/*
* 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 *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 );
}
/*
* 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)
/*
* 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 );
}
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 */
