window size used. */
 #endif /* MBEDTLS_ECP_WINDOW_SIZE */
 
 #if !defined(MBEDTLS_ECP_FIXED_POINT_OPTIM)
@@ -286,21 +297,41 @@
  *
  * Change this value to 0 to reduce peak memory usage.
  */
-#define MBEDTLS_ECP_FIXED_POINT_OPTIM  1   /**< Enable fixed-point speed-up */
+#define MBEDTLS_ECP_FIXED_POINT_OPTIM  1   /**< Enable fixed-point speed-up. */
 #endif /* MBEDTLS_ECP_FIXED_POINT_OPTIM */
 
 /* \} name SECTION: Module settings */
 
+#else  /* MBEDTLS_ECP_ALT */
+#include "ecp_alt.h"
+#endif /* MBEDTLS_ECP_ALT */
+
+/**
+ * \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_group grp;      /*!<  Elliptic curve and base point     */
+    mbedtls_mpi d;              /*!<  our secret value                  */
+    mbedtls_ecp_point Q;        /*!<  our public value                  */
+}
+mbedtls_ecp_keypair;
+
 /*
  * 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 */
+#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 */
+#define MBEDTLS_ECP_TLS_NAMED_CURVE    3   /**< The named_curve of ECCurveType. */
 
 #if defined(MBEDTLS_ECP_RESTARTABLE)
 /**
@@ -362,16 +393,18 @@
 #endif /* MBEDTLS_ECP_RESTARTABLE */
 
 /**
- * \brief           Get the list of supported curves in order of preferrence
- *                  (full information)
+ * \brief           This function retrieves the information defined in
+ *                  mbedtls_ecp_curve_info() for all supported curves in order
+ *                  of preference.
  *
- * \return          A statically allocated array, the last entry is 0.
+ * \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)
+ * \brief           This function retrieves the list of internal group
+ *                  identifiers of all supported curves in the order of
+ *                  preference.
  *
  * \return          A statically allocated array,
  *                  terminated with MBEDTLS_ECP_DP_NONE.
@@ -379,59 +412,79 @@
 const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list( void );
 
 /**
- * \brief           Get curve information from an internal group identifier
+ * \brief           This function retrieves curve information from an internal
+ *                  group identifier.
  *
- * \param grp_id    A MBEDTLS_ECP_DP_XXX value
+ * \param grp_id    An \c MBEDTLS_ECP_DP_XXX value.
  *
- * \return          The associated curve information or NULL
+ * \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           Get curve information from a TLS NamedCurve value
+ * \brief           This function retrieves curve information from a TLS
+ *                  NamedCurve value.
  *
- * \param tls_id    A MBEDTLS_ECP_DP_XXX value
+ * \param tls_id    An \c MBEDTLS_ECP_DP_XXX value.
  *
- * \return          The associated curve information or NULL
+ * \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           Get curve information from a human-readable name
+ * \brief           This function retrieves curve information from a
+ *                  human-readable name.
  *
- * \param name      The name
+ * \param name      The human-readable name.
  *
- * \return          The associated curve information or NULL
+ * \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           Initialize a point (as zero)
+ * \brief           This function initializes a point as zero.
+ *
+ * \param pt        The point to initialize.
  */
 void mbedtls_ecp_point_init( mbedtls_ecp_point *pt );
 
 /**
- * \brief           Initialize a group (to something meaningless)
+ * \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_load() or mbedtls_ecp_tls_read_group()
+ *                  functions.
  */
 void mbedtls_ecp_group_init( mbedtls_ecp_group *grp );
 
 /**
- * \brief           Initialize a key pair (as an invalid one)
+ * \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           Free the components of a point
+ * \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           Free the components of an ECP group
+ * \brief           This function frees the components of an ECP group.
+ * \param grp       The group to free.
  */
 void mbedtls_ecp_group_free( mbedtls_ecp_group *grp );
 
