/* * Multi-precision integer library * * 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. */ /* * The following sources were referenced in the design of this Multi-precision * Integer library: * * [1] Handbook of Applied Cryptography - 1997 * Menezes, van Oorschot and Vanstone * * [2] Multi-Precision Math * Tom St Denis * https://github.com/libtom/libtommath/blob/develop/tommath.pdf * * [3] GNU Multi-Precision Arithmetic Library * https://gmplib.org/manual/index.html * */ #include "common.h" #if defined(MBEDTLS_BIGNUM_C) #include "mbedtls/bignum.h" #include "bignum_core.h" #include "bn_mul.h" #include "mbedtls/platform_util.h" #include "mbedtls/error.h" #include "constant_time_internal.h" #include #include #include "mbedtls/platform.h" #define MPI_VALIDATE_RET(cond) \ MBEDTLS_INTERNAL_VALIDATE_RET(cond, MBEDTLS_ERR_MPI_BAD_INPUT_DATA) #define MPI_VALIDATE(cond) \ MBEDTLS_INTERNAL_VALIDATE(cond) /* * Compare signed values in constant time */ int mbedtls_mpi_lt_mpi_ct(const mbedtls_mpi *X, const mbedtls_mpi *Y, unsigned *ret) { mbedtls_ct_condition_t different_sign, X_is_negative, Y_is_negative, result; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(Y != NULL); MPI_VALIDATE_RET(ret != NULL); if (X->n != Y->n) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } /* * Set N_is_negative to MBEDTLS_CT_FALSE if N >= 0, MBEDTLS_CT_TRUE if N < 0. * We know that N->s == 1 if N >= 0 and N->s == -1 if N < 0. */ X_is_negative = mbedtls_ct_bool((X->s & 2) >> 1); Y_is_negative = mbedtls_ct_bool((Y->s & 2) >> 1); /* * If the signs are different, then the positive operand is the bigger. * That is if X is negative (X_is_negative == 1), then X < Y is true and it * is false if X is positive (X_is_negative == 0). */ different_sign = mbedtls_ct_bool_ne(X_is_negative, Y_is_negative); // true if different sign result = mbedtls_ct_bool_and(different_sign, X_is_negative); /* * Assuming signs are the same, compare X and Y. We switch the comparison * order if they are negative so that we get the right result, regardles of * sign. */ /* This array is used to conditionally swap the pointers in const time */ void * const p[2] = { X->p, Y->p }; size_t i = mbedtls_ct_size_if_else_0(X_is_negative, 1); mbedtls_ct_condition_t lt = mbedtls_mpi_core_lt_ct(p[i], p[i ^ 1], X->n); /* * Store in result iff the signs are the same (i.e., iff different_sign == false). If * the signs differ, result has already been set, so we don't change it. */ result = mbedtls_ct_bool_or(result, mbedtls_ct_bool_and(mbedtls_ct_bool_not(different_sign), lt)); *ret = mbedtls_ct_uint_if_else_0(result, 1); return 0; } /* * Conditionally assign X = Y, without leaking information * about whether the assignment was made or not. * (Leaking information about the respective sizes of X and Y is ok however.) */ #if defined(_MSC_VER) && defined(_M_ARM64) && (_MSC_FULL_VER < 193131103) /* * MSVC miscompiles this function if it's inlined prior to Visual Studio 2022 version 17.1. See: * https://developercommunity.visualstudio.com/t/c-compiler-miscompiles-part-of-mbedtls-library-on/1646989 */ __declspec(noinline) #endif int mbedtls_mpi_safe_cond_assign(mbedtls_mpi *X, const mbedtls_mpi *Y, unsigned char assign) { int ret = 0; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(Y != NULL); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, Y->n)); { mbedtls_ct_condition_t do_assign = mbedtls_ct_bool(assign); X->s = (int) mbedtls_ct_uint_if(do_assign, Y->s, X->s); mbedtls_mpi_core_cond_assign(X->p, Y->p, Y->n, do_assign); mbedtls_ct_condition_t do_not_assign = mbedtls_ct_bool_not(do_assign); for (size_t i = Y->n; i < X->n; i++) { X->p[i] = mbedtls_ct_mpi_uint_if_else_0(do_not_assign, X->p[i]); } } cleanup: return ret; } /* * Conditionally swap X and Y, without leaking information * about whether the swap was made or not. * Here it is not ok to simply swap the pointers, which would lead to * different memory access patterns when X and Y are used afterwards. */ int mbedtls_mpi_safe_cond_swap(mbedtls_mpi *X, mbedtls_mpi *Y, unsigned char swap) { int ret = 0; int s; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(Y != NULL); if (X == Y) { return 0; } mbedtls_ct_condition_t do_swap = mbedtls_ct_bool(swap); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, Y->n)); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(Y, X->n)); s = X->s; X->s = (int) mbedtls_ct_uint_if(do_swap, Y->s, X->s); Y->s = (int) mbedtls_ct_uint_if(do_swap, s, Y->s); mbedtls_mpi_core_cond_swap(X->p, Y->p, X->n, do_swap); cleanup: return ret; } /* Implementation that should never be optimized out by the compiler */ #define mbedtls_mpi_zeroize_and_free(v, n) mbedtls_zeroize_and_free(v, ciL * (n)) /* * Initialize one MPI */ void mbedtls_mpi_init(mbedtls_mpi *X) { MPI_VALIDATE(X != NULL); X->s = 1; X->n = 0; X->p = NULL; } /* * Unallocate one MPI */ void mbedtls_mpi_free(mbedtls_mpi *X) { if (X == NULL) { return; } if (X->p != NULL) { mbedtls_mpi_zeroize_and_free(X->p, X->n); } X->s = 1; X->n = 0; X->p = NULL; } /* * Enlarge to the specified number of limbs */ int mbedtls_mpi_grow(mbedtls_mpi *X, size_t nblimbs) { mbedtls_mpi_uint *p; MPI_VALIDATE_RET(X != NULL); if (nblimbs > MBEDTLS_MPI_MAX_LIMBS) { return MBEDTLS_ERR_MPI_ALLOC_FAILED; } if (X->n < nblimbs) { if ((p = (mbedtls_mpi_uint *) mbedtls_calloc(nblimbs, ciL)) == NULL) { return MBEDTLS_ERR_MPI_ALLOC_FAILED; } if (X->p != NULL) { memcpy(p, X->p, X->n * ciL); mbedtls_mpi_zeroize_and_free(X->p, X->n); } /* nblimbs fits in n because we ensure that MBEDTLS_MPI_MAX_LIMBS * fits, and we've checked that nblimbs <= MBEDTLS_MPI_MAX_LIMBS. */ X->n = (unsigned short) nblimbs; X->p = p; } return 0; } /* * Resize down as much as possible, * while keeping at least the specified number of limbs */ int mbedtls_mpi_shrink(mbedtls_mpi *X, size_t nblimbs) { mbedtls_mpi_uint *p; size_t i; MPI_VALIDATE_RET(X != NULL); if (nblimbs > MBEDTLS_MPI_MAX_LIMBS) { return MBEDTLS_ERR_MPI_ALLOC_FAILED; } /* Actually resize up if there are currently fewer than nblimbs limbs. */ if (X->n <= nblimbs) { return mbedtls_mpi_grow(X, nblimbs); } /* After this point, then X->n > nblimbs and in particular X->n > 0. */ for (i = X->n - 1; i > 0; i--) { if (X->p[i] != 0) { break; } } i++; if (i < nblimbs) { i = nblimbs; } if ((p = (mbedtls_mpi_uint *) mbedtls_calloc(i, ciL)) == NULL) { return MBEDTLS_ERR_MPI_ALLOC_FAILED; } if (X->p != NULL) { memcpy(p, X->p, i * ciL); mbedtls_mpi_zeroize_and_free(X->p, X->n); } /* i fits in n because we ensure that MBEDTLS_MPI_MAX_LIMBS * fits, and we've checked that i <= nblimbs <= MBEDTLS_MPI_MAX_LIMBS. */ X->n = (unsigned short) i; X->p = p; return 0; } /* Resize X to have exactly n limbs and set it to 0. */ static int mbedtls_mpi_resize_clear(mbedtls_mpi *X, size_t limbs) { if (limbs == 0) { mbedtls_mpi_free(X); return 0; } else if (X->n == limbs) { memset(X->p, 0, limbs * ciL); X->s = 1; return 0; } else { mbedtls_mpi_free(X); return mbedtls_mpi_grow(X, limbs); } } /* * Copy the contents of Y into X. * * This function is not constant-time. Leading zeros in Y may be removed. * * Ensure that X does not shrink. This is not guaranteed by the public API, * but some code in the bignum module relies on this property, for example * in mbedtls_mpi_exp_mod(). */ int mbedtls_mpi_copy(mbedtls_mpi *X, const mbedtls_mpi *Y) { int ret = 0; size_t i; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(Y != NULL); if (X == Y) { return 0; } if (Y->n == 0) { if (X->n != 0) { X->s = 1; memset(X->p, 0, X->n * ciL); } return 0; } for (i = Y->n - 1; i > 0; i--) { if (Y->p[i] != 0) { break; } } i++; X->s = Y->s; if (X->n < i) { MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, i)); } else { memset(X->p + i, 0, (X->n - i) * ciL); } memcpy(X->p, Y->p, i * ciL); cleanup: return ret; } /* * Swap the contents of X and Y */ void mbedtls_mpi_swap(mbedtls_mpi *X, mbedtls_mpi *Y) { mbedtls_mpi T; MPI_VALIDATE(X != NULL); MPI_VALIDATE(Y != NULL); memcpy(&T, X, sizeof(mbedtls_mpi)); memcpy(X, Y, sizeof(mbedtls_mpi)); memcpy(Y, &T, sizeof(mbedtls_mpi)); } static