library/rsa_internal.c - security/mbedtls - Rivoreo Source Code Repositories

 /*
  *  Helper functions for the RSA module
  *
  *  Copyright (C) 2006-2017, 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)
  *
  */

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

 #if defined(MBEDTLS_RSA_C)

 #include "mbedtls/rsa.h"
 #include "mbedtls/bignum.h"
 #include "mbedtls/rsa_internal.h"

 /*
  * Compute RSA prime factors from public and private exponents
  *
  * Summary of algorithm:
  * Setting F := lcm(P-1,Q-1), the idea is as follows:
  *
  * (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2)
  *     is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the
  *     square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four
  *     possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1)
  *     or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime
  *     factors of N.
  *
  * (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same
  *     construction still applies since (-)^K is the identity on the set of
  *     roots of 1 in Z/NZ.
  *
  * The public and private key primitives (-)^E and (-)^D are mutually inverse
  * bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e.
  * if and only if DE - 1 is a multiple of F, say DE - 1 = F * L.
  * Splitting L = 2^t * K with K odd, we have
  *
  *   DE - 1 = FL = (F/2) * (2^(t+1)) * K,
  *
  * so (F / 2) * K is among the numbers
  *
  *   (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord
  *
  * where ord is the order of 2 in (DE - 1).
  * We can therefore iterate through these numbers apply the construction
  * of (a) and (b) above to attempt to factor N.
  *
  */
 int mbedtls_rsa_deduce_primes( mbedtls_mpi const *N,
                      mbedtls_mpi const *D, mbedtls_mpi const *E,
                      mbedtls_mpi *P, mbedtls_mpi *Q )
 {
     int ret = 0;

     uint16_t attempt;  /* Number of current attempt  */
     uint16_t iter;     /* Number of squares computed in the current attempt */

     uint16_t order;    /* Order of 2 in DE - 1 */

     mbedtls_mpi T;  /* Holds largest odd divisor of DE - 1     */
     mbedtls_mpi K;  /* Temporary holding the current candidate */

     const unsigned int primes[] = { 2,
            3,    5,    7,   11,   13,   17,   19,   23,
           29,   31,   37,   41,   43,   47,   53,   59,
           61,   67,   71,   73,   79,   83,   89,   97,
          101,  103,  107,  109,  113,  127,  131,  137,
          139,  149,  151,  157,  163,  167,  173,  179,
          181,  191,  193,  197,  199,  211,  223,  227,
          229,  233,  239,  241,  251,  257,  263,  269,
          271,  277,  281,  283,  293,  307,  311,  313
     };

     const size_t num_primes = sizeof( primes ) / sizeof( *primes );

     if( P == NULL || Q == NULL || P->p != NULL || Q->p != NULL )
         return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );

     if( mbedtls_mpi_cmp_int( N, 0 ) <= 0 ||
         mbedtls_mpi_cmp_int( D, 1 ) <= 0 ||
         mbedtls_mpi_cmp_mpi( D, N ) >= 0 ||
         mbedtls_mpi_cmp_int( E, 1 ) <= 0 ||
         mbedtls_mpi_cmp_mpi( E, N ) >= 0 )
     {
         return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
     }

     /*
      * Initializations and temporary changes
      */

     mbedtls_mpi_init( &K );
     mbedtls_mpi_init( &T );

     /* T := DE - 1 */
     MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, D,  E ) );
     MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &T, &T, 1 ) );

     if( ( order = mbedtls_mpi_lsb( &T ) ) == 0 )
     {
         ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
         goto cleanup;
     }

     /* After this operation, T holds the largest odd divisor of DE - 1. */
     MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &T, order ) );

     /*
      * Actual work
      */

     /* Skip trying 2 if N == 1 mod 8 */
     attempt = 0;
     if( N->p[0] % 8 == 1 )
         attempt = 1;

     for( ; attempt < num_primes; ++attempt )
     {
         mbedtls_mpi_lset( &K, primes[attempt] );

         /* Check if gcd(K,N) = 1 */
         MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );
         if( mbedtls_mpi_cmp_int( P, 1 ) != 0 )
             continue;

         /* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
          * and check whether they have nontrivial GCD with N. */
         MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &K, &K, &T, N,
                              Q /* temporarily use Q for storing Montgomery
                                 * multiplication helper values */ ) );

         for( iter = 1; iter < order; ++iter )
         {
             MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( &K, &K, 1 ) );
             MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );

             if( mbedtls_mpi_cmp_int( P, 1 ) ==  1 &&
                 mbedtls_mpi_cmp_mpi( P, N ) == -1 )
             {
                 /*
                  * Have found a nontrivial divisor P of N.
                  * Set Q := N / P.
                  */

