/* * 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 "mbedtls/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) #define ciL (sizeof(mbedtls_mpi_uint)) /* chars in limb */ #define biL (ciL << 3) /* bits in limb */ #define biH (ciL << 2) /* half limb size */ #define MPI_SIZE_T_MAX ((size_t) -1) /* SIZE_T_MAX is not standard */ /* * Convert between bits/chars and number of limbs * Divide first in order to avoid potential overflows */ #define BITS_TO_LIMBS(i) ((i) / biL + ((i) % biL != 0)) #define CHARS_TO_LIMBS(i) ((i) / ciL + ((i) % ciL != 0)) /* Implementation that should never be optimized out by the compiler */ static void mbedtls_mpi_zeroize(mbedtls_mpi_uint *v, size_t n) { mbedtls_platform_zeroize(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(X->p, X->n); mbedtls_free(X->p); } 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(X->p, X->n); mbedtls_free(X->p); } X->n = 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(X->p, X->n); mbedtls_free(X->p); } X->n = 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; } /* * 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 = (z < 0) ? -1 : 1; 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; } /* Get a specific byte, without range checks. */ #define GET_BYTE(X, i) \ (((X)->p[(i) / ciL] >> (((i) % ciL) * 8)) & 0xff) /* * 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, j, count = 0; MBEDTLS_INTERNAL_VALIDATE_RET(X != NULL, 0); for (i = 0; i < X->n; i++) { for (j = 0; j < biL; j++, count++) { if (((X->p[i] >> j) & 1) != 0) { return count; } } } return 0; } /* * Count leading zero bits in a given integer */ static size_t mbedtls_clz(const mbedtls_mpi_uint x) { size_t j; mbedtls_mpi_uint mask = (mbedtls_mpi_uint) 1 << (biL - 1); for (j = 0; j < biL; j++) { if (x & mask) { break; } mask >>= 1; } return j; } /* * Return the number of bits */ size_t mbedtls_mpi_bitlen(const mbedtls_mpi *X) { size_t i, j; if (X->n == 0) { return 0; } for (i = X->n - 1; i > 0; i--) { if (X->p[i] != 0) { break; } } j = biL - mbedtls_clz(X->p[i]); return (i * biL) + j; } /* * 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 > MPI_SIZE_T_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 */ /* Convert a big-endian byte array aligned to the size of mbedtls_mpi_uint * into the storage form used by mbedtls_mpi. */ static mbedtls_mpi_uint mpi_uint_bigendian_to_host_c(mbedtls_mpi_uint x) { uint8_t i; unsigned char *x_ptr; mbedtls_mpi_uint tmp = 0; for (i = 0, x_ptr = (unsigned char *) &x; i < ciL; i++, x_ptr++) { tmp <<= CHAR_BIT; tmp |= (mbedtls_mpi_uint) *x_ptr; } return tmp; } static mbedtls_mpi_uint mpi_uint_bigendian_to_host(mbedtls_mpi_uint x) { #if defined(__BYTE_ORDER__) /* Nothing to do on bigendian systems. */ #if (__BYTE_ORDER__ == __ORDER_BIG_ENDIAN__) return x; #endif /* __BYTE_ORDER__ == __ORDER_BIG_ENDIAN__ */ #if (__BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__) /* For GCC and Clang, have builtins for byte swapping. */ #if defined(__GNUC__) && defined(__GNUC_PREREQ) #if __GNUC_PREREQ(4, 3) #define have_bswap #endif #endif #if defined(__clang__) && defined(__has_builtin) #if __has_builtin(__builtin_bswap32) && \ __has_builtin(__builtin_bswap64) #define have_bswap #endif #endif #if defined(have_bswap) /* The compiler is hopefully able to statically evaluate this! */ switch (sizeof(mbedtls_mpi_uint)) { case 4: return __builtin_bswap32(x); case 8: return __builtin_bswap64(x); } #endif #endif /* __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__ */ #endif /* __BYTE_ORDER__ */ /* Fall back to C-based reordering if we don't know the byte order * or we couldn't use a compiler-specific builtin. */ return mpi_uint_bigendian_to_host_c(x); } static void mpi_bigendian_to_host(mbedtls_mpi_uint * const p, size_t limbs) { mbedtls_mpi_uint *cur_limb_left; mbedtls_mpi_uint *cur_limb_right; if (limbs == 0) { return; } /* * Traverse limbs and * - adapt byte-order in each limb * - swap the limbs themselves. * For that, simultaneously traverse the limbs from left to right * and from right to left, as long as the left index is not bigger * than the