459 lines
14 KiB
C
459 lines
14 KiB
C
/*
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* Helper functions for the RSA module
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*
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* Copyright The Mbed TLS Contributors
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* SPDX-License-Identifier: Apache-2.0
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*
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* Licensed under the Apache License, Version 2.0 (the "License"); you may
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* not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
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* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*
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*/
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#include "common.h"
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#if defined(MBEDTLS_RSA_C)
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#include "mbedtls/rsa.h"
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#include "mbedtls/bignum.h"
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#include "rsa_alt_helpers.h"
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/*
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* Compute RSA prime factors from public and private exponents
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*
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* Summary of algorithm:
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* Setting F := lcm(P-1,Q-1), the idea is as follows:
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*
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* (a) For any 1 <= X < N with gcd(X,N)=1, we have X^F = 1 modulo N, so X^(F/2)
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* is a square root of 1 in Z/NZ. Since Z/NZ ~= Z/PZ x Z/QZ by CRT and the
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* square roots of 1 in Z/PZ and Z/QZ are +1 and -1, this leaves the four
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* possibilities X^(F/2) = (+-1, +-1). If it happens that X^(F/2) = (-1,+1)
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* or (+1,-1), then gcd(X^(F/2) + 1, N) will be equal to one of the prime
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* factors of N.
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*
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* (b) If we don't know F/2 but (F/2) * K for some odd (!) K, then the same
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* construction still applies since (-)^K is the identity on the set of
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* roots of 1 in Z/NZ.
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*
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* The public and private key primitives (-)^E and (-)^D are mutually inverse
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* bijections on Z/NZ if and only if (-)^(DE) is the identity on Z/NZ, i.e.
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* if and only if DE - 1 is a multiple of F, say DE - 1 = F * L.
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* Splitting L = 2^t * K with K odd, we have
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*
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* DE - 1 = FL = (F/2) * (2^(t+1)) * K,
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*
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* so (F / 2) * K is among the numbers
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*
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* (DE - 1) >> 1, (DE - 1) >> 2, ..., (DE - 1) >> ord
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*
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* where ord is the order of 2 in (DE - 1).
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* We can therefore iterate through these numbers apply the construction
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* of (a) and (b) above to attempt to factor N.
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*
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*/
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int mbedtls_rsa_deduce_primes(mbedtls_mpi const *N,
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mbedtls_mpi const *E, mbedtls_mpi const *D,
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mbedtls_mpi *P, mbedtls_mpi *Q)
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{
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int ret = 0;
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uint16_t attempt; /* Number of current attempt */
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uint16_t iter; /* Number of squares computed in the current attempt */
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uint16_t order; /* Order of 2 in DE - 1 */
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mbedtls_mpi T; /* Holds largest odd divisor of DE - 1 */
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mbedtls_mpi K; /* Temporary holding the current candidate */
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const unsigned char primes[] = { 2,
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3, 5, 7, 11, 13, 17, 19, 23,
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29, 31, 37, 41, 43, 47, 53, 59,
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61, 67, 71, 73, 79, 83, 89, 97,
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101, 103, 107, 109, 113, 127, 131, 137,
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139, 149, 151, 157, 163, 167, 173, 179,
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181, 191, 193, 197, 199, 211, 223, 227,
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229, 233, 239, 241, 251 };
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const size_t num_primes = sizeof(primes) / sizeof(*primes);
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if (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) {
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return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
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}
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if (mbedtls_mpi_cmp_int(N, 0) <= 0 ||
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mbedtls_mpi_cmp_int(D, 1) <= 0 ||
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mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
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mbedtls_mpi_cmp_int(E, 1) <= 0 ||
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mbedtls_mpi_cmp_mpi(E, N) >= 0) {
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return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
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}
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/*
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* Initializations and temporary changes
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*/
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mbedtls_mpi_init(&K);
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mbedtls_mpi_init(&T);
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/* T := DE - 1 */
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MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, D, E));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&T, &T, 1));
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if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) {
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ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
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goto cleanup;
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}
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/* After this operation, T holds the largest odd divisor of DE - 1. */
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MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&T, order));
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/*
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* Actual work
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*/
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/* Skip trying 2 if N == 1 mod 8 */
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attempt = 0;
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if (N->p[0] % 8 == 1) {
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attempt = 1;
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}
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for (; attempt < num_primes; ++attempt) {
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mbedtls_mpi_lset(&K, primes[attempt]);
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/* Check if gcd(K,N) = 1 */
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MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
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if (mbedtls_mpi_cmp_int(P, 1) != 0) {
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continue;
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}
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/* Go through K^T + 1, K^(2T) + 1, K^(4T) + 1, ...