inline mbedtls_mpi_uint mpi_sint_abs(mbedtls_mpi_sint z) { if (z >= 0) { return z; } /* Take care to handle the most negative value (-2^(biL-1)) correctly. * A naive -z would have undefined behavior. * Write this in a way that makes popular compilers happy (GCC, Clang, * MSVC). */ return (mbedtls_mpi_uint) 0 - (mbedtls_mpi_uint) z; } /* Convert x to a sign, i.e. to 1, if x is positive, or -1, if x is negative. * This looks awkward but generates smaller code than (x < 0 ? -1 : 1) */ #define TO_SIGN(x) ((mbedtls_mpi_sint) (((mbedtls_mpi_uint) x) >> (biL - 1)) * -2 + 1) /* * Set value from integer */ int mbedtls_mpi_lset(mbedtls_mpi *X, mbedtls_mpi_sint z) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MPI_VALIDATE_RET(X != NULL); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, 1)); memset(X->p, 0, X->n * ciL); X->p[0] = mpi_sint_abs(z); X->s = TO_SIGN(z); cleanup: return ret; } /* * Get a specific bit */ int mbedtls_mpi_get_bit(const mbedtls_mpi *X, size_t pos) { MPI_VALIDATE_RET(X != NULL); if (X->n * biL <= pos) { return 0; } return (X->p[pos / biL] >> (pos % biL)) & 0x01; } /* * Set a bit to a specific value of 0 or 1 */ int mbedtls_mpi_set_bit(mbedtls_mpi *X, size_t pos, unsigned char val) { int ret = 0; size_t off = pos / biL; size_t idx = pos % biL; MPI_VALIDATE_RET(X != NULL); if (val != 0 && val != 1) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } if (X->n * biL <= pos) { if (val == 0) { return 0; } MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, off + 1)); } X->p[off] &= ~((mbedtls_mpi_uint) 0x01 << idx); X->p[off] |= (mbedtls_mpi_uint) val << idx; cleanup: return ret; } /* * Return the number of less significant zero-bits */ size_t mbedtls_mpi_lsb(const mbedtls_mpi *X) { size_t i; MBEDTLS_INTERNAL_VALIDATE_RET(X != NULL, 0); #if defined(__has_builtin) #if (MBEDTLS_MPI_UINT_MAX == UINT_MAX) && __has_builtin(__builtin_ctz) #define mbedtls_mpi_uint_ctz __builtin_ctz #elif (MBEDTLS_MPI_UINT_MAX == ULONG_MAX) && __has_builtin(__builtin_ctzl) #define mbedtls_mpi_uint_ctz __builtin_ctzl #elif (MBEDTLS_MPI_UINT_MAX == ULLONG_MAX) && __has_builtin(__builtin_ctzll) #define mbedtls_mpi_uint_ctz __builtin_ctzll #endif #endif #if defined(mbedtls_mpi_uint_ctz) for (i = 0; i < X->n; i++) { if (X->p[i] != 0) { return i * biL + mbedtls_mpi_uint_ctz(X->p[i]); } } #else size_t count = 0; for (i = 0; i < X->n; i++) { for (size_t j = 0; j < biL; j++, count++) { if (((X->p[i] >> j) & 1) != 0) { return count; } } } #endif return 0; } /* * Return the number of bits */ size_t mbedtls_mpi_bitlen(const mbedtls_mpi *X) { return mbedtls_mpi_core_bitlen(X->p, X->n); } /* * Return the total size in bytes */ size_t mbedtls_mpi_size(const mbedtls_mpi *X) { return (mbedtls_mpi_bitlen(X) + 7) >> 3; } /* * Convert an ASCII character to digit value */ static int mpi_get_digit(mbedtls_mpi_uint *d, int radix, char c) { *d = 255; if (c >= 0x30 && c <= 0x39) { *d = c - 0x30; } if (c >= 0x41 && c <= 0x46) { *d = c - 0x37; } if (c >= 0x61 && c <= 0x66) { *d = c - 0x57; } if (*d >= (mbedtls_mpi_uint) radix) { return MBEDTLS_ERR_MPI_INVALID_CHARACTER; } return 0; } /* * Import from an ASCII string */ int mbedtls_mpi_read_string(mbedtls_mpi *X, int radix, const char *s) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, j, slen, n; int sign = 1; mbedtls_mpi_uint d; mbedtls_mpi T; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(s != NULL); if (radix < 2 || radix > 16) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } mbedtls_mpi_init(&T); if (s[0] == 0) { mbedtls_mpi_free(X); return 0; } if (s[0] == '-') { ++s; sign = -1; } slen = strlen(s); if (radix == 16) { if (slen > SIZE_MAX >> 2) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } n = BITS_TO_LIMBS(slen << 2); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, n)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, 0)); for (i = slen, j = 0; i > 0; i--, j++) { MBEDTLS_MPI_CHK(mpi_get_digit(&d, radix, s[i - 1])); X->p[j / (2 * ciL)] |= d << ((j % (2 * ciL)) << 2); } } else { MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, 0)); for (i = 0; i < slen; i++) { MBEDTLS_MPI_CHK(mpi_get_digit(&d, radix, s[i])); MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&T, X, radix)); MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(X, &T, d)); } } if (sign < 0 && mbedtls_mpi_bitlen(X) != 0) { X->s = -1; } cleanup: mbedtls_mpi_free(&T); return ret; } /* * Helper to write the digits high-order first. */ static int mpi_write_hlp(mbedtls_mpi *X, int radix, char **p, const size_t buflen) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi_uint r; size_t length = 0; char *p_end = *p + buflen; do { if (length >= buflen) { return MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL; } MBEDTLS_MPI_CHK(mbedtls_mpi_mod_int(&r, X, radix)); MBEDTLS_MPI_CHK(mbedtls_mpi_div_int(X, NULL, X, radix)); /* * Write the residue in the current position, as an ASCII character. */ if (r < 0xA) { *(--p_end) = (char) ('0' + r); } else { *(--p_end) = (char) ('A' + (r - 0xA)); } length++; } while (mbedtls_mpi_cmp_int(X, 0) != 0); memmove(*p, p_end, length); *p += length; cleanup: return ret; } /* * Export into an ASCII string */ int mbedtls_mpi_write_string(const mbedtls_mpi *X, int radix, char *buf, size_t buflen, size_t *olen) { int ret = 0; size_t n; char *p; mbedtls_mpi T; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(olen != NULL); MPI_VALIDATE_RET(buflen == 0 || buf != NULL); if (radix < 2 || radix > 16) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } n = mbedtls_mpi_bitlen(X); /* Number of bits necessary to present `n`. */ if (radix >= 4) { n >>= 1; /* Number of 4-adic digits necessary to present * `n`. If radix > 4, this might be a strict * overapproximation of the number of * radix-adic digits needed to present `n`. */ } if (radix >= 16) { n >>= 1; /* Number of hexadecimal digits necessary to * present `n`. */ } n += 1; /* Terminating null byte */ n += 1; /* Compensate for the divisions above, which round down `n` * in case it's not even. */ n += 1; /* Potential '-'-sign. */ n += (n & 1); /* Make n even to have enough space for hexadecimal writing, * which always uses an even number of hex-digits. */ if (buflen < n) { *olen = n; return MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL; } p = buf; mbedtls_mpi_init(&T); if (X->s == -1) { *p++ = '-'; buflen--; } if (radix == 16) { int c; size_t i, j, k; for (i = X->n, k = 0; i > 0; i--) { for (j = ciL; j > 0; j--) { c = (X->p[i - 1] >> ((j - 1) << 3)) & 0xFF; if (c == 0 && k == 0 && (i + j) != 2) { continue; } *(p++) = "0123456789ABCDEF" [c / 16]; *(p++) = "0123456789ABCDEF" [c % 16]; k = 1; } } } else { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&T, X)); if (T.s == -1) { T.s = 1; } MBEDTLS_MPI_CHK(mpi_write_hlp(&T, radix, &p, buflen)); } *p++ = '\0'; *olen = p - buf; cleanup: mbedtls_mpi_free(&T); return ret; } #if defined(MBEDTLS_FS_IO) /* * Read X from an opened file */ int mbedtls_mpi_read_file(mbedtls_mpi *X, int radix, FILE *fin) { mbedtls_mpi_uint d; size_t slen; char *p; /* * Buffer should have space for (short) label and decimal formatted MPI, * newline characters and '\0' */ char s[MBEDTLS_MPI_RW_BUFFER_SIZE]; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(fin != NULL); if (radix < 2 || radix > 16) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } memset(s, 0, sizeof(s)); if (fgets(s, sizeof(s) - 1, fin) == NULL) { return MBEDTLS_ERR_MPI_FILE_IO_ERROR; } slen = strlen(s); if (slen == sizeof(s) - 2) { return MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL; } if (slen > 0 && s[slen - 1] == '\n') { slen--; s[slen] = '\0'; } if (slen > 0 && s[slen - 1] == '\r') { slen--; s[slen] = '\0'; } p = s + slen; while (p-- > s) { if (mpi_get_digit(&d, radix, *p) != 0) { break; } } return mbedtls_mpi_read_string(X, radix, p + 1); } /* * Write X into an opened file (or stdout if fout == NULL) */ int mbedtls_mpi_write_file(const char *p, const mbedtls_mpi *X, int radix, FILE *fout) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t n, slen, plen; /* * Buffer should have space for (short) label and decimal formatted MPI, * newline characters and '\0' */ char s[MBEDTLS_MPI_RW_BUFFER_SIZE]; MPI_VALIDATE_RET(X != NULL); if (radix < 2 || radix > 16) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } memset(s, 0, sizeof(s)); MBEDTLS_MPI_CHK(mbedtls_mpi_write_string(X, radix, s, sizeof(s) - 2, &n)); if (p == NULL) { p = ""; } plen = strlen(p); slen = strlen(s); s[slen++] = '\r'; s[slen++] = '\n'; if (fout != NULL) { if (fwrite(p, 1, plen, fout) != plen || fwrite(s, 1, slen, fout) != slen) { return MBEDTLS_ERR_MPI_FILE_IO_ERROR; } } else { mbedtls_printf("%s%s", p, s); } cleanup: return ret; } #endif /* MBEDTLS_FS_IO */ /* * Import X from unsigned binary data, little endian * * This function is guaranteed to return an MPI with exactly the necessary * number of limbs (in particular, it does not skip 0s in the input). */ int mbedtls_mpi_read_binary_le(mbedtls_mpi *X, const unsigned char *buf, size_t buflen) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; const size_t limbs = CHARS_TO_LIMBS(buflen); /* Ensure that target MPI has exactly the necessary number of limbs */ MBEDTLS_MPI_CHK(mbedtls_mpi_resize_clear(X, limbs)); MBEDTLS_MPI_CHK(mbedtls_mpi_core_read_le(X->p, X->n, buf, buflen)); cleanup: /* * This function is also used to import keys. However, wiping the buffers * upon failure is not necessary because failure only can happen before any * input is copied. */ return ret; } /* * Import X from unsigned binary data, big endian * * This function is guaranteed to return an MPI with exactly the necessary * number of limbs (in particular, it does not skip 0s in the input). */ int mbedtls_mpi_read_binary(mbedtls_mpi *X, const unsigned char *buf, size_t buflen) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; const size_t limbs = CHARS_TO_LIMBS(buflen); MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(buflen == 0 || buf != NULL); /* Ensure that target MPI has exactly the necessary number of limbs */ MBEDTLS_MPI_CHK(mbedtls_mpi_resize_clear(X, limbs)); MBEDTLS_MPI_CHK(mbedtls_mpi_core_read_be(X->p, X->n, buf, buflen)); cleanup: /* * This function is also used to import keys. However, wiping the buffers * upon failure is not necessary because failure only can happen before any * input is copied. */ return ret; } /* * Export X into unsigned binary data, little endian */ int mbedtls_mpi_write_binary_le(const mbedtls_mpi *X, unsigned char *buf, size_t buflen) { return mbedtls_mpi_core_write_le(X->p, X->n, buf, buflen); } /* * Export X into unsigned binary data, big endian */ int mbedtls_mpi_write_binary(const mbedtls_mpi *X, unsigned char *buf, size_t buflen) { return mbedtls_mpi_core_write_be(X->p, X->n, buf, buflen); } /* * Left-shift: X <<= count */ int mbedtls_mpi_shift_l(mbedtls_mpi *X, size_t count) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i; MPI_VALIDATE_RET(X != NULL); i = mbedtls_mpi_bitlen(X) + count; if (X->n * biL < i) { MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, BITS_TO_LIMBS(i))); } ret = 0; mbedtls_mpi_core_shift_l(X->p, X->n, count); cleanup: return ret; } /* * Right-shift: X >>= count */ int mbedtls_mpi_shift_r(mbedtls_mpi *X, size_t count) { MPI_VALIDATE_RET(X != NULL); if (X->n != 0) { mbedtls_mpi_core_shift_r(X->p, X->n, count); } return 0; } /* * Compare unsigned values */ int mbedtls_mpi_cmp_abs(const mbedtls_mpi *X, const mbedtls_mpi *Y) { size_t i, j; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(Y != NULL); for (i = X->n; i > 0; i--) { if (X->p[i - 1] != 0) { break; } } for (j = Y->n; j > 0; j--) { if (Y->p[j - 1] != 0) { break; } } /* If i == j == 0, i.e. abs(X) == abs(Y), * we end up returning 0 at the end of the function. */ if (i > j) { return 1; } if (j > i) { return -1; } for (; i > 0; i--) { if (X->p[i - 1] > Y->p[i - 1]) { return 1; } if (X->p[i - 1] < Y->p[i - 1]) { return -1; } } return 0; } /* * Compare signed values */ int mbedtls_mpi_cmp_mpi(const mbedtls_mpi *X, const mbedtls_mpi *Y) { size_t i, j; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(Y != NULL); for (i = X->n; i > 0; i--) { if (X->p[i - 1] != 0) { break; } } for (j = Y->n; j > 0; j--) { if (Y->p[j - 1] != 0) { break; } } if (i == 0 && j == 0) { return 0; } if (i > j) { return X->s; } if (j > i) { return -Y->s; } if (X->s > 0 && Y->s < 0) { return 1; } if (Y->s > 0 && X->s < 0) { return -1; } for (; i > 0; i--) { if (X->p[i - 1] > Y->p[i - 1]) { return X->s; } if (X->p[i - 1] < Y->p[i - 1]) { return -X->s; } } return 0; } /* * Compare signed values */ int mbedtls_mpi_cmp_int(const mbedtls_mpi *X, mbedtls_mpi_sint z) { mbedtls_mpi Y; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET(X != NULL); *p = mpi_sint_abs(z); Y.s = TO_SIGN(z); Y.n = 1; Y.p = p; return mbedtls_mpi_cmp_mpi(X, &Y); } /* * Unsigned addition: X = |A| + |B| (HAC 14.7) */ int mbedtls_mpi_add_abs(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t j; mbedtls_mpi_uint *p; mbedtls_mpi_uint c; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(B != NULL); if (X == B) { const mbedtls_mpi *T = A; A = X; B = T; } if (X != A) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A)); } /* * X must always be positive as a result of unsigned additions. */ X->s = 1; for (j = B->n; j > 0; j--) { if (B->p[j - 1] != 0) { break; } } /* Exit early to avoid undefined behavior on NULL+0 when X->n == 0 * and B is 0 (of any size). */ if (j == 0) { return 0; } MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, j)); /* j is the number of non-zero limbs of B. Add those to X. */ p = X->p; c = mbedtls_mpi_core_add(p, p, B->p, j); p += j; /* Now propagate any carry */ while (c != 0) { if (j >= X->n) { MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, j + 1)); p = X->p + j; } *p += c; c = (*p < c); j++; p++; } cleanup: return ret; } /* * Unsigned subtraction: X = |A| - |B| (HAC 14.9, 14.10) */ int mbedtls_mpi_sub_abs(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t n; mbedtls_mpi_uint carry; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(B != NULL); for (n = B->n; n > 0; n--) { if (B->p[n - 1] != 0) { break; } } if (n > A->n) { /* B >= (2^ciL)^n > A */ ret = MBEDTLS_ERR_MPI_NEGATIVE_VALUE; goto cleanup; } MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, A->n)); /* Set the high limbs of X to match A. Don't touch the lower limbs * because X might be aliased to B, and we must not overwrite the * significant digits of B. */ if (A->n > n && A != X) { memcpy(X->p + n, A->p + n, (A->n - n) * ciL); } if (X->n > A->n) { memset(X->p + A->n, 0, (X->n - A->n) * ciL); } carry = mbedtls_mpi_core_sub(X->p, A->p, B->p, n); if (carry != 0) { /* Propagate the carry through the rest of X. */ carry = mbedtls_mpi_core_sub_int(X->p + n, X->p + n, carry, X->n - n); /* If we have further carry/borrow, the result is negative. */ if (carry != 0) { ret = MBEDTLS_ERR_MPI_NEGATIVE_VALUE; goto cleanup; } } /* X should always be positive as a result of unsigned subtractions. */ X->s = 1; cleanup: return ret; } /* Common function for signed addition and subtraction. * Calculate A + B * flip_B where flip_B is 1 or -1. */ static int add_sub_mpi(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B, int flip_B) { int ret, s; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(B != NULL); s = A->s; if (A->s * B->s * flip_B < 0) { int cmp = mbedtls_mpi_cmp_abs(A, B); if (cmp >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(X, A, B)); /* If |A| = |B|, the result is 0 and we must set the sign bit * to +1 regardless of which of A or B was negative. Otherwise, * since |A| > |B|, the sign is the sign of A. */ X->s = cmp == 0 ? 