right index (it's not a problem if limbs is odd and the * indices coincide in the last iteration). */ for (cur_limb_left = p, cur_limb_right = p + (limbs - 1); cur_limb_left <= cur_limb_right; cur_limb_left++, cur_limb_right--) { mbedtls_mpi_uint tmp; /* Note that if cur_limb_left == cur_limb_right, * this code effectively swaps the bytes only once. */ tmp = mpi_uint_bigendian_to_host(*cur_limb_left); *cur_limb_left = mpi_uint_bigendian_to_host(*cur_limb_right); *cur_limb_right = tmp; } } /* * Import X from unsigned binary data, little endian */ int mbedtls_mpi_read_binary_le(mbedtls_mpi *X, const unsigned char *buf, size_t buflen) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i; size_t const 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)); for (i = 0; i < buflen; i++) { X->p[i / ciL] |= ((mbedtls_mpi_uint) buf[i]) << ((i % ciL) << 3); } 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 */ int mbedtls_mpi_read_binary(mbedtls_mpi *X, const unsigned char *buf, size_t buflen) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t const limbs = CHARS_TO_LIMBS(buflen); size_t const overhead = (limbs * ciL) - buflen; unsigned char *Xp; 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)); /* Avoid calling `memcpy` with NULL source or destination argument, * even if buflen is 0. */ if (buflen != 0) { Xp = (unsigned char *) X->p; memcpy(Xp + overhead, buf, buflen); mpi_bigendian_to_host(X->p, limbs); } 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) { size_t stored_bytes = X->n * ciL; size_t bytes_to_copy; size_t i; if (stored_bytes < buflen) { bytes_to_copy = stored_bytes; } else { bytes_to_copy = buflen; /* The output buffer is smaller than the allocated size of X. * However X may fit if its leading bytes are zero. */ for (i = bytes_to_copy; i < stored_bytes; i++) { if (GET_BYTE(X, i) != 0) { return MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL; } } } for (i = 0; i < bytes_to_copy; i++) { buf[i] = GET_BYTE(X, i); } if (stored_bytes < buflen) { /* Write trailing 0 bytes */ memset(buf + stored_bytes, 0, buflen - stored_bytes); } return 0; } /* * Export X into unsigned binary data, big endian */ int mbedtls_mpi_write_binary(const mbedtls_mpi *X, unsigned char *buf, size_t buflen) { size_t stored_bytes; size_t bytes_to_copy; unsigned char *p; size_t i; MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(buflen == 0 || buf != NULL); stored_bytes = X->n * ciL; if (stored_bytes < buflen) { /* There is enough space in the output buffer. Write initial * null bytes and record the position at which to start * writing the significant bytes. In this case, the execution * trace of this function does not depend on the value of the * number. */ bytes_to_copy = stored_bytes; p = buf + buflen - stored_bytes; memset(buf, 0, buflen - stored_bytes); } else { /* The output buffer is smaller than the allocated size of X. * However X may fit if its leading bytes are zero. */ bytes_to_copy = buflen; p = buf; for (i = bytes_to_copy; i < stored_bytes; i++) { if (GET_BYTE(X, i) != 0) { return MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL; } } } for (i = 0; i < bytes_to_copy; i++) { p[bytes_to_copy - i - 1] = GET_BYTE(X, i); } return 0; } /* * 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, v0, t1; mbedtls_mpi_uint r0 = 0, r1; MPI_VALIDATE_RET(X != NULL); v0 = count / (biL); t1 = count & (biL - 1); i = mbedtls_mpi_bitlen(X) + count; if (X->n * biL < i) { MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, BITS_TO_LIMBS(i))); } ret = 0; /* * shift by count / limb_size */ if (v0 > 0) { for (i = X->n; i > v0; i--) { X->p[i - 1] = X->p[i - v0 - 1]; } for (; i > 0; i--) { X->p[i - 1] = 0; } } /* * shift by count % limb_size */ if (t1 > 0) { for (i = v0; i < X->n; i++) { r1 = X->p[i] >> (biL - t1); X->p[i] <<= t1; X->p[i] |= r0; r0 = r1; } } cleanup: return ret; } /* * Right-shift: X >>= count */ int mbedtls_mpi_shift_r(mbedtls_mpi *X, size_t count) { size_t i, v0, v1; mbedtls_mpi_uint r0 = 0, r1; MPI_VALIDATE_RET(X != NULL); v0 = count / biL; v1 = count & (biL - 1); if (v0 > X->n || (v0 == X->n && v1 > 0)) { return mbedtls_mpi_lset(X, 0); } /* * shift by count / limb_size */ if (v0 > 0) { for (i = 0; i < X->n - v0; i++) { X->p[i] = X->p[i + v0]; } for (; i < X->n; i++) { X->p[i] = 0; } } /* * shift by count % limb_size */ if (v1 > 0) { for (i = X->n; i > 0; i--) { r1 = X->p[i - 1] << (biL - v1); X->p[i - 1] >>= v1; X->p[i - 1] |= r0; r0 = r1; } } 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 == 0 && j == 0) { return 0; } 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 = (z < 0) ? -1 : 1; 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 i, j; mbedtls_mpi_uint *o, *p, c, tmp; 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 should 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)); o = B->p; p = X->p; c = 0; /* * tmp is used because it might happen that p == o */ for (i = 0; i < j; i++, o++, p++) { tmp = *o; *p += c; c = (*p < c); *p += tmp; c += (*p < tmp); } while (c != 0) { if (i >= X->n) { MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, i + 1)); p = X->p + i; } *p += c; c = (*p < c); i++; p++; } cleanup: return ret; } /** * Helper for mbedtls_mpi subtraction. * * Calculate l - r where l and r have the same size. * This function operates modulo (2^ciL)^n and returns the carry * (1 if there was a wraparound, i.e. if `l < r`, and 0 otherwise). * * d may be aliased to l or r. * * \param n Number of limbs of \p d, \p l and \p r. * \param[out] d The result of the subtraction. * \param[in] l The left operand. * \param[in] r The right operand. * * \return 1 if `l < r`. * 0 if `l >= r`. */ static mbedtls_mpi_uint mpi_sub_hlp(size_t n, mbedtls_mpi_uint *d, const mbedtls_mpi_uint *l, const mbedtls_mpi_uint *r) { size_t i; mbedtls_mpi_uint c = 0, t, z; for (i = 0; i < n; i++) { z = (l[i] < c); t = l[i] - c; c = (t < r[i]) + z; d[i] = t - r[i]; } return c; } /* * 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 = mpi_sub_hlp(n, X->p, A->p, B->p); if (carry != 0) { /* Propagate the carry to the first nonzero limb of X. */ for (; n < X->n && X->p[n] == 0; n++) { --X->p[n]; } /* If we ran out of space for the carry, it means that the result * is negative. */ if (n == X->n) { ret = MBEDTLS_ERR_MPI_NEGATIVE_VALUE; goto cleanup; } --X->p[n]; } /* 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 = (b < 0) ? -1 : 1; 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 = (b < 0) ? -1 : 1; B.n = 1; B.p = p; return mbedtls_mpi_sub_mpi(X, A, &B); } /** Helper for mbedtls_mpi multiplication. * * Add \p b * \p s to \p d. * * \param i The number of limbs of \p s. * \param[in] s A bignum to multiply, of size \p i. * It may overlap with \p d, but only if * \p d <= \p s. * Its leading limb must not be \c 0. * \param[in,out] d The bignum to add to. * It must be sufficiently large to store the * result of the multiplication. This means * \p i + 1 limbs if \p d[\p i - 1] started as 0 and \p b * is not known a priori. * \param b A scalar to multiply. */ static #if defined(__APPLE__) && defined(__arm__) /* * Apple LLVM version 4.2 (clang-425.0.24) (based on LLVM 3.2svn) * appears to need this to prevent bad ARM code generation at -O3. */ __attribute__((noinline)) #endif void mpi_mul_hlp(size_t i, const mbedtls_mpi_uint *s, mbedtls_mpi_uint *d, mbedtls_mpi_uint b) { mbedtls_mpi_uint c = 0, t = 0; (void) t; /* Unused in some architectures */ #if defined(MULADDC_HUIT) for (; i >= 8; i -= 8) { MULADDC_INIT MULADDC_HUIT MULADDC_STOP } for (; i > 0; i--) { MULADDC_INIT MULADDC_CORE MULADDC_STOP } #else /* MULADDC_HUIT */ for (; i >= 16; i -= 16) { MULADDC_INIT MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_STOP } for (; i >= 8; i -= 8) { MULADDC_INIT MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_STOP } for (; i > 0; i--) { MULADDC_INIT MULADDC_CORE MULADDC_STOP } #endif /* MULADDC_HUIT */ while (c != 0) { *d += c; c = (*d < c); d++; } } /* * 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)); for (; j > 0; j--) { mpi_mul_hlp(i, A->p, X->p + j - 1, B->p[j - 1]); } /* 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); /* mpi_mul_hlp can't deal with a leading 0. */ size_t n = A->n; while (n > 0 && A->p[n - 1] == 0) { --n; } /* The general method below doesn't work if n==0 or b==0. By chance * calculating the result is trivial in those cases. */ if (b == 0 || n == 0) { return mbedtls_mpi_lset(X, 0); } /* Calculate A*b as A + A*(b-1) to take advantage of mpi_mul_hlp */ 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. */ MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, n + 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A)); mpi_mul_hlp(n, A->p, X->p, 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_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 = (b < 0) ? -1 : 1; 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; } /* * Fast Montgomery initialization (thanks to Tom St Denis) */ static void mpi_montg_init(mbedtls_mpi_uint *mm, const mbedtls_mpi *N) { mbedtls_mpi_uint x, m0 = N->p[0]; unsigned int i; x = m0; x += ((m0 + 2) & 4) << 1; for (i = biL; i >= 8; i /= 2) { x *= (2 - (m0 * x)); } *mm = ~x + 1; } /** 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 modulo. 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 2 * (T->n >= 2 * (N->n + 1)). * Its initial content is unused and * its final content is indeterminate. * Note that unlike the usual convention in the library * for `const mbedtls_mpi*`, the content of T can change. */ static void mpi_montmul(mbedtls_mpi *A, const mbedtls_mpi *B, const mbedtls_mpi *N, mbedtls_mpi_uint mm, const mbedtls_mpi *T) { size_t i, n, m; mbedtls_mpi_uint u0, u1, *d; memset(T->p, 0, T->n * ciL); d = T->p; n = N->n; m = (B->n < n) ? B->n : n; for (i = 0; i < n; i++) { /* * T = (T + u0*B + u1*N) / 2^biL */ u0 = A->p[i]; u1 = (d[0] + u0 * B->p[0]) * mm; mpi_mul_hlp(m, B->p, d, u0); mpi_mul_hlp(n, N->p, d, u1); *d++ = u0; d[n + 1] = 0; } /* At this point, d is either the desired result or the desired result * plus N. We now potentially subtract N, avoiding leaking whether the * subtraction is performed through side channels. */ /* Copy the n least significant limbs of d to A, so that * A = d if d < N (recall that N has n limbs). */ memcpy(A->p, d, n * ciL); /* If d >= N then we want to set A to d - N. To prevent timing attacks, * do the calculation without using conditional tests. */ /* Set d to d0 + (2^biL)^n - N where d0 is the current value of d. */ d[n] += 1; d[n] -= mpi_sub_hlp(n, d, d, N->p); /* If d0 < N then d < (2^biL)^n * so d[n] == 0 and we want to keep A as it is. * If d0 >= N then d >= (2^biL)^n, and d <= (2^biL)^n + N < 2 * (2^biL)^n * so d[n] == 1 and we want to set A to the result of the subtraction * which is d - (2^biL)^n, i.e. the n least significant limbs of d. * This exactly corresponds to a conditional assignment. */ mbedtls_ct_mpi_uint_cond_assign(n, A->p, d, (unsigned char) d[n]); } /* * 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, const mbedtls_mpi *T) { mbedtls_mpi_uint z = 1; mbedtls_mpi U; U.n = U.s = (int) z; 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_size_bool_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 X in the table as * well, and also do a constant-trace table lookup to multiply by X. * * 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. * * To achieve this, we make a copy of X and we use the table entry in each * calculation from this point on. */ const size_t x_index = 0; mbedtls_mpi_init(&W[x_index]); mbedtls_mpi_copy(&W[x_index], X); j = N->n + 1; /* All W[i] and X must have at least N->n limbs for the mpi_montmul() * and mpi_montred() calls later. Here we ensure that W[1] and X are * large enough, and 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_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 n_bytes random bytes. * X must already have room for those bytes. * The ordering of the bytes returned from the RNG is suitable for * deterministic ECDSA (see RFC 6979 §3.3 and mbedtls_mpi_random()). * The size and sign of X are unchanged. * n_bytes must not be 0. */ static int mpi_fill_random_internal( mbedtls_mpi *X, size_t n_bytes, 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(n_bytes); const size_t overhead = (limbs * ciL) - n_bytes; if (X->n < limbs) { return