text: /*
   *  Elliptic curves over GF(p): generic functions
   *
   *  Copyright (C) 2006-2015, ARM Limited, All Rights Reserved
   *  SPDX-License-Identifier: Apache-2.0
   *
   *  Licensed under the Apache License, Version 2.0 (the "License"); you may
   *  not use this file except in compliance with the License.
   *  You may obtain a copy of the License at
   *
   *  http://www.apache.org/licenses/LICENSE-2.0
   *
   *  Unless required by applicable law or agreed to in writing, software
   *  distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
   *  WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   *  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)
   */
  
  /*
   * References:
   *
   * SEC1 http://www.secg.org/index.php?action=secg,docs_secg
   * GECC = Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
   * FIPS 186-3 http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf
   * RFC 4492 for the related TLS structures and constants
   *
   * [Curve25519] http://cr.yp.to/ecdh/curve25519-20060209.pdf
   *
   * [2] CORON, Jean-S'ebastien. Resistance against differential power analysis
   *     for elliptic curve cryptosystems. In : Cryptographic Hardware and
   *     Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302.
   *     <http://link.springer.com/chapter/10.1007/3-540-48059-5_25>
   *
   * [3] HEDABOU, Mustapha, PINEL, Pierre, et B'EN'ETEAU, Lucien. A comb method to
   *     render ECC resistant against Side Channel Attacks. IACR Cryptology
   *     ePrint Archive, 2004, vol. 2004, p. 342.
   *     <http://eprint.iacr.org/2004/342.pdf>
   */
  
  #if !defined(MBEDTLS_CONFIG_FILE)
  #include "mbedtls/config.h"
  #else
  #include MBEDTLS_CONFIG_FILE
  #endif
  
  #if defined(MBEDTLS_ECP_C)
  
  #include "mbedtls/ecp.h"
  #include "mbedtls/threading.h"
  
  #include <string.h>
  
  #if !defined(MBEDTLS_ECP_ALT)
  
  #if defined(MBEDTLS_PLATFORM_C)
  #include "mbedtls/platform.h"
  #else
  #include <stdlib.h>
  #include <stdio.h>
  #define mbedtls_printf     printf
  #define mbedtls_calloc    calloc
  #define mbedtls_free       free
  #endif
  
  #include "mbedtls/ecp_internal.h"
  
  #if ( defined(__ARMCC_VERSION) || defined(_MSC_VER) ) && \
      !defined(inline) && !defined(__cplusplus)
  #define inline __inline
  #endif
  
  /* Implementation that should never be optimized out by the compiler */
  static void mbedtls_zeroize( void *v, size_t n ) {
      volatile unsigned char *p = v; while( n-- ) *p++ = 0;
  }
  
  #if defined(MBEDTLS_SELF_TEST)
  /*
   * Counts of point addition and doubling, and field multiplications.
   * Used to test resistance of point multiplication to simple timing attacks.
   */
  static unsigned long add_count, dbl_count, mul_count;
  #endif
  
  #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) ||   \
      defined(MBEDTLS_ECP_DP_SECP224R1_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_SECP224K1_ENABLED) ||   \
      defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
  #define ECP_SHORTWEIERSTRASS
  #endif
  
  #if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED)
  #define ECP_MONTGOMERY
  #endif
  
  /*
   * Curve types: internal for now, might be exposed later
   */
  typedef enum
  {
      ECP_TYPE_NONE = 0,
      ECP_TYPE_SHORT_WEIERSTRASS,    /* y^2 = x^3 + a x + b      */
      ECP_TYPE_MONTGOMERY,           /* y^2 = x^3 + a x^2 + x    */
  } ecp_curve_type;
  
  /*
   * List of supported curves:
   *  - internal ID
   *  - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2)
   *  - size in bits
   *  - readable name
   *
   * Curves are listed in order: largest curves first, and for a given size,
   * fastest curves first. This provides the default order for the SSL module.
   *
   * Reminder: update profiles in x509_crt.c when adding a new curves!
   */
  static const mbedtls_ecp_curve_info ecp_supported_curves[] =
  {
  #if defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED)
      { MBEDTLS_ECP_DP_SECP521R1,    25,     521,    "secp521r1"         },
  #endif
  #if defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)
      { MBEDTLS_ECP_DP_BP512R1,      28,     512,    "brainpoolP512r1"   },
  #endif
  #if defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED)
      { MBEDTLS_ECP_DP_SECP384R1,    24,     384,    "secp384r1"         },
  #endif
  #if defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)
      { MBEDTLS_ECP_DP_BP384R1,      27,     384,    "brainpoolP384r1"   },
  #endif
  #if defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED)
      { MBEDTLS_ECP_DP_SECP256R1,    23,     256,    "secp256r1"         },
  #endif
  #if defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
      { MBEDTLS_ECP_DP_SECP256K1,    22,     256,    "secp256k1"         },
  #endif
  #if defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)
      { MBEDTLS_ECP_DP_BP256R1,      26,     256,    "brainpoolP256r1"   },
  #endif
  #if defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED)
      { MBEDTLS_ECP_DP_SECP224R1,    21,     224,    "secp224r1"         },
  #endif
  #if defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED)
      { MBEDTLS_ECP_DP_SECP224K1,    20,     224,    "secp224k1"         },
  #endif
  #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
      { MBEDTLS_ECP_DP_SECP192R1,    19,     192,    "secp192r1"         },
  #endif
  #if defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED)
      { MBEDTLS_ECP_DP_SECP192K1,    18,     192,    "secp192k1"         },
  #endif
      { MBEDTLS_ECP_DP_NONE,          0,     0,      NULL                },
  };
  
  #define ECP_NB_CURVES   sizeof( ecp_supported_curves ) /    \
                          sizeof( ecp_supported_curves[0] )
  
  static mbedtls_ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES];
  
  /*
   * List of supported curves and associated info
   */
  const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list( void )
  {
      return( ecp_supported_curves );
  }
  
  /*
   * List of supported curves, group ID only
   */
  const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list( void )
  {
      static int init_done = 0;
  
      if( ! init_done )
      {
          size_t i = 0;
          const mbedtls_ecp_curve_info *curve_info;
  
          for( curve_info = mbedtls_ecp_curve_list();
               curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
               curve_info++ )
          {
              ecp_supported_grp_id[i++] = curve_info->grp_id;
          }
          ecp_supported_grp_id[i] = MBEDTLS_ECP_DP_NONE;
  
          init_done = 1;
      }
  
      return( ecp_supported_grp_id );
  }
  
  /*
   * Get the curve info for the internal identifier
   */
  const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id( mbedtls_ecp_group_id grp_id )
  {
      const mbedtls_ecp_curve_info *curve_info;
  
      for( curve_info = mbedtls_ecp_curve_list();
           curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
           curve_info++ )
      {
          if( curve_info->grp_id == grp_id )
              return( curve_info );
      }
  
      return( NULL );
  }
  
  /*
   * Get the curve info from the TLS identifier
   */
  const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id( uint16_t tls_id )
  {
      const mbedtls_ecp_curve_info *curve_info;
  
      for( curve_info = mbedtls_ecp_curve_list();
           curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
           curve_info++ )
      {
          if( curve_info->tls_id == tls_id )
              return( curve_info );
      }
  
      return( NULL );
  }
  
  /*
   * Get the curve info from the name
   */
  const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name( const char *name )
  {
      const mbedtls_ecp_curve_info *curve_info;
  
      for( curve_info = mbedtls_ecp_curve_list();
           curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
           curve_info++ )
      {
          if( strcmp( curve_info->name, name ) == 0 )
              return( curve_info );
      }
  
      return( NULL );
  }
  
  /*
   * Get the type of a curve
   */
  static inline ecp_curve_type ecp_get_type( const mbedtls_ecp_group *grp )
  {
      if( grp->G.X.p == NULL )
          return( ECP_TYPE_NONE );
  
      if( grp->G.Y.p == NULL )
          return( ECP_TYPE_MONTGOMERY );
      else
          return( ECP_TYPE_SHORT_WEIERSTRASS );
  }
  
  /*
   * Initialize (the components of) a point
   */
  void mbedtls_ecp_point_init( mbedtls_ecp_point *pt )
  {
      if( pt == NULL )
          return;
  
      mbedtls_mpi_init( &pt->X );
      mbedtls_mpi_init( &pt->Y );
      mbedtls_mpi_init( &pt->Z );
  }
  
  /*
   * Initialize (the components of) a group
   */
  void mbedtls_ecp_group_init( mbedtls_ecp_group *grp )
  {
      if( grp == NULL )
          return;
  
      memset( grp, 0, sizeof( mbedtls_ecp_group ) );
  }
  
  /*
   * Initialize (the components of) a key pair
   */
  void mbedtls_ecp_keypair_init( mbedtls_ecp_keypair *key )
  {
      if( key == NULL )
          return;
  
      mbedtls_ecp_group_init( &key->grp );
      mbedtls_mpi_init( &key->d );
      mbedtls_ecp_point_init( &key->Q );
  }
  