1 : s; } else { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(X, B, A)); /* Since |A| < |B|, the sign is the opposite of A. */ X->s = -s; } } else { MBEDTLS_MPI_CHK(mbedtls_mpi_add_abs(X, A, B)); X->s = s; } cleanup: return ret; } /* * Signed addition: X = A + B */ int mbedtls_mpi_add_mpi(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { return add_sub_mpi(X, A, B, 1); } /* * Signed subtraction: X = A - B */ int mbedtls_mpi_sub_mpi(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { return add_sub_mpi(X, A, B, -1); } /* * Signed addition: X = A + b */ int mbedtls_mpi_add_int(mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_sint b) { mbedtls_mpi B; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); p[0] = mpi_sint_abs(b); B.s = TO_SIGN(b); B.n = 1; B.p = p; return mbedtls_mpi_add_mpi(X, A, &B); } /* * Signed subtraction: X = A - b */ int mbedtls_mpi_sub_int(mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_sint b) { mbedtls_mpi B; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); p[0] = mpi_sint_abs(b); B.s = TO_SIGN(b); B.n = 1; B.p = p; return mbedtls_mpi_sub_mpi(X, A, &B); } /* * Baseline multiplication: X = A * B (HAC 14.12) */ int mbedtls_mpi_mul_mpi(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, j; mbedtls_mpi TA, TB; int result_is_zero = 0; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(B != NULL); mbedtls_mpi_init(&TA); mbedtls_mpi_init(&TB); if (X == A) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TA, A)); A = &TA; } if (X == B) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TB, B)); B = &TB; } for (i = A->n; i > 0; i--) { if (A->p[i - 1] != 0) { break; } } if (i == 0) { result_is_zero = 1; } for (j = B->n; j > 0; j--) { if (B->p[j - 1] != 0) { break; } } if (j == 0) { result_is_zero = 1; } MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, i + j)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, 0)); mbedtls_mpi_core_mul(X->p, A->p, i, B->p, j); /* If the result is 0, we don't shortcut the operation, which reduces * but does not eliminate side channels leaking the zero-ness. We do * need to take care to set the sign bit properly since the library does * not fully support an MPI object with a value of 0 and s == -1. */ if (result_is_zero) { X->s = 1; } else { X->s = A->s * B->s; } cleanup: mbedtls_mpi_free(&TB); mbedtls_mpi_free(&TA); return ret; } /* * Baseline multiplication: X = A * b */ int mbedtls_mpi_mul_int(mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_uint b) { MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); size_t n = A->n; while (n > 0 && A->p[n - 1] == 0) { --n; } /* The general method below doesn't work if b==0. */ if (b == 0 || n == 0) { return mbedtls_mpi_lset(X, 0); } /* Calculate A*b as A + A*(b-1) to take advantage of mbedtls_mpi_core_mla */ int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; /* In general, A * b requires 1 limb more than b. If * A->p[n - 1] * b / b == A->p[n - 1], then A * b fits in the same * number of limbs as A and the call to grow() is not required since * copy() will take care of the growth if needed. However, experimentally, * making the call to grow() unconditional causes slightly fewer * calls to calloc() in ECP code, presumably because it reuses the * same mpi for a while and this way the mpi is more likely to directly * grow to its final size. * * Note that calculating A*b as 0 + A*b doesn't work as-is because * A,X can be the same. */ MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, n + 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A)); mbedtls_mpi_core_mla(X->p, X->n, A->p, n, b - 1); cleanup: return ret; } /* * Unsigned integer divide - double mbedtls_mpi_uint dividend, u1/u0, and * mbedtls_mpi_uint divisor, d */ static mbedtls_mpi_uint mbedtls_int_div_int(mbedtls_mpi_uint u1, mbedtls_mpi_uint u0, mbedtls_mpi_uint d, mbedtls_mpi_uint *r) { #if defined(MBEDTLS_HAVE_UDBL) mbedtls_t_udbl dividend, quotient; #else const mbedtls_mpi_uint radix = (mbedtls_mpi_uint) 1 << biH; const mbedtls_mpi_uint uint_halfword_mask = ((mbedtls_mpi_uint) 1 << biH) - 1; mbedtls_mpi_uint d0, d1, q0, q1, rAX, r0, quotient; mbedtls_mpi_uint u0_msw, u0_lsw; size_t s; #endif /* * Check for overflow */ if (0 == d || u1 >= d) { if (r != NULL) { *r = ~(mbedtls_mpi_uint) 0u; } return ~(mbedtls_mpi_uint) 0u; } #if defined(MBEDTLS_HAVE_UDBL) dividend = (mbedtls_t_udbl) u1 << biL; dividend |= (mbedtls_t_udbl) u0; quotient = dividend / d; if (quotient > ((mbedtls_t_udbl) 1 << biL) - 1) { quotient = ((mbedtls_t_udbl) 1 << biL) - 1; } if (r != NULL) { *r = (mbedtls_mpi_uint) (dividend - (quotient * d)); } return (mbedtls_mpi_uint) quotient; #else /* * Algorithm D, Section 4.3.1 - The Art of Computer Programming * Vol. 2 - Seminumerical Algorithms, Knuth */ /* * Normalize the divisor, d, and dividend, u0, u1 */ s = mbedtls_mpi_core_clz(d); d = d << s; u1 = u1 << s; u1 |= (u0 >> (biL - s)) & (-(mbedtls_mpi_sint) s >> (biL - 1)); u0 = u0 << s; d1 = d >> biH; d0 = d & uint_halfword_mask; u0_msw = u0 >> biH; u0_lsw = u0 & uint_halfword_mask; /* * Find the first quotient and remainder */ q1 = u1 / d1; r0 = u1 - d1 * q1; while (q1 >= radix || (q1 * d0 > radix * r0 + u0_msw)) { q1 -= 1; r0 += d1; if (r0 >= radix) { break; } } rAX = (u1 * radix) + (u0_msw - q1 * d); q0 = rAX / d1; r0 = rAX - q0 * d1; while (q0 >= radix || (q0 * d0 > radix * r0 + u0_lsw)) { q0 -= 1; r0 += d1; if (r0 >= radix) { break; } } if (r != NULL) { *r = (rAX * radix + u0_lsw - q0 * d) >> s; } quotient = q1 * radix + q0; return quotient; #endif } /* * Division by mbedtls_mpi: A = Q * B + R (HAC 14.20) */ int mbedtls_mpi_div_mpi(mbedtls_mpi *Q, mbedtls_mpi *R, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, n, t, k; mbedtls_mpi X, Y, Z, T1, T2; mbedtls_mpi_uint TP2[3]; MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(B != NULL); if (mbedtls_mpi_cmp_int(B, 0) == 0) { return MBEDTLS_ERR_MPI_DIVISION_BY_ZERO; } mbedtls_mpi_init(&X); mbedtls_mpi_init(&Y); mbedtls_mpi_init(&Z); mbedtls_mpi_init(&T1); /* * Avoid dynamic memory allocations for constant-size T2. * * T2 is used for comparison only and the 3 limbs are assigned explicitly, * so nobody increase the size of the MPI and we're safe to use an on-stack * buffer. */ T2.s = 1; T2.n = sizeof(TP2) / sizeof(*TP2); T2.p = TP2; if (mbedtls_mpi_cmp_abs(A, B) < 0) { if (Q != NULL) { MBEDTLS_MPI_CHK(mbedtls_mpi_lset(Q, 0)); } if (R != NULL) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(R, A)); } return 0; } MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&X, A)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&Y, B)); X.s = Y.s = 1; MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&Z, A->n + 2)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&Z, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&T1, A->n + 2)); k = mbedtls_mpi_bitlen(&Y) % biL; if (k < biL - 1) { k = biL - 1 - k; MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&X, k)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&Y, k)); } else { k = 0; } n = X.n - 1; t = Y.n - 1; MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&Y, biL * (n - t))); while (mbedtls_mpi_cmp_mpi(&X, &Y) >= 0) { Z.p[n - t]++; MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&X, &X, &Y)); } MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&Y, biL * (n - t))); for (i = n; i > t; i--) { if (X.p[i] >= Y.p[t]) { Z.p[i - t - 1] = ~(mbedtls_mpi_uint) 0u; } else { Z.p[i - t - 1] = mbedtls_int_div_int(X.p[i], X.p[i - 1], Y.p[t], NULL); } T2.p[0] = (i < 2) ? 0 : X.p[i - 2]; T2.p[1] = (i < 1) ? 0 : X.p[i - 1]; T2.p[2] = X.p[i]; Z.p[i - t - 1]++; do { Z.p[i - t - 1]--; MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&T1, 0)); T1.p[0] = (t < 1) ? 