library/ecdsa.c

/*
 *  Elliptic curve DSA
 *
 *  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
 */

#if !defined(MBEDTLS_CONFIG_FILE)
#include "mbedtls/config.h"
#else
#include MBEDTLS_CONFIG_FILE
#endif

#if defined(MBEDTLS_ECDSA_C)

#include "mbedtls/ecdsa.h"
#include "mbedtls/asn1write.h"

#include <string.h>

#if defined(MBEDTLS_ECDSA_DETERMINISTIC)
#include "mbedtls/hmac_drbg.h"
#endif

#if defined(MBEDTLS_PLATFORM_C)
#include "mbedtls/platform.h"
#else
#include <stdlib.h>
#define mbedtls_calloc    calloc
#define mbedtls_free       free
#endif

#include "mbedtls/platform_util.h"

/* Parameter validation macros based on platform_util.h */
#define ECDSA_VALIDATE_RET( cond )    \
    MBEDTLS_INTERNAL_VALIDATE_RET( cond, MBEDTLS_ERR_ECP_BAD_INPUT_DATA )
#define ECDSA_VALIDATE( cond )        \
    MBEDTLS_INTERNAL_VALIDATE( cond )

#if defined(MBEDTLS_ECP_RESTARTABLE)

/*
 * Sub-context for ecdsa_verify()
 */
struct mbedtls_ecdsa_restart_ver
{
    mbedtls_mpi u1, u2;     /* intermediate values  */
    enum {                  /* what to do next?     */
        ecdsa_ver_init = 0, /* getting started      */
        ecdsa_ver_muladd,   /* muladd step          */
    } state;
};

/*
 * Init verify restart sub-context
 */
static void ecdsa_restart_ver_init( mbedtls_ecdsa_restart_ver_ctx *ctx )
{
    mbedtls_mpi_init( &ctx->u1 );
    mbedtls_mpi_init( &ctx->u2 );
    ctx->state = ecdsa_ver_init;
}

/*
 * Free the components of a verify restart sub-context
 */
static void ecdsa_restart_ver_free( mbedtls_ecdsa_restart_ver_ctx *ctx )
{
    if( ctx == NULL )
        return;

    mbedtls_mpi_free( &ctx->u1 );
    mbedtls_mpi_free( &ctx->u2 );

    ecdsa_restart_ver_init( ctx );
}

/*
 * Sub-context for ecdsa_sign()
 */
struct mbedtls_ecdsa_restart_sig
{
    int sign_tries;
    int key_tries;
    mbedtls_mpi k;          /* per-signature random */
    mbedtls_mpi r;          /* r value              */
    enum {                  /* what to do next?     */
        ecdsa_sig_init = 0, /* getting started      */
        ecdsa_sig_mul,      /* doing ecp_mul()      */
        ecdsa_sig_modn,     /* mod N computations   */
    } state;
};

/*
 * Init verify sign sub-context
 */
static void ecdsa_restart_sig_init( mbedtls_ecdsa_restart_sig_ctx *ctx )
{
    ctx->sign_tries = 0;
    ctx->key_tries = 0;
    mbedtls_mpi_init( &ctx->k );
    mbedtls_mpi_init( &ctx->r );
    ctx->state = ecdsa_sig_init;
}

/*
 * Free the components of a sign restart sub-context
 */
static void ecdsa_restart_sig_free( mbedtls_ecdsa_restart_sig_ctx *ctx )
{
    if( ctx == NULL )
        return;

    mbedtls_mpi_free( &ctx->k );
    mbedtls_mpi_free( &ctx->r );
}

#if defined(MBEDTLS_ECDSA_DETERMINISTIC)
/*
 * Sub-context for ecdsa_sign_det()
 */
struct mbedtls_ecdsa_restart_det
{
    mbedtls_hmac_drbg_context rng_ctx;  /* DRBG state   */
    enum {                      /* what to do next?     */
        ecdsa_det_init = 0,     /* getting started      */
        ecdsa_det_sign,         /* make signature       */
    } state;
};

/*
 * Init verify sign_det sub-context
 */
static void ecdsa_restart_det_init( mbedtls_ecdsa_restart_det_ctx *ctx )
{
    mbedtls_hmac_drbg_init( &ctx->rng_ctx );
    ctx->state = ecdsa_det_init;
}

