diff --git a/include/mbedtls/ecp.h b/include/mbedtls/ecp.h
index 2fb1af4..2401778 100644
--- a/include/mbedtls/ecp.h
+++ b/include/mbedtls/ecp.h
@@ -36,6 +36,12 @@
 #ifndef MBEDTLS_ECP_H
 #define MBEDTLS_ECP_H
 
+#if !defined(MBEDTLS_CONFIG_FILE)
+#include "config.h"
+#else
+#include MBEDTLS_CONFIG_FILE
+#endif
+
 #include "bignum.h"
 
 /*
@@ -159,6 +165,10 @@
  * additions or subtractions. Therefore, it is only an approximative modular
  * reduction. It must return 0 on success and non-zero on failure.
  *
+ * \note        Alternative implementations must keep the group IDs distinct. If
+ *              two group structures have the same ID, then they must be
+ *              identical.
+ *
  */
 typedef struct mbedtls_ecp_group
 {
@@ -185,6 +195,68 @@
 }
 mbedtls_ecp_group;
 
+/**
+ * \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 using the compiler command line.
+ * \{
+ */
+
+#if !defined(MBEDTLS_ECP_MAX_BITS)
+/**
+ * The maximum size of the groups, that is, of \c N and \c P.
+ */
+#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 )
+#define MBEDTLS_ECP_MAX_PT_LEN   ( 2 * MBEDTLS_ECP_MAX_BYTES + 1 )
+
+#if !defined(MBEDTLS_ECP_WINDOW_SIZE)
+/*
+ * Maximum "window" size used for point multiplication.
+ * Default: 6.
+ * Minimum value: 2. Maximum value: 7.
+ *
+ * Result is an array of at most ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) )
+ * points used for point multiplication. This value is directly tied to EC
+ * peak memory usage, so decreasing it by one should roughly cut memory usage
+ * by two (if large curves are in use).
+ *
+ * Reduction in size may reduce speed, but larger curves are impacted first.
+ * Sample performances (in ECDHE handshakes/s, with FIXED_POINT_OPTIM = 1):
+ *      w-size:     6       5       4       3       2
+ *      521       145     141     135     120      97
+ *      384       214     209     198     177     146
+ *      256       320     320     303     262     226
+ *      224       475     475     453     398     342
+ *      192       640     640     633     587     476
+ */
+#define MBEDTLS_ECP_WINDOW_SIZE    6   /**< The maximum window size used. */
+#endif /* MBEDTLS_ECP_WINDOW_SIZE */
+
+#if !defined(MBEDTLS_ECP_FIXED_POINT_OPTIM)
+/*
+ * Trade memory for speed on fixed-point multiplication.
+ *
+ * This speeds up repeated multiplication of the generator (that is, the
+ * multiplication in ECDSA signatures, and half of the multiplications in
+ * ECDSA verification and ECDHE) by a factor roughly 3 to 4.
+ *
+ * The cost is increasing EC peak memory usage by a factor roughly 2.
+ *
+ * Change this value to 0 to reduce peak memory usage.
+ */
+#define MBEDTLS_ECP_FIXED_POINT_OPTIM  1   /**< Enable fixed-point speed-up. */
+#endif /* MBEDTLS_ECP_FIXED_POINT_OPTIM */
+
+/* \} name SECTION: Module settings */
+
+#else  /* MBEDTLS_ECP_ALT */
+#include "ecp_alt.h"
+#endif /* MBEDTLS_ECP_ALT */
+
 #if defined(MBEDTLS_ECP_RESTARTABLE)
 