0 : Y.p[t - 1]; T1.p[1] = Y.p[t]; MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&T1, &T1, Z.p[i - t - 1])); } while (mbedtls_mpi_cmp_mpi(&T1, &T2) > 0); MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&T1, &Y, Z.p[i - t - 1])); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&T1, biL * (i - t - 1))); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&X, &X, &T1)); if (mbedtls_mpi_cmp_int(&X, 0) < 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&T1, &Y)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&T1, biL * (i - t - 1))); MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&X, &X, &T1)); Z.p[i - t - 1]--; } } if (Q != NULL) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(Q, &Z)); Q->s = A->s * B->s; } if (R != NULL) { MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&X, k)); X.s = A->s; MBEDTLS_MPI_CHK(mbedtls_mpi_copy(R, &X)); if (mbedtls_mpi_cmp_int(R, 0) == 0) { R->s = 1; } } cleanup: mbedtls_mpi_free(&X); mbedtls_mpi_free(&Y); mbedtls_mpi_free(&Z); mbedtls_mpi_free(&T1); mbedtls_platform_zeroize(TP2, sizeof(TP2)); return ret; } /* * Division by int: A = Q * b + R */ int mbedtls_mpi_div_int(mbedtls_mpi *Q, mbedtls_mpi *R, const mbedtls_mpi *A, mbedtls_mpi_sint b) { mbedtls_mpi B; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET(A != NULL); p[0] = mpi_sint_abs(b); B.s = TO_SIGN(b); B.n = 1; B.p = p; return mbedtls_mpi_div_mpi(Q, R, A, &B); } /* * Modulo: R = A mod B */ int mbedtls_mpi_mod_mpi(mbedtls_mpi *R, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MPI_VALIDATE_RET(R != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(B != NULL); if (mbedtls_mpi_cmp_int(B, 0) < 0) { return MBEDTLS_ERR_MPI_NEGATIVE_VALUE; } MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(NULL, R, A, B)); while (mbedtls_mpi_cmp_int(R, 0) < 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(R, R, B)); } while (mbedtls_mpi_cmp_mpi(R, B) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(R, R, B)); } cleanup: return ret; } /* * Modulo: r = A mod b */ int mbedtls_mpi_mod_int(mbedtls_mpi_uint *r, const mbedtls_mpi *A, mbedtls_mpi_sint b) { size_t i; mbedtls_mpi_uint x, y, z; MPI_VALIDATE_RET(r != NULL); MPI_VALIDATE_RET(A != NULL); if (b == 0) { return MBEDTLS_ERR_MPI_DIVISION_BY_ZERO; } if (b < 0) { return MBEDTLS_ERR_MPI_NEGATIVE_VALUE; } /* * handle trivial cases */ if (b == 1 || A->n == 0) { *r = 0; return 0; } if (b == 2) { *r = A->p[0] & 1; return 0; } /* * general case */ for (i = A->n, y = 0; i > 0; i--) { x = A->p[i - 1]; y = (y << biH) | (x >> biH); z = y / b; y -= z * b; x <<= biH; y = (y << biH) | (x >> biH); z = y / b; y -= z * b; } /* * If A is negative, then the current y represents a negative value. * Flipping it to the positive side. */ if (A->s < 0 && y != 0) { y = b - y; } *r = y; return 0; } static void mpi_montg_init(mbedtls_mpi_uint *mm, const mbedtls_mpi *N) { *mm = mbedtls_mpi_core_montmul_init(N->p); } /** Montgomery multiplication: A = A * B * R^-1 mod N (HAC 14.36) * * \param[in,out] A One of the numbers to multiply. * It must have at least as many limbs as N * (A->n >= N->n), and any limbs beyond n are ignored. * On successful completion, A contains the result of * the multiplication A * B * R^-1 mod N where * R = (2^ciL)^n. * \param[in] B One of the numbers to multiply. * It must be nonzero and must not have more limbs than N * (B->n <= N->n). * \param[in] N The modulus. \p N must be odd. * \param mm The value calculated by `mpi_montg_init(&mm, N)`. * This is -N^-1 mod 2^ciL. * \param[in,out] T A bignum for temporary storage. * It must be at least twice the limb size of N plus 1 * (T->n >= 2 * N->n + 1). * Its initial content is unused and * its final content is indeterminate. * It does not get reallocated. */ static void mpi_montmul(mbedtls_mpi *A, const mbedtls_mpi *B, const mbedtls_mpi *N, mbedtls_mpi_uint mm, mbedtls_mpi *T) { mbedtls_mpi_core_montmul(A->p, A->p, B->p, B->n, N->p, N->n, mm, T->p); } /* * Montgomery reduction: A = A * R^-1 mod N * * See mpi_montmul() regarding constraints and guarantees on the parameters. */ static void mpi_montred(mbedtls_mpi *A, const mbedtls_mpi *N, mbedtls_mpi_uint mm, mbedtls_mpi *T) { mbedtls_mpi_uint z = 1; mbedtls_mpi U; U.n = 1; U.s = 1; U.p = &z; mpi_montmul(A, &U, N, mm, T); } /** * Select an MPI from a table without leaking the index. * * This is functionally equivalent to mbedtls_mpi_copy(R, T[idx]) except it * reads the entire table in order to avoid leaking the value of idx to an * attacker able to observe memory access patterns. * * \param[out] R Where to write the selected MPI. * \param[in] T The table to read from. * \param[in] T_size The number of elements in the table. * \param[in] idx The index of the element to select; * this must satisfy 0 <= idx < T_size. * * \return \c 0 on success, or a negative error code. */ static int mpi_select(mbedtls_mpi *R, const mbedtls_mpi *T, size_t T_size, size_t idx) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; for (size_t i = 0; i < T_size; i++) { MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(R, &T[i], (unsigned char) mbedtls_ct_uint_eq(i, idx))); } cleanup: return ret; } /* * Sliding-window exponentiation: X = A^E mod N (HAC 14.85) */ int mbedtls_mpi_exp_mod(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *E, const mbedtls_mpi *N, mbedtls_mpi *prec_RR) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t window_bitsize; size_t i, j, nblimbs; size_t bufsize, nbits; size_t exponent_bits_in_window = 0; mbedtls_mpi_uint ei, mm, state; mbedtls_mpi RR, T, W[(size_t) 1 << MBEDTLS_MPI_WINDOW_SIZE], WW, Apos; int neg; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(E != NULL); MPI_VALIDATE_RET(N != NULL); if (mbedtls_mpi_cmp_int(N, 0) <= 0 || (N->p[0] & 1) == 0) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } if (mbedtls_mpi_cmp_int(E, 0) < 0) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } if (mbedtls_mpi_bitlen(E) > MBEDTLS_MPI_MAX_BITS || mbedtls_mpi_bitlen(N) > MBEDTLS_MPI_MAX_BITS) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } /* * Init temps and window size */ mpi_montg_init(&mm, N); mbedtls_mpi_init(&RR); mbedtls_mpi_init(&T); mbedtls_mpi_init(&Apos); mbedtls_mpi_init(&WW); memset(W, 0, sizeof(W)); i = mbedtls_mpi_bitlen(E); window_bitsize = (i > 671) ? 6 : (i > 239) ? 5 : (i > 79) ? 4 : (i > 23) ? 3 : 1; #if (MBEDTLS_MPI_WINDOW_SIZE < 6) if (window_bitsize > MBEDTLS_MPI_WINDOW_SIZE) { window_bitsize = MBEDTLS_MPI_WINDOW_SIZE; } #endif const size_t w_table_used_size = (size_t) 1 << window_bitsize; /* * This function is not constant-trace: its memory accesses depend on the * exponent value. To defend against timing attacks, callers (such as RSA * and DHM) should use exponent blinding. However this is not enough if the * adversary can find the exponent in a single trace, so this function * takes extra precautions against adversaries who can observe memory * access patterns. * * This function performs a series of multiplications by table elements and * squarings, and we want the prevent the adversary from finding out which * table element was used, and from distinguishing between multiplications * and squarings. Firstly, when multiplying by an element of the window * W[i], we do a constant-trace table lookup to obfuscate i. This leaves * squarings as having a different memory access patterns from other * multiplications. So secondly, we put the accumulator in the table as * well, and also do a constant-trace table lookup to multiply by the * accumulator which is W[x_index]. * * This way, all multiplications take the form of a lookup-and-multiply. * The number of lookup-and-multiply operations inside each iteration of * the main loop still depends on the bits of the exponent, but since the * other operations in the loop don't have an easily recognizable memory * trace, an adversary is unlikely to be able to observe the exact * patterns. * * An adversary may still be able to recover the exponent if they can * observe both memory accesses and branches. However, branch prediction * exploitation typically requires many traces of execution over the same * data, which is defeated by randomized blinding. */ const size_t x_index = 0; mbedtls_mpi_init(&W[x_index]); j = N->n + 1; /* All W[i] including the accumulator must have at least N->n limbs for * the mpi_montmul() and mpi_montred() calls later. Here we ensure that * W[1] and the accumulator W[x_index] are large enough. later we'll grow * other W[i] to the same length. They must not be shrunk midway through * this function! */ MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&W[x_index], j)); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&W[1], j)); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&T, j * 2)); /* * Compensate for negative A (and correct at the end) */ neg = (A->s == -1); if (neg) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&Apos, A)); Apos.s = 1; A = &Apos; } /* * If 1st call, pre-compute R^2 mod N */ if (prec_RR == NULL || prec_RR->p == NULL) { MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&RR, 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&RR, N->n * 2 * biL)); MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&RR, &RR, N)); if (prec_RR != NULL) { memcpy(prec_RR, &RR, sizeof(mbedtls_mpi)); } } else { memcpy(&RR, prec_RR, sizeof(mbedtls_mpi)); } /* * W[1] = A * R^2 * R^-1 mod N = A * R mod N */ if (mbedtls_mpi_cmp_mpi(A, N) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&W[1], A, N)); /* This should be a no-op because W[1] is already that large before * mbedtls_mpi_mod_mpi(), but it's necessary to avoid an overflow * in mpi_montmul() below, so let's make sure. */ MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&W[1], N->n + 1)); } else { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&W[1], A)); } /* Note that this is safe because W[1] always has at least N->n limbs * (it grew above and was preserved by mbedtls_mpi_copy()). */ mpi_montmul(&W[1], &RR, N, mm, &T); /* * W[x_index] = R^2 * R^-1 mod N = R mod N */ MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&W[x_index], &RR)); mpi_montred(&W[x_index], N, mm, &T); if (window_bitsize > 1) { /* * W[i] = W[1] ^ i * * The first bit of the sliding window is always 1 and therefore we * only need to store the second half of the table. * * (There are two special elements in the table: W[0] for the * accumulator/result and W[1] for A in Montgomery form. Both of these * are already set at this point.) */ j = w_table_used_size / 2; MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&W[j], N->n + 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&W[j], &W[1])); for (i = 0; i < window_bitsize - 1; i++) { mpi_montmul(&W[j], &W[j], N, mm, &T); } /* * W[i] = W[i - 1] * W[1] */ for (i = j + 1; i < w_table_used_size; i++) { MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&W[i], N->n + 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&W[i], &W[i - 1])); mpi_montmul(&W[i], &W[1], N, mm, &T); } } nblimbs = E->n; bufsize = 0; nbits = 0; state = 0; while (1) { if (bufsize == 0) { if (nblimbs == 0) { break; } nblimbs--; bufsize = sizeof(mbedtls_mpi_uint) << 3; } bufsize--; ei = (E->p[nblimbs] >> bufsize) & 1; /* * skip leading 0s */ if (ei == 0 && state == 0) { continue; } if (ei == 0 && state == 1) { /* * out of window, square W[x_index] */ MBEDTLS_MPI_CHK(mpi_select(&WW, W, w_table_used_size, x_index)); mpi_montmul(&W[x_index], &WW, N, mm, &T); continue; } /* * add ei to current window */ state = 2; nbits++; exponent_bits_in_window |= (ei << (window_bitsize - nbits)); if (nbits == window_bitsize) { /* * W[x_index] = W[x_index]^window_bitsize R^-1 mod N */ for (i = 0; i < window_bitsize; i++) { MBEDTLS_MPI_CHK(mpi_select(&WW, W, w_table_used_size, x_index)); mpi_montmul(&W[x_index], &WW, N, mm, &T); } /* * W[x_index] = W[x_index] * W[exponent_bits_in_window] R^-1 mod N */ MBEDTLS_MPI_CHK(mpi_select(&WW, W, w_table_used_size, exponent_bits_in_window)); mpi_montmul(&W[x_index], &WW, N, mm, &T); state--; nbits = 0; exponent_bits_in_window = 0; } } /* * process the remaining bits */ for (i = 0; i < nbits; i++) { MBEDTLS_MPI_CHK(mpi_select(&WW, W, w_table_used_size, x_index)); mpi_montmul(&W[x_index], &WW, N, mm, &T); exponent_bits_in_window <<= 1; if ((exponent_bits_in_window & ((size_t) 1 << window_bitsize)) != 0) { MBEDTLS_MPI_CHK(mpi_select(&WW, W, w_table_used_size, 1)); mpi_montmul(&W[x_index], &WW, N, mm, &T); } } /* * W[x_index] = A^E * R * R^-1 mod N = A^E mod N */ mpi_montred(&W[x_index], N, mm, &T); if (neg && E->n != 0 && (E->p[0] & 1) != 0) { W[x_index].s = -1; MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&W[x_index], N, &W[x_index])); } /* * Load the result in the output variable. */ MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, &W[x_index])); cleanup: /* The first bit of the sliding window is always 1 and therefore the first * half of the table was unused. */ for (i = w_table_used_size/2; i < w_table_used_size; i++) { mbedtls_mpi_free(&W[i]); } mbedtls_mpi_free(&W[x_index]); mbedtls_mpi_free(&W[1]); mbedtls_mpi_free(&T); mbedtls_mpi_free(&Apos); mbedtls_mpi_free(&WW); if (prec_RR == NULL || prec_RR->p == NULL) { mbedtls_mpi_free(&RR); } return ret; } /* * Greatest common divisor: G = gcd(A, B) (HAC 14.54) */ int mbedtls_mpi_gcd(mbedtls_mpi *G, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t lz, lzt; mbedtls_mpi TA, TB; MPI_VALIDATE_RET(G != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(B != NULL); mbedtls_mpi_init(&TA); mbedtls_mpi_init(&TB); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TA, A)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TB, B)); lz = mbedtls_mpi_lsb(&TA); lzt = mbedtls_mpi_lsb(&TB); /* The loop below gives the correct result when A==0 but not when B==0. * So have a special case for B==0. Leverage the fact that we just * calculated the lsb and lsb(B)==0 iff B is odd or 0 to make the test * slightly more efficient than cmp_int(). */ if (lzt == 0 && mbedtls_mpi_get_bit(&TB, 0) == 0) { ret = mbedtls_mpi_copy(G, A); goto cleanup; } if (lzt < lz) { lz = lzt; } TA.s = TB.s = 1; /* We mostly follow the procedure described in HAC 14.54, but with some * minor differences: * - Sequences of multiplications or divisions by 2 are grouped into a * single shift operation. * - The procedure in HAC assumes that 0 < TB <= TA. * - The condition TB <= TA is not actually necessary for correctness. * TA and TB have symmetric roles except for the loop termination * condition, and the shifts at the beginning of the loop body * remove any significance from the ordering of TA vs TB before * the shifts. * - If TA = 0, the loop goes through 0 iterations and the result is * correctly TB. * - The case TB = 0 was short-circuited above. * * For the correctness proof below, decompose the original values of * A and B as * A = sa * 2^a * A' with A'=0 or A' odd, and sa = +-1 * B = sb * 2^b * B' with B'=0 or B' odd, and sb = +-1 * Then gcd(A, B) = 2^{min(a,b)} * gcd(A',B'), * and gcd(A',B') is odd or 0. * * At the beginning, we have TA = |A| and TB = |B| so gcd(A,B) = gcd(TA,TB). * The code maintains the following invariant: * gcd(A,B) = 2^k * gcd(TA,TB) for some k (I) */ /* Proof that the loop terminates: * At each iteration, either the right-shift by 1 is made on a nonzero * value and the nonnegative integer bitlen(TA) + bitlen(TB) decreases * by at least 1, or the right-shift by 1 is made on zero and then * TA becomes 0 which ends the loop (TB cannot be 0 if it is right-shifted * since in that case TB is calculated from TB-TA with the condition TB>TA). */ while (mbedtls_mpi_cmp_int(&TA, 0) != 0) { /* Divisions by 2 preserve the invariant (I). */ MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TA, mbedtls_mpi_lsb(&TA))); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TB, mbedtls_mpi_lsb(&TB))); /* Set either TA or TB to |TA-TB|/2. Since TA and TB are both odd, * TA-TB is even so the division by 2 has an integer result. * Invariant (I) is preserved since any odd divisor of both TA and TB * also divides |TA-TB|/2, and any odd divisor of both TA and |TA-TB|/2 * also divides TB, and any odd divisor of both TB and |TA-TB|/2 also * divides TA. */ if (mbedtls_mpi_cmp_mpi(&TA, &TB) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(&TA, &TA, &TB)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TA, 1)); } else { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(&TB, &TB, &TA)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TB, 1)); } /* Note that one of TA or TB is still odd. */ } /* By invariant (I), gcd(A,B) = 2^k * gcd(TA,TB) for some k. * At the loop exit, TA = 0, so gcd(TA,TB) = TB. * - If there was at least one loop iteration, then one of TA or TB is odd, * and TA = 0, so TB