  /*
   * Unallocate (the components of) a point
   */
  void mbedtls_ecp_point_free( mbedtls_ecp_point *pt )
  {
      if( pt == NULL )
          return;
  
      mbedtls_mpi_free( &( pt->X ) );
      mbedtls_mpi_free( &( pt->Y ) );
      mbedtls_mpi_free( &( pt->Z ) );
  }
  
  /*
   * Unallocate (the components of) a group
   */
  void mbedtls_ecp_group_free( mbedtls_ecp_group *grp )
  {
      size_t i;
  
      if( grp == NULL )
          return;
  
      if( grp->h != 1 )
      {
          mbedtls_mpi_free( &grp->P );
          mbedtls_mpi_free( &grp->A );
          mbedtls_mpi_free( &grp->B );
          mbedtls_ecp_point_free( &grp->G );
          mbedtls_mpi_free( &grp->N );
      }
  
      if( grp->T != NULL )
      {
          for( i = 0; i < grp->T_size; i++ )
              mbedtls_ecp_point_free( &grp->T[i] );
          mbedtls_free( grp->T );
      }
  
      mbedtls_zeroize( grp, sizeof( mbedtls_ecp_group ) );
  }
  
  /*
   * Unallocate (the components of) a key pair
   */
  void mbedtls_ecp_keypair_free( mbedtls_ecp_keypair *key )
  {
      if( key == NULL )
          return;
  
      mbedtls_ecp_group_free( &key->grp );
      mbedtls_mpi_free( &key->d );
      mbedtls_ecp_point_free( &key->Q );
  }
  
  /*
   * Copy the contents of a point
   */
  int mbedtls_ecp_copy( mbedtls_ecp_point *P, const mbedtls_ecp_point *Q )
  {
      int ret;
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->X, &Q->X ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->Y, &Q->Y ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->Z, &Q->Z ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Copy the contents of a group object
   */
  int mbedtls_ecp_group_copy( mbedtls_ecp_group *dst, const mbedtls_ecp_group *src )
  {
      return mbedtls_ecp_group_load( dst, src->id );
  }
  
  /*
   * Set point to zero
   */
  int mbedtls_ecp_set_zero( mbedtls_ecp_point *pt )
  {
      int ret;
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->X , 1 ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Y , 1 ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Z , 0 ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Tell if a point is zero
   */
  int mbedtls_ecp_is_zero( mbedtls_ecp_point *pt )
  {
      return( mbedtls_mpi_cmp_int( &pt->Z, 0 ) == 0 );
  }
  
  /*
   * Compare two points lazily
   */
  int mbedtls_ecp_point_cmp( const mbedtls_ecp_point *P,
                             const mbedtls_ecp_point *Q )
  {
      if( mbedtls_mpi_cmp_mpi( &P->X, &Q->X ) == 0 &&
          mbedtls_mpi_cmp_mpi( &P->Y, &Q->Y ) == 0 &&
          mbedtls_mpi_cmp_mpi( &P->Z, &Q->Z ) == 0 )
      {
          return( 0 );
      }
  
      return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  }
  
  /*
   * Import a non-zero point from ASCII strings
   */
  int mbedtls_ecp_point_read_string( mbedtls_ecp_point *P, int radix,
                             const char *x, const char *y )
  {
      int ret;
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &P->X, radix, x ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &P->Y, radix, y ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &P->Z, 1 ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Export a point into unsigned binary data (SEC1 2.3.3)
   */
  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 )
  {
      int ret = 0;
      size_t plen;
  
      if( format != MBEDTLS_ECP_PF_UNCOMPRESSED &&
          format != MBEDTLS_ECP_PF_COMPRESSED )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      /*
       * Common case: P == 0
       */
      if( mbedtls_mpi_cmp_int( &P->Z, 0 ) == 0 )
      {
          if( buflen < 1 )
              return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
  
          buf[0] = 0x00;
          *olen = 1;
  
          return( 0 );
      }
  
      plen = mbedtls_mpi_size( &grp->P );
  
      if( format == MBEDTLS_ECP_PF_UNCOMPRESSED )
      {
          *olen = 2 * plen + 1;
  
          if( buflen < *olen )
              return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
  
          buf[0] = 0x04;
          MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P->X, buf + 1, plen ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P->Y, buf + 1 + plen, plen ) );
      }
      else if( format == MBEDTLS_ECP_PF_COMPRESSED )
      {
          *olen = plen + 1;
  
          if( buflen < *olen )
              return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
  
          buf[0] = 0x02 + mbedtls_mpi_get_bit( &P->Y, 0 );
          MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P->X, buf + 1, plen ) );
      }
  
  cleanup:
      return( ret );
  }
  
  /*
   * Import a point from unsigned binary data (SEC1 2.3.4)
   */
  int mbedtls_ecp_point_read_binary( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
                             const unsigned char *buf, size_t ilen )
  {
      int ret;
      size_t plen;
  
      if( ilen < 1 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      if( buf[0] == 0x00 )
      {
          if( ilen == 1 )
              return( mbedtls_ecp_set_zero( pt ) );
          else
              return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
      }
  
      plen = mbedtls_mpi_size( &grp->P );
  
      if( buf[0] != 0x04 )
          return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE );
  
      if( ilen != 2 * plen + 1 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_read_binary( &pt->X, buf + 1, plen ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_read_binary( &pt->Y, buf + 1 + plen, plen ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Z, 1 ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Import a point from a TLS ECPoint record (RFC 4492)
   *      struct {
   *          opaque point <1..2^8-1>;
   *      } ECPoint;
   */
  int mbedtls_ecp_tls_read_point( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
                          const unsigned char **buf, size_t buf_len )
  {
      unsigned char data_len;
      const unsigned char *buf_start;
  
      /*
       * We must have at least two bytes (1 for length, at least one for data)
       */
      if( buf_len < 2 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      data_len = *(*buf)++;
      if( data_len < 1 || data_len > buf_len - 1 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      /*
       * Save buffer start for read_binary and update buf
       */
      buf_start = *buf;
      *buf += data_len;
  
      return mbedtls_ecp_point_read_binary( grp, pt, buf_start, data_len );
  }
  
  /*
   * Export a point as a TLS ECPoint record (RFC 4492)
   *      struct {
   *          opaque point <1..2^8-1>;
   *      } ECPoint;
   */
  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 ret;
  
      /*
       * buffer length must be at least one, for our length byte
       */
      if( blen < 1 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      if( ( ret = mbedtls_ecp_point_write_binary( grp, pt, format,
                      olen, buf + 1, blen - 1) ) != 0 )
          return( ret );
  
      /*
       * write length to the first byte and update total length
       */
      buf[0] = (unsigned char) *olen;
      ++*olen;
  
      return( 0 );
  }
  
  /*
   * Set a group from an ECParameters record (RFC 4492)
   */
  int mbedtls_ecp_tls_read_group( mbedtls_ecp_group *grp, const unsigned char **buf, size_t len )
  {
      uint16_t tls_id;
      const mbedtls_ecp_curve_info *curve_info;
  
      /*
       * We expect at least three bytes (see below)
       */
      if( len < 3 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      /*
       * First byte is curve_type; only named_curve is handled
       */
      if( *(*buf)++ != MBEDTLS_ECP_TLS_NAMED_CURVE )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      /*
       * Next two bytes are the namedcurve value
       */
      tls_id = *(*buf)++;
      tls_id <<= 8;
      tls_id |= *(*buf)++;
  
      if( ( curve_info = mbedtls_ecp_curve_info_from_tls_id( tls_id ) ) == NULL )
          return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE );
  
      return mbedtls_ecp_group_load( grp, curve_info->grp_id );
  }
  
  /*
   * Write the ECParameters record corresponding to a group (RFC 4492)
   */
  int mbedtls_ecp_tls_write_group( const mbedtls_ecp_group *grp, size_t *olen,
                           unsigned char *buf, size_t blen )
  {
      const mbedtls_ecp_curve_info *curve_info;
  
      if( ( curve_info = mbedtls_ecp_curve_info_from_grp_id( grp->id ) ) == NULL )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      /*
       * We are going to write 3 bytes (see below)
       */
      *olen = 3;
      if( blen < *olen )
          return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
  
      /*
       * First byte is curve_type, always named_curve
       */
      *buf++ = MBEDTLS_ECP_TLS_NAMED_CURVE;
  
      /*
       * Next two bytes are the namedcurve value
       */
      buf[0] = curve_info->tls_id >> 8;
      buf[1] = curve_info->tls_id & 0xFF;
  
      return( 0 );
  }
  
  /*
   * Wrapper around fast quasi-modp functions, with fall-back to mbedtls_mpi_mod_mpi.
   * See the documentation of struct mbedtls_ecp_group.
   *
   * This function is in the critial loop for mbedtls_ecp_mul, so pay attention to perf.
   */
  static int ecp_modp( mbedtls_mpi *N, const mbedtls_ecp_group *grp )
  {
      int ret;
  
      if( grp->modp == NULL )
          return( mbedtls_mpi_mod_mpi( N, N, &grp->P ) );
  