                 MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( Q, NULL, N, P ) );
                 goto cleanup;
             }

             MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
             MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &K ) );
             MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, N ) );
         }

         if( mbedtls_mpi_cmp_int( &K, 1 ) != 0 )
         {
             break;
         }
     }

     ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;

 cleanup:

     mbedtls_mpi_free( &K );
     mbedtls_mpi_free( &T );
     return( ret );
 }

 /*
  * Given P, Q and the public exponent E, deduce D.
  * This is essentially a modular inversion.
  */
 int mbedtls_rsa_deduce_private_exponent( mbedtls_mpi const *P,
                                          mbedtls_mpi const *Q,
                                          mbedtls_mpi const *E,
                                          mbedtls_mpi *D )
 {
     int ret = 0;
     mbedtls_mpi K, L;

     if( D == NULL || mbedtls_mpi_cmp_int( D, 0 ) != 0 )
         return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );

     if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 ||
         mbedtls_mpi_cmp_int( Q, 1 ) <= 0 ||
         mbedtls_mpi_cmp_int( E, 0 ) == 0 )
     {
         return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
     }

     mbedtls_mpi_init( &K );
     mbedtls_mpi_init( &L );

     /* Temporarily put K := P-1 and L := Q-1 */
     MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
     MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );

     /* Temporarily put D := gcd(P-1, Q-1) */
     MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( D, &K, &L ) );

     /* K := LCM(P-1, Q-1) */
     MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &L ) );
     MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( &K, NULL, &K, D ) );

     /* Compute modular inverse of E in LCM(P-1, Q-1) */
     MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( D, E, &K ) );

 cleanup:

     mbedtls_mpi_free( &K );
     mbedtls_mpi_free( &L );

     return( ret );
 }

 /*
  * Check that RSA CRT parameters are in accordance with core parameters.
  */
 int mbedtls_rsa_validate_crt( const mbedtls_mpi *P,  const mbedtls_mpi *Q,
                               const mbedtls_mpi *D,  const mbedtls_mpi *DP,
                               const mbedtls_mpi *DQ, const mbedtls_mpi *QP )
 {
     int ret = 0;

     mbedtls_mpi K, L;
     mbedtls_mpi_init( &K );
     mbedtls_mpi_init( &L );

     /* Check that DP - D == 0 mod P - 1 */
     if( DP != NULL )
     {
         if( P == NULL )
         {
             ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
             goto cleanup;
         }

         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DP, D ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );

         if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }
     }

     /* Check that DQ - D == 0 mod Q - 1 */
     if( DQ != NULL )
     {
         if( Q == NULL )
         {
             ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
             goto cleanup;
         }

         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DQ, D ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );

         if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }
     }

     /* Check that QP * Q - 1 == 0 mod P */
     if( QP != NULL )
     {
         if( P == NULL || Q == NULL )
         {
             ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
             goto cleanup;
         }

         MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, QP, Q ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, P ) );
         if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }
     }

 cleanup:

     /* Wrap MPI error codes by RSA check failure error code */
     if( ret != 0 &&
         ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
         ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA )
     {
         ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
     }

     mbedtls_mpi_free( &K );
     mbedtls_mpi_free( &L );

     return( ret );
 }

 /*
  * Check that core RSA parameters are sane.
  */
 int mbedtls_rsa_validate_params( const mbedtls_mpi *N, const mbedtls_mpi *P,
                                  const mbedtls_mpi *Q, const mbedtls_mpi *D,
                                  const mbedtls_mpi *E,
                                  int (*f_rng)(void *, unsigned char *, size_t),
                                  void *p_rng )
 {
     int ret = 0;
     mbedtls_mpi K, L;

     mbedtls_mpi_init( &K );
     mbedtls_mpi_init( &L );

     /*
      * Step 1: If PRNG provided, check that P and Q are prime
      */

 #if defined(MBEDTLS_GENPRIME)
     if( f_rng != NULL && P != NULL &&
         ( ret = mbedtls_mpi_is_prime( P, f_rng, p_rng ) ) != 0 )
     {
         ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
         goto cleanup;
     }

     if( f_rng != NULL && Q != NULL &&
         ( ret = mbedtls_mpi_is_prime( Q, f_rng, p_rng ) ) != 0 )
     {
         ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
         goto cleanup;
     }
 #else
     ((void) f_rng);
     ((void) p_rng);
 #endif /* MBEDTLS_GENPRIME */