 /**
- * \brief           Free the components of a key pair
+ * \brief           This function frees the components of a key pair.
+ * \param key       The key pair to free.
  */
 void mbedtls_ecp_keypair_free( mbedtls_ecp_keypair *key );
 
@@ -448,236 +501,253 @@
 #endif /* MBEDTLS_ECP_RESTARTABLE */
 
 /**
- * \brief           Copy the contents of point Q into P
+ * \brief           This function copies the contents of point \p Q into
+ *                  point \p P.
  *
- * \param P         Destination point
- * \param Q         Source point
+ * \param P         The destination point.
+ * \param Q         The source point.
  *
- * \return          0 if successful,
- *                  MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed
+ * \return          \c 0 on success.
+ * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  */
 int mbedtls_ecp_copy( mbedtls_ecp_point *P, const mbedtls_ecp_point *Q );
 
 /**
- * \brief           Copy the contents of a group object
+ * \brief           This function copies the contents of group \p src into
+ *                  group \p dst.
  *
- * \param dst       Destination group
- * \param src       Source group
+ * \param dst       The destination group.
+ * \param src       The source group.
  *
- * \return          0 if successful,
- *                  MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed
+ * \return          \c 0 on success.
+ * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  */
 int mbedtls_ecp_group_copy( mbedtls_ecp_group *dst, const mbedtls_ecp_group *src );
 
 /**
- * \brief           Set a point to zero
+ * \brief           This function sets a point to zero.
  *
- * \param pt        Destination point
+ * \param pt        The point to set.
  *
- * \return          0 if successful,
- *                  MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed
+ * \return          \c 0 on success.
+ * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
  */
 int mbedtls_ecp_set_zero( mbedtls_ecp_point *pt );
 
 /**
- * \brief           Tell if a point is zero
+ * \brief           This function checks if a point is zero.
  *
- * \param pt        Point to test
+ * \param pt        The point to test.
  *
- * \return          1 if point is zero, 0 otherwise
+ * \return          \c 1 if the point is zero.
+ * \return          \c 0 if the point is non-zero.
  */
 int mbedtls_ecp_is_zero( mbedtls_ecp_point *pt );
 