      /* N->s < 0 is a much faster test, which fails only if N is 0 */
      if( ( N->s < 0 && mbedtls_mpi_cmp_int( N, 0 ) != 0 ) ||
          mbedtls_mpi_bitlen( N ) > 2 * grp->pbits )
      {
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
      }
  
      MBEDTLS_MPI_CHK( grp->modp( N ) );
  
      /* N->s < 0 is a much faster test, which fails only if N is 0 */
      while( N->s < 0 && mbedtls_mpi_cmp_int( N, 0 ) != 0 )
          MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( N, N, &grp->P ) );
  
      while( mbedtls_mpi_cmp_mpi( N, &grp->P ) >= 0 )
          /* we known P, N and the result are positive */
          MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( N, N, &grp->P ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Fast mod-p functions expect their argument to be in the 0..p^2 range.
   *
   * In order to guarantee that, we need to ensure that operands of
   * mbedtls_mpi_mul_mpi are in the 0..p range. So, after each operation we will
   * bring the result back to this range.
   *
   * The following macros are shortcuts for doing that.
   */
  
  /*
   * Reduce a mbedtls_mpi mod p in-place, general case, to use after mbedtls_mpi_mul_mpi
   */
  #if defined(MBEDTLS_SELF_TEST)
  #define INC_MUL_COUNT   mul_count++;
  #else
  #define INC_MUL_COUNT
  #endif
  
  #define MOD_MUL( N )    do { MBEDTLS_MPI_CHK( ecp_modp( &N, grp ) ); INC_MUL_COUNT } \
                          while( 0 )
  
  /*
   * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_sub_mpi
   * N->s < 0 is a very fast test, which fails only if N is 0
   */
  #define MOD_SUB( N )                                \
      while( N.s < 0 && mbedtls_mpi_cmp_int( &N, 0 ) != 0 )   \
          MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &N, &N, &grp->P ) )
  
  /*
   * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_add_mpi and mbedtls_mpi_mul_int.
   * We known P, N and the result are positive, so sub_abs is correct, and
   * a bit faster.
   */
  #define MOD_ADD( N )                                \
      while( mbedtls_mpi_cmp_mpi( &N, &grp->P ) >= 0 )        \
          MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( &N, &N, &grp->P ) )
  
  #if defined(ECP_SHORTWEIERSTRASS)
  /*
   * For curves in short Weierstrass form, we do all the internal operations in
   * Jacobian coordinates.
   *
   * For multiplication, we'll use a comb method with coutermeasueres against
   * SPA, hence timing attacks.
   */
  
  /*
   * Normalize jacobian coordinates so that Z == 0 || Z == 1  (GECC 3.2.1)
   * Cost: 1N := 1I + 3M + 1S
   */
  static int ecp_normalize_jac( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt )
  {
      int ret;
      mbedtls_mpi Zi, ZZi;
  
      if( mbedtls_mpi_cmp_int( &pt->Z, 0 ) == 0 )
          return( 0 );
  
  #if defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_normalize_jac( grp, pt );
      }
  #endif /* MBEDTLS_ECP_NORMALIZE_JAC_ALT */
      mbedtls_mpi_init( &Zi ); mbedtls_mpi_init( &ZZi );
  
      /*
       * X = X / Z^2  mod p
       */
      MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &Zi,      &pt->Z,     &grp->P ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ZZi,     &Zi,        &Zi     ) ); MOD_MUL( ZZi );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->X,   &pt->X,     &ZZi    ) ); MOD_MUL( pt->X );
  
      /*
       * Y = Y / Z^3  mod p
       */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Y,   &pt->Y,     &ZZi    ) ); MOD_MUL( pt->Y );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Y,   &pt->Y,     &Zi     ) ); MOD_MUL( pt->Y );
  
      /*
       * Z = 1
       */
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Z, 1 ) );
  
  cleanup:
  
      mbedtls_mpi_free( &Zi ); mbedtls_mpi_free( &ZZi );
  
      return( ret );
  }
  
  /*
   * Normalize jacobian coordinates of an array of (pointers to) points,
   * using Montgomery's trick to perform only one inversion mod P.
   * (See for example Cohen's "A Course in Computational Algebraic Number
   * Theory", Algorithm 10.3.4.)
   *
   * Warning: fails (returning an error) if one of the points is zero!
   * This should never happen, see choice of w in ecp_mul_comb().
   *
   * Cost: 1N(t) := 1I + (6t - 3)M + 1S
   */
  static int ecp_normalize_jac_many( const mbedtls_ecp_group *grp,
                                     mbedtls_ecp_point *T[], size_t t_len )
  {
      int ret;
      size_t i;
      mbedtls_mpi *c, u, Zi, ZZi;
  
      if( t_len < 2 )
          return( ecp_normalize_jac( grp, *T ) );
  
  #if defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_normalize_jac_many(grp, T, t_len);
      }
  #endif
  
      if( ( c = mbedtls_calloc( t_len, sizeof( mbedtls_mpi ) ) ) == NULL )
          return( MBEDTLS_ERR_ECP_ALLOC_FAILED );
  
      mbedtls_mpi_init( &u ); mbedtls_mpi_init( &Zi ); mbedtls_mpi_init( &ZZi );
  
      /*
       * c[i] = Z_0 * ... * Z_i
       */
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &c[0], &T[0]->Z ) );
      for( i = 1; i < t_len; i++ )
      {
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &c[i], &c[i-1], &T[i]->Z ) );
          MOD_MUL( c[i] );
      }
  
      /*
       * u = 1 / (Z_0 * ... * Z_n) mod P
       */
      MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &u, &c[t_len-1], &grp->P ) );
  
      for( i = t_len - 1; ; i-- )
      {
          /*
           * Zi = 1 / Z_i mod p
           * u = 1 / (Z_0 * ... * Z_i) mod P
           */
          if( i == 0 ) {
              MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &Zi, &u ) );
          }
          else
          {
              MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &Zi, &u, &c[i-1]  ) ); MOD_MUL( Zi );
              MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &u,  &u, &T[i]->Z ) ); MOD_MUL( u );
          }
  
          /*
           * proceed as in normalize()
           */
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ZZi,     &Zi,      &Zi  ) ); MOD_MUL( ZZi );
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T[i]->X, &T[i]->X, &ZZi ) ); MOD_MUL( T[i]->X );
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &ZZi ) ); MOD_MUL( T[i]->Y );
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &Zi  ) ); MOD_MUL( T[i]->Y );
  
          /*
           * Post-precessing: reclaim some memory by shrinking coordinates
           * - not storing Z (always 1)
           * - shrinking other coordinates, but still keeping the same number of
           *   limbs as P, as otherwise it will too likely be regrown too fast.
           */
          MBEDTLS_MPI_CHK( mbedtls_mpi_shrink( &T[i]->X, grp->P.n ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_shrink( &T[i]->Y, grp->P.n ) );
          mbedtls_mpi_free( &T[i]->Z );
  
          if( i == 0 )
              break;
      }
  
  cleanup:
  
      mbedtls_mpi_free( &u ); mbedtls_mpi_free( &Zi ); mbedtls_mpi_free( &ZZi );
      for( i = 0; i < t_len; i++ )
          mbedtls_mpi_free( &c[i] );
      mbedtls_free( c );
  
      return( ret );
  }
  
  /*
   * Conditional point inversion: Q -> -Q = (Q.X, -Q.Y, Q.Z) without leak.
   * "inv" must be 0 (don't invert) or 1 (invert) or the result will be invalid
   */
  static int ecp_safe_invert_jac( const mbedtls_ecp_group *grp,
                              mbedtls_ecp_point *Q,
                              unsigned char inv )
  {
      int ret;
      unsigned char nonzero;
      mbedtls_mpi mQY;
  
      mbedtls_mpi_init( &mQY );
  
      /* Use the fact that -Q.Y mod P = P - Q.Y unless Q.Y == 0 */
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &mQY, &grp->P, &Q->Y ) );
      nonzero = mbedtls_mpi_cmp_int( &Q->Y, 0 ) != 0;
      MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &Q->Y, &mQY, inv & nonzero ) );
  
  cleanup:
      mbedtls_mpi_free( &mQY );
  
      return( ret );
  }
  
  /*
   * Point doubling R = 2 P, Jacobian coordinates
   *
   * Based on http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-1998-cmo-2 .
   *
   * We follow the variable naming fairly closely. The formula variations that trade a MUL for a SQR
   * (plus a few ADDs) aren't useful as our bignum implementation doesn't distinguish squaring.
   *
   * Standard optimizations are applied when curve parameter A is one of { 0, -3 }.
   *
   * Cost: 1D := 3M + 4S          (A ==  0)
   *             4M + 4S          (A == -3)
   *             3M + 6S + 1a     otherwise
   */
  static int ecp_double_jac( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                             const mbedtls_ecp_point *P )
  {
      int ret;
      mbedtls_mpi M, S, T, U;
  