MBEDTLS_ERR_MPI_BAD_INPUT_DATA; } memset(X->p, 0, overhead); memset((unsigned char *) X->p + limbs * ciL, 0, (X->n - limbs) * ciL); MBEDTLS_MPI_CHK(f_rng(p_rng, (unsigned char *) X->p + overhead, n_bytes)); mpi_bigendian_to_host(X->p, limbs); cleanup: return ret; } /* * Fill X with size bytes of random. * * Use a temporary bytes representation to make sure the result is the same * regardless of the platform endianness (useful when f_rng is actually * deterministic, eg for tests). */ 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; size_t const 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 = mpi_fill_random_internal(X, 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) { int ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; int count; unsigned lt_lower = 1, lt_upper = 0; size_t n_bits = mbedtls_mpi_bitlen(N); size_t n_bytes = (n_bits + 7) / 8; mbedtls_mpi lower_bound; 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; } /* * When min == 0, each try has at worst a probability 1/2 of failing * (the msb has a probability 1/2 of being 0, and then the result will * be < N), so after 30 tries failure probability is a most 2**(-30). * * When N is just below a power of 2, as is the case when generating * a random scalar on most elliptic curves, 1 try is enough with * overwhelming probability. When N is just above a power of 2, * as when generating a random scalar on secp224k1, each try has * a probability of failing that is almost 1/2. * * The probabilities are almost the same if min is nonzero but negligible * compared to N. This is always the case when N is crypto-sized, but * it's convenient to support small N for testing purposes. When N * is small, use a higher repeat count, otherwise the probability of * failure is macroscopic. */ count = (n_bytes > 4 ? 30 : 250); mbedtls_mpi_init(&lower_bound); /* 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 the mbedtls_mpi_lt_mpi_ct() check. */ MBEDTLS_MPI_CHK(mbedtls_mpi_resize_clear(X, N->n)); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&lower_bound, N->n)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&lower_bound, min)); /* * Match the procedure given in RFC 6979 §3.3 (deterministic ECDSA) * when f_rng is a suitably parametrized instance of HMAC_DRBG: * - use the same byte ordering; * - keep the leftmost n_bits bits of the generated octet string; * - try until result is in the desired range. * This also avoids any bias, which is especially important for ECDSA. */ do { MBEDTLS_MPI_CHK(mpi_fill_random_internal(X, n_bytes, f_rng, p_rng)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(X, 8 * n_bytes - n_bits)); if (--count == 0) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; goto cleanup; } MBEDTLS_MPI_CHK(mbedtls_mpi_lt_mpi_ct(X, &lower_bound, <_lower)); MBEDTLS_MPI_CHK(mbedtls_mpi_lt_mpi_ct(X, N, <_upper)); } while (lt_lower != 0 || lt_upper == 0); cleanup: mbedtls_mpi_free(&lower_bound); return ret; } /* * 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) static const int small_prime[] = { 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, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, -103 }; /* * 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; if ((X->p[0] & 1) == 0) { return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; } for (i = 0; small_prime[i] > 0; i++) { if (mbedtls_mpi_cmp_int(X, small_prime[i]) <= 0) { return 1; } MBEDTLS_MPI_CHK(mbedtls_mpi_mod_int(&r, X, small_prime[i])); if (r == 0) { 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); } #if !defined(MBEDTLS_DEPRECATED_REMOVED) /* * Pseudo-primality test, error probability 2^-80 */ int mbedtls_mpi_is_prime(const mbedtls_mpi *X, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { MPI_VALIDATE_RET(X != NULL); MPI_VALIDATE_RET(f_rng != NULL); /* * In the past our key generation aimed for an error rate of at most * 2^-80. Since this function is deprecated, aim for the same certainty * here as well. */ return mbedtls_mpi_is_prime_ext(X, 40, f_rng, p_rng); } #endif /* * 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 */