 /**
@@ -250,68 +322,6 @@
 #endif /* MBEDTLS_ECP_RESTARTABLE */
 
 /**
- * \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 using the compiler command line.
- * \{
- */
-
-#if !defined(MBEDTLS_ECP_MAX_BITS)
-/**
- * The maximum size of the groups, that is, of \c N and \c P.
- */
-#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 )
-#define MBEDTLS_ECP_MAX_PT_LEN   ( 2 * MBEDTLS_ECP_MAX_BYTES + 1 )
-
-#if !defined(MBEDTLS_ECP_WINDOW_SIZE)
-/*
- * Maximum "window" size used for point multiplication.
- * Default: 6.
- * Minimum value: 2. Maximum value: 7.
- *
- * Result is an array of at most ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) )
- * points used for point multiplication. This value is directly tied to EC
- * peak memory usage, so decreasing it by one should roughly cut memory usage
- * by two (if large curves are in use).
- *
- * Reduction in size may reduce speed, but larger curves are impacted first.
- * Sample performances (in ECDHE handshakes/s, with FIXED_POINT_OPTIM = 1):
- *      w-size:     6       5       4       3       2
- *      521       145     141     135     120      97
- *      384       214     209     198     177     146
- *      256       320     320     303     262     226
- *      224       475     475     453     398     342
- *      192       640     640     633     587     476
- */
-#define MBEDTLS_ECP_WINDOW_SIZE    6   /**< The maximum window size used. */
-#endif /* MBEDTLS_ECP_WINDOW_SIZE */
-
-#if !defined(MBEDTLS_ECP_FIXED_POINT_OPTIM)
-/*
- * Trade memory for speed on fixed-point multiplication.
- *
- * This speeds up repeated multiplication of the generator (that is, the
- * multiplication in ECDSA signatures, and half of the multiplications in
- * ECDSA verification and ECDHE) by a factor roughly 3 to 4.
- *
- * The cost is increasing EC peak memory usage by a factor roughly 2.
- *
- * Change this value to 0 to reduce peak memory usage.
- */
-#define MBEDTLS_ECP_FIXED_POINT_OPTIM  1   /**< Enable fixed-point speed-up. */
-#endif /* MBEDTLS_ECP_FIXED_POINT_OPTIM */
-
-/* \} name SECTION: Module settings */
-
-#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.
@@ -493,24 +503,37 @@
 
 /**
  * \brief           This function frees the components of an ECP group.
- * \param grp       The group to free.
+ *
+ * \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.
+ *
+ * \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
+ * \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
+ * \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 */
@@ -519,11 +542,12 @@
  * \brief           This function copies the contents of point \p Q into
  *                  point \p P.
  *
- * \param P         The destination point.
- * \param Q         The source point.
+ * \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 );
 
@@ -531,31 +555,35 @@
  * \brief           This function copies the contents of group \p src into
  *                  group \p dst.
  *
- * \param dst       The destination group.
- * \param src       The source group.
+ * \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 );
+int mbedtls_ecp_group_copy( mbedtls_ecp_group *dst,
+                            const mbedtls_ecp_group *src );
 
 /**
- * \brief           This function sets a point to zero.
+ * \brief           This function sets a point to the point at infinity.
  *
- * \param pt        The point to set.
+ * \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 zero.
+ * \brief           This function checks if a point is the point at infinity.
  *
- * \param pt        The point to test.
+ * \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 );
 
@@ -565,8 +593,8 @@
  * \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.
- * \param Q         The second point to compare.
+ * \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.
@@ -578,7 +606,7 @@
  * \brief           This function imports a non-zero point from two ASCII
  *                  strings.
  *
- * \param P         The destination point.
+ * \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.
@@ -593,15 +621,21 @@
  * \brief           This function exports a point into unsigned binary data.
  *
  * \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.
+ *                  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.
+ * \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_BAD_INPUT_DATA
- *                  or #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL on failure.
+ * \return          #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL if the output buffer
+ *                  is too small to hold the point.
+ * \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,
@@ -615,108 +649,158 @@
  *                  for that.
  *
  * \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.
+ *                  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 input is invalid.
+ * \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 point format
  *                  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 );
+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 to immediately
+ * \note            On function return, \p *buf is updated to point immediately
  *                  after the ECPoint record.
  *
- * \param grp       The ECP group used.
+ * \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          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 );
+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.
+ * \brief           This function exports a point as a TLS ECPoint record
+ *                  defined in RFC 4492, Section 5.4.
  *
- * \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.
+ * \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 or
- *                  #MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL on failure.
+ * \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 );
+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 a group using standardized domain parameters.
+ * \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 destination group.
+ * \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          An \c MBEDTLS_ERR_MPI_XXX error code on initialization failure.
- * \return          #MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE for unkownn groups.
-
+ * \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 a group from a TLS ECParameters record.
+ * \brief           This function sets up an ECP group context from a TLS
+ *                  ECParameters record as defined in RFC 4492, Section 5.4.
  *
- * \note            \p buf is updated to point right after the ECParameters record
- *                  on exit.
+ * \note            The read pointer \p buf is updated to point right after
+ *                  the ECParameters record on exit.
  *
- * \param grp       The destination group.
+ * \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 buffer.
+ * \param len       The length of the input buffer \c *buf in Bytes.
  *
  * \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.
+ * \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 );
+int mbedtls_ecp_tls_read_group( mbedtls_ecp_group *grp,
+                                const unsigned char **buf, size_t len );
 