     /*
      * Step 2: Check that 1 < N = PQ
      */

     if( P != NULL && Q != NULL && N != NULL )
     {
         MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, P, Q ) );
         if( mbedtls_mpi_cmp_int( N, 1 )  <= 0 ||
             mbedtls_mpi_cmp_mpi( &K, N ) != 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }
     }

     /*
      * Step 3: Check and 1 < D, E < N if present.
      */

     if( N != NULL && D != NULL && E != NULL )
     {
         if ( mbedtls_mpi_cmp_int( D, 1 ) <= 0 ||
              mbedtls_mpi_cmp_int( E, 1 ) <= 0 ||
              mbedtls_mpi_cmp_mpi( D, N ) >= 0 ||
              mbedtls_mpi_cmp_mpi( E, N ) >= 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }
     }

     /*
      * Step 4: Check that D, E are inverse modulo P-1 and Q-1
      */

     if( P != NULL && Q != NULL && D != NULL && E != NULL )
     {
         if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 ||
             mbedtls_mpi_cmp_int( Q, 1 ) <= 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }

         /* Compute DE-1 mod P-1 */
         MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, P, 1 ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
         if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }

         /* Compute DE-1 mod Q-1 */
         MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
         if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
         {
             ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
             goto cleanup;
         }
     }

 cleanup:

     mbedtls_mpi_free( &K );
     mbedtls_mpi_free( &L );

     /* Wrap MPI error codes by RSA check failure error code */
     if( ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED )
     {
         ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
     }

     return( ret );
 }

 int mbedtls_rsa_deduce_crt( const mbedtls_mpi *P, const mbedtls_mpi *Q,
                             const mbedtls_mpi *D, mbedtls_mpi *DP,
                             mbedtls_mpi *DQ, mbedtls_mpi *QP )
 {
     int ret = 0;
     mbedtls_mpi K;
     mbedtls_mpi_init( &K );

     /* DP = D mod P-1 */
     if( DP != NULL )
     {
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1  ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DP, D, &K ) );
     }

     /* DQ = D mod Q-1 */
     if( DQ != NULL )
     {
         MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1  ) );
         MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DQ, D, &K ) );
     }

     /* QP = Q^{-1} mod P */
     if( QP != NULL )
     {
         MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( QP, Q, P ) );
     }

 cleanup:
     mbedtls_mpi_free( &K );

     return( ret );
 }

 #endif /* MBEDTLS_RSA_C */
	/*
	* Helper functions for the RSA module
	*
	* Copyright (C) 2006-2017, 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)
	*
	*/

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

	#if defined(MBEDTLS_RSA_C)

	#include "mbedtls/rsa.h"
	#include "mbedtls/bignum.h"
	#include "mbedtls/rsa_internal.h"

	/*
	* Compute RSA prime factors from public and private exponents
	*
	* Summary of algorithm:
	* Setting F := lcm(P-1,Q-1), the idea is as follows:
	*
	* (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2)
	* is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the
	* square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four
	* possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1)
	* or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime
	* factors of N.
	*
	* (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same
	* construction still applies since (-)^K is the identity on the set of
	* roots of 1 in Z/NZ.
	*
	* The public and private key primitives (-)^E and (-)^D are mutually inverse
	* bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e.
	* if and only if DE - 1 is a multiple of F, say DE - 1 = F * L.
	* Splitting L = 2^t * K with K odd, we have
	*
	* DE - 1 = FL = (F/2) * (2^(t+1)) * K,
	*
	* so (F / 2) * K is among the numbers
	*
	* (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord
	*
	* where ord is the order of 2 in (DE - 1).
	* We can therefore iterate through these numbers apply the construction
	* of (a) and (b) above to attempt to factor N.
	*
	*/
	int mbedtls_rsa_deduce_primes( mbedtls_mpi const *N,
	mbedtls_mpi const D, mbedtls_mpi const E,
	mbedtls_mpi P, mbedtls_mpi Q )
	{
	int ret = 0;

	uint16_t attempt; /* Number of current attempt */
	uint16_t iter; /* Number of squares computed in the current attempt */

	uint16_t order; /* Order of 2 in DE - 1 */

	mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */
	mbedtls_mpi K; /* Temporary holding the current candidate */

