quickjs-tart

bignum.c (66561B)

---

 /*
  *  Multi-precision integer library
  *
  *  Copyright The Mbed TLS Contributors
  *  SPDX-License-Identifier: Apache-2.0 OR GPL-2.0-or-later
  */
 
 /*
  *  The following sources were referenced in the design of this Multi-precision
  *  Integer library:
  *
  *  [1] Handbook of Applied Cryptography - 1997
  *      Menezes, van Oorschot and Vanstone
  *
  *  [2] Multi-Precision Math
  *      Tom St Denis
  *      https://github.com/libtom/libtommath/blob/develop/tommath.pdf
  *
  *  [3] GNU Multi-Precision Arithmetic Library
  *      https://gmplib.org/manual/index.html
  *
  */
 
 #include "common.h"
 
 #if defined(MBEDTLS_BIGNUM_C)
 
 #include "mbedtls/bignum.h"
 #include "bignum_core.h"
 #include "bignum_internal.h"
 #include "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"
 
 
 
 /*
  * Conditionally select an MPI sign in constant time.
  * (MPI sign is the field s in mbedtls_mpi. It is unsigned short and only 1 and -1 are valid
  * values.)
  */
 static inline signed short mbedtls_ct_mpi_sign_if(mbedtls_ct_condition_t cond,
                                                   signed short sign1, signed short sign2)
 {
     return (signed short) mbedtls_ct_uint_if(cond, sign1 + 1, sign2 + 1) - 1;
 }
 
 /*
  * Compare signed values in constant time
  */
 int mbedtls_mpi_lt_mpi_ct(const mbedtls_mpi *X,
                           const mbedtls_mpi *Y,
                           unsigned *ret)
 {
     mbedtls_ct_condition_t different_sign, X_is_negative, Y_is_negative, result;
 
     if (X->n != Y->n) {
         return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
     }
 
     /*
      * Set N_is_negative to MBEDTLS_CT_FALSE if N >= 0, MBEDTLS_CT_TRUE if N < 0.
      * We know that N->s == 1 if N >= 0 and N->s == -1 if N < 0.
      */
     X_is_negative = mbedtls_ct_bool((X->s & 2) >> 1);
     Y_is_negative = mbedtls_ct_bool((Y->s & 2) >> 1);
 
     /*
      * If the signs are different, then the positive operand is the bigger.
      * That is if X is negative (X_is_negative == 1), then X < Y is true and it
      * is false if X is positive (X_is_negative == 0).
      */
     different_sign = mbedtls_ct_bool_ne(X_is_negative, Y_is_negative); // true if different sign
     result = mbedtls_ct_bool_and(different_sign, X_is_negative);
 
     /*
      * Assuming signs are the same, compare X and Y. We switch the comparison
      * order if they are negative so that we get the right result, regardles of
      * sign.
      */
 
     /* This array is used to conditionally swap the pointers in const time */
     void * const p[2] = { X->p, Y->p };
     size_t i = mbedtls_ct_size_if_else_0(X_is_negative, 1);
     mbedtls_ct_condition_t lt = mbedtls_mpi_core_lt_ct(p[i], p[i ^ 1], X->n);
 
     /*
      * Store in result iff the signs are the same (i.e., iff different_sign == false). If
      * the signs differ, result has already been set, so we don't change it.
      */
     result = mbedtls_ct_bool_or(result,
                                 mbedtls_ct_bool_and(mbedtls_ct_bool_not(different_sign), lt));
 
     *ret = mbedtls_ct_uint_if_else_0(result, 1);
 
     return 0;
 }
 
 /*
  * Conditionally assign X = Y, without leaking information
  * about whether the assignment was made or not.
  * (Leaking information about the respective sizes of X and Y is ok however.)
  */
 #if defined(_MSC_VER) && defined(MBEDTLS_PLATFORM_IS_WINDOWS_ON_ARM64) && \
     (_MSC_FULL_VER < 193131103)
 /*
  * MSVC miscompiles this function if it's inlined prior to Visual Studio 2022 version 17.1. See:
  * https://developercommunity.visualstudio.com/t/c-compiler-miscompiles-part-of-mbedtls-library-on/1646989
  */
 __declspec(noinline)
 #endif
 int mbedtls_mpi_safe_cond_assign(mbedtls_mpi *X,
                                  const mbedtls_mpi *Y,
                                  unsigned char assign)
 {
     int ret = 0;
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, Y->n));
 