 /**
- * \brief           Compare two points
+ * \brief           This function compares two points.
  *
- * \note            This assumes the points are normalized. Otherwise,
+ * \note            This assumes that 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
+ * \param P         The first point to compare.
+ * \param Q         The second point to compare.
  *
- * \return          0 if the points are equal,
- *                  MBEDTLS_ERR_ECP_BAD_INPUT_DATA otherwise
+ * \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           Import a non-zero point from two ASCII strings
+ * \brief           This function imports 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
+ * \param P         The destination point.
+ * \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          0 if successful, or a MBEDTLS_ERR_MPI_XXX error code
+ * \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           Export a point into unsigned binary data
+ * \brief           This function exports 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
+ * \param grp       The group to which the point should belong.
+ * \param P         The point to export.
+ * \param format    The point format. Should be an \c MBEDTLS_ECP_PF_XXX macro.
+ * \param olen      The length of the output.
+ * \param buf       The output buffer.
+ * \param buflen    The length of the output buffer.
  *
- * \return          0 if successful,
- *                  or MBEDTLS_ERR_ECP_BAD_INPUT_DATA
- *                  or MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL
+ * \return          \c 0 on success.
+ * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA
+ *                  or #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL on 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           Import a point from unsigned binary data
+ * \brief           This function imports 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
+ * \note            This function does not check that the point actually
+ *                  belongs to the given group, see mbedtls_ecp_check_pubkey()
+ *                  for that.
  *
- * \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
+ * \param grp       The group to which the point should belong.
+ * \param P         The point to import.
+ * \param buf       The input buffer.
+ * \param ilen      The length of the input.
+ *
+ * \return          \c 0 on success.
+ * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid.
+ * \return          #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure.
+ * \return          #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
+ * \brief           This function imports 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            On function return, \p buf is updated to point to immediately
+ *                  after the ECPoint record.
  *
- * \note            buf is updated to point right after the ECPoint on exit
+ * \param grp       The ECP group used.
+ * \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          0 if successful,
- *                  MBEDTLS_ERR_MPI_XXX if initialization failed
- *                  MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid
+ * \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           Export a point as a TLS ECPoint record
+ * \brief           This function exports 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
+ * \param grp       The ECP group used.
+ * \param pt        The point format to export to. The point format is an
+ *                  \c MBEDTLS_ECP_PF_XXX constant.
+ * \param format    The export format.
+ * \param olen      The length of the data written.
+ * \param buf       The buffer to write to.
+ * \param blen      The length of the buffer.
  *
- * \return          0 if successful,
- *                  or MBEDTLS_ERR_ECP_BAD_INPUT_DATA
- *                  or MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL
+ * \return          \c 0 on success.
+ * \return          #MBEDTLS_ERR_ECP_BAD_INPUT_DATA or
+ *                  #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL on 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           Set a group using well-known domain parameters
+ * \brief           This function sets a group using standardized domain parameters.
  *
- * \param grp       Destination group
- * \param index     Index in the list of well-known 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.
  *
- * \return          0 if successful,
- *                  MBEDTLS_ERR_MPI_XXX if initialization failed
- *                  MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE for unkownn groups
+ * \param grp       The destination group.
+ * \param id        The identifier of the domain parameter set to load.
  *
- * \note            Index should be a value of RFC 4492's enum NamedCurve,
- *                  usually in the form of a MBEDTLS_ECP_DP_XXX macro.
+ * \return          \c 0 on success,
+ * \return          An \c MBEDTLS_ERR_MPI_XXX error code on initialization failure.
+ * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE for unkownn groups.
+
  */
-int mbedtls_ecp_group_load( mbedtls_ecp_group *grp, mbedtls_ecp_group_id index );
+int mbedtls_ecp_group_load( mbedtls_ecp_group *grp, mbedtls_ecp_group_id id );
 
 /**
- * \brief           Set a group from a TLS ECParameters record
+ * \brief           This function sets a group from a TLS ECParameters record.
  *
- * \param grp       Destination group
- * \param buf       &(Start of input buffer)
- * \param len       Buffer length
+ * \note            \p buf is updated to point right after the ECParameters record
+ *                  on exit.
  *
- * \note            buf is updated to point right after ECParameters on exit
+ * \param grp       The destination group.
+ * \param buf       The address of the pointer to the start of the input buffer.
+ * \param len       The length of the buffer.
  *
- * \return          0 if successful,
- *                  MBEDTLS_ERR_MPI_XXX if initialization failed
- *                  MBEDTLS_ERR_ECP_BAD_INPUT_DATA if input is invalid
+ * \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_group( mbedtls_ecp_group *grp, const unsigned char **buf, size_t len );
 