  #if defined(MBEDTLS_SELF_TEST)
      dbl_count++;
  #endif
  
  #if defined(MBEDTLS_ECP_DOUBLE_JAC_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_double_jac( grp, R, P );
      }
  #endif /* MBEDTLS_ECP_DOUBLE_JAC_ALT */
  
      mbedtls_mpi_init( &M ); mbedtls_mpi_init( &S ); mbedtls_mpi_init( &T ); mbedtls_mpi_init( &U );
  
      /* Special case for A = -3 */
      if( grp->A.p == NULL )
      {
          /* M = 3(X + Z^2)(X - Z^2) */
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S,  &P->Z,  &P->Z   ) ); MOD_MUL( S );
          MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &T,  &P->X,  &S      ) ); MOD_ADD( T );
          MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &U,  &P->X,  &S      ) ); MOD_SUB( U );
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S,  &T,     &U      ) ); MOD_MUL( S );
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &M,  &S,     3       ) ); MOD_ADD( M );
      }
      else
      {
          /* M = 3.X^2 */
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S,  &P->X,  &P->X   ) ); MOD_MUL( S );
          MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &M,  &S,     3       ) ); MOD_ADD( M );
  
          /* Optimize away for "koblitz" curves with A = 0 */
          if( mbedtls_mpi_cmp_int( &grp->A, 0 ) != 0 )
          {
              /* M += A.Z^4 */
              MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S,  &P->Z,  &P->Z   ) ); MOD_MUL( S );
              MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T,  &S,     &S      ) ); MOD_MUL( T );
              MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S,  &T,     &grp->A ) ); MOD_MUL( S );
              MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &M,  &M,     &S      ) ); MOD_ADD( M );
          }
      }
  
      /* S = 4.X.Y^2 */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T,  &P->Y,  &P->Y   ) ); MOD_MUL( T );
      MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &T,  1               ) ); MOD_ADD( T );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S,  &P->X,  &T      ) ); MOD_MUL( S );
      MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &S,  1               ) ); MOD_ADD( S );
  
      /* U = 8.Y^4 */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &U,  &T,     &T      ) ); MOD_MUL( U );
      MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &U,  1               ) ); MOD_ADD( U );
  
      /* T = M^2 - 2.S */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T,  &M,     &M      ) ); MOD_MUL( T );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T,  &T,     &S      ) ); MOD_SUB( T );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T,  &T,     &S      ) ); MOD_SUB( T );
  
      /* S = M(S - T) - U */
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S,  &S,     &T      ) ); MOD_SUB( S );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S,  &S,     &M      ) ); MOD_MUL( S );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S,  &S,     &U      ) ); MOD_SUB( S );
  
      /* U = 2.Y.Z */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &U,  &P->Y,  &P->Z   ) ); MOD_MUL( U );
      MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &U,  1               ) ); MOD_ADD( U );
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->X, &T ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Y, &S ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Z, &U ) );
  
  cleanup:
      mbedtls_mpi_free( &M ); mbedtls_mpi_free( &S ); mbedtls_mpi_free( &T ); mbedtls_mpi_free( &U );
  
      return( ret );
  }
  
  /*
   * Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22)
   *
   * The coordinates of Q must be normalized (= affine),
   * but those of P don't need to. R is not normalized.
   *
   * Special cases: (1) P or Q is zero, (2) R is zero, (3) P == Q.
   * None of these cases can happen as intermediate step in ecp_mul_comb():
   * - at each step, P, Q and R are multiples of the base point, the factor
   *   being less than its order, so none of them is zero;
   * - Q is an odd multiple of the base point, P an even multiple,
   *   due to the choice of precomputed points in the modified comb method.
   * So branches for these cases do not leak secret information.
   *
   * We accept Q->Z being unset (saving memory in tables) as meaning 1.
   *
   * Cost: 1A := 8M + 3S
   */
  static int ecp_add_mixed( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                            const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q )
  {
      int ret;
      mbedtls_mpi T1, T2, T3, T4, X, Y, Z;
  
  #if defined(MBEDTLS_SELF_TEST)
      add_count++;
  #endif
  
  #if defined(MBEDTLS_ECP_ADD_MIXED_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_add_mixed( grp, R, P, Q );
      }
  #endif /* MBEDTLS_ECP_ADD_MIXED_ALT */
  
      /*
       * Trivial cases: P == 0 or Q == 0 (case 1)
       */
      if( mbedtls_mpi_cmp_int( &P->Z, 0 ) == 0 )
          return( mbedtls_ecp_copy( R, Q ) );
  
      if( Q->Z.p != NULL && mbedtls_mpi_cmp_int( &Q->Z, 0 ) == 0 )
          return( mbedtls_ecp_copy( R, P ) );
  
      /*
       * Make sure Q coordinates are normalized
       */
      if( Q->Z.p != NULL && mbedtls_mpi_cmp_int( &Q->Z, 1 ) != 0 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      mbedtls_mpi_init( &T1 ); mbedtls_mpi_init( &T2 ); mbedtls_mpi_init( &T3 ); mbedtls_mpi_init( &T4 );
      mbedtls_mpi_init( &X ); mbedtls_mpi_init( &Y ); mbedtls_mpi_init( &Z );
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T1,  &P->Z,  &P->Z ) );  MOD_MUL( T1 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T2,  &T1,    &P->Z ) );  MOD_MUL( T2 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T1,  &T1,    &Q->X ) );  MOD_MUL( T1 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T2,  &T2,    &Q->Y ) );  MOD_MUL( T2 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T1,  &T1,    &P->X ) );  MOD_SUB( T1 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T2,  &T2,    &P->Y ) );  MOD_SUB( T2 );
  
      /* Special cases (2) and (3) */
      if( mbedtls_mpi_cmp_int( &T1, 0 ) == 0 )
      {
          if( mbedtls_mpi_cmp_int( &T2, 0 ) == 0 )
          {
              ret = ecp_double_jac( grp, R, P );
              goto cleanup;
          }
          else
          {
              ret = mbedtls_ecp_set_zero( R );
              goto cleanup;
          }
      }
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &Z,   &P->Z,  &T1   ) );  MOD_MUL( Z  );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3,  &T1,    &T1   ) );  MOD_MUL( T3 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T4,  &T3,    &T1   ) );  MOD_MUL( T4 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3,  &T3,    &P->X ) );  MOD_MUL( T3 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &T1,  &T3,    2     ) );  MOD_ADD( T1 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &X,   &T2,    &T2   ) );  MOD_MUL( X  );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X,   &X,     &T1   ) );  MOD_SUB( X  );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X,   &X,     &T4   ) );  MOD_SUB( X  );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T3,  &T3,    &X    ) );  MOD_SUB( T3 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3,  &T3,    &T2   ) );  MOD_MUL( T3 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T4,  &T4,    &P->Y ) );  MOD_MUL( T4 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &Y,   &T3,    &T4   ) );  MOD_SUB( Y  );
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->X, &X ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Y, &Y ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Z, &Z ) );
  
  cleanup:
  
      mbedtls_mpi_free( &T1 ); mbedtls_mpi_free( &T2 ); mbedtls_mpi_free( &T3 ); mbedtls_mpi_free( &T4 );
      mbedtls_mpi_free( &X ); mbedtls_mpi_free( &Y ); mbedtls_mpi_free( &Z );
  
      return( ret );
  }
  
  /*
   * Randomize jacobian coordinates:
   * (X, Y, Z) -> (l^2 X, l^3 Y, l Z) for random l
   * This is sort of the reverse operation of ecp_normalize_jac().
   *
   * This countermeasure was first suggested in [2].
   */
  static int ecp_randomize_jac( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
                  int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
  {
      int ret;
      mbedtls_mpi l, ll;
      size_t p_size;
      int count = 0;
  
  #if defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_randomize_jac( grp, pt, f_rng, p_rng );
      }
  #endif /* MBEDTLS_ECP_RANDOMIZE_JAC_ALT */
  
      p_size = ( grp->pbits + 7 ) / 8;
      mbedtls_mpi_init( &l ); mbedtls_mpi_init( &ll );
  
      /* Generate l such that 1 < l < p */
      do
      {
          MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &l, p_size, f_rng, p_rng ) );
  
          while( mbedtls_mpi_cmp_mpi( &l, &grp->P ) >= 0 )
              MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &l, 1 ) );
  
          if( count++ > 10 )
              return( MBEDTLS_ERR_ECP_RANDOM_FAILED );
      }
      while( mbedtls_mpi_cmp_int( &l, 1 ) <= 0 );
  
      /* Z = l * Z */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Z,   &pt->Z,     &l  ) ); MOD_MUL( pt->Z );
  
      /* X = l^2 * X */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ll,      &l,         &l  ) ); MOD_MUL( ll );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->X,   &pt->X,     &ll ) ); MOD_MUL( pt->X );
  
      /* Y = l^3 * Y */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ll,      &ll,        &l  ) ); MOD_MUL( ll );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Y,   &pt->Y,     &ll ) ); MOD_MUL( pt->Y );
  
  cleanup:
      mbedtls_mpi_free( &l ); mbedtls_mpi_free( &ll );
  
      return( ret );
  }
  
  /*
   * Check and define parameters used by the comb method (see below for details)
   */
  #if MBEDTLS_ECP_WINDOW_SIZE < 2 || MBEDTLS_ECP_WINDOW_SIZE > 7
  #error "MBEDTLS_ECP_WINDOW_SIZE out of bounds"
  #endif
  