     {
         mbedtls_ct_condition_t do_assign = mbedtls_ct_bool(assign);
 
         X->s = mbedtls_ct_mpi_sign_if(do_assign, Y->s, X->s);
 
         mbedtls_mpi_core_cond_assign(X->p, Y->p, Y->n, do_assign);
 
         mbedtls_ct_condition_t do_not_assign = mbedtls_ct_bool_not(do_assign);
         for (size_t i = Y->n; i < X->n; i++) {
             X->p[i] = mbedtls_ct_mpi_uint_if_else_0(do_not_assign, X->p[i]);
         }
     }
 
 cleanup:
     return ret;
 }
 
 /*
  * Conditionally swap X and Y, without leaking information
  * about whether the swap was made or not.
  * Here it is not ok to simply swap the pointers, which would lead to
  * different memory access patterns when X and Y are used afterwards.
  */
 int mbedtls_mpi_safe_cond_swap(mbedtls_mpi *X,
                                mbedtls_mpi *Y,
                                unsigned char swap)
 {
     int ret = 0;
     int s;
 
     if (X == Y) {
         return 0;
     }
 
     mbedtls_ct_condition_t do_swap = mbedtls_ct_bool(swap);
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, Y->n));
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(Y, X->n));
 
     s = X->s;
     X->s = mbedtls_ct_mpi_sign_if(do_swap, Y->s, X->s);
     Y->s = mbedtls_ct_mpi_sign_if(do_swap, s, Y->s);
 
     mbedtls_mpi_core_cond_swap(X->p, Y->p, X->n, do_swap);
 
 cleanup:
     return ret;
 }
 
 /* Implementation that should never be optimized out by the compiler */
 #define mbedtls_mpi_zeroize_and_free(v, n) mbedtls_zeroize_and_free(v, ciL * (n))
 
 /*
  * Initialize one MPI
  */
 void mbedtls_mpi_init(mbedtls_mpi *X)
 {
     X->s = 1;
     X->n = 0;
     X->p = NULL;
 }
 
 /*
  * Unallocate one MPI
  */
 void mbedtls_mpi_free(mbedtls_mpi *X)
 {
     if (X == NULL) {
         return;
     }
 
     if (X->p != NULL) {
         mbedtls_mpi_zeroize_and_free(X->p, X->n);
     }
 
     X->s = 1;
     X->n = 0;
     X->p = NULL;
 }
 
 /*
  * Enlarge to the specified number of limbs
  */
 int mbedtls_mpi_grow(mbedtls_mpi *X, size_t nblimbs)
 {
     mbedtls_mpi_uint *p;
 
     if (nblimbs > MBEDTLS_MPI_MAX_LIMBS) {
         return MBEDTLS_ERR_MPI_ALLOC_FAILED;
     }
 
     if (X->n < nblimbs) {
         if ((p = (mbedtls_mpi_uint *) mbedtls_calloc(nblimbs, ciL)) == NULL) {
             return MBEDTLS_ERR_MPI_ALLOC_FAILED;
         }
 
         if (X->p != NULL) {
             memcpy(p, X->p, X->n * ciL);
             mbedtls_mpi_zeroize_and_free(X->p, X->n);
         }
 
         /* nblimbs fits in n because we ensure that MBEDTLS_MPI_MAX_LIMBS
          * fits, and we've checked that nblimbs <= MBEDTLS_MPI_MAX_LIMBS. */
         X->n = (unsigned short) nblimbs;
         X->p = p;
     }
 
     return 0;
 }
 
 /*
  * Resize down as much as possible,
  * while keeping at least the specified number of limbs
  */
 int mbedtls_mpi_shrink(mbedtls_mpi *X, size_t nblimbs)
 {
     mbedtls_mpi_uint *p;
     size_t i;
 
     if (nblimbs > MBEDTLS_MPI_MAX_LIMBS) {
         return MBEDTLS_ERR_MPI_ALLOC_FAILED;
     }
 
     /* Actually resize up if there are currently fewer than nblimbs limbs. */
     if (X->n <= nblimbs) {
         return mbedtls_mpi_grow(X, nblimbs);
     }
     /* After this point, then X->n > nblimbs and in particular X->n > 0. */
 
     for (i = X->n - 1; i > 0; i--) {
         if (X->p[i] != 0) {
             break;
         }
     }
     i++;
 
     if (i < nblimbs) {
         i = nblimbs;
     }
 
     if ((p = (mbedtls_mpi_uint *) mbedtls_calloc(i, ciL)) == NULL) {
         return MBEDTLS_ERR_MPI_ALLOC_FAILED;
     }
 
     if (X->p != NULL) {
         memcpy(p, X->p, i * ciL);
         mbedtls_mpi_zeroize_and_free(X->p, X->n);
     }
 
