bignum.c

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/*
 *  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 <limits.h>
#include <string.h>
 
#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, &lt_lower));
        MBEDTLS_MPI_CHK(mbedtls_mpi_lt_mpi_ct(X, N, &lt_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 */