/*
 * Free the components of a sign_det restart sub-context
 */
static void ecdsa_restart_det_free( mbedtls_ecdsa_restart_det_ctx *ctx )
{
    if( ctx == NULL )
        return;

    mbedtls_hmac_drbg_free( &ctx->rng_ctx );

    ecdsa_restart_det_init( ctx );
}
#endif /* MBEDTLS_ECDSA_DETERMINISTIC */

#define ECDSA_RS_ECP    &rs_ctx->ecp

/* Utility macro for checking and updating ops budget */
#define ECDSA_BUDGET( ops )   \
    MBEDTLS_MPI_CHK( mbedtls_ecp_check_budget( grp, &rs_ctx->ecp, ops ) );

/* Call this when entering a function that needs its own sub-context */
#define ECDSA_RS_ENTER( SUB )   do {                                 \
    /* reset ops count for this call if top-level */                 \
    if( rs_ctx != NULL && rs_ctx->ecp.depth++ == 0 )                 \
        rs_ctx->ecp.ops_done = 0;                                    \
                                                                     \
    /* set up our own sub-context if needed */                       \
    if( mbedtls_ecp_restart_is_enabled() &&                          \
        rs_ctx != NULL && rs_ctx->SUB == NULL )                      \
    {                                                                \
        rs_ctx->SUB = mbedtls_calloc( 1, sizeof( *rs_ctx->SUB ) );   \
        if( rs_ctx->SUB == NULL )                                    \
            return( MBEDTLS_ERR_ECP_ALLOC_FAILED );                  \
                                                                     \
        ecdsa_restart_## SUB ##_init( rs_ctx->SUB );                 \
    }                                                                \
} while( 0 )

/* Call this when leaving a function that needs its own sub-context */
#define ECDSA_RS_LEAVE( SUB )   do {                                 \
    /* clear our sub-context when not in progress (done or error) */ \
    if( rs_ctx != NULL && rs_ctx->SUB != NULL &&                     \
        ret != MBEDTLS_ERR_ECP_IN_PROGRESS )                         \
    {                                                                \
        ecdsa_restart_## SUB ##_free( rs_ctx->SUB );                 \
        mbedtls_free( rs_ctx->SUB );                                 \
        rs_ctx->SUB = NULL;                                          \
    }                                                                \
                                                                     \
    if( rs_ctx != NULL )                                             \
        rs_ctx->ecp.depth--;                                         \
} while( 0 )

#else /* MBEDTLS_ECP_RESTARTABLE */

#define ECDSA_RS_ECP    NULL

#define ECDSA_BUDGET( ops )   /* no-op; for compatibility */

#define ECDSA_RS_ENTER( SUB )   (void) rs_ctx
#define ECDSA_RS_LEAVE( SUB )   (void) rs_ctx

#endif /* MBEDTLS_ECP_RESTARTABLE */

/*
 * Derive a suitable integer for group grp from a buffer of length len
 * SEC1 4.1.3 step 5 aka SEC1 4.1.4 step 3
 */
static int derive_mpi( const mbedtls_ecp_group *grp, mbedtls_mpi *x,
                       const unsigned char *buf, size_t blen )
{
    int ret;
    size_t n_size = ( grp->nbits + 7 ) / 8;
    size_t use_size = blen > n_size ? n_size : blen;

    MBEDTLS_MPI_CHK( mbedtls_mpi_read_binary( x, buf, use_size ) );
    if( use_size * 8 > grp->nbits )
        MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( x, use_size * 8 - grp->nbits ) );

    /* While at it, reduce modulo N */
    if( mbedtls_mpi_cmp_mpi( x, &grp->N ) >= 0 )
        MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( x, x, &grp->N ) );

cleanup:
    return( ret );
}

#if !defined(MBEDTLS_ECDSA_SIGN_ALT)
/*
 * Compute ECDSA signature of a hashed message (SEC1 4.1.3)
 * Obviously, compared to SEC1 4.1.3, we skip step 4 (hash message)
 */
static int ecdsa_sign_restartable( mbedtls_ecp_group *grp,
                mbedtls_mpi *r, mbedtls_mpi *s,
                const mbedtls_mpi *d, const unsigned char *buf, size_t blen,
                int (*f_rng)(void *, unsigned char *, size_t), void *p_rng,
                int (*f_rng_blind)(void *, unsigned char *, size_t),
                void *p_rng_blind,
                mbedtls_ecdsa_restart_ctx *rs_ctx )
{
    int ret, key_tries, sign_tries;
    int *p_sign_tries = &sign_tries, *p_key_tries = &key_tries;
    mbedtls_ecp_point R;
    mbedtls_mpi k, e, t;
    mbedtls_mpi *pk = &k, *pr = r;