is odd and gcd(TA,TB) = gcd(A',B'). In this case, * lz = min(a,b) so gcd(A,B) = 2^lz * TB. * - If there was no loop iteration, then A was 0, and gcd(A,B) = B. * In this case, lz = 0 and B = TB so gcd(A,B) = B = 2^lz * TB as well. */ MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&TB, lz)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(G, &TB)); cleanup: mbedtls_mpi_free(&TA); mbedtls_mpi_free(&TB); return ret; } /* * Fill X with size bytes of random. * The bytes returned from the RNG are used in a specific order which * is suitable for deterministic ECDSA (see the specification of * mbedtls_mpi_random() and the implementation in mbedtls_mpi_fill_random()). */ int mbedtls_mpi_fill_random(mbedtls_mpi *X, size_t size, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; const size_t limbs = CHARS_TO_LIMBS(size); MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(f_rng != NULL); /* Ensure that target MPI has exactly the necessary number of limbs */ MBEDTLS_MPI_CHK(mbedtls_mpi_resize_clear(X, limbs)); if (size == 0) { return 0; } ret = mbedtls_mpi_core_fill_random(X->p, X->n, size, f_rng, p_rng); cleanup: return ret; } int mbedtls_mpi_random(mbedtls_mpi *X, mbedtls_mpi_sint min, const mbedtls_mpi *N, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { if (min < 0) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } if (mbedtls_mpi_cmp_int(N, min) <= 0) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } /* Ensure that target MPI has exactly the same number of limbs * as the upper bound, even if the upper bound has leading zeros. * This is necessary for mbedtls_mpi_core_random. */ int ret = mbedtls_mpi_resize_clear(X, N->n); if (ret != 0) { return ret; } return mbedtls_mpi_core_random(X->p, min, N->p, X->n, f_rng, p_rng); } /* * Modular inverse: X = A^-1 mod N (HAC 14.61 / 14.64) */ int mbedtls_mpi_inv_mod(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *N) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi G, TA, TU, U1, U2, TB, TV, V1, V2; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(A != NULL); MPI_VALIDATE_RET(N != NULL); if (mbedtls_mpi_cmp_int(N, 1) <= 0) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } mbedtls_mpi_init(&TA); mbedtls_mpi_init(&TU); mbedtls_mpi_init(&U1); mbedtls_mpi_init(&U2); mbedtls_mpi_init(&G); mbedtls_mpi_init(&TB); mbedtls_mpi_init(&TV); mbedtls_mpi_init(&V1); mbedtls_mpi_init(&V2); MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(&G, A, N)); if (mbedtls_mpi_cmp_int(&G, 1) != 0) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; goto cleanup; } MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&TA, A, N)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TU, &TA)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TB, N)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TV, N)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&U1, 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&U2, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&V1, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&V2, 1)); do { while ((TU.p[0] & 1) == 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TU, 1)); if ((U1.p[0] & 1) != 0 || (U2.p[0] & 1) != 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&U1, &U1, &TB)); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U2, &U2, &TA)); } MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&U1, 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&U2, 1)); } while ((TV.p[0] & 1) == 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TV, 1)); if ((V1.p[0] & 1) != 0 || (V2.p[0] & 1) != 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&V1, &V1, &TB)); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V2, &V2, &TA)); } MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&V1, 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&V2, 1)); } if (mbedtls_mpi_cmp_mpi(&TU, &TV) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&TU, &TU, &TV)); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U1, &U1, &V1)); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U2, &U2, &V2)); } else { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&TV, &TV, &TU)); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V1, &V1, &U1)); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V2, &V2, &U2)); } } while (mbedtls_mpi_cmp_int(&TU, 0) != 0); while (mbedtls_mpi_cmp_int(&V1, 0) < 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&V1, &V1, N)); } while (mbedtls_mpi_cmp_mpi(&V1, N) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V1, &V1, N)); } MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, &V1)); cleanup: mbedtls_mpi_free(&TA); mbedtls_mpi_free(&TU); mbedtls_mpi_free(&U1); mbedtls_mpi_free(&U2); mbedtls_mpi_free(&G); mbedtls_mpi_free(&TB); mbedtls_mpi_free(&TV); mbedtls_mpi_free(&V1); mbedtls_mpi_free(&V2); return ret; } #if defined(MBEDTLS_GENPRIME) /* Gaps between primes, starting at 3. https://oeis.org/A001223 */ static const unsigned char small_prime_gaps[] = { 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10, 2, 6, 4, 6, 8, 4, 2, 4, 12, 8, 4, 8, 4, 6, 12, 2, 18, 6, 10, 6, 6, 2, 6, 10, 6, 6, 2, 6, 6, 4, 2, 12, 10, 2, 4, 6, 6, 2, 12, 4, 6, 8, 10, 8, 10, 8, 6, 6, 4, 8, 6, 4, 8, 4, 14, 10, 12, 2, 10, 2, 4, 2, 10, 14, 4, 2, 4, 14, 4, 2, 4, 20, 4, 8, 10, 8, 4, 6, 6, 14, 4, 6, 6, 8, 6, /*reaches 997*/ 0 /* the last entry is effectively unused */ }; /* * Small divisors test (X must be positive) * * Return values: * 0: no small factor (possible prime, more tests needed) * 1: certain prime * MBEDTLS_ERR_MPI_NOT_ACCEPTABLE: certain non-prime * other negative: error */ static int mpi_check_small_factors(const mbedtls_mpi *X) { int ret = 0; size_t i; mbedtls_mpi_uint r; unsigned p = 3; /* The first odd prime */ if ((X->p[0] & 1) == 0) { return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; } for (i = 0; i < sizeof(small_prime_gaps); p += small_prime_gaps[i], i++) { MBEDTLS_MPI_CHK(mbedtls_mpi_mod_int(&r, X, p)); if (r == 0) { if (mbedtls_mpi_cmp_int(X, p) == 0) { return 1; } else { return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; } } } cleanup: return ret; } /* * Miller-Rabin pseudo-primality test (HAC 4.24) */ static int mpi_miller_rabin(const mbedtls_mpi *X, size_t rounds, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret, count; size_t i, j, k, s; mbedtls_mpi W, R, T, A, RR; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(f_rng != NULL); mbedtls_mpi_init(&W); mbedtls_mpi_init(&R); mbedtls_mpi_init(&T); mbedtls_mpi_init(&A); mbedtls_mpi_init(&RR); /* * W = |X| - 1 * R = W >> lsb( W ) */ MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&W, X, 1)); s = mbedtls_mpi_lsb(&W); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R, &W)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&R, s)); for (i = 0; i < rounds; i++) { /* * pick a random A, 1 < A < |X| - 1 */ count = 0; do { MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(&A, X->n * ciL, f_rng, p_rng)); j = mbedtls_mpi_bitlen(&A); k = mbedtls_mpi_bitlen(&W); if (j > k) { A.p[A.n - 1] &= ((mbedtls_mpi_uint) 1 << (k - (A.n - 1) * biL - 1)) - 1; } if (count++ > 30) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; goto cleanup; } } while (mbedtls_mpi_cmp_mpi(&A, &W) >= 0 || mbedtls_mpi_cmp_int(&A, 1) <= 0); /* * A = A^R mod |X| */ MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&A, &A, &R, X, &RR)); if (mbedtls_mpi_cmp_mpi(&A, &W) == 0 || mbedtls_mpi_cmp_int(&A, 1) == 0) { continue; } j = 1; while (j < s && mbedtls_mpi_cmp_mpi(&A, &W) != 0) { /* * A = A * A mod |X| */ MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, &A, &A)); MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&A, &T, X)); if (mbedtls_mpi_cmp_int(&A, 1) == 0) { break; } j++; } /* * not prime if A != |X| - 1 or A == 1 */ if (mbedtls_mpi_cmp_mpi(&A, &W) != 0 || mbedtls_mpi_cmp_int(&A, 1) == 0) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; break; } } cleanup: mbedtls_mpi_free(&W); mbedtls_mpi_free(&R); mbedtls_mpi_free(&T); mbedtls_mpi_free(&A); mbedtls_mpi_free(&RR); return