library/rsa_alt_helpers.c - mirror/mbed-tls - TrustedFirmware Git Browser

 /*
  *  Helper functions for the RSA module
  *
  *  Copyright The Mbed TLS Contributors
  *  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.
  *
  */

 #include "common.h"

 #if defined(MBEDTLS_RSA_C)

 #include "mbedtls/rsa.h"
 #include "mbedtls/bignum.h"
 #include "rsa_alt_helpers.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 *E, mbedtls_mpi const *D,
                               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 char 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 };

     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 = (uint16_t) 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) {
             /* If we reach 1 prematurely, there's no point
              * in continuing to square K */
             if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
                 break;
             }

             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 we get here, then either we prematurely aborted the loop because
          * we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
          * be 1 if D,E,N were consistent.
          * Check if that's the case and abort if not, to avoid very long,
          * yet eventually failing, computations if N,D,E were not sane.
          */
         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;
 }

 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;
 }

 /*
  * 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)
     /*
      * When generating keys, the strongest security we support aims for an error
      * rate of at most 2^-100 and we are aiming for the same certainty here as
      * well.
      */
     if (f_rng != NULL && P != NULL &&
         (ret = mbedtls_mpi_is_prime_ext(P, 50, 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_ext(Q, 50, 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 = P * Q
      */

     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;
 }

 /*
  * 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;
 }

 #endif /* MBEDTLS_RSA_C */
	/*
	* Helper functions for the RSA module
	*
	* Copyright The Mbed TLS Contributors
	* 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.
	*
	*/

	#include "common.h"

	#if defined(MBEDTLS_RSA_C)

	#include "mbedtls/rsa.h"
	#include "mbedtls/bignum.h"
	#include "rsa_alt_helpers.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 E, mbedtls_mpi const D,
	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 char 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 };

	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 = (uint16_t) 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) {
	/* If we reach 1 prematurely, there's no point
	* in continuing to square K */
	if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
	break;
	}

	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 we get here, then either we prematurely aborted the loop because
	* we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
	* be 1 if D,E,N were consistent.
	* Check if that's the case and abort if not, to avoid very long,
	* yet eventually failing, computations if N,D,E were not sane.
	*/
	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;
	}

	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;
	}

	/*
	* 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)
	/*
	* When generating keys, the strongest security we support aims for an error
	* rate of at most 2^-100 and we are aiming for the same certainty here as
	* well.
	*/
	if (f_rng != NULL && P != NULL &&
	(ret = mbedtls_mpi_is_prime_ext(P, 50, 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_ext(Q, 50, 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 = P * Q
	*/

	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;
	}

	/*
	* 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;
	}

	#endif /* MBEDTLS_RSA_C */