mbed TLS Library
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Outline
#include "common.h"
#include "mbedtls/rsa.h"
#include "mbedtls/bignum.h"
#include "mbedtls/rsa_internal.h"
mbedtls_rsa_deduce_primes(const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, mbedtls_mpi *, mbedtls_mpi *)
mbedtls_rsa_deduce_private_exponent(const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, mbedtls_mpi *)
mbedtls_rsa_validate_crt(const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *)
mbedtls_rsa_validate_params(const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, int (*)(void *, unsigned char *, size_t), void *)
mbedtls_rsa_deduce_crt(const mbedtls_mpi *, const mbedtls_mpi *, const mbedtls_mpi *, mbedtls_mpi *, mbedtls_mpi *, mbedtls_mpi *)
Files
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SourceVuRaspberry Pi Pico SDK and Examplesmbed TLS Librarylibrary/rsa_internal.c
 
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/* * Helper functions for the RSA module * * Copyright The Mbed TLS Contributors * SPDX-License-Identifier: Apache-2.0 OR GPL-2.0-or-later * *//* ... */ #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 (P == NULL || Q == NULL || P->p != NULL || Q->p != NULL) { ... } 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; }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) { ... } /* * 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; }if ((order = (uint16_t) mbedtls_mpi_lsb(&T)) == 0) { ... } /* 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; }if (N->p[0] % 8 == 1) { ... } for (; attempt < num_primes; ++attempt) { MBEDTLS_MPI_CHK(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; }if (mbedtls_mpi_cmp_int(P, 1) != 0) { ... } /* 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; }if (mbedtls_mpi_cmp_int(&K, 1) == 0) { ... } 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; }if (mbedtls_mpi_cmp_int(P, 1) == 1 && mbedtls_mpi_cmp_mpi(P, N) == -1) { ... } 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)); }for (iter = 1; iter <= order; ++iter) { ... } /* * 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; }if (mbedtls_mpi_cmp_int(&K, 1) != 0) { ... } }for (; attempt < num_primes; ++attempt) { ... } 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 (D == NULL || mbedtls_mpi_cmp_int(D, 0) != 0) { ... } 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; }if (mbedtls_mpi_cmp_int(P, 1) <= 0 || mbedtls_mpi_cmp_int(Q, 1) <= 0 || mbedtls_mpi_cmp_int(E, 0) == 0) { ... } 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; }if (P == NULL) { ... } 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; }if (mbedtls_mpi_cmp_int(&L, 0) != 0) { ... } }if (DP != NULL) { ... } /* Check that DQ - D == 0 mod Q - 1 */ if (DQ != NULL) { if (Q == NULL) { ret = MBEDTLS_ERR_RSA_BAD_INPUT_DATA; goto cleanup; }if (Q == NULL) { ... } 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; }if (mbedtls_mpi_cmp_int(&L, 0) != 0) { ... } }if (DQ != NULL) { ... } /* 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; }if (P == NULL || Q == NULL) { ... } 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; }if (mbedtls_mpi_cmp_int(&K, 0) != 0) { ... } }if (QP != NULL) { ... } 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; }if (ret != 0 && ret != MBEDTLS_ERR_RSA_KEY_CHECK_FAILED && ret != MBEDTLS_ERR_RSA_BAD_INPUT_DATA) { ... } 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 && P != NULL && (ret = mbedtls_mpi_is_prime_ext(P, 50, f_rng, p_rng)) != 0) { ... } 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; }if (f_rng != NULL && Q != NULL && (ret = mbedtls_mpi_is_prime_ext(Q, 50, f_rng, p_rng)) != 0) { ... } /* ... */#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; }if (mbedtls_mpi_cmp_int(N, 1) <= 0 || mbedtls_mpi_cmp_mpi(&K, N) != 0) { ... } }if (P != NULL && Q != NULL && N != NULL) { ... } /* * 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; }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) { ... } }if (N != NULL && D != NULL && E != NULL) { ... } /* * 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; }if (mbedtls_mpi_cmp_int(P, 1) <= 0 || mbedtls_mpi_cmp_int(Q, 1) <= 0) { ... } /* 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; }if (mbedtls_mpi_cmp_int(&K, 0) != 0) { ... } /* 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; }if (mbedtls_mpi_cmp_int(&K, 0) != 0) { ... } }if (P != NULL && Q != NULL && D != NULL && E != NULL) { ... } 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; }if (ret != 0 && 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)); }if (DP != NULL) { ... } /* 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)); }if (DQ != NULL) { ... } /* QP = Q^{-1} mod P */ if (QP != NULL) { MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(QP, Q, P)); }if (QP != NULL) { ... } cleanup: mbedtls_mpi_free(&K); return ret; }{ ... } /* ... */#endif /* MBEDTLS_RSA_C */
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