    /* Fail cleanly on curves such as Curve25519 that can't be used for ECDSA */
    if( grp->N.p == NULL )
        return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );

    /* Make sure d is in range 1..n-1 */
    if( mbedtls_mpi_cmp_int( d, 1 ) < 0 || mbedtls_mpi_cmp_mpi( d, &grp->N ) >= 0 )
        return( MBEDTLS_ERR_ECP_INVALID_KEY );

    mbedtls_ecp_point_init( &R );
    mbedtls_mpi_init( &k ); mbedtls_mpi_init( &e ); mbedtls_mpi_init( &t );

    ECDSA_RS_ENTER( sig );

#if defined(MBEDTLS_ECP_RESTARTABLE)
    if( rs_ctx != NULL && rs_ctx->sig != NULL )
    {
        /* redirect to our context */
        p_sign_tries = &rs_ctx->sig->sign_tries;
        p_key_tries = &rs_ctx->sig->key_tries;
        pk = &rs_ctx->sig->k;
        pr = &rs_ctx->sig->r;

        /* jump to current step */
        if( rs_ctx->sig->state == ecdsa_sig_mul )
            goto mul;
        if( rs_ctx->sig->state == ecdsa_sig_modn )
            goto modn;
    }
#endif /* MBEDTLS_ECP_RESTARTABLE */

    *p_sign_tries = 0;
    do
    {
        if( *p_sign_tries++ > 10 )
        {
            ret = MBEDTLS_ERR_ECP_RANDOM_FAILED;
            goto cleanup;
        }

        /*
         * Steps 1-3: generate a suitable ephemeral keypair
         * and set r = xR mod n
         */
        *p_key_tries = 0;
        do
        {
            if( *p_key_tries++ > 10 )
            {
                ret = MBEDTLS_ERR_ECP_RANDOM_FAILED;
                goto cleanup;
            }

            MBEDTLS_MPI_CHK( mbedtls_ecp_gen_privkey( grp, pk, f_rng, p_rng ) );

#if defined(MBEDTLS_ECP_RESTARTABLE)
            if( rs_ctx != NULL && rs_ctx->sig != NULL )
                rs_ctx->sig->state = ecdsa_sig_mul;

mul:
#endif
            MBEDTLS_MPI_CHK( mbedtls_ecp_mul_restartable( grp, &R, pk, &grp->G,
                                                          f_rng_blind,
                                                          p_rng_blind,
                                                          ECDSA_RS_ECP ) );
            MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( pr, &R.X, &grp->N ) );
        }
        while( mbedtls_mpi_cmp_int( pr, 0 ) == 0 );

#if defined(MBEDTLS_ECP_RESTARTABLE)
        if( rs_ctx != NULL && rs_ctx->sig != NULL )
            rs_ctx->sig->state = ecdsa_sig_modn;

modn:
#endif
        /*
         * Accounting for everything up to the end of the loop
         * (step 6, but checking now avoids saving e and t)
         */
        ECDSA_BUDGET( MBEDTLS_ECP_OPS_INV + 4 );

        /*
         * Step 5: derive MPI from hashed message
         */
        MBEDTLS_MPI_CHK( derive_mpi( grp, &e, buf, blen ) );

        /*
         * Generate a random value to blind inv_mod in next step,
         * avoiding a potential timing leak.
         */
        MBEDTLS_MPI_CHK( mbedtls_ecp_gen_privkey( grp, &t, f_rng_blind,
                                                  p_rng_blind ) );

        /*
         * Step 6: compute s = (e + r * d) / k = t (e + rd) / (kt) mod n
         */
        MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( s, pr, d ) );
        MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &e, &e, s ) );
        MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &e, &e, &t ) );
        MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( pk, pk, &t ) );
        MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( s, pk, &grp->N ) );
        MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( s, s, &e ) );
        MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( s, s, &grp->N ) );
    }
    while( mbedtls_mpi_cmp_int( s, 0 ) == 0 );

#if defined(MBEDTLS_ECP_RESTARTABLE)
    if( rs_ctx != NULL && rs_ctx->sig != NULL )
        mbedtls_mpi_copy( r, pr );
#endif

cleanup:
    mbedtls_ecp_point_free( &R );
    mbedtls_mpi_free( &k ); mbedtls_mpi_free( &e ); mbedtls_mpi_free( &t );

