/* * Multi-precision integer library * * Copyright (C) 2006-2015, ARM Limited, All Rights Reserved * SPDX-License-Identifier: GPL-2.0 * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License along * with this program; if not, write to the Free Software Foundation, Inc., * 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. * * This file is part of mbed TLS (https://tls.mbed.org) */ /* * 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 * */ #if !defined(MBEDTLS_CONFIG_FILE) #include "mbedtls/config.h" #else #include MBEDTLS_CONFIG_FILE #endif #if defined(MBEDTLS_BIGNUM_C) #include "mbedtls/bignum.h" #include "mbedtls/bn_mul.h" #include "mbedtls/platform_util.h" #include #if defined(MBEDTLS_PLATFORM_C) #include "mbedtls/platform.h" #else #include #include #define mbedtls_printf printf #define mbedtls_calloc calloc #define mbedtls_free free #endif #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) { if (X == NULL) return; 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; 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; /* Actually resize up in this case */ if (X->n <= nblimbs) return (mbedtls_mpi_grow(X, nblimbs)); 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); } /* * Copy the contents of Y into X */ int mbedtls_mpi_copy(mbedtls_mpi *X, const mbedtls_mpi *Y) { int ret = 0; size_t i; if (X == Y) return (0); if (Y->p == NULL) { mbedtls_mpi_free(X); 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; memcpy(&T, X, sizeof(mbedtls_mpi)); memcpy(X, Y, sizeof(mbedtls_mpi)); memcpy(Y, &T, sizeof(mbedtls_mpi)); } /* * 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.) */ int mbedtls_mpi_safe_cond_assign(mbedtls_mpi *X, const mbedtls_mpi *Y, unsigned char assign) { int ret = 0; size_t i; /* make sure assign is 0 or 1 in a time-constant manner */ assign = (assign | (unsigned char) - assign) >> 7; MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, Y->n)); X->s = X->s * (1 - assign) + Y->s * assign; for (i = 0; i < Y->n; i++) X->p[i] = X->p[i] * (1 - assign) + Y->p[i] * assign; for (; i < X->n; i++) X->p[i] *= (1 - assign); 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 whould 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, s; size_t i; mbedtls_mpi_uint tmp; if (X == Y) return (0); /* make sure swap is 0 or 1 in a time-constant manner */ swap = (swap | (unsigned char) - swap) >> 7; MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, Y->n)); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(Y, X->n)); s = X->s; X->s = X->s * (1 - swap) + Y->s * swap; Y->s = Y->s * (1 - swap) + s * swap; for (i = 0; i < X->n; i++) { tmp = X->p[i]; X->p[i] = X->p[i] * (1 - swap) + Y->p[i] * swap; Y->p[i] = Y->p[i] * (1 - swap) + tmp * swap; } cleanup: return (ret); } /* * Set value from integer */ int mbedtls_mpi_lset(mbedtls_mpi *X, mbedtls_mpi_sint z) { int ret; MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, 1)); memset(X->p, 0, X->n * ciL); X->p[0] = (z < 0) ? -z : 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) { 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; 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; 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; size_t i, j, slen, n; mbedtls_mpi_uint d; mbedtls_mpi T; if (radix < 2 || radix > 16) return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA); mbedtls_mpi_init(&T); 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++) { if (i == 1 && s[i - 1] == '-') { X->s = -1; break; } 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++) { if (i == 0 && s[i] == '-') { X->s = -1; continue; } MBEDTLS_MPI_CHK(mpi_get_digit(&d, radix, s[i])); MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&T, X, radix)); if (X->s == 1) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(X, &T, d)); } else { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(X, &T, d)); } } } 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) { int ret; mbedtls_mpi_uint r; if (radix < 2 || radix > 16) return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA); MBEDTLS_MPI_CHK(mbedtls_mpi_mod_int(&r, X, radix)); MBEDTLS_MPI_CHK(mbedtls_mpi_div_int(X, NULL, X, radix)); if (mbedtls_mpi_cmp_int(X, 0) != 0) MBEDTLS_MPI_CHK(mpi_write_hlp(X, radix, p)); if (r < 10) *(*p)++ = (char)(r + 0x30); else *(*p)++ = (char)(r + 0x37); 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; if (radix < 2 || radix > 16) return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA); n = mbedtls_mpi_bitlen(X); if (radix >= 4) n >>= 1; if (radix >= 16) n >>= 1; /* * Round up the buffer length to an even value to ensure that there is * enough room for hexadecimal values that can be represented in an odd * number of digits. */ n += 3 + ((n + 1) & 1); if (buflen < n) { *olen = n; return (MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL); } p = buf; mbedtls_mpi_init(&T); if (X->s == -1) *p++ = '-'; 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)); } *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 ]; 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; 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 ]; 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, big endian */ int mbedtls_mpi_read_binary(mbedtls_mpi *X, const unsigned char *buf, size_t buflen) { int ret; size_t i, j; size_t const limbs = CHARS_TO_LIMBS(buflen); /* Ensure that target MPI has exactly the necessary number of limbs */ if (X->n != limbs) { mbedtls_mpi_free(X); mbedtls_mpi_init(X); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, limbs)); } MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, 0)); for (i = buflen, j = 0; i > 0; i--, j++) X->p[j / ciL] |= ((mbedtls_mpi_uint) buf[i - 1]) << ((j % ciL) << 3); cleanup: return (ret); } /* * 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 i, j, n; n = mbedtls_mpi_size(X); if (buflen < n) return (MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL); memset(buf, 0, buflen); for (i = buflen - 1, j = 0; n > 0; i--, j++, n--) buf[i] = (unsigned char)(X->p[j / ciL] >> ((j % ciL) << 3)); return (0); } /* * Left-shift: X <<= count */ int mbedtls_mpi_shift_l(mbedtls_mpi *X, size_t count) { int ret; size_t i, v0, t1; mbedtls_mpi_uint r0 = 0, r1; 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; 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; 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; 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]; *p = (z < 0) ? -z : 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; size_t i, j; mbedtls_mpi_uint *o, *p, c, tmp; 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; 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 */ static void mpi_sub_hlp(size_t n, mbedtls_mpi_uint *s, mbedtls_mpi_uint *d) { size_t i; mbedtls_mpi_uint c, z; for (i = c = 0; i < n; i++, s++, d++) { z = (*d < c); *d -= c; c = (*d < *s) + z; *d -= *s; } while (c != 0) { z = (*d < c); *d -= c; c = z; d++; } } /* * Unsigned subtraction: X = |A| - |B| (HAC 14.9) */ int mbedtls_mpi_sub_abs(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { mbedtls_mpi TB; int ret; size_t n; if (mbedtls_mpi_cmp_abs(A, B) < 0) return (MBEDTLS_ERR_MPI_NEGATIVE_VALUE); mbedtls_mpi_init(&TB); if (X == B) { MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TB, B)); B = &TB; } if (X != A) MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A)); /* * X should always be positive as a result of unsigned subtractions. */ X->s = 1; ret = 0; for (n = B->n; n > 0; n--) if (B->p[n - 1] != 0) break; mpi_sub_hlp(n, B->p, X->p); cleanup: mbedtls_mpi_free(&TB); return (ret); } /* * Signed addition: X = A + B */ int mbedtls_mpi_add_mpi(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret, s = A->s; if (A->s * B->s < 0) { if (mbedtls_mpi_cmp_abs(A, B) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(X, A, B)); X->s = s; } else { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(X, B, A)); X->s = -s; } } else { MBEDTLS_MPI_CHK(mbedtls_mpi_add_abs(X, A, B)); X->s = s; } cleanup: return (ret); } /* * Signed subtraction: X = A - B */ int mbedtls_mpi_sub_mpi(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret, s = A->s; if (A->s * B->s > 0) { if (mbedtls_mpi_cmp_abs(A, B) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(X, A, B)); X->s = s; } else { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(X, B, 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_int(mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_sint b) { mbedtls_mpi _B; mbedtls_mpi_uint p[1]; p[0] = (b < 0) ? -b : 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]; p[0] = (b < 0) ? -b : 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 */ 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, mbedtls_mpi_uint *s, mbedtls_mpi_uint *d, mbedtls_mpi_uint b) { mbedtls_mpi_uint c = 0, t = 0; #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 */ t++; do { *d += c; c = (*d < c); d++; } while (c != 0); } /* * 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; size_t i, j; mbedtls_mpi TA, TB; 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; for (j = B->n; j > 0; j--) if (B->p[j - 1] != 0) break; 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]); 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) { mbedtls_mpi _B; mbedtls_mpi_uint p[1]; _B.s = 1; _B.n = 1; _B.p = p; p[0] = b; return (mbedtls_mpi_mul_mpi(X, A, &_B)); } /* * 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 = ~0; return (~0); } #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; size_t i, n, t, k; mbedtls_mpi X, Y, Z, T1, T2; 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); mbedtls_mpi_init(&T2); 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, 2)); MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&T2, 3)); 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] = ~0; else { Z.p[i - t - 1] = mbedtls_int_div_int(X.p[i], X.p[i - 1], Y.p[t], NULL); } 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])); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&T2, 0)); 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]; } 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_mpi_free(&T2); 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]; p[0] = (b < 0) ? -b : 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; 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; 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) { *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) */ static int 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; if (T->n < N->n + 1 || T->p == NULL) return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA); 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; } memcpy(A->p, d, (n + 1) * ciL); if (mbedtls_mpi_cmp_abs(A, N) >= 0) mpi_sub_hlp(n, N->p, A->p); else /* prevent timing attacks */ mpi_sub_hlp(n, A->p, T->p); return (0); } /* * Montgomery reduction: A = A * R^-1 mod N */ static int 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; return (mpi_montmul(A, &U, N, mm, T)); } /* * 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 *_RR) { int ret; size_t wbits, wsize, one = 1; size_t i, j, nblimbs; size_t bufsize, nbits; mbedtls_mpi_uint ei, mm, state; mbedtls_mpi RR, T, W[ 2 << MBEDTLS_MPI_WINDOW_SIZE ], Apos; int neg; 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); /* * Init temps and window size */ mpi_montg_init(&mm, N); mbedtls_mpi_init(&RR); mbedtls_mpi_init(&T); mbedtls_mpi_init(&Apos); memset(W, 0, sizeof(W)); i = mbedtls_mpi_bitlen(E); wsize = (i > 671) ? 6 : (i > 239) ? 5 : (i > 79) ? 4 : (i > 23) ? 3 : 1; if (wsize > MBEDTLS_MPI_WINDOW_SIZE) wsize = MBEDTLS_MPI_WINDOW_SIZE; j = N->n + 1; MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, 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 (_RR == NULL || _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 (_RR != NULL) memcpy(_RR, &RR, sizeof(mbedtls_mpi)); } else memcpy(&RR, _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)); else MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&W[1], A)); MBEDTLS_MPI_CHK(mpi_montmul(&W[1], &RR, N, mm, &T)); /* * X = R^2 * R^-1 mod N = R mod N */ MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, &RR)); MBEDTLS_MPI_CHK(mpi_montred(X, N, mm, &T)); if (wsize > 1) { /* * W[1 << (wsize - 1)] = W[1] ^ (wsize - 1) */ j = one << (wsize - 1); 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 < wsize - 1; i++) MBEDTLS_MPI_CHK(mpi_montmul(&W[j], &W[j], N, mm, &T)); /* * W[i] = W[i - 1] * W[1] */ for (i = j + 1; i < (one << wsize); i++) { MBEDTLS_MPI_CHK(mbedtls_mpi_grow(&W[i], N->n + 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&W[i], &W[i - 1])); MBEDTLS_MPI_CHK(mpi_montmul(&W[i], &W[1], N, mm, &T)); } } nblimbs = E->n; bufsize = 0; nbits = 0; wbits = 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 X */ MBEDTLS_MPI_CHK(mpi_montmul(X, X, N, mm, &T)); continue; } /* * add ei to