  /* d = ceil( n / w ) */
  #define COMB_MAX_D      ( MBEDTLS_ECP_MAX_BITS + 1 ) / 2
  
  /* number of precomputed points */
  #define COMB_MAX_PRE    ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) )
  
  /*
   * Compute the representation of m that will be used with our comb method.
   *
   * The basic comb method is described in GECC 3.44 for example. We use a
   * modified version that provides resistance to SPA by avoiding zero
   * digits in the representation as in [3]. We modify the method further by
   * requiring that all K_i be odd, which has the small cost that our
   * representation uses one more K_i, due to carries.
   *
   * Also, for the sake of compactness, only the seven low-order bits of x[i]
   * are used to represent K_i, and the msb of x[i] encodes the the sign (s_i in
   * the paper): it is set if and only if if s_i == -1;
   *
   * Calling conventions:
   * - x is an array of size d + 1
   * - w is the size, ie number of teeth, of the comb, and must be between
   *   2 and 7 (in practice, between 2 and MBEDTLS_ECP_WINDOW_SIZE)
   * - m is the MPI, expected to be odd and such that bitlength(m) <= w * d
   *   (the result will be incorrect if these assumptions are not satisfied)
   */
  static void ecp_comb_fixed( unsigned char x[], size_t d,
                              unsigned char w, const mbedtls_mpi *m )
  {
      size_t i, j;
      unsigned char c, cc, adjust;
  
      memset( x, 0, d+1 );
  
      /* First get the classical comb values (except for x_d = 0) */
      for( i = 0; i < d; i++ )
          for( j = 0; j < w; j++ )
              x[i] |= mbedtls_mpi_get_bit( m, i + d * j ) << j;
  
      /* Now make sure x_1 .. x_d are odd */
      c = 0;
      for( i = 1; i <= d; i++ )
      {
          /* Add carry and update it */
          cc   = x[i] & c;
          x[i] = x[i] ^ c;
          c = cc;
  
          /* Adjust if needed, avoiding branches */
          adjust = 1 - ( x[i] & 0x01 );
          c   |= x[i] & ( x[i-1] * adjust );
          x[i] = x[i] ^ ( x[i-1] * adjust );
          x[i-1] |= adjust << 7;
      }
  }
  
  /*
   * Precompute points for the comb method
   *
   * If i = i_{w-1} ... i_1 is the binary representation of i, then
   * T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P
   *
   * T must be able to hold 2^{w - 1} elements
   *
   * Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1)
   */
  static int ecp_precompute_comb( const mbedtls_ecp_group *grp,
                                  mbedtls_ecp_point T[], const mbedtls_ecp_point *P,
                                  unsigned char w, size_t d )
  {
      int ret;
      unsigned char i, k;
      size_t j;
      mbedtls_ecp_point *cur, *TT[COMB_MAX_PRE - 1];
  
      /*
       * Set T[0] = P and
       * T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value)
       */
      MBEDTLS_MPI_CHK( mbedtls_ecp_copy( &T[0], P ) );
  
      k = 0;
      for( i = 1; i < ( 1U << ( w - 1 ) ); i <<= 1 )
      {
          cur = T + i;
          MBEDTLS_MPI_CHK( mbedtls_ecp_copy( cur, T + ( i >> 1 ) ) );
          for( j = 0; j < d; j++ )
              MBEDTLS_MPI_CHK( ecp_double_jac( grp, cur, cur ) );
  
          TT[k++] = cur;
      }
  
      MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp, TT, k ) );
  
      /*
       * Compute the remaining ones using the minimal number of additions
       * Be careful to update T[2^l] only after using it!
       */
      k = 0;
      for( i = 1; i < ( 1U << ( w - 1 ) ); i <<= 1 )
      {
          j = i;
          while( j-- )
          {
              MBEDTLS_MPI_CHK( ecp_add_mixed( grp, &T[i + j], &T[j], &T[i] ) );
              TT[k++] = &T[i + j];
          }
      }
  
      MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp, TT, k ) );
  
  cleanup:
  
      return( ret );
  }
  
  /*
   * Select precomputed point: R = sign(i) * T[ abs(i) / 2 ]
   */
  static int ecp_select_comb( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                              const mbedtls_ecp_point T[], unsigned char t_len,
                              unsigned char i )
  {
      int ret;
      unsigned char ii, j;
  
      /* Ignore the "sign" bit and scale down */
      ii =  ( i & 0x7Fu ) >> 1;
  
      /* Read the whole table to thwart cache-based timing attacks */
      for( j = 0; j < t_len; j++ )
      {
          MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &R->X, &T[j].X, j == ii ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &R->Y, &T[j].Y, j == ii ) );
      }
  
      /* Safely invert result if i is "negative" */
      MBEDTLS_MPI_CHK( ecp_safe_invert_jac( grp, R, i >> 7 ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Core multiplication algorithm for the (modified) comb method.
   * This part is actually common with the basic comb method (GECC 3.44)
   *
   * Cost: d A + d D + 1 R
   */
  static int ecp_mul_comb_core( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
                                const mbedtls_ecp_point T[], unsigned char t_len,
                                const unsigned char x[], size_t d,
                                int (*f_rng)(void *, unsigned char *, size_t),
                                void *p_rng )
  {
      int ret;
      mbedtls_ecp_point Txi;
      size_t i;
  
      mbedtls_ecp_point_init( &Txi );
  
      /* Start with a non-zero point and randomize its coordinates */
      i = d;
      MBEDTLS_MPI_CHK( ecp_select_comb( grp, R, T, t_len, x[i] ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R->Z, 1 ) );
      if( f_rng != 0 )
          MBEDTLS_MPI_CHK( ecp_randomize_jac( grp, R, f_rng, p_rng ) );
  
      while( i-- != 0 )
      {
          MBEDTLS_MPI_CHK( ecp_double_jac( grp, R, R ) );
          MBEDTLS_MPI_CHK( ecp_select_comb( grp, &Txi, T, t_len, x[i] ) );
          MBEDTLS_MPI_CHK( ecp_add_mixed( grp, R, R, &Txi ) );
      }
  
  cleanup:
  
      mbedtls_ecp_point_free( &Txi );
  
      return( ret );
  }
  
  /*
   * Multiplication using the comb method,
   * for curves in short Weierstrass form
   */
  static int ecp_mul_comb( 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 )
  {
      int ret;
      unsigned char w, m_is_odd, p_eq_g, pre_len, i;
      size_t d;
      unsigned char k[COMB_MAX_D + 1];
      mbedtls_ecp_point *T;
      mbedtls_mpi M, mm;
  
      mbedtls_mpi_init( &M );
      mbedtls_mpi_init( &mm );
  
      /* we need N to be odd to trnaform m in an odd number, check now */
      if( mbedtls_mpi_get_bit( &grp->N, 0 ) != 1 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      /*
       * Minimize the number of multiplications, that is minimize
       * 10 * d * w + 18 * 2^(w-1) + 11 * d + 7 * w, with d = ceil( nbits / w )
       * (see costs of the various parts, with 1S = 1M)
       */
      w = grp->nbits >= 384 ? 5 : 4;
  
      /*
       * If P == G, pre-compute a bit more, since this may be re-used later.
       * Just adding one avoids upping the cost of the first mul too much,
       * and the memory cost too.
       */
  #if MBEDTLS_ECP_FIXED_POINT_OPTIM == 1
      p_eq_g = ( mbedtls_mpi_cmp_mpi( &P->Y, &grp->G.Y ) == 0 &&
                 mbedtls_mpi_cmp_mpi( &P->X, &grp->G.X ) == 0 );
      if( p_eq_g )
          w++;
  #else
      p_eq_g = 0;
  #endif
  
      /*
       * Make sure w is within bounds.
       * (The last test is useful only for very small curves in the test suite.)
       */
      if( w > MBEDTLS_ECP_WINDOW_SIZE )
          w = MBEDTLS_ECP_WINDOW_SIZE;
      if( w >= grp->nbits )
          w = 2;
  
      /* Other sizes that depend on w */
      pre_len = 1U << ( w - 1 );
      d = ( grp->nbits + w - 1 ) / w;
  
      /*
       * Prepare precomputed points: if P == G we want to
       * use grp->T if already initialized, or initialize it.
       */
      T = p_eq_g ? grp->T : NULL;
  
      if( T == NULL )
      {
          T = mbedtls_calloc( pre_len, sizeof( mbedtls_ecp_point ) );
          if( T == NULL )
          {
              ret = MBEDTLS_ERR_ECP_ALLOC_FAILED;
              goto cleanup;
          }
  
          MBEDTLS_MPI_CHK( ecp_precompute_comb( grp, T, P, w, d ) );
  
          if( p_eq_g )
          {
              grp->T = T;
              grp->T_size = pre_len;
          }
      }
  