 /**
- * \brief           This function writes the TLS ECParameters record for a group.
+ * \brief           This function extracts an elliptic curve group ID from a
+ *                  TLS ECParameters record as defined in RFC 4492, Section 5.4.
  *
- * \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.
+ * \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_BUFFER_TOO_SMALL on failure.
+ * \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_write_group( const mbedtls_ecp_group *grp, size_t *olen,
-                         unsigned char *buf, size_t blen );
+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 multiplication of a point by
- *                  an integer: \p R = \p m * \p P.
+ * \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.
  *
@@ -730,17 +814,22 @@
  *                  targeting these results. We recommend always providing
  *                  a non-NULL \p f_rng. The overhead is negligible.
  *
- * \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 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 may be \c NULL if randomization
+ *                  of intermediate results isn't desired (discouraged).
+ * \param p_rng     The RNG context to be passed to \p p_rng.
  *
  * \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,
@@ -756,12 +845,16 @@
  *                  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 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 may be \c NULL if randomization
+ *                  of intermediate results isn't desired (discouraged).
+ * \param p_rng     The RNG context to be passed to \p p_rng.
  * \param rs_ctx    The restart context (NULL disables restart).
  *
  * \return          \c 0 on success.
@@ -770,6 +863,7 @@
  * \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,
@@ -785,18 +879,25 @@
  * \note            In contrast to mbedtls_ecp_mul(), this function does not
  *                  guarantee a constant execution flow and timing.
  *
- * \param grp       The ECP group.
- * \param R         The destination point.
+ * \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.
- * \param P         The point to multiply by \p m.
+ *                  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          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,
@@ -813,12 +914,18 @@
  *                  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 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.
- * \param P         The point to multiply by \p m.
+ *                  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.
@@ -828,6 +935,7 @@
  * \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_muladd_restartable(
              mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
@@ -852,38 +960,51 @@
  *                  structures, such as ::mbedtls_ecdh_context or
  *                  ::mbedtls_ecdsa_context.
  *
- * \param grp       The curve the point should lie on.
- * \param pt        The point to check.
+ * \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 on failure.
+ * \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 );
+int mbedtls_ecp_check_pubkey( const mbedtls_ecp_group *grp,
+                              const mbedtls_ecp_point *pt );
 
 /**
- * \brief           This function checks that an \p 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.
  *
  * \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 group used.
- * \param d         The integer to check.
+ * \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 on failure.
+ * \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 );
+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.
- * \param d         The destination MPI (secret part).
- * \param f_rng     The RNG function.
- * \param p_rng     The RNG parameter.
+ * \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
@@ -903,22 +1024,29 @@
  *                  structures, such as ::mbedtls_ecdh_context or
  *                  ::mbedtls_ecdsa_context.
  *
- * \param grp       The ECP group.
- * \param G         The chosen base point.
+ * \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).
- * \param f_rng     The RNG function.
- * \param p_rng     The RNG context.
+ *                  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 );
+                                  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.
@@ -928,34 +1056,42 @@
  *                  structures, such as ::mbedtls_ecdh_context or
  *                  ::mbedtls_ecdsa_context.
  *
- * \param grp       The ECP group.
+ * \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).
- * \param f_rng     The RNG function.
- * \param p_rng     The RNG context.
+ *                  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 );
+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.
- * \param f_rng     The RNG function.
- * \param p_rng     The RNG context.
+ * \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 );
+                         int (*f_rng)(void *, unsigned char *, size_t),
+                         void *p_rng );
 
 /**
  * \brief           This function checks that the keypair objects
@@ -963,16 +1099,19 @@
  *                  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.
- *                  If it contains a private key, that part is ignored.
+ * \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.
  *
  * \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 mbedtls_ecp_check_pub_priv( const mbedtls_ecp_keypair *pub,
+                                const mbedtls_ecp_keypair *prv );
 
 #if defined(MBEDTLS_SELF_TEST)