 /**
- * \brief           Write the TLS ECParameters record for a group
+ * \brief           This function writes 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
+ * \param grp       The ECP group used.
+ * \param olen      The number of Bytes written.
+ * \param buf       The buffer to write to.
+ * \param blen      The length of the buffer.
  *
- * \return          0 if successful,
- *                  or MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL
+ * \return          \c 0 on success.
+ * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL on failure.
  */
 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)
+ * \brief           This function performs multiplication of a point by
+ *                  an integer: \p R = \p m * \p P.
  *
- * \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.
+ *                  It is not thread-safe to use same group in multiple threads.
  *
- * \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).
+ * \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.
  *
- * \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
+ * \note            If \p f_rng is not NULL, it is used to randomize
+ *                  intermediate results to prevent potential timing attacks
+ *                  targeting these results. We recommend always providing
+ *                  a non-NULL \p f_rng. The overhead is negligible.
  *
- * \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.
+ * \param grp       The ECP group.
+ * \param R         The destination point.
+ * \param m         The integer by which to multiply.
+ * \param P         The point to multiply.
+ * \param f_rng     The RNG function.
+ * \param p_rng     The RNG 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.
  */
 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           Restartable version of \c mbedtls_ecp_mul()
+ * \brief           This function performs multiplication of a point by
+ *                  an integer: \p R = \p m * \p P in a restartable way.
  *
- * \note            Performs the same job as \c mbedtls_ecp_mul(), but can
- *                  return early and restart according to the limit set with
- *                  \c mbedtls_ecp_set_max_ops() to reduce blocking.
+ *                  \c mbedtls_ecp_mul()
  *
- * \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
- * \param rs_ctx    Restart context
+ * \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.
+ * \param R         The destination point.
+ * \param m         The integer by which to multiply.
+ * \param P         The point to multiply.
+ * \param f_rng     The RNG function.
+ * \param p_rng     The RNG context.
+ * \param rs_ctx    The restart context.
  *
  * \return          See \c mbedtls_ecp_mul(), or
  *                  MBEDTLS_ERR_ECP_IN_PROGRESS if maximum number of
@@ -689,46 +759,56 @@
              mbedtls_ecp_restart_ctx *rs_ctx );
 
 /**
- * \brief           Multiplication and addition of two points by integers:
- *                  R = m * P + n * Q
- *                  (Not thread-safe to use same group in multiple threads)
+ * \brief           This function performs multiplication and addition of two
+ *                  points by integers: \p R = \p m * \p P + \p n * \p Q
  *
- * \note            In contrast to mbedtls_ecp_mul(), this function does not guarantee
- *                  a constant execution flow and timing.
+ *                  It is not thread-safe to use same group in multiple threads.
  *
- * \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
+ * \note            In contrast to mbedtls_ecp_mul(), this function does not
+ *                  guarantee a constant execution flow and timing.
  *
- * \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
+ * \param grp       The ECP group.
+ * \param R         The destination point.
+ * \param m         The integer by which to multiply \p P.
+ * \param P         The point to multiply by \p m.
+ * \param n         The integer by which to multiply \p Q.
+ * \param Q         The point to be multiplied by \p n.
+ *
+ * \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.
  */
 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           Restartable version of \c mbedtls_ecp_muladd()
+ * \brief           This function performs multiplication and addition of two
+ *                  points by integers: \p R = \p m * \p P + \p n * \p Q in a
+ *                  reastartable way.
  *
- * \note            Performs the same job as \c mbedtls_ecp_muladd(), but can
- *                  return early and restart according to the limit set with
- *                  \c mbedtls_ecp_set_max_ops() to reduce blocking.
+ * \see             \c mbedtls_ecp_muladd()
  *
- * \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
- * \param rs_ctx    Restart context
+ * \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.
  *
- * \return          See \c mbedtls_ecp_muladd(), or
- *                  MBEDTLS_ERR_ECP_IN_PROGRESS if maximum number of
+ * \param grp       The ECP group.
+ * \param R         The destination point.
+ * \param m         The integer by which to multiply \p P.
+ * \param P         The point to multiply by \p m.
+ * \param n         The integer by which to multiply \p Q.
+ * \param Q         The point to be multiplied by \p n.
+ * \param rs_ctx    The restart context.
+ *
+ * \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_IN_PROGRESS if maximum number of
  *                  operations was reached: see \c mbedtls_ecp_set_max_ops().
  */
 int mbedtls_ecp_muladd_restartable(
@@ -738,53 +818,58 @@
              mbedtls_ecp_restart_ctx *rs_ctx );
 