      /*
       * Make sure M is odd (M = m or M = N - m, since N is odd)
       * using the fact that m * P = - (N - m) * P
       */
      m_is_odd = ( mbedtls_mpi_get_bit( m, 0 ) == 1 );
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &M, m ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &mm, &grp->N, m ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &M, &mm, ! m_is_odd ) );
  
      /*
       * Go for comb multiplication, R = M * P
       */
      ecp_comb_fixed( k, d, w, &M );
      MBEDTLS_MPI_CHK( ecp_mul_comb_core( grp, R, T, pre_len, k, d, f_rng, p_rng ) );
  
      /*
       * Now get m * P from M * P and normalize it
       */
      MBEDTLS_MPI_CHK( ecp_safe_invert_jac( grp, R, ! m_is_odd ) );
      MBEDTLS_MPI_CHK( ecp_normalize_jac( grp, R ) );
  
  cleanup:
  
      /* There are two cases where T is not stored in grp:
       * - P != G
       * - An intermediate operation failed before setting grp->T
       * In either case, T must be freed.
       */
      if( T != NULL && T != grp->T )
      {
          for( i = 0; i < pre_len; i++ )
              mbedtls_ecp_point_free( &T[i] );
          mbedtls_free( T );
      }
  
      mbedtls_mpi_free( &M );
      mbedtls_mpi_free( &mm );
  
      if( ret != 0 )
          mbedtls_ecp_point_free( R );
  
      return( ret );
  }
  
  #endif /* ECP_SHORTWEIERSTRASS */
  
  #if defined(ECP_MONTGOMERY)
  /*
   * For Montgomery curves, we do all the internal arithmetic in projective
   * coordinates. Import/export of points uses only the x coordinates, which is
   * internaly represented as X / Z.
   *
   * For scalar multiplication, we'll use a Montgomery ladder.
   */
  
  /*
   * Normalize Montgomery x/z coordinates: X = X/Z, Z = 1
   * Cost: 1M + 1I
   */
  static int ecp_normalize_mxz( const mbedtls_ecp_group *grp, mbedtls_ecp_point *P )
  {
      int ret;
  
  #if defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_normalize_mxz( grp, P );
      }
  #endif /* MBEDTLS_ECP_NORMALIZE_MXZ_ALT */
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &P->Z, &P->Z, &grp->P ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P->X, &P->X, &P->Z ) ); MOD_MUL( P->X );
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &P->Z, 1 ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Randomize projective x/z coordinates:
   * (X, Z) -> (l X, l Z) for random l
   * This is sort of the reverse operation of ecp_normalize_mxz().
   *
   * This countermeasure was first suggested in [2].
   * Cost: 2M
   */
  static int ecp_randomize_mxz( const mbedtls_ecp_group *grp, mbedtls_ecp_point *P,
                  int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
  {
      int ret;
      mbedtls_mpi l;
      size_t p_size;
      int count = 0;
  
  #if defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_randomize_mxz( grp, P, f_rng, p_rng );
      }
  #endif /* MBEDTLS_ECP_RANDOMIZE_MXZ_ALT */
  
      p_size = ( grp->pbits + 7 ) / 8;
      mbedtls_mpi_init( &l );
  
      /* Generate l such that 1 < l < p */
      do
      {
          MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &l, p_size, f_rng, p_rng ) );
  
          while( mbedtls_mpi_cmp_mpi( &l, &grp->P ) >= 0 )
              MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &l, 1 ) );
  
          if( count++ > 10 )
              return( MBEDTLS_ERR_ECP_RANDOM_FAILED );
      }
      while( mbedtls_mpi_cmp_int( &l, 1 ) <= 0 );
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P->X, &P->X, &l ) ); MOD_MUL( P->X );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P->Z, &P->Z, &l ) ); MOD_MUL( P->Z );
  
  cleanup:
      mbedtls_mpi_free( &l );
  
      return( ret );
  }
  
  /*
   * Double-and-add: R = 2P, S = P + Q, with d = X(P - Q),
   * for Montgomery curves in x/z coordinates.
   *
   * http://www.hyperelliptic.org/EFD/g1p/auto-code/montgom/xz/ladder/mladd-1987-m.op3
   * with
   * d =  X1
   * P = (X2, Z2)
   * Q = (X3, Z3)
   * R = (X4, Z4)
   * S = (X5, Z5)
   * and eliminating temporary variables tO, ..., t4.
   *
   * Cost: 5M + 4S
   */
  static int ecp_double_add_mxz( const mbedtls_ecp_group *grp,
                                 mbedtls_ecp_point *R, mbedtls_ecp_point *S,
                                 const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q,
                                 const mbedtls_mpi *d )
  {
      int ret;
      mbedtls_mpi A, AA, B, BB, E, C, D, DA, CB;
  
  #if defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT)
      if ( mbedtls_internal_ecp_grp_capable( grp ) )
      {
          return mbedtls_internal_ecp_double_add_mxz( grp, R, S, P, Q, d );
      }
  #endif /* MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT */
  
      mbedtls_mpi_init( &A ); mbedtls_mpi_init( &AA ); mbedtls_mpi_init( &B );
      mbedtls_mpi_init( &BB ); mbedtls_mpi_init( &E ); mbedtls_mpi_init( &C );
      mbedtls_mpi_init( &D ); mbedtls_mpi_init( &DA ); mbedtls_mpi_init( &CB );
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &A,    &P->X,   &P->Z ) ); MOD_ADD( A    );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &AA,   &A,      &A    ) ); MOD_MUL( AA   );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &B,    &P->X,   &P->Z ) ); MOD_SUB( B    );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &BB,   &B,      &B    ) ); MOD_MUL( BB   );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &E,    &AA,     &BB   ) ); MOD_SUB( E    );
      MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &C,    &Q->X,   &Q->Z ) ); MOD_ADD( C    );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &D,    &Q->X,   &Q->Z ) ); MOD_SUB( D    );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &DA,   &D,      &A    ) ); MOD_MUL( DA   );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &CB,   &C,      &B    ) ); MOD_MUL( CB   );
      MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &S->X, &DA,     &CB   ) ); MOD_MUL( S->X );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S->X, &S->X,   &S->X ) ); MOD_MUL( S->X );
      MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S->Z, &DA,     &CB   ) ); MOD_SUB( S->Z );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S->Z, &S->Z,   &S->Z ) ); MOD_MUL( S->Z );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S->Z, d,       &S->Z ) ); MOD_MUL( S->Z );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R->X, &AA,     &BB   ) ); MOD_MUL( R->X );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R->Z, &grp->A, &E    ) ); MOD_MUL( R->Z );
      MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &R->Z, &BB,     &R->Z ) ); MOD_ADD( R->Z );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R->Z, &E,      &R->Z ) ); MOD_MUL( R->Z );
  
  cleanup:
      mbedtls_mpi_free( &A ); mbedtls_mpi_free( &AA ); mbedtls_mpi_free( &B );
      mbedtls_mpi_free( &BB ); mbedtls_mpi_free( &E ); mbedtls_mpi_free( &C );
      mbedtls_mpi_free( &D ); mbedtls_mpi_free( &DA ); mbedtls_mpi_free( &CB );
  
      return( ret );
  }
  
  /*
   * Multiplication with Montgomery ladder in x/z coordinates,
   * for curves in Montgomery form
   */
  static int ecp_mul_mxz( 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 )
  {
      int ret;
      size_t i;
      unsigned char b;
      mbedtls_ecp_point RP;
      mbedtls_mpi PX;
  
      mbedtls_ecp_point_init( &RP ); mbedtls_mpi_init( &PX );
  
      /* Save PX and read from P before writing to R, in case P == R */
      MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &PX, &P->X ) );
      MBEDTLS_MPI_CHK( mbedtls_ecp_copy( &RP, P ) );
  
      /* Set R to zero in modified x/z coordinates */
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R->X, 1 ) );
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R->Z, 0 ) );
      mbedtls_mpi_free( &R->Y );
  
      /* RP.X might be sligtly larger than P, so reduce it */
      MOD_ADD( RP.X );
  
      /* Randomize coordinates of the starting point */
      if( f_rng != NULL )
          MBEDTLS_MPI_CHK( ecp_randomize_mxz( grp, &RP, f_rng, p_rng ) );
  
      /* Loop invariant: R = result so far, RP = R + P */
      i = mbedtls_mpi_bitlen( m ); /* one past the (zero-based) most significant bit */
      while( i-- > 0 )
      {
          b = mbedtls_mpi_get_bit( m, i );
          /*
           *  if (b) R = 2R + P else R = 2R,
           * which is:
           *  if (b) double_add( RP, R, RP, R )
           *  else   double_add( R, RP, R, RP )
           * but using safe conditional swaps to avoid leaks
           */
          MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->X, &RP.X, b ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->Z, &RP.Z, b ) );
          MBEDTLS_MPI_CHK( ecp_double_add_mxz( grp, R, &RP, R, &RP, &PX ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->X, &RP.X, b ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->Z, &RP.Z, b ) );
      }
  