current window */ state = 2; nbits++; wbits |= (ei << (wsize - nbits)); if (nbits == wsize) { /* * X = X^wsize R^-1 mod N */ for (i = 0; i < wsize; i++) MBEDTLS_MPI_CHK(mpi_montmul(X, X, N, mm, &T)); /* * X = X * W[wbits] R^-1 mod N */ MBEDTLS_MPI_CHK(mpi_montmul(X, &W[wbits], N, mm, &T)); state--; nbits = 0; wbits = 0; } } /* * process the remaining bits */ for (i = 0; i < nbits; i++) { MBEDTLS_MPI_CHK(mpi_montmul(X, X, N, mm, &T)); wbits <<= 1; if ((wbits & (one << wsize)) != 0) MBEDTLS_MPI_CHK(mpi_montmul(X, &W[1], N, mm, &T)); } /* * X = A^E * R * R^-1 mod N = A^E mod N */ MBEDTLS_MPI_CHK(mpi_montred(X, N, mm, &T)); if (neg && E->n != 0 && (E->p[0] & 1) != 0) { X->s = -1; MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(X, N, X)); } cleanup: for (i = (one << (wsize - 1)); i < (one << wsize); i++) mbedtls_mpi_free(&W[i]); mbedtls_mpi_free(&W[1]); mbedtls_mpi_free(&T); mbedtls_mpi_free(&Apos); if (_RR == NULL || _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; size_t lz, lzt; mbedtls_mpi TG, TA, TB; mbedtls_mpi_init(&TG); 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); if (lzt < lz) lz = lzt; MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TA, lz)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TB, lz)); TA.s = TB.s = 1; while (mbedtls_mpi_cmp_int(&TA, 0) != 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TA, mbedtls_mpi_lsb(&TA))); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TB, mbedtls_mpi_lsb(&TB))); 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)); } } MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&TB, lz)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(G, &TB)); cleanup: mbedtls_mpi_free(&TG); mbedtls_mpi_free(&TA); mbedtls_mpi_free(&TB); 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; unsigned char buf[MBEDTLS_MPI_MAX_SIZE]; if (size > MBEDTLS_MPI_MAX_SIZE) return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA); MBEDTLS_MPI_CHK(f_rng(p_rng, buf, size)); MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary(X, buf, size)); cleanup: mbedtls_platform_zeroize(buf, sizeof(buf)); 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_mpi G, TA, TU, U1, U2, TB, TV, V1, V2; 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, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret, count; size_t i, j, k, n, s; mbedtls_mpi W, R, T, A, RR; 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)); i = mbedtls_mpi_bitlen(X); /* * HAC, table 4.4 */ n = ((i >= 1300) ? 2 : (i >= 850) ? 3 : (i >= 650) ? 4 : (i >= 350) ? 8 : (i >= 250) ? 12 : (i >= 150) ? 18 : 27); for (i = 0; i < n; i++) { /* * pick a random A, 1 < A < |X| - 1 */ MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(&A, X->n * ciL, f_rng, p_rng)); if (mbedtls_mpi_cmp_mpi(&A, &W) >= 0) { j = mbedtls_mpi_bitlen(&A) - mbedtls_mpi_bitlen(&W); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&A, j + 1)); } A.p[0] |= 3; 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) { MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&A, j - k)); } if (count++ > 30) { return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; } } 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(const mbedtls_mpi *X, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret; mbedtls_mpi XX; 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, f_rng, p_rng)); } /* * Prime number generation * * If dh_flag is 0 and nbits is at least 1024, then the procedure * follows the RSA probably-prime generation method of FIPS 186-4. * NB. FIPS 186-4 only allows the specific bit lengths of 1024 and 1536. */ int mbedtls_mpi_gen_prime(mbedtls_mpi *X, size_t nbits, int dh_flag, 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; mbedtls_mpi_uint r; mbedtls_mpi Y; if (nbits < 3 || nbits > MBEDTLS_MPI_MAX_BITS) return (MBEDTLS_ERR_MPI_BAD_INPUT_DATA); mbedtls_mpi_init(&Y); n = BITS_TO_LIMBS(nbits); 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 (dh_flag == 0) { ret = mbedtls_mpi_is_prime(X, f_rng, p_rng); if (ret != MBEDTLS_ERR_MPI_NOT_ACCEPTABLE) goto cleanup; } else { /* * An 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, f_rng, p_rng)) == 0 && (ret = mpi_miller_rabin(&Y, 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", 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 */