    ECDSA_RS_LEAVE( sig );

    return( ret );
}

/*
 * Compute ECDSA signature of a hashed message
 */
int mbedtls_ecdsa_sign( mbedtls_ecp_group *grp, mbedtls_mpi *r, mbedtls_mpi *s,
                const mbedtls_mpi *d, const unsigned char *buf, size_t blen,
                int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
    ECDSA_VALIDATE_RET( grp   != NULL );
    ECDSA_VALIDATE_RET( r     != NULL );
    ECDSA_VALIDATE_RET( s     != NULL );
    ECDSA_VALIDATE_RET( d     != NULL );
    ECDSA_VALIDATE_RET( f_rng != NULL );
    ECDSA_VALIDATE_RET( buf   != NULL || blen == 0 );

    /* Use the same RNG for both blinding and ephemeral key generation */
    return( ecdsa_sign_restartable( grp, r, s, d, buf, blen,
                                    f_rng, p_rng, f_rng, p_rng, NULL ) );
}
#endif /* !MBEDTLS_ECDSA_SIGN_ALT */

#if defined(MBEDTLS_ECDSA_DETERMINISTIC)
/*
 * Deterministic signature wrapper
 */
static int ecdsa_sign_det_restartable( mbedtls_ecp_group *grp,
                    mbedtls_mpi *r, mbedtls_mpi *s,
                    const mbedtls_mpi *d, const unsigned char *buf, size_t blen,
                    mbedtls_md_type_t md_alg,
                    int (*f_rng_blind)(void *, unsigned char *, size_t),
                    void *p_rng_blind,
                    mbedtls_ecdsa_restart_ctx *rs_ctx )
{
    int ret;
    mbedtls_hmac_drbg_context rng_ctx;
    mbedtls_hmac_drbg_context *p_rng = &rng_ctx;
    unsigned char data[2 * MBEDTLS_ECP_MAX_BYTES];
    size_t grp_len = ( grp->nbits + 7 ) / 8;
    const mbedtls_md_info_t *md_info;
    mbedtls_mpi h;

    if( ( md_info = mbedtls_md_info_from_type( md_alg ) ) == NULL )
        return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );

    mbedtls_mpi_init( &h );
    mbedtls_hmac_drbg_init( &rng_ctx );

    ECDSA_RS_ENTER( det );

#if defined(MBEDTLS_ECP_RESTARTABLE)
    if( rs_ctx != NULL && rs_ctx->det != NULL )
    {
        /* redirect to our context */
        p_rng = &rs_ctx->det->rng_ctx;

        /* jump to current step */
        if( rs_ctx->det->state == ecdsa_det_sign )
            goto sign;
    }
#endif /* MBEDTLS_ECP_RESTARTABLE */

    /* Use private key and message hash (reduced) to initialize HMAC_DRBG */
    MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( d, data, grp_len ) );
    MBEDTLS_MPI_CHK( derive_mpi( grp, &h, buf, blen ) );
    MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &h, data + grp_len, grp_len ) );
    mbedtls_hmac_drbg_seed_buf( p_rng, md_info, data, 2 * grp_len );

#if defined(MBEDTLS_ECP_RESTARTABLE)
    if( rs_ctx != NULL && rs_ctx->det != NULL )
        rs_ctx->det->state = ecdsa_det_sign;

sign:
#endif
#if defined(MBEDTLS_ECDSA_SIGN_ALT)
    ret = mbedtls_ecdsa_sign( grp, r, s, d, buf, blen,
                              mbedtls_hmac_drbg_random, p_rng );
#else
    if( f_rng_blind != NULL )
        ret = ecdsa_sign_restartable( grp, r, s, d, buf, blen,
                                      mbedtls_hmac_drbg_random, p_rng,
                                      f_rng_blind, p_rng_blind, rs_ctx );
    else
        /*
         * Use the same RNG for both blinding and ephemeral key generation.
         * Since the RNG output is always the same for the same key and message,
         * this limits the efficiency of blinding and leaks information through
         * side channels.
         */
        ret = ecdsa_sign_restartable( grp, r, s, d, buf, blen,
                                      mbedtls_hmac_drbg_random, p_rng,
                                      mbedtls_hmac_drbg_random, p_rng, rs_ctx );
#endif /* MBEDTLS_ECDSA_SIGN_ALT */

cleanup:
    mbedtls_hmac_drbg_free( &rng_ctx );
    mbedtls_mpi_free( &h );