      MBEDTLS_MPI_CHK( ecp_normalize_mxz( grp, R ) );
  
  cleanup:
      mbedtls_ecp_point_free( &RP ); mbedtls_mpi_free( &PX );
  
      return( ret );
  }
  
  #endif /* ECP_MONTGOMERY */
  
  /*
   * Multiplication R = m * P
   */
  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 )
  {
      int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
  #if defined(MBEDTLS_ECP_INTERNAL_ALT)
      char is_grp_capable = 0;
  #endif
  
      /* Common sanity checks */
      if( mbedtls_mpi_cmp_int( &P->Z, 1 ) != 0 )
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  
      if( ( ret = mbedtls_ecp_check_privkey( grp, m ) ) != 0 ||
          ( ret = mbedtls_ecp_check_pubkey( grp, P ) ) != 0 )
          return( ret );
  
  #if defined(MBEDTLS_ECP_INTERNAL_ALT)
      if ( is_grp_capable = mbedtls_internal_ecp_grp_capable( grp )  )
      {
          MBEDTLS_MPI_CHK( mbedtls_internal_ecp_init( grp ) );
      }
  
  #endif /* MBEDTLS_ECP_INTERNAL_ALT */
  #if defined(ECP_MONTGOMERY)
      if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
          ret = ecp_mul_mxz( grp, R, m, P, f_rng, p_rng );
  
  #endif
  #if defined(ECP_SHORTWEIERSTRASS)
      if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
          ret = ecp_mul_comb( grp, R, m, P, f_rng, p_rng );
  
  #endif
  #if defined(MBEDTLS_ECP_INTERNAL_ALT)
  cleanup:
  
      if ( is_grp_capable )
      {
          mbedtls_internal_ecp_free( grp );
      }
  
  #endif /* MBEDTLS_ECP_INTERNAL_ALT */
      return( ret );
  }
  
  #if defined(ECP_SHORTWEIERSTRASS)
  /*
   * Check that an affine point is valid as a public key,
   * short weierstrass curves (SEC1 3.2.3.1)
   */
  static int ecp_check_pubkey_sw( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt )
  {
      int ret;
      mbedtls_mpi YY, RHS;
  
      /* pt coordinates must be normalized for our checks */
      if( mbedtls_mpi_cmp_int( &pt->X, 0 ) < 0 ||
          mbedtls_mpi_cmp_int( &pt->Y, 0 ) < 0 ||
          mbedtls_mpi_cmp_mpi( &pt->X, &grp->P ) >= 0 ||
          mbedtls_mpi_cmp_mpi( &pt->Y, &grp->P ) >= 0 )
          return( MBEDTLS_ERR_ECP_INVALID_KEY );
  
      mbedtls_mpi_init( &YY ); mbedtls_mpi_init( &RHS );
  
      /*
       * YY = Y^2
       * RHS = X (X^2 + A) + B = X^3 + A X + B
       */
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &YY,  &pt->Y,   &pt->Y  ) );  MOD_MUL( YY  );
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &RHS, &pt->X,   &pt->X  ) );  MOD_MUL( RHS );
  
      /* Special case for A = -3 */
      if( grp->A.p == NULL )
      {
          MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &RHS, &RHS, 3       ) );  MOD_SUB( RHS );
      }
      else
      {
          MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &RHS, &RHS, &grp->A ) );  MOD_ADD( RHS );
      }
  
      MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &RHS, &RHS,     &pt->X  ) );  MOD_MUL( RHS );
      MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &RHS, &RHS,     &grp->B ) );  MOD_ADD( RHS );
  
      if( mbedtls_mpi_cmp_mpi( &YY, &RHS ) != 0 )
          ret = MBEDTLS_ERR_ECP_INVALID_KEY;
  
  cleanup:
  
      mbedtls_mpi_free( &YY ); mbedtls_mpi_free( &RHS );
  
      return( ret );
  }
  #endif /* ECP_SHORTWEIERSTRASS */
  
  /*
   * R = m * P with shortcuts for m == 1 and m == -1
   * NOT constant-time - ONLY for short Weierstrass!
   */
  static int mbedtls_ecp_mul_shortcuts( mbedtls_ecp_group *grp,
                                        mbedtls_ecp_point *R,
                                        const mbedtls_mpi *m,
                                        const mbedtls_ecp_point *P )
  {
      int ret;
  
      if( mbedtls_mpi_cmp_int( m, 1 ) == 0 )
      {
          MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R, P ) );
      }
      else if( mbedtls_mpi_cmp_int( m, -1 ) == 0 )
      {
          MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R, P ) );
          if( mbedtls_mpi_cmp_int( &R->Y, 0 ) != 0 )
              MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &R->Y, &grp->P, &R->Y ) );
      }
      else
      {
          MBEDTLS_MPI_CHK( mbedtls_ecp_mul( grp, R, m, P, NULL, NULL ) );
      }
  
  cleanup:
      return( ret );
  }
  
  /*
   * Linear combination
   * NOT constant-time
   */
  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 )
  {
      int ret;
      mbedtls_ecp_point mP;
  #if defined(MBEDTLS_ECP_INTERNAL_ALT)
      char is_grp_capable = 0;
  #endif
  
      if( ecp_get_type( grp ) != ECP_TYPE_SHORT_WEIERSTRASS )
          return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE );
  
      mbedtls_ecp_point_init( &mP );
  
      MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp, &mP, m, P ) );
      MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp, R,   n, Q ) );
  
  #if defined(MBEDTLS_ECP_INTERNAL_ALT)
      if (  is_grp_capable = mbedtls_internal_ecp_grp_capable( grp )  )
      {
          MBEDTLS_MPI_CHK( mbedtls_internal_ecp_init( grp ) );
      }
  
  #endif /* MBEDTLS_ECP_INTERNAL_ALT */
      MBEDTLS_MPI_CHK( ecp_add_mixed( grp, R, &mP, R ) );
      MBEDTLS_MPI_CHK( ecp_normalize_jac( grp, R ) );
  
  cleanup:
  
  #if defined(MBEDTLS_ECP_INTERNAL_ALT)
      if ( is_grp_capable )
      {
          mbedtls_internal_ecp_free( grp );
      }
  
  #endif /* MBEDTLS_ECP_INTERNAL_ALT */
      mbedtls_ecp_point_free( &mP );
  
      return( ret );
  }
  
  
  #if defined(ECP_MONTGOMERY)
  /*
   * Check validity of a public key for Montgomery curves with x-only schemes
   */
  static int ecp_check_pubkey_mx( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt )
  {
      /* [Curve25519 p. 5] Just check X is the correct number of bytes */
      if( mbedtls_mpi_size( &pt->X ) > ( grp->nbits + 7 ) / 8 )
          return( MBEDTLS_ERR_ECP_INVALID_KEY );
  
      return( 0 );
  }
  #endif /* ECP_MONTGOMERY */
  
  /*
   * Check that a point is valid as a public key
   */
  int mbedtls_ecp_check_pubkey( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt )
  {
      /* Must use affine coordinates */
      if( mbedtls_mpi_cmp_int( &pt->Z, 1 ) != 0 )
          return( MBEDTLS_ERR_ECP_INVALID_KEY );
  
  #if defined(ECP_MONTGOMERY)
      if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
          return( ecp_check_pubkey_mx( grp, pt ) );
  #endif
  #if defined(ECP_SHORTWEIERSTRASS)
      if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
          return( ecp_check_pubkey_sw( grp, pt ) );
  #endif
      return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  }
  
  /*
   * Check that an mbedtls_mpi is valid as a private key
   */
  int mbedtls_ecp_check_privkey( const mbedtls_ecp_group *grp, const mbedtls_mpi *d )
  {
  #if defined(ECP_MONTGOMERY)
      if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
      {
          /* see [Curve25519] page 5 */
          if( mbedtls_mpi_get_bit( d, 0 ) != 0 ||
              mbedtls_mpi_get_bit( d, 1 ) != 0 ||
              mbedtls_mpi_get_bit( d, 2 ) != 0 ||
              mbedtls_mpi_bitlen( d ) - 1 != grp->nbits ) /* mbedtls_mpi_bitlen is one-based! */
              return( MBEDTLS_ERR_ECP_INVALID_KEY );
          else
              return( 0 );
      }
  #endif /* ECP_MONTGOMERY */
  #if defined(ECP_SHORTWEIERSTRASS)
      if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
      {
          /* see SEC1 3.2 */
          if( mbedtls_mpi_cmp_int( d, 1 ) < 0 ||
              mbedtls_mpi_cmp_mpi( d, &grp->N ) >= 0 )
              return( MBEDTLS_ERR_ECP_INVALID_KEY );
          else
              return( 0 );
      }
  #endif /* ECP_SHORTWEIERSTRASS */
  
      return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
  }
  
  /*
   * Generate a private key
   */
  int mbedtls_ecp_gen_privkey( const mbedtls_ecp_group *grp,
                       mbedtls_mpi *d,
                       int (*f_rng)(void *, unsigned char *, size_t),
                       void *p_rng )
  {
      int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
      size_t n_size = ( grp->nbits + 7 ) / 8;
  