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* and check whether they have nontrivial GCD with N. */
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MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&K, &K, &T, N,
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Q /* temporarily use Q for storing Montgomery
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* multiplication helper values */));
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for (iter = 1; iter <= order; ++iter) {
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/* If we reach 1 prematurely, there's no point
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* in continuing to square K */
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if (mbedtls_mpi_cmp_int(&K, 1) == 0) {
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break;
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}
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MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&K, &K, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(P, &K, N));
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if (mbedtls_mpi_cmp_int(P, 1) == 1 &&
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mbedtls_mpi_cmp_mpi(P, N) == -1) {
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/*
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* Have found a nontrivial divisor P of N.
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* Set Q := N / P.
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*/
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MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(Q, NULL, N, P));
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goto cleanup;
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}
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &K));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, N));
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}
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/*
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* If we get here, then either we prematurely aborted the loop because
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* we reached 1, or K holds primes[attempt]^(DE - 1) mod N, which must
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* be 1 if D,E,N were consistent.
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* Check if that's the case and abort if not, to avoid very long,
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* yet eventually failing, computations if N,D,E were not sane.
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*/
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if (mbedtls_mpi_cmp_int(&K, 1) != 0) {
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break;
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}
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}
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ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
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cleanup:
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mbedtls_mpi_free(&K);
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mbedtls_mpi_free(&T);
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return ret;
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}
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/*
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* Given P, Q and the public exponent E, deduce D.
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* This is essentially a modular inversion.
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*/
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int mbedtls_rsa_deduce_private_exponent(mbedtls_mpi const *P,
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mbedtls_mpi const *Q,
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mbedtls_mpi const *E,
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mbedtls_mpi *D)
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{
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int ret = 0;
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mbedtls_mpi K, L;
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if (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) {
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return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
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}
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if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
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mbedtls_mpi_cmp_int(Q, 1) <= 0 ||
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mbedtls_mpi_cmp_int(E, 0) == 0) {
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return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
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}
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mbedtls_mpi_init(&K);
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mbedtls_mpi_init(&L);
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/* Temporarily put K := P-1 and L := Q-1 */
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
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/* Temporarily put D := gcd(P-1, Q-1) */
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MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(D, &K, &L));
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/* K := LCM(P-1, Q-1) */
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MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, &K, &L));
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MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&K, NULL, &K, D));
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/* Compute modular inverse of E in LCM(P-1, Q-1) */
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MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(D, E, &K));
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cleanup:
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mbedtls_mpi_free(&K);
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mbedtls_mpi_free(&L);
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return ret;
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}
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int mbedtls_rsa_deduce_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
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const mbedtls_mpi *D, mbedtls_mpi *DP,
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mbedtls_mpi *DQ, mbedtls_mpi *QP)
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{
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int ret = 0;
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mbedtls_mpi K;
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mbedtls_mpi_init(&K);
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/* DP = D mod P-1 */
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if (DP != NULL) {
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DP, D, &K));
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}
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/* DQ = D mod Q-1 */
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if (DQ != NULL) {
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(DQ, D, &K));
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}
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/* QP = Q^{-1} mod P */
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if (QP != NULL) {
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MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P));
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}
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cleanup:
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mbedtls_mpi_free(&K);
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return ret;
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}
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/*
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* Check that core RSA parameters are sane.