	const unsigned int primes[] = { 2,
	3, 5, 7, 11, 13, 17, 19, 23,
	29, 31, 37, 41, 43, 47, 53, 59,
	61, 67, 71, 73, 79, 83, 89, 97,
	101, 103, 107, 109, 113, 127, 131, 137,
	139, 149, 151, 157, 163, 167, 173, 179,
	181, 191, 193, 197, 199, 211, 223, 227,
	229, 233, 239, 241, 251, 257, 263, 269,
	271, 277, 281, 283, 293, 307, 311, 313
	};

	const size_t num_primes = sizeof( primes ) / sizeof( *primes );

	if( P == NULL \|\| Q == NULL \|\| P->p != NULL \|\| Q->p != NULL )
	return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );

	if( mbedtls_mpi_cmp_int( N, 0 ) <= 0 \|\|
	mbedtls_mpi_cmp_int( D, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_mpi( D, N ) >= 0 \|\|
	mbedtls_mpi_cmp_int( E, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_mpi( E, N ) >= 0 )
	{
	return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
	}

	/*
	* Initializations and temporary changes
	*/

	mbedtls_mpi_init( &K );
	mbedtls_mpi_init( &T );

	/* T := DE - 1 */
	MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, D, E ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &T, &T, 1 ) );

	if( ( order = mbedtls_mpi_lsb( &T ) ) == 0 )
	{
	ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
	goto cleanup;
	}

	/* After this operation, T holds the largest odd divisor of DE - 1. */
	MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &T, order ) );

	/*
	* Actual work
	*/

	/* Skip trying 2 if N == 1 mod 8 */
	attempt = 0;
	if( N->p[0] % 8 == 1 )
	attempt = 1;

	for( ; attempt < num_primes; ++attempt )
	{
	mbedtls_mpi_lset( &K, primes[attempt] );

	/* Check if gcd(K,N) = 1 */
	MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );
	if( mbedtls_mpi_cmp_int( P, 1 ) != 0 )
	continue;

	/* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
	* and check whether they have nontrivial GCD with N. */
	MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &K, &K, &T, N,
	Q /* temporarily use Q for storing Montgomery
	* multiplication helper values */ ) );

	for( iter = 1; iter < order; ++iter )
	{
	MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( &K, &K, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( P, &K, N ) );

	if( mbedtls_mpi_cmp_int( P, 1 ) == 1 &&
	mbedtls_mpi_cmp_mpi( P, N ) == -1 )
	{
	/*
	* Have found a nontrivial divisor P of N.
	* Set Q := N / P.
	*/

	MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( Q, NULL, N, P ) );
	goto cleanup;
	}

	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &K ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, N ) );
	}

	if( mbedtls_mpi_cmp_int( &K, 1 ) != 0 )
	{
	break;
	}
	}

	ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;

	cleanup:

	mbedtls_mpi_free( &K );
	mbedtls_mpi_free( &T );
	return( ret );
	}

	/*
	* Given P, Q and the public exponent E, deduce D.
	* This is essentially a modular inversion.
	*/
	int mbedtls_rsa_deduce_private_exponent( mbedtls_mpi const *P,
	mbedtls_mpi const *Q,
	mbedtls_mpi const *E,
	mbedtls_mpi *D )
	{
	int ret = 0;
	mbedtls_mpi K, L;

	if( D == NULL \|\| mbedtls_mpi_cmp_int( D, 0 ) != 0 )
	return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );

	if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_int( Q, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_int( E, 0 ) == 0 )
	{
	return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA );
	}

	mbedtls_mpi_init( &K );
	mbedtls_mpi_init( &L );

	/* Temporarily put K := P-1 and L := Q-1 */
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );

	/* Temporarily put D := gcd(P-1, Q-1) */
	MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( D, &K, &L ) );

	/* K := LCM(P-1, Q-1) */
	MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, &K, &L ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( &K, NULL, &K, D ) );

	/* Compute modular inverse of E in LCM(P-1, Q-1) */
	MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( D, E, &K ) );

	cleanup:

	mbedtls_mpi_free( &K );
	mbedtls_mpi_free( &L );

	return( ret );
	}

	/*
	* Check that RSA CRT parameters are in accordance with core parameters.
	*/
	int mbedtls_rsa_validate_crt( const mbedtls_mpi P, const mbedtls_mpi Q,
	const mbedtls_mpi D, const mbedtls_mpi DP,
	const mbedtls_mpi DQ, const mbedtls_mpi QP )
	{
	int ret = 0;

	mbedtls_mpi K, L;
	mbedtls_mpi_init( &K );
	mbedtls_mpi_init( &L );