  #if defined(ECP_MONTGOMERY)
      if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
      {
          /* [M225] page 5 */
          size_t b;
  
          do {
              MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d, n_size, f_rng, p_rng ) );
          } while( mbedtls_mpi_bitlen( d ) == 0);
  
          /* Make sure the most significant bit is nbits */
          b = mbedtls_mpi_bitlen( d ) - 1; /* mbedtls_mpi_bitlen is one-based */
          if( b > grp->nbits )
              MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d, b - grp->nbits ) );
          else
              MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, grp->nbits, 1 ) );
  
          /* Make sure the last three bits are unset */
          MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 0, 0 ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 1, 0 ) );
          MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 2, 0 ) );
      }
  #endif /* ECP_MONTGOMERY */
  
  #if defined(ECP_SHORTWEIERSTRASS)
      if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
      {
          /* SEC1 3.2.1: Generate d such that 1 <= n < N */
          int count = 0;
          unsigned cmp = 0;
  
          /*
           * Match the procedure given in RFC 6979 (deterministic ECDSA):
           * - use the same byte ordering;
           * - keep the leftmost nbits bits of the generated octet string;
           * - try until result is in the desired range.
           * This also avoids any biais, which is especially important for ECDSA.
           */
          do
          {
              MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d, n_size, f_rng, p_rng ) );
              MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d, 8 * n_size - grp->nbits ) );
  
              /*
               * Each try has at worst a probability 1/2 of failing (the msb has
               * a probability 1/2 of being 0, and then the result will be < N),
               * so after 30 tries failure probability is a most 2**(-30).
               *
               * For most curves, 1 try is enough with overwhelming probability,
               * since N starts with a lot of 1s in binary, but some curves
               * such as secp224k1 are actually very close to the worst case.
               */
              if( ++count > 30 )
                  return( MBEDTLS_ERR_ECP_RANDOM_FAILED );
  
              ret = mbedtls_mpi_lt_mpi_ct( d, &grp->N, &cmp );
              if( ret != 0 )
              {
                  goto cleanup;
              }
          }
          while( mbedtls_mpi_cmp_int( d, 1 ) < 0 || cmp != 1 );
      }
  #endif /* ECP_SHORTWEIERSTRASS */
  
  cleanup:
      return( ret );
  }
  
  /*
   * Generate a keypair with configurable base point
   */
  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 )
  {
      int ret;
  
      MBEDTLS_MPI_CHK( mbedtls_ecp_gen_privkey( grp, d, f_rng, p_rng ) );
      MBEDTLS_MPI_CHK( mbedtls_ecp_mul( grp, Q, d, G, f_rng, p_rng ) );
  
  cleanup:
      return( ret );
  }
  
  /*
   * Generate key pair, wrapper for conventional base point
   */
  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 )
  {
      return( mbedtls_ecp_gen_keypair_base( grp, &grp->G, d, Q, f_rng, p_rng ) );
  }
  
  /*
   * Generate a keypair, prettier wrapper
   */
  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 ret;
  
      if( ( ret = mbedtls_ecp_group_load( &key->grp, grp_id ) ) != 0 )
          return( ret );
  
      return( mbedtls_ecp_gen_keypair( &key->grp, &key->d, &key->Q, f_rng, p_rng ) );
  }
  
  /*
   * Check a public-private key pair
   */
  int mbedtls_ecp_check_pub_priv( const mbedtls_ecp_keypair *pub, const mbedtls_ecp_keypair *prv )
  {
      int ret;
      mbedtls_ecp_point Q;
      mbedtls_ecp_group grp;
  
      if( pub->grp.id == MBEDTLS_ECP_DP_NONE ||
          pub->grp.id != prv->grp.id ||
          mbedtls_mpi_cmp_mpi( &pub->Q.X, &prv->Q.X ) ||
          mbedtls_mpi_cmp_mpi( &pub->Q.Y, &prv->Q.Y ) ||
          mbedtls_mpi_cmp_mpi( &pub->Q.Z, &prv->Q.Z ) )
      {
          return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
      }
  
      mbedtls_ecp_point_init( &Q );
      mbedtls_ecp_group_init( &grp );
  
      /* mbedtls_ecp_mul() needs a non-const group... */
      mbedtls_ecp_group_copy( &grp, &prv->grp );
  
      /* Also checks d is valid */
      MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &Q, &prv->d, &prv->grp.G, NULL, NULL ) );
  
      if( mbedtls_mpi_cmp_mpi( &Q.X, &prv->Q.X ) ||
          mbedtls_mpi_cmp_mpi( &Q.Y, &prv->Q.Y ) ||
          mbedtls_mpi_cmp_mpi( &Q.Z, &prv->Q.Z ) )
      {
          ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
          goto cleanup;
      }
  
  cleanup:
      mbedtls_ecp_point_free( &Q );
      mbedtls_ecp_group_free( &grp );
  
      return( ret );
  }
  
  #if defined(MBEDTLS_SELF_TEST)
  
  /*
   * Checkup routine
   */
  int mbedtls_ecp_self_test( int verbose )
  {
      int ret;
      size_t i;
      mbedtls_ecp_group grp;
      mbedtls_ecp_point R, P;
      mbedtls_mpi m;
      unsigned long add_c_prev, dbl_c_prev, mul_c_prev;
      /* exponents especially adapted for secp192r1 */
      const char *exponents[] =
      {
          "000000000000000000000000000000000000000000000001", /* one */
          "FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22830", /* N - 1 */
          "5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */
          "400000000000000000000000000000000000000000000000", /* one and zeros */
          "7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF", /* all ones */
          "555555555555555555555555555555555555555555555555", /* 101010... */
      };
  
      mbedtls_ecp_group_init( &grp );
      mbedtls_ecp_point_init( &R );
      mbedtls_ecp_point_init( &P );
      mbedtls_mpi_init( &m );
  
      /* Use secp192r1 if available, or any available curve */
  #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
      MBEDTLS_MPI_CHK( mbedtls_ecp_group_load( &grp, MBEDTLS_ECP_DP_SECP192R1 ) );
  #else
      MBEDTLS_MPI_CHK( mbedtls_ecp_group_load( &grp, mbedtls_ecp_curve_list()->grp_id ) );
  #endif
  
      if( verbose != 0 )
          mbedtls_printf( "  ECP test #1 (constant op_count, base point G): " );
  
      /* Do a dummy multiplication first to trigger precomputation */
      MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &m, 2 ) );
      MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &P, &m, &grp.G, NULL, NULL ) );
  
      add_count = 0;
      dbl_count = 0;
      mul_count = 0;
      MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[0] ) );
      MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &grp.G, NULL, NULL ) );
  
      for( i = 1; i < sizeof( exponents ) / sizeof( exponents[0] ); i++ )
      {
          add_c_prev = add_count;
          dbl_c_prev = dbl_count;
          mul_c_prev = mul_count;
          add_count = 0;
          dbl_count = 0;
          mul_count = 0;
  
          MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[i] ) );
          MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &grp.G, NULL, NULL ) );
  
          if( add_count != add_c_prev ||
              dbl_count != dbl_c_prev ||
              mul_count != mul_c_prev )
          {
              if( verbose != 0 )
                  mbedtls_printf( "failed (%u)\n", (unsigned int) i );
  
              ret = 1;
              goto cleanup;
          }
      }
  
      if( verbose != 0 )
          mbedtls_printf( "passed\n" );
  
      if( verbose != 0 )
          mbedtls_printf( "  ECP test #2 (constant op_count, other point): " );
      /* We computed P = 2G last time, use it */
  
      add_count = 0;
      dbl_count = 0;
      mul_count = 0;
      MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[0] ) );
      MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &P, NULL, NULL ) );
  
      for( i = 1; i < sizeof( exponents ) / sizeof( exponents[0] ); i++ )
      {
          add_c_prev = add_count;
          dbl_c_prev = dbl_count;
          mul_c_prev = mul_count;
          add_count = 0;
          dbl_count = 0;
          mul_count = 0;
  
          MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[i] ) );
          MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &P, NULL, NULL ) );
  
          if( add_count != add_c_prev ||
              dbl_count != dbl_c_prev ||
              mul_count != mul_c_prev )
          {
              if( verbose != 0 )
                  mbedtls_printf( "failed (%u)\n", (unsigned int) i );
  
              ret = 1;
              goto cleanup;
          }
      }
  
      if( verbose != 0 )
          mbedtls_printf( "passed\n" );
  
  cleanup:
  
      if( ret < 0 && verbose != 0 )
          mbedtls_printf( "Unexpected error, return code = %08X\n", ret );
  
      mbedtls_ecp_group_free( &grp );
      mbedtls_ecp_point_free( &R );
      mbedtls_ecp_point_free( &P );
      mbedtls_mpi_free( &m );
  
      if( verbose != 0 )
          mbedtls_printf( "\n" );
  
      return( ret );
  }
  
  #endif /* MBEDTLS_SELF_TEST */
  
  #endif /* !MBEDTLS_ECP_ALT */
  
  #endif /* MBEDTLS_ECP_C */
  