    ECDSA_RS_LEAVE( det );

    return( ret );
}

/*
 * Deterministic signature wrappers
 */
int mbedtls_ecdsa_sign_det( mbedtls_ecp_group *grp, mbedtls_mpi *r,
                            mbedtls_mpi *s, const mbedtls_mpi *d,
                            const unsigned char *buf, size_t blen,
                            mbedtls_md_type_t md_alg )
{
    ECDSA_VALIDATE_RET( grp   != NULL );
    ECDSA_VALIDATE_RET( r     != NULL );
    ECDSA_VALIDATE_RET( s     != NULL );
    ECDSA_VALIDATE_RET( d     != NULL );
    ECDSA_VALIDATE_RET( buf   != NULL || blen == 0 );

    return( ecdsa_sign_det_restartable( grp, r, s, d, buf, blen, md_alg,
                                        NULL, NULL, NULL ) );
}

int mbedtls_ecdsa_sign_det_ext( mbedtls_ecp_group *grp, mbedtls_mpi *r,
                                mbedtls_mpi *s, const mbedtls_mpi *d,
                                const unsigned char *buf, size_t blen,
                                mbedtls_md_type_t md_alg,
                                int (*f_rng_blind)(void *, unsigned char *,
                                                   size_t),
                                void *p_rng_blind )
{
    ECDSA_VALIDATE_RET( grp   != NULL );
    ECDSA_VALIDATE_RET( r     != NULL );
    ECDSA_VALIDATE_RET( s     != NULL );
    ECDSA_VALIDATE_RET( d     != NULL );
    ECDSA_VALIDATE_RET( buf   != NULL || blen == 0 );
    ECDSA_VALIDATE_RET( f_rng_blind != NULL );

    return( ecdsa_sign_det_restartable( grp, r, s, d, buf, blen, md_alg,
                                        f_rng_blind, p_rng_blind, NULL ) );
}
#endif /* MBEDTLS_ECDSA_DETERMINISTIC */

#if !defined(MBEDTLS_ECDSA_VERIFY_ALT)
/*
 * Verify ECDSA signature of hashed message (SEC1 4.1.4)
 * Obviously, compared to SEC1 4.1.3, we skip step 2 (hash message)
 */
static int ecdsa_verify_restartable( mbedtls_ecp_group *grp,
                                     const unsigned char *buf, size_t blen,
                                     const mbedtls_ecp_point *Q,
                                     const mbedtls_mpi *r, const mbedtls_mpi *s,
                                     mbedtls_ecdsa_restart_ctx *rs_ctx )
{
    int ret;
    mbedtls_mpi e, s_inv, u1, u2;
    mbedtls_ecp_point R;
    mbedtls_mpi *pu1 = &u1, *pu2 = &u2;

    mbedtls_ecp_point_init( &R );
    mbedtls_mpi_init( &e ); mbedtls_mpi_init( &s_inv );
    mbedtls_mpi_init( &u1 ); mbedtls_mpi_init( &u2 );

    /* Fail cleanly on curves such as Curve25519 that can't be used for ECDSA */
    if( grp->N.p == NULL )
        return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );

    ECDSA_RS_ENTER( ver );

#if defined(MBEDTLS_ECP_RESTARTABLE)
    if( rs_ctx != NULL && rs_ctx->ver != NULL )
    {
        /* redirect to our context */
        pu1 = &rs_ctx->ver->u1;
        pu2 = &rs_ctx->ver->u2;

        /* jump to current step */
        if( rs_ctx->ver->state == ecdsa_ver_muladd )
            goto muladd;
    }
#endif /* MBEDTLS_ECP_RESTARTABLE */

    /*
     * Step 1: make sure r and s are in range 1..n-1
     */
    if( mbedtls_mpi_cmp_int( r, 1 ) < 0 || mbedtls_mpi_cmp_mpi( r, &grp->N ) >= 0 ||
        mbedtls_mpi_cmp_int( s, 1 ) < 0 || mbedtls_mpi_cmp_mpi( s, &grp->N ) >= 0 )
    {
        ret = MBEDTLS_ERR_ECP_VERIFY_FAILED;
        goto cleanup;
    }

    /*
     * Step 3: derive MPI from hashed message
     */
    MBEDTLS_MPI_CHK( derive_mpi( grp, &e, buf, blen ) );

    /*
     * Step 4: u1 = e / s mod n, u2 = r / s mod n
     */
    ECDSA_BUDGET( MBEDTLS_ECP_OPS_CHK + MBEDTLS_ECP_OPS_INV + 2 );

    MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &s_inv, s, &grp->N ) );

    MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( pu1, &e, &s_inv ) );
    MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( pu1, pu1, &grp->N ) );

    MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( pu2, r, &s_inv ) );
    MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( pu2, pu2, &grp->N ) );

