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 /* * 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 "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 */