/* | |
* 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 */ |