#if defined(MBEDTLS_ECP_RESTARTABLE)
    if( rs_ctx != NULL && rs_ctx->ver != NULL )
        rs_ctx->ver->state = ecdsa_ver_muladd;

muladd:
#endif
    /*
     * Step 5: R = u1 G + u2 Q
     */
    MBEDTLS_MPI_CHK( mbedtls_ecp_muladd_restartable( grp,
                     &R, pu1, &grp->G, pu2, Q, ECDSA_RS_ECP ) );

    if( mbedtls_ecp_is_zero( &R ) )
    {
        ret = MBEDTLS_ERR_ECP_VERIFY_FAILED;
        goto cleanup;
    }

    /*
     * Step 6: convert xR to an integer (no-op)
     * Step 7: reduce xR mod n (gives v)
     */
    MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &R.X, &R.X, &grp->N ) );

    /*
     * Step 8: check if v (that is, R.X) is equal to r
     */
    if( mbedtls_mpi_cmp_mpi( &R.X, r ) != 0 )
    {
        ret = MBEDTLS_ERR_ECP_VERIFY_FAILED;
        goto cleanup;
    }

cleanup:
    mbedtls_ecp_point_free( &R );
    mbedtls_mpi_free( &e ); mbedtls_mpi_free( &s_inv );
    mbedtls_mpi_free( &u1 ); mbedtls_mpi_free( &u2 );

    ECDSA_RS_LEAVE( ver );

    return( ret );
}

/*
 * Verify ECDSA signature of hashed message
 */
int mbedtls_ecdsa_verify( mbedtls_ecp_group *grp,
                          const unsigned char *buf, size_t blen,
                          const mbedtls_ecp_point *Q,
                          const mbedtls_mpi *r,
                          const mbedtls_mpi *s)
{
    ECDSA_VALIDATE_RET( grp != NULL );
    ECDSA_VALIDATE_RET( Q   != NULL );
    ECDSA_VALIDATE_RET( r   != NULL );
    ECDSA_VALIDATE_RET( s   != NULL );
    ECDSA_VALIDATE_RET( buf != NULL || blen == 0 );

    return( ecdsa_verify_restartable( grp, buf, blen, Q, r, s, NULL ) );
}
#endif /* !MBEDTLS_ECDSA_VERIFY_ALT */

/*
 * Convert a signature (given by context) to ASN.1
 */
static int ecdsa_signature_to_asn1( const mbedtls_mpi *r, const mbedtls_mpi *s,
                                    unsigned char *sig, size_t *slen )
{
    int ret;
    unsigned char buf[MBEDTLS_ECDSA_MAX_LEN];
    unsigned char *p = buf + sizeof( buf );
    size_t len = 0;

    MBEDTLS_ASN1_CHK_ADD( len, mbedtls_asn1_write_mpi( &p, buf, s ) );
    MBEDTLS_ASN1_CHK_ADD( len, mbedtls_asn1_write_mpi( &p, buf, r ) );

    MBEDTLS_ASN1_CHK_ADD( len, mbedtls_asn1_write_len( &p, buf, len ) );
    MBEDTLS_ASN1_CHK_ADD( len, mbedtls_asn1_write_tag( &p, buf,
                                       MBEDTLS_ASN1_CONSTRUCTED | MBEDTLS_ASN1_SEQUENCE ) );

    memcpy( sig, p, len );
    *slen = len;

    return( 0 );
}

/*
 * Compute and write signature
 */
int mbedtls_ecdsa_write_signature_restartable( mbedtls_ecdsa_context *ctx,
                           mbedtls_md_type_t md_alg,
                           const unsigned char *hash, size_t hlen,
                           unsigned char *sig, size_t *slen,
                           int (*f_rng)(void *, unsigned char *, size_t),
                           void *p_rng,
                           mbedtls_ecdsa_restart_ctx *rs_ctx )
{
    int ret;
    mbedtls_mpi r, s;
    ECDSA_VALIDATE_RET( ctx  != NULL );
    ECDSA_VALIDATE_RET( hash != NULL );
    ECDSA_VALIDATE_RET( sig  != NULL );
    ECDSA_VALIDATE_RET( slen != NULL );

    mbedtls_mpi_init( &r );
    mbedtls_mpi_init( &s );

#if defined(MBEDTLS_ECDSA_DETERMINISTIC)
    MBEDTLS_MPI_CHK( ecdsa_sign_det_restartable( &ctx->grp, &r, &s, &ctx->d,
                                                 hash, hlen, md_alg, f_rng,
                                                 p_rng, rs_ctx ) );
#else
    (void) md_alg;