	/* Check that DP - D == 0 mod P - 1 */
	if( DP != NULL )
	{
	if( P == NULL )
	{
	ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
	goto cleanup;
	}

	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DP, D ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );

	if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}
	}

	/* Check that DQ - D == 0 mod Q - 1 */
	if( DQ != NULL )
	{
	if( Q == NULL )
	{
	ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
	goto cleanup;
	}

	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &L, DQ, D ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &L, &L, &K ) );

	if( mbedtls_mpi_cmp_int( &L, 0 ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}
	}

	/* Check that QP * Q - 1 == 0 mod P */
	if( QP != NULL )
	{
	if( P == NULL \|\| Q == NULL )
	{
	ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
	goto cleanup;
	}

	MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, QP, Q ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, P ) );
	if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}
	}

	cleanup:

	/* Wrap MPI error codes by RSA check failure error code */
	if( ret != 0 &&
	ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
	ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA )
	{
	ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	}

	mbedtls_mpi_free( &K );
	mbedtls_mpi_free( &L );

	return( ret );
	}

	/*
	* Check that core RSA parameters are sane.
	*/
	int mbedtls_rsa_validate_params( const mbedtls_mpi N, const mbedtls_mpi P,
	const mbedtls_mpi Q, const mbedtls_mpi D,
	const mbedtls_mpi *E,
	int (f_rng)(void , unsigned char *, size_t),
	void *p_rng )
	{
	int ret = 0;
	mbedtls_mpi K, L;

	mbedtls_mpi_init( &K );
	mbedtls_mpi_init( &L );

	/*
	* Step 1: If PRNG provided, check that P and Q are prime
	*/

	#if defined(MBEDTLS_GENPRIME)
	if( f_rng != NULL && P != NULL &&
	( ret = mbedtls_mpi_is_prime( P, f_rng, p_rng ) ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}

	if( f_rng != NULL && Q != NULL &&
	( ret = mbedtls_mpi_is_prime( Q, f_rng, p_rng ) ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}
	#else
	((void) f_rng);
	((void) p_rng);
	#endif /* MBEDTLS_GENPRIME */

	/*
	* Step 2: Check that 1 < N = PQ
	*/

	if( P != NULL && Q != NULL && N != NULL )
	{
	MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, P, Q ) );
	if( mbedtls_mpi_cmp_int( N, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_mpi( &K, N ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}
	}

	/*
	* Step 3: Check and 1 < D, E < N if present.
	*/

	if( N != NULL && D != NULL && E != NULL )
	{
	if ( mbedtls_mpi_cmp_int( D, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_int( E, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_mpi( D, N ) >= 0 \|\|
	mbedtls_mpi_cmp_mpi( E, N ) >= 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}
	}

	/*
	* Step 4: Check that D, E are inverse modulo P-1 and Q-1
	*/

	if( P != NULL && Q != NULL && D != NULL && E != NULL )
	{
	if( mbedtls_mpi_cmp_int( P, 1 ) <= 0 \|\|
	mbedtls_mpi_cmp_int( Q, 1 ) <= 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}

	/* Compute DE-1 mod P-1 */
	MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, P, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
	if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}

	/* Compute DE-1 mod Q-1 */
	MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &K, D, E ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, &K, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &L, Q, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &K, &K, &L ) );
	if( mbedtls_mpi_cmp_int( &K, 0 ) != 0 )
	{
	ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	goto cleanup;
	}
	}

	cleanup:

	mbedtls_mpi_free( &K );
	mbedtls_mpi_free( &L );

	/* Wrap MPI error codes by RSA check failure error code */
	if( ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED )
	{
	ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
	}

	return( ret );
	}

	int mbedtls_rsa_deduce_crt( const mbedtls_mpi P, const mbedtls_mpi Q,
	const mbedtls_mpi D, mbedtls_mpi DP,
	mbedtls_mpi DQ, mbedtls_mpi QP )
	{
	int ret = 0;
	mbedtls_mpi K;
	mbedtls_mpi_init( &K );

	/* DP = D mod P-1 */
	if( DP != NULL )
	{
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, P, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DP, D, &K ) );
	}

	/* DQ = D mod Q-1 */
	if( DQ != NULL )
	{
	MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &K, Q, 1 ) );
	MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( DQ, D, &K ) );
	}

	/* QP = Q^{-1} mod P */
	if( QP != NULL )
	{
	MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( QP, Q, P ) );
	}

	cleanup:
	mbedtls_mpi_free( &K );

	return( ret );
	}

	#endif /* MBEDTLS_RSA_C */