ret; } /* * Pseudo-primality test: small factors, then Miller-Rabin */ int mbedtls_mpi_is_prime_ext(const mbedtls_mpi *X, int rounds, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi XX; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(f_rng != NULL); XX.s = 1; XX.n = X->n; XX.p = X->p; if (mbedtls_mpi_cmp_int(&XX, 0) == 0 || mbedtls_mpi_cmp_int(&XX, 1) == 0) { return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; } if (mbedtls_mpi_cmp_int(&XX, 2) == 0) { return 0; } if ((ret = mpi_check_small_factors(&XX)) != 0) { if (ret == 1) { return 0; } return ret; } return mpi_miller_rabin(&XX, rounds, f_rng, p_rng); } /* * Prime number generation * * To generate an RSA key in a way recommended by FIPS 186-4, both primes must * be either 1024 bits or 1536 bits long, and flags must contain * MBEDTLS_MPI_GEN_PRIME_FLAG_LOW_ERR. */ int mbedtls_mpi_gen_prime(mbedtls_mpi *X, size_t nbits, int flags, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { #ifdef MBEDTLS_HAVE_INT64 // ceil(2^63.5) #define CEIL_MAXUINT_DIV_SQRT2 0xb504f333f9de6485ULL #else // ceil(2^31.5) #define CEIL_MAXUINT_DIV_SQRT2 0xb504f334U #endif int ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; size_t k, n; int rounds; mbedtls_mpi_uint r; mbedtls_mpi Y; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(f_rng != NULL); if (nbits < 3 || nbits > MBEDTLS_MPI_MAX_BITS) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } mbedtls_mpi_init(&Y); n = BITS_TO_LIMBS(nbits); if ((flags & MBEDTLS_MPI_GEN_PRIME_FLAG_LOW_ERR) == 0) { /* * 2^-80 error probability, number of rounds chosen per HAC, table 4.4 */ rounds = ((nbits >= 1300) ? 2 : (nbits >= 850) ? 3 : (nbits >= 650) ? 4 : (nbits >= 350) ? 8 : (nbits >= 250) ? 12 : (nbits >= 150) ? 18 : 27); } else { /* * 2^-100 error probability, number of rounds computed based on HAC, * fact 4.48 */ rounds = ((nbits >= 1450) ? 4 : (nbits >= 1150) ? 5 : (nbits >= 1000) ? 6 : (nbits >= 850) ? 7 : (nbits >= 750) ? 8 : (nbits >= 500) ? 13 : (nbits >= 250) ? 28 : (nbits >= 150) ? 40 : 51); } while (1) { MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(X, n * ciL, f_rng, p_rng)); /* make sure generated number is at least (nbits-1)+0.5 bits (FIPS 186-4 §B.3.3 steps 4.4, 5.5) */ if (X->p[n-1] < CEIL_MAXUINT_DIV_SQRT2) { continue; } k = n * biL; if (k > nbits) { MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(X, k - nbits)); } X->p[0] |= 1; if ((flags & MBEDTLS_MPI_GEN_PRIME_FLAG_DH) == 0) { ret = mbedtls_mpi_is_prime_ext(X, rounds, f_rng, p_rng); if (ret != MBEDTLS_ERR_MPI_NOT_ACCEPTABLE) { goto cleanup; } } else { /* * A necessary condition for Y and X = 2Y + 1 to be prime * is X = 2 mod 3 (which is equivalent to Y = 2 mod 3). * Make sure it is satisfied, while keeping X = 3 mod 4 */ X->p[0] |= 2; MBEDTLS_MPI_CHK(mbedtls_mpi_mod_int(&r, X, 3)); if (r == 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(X, X, 8)); } else if (r == 1) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(X, X, 4)); } /* Set Y = (X-1) / 2, which is X / 2 because X is odd */ MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&Y, X)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&Y, 1)); while (1) { /* * First, check small factors for X and Y * before doing Miller-Rabin on any of them */ if ((ret = mpi_check_small_factors(X)) == 0 && (ret = mpi_check_small_factors(&Y)) == 0 && (ret = mpi_miller_rabin(X, rounds, f_rng, p_rng)) == 0 && (ret = mpi_miller_rabin(&Y, rounds, f_rng, p_rng)) == 0) { goto cleanup; } if (ret != MBEDTLS_ERR_MPI_NOT_ACCEPTABLE) { goto cleanup; } /* * Next candidates. We want to preserve Y = (X-1) / 2 and * Y = 1 mod 2 and Y = 2 mod 3 (eq X = 3 mod 4 and X = 2 mod 3) * so up Y by 6 and X by 12. */ MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(X, X, 12)); MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&Y, &Y, 6)); } } } cleanup: mbedtls_mpi_free(&Y); return ret; } #endif /* MBEDTLS_GENPRIME */ #if defined(MBEDTLS_SELF_TEST) #define GCD_PAIR_COUNT 3 static const int gcd_pairs[GCD_PAIR_COUNT][3] = { { 693, 609, 21 }, { 1764, 868, 28 }, { 768454923, 542167814, 1 } }; /* * Checkup routine */ int mbedtls_mpi_self_test(int verbose) { int ret, i; mbedtls_mpi A, E, N, X, Y, U, V; mbedtls_mpi_init(&A); mbedtls_mpi_init(&E); mbedtls_mpi_init(&N); mbedtls_mpi_init(&X); mbedtls_mpi_init(&Y); mbedtls_mpi_init(&U); mbedtls_mpi_init(&V); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&A, 16, "EFE021C2645FD1DC586E69184AF4A31E" \ "D5F53E93B5F123FA41680867BA110131" \ "944FE7952E2517337780CB0DB80E61AA" \ "E7C8DDC6C5C6AADEB34EB38A2F40D5E6")); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&E, 16, "B2E7EFD37075B9F03FF989C7C5051C20" \ "34D2A323810251127E7BF8625A4F49A5" \ "F3E27F4DA8BD59C47D6DAABA4C8127BD" \ "5B5C25763222FEFCCFC38B832366C29E")); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&N, 16, "0066A198186C18C10B2F5ED9B522752A" \ "9830B69916E535C8F047518A889A43A5" \ "94B6BED27A168D31D4A52F88925AA8F5")); MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&X, &A, &N)); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&U, 16, "602AB7ECA597A3D6B56FF9829A5E8B85" \ "9E857EA95A03512E2BAE7391688D264A" \ "A5663B0341DB9CCFD2C4C5F421FEC814" \ "8001B72E848A38CAE1C65F78E56ABDEF" \ "E12D3C039B8A02D6BE593F0BBBDA56F1" \ "ECF677152EF804370C1A305CAF3B5BF1" \ "30879B56C61DE584A0F53A2447A51E")); if (verbose != 0) { mbedtls_printf(" MPI test #1 (mul_mpi): "); } if (mbedtls_mpi_cmp_mpi(&X, &U) != 0) { if (verbose != 0) { mbedtls_printf("failed\n"); } ret = 1; goto cleanup; } if (verbose != 0) { mbedtls_printf("passed\n"); } MBEDTLS_MPI_CHK(mbedtls_mpi_div_mpi(&X, &Y, &A, &N)); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&U, 16, "256567336059E52CAE22925474705F39A94")); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&V, 16, "6613F26162223DF488E9CD48CC132C7A" \ "0AC93C701B001B092E4E5B9F73BCD27B" \ "9EE50D0657C77F374E903CDFA4C642")); if (verbose != 0) { mbedtls_printf(" MPI test #2 (div_mpi): "); } if (mbedtls_mpi_cmp_mpi(&X, &U) != 0 || mbedtls_mpi_cmp_mpi(&Y, &V) != 0) { if (verbose != 0) { mbedtls_printf("failed\n"); } ret = 1; goto cleanup; } if (verbose != 0) { mbedtls_printf("passed\n"); } MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(&X, &A, &E, &N, NULL)); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&U, 16, "36E139AEA55215609D2816998ED020BB" \ "BD96C37890F65171D948E9BC7CBAA4D9" \ "325D24D6A3C12710F10A09FA08AB87")); if (verbose != 0) { mbedtls_printf(" MPI test #3 (exp_mod): "); } if (mbedtls_mpi_cmp_mpi(&X, &U) != 0) { if (verbose != 0) { mbedtls_printf("failed\n"); } ret = 1; goto cleanup; } if (verbose != 0) { mbedtls_printf("passed\n"); } MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(&X, &A, &N)); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&U, 16, "003A0AAEDD7E784FC07D8F9EC6E3BFD5" \ "C3DBA76456363A10869622EAC2DD84EC" \ "C5B8A74DAC4D09E03B5E0BE779F2DF61")); if (verbose != 0) { mbedtls_printf(" MPI test #4 (inv_mod): "); } if (mbedtls_mpi_cmp_mpi(&X, &U) != 0) { if (verbose != 0) { mbedtls_printf("failed\n"); } ret = 1; goto cleanup; } if (verbose != 0) { mbedtls_printf("passed\n"); } if (verbose != 0) { mbedtls_printf(" MPI test #5 (simple gcd): "); } for (i = 0; i < GCD_PAIR_COUNT; i++) { MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&X, gcd_pairs[i][0])); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&Y, gcd_pairs[i][1])); MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(&A, &X, &Y)); if (mbedtls_mpi_cmp_int(&A, gcd_pairs[i][2]) != 0) { if (verbose != 0) { mbedtls_printf("failed at %d\n", i); } ret = 1; goto cleanup; } } if (verbose != 0) { mbedtls_printf("passed\n"); } cleanup: if (ret != 0 && verbose != 0) { mbedtls_printf("Unexpected error, return code = %08X\n", (unsigned int) ret); } mbedtls_mpi_free(&A); mbedtls_mpi_free(&E); mbedtls_mpi_free(&N); mbedtls_mpi_free(&X); mbedtls_mpi_free(&Y); mbedtls_mpi_free(&U); mbedtls_mpi_free(&V); if (verbose != 0) { mbedtls_printf("\n"); } return ret; } #endif /* MBEDTLS_SELF_TEST */ #endif /* MBEDTLS_BIGNUM_C */