#if defined(MBEDTLS_ECDSA_SIGN_ALT)
    MBEDTLS_MPI_CHK( mbedtls_ecdsa_sign( &ctx->grp, &r, &s, &ctx->d,
                         hash, hlen, f_rng, p_rng ) );
#else
    /* Use the same RNG for both blinding and ephemeral key generation */
    MBEDTLS_MPI_CHK( ecdsa_sign_restartable( &ctx->grp, &r, &s, &ctx->d,
                                             hash, hlen, f_rng, p_rng, f_rng,
                                             p_rng, rs_ctx ) );
#endif /* MBEDTLS_ECDSA_SIGN_ALT */
#endif /* MBEDTLS_ECDSA_DETERMINISTIC */

    MBEDTLS_MPI_CHK( ecdsa_signature_to_asn1( &r, &s, sig, slen ) );

cleanup:
    mbedtls_mpi_free( &r );
    mbedtls_mpi_free( &s );

    return( ret );
}

/*
 * Compute and write signature
 */
int mbedtls_ecdsa_write_signature( mbedtls_ecdsa_context *ctx,
                                 mbedtls_md_type_t md_alg,
                                 const unsigned char *hash, size_t hlen,
                                 unsigned char *sig, size_t *slen,
                                 int (*f_rng)(void *, unsigned char *, size_t),
                                 void *p_rng )
{
    ECDSA_VALIDATE_RET( ctx  != NULL );
    ECDSA_VALIDATE_RET( hash != NULL );
    ECDSA_VALIDATE_RET( sig  != NULL );
    ECDSA_VALIDATE_RET( slen != NULL );
    return( mbedtls_ecdsa_write_signature_restartable(
                ctx, md_alg, hash, hlen, sig, slen, f_rng, p_rng, NULL ) );
}

#if !defined(MBEDTLS_DEPRECATED_REMOVED) && \
    defined(MBEDTLS_ECDSA_DETERMINISTIC)
int mbedtls_ecdsa_write_signature_det( mbedtls_ecdsa_context *ctx,
                               const unsigned char *hash, size_t hlen,
                               unsigned char *sig, size_t *slen,
                               mbedtls_md_type_t md_alg )
{
    ECDSA_VALIDATE_RET( ctx  != NULL );
    ECDSA_VALIDATE_RET( hash != NULL );
    ECDSA_VALIDATE_RET( sig  != NULL );
    ECDSA_VALIDATE_RET( slen != NULL );
    return( mbedtls_ecdsa_write_signature( ctx, md_alg, hash, hlen, sig, slen,
                                   NULL, NULL ) );
}
#endif

/*
 * Read and check signature
 */
int mbedtls_ecdsa_read_signature( mbedtls_ecdsa_context *ctx,
                          const unsigned char *hash, size_t hlen,
                          const unsigned char *sig, size_t slen )
{
    ECDSA_VALIDATE_RET( ctx  != NULL );
    ECDSA_VALIDATE_RET( hash != NULL );
    ECDSA_VALIDATE_RET( sig  != NULL );
    return( mbedtls_ecdsa_read_signature_restartable(
                ctx, hash, hlen, sig, slen, NULL ) );
}

/*
 * Restartable read and check signature
 */
int mbedtls_ecdsa_read_signature_restartable( mbedtls_ecdsa_context *ctx,
                          const unsigned char *hash, size_t hlen,
                          const unsigned char *sig, size_t slen,
                          mbedtls_ecdsa_restart_ctx *rs_ctx )
{
    int ret;
    unsigned char *p = (unsigned char *) sig;
    const unsigned char *end = sig + slen;
    size_t len;
    mbedtls_mpi r, s;
    ECDSA_VALIDATE_RET( ctx  != NULL );
    ECDSA_VALIDATE_RET( hash != NULL );
    ECDSA_VALIDATE_RET( sig  != NULL );

    mbedtls_mpi_init( &r );
    mbedtls_mpi_init( &s );

    if( ( ret = mbedtls_asn1_get_tag( &p, end, &len,
                    MBEDTLS_ASN1_CONSTRUCTED | MBEDTLS_ASN1_SEQUENCE ) ) != 0 )
    {
        ret += MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
        goto cleanup;
    }

    if( p + len != end )
    {
        ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA +
              MBEDTLS_ERR_ASN1_LENGTH_MISMATCH;
        goto cleanup;
    }