 /**
- * \brief           Check that a point is a valid public key on this curve
+ * \brief           This function checks 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
+ *                  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 \p 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.
  *
- * \return          0 if point is a valid public key,
- *                  MBEDTLS_ERR_ECP_INVALID_KEY otherwise.
+ * \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.
  *
- * \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.
+ * \param grp       The curve the point should lie on.
+ * \param pt        The point to check.
  *
- * \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.
+ * \return          \c 0 if the point is a valid public key.
+ * \return          #MBEDTLS_ERR_ECP_INVALID_KEY on failure.
  */
 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
+ * \brief           This function checks that an \p mbedtls_mpi is a valid private
+ *                  key for this curve.
  *
- * \param grp       Group used
- * \param d         Integer to check
+ * \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.
  *
- * \return          0 if point is a valid private key,
- *                  MBEDTLS_ERR_ECP_INVALID_KEY otherwise.
+ * \param grp       The group used.
+ * \param d         The integer to check.
  *
- * \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.
+ * \return          \c 0 if the point is a valid private key.
+ * \return          #MBEDTLS_ERR_ECP_INVALID_KEY on failure.
  */
 int mbedtls_ecp_check_privkey( const mbedtls_ecp_group *grp, const mbedtls_mpi *d );
 
 /**
- * \brief           Generate a private key
+ * \brief           This function generates a private key.
  *
- * \param grp       ECP group
- * \param d         Destination MPI (secret part)
- * \param f_rng     RNG function
- * \param p_rng     RNG parameter
+ * \param grp       The ECP group.
+ * \param d         The Destination MPI (secret part).
+ * \param f_rng     The RNG function.
+ * \param p_rng     The RNG parameter.
  *
- * \return          0 if successful,
- *                  or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code
+ * \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,
@@ -792,21 +877,24 @@
                      void *p_rng );
 
 /**
- * \brief           Generate a keypair with configurable base point
+ * \brief           This function generates a keypair with a 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
+ * \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.
  *
- * \return          0 if successful,
- *                  or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code
+ * \param grp       The ECP group.
+ * \param G         The chosen base point.
+ * \param d         The destination MPI (secret part).
+ * \param Q         The destination point (public part).
+ * \param f_rng     The RNG function.
+ * \param p_rng     The RNG context.
  *
- * \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.
+ * \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,
@@ -815,57 +903,66 @@
                      void *p_rng );
 
 /**
- * \brief           Generate a keypair
+ * \brief           This function generates an ECP 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
+ * \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.
  *
- * \return          0 if successful,
- *                  or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code
+ * \param grp       The ECP group.
+ * \param d         The destination MPI (secret part).
+ * \param Q         The destination point (public part).
+ * \param f_rng     The RNG function.
+ * \param p_rng     The RNG context.
  *
- * \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.
+ * \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           Generate a keypair
+ * \brief           This function generates an ECP key.
  *
- * \param grp_id    ECP group identifier
- * \param key       Destination keypair
- * \param f_rng     RNG function
- * \param p_rng     RNG parameter
+ * \param grp_id    The ECP group identifier.
+ * \param key       The destination key.
+ * \param f_rng     The RNG function.
+ * \param p_rng     The RNG context.
  *
- * \return          0 if successful,
- *                  or a MBEDTLS_ERR_ECP_XXX or MBEDTLS_MPI_XXX error code
+ * \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           Check a public-private key pair
+ * \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       Keypair structure holding a public key
- * \param prv       Keypair structure holding a private (plus public) key
+ * \param pub       The keypair structure holding the public key.
+ *                  If it contains a private key, that part is ignored.
+ * \param prv       The keypair structure holding the full keypair.
  *
- * \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.
+ * \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 );
 
 #if defined(MBEDTLS_SELF_TEST)
 
 /**
- * \brief          Checkup routine
+ * \brief          The ECP checkup routine.
  *
- * \return         0 if successful, or 1 if a test failed
+ * \return         \c 0 on success.
+ * \return         \c 1 on failure.
  */
 int mbedtls_ecp_self_test( int verbose );
 
@@ -875,8 +972,4 @@
 }
 #endif
 
-#else  /* MBEDTLS_ECP_ALT */
-#include "ecp_alt.h"
-#endif /* MBEDTLS_ECP_ALT */
-
 #endif /* ecp.h */