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*/
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int mbedtls_rsa_validate_params(const mbedtls_mpi *N, const mbedtls_mpi *P,
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const mbedtls_mpi *Q, const mbedtls_mpi *D,
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const mbedtls_mpi *E,
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int (*f_rng)(void *, unsigned char *, size_t),
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void *p_rng)
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{
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int ret = 0;
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mbedtls_mpi K, L;
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mbedtls_mpi_init(&K);
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mbedtls_mpi_init(&L);
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/*
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* Step 1: If PRNG provided, check that P and Q are prime
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*/
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#if defined(MBEDTLS_GENPRIME)
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/*
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* When generating keys, the strongest security we support aims for an error
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* rate of at most 2^-100 and we are aiming for the same certainty here as
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* well.
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*/
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if (f_rng != NULL && P != NULL &&
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(ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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if (f_rng != NULL && Q != NULL &&
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(ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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#else
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((void) f_rng);
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((void) p_rng);
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#endif /* MBEDTLS_GENPRIME */
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/*
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* Step 2: Check that 1 < N = P * Q
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*/
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if (P != NULL && Q != NULL && N != NULL) {
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MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, P, Q));
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if (mbedtls_mpi_cmp_int(N, 1) <= 0 ||
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mbedtls_mpi_cmp_mpi(&K, N) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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}
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/*
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* Step 3: Check and 1 < D, E < N if present.
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*/
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if (N != NULL && D != NULL && E != NULL) {
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if (mbedtls_mpi_cmp_int(D, 1) <= 0 ||
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mbedtls_mpi_cmp_int(E, 1) <= 0 ||
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mbedtls_mpi_cmp_mpi(D, N) >= 0 ||
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mbedtls_mpi_cmp_mpi(E, N) >= 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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}
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/*
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* Step 4: Check that D, E are inverse modulo P-1 and Q-1
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*/
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if (P != NULL && Q != NULL && D != NULL && E != NULL) {
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if (mbedtls_mpi_cmp_int(P, 1) <= 0 ||
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mbedtls_mpi_cmp_int(Q, 1) <= 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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/* Compute DE-1 mod P-1 */
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MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, P, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
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if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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/* Compute DE-1 mod Q-1 */
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MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, D, E));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&L, Q, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, &L));
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if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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}
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cleanup:
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mbedtls_mpi_free(&K);
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mbedtls_mpi_free(&L);
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/* Wrap MPI error codes by RSA check failure error code */
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if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED) {
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ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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}
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return ret;
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}
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/*
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* Check that RSA CRT parameters are in accordance with core parameters.
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*/
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int mbedtls_rsa_validate_crt(const mbedtls_mpi *P, const mbedtls_mpi *Q,
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const mbedtls_mpi *D, const mbedtls_mpi *DP,
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const mbedtls_mpi *DQ, const mbedtls_mpi *QP)
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{
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int ret = 0;
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mbedtls_mpi K, L;
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mbedtls_mpi_init(&K);
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mbedtls_mpi_init(&L);
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/* Check that DP - D == 0 mod P - 1 */
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if (DP != NULL) {
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if (P == NULL) {
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ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
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goto cleanup;
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}
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, P, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DP, D));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
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if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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}
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/* Check that DQ - D == 0 mod Q - 1 */
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if (DQ != NULL) {
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if (Q == NULL) {
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ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
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goto cleanup;
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}
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, Q, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&L, DQ, D));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&L, &L, &K));
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if (mbedtls_mpi_cmp_int(&L, 0) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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}
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/* Check that QP * Q - 1 == 0 mod P */
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if (QP != NULL) {
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if (P == NULL || Q == NULL) {
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ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA;
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goto cleanup;
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}
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MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&K, QP, Q));
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MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&K, &K, 1));
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MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&K, &K, P));
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if (mbedtls_mpi_cmp_int(&K, 0) != 0) {
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ret = MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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goto cleanup;
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}
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}
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cleanup:
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/* Wrap MPI error codes by RSA check failure error code */
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if (ret != 0 &&
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ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED &&
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ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) {
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ret += MBEDTLS_ERR_RSA_KEY_CHECK_FAILED;
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}
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mbedtls_mpi_free(&K);
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mbedtls_mpi_free(&L);
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return ret;
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}
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#endif /* MBEDTLS_RSA_C */
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