    if( ( ret = mbedtls_asn1_get_mpi( &p, end, &r ) ) != 0 ||
        ( ret = mbedtls_asn1_get_mpi( &p, end, &s ) ) != 0 )
    {
        ret += MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
        goto cleanup;
    }
#if defined(MBEDTLS_ECDSA_VERIFY_ALT)
    if( ( ret = mbedtls_ecdsa_verify( &ctx->grp, hash, hlen,
                                      &ctx->Q, &r, &s ) ) != 0 )
        goto cleanup;
#else
    if( ( ret = ecdsa_verify_restartable( &ctx->grp, hash, hlen,
                              &ctx->Q, &r, &s, rs_ctx ) ) != 0 )
        goto cleanup;
#endif /* MBEDTLS_ECDSA_VERIFY_ALT */

    /* At this point we know that the buffer starts with a valid signature.
     * Return 0 if the buffer just contains the signature, and a specific
     * error code if the valid signature is followed by more data. */
    if( p != end )
        ret = MBEDTLS_ERR_ECP_SIG_LEN_MISMATCH;

cleanup:
    mbedtls_mpi_free( &r );
    mbedtls_mpi_free( &s );

    return( ret );
}

#if !defined(MBEDTLS_ECDSA_GENKEY_ALT)
/*
 * Generate key pair
 */
int mbedtls_ecdsa_genkey( mbedtls_ecdsa_context *ctx, mbedtls_ecp_group_id gid,
                  int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
    int ret = 0;
    ECDSA_VALIDATE_RET( ctx   != NULL );
    ECDSA_VALIDATE_RET( f_rng != NULL );

    ret = mbedtls_ecp_group_load( &ctx->grp, gid );
    if( ret != 0 )
        return( ret );

   return( mbedtls_ecp_gen_keypair( &ctx->grp, &ctx->d,
                                    &ctx->Q, f_rng, p_rng ) );
}
#endif /* !MBEDTLS_ECDSA_GENKEY_ALT */

/*
 * Set context from an mbedtls_ecp_keypair
 */
int mbedtls_ecdsa_from_keypair( mbedtls_ecdsa_context *ctx, const mbedtls_ecp_keypair *key )
{
    int ret;
    ECDSA_VALIDATE_RET( ctx != NULL );
    ECDSA_VALIDATE_RET( key != NULL );

    if( ( ret = mbedtls_ecp_group_copy( &ctx->grp, &key->grp ) ) != 0 ||
        ( ret = mbedtls_mpi_copy( &ctx->d, &key->d ) ) != 0 ||
        ( ret = mbedtls_ecp_copy( &ctx->Q, &key->Q ) ) != 0 )
    {
        mbedtls_ecdsa_free( ctx );
    }

    return( ret );
}

/*
 * Initialize context
 */
void mbedtls_ecdsa_init( mbedtls_ecdsa_context *ctx )
{
    ECDSA_VALIDATE( ctx != NULL );

    mbedtls_ecp_keypair_init( ctx );
}

/*
 * Free context
 */
void mbedtls_ecdsa_free( mbedtls_ecdsa_context *ctx )
{
    if( ctx == NULL )
        return;

    mbedtls_ecp_keypair_free( ctx );
}

#if defined(MBEDTLS_ECP_RESTARTABLE)
/*
 * Initialize a restart context
 */
void mbedtls_ecdsa_restart_init( mbedtls_ecdsa_restart_ctx *ctx )
{
    ECDSA_VALIDATE( ctx != NULL );

    mbedtls_ecp_restart_init( &ctx->ecp );

    ctx->ver = NULL;
    ctx->sig = NULL;
#if defined(MBEDTLS_ECDSA_DETERMINISTIC)
    ctx->det = NULL;
#endif
}

/*
 * Free the components of a restart context
 */
void mbedtls_ecdsa_restart_free( mbedtls_ecdsa_restart_ctx *ctx )
{
    if( ctx == NULL )
        return;

    mbedtls_ecp_restart_free( &ctx->ecp );

    ecdsa_restart_ver_free( ctx->ver );
    mbedtls_free( ctx->ver );
    ctx->ver = NULL;

    ecdsa_restart_sig_free( ctx->sig );
    mbedtls_free( ctx->sig );
    ctx->sig = NULL;

#if defined(MBEDTLS_ECDSA_DETERMINISTIC)
    ecdsa_restart_det_free( ctx->det );
    mbedtls_free( ctx->det );
    ctx->det = NULL;
#endif
}
#endif /* MBEDTLS_ECP_RESTARTABLE */

#endif /* MBEDTLS_ECDSA_C */