     /* i fits in n because we ensure that MBEDTLS_MPI_MAX_LIMBS
      * fits, and we've checked that i <= nblimbs <= MBEDTLS_MPI_MAX_LIMBS. */
     X->n = (unsigned short) i;
     X->p = p;
 
     return 0;
 }
 
 /* Resize X to have exactly n limbs and set it to 0. */
 static int mbedtls_mpi_resize_clear(mbedtls_mpi *X, size_t limbs)
 {
     if (limbs == 0) {
         mbedtls_mpi_free(X);
         return 0;
     } else if (X->n == limbs) {
         memset(X->p, 0, limbs * ciL);
         X->s = 1;
         return 0;
     } else {
         mbedtls_mpi_free(X);
         return mbedtls_mpi_grow(X, limbs);
     }
 }
 
 /*
  * Copy the contents of Y into X.
  *
  * This function is not constant-time. Leading zeros in Y may be removed.
  *
  * Ensure that X does not shrink. This is not guaranteed by the public API,
  * but some code in the bignum module might still rely on this property.
  */
 int mbedtls_mpi_copy(mbedtls_mpi *X, const mbedtls_mpi *Y)
 {
     int ret = 0;
     size_t i;
 
     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;
 
     memcpy(&T,  X, sizeof(mbedtls_mpi));
     memcpy(X,  Y, sizeof(mbedtls_mpi));
     memcpy(Y, &T, sizeof(mbedtls_mpi));
 }
 
 static inline mbedtls_mpi_uint mpi_sint_abs(mbedtls_mpi_sint z)
 {
     if (z >= 0) {
         return z;
     }
     /* Take care to handle the most negative value (-2^(biL-1)) correctly.
      * A naive -z would have undefined behavior.
      * Write this in a way that makes popular compilers happy (GCC, Clang,
      * MSVC). */
     return (mbedtls_mpi_uint) 0 - (mbedtls_mpi_uint) z;
 }
 
 /* Convert x to a sign, i.e. to 1, if x is positive, or -1, if x is negative.
  * This looks awkward but generates smaller code than (x < 0 ? -1 : 1) */
 #define TO_SIGN(x) ((mbedtls_mpi_sint) (((mbedtls_mpi_uint) x) >> (biL - 1)) * -2 + 1)
 
 /*
  * Set value from integer
  */
 int mbedtls_mpi_lset(mbedtls_mpi *X, mbedtls_mpi_sint z)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, 1));
     memset(X->p, 0, X->n * ciL);
 
     X->p[0] = mpi_sint_abs(z);
     X->s    = TO_SIGN(z);
 
 cleanup:
 
     return ret;
 }
 
 /*
  * Get a specific bit
  */
 int mbedtls_mpi_get_bit(const mbedtls_mpi *X, size_t pos)
 {
     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;
 
 #if defined(__has_builtin)
 #if (MBEDTLS_MPI_UINT_MAX == UINT_MAX) && __has_builtin(__builtin_ctz)
     #define mbedtls_mpi_uint_ctz __builtin_ctz
 #elif (MBEDTLS_MPI_UINT_MAX == ULONG_MAX) && __has_builtin(__builtin_ctzl)
     #define mbedtls_mpi_uint_ctz __builtin_ctzl
 #elif (MBEDTLS_MPI_UINT_MAX == ULLONG_MAX) && __has_builtin(__builtin_ctzll)
     #define mbedtls_mpi_uint_ctz __builtin_ctzll
 #endif
 #endif
 
 #if defined(mbedtls_mpi_uint_ctz)
     for (i = 0; i < X->n; i++) {
         if (X->p[i] != 0) {
             return i * biL + mbedtls_mpi_uint_ctz(X->p[i]);
         }
     }
 #else
     size_t count = 0;
     for (i = 0; i < X->n; i++) {
         for (size_t j = 0; j < biL; j++, count++) {
             if (((X->p[i] >> j) & 1) != 0) {
                 return count;
             }
         }
     }
 #endif
 
     return 0;
 }
 
 /*
  * Return the number of bits
  */
 size_t mbedtls_mpi_bitlen(const mbedtls_mpi *X)
 {
     return mbedtls_mpi_core_bitlen(X->p, X->n);
 }
 
 /*
  * Return the total size in bytes
  */
 size_t mbedtls_mpi_size(const mbedtls_mpi *X)
 {
     return (mbedtls_mpi_bitlen(X) + 7) >> 3;
 }
 
 /*
  * Convert an ASCII character to digit value
  */
 static int mpi_get_digit(mbedtls_mpi_uint *d, int radix, char c)
 {
     *d = 255;
 
     if (c >= 0x30 && c <= 0x39) {
         *d = c - 0x30;
     }
     if (c >= 0x41 && c <= 0x46) {
         *d = c - 0x37;
     }
     if (c >= 0x61 && c <= 0x66) {
         *d = c - 0x57;
     }
 
     if (*d >= (mbedtls_mpi_uint) radix) {
         return MBEDTLS_ERR_MPI_INVALID_CHARACTER;
     }
 
     return 0;
 }
 
 /*
  * Import from an ASCII string
  */
 int mbedtls_mpi_read_string(mbedtls_mpi *X, int radix, const char *s)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     size_t i, j, slen, n;
     int sign = 1;
     mbedtls_mpi_uint d;
     mbedtls_mpi T;
 
     if (radix < 2 || radix > 16) {
         return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
     }
 
     mbedtls_mpi_init(&T);
 
     if (s[0] == 0) {
         mbedtls_mpi_free(X);
         return 0;
     }
 
     if (s[0] == '-') {
         ++s;
         sign = -1;
     }
 
     slen = strlen(s);
 
     if (radix == 16) {
         if (slen > SIZE_MAX >> 2) {
             return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
         }
 
         n = BITS_TO_LIMBS(slen << 2);
 
         MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, n));
         MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, 0));
 
         for (i = slen, j = 0; i > 0; i--, j++) {
             MBEDTLS_MPI_CHK(mpi_get_digit(&d, radix, s[i - 1]));
             X->p[j / (2 * ciL)] |= d << ((j % (2 * ciL)) << 2);
         }
     } else {
         MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, 0));
 
         for (i = 0; i < slen; i++) {
             MBEDTLS_MPI_CHK(mpi_get_digit(&d, radix, s[i]));
             MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&T, X, radix));
             MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(X, &T, d));
         }
     }
 
     if (sign < 0 && mbedtls_mpi_bitlen(X) != 0) {
         X->s = -1;
     }
 
 cleanup:
 
     mbedtls_mpi_free(&T);
 
     return ret;
 }
 
 /*
  * Helper to write the digits high-order first.
  */
 static int mpi_write_hlp(mbedtls_mpi *X, int radix,
                          char **p, const size_t buflen)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     mbedtls_mpi_uint r;
     size_t length = 0;
     char *p_end = *p + buflen;
 
     do {
         if (length >= buflen) {
             return MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL;
         }
 
         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_int(&r, X, radix));
         MBEDTLS_MPI_CHK(mbedtls_mpi_div_int(X, NULL, X, radix));
         /*
          * Write the residue in the current position, as an ASCII character.
          */
         if (r < 0xA) {
             *(--p_end) = (char) ('0' + r);
         } else {
             *(--p_end) = (char) ('A' + (r - 0xA));
         }
 
         length++;
     } while (mbedtls_mpi_cmp_int(X, 0) != 0);
 
     memmove(*p, p_end, length);
     *p += length;
 
 cleanup:
 
     return ret;
 }
 
 /*
  * Export into an ASCII string
  */
 int mbedtls_mpi_write_string(const mbedtls_mpi *X, int radix,
                              char *buf, size_t buflen, size_t *olen)
 {
     int ret = 0;
     size_t n;
     char *p;
     mbedtls_mpi T;
 
     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 = (size_t) (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];
 
     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];
 
     if (radix < 2 || radix > 16) {
         return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
     }
 
     memset(s, 0, sizeof(s));
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_write_string(X, radix, s, sizeof(s) - 2, &n));
 
     if (p == NULL) {
         p = "";
     }
 
     plen = strlen(p);
     slen = strlen(s);
     s[slen++] = '\r';
     s[slen++] = '\n';
 
     if (fout != NULL) {
         if (fwrite(p, 1, plen, fout) != plen ||
             fwrite(s, 1, slen, fout) != slen) {
             return MBEDTLS_ERR_MPI_FILE_IO_ERROR;
         }
     } else {
         mbedtls_printf("%s%s", p, s);
     }
 
 cleanup:
 
     return ret;
 }
 #endif /* MBEDTLS_FS_IO */
 
 /*
  * Import X from unsigned binary data, little endian
  *
  * This function is guaranteed to return an MPI with exactly the necessary
  * number of limbs (in particular, it does not skip 0s in the input).
  */
 int mbedtls_mpi_read_binary_le(mbedtls_mpi *X,
                                const unsigned char *buf, size_t buflen)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     const size_t limbs = CHARS_TO_LIMBS(buflen);
 
     /* Ensure that target MPI has exactly the necessary number of limbs */
     MBEDTLS_MPI_CHK(mbedtls_mpi_resize_clear(X, limbs));
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_core_read_le(X->p, X->n, buf, buflen));
 
 cleanup:
 
     /*
      * This function is also used to import keys. However, wiping the buffers
      * upon failure is not necessary because failure only can happen before any
      * input is copied.
      */
     return ret;
 }
 
 /*
  * Import X from unsigned binary data, big endian
  *
  * This function is guaranteed to return an MPI with exactly the necessary
  * number of limbs (in particular, it does not skip 0s in the input).
  */
 int mbedtls_mpi_read_binary(mbedtls_mpi *X, const unsigned char *buf, size_t buflen)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     const size_t limbs = CHARS_TO_LIMBS(buflen);
 
     /* Ensure that target MPI has exactly the necessary number of limbs */
     MBEDTLS_MPI_CHK(mbedtls_mpi_resize_clear(X, limbs));
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_core_read_be(X->p, X->n, buf, buflen));
 
 cleanup:
 
     /*
      * This function is also used to import keys. However, wiping the buffers
      * upon failure is not necessary because failure only can happen before any
      * input is copied.
      */
     return ret;
 }
 
 /*
  * Export X into unsigned binary data, little endian
  */
 int mbedtls_mpi_write_binary_le(const mbedtls_mpi *X,
                                 unsigned char *buf, size_t buflen)
 {
     return mbedtls_mpi_core_write_le(X->p, X->n, buf, buflen);
 }
 
 /*
  * Export X into unsigned binary data, big endian
  */
 int mbedtls_mpi_write_binary(const mbedtls_mpi *X,
                              unsigned char *buf, size_t buflen)
 {
     return mbedtls_mpi_core_write_be(X->p, X->n, buf, buflen);
 }
 
 /*
  * Left-shift: X <<= count
  */
 int mbedtls_mpi_shift_l(mbedtls_mpi *X, size_t count)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     size_t i;
 
     i = mbedtls_mpi_bitlen(X) + count;
 
     if (X->n * biL < i) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, BITS_TO_LIMBS(i)));
     }
 
     ret = 0;
 
     mbedtls_mpi_core_shift_l(X->p, X->n, count);
 cleanup:
 
     return ret;
 }
 
 /*
  * Right-shift: X >>= count
  */
 int mbedtls_mpi_shift_r(mbedtls_mpi *X, size_t count)
 {
     if (X->n != 0) {
         mbedtls_mpi_core_shift_r(X->p, X->n, count);
     }
     return 0;
 }
 
 /*
  * Compare unsigned values
  */
 int mbedtls_mpi_cmp_abs(const mbedtls_mpi *X, const mbedtls_mpi *Y)
 {
     size_t i, j;
 
     for (i = X->n; i > 0; i--) {
         if (X->p[i - 1] != 0) {
             break;
         }
     }
 
     for (j = Y->n; j > 0; j--) {
         if (Y->p[j - 1] != 0) {
             break;
         }
     }
 
     /* If i == j == 0, i.e. abs(X) == abs(Y),
      * we end up returning 0 at the end of the function. */
 
     if (i > j) {
         return 1;
     }
     if (j > i) {
         return -1;
     }
 
     for (; i > 0; i--) {
         if (X->p[i - 1] > Y->p[i - 1]) {
             return 1;
         }
         if (X->p[i - 1] < Y->p[i - 1]) {
             return -1;
         }
     }
 
     return 0;
 }
 
 /*
  * Compare signed values
  */
 int mbedtls_mpi_cmp_mpi(const mbedtls_mpi *X, const mbedtls_mpi *Y)
 {
     size_t i, j;
 
     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  = mpi_sint_abs(z);
     Y.s = TO_SIGN(z);
     Y.n = 1;
     Y.p = p;
 
     return mbedtls_mpi_cmp_mpi(X, &Y);
 }
 
 /*
  * Unsigned addition: X = |A| + |B|  (HAC 14.7)
  */
 int mbedtls_mpi_add_abs(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     size_t j;
     mbedtls_mpi_uint *p;
     mbedtls_mpi_uint c;
 
     if (X == B) {
         const mbedtls_mpi *T = A; A = X; B = T;
     }
 
     if (X != A) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A));
     }
 
     /*
      * X must always be positive as a result of unsigned additions.
      */
     X->s = 1;
 
     for (j = B->n; j > 0; j--) {
         if (B->p[j - 1] != 0) {
             break;
         }
     }
 
     /* Exit early to avoid undefined behavior on NULL+0 when X->n == 0
      * and B is 0 (of any size). */
     if (j == 0) {
         return 0;
     }
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, j));
 
     /* j is the number of non-zero limbs of B. Add those to X. */
 
     p = X->p;
 
     c = mbedtls_mpi_core_add(p, p, B->p, j);
 
     p += j;
 
     /* Now propagate any carry */
 
     while (c != 0) {
         if (j >= X->n) {
             MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, j + 1));
             p = X->p + j;
         }
 
         *p += c; c = (*p < c); j++; p++;
     }
 
 cleanup:
 
     return ret;
 }
 
 /*
  * Unsigned subtraction: X = |A| - |B|  (HAC 14.9, 14.10)
  */
 int mbedtls_mpi_sub_abs(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     size_t n;
     mbedtls_mpi_uint carry;
 
     for (n = B->n; n > 0; n--) {
         if (B->p[n - 1] != 0) {
             break;
         }
     }
     if (n > A->n) {
         /* B >= (2^ciL)^n > A */
         ret = MBEDTLS_ERR_MPI_NEGATIVE_VALUE;
         goto cleanup;
     }
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, A->n));
 
     /* Set the high limbs of X to match A. Don't touch the lower limbs
      * because X might be aliased to B, and we must not overwrite the
      * significant digits of B. */
     if (A->n > n && A != X) {
         memcpy(X->p + n, A->p + n, (A->n - n) * ciL);
     }
     if (X->n > A->n) {
         memset(X->p + A->n, 0, (X->n - A->n) * ciL);
     }
 
     carry = mbedtls_mpi_core_sub(X->p, A->p, B->p, n);
     if (carry != 0) {
         /* Propagate the carry through the rest of X. */
         carry = mbedtls_mpi_core_sub_int(X->p + n, X->p + n, carry, X->n - n);
 
         /* If we have further carry/borrow, the result is negative. */
         if (carry != 0) {
             ret = MBEDTLS_ERR_MPI_NEGATIVE_VALUE;
             goto cleanup;
         }
     }
 
     /* X should always be positive as a result of unsigned subtractions. */
     X->s = 1;
 
 cleanup:
     return ret;
 }
 
 /* Common function for signed addition and subtraction.
  * Calculate A + B * flip_B where flip_B is 1 or -1.
  */
 static int add_sub_mpi(mbedtls_mpi *X,
                        const mbedtls_mpi *A, const mbedtls_mpi *B,
                        int flip_B)
 {
     int ret, s;
 
     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];
 
     p[0] = mpi_sint_abs(b);
     B.s = TO_SIGN(b);
     B.n = 1;
     B.p = p;
 
     return mbedtls_mpi_add_mpi(X, A, &B);
 }
 
 /*
  * Signed subtraction: X = A - b
  */
 int mbedtls_mpi_sub_int(mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_sint b)
 {
     mbedtls_mpi B;
     mbedtls_mpi_uint p[1];
 
     p[0] = mpi_sint_abs(b);
     B.s = TO_SIGN(b);
     B.n = 1;
     B.p = p;
 
     return mbedtls_mpi_sub_mpi(X, A, &B);
 }
 
 /*
  * Baseline multiplication: X = A * B  (HAC 14.12)
  */
 int mbedtls_mpi_mul_mpi(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     size_t i, j;
     mbedtls_mpi TA, TB;
     int result_is_zero = 0;
 
     mbedtls_mpi_init(&TA);
     mbedtls_mpi_init(&TB);
 
     if (X == A) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TA, A)); A = &TA;
     }
     if (X == B) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TB, B)); B = &TB;
     }
 
     for (i = A->n; i > 0; i--) {
         if (A->p[i - 1] != 0) {
             break;
         }
     }
     if (i == 0) {
         result_is_zero = 1;
     }
 
     for (j = B->n; j > 0; j--) {
         if (B->p[j - 1] != 0) {
             break;
         }
     }
     if (j == 0) {
         result_is_zero = 1;
     }
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, i + j));
     MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, 0));
 
     mbedtls_mpi_core_mul(X->p, A->p, i, B->p, j);
 
     /* If the result is 0, we don't shortcut the operation, which reduces
      * but does not eliminate side channels leaking the zero-ness. We do
      * need to take care to set the sign bit properly since the library does
      * not fully support an MPI object with a value of 0 and s == -1. */
     if (result_is_zero) {
         X->s = 1;
     } else {
         X->s = A->s * B->s;
     }
 
 cleanup:
 
     mbedtls_mpi_free(&TB); mbedtls_mpi_free(&TA);
 
     return ret;
 }
 
 /*
  * Baseline multiplication: X = A * b
  */
 int mbedtls_mpi_mul_int(mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_uint b)
 {
     size_t n = A->n;
     while (n > 0 && A->p[n - 1] == 0) {
         --n;
     }
 
     /* The general method below doesn't work if b==0. */
     if (b == 0 || n == 0) {
         return mbedtls_mpi_lset(X, 0);
     }
 
     /* Calculate A*b as A + A*(b-1) to take advantage of mbedtls_mpi_core_mla */
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     /* In general, A * b requires 1 limb more than b. If
      * A->p[n - 1] * b / b == A->p[n - 1], then A * b fits in the same
      * number of limbs as A and the call to grow() is not required since
      * copy() will take care of the growth if needed. However, experimentally,
      * making the call to grow() unconditional causes slightly fewer
      * calls to calloc() in ECP code, presumably because it reuses the
      * same mpi for a while and this way the mpi is more likely to directly
      * grow to its final size.
      *
      * Note that calculating A*b as 0 + A*b doesn't work as-is because
      * A,X can be the same. */
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, n + 1));
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A));
     mbedtls_mpi_core_mla(X->p, X->n, A->p, n, b - 1);
 
 cleanup:
     return ret;
 }
 
 /*
  * Unsigned integer divide - double mbedtls_mpi_uint dividend, u1/u0, and
  * mbedtls_mpi_uint divisor, d
  */
 static mbedtls_mpi_uint mbedtls_int_div_int(mbedtls_mpi_uint u1,
                                             mbedtls_mpi_uint u0,
                                             mbedtls_mpi_uint d,
                                             mbedtls_mpi_uint *r)
 {
 #if defined(MBEDTLS_HAVE_UDBL)
     mbedtls_t_udbl dividend, quotient;
 #else
     const mbedtls_mpi_uint radix = (mbedtls_mpi_uint) 1 << biH;
     const mbedtls_mpi_uint uint_halfword_mask = ((mbedtls_mpi_uint) 1 << biH) - 1;
     mbedtls_mpi_uint d0, d1, q0, q1, rAX, r0, quotient;
     mbedtls_mpi_uint u0_msw, u0_lsw;
     size_t s;
 #endif
 
     /*
      * Check for overflow
      */
     if (0 == d || u1 >= d) {
         if (r != NULL) {
             *r = ~(mbedtls_mpi_uint) 0u;
         }
 
         return ~(mbedtls_mpi_uint) 0u;
     }
 
 #if defined(MBEDTLS_HAVE_UDBL)
     dividend  = (mbedtls_t_udbl) u1 << biL;
     dividend |= (mbedtls_t_udbl) u0;
     quotient = dividend / d;
     if (quotient > ((mbedtls_t_udbl) 1 << biL) - 1) {
         quotient = ((mbedtls_t_udbl) 1 << biL) - 1;
     }
 
     if (r != NULL) {
         *r = (mbedtls_mpi_uint) (dividend - (quotient * d));
     }
 
     return (mbedtls_mpi_uint) quotient;
 #else
 
     /*
      * Algorithm D, Section 4.3.1 - The Art of Computer Programming
      *   Vol. 2 - Seminumerical Algorithms, Knuth
      */
 
     /*
      * Normalize the divisor, d, and dividend, u0, u1
      */
     s = mbedtls_mpi_core_clz(d);
     d = d << s;
 
     u1 = u1 << s;
     u1 |= (u0 >> (biL - s)) & (-(mbedtls_mpi_sint) s >> (biL - 1));
     u0 =  u0 << s;
 
     d1 = d >> biH;
     d0 = d & uint_halfword_mask;
 
     u0_msw = u0 >> biH;
     u0_lsw = u0 & uint_halfword_mask;
 
     /*
      * Find the first quotient and remainder
      */
     q1 = u1 / d1;
     r0 = u1 - d1 * q1;
 
     while (q1 >= radix || (q1 * d0 > radix * r0 + u0_msw)) {
         q1 -= 1;
         r0 += d1;
 
         if (r0 >= radix) {
             break;
         }
     }
 
     rAX = (u1 * radix) + (u0_msw - q1 * d);
     q0 = rAX / d1;
     r0 = rAX - q0 * d1;
 
     while (q0 >= radix || (q0 * d0 > radix * r0 + u0_lsw)) {
         q0 -= 1;
         r0 += d1;
 
         if (r0 >= radix) {
             break;
         }
     }
 
     if (r != NULL) {
         *r = (rAX * radix + u0_lsw - q0 * d) >> s;
     }
 
     quotient = q1 * radix + q0;
 
     return quotient;
 #endif
 }
 
 /*
  * Division by mbedtls_mpi: A = Q * B + R  (HAC 14.20)
  */
 int mbedtls_mpi_div_mpi(mbedtls_mpi *Q, mbedtls_mpi *R, const mbedtls_mpi *A,
                         const mbedtls_mpi *B)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     size_t i, n, t, k;
     mbedtls_mpi X, Y, Z, T1, T2;
     mbedtls_mpi_uint TP2[3];
 
     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];
 
     p[0] = mpi_sint_abs(b);
     B.s = TO_SIGN(b);
     B.n = 1;
     B.p = p;
 
     return mbedtls_mpi_div_mpi(Q, R, A, &B);
 }
 
 /*
  * Modulo: R = A mod B
  */
 int mbedtls_mpi_mod_mpi(mbedtls_mpi *R, const mbedtls_mpi *A, const mbedtls_mpi *B)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
 
     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 || 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;
 }
 
 /*
  * Warning! If the parameter E_public has MBEDTLS_MPI_IS_PUBLIC as its value,
  * this function is not constant time with respect to the exponent (parameter E).
  */
 static int mbedtls_mpi_exp_mod_optionally_safe(mbedtls_mpi *X, const mbedtls_mpi *A,
                                                const mbedtls_mpi *E, int E_public,
                                                const mbedtls_mpi *N, mbedtls_mpi *prec_RR)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
 
     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;
     }
 
     /*
      * Ensure that the exponent that we are passing to the core is not NULL.
      */
     if (E->n == 0) {
         ret = mbedtls_mpi_lset(X, 1);
         return ret;
     }
 
     /*
      * Allocate working memory for mbedtls_mpi_core_exp_mod()
      */
     size_t T_limbs = mbedtls_mpi_core_exp_mod_working_limbs(N->n, E->n);
     mbedtls_mpi_uint *T = (mbedtls_mpi_uint *) mbedtls_calloc(T_limbs, sizeof(mbedtls_mpi_uint));
     if (T == NULL) {
         return MBEDTLS_ERR_MPI_ALLOC_FAILED;
     }
 
     mbedtls_mpi RR;
     mbedtls_mpi_init(&RR);
 
     /*
      * If 1st call, pre-compute R^2 mod N
      */
     if (prec_RR == NULL || prec_RR->p == NULL) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_core_get_mont_r2_unsafe(&RR, N));
 
         if (prec_RR != NULL) {
             *prec_RR = RR;
         }
     } else {
         MBEDTLS_MPI_CHK(mbedtls_mpi_grow(prec_RR, N->n));
         RR = *prec_RR;
     }
 
     /*
      * To preserve constness we need to make a copy of A. Using X for this to
      * save memory.
      */
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A));
 
     /*
      * Compensate for negative A (and correct at the end).
      */
     X->s = 1;
 
     /*
      * Make sure that X is in a form that is safe for consumption by
      * the core functions.
      *
      * - The core functions will not touch the limbs of X above N->n. The
      *   result will be correct if those limbs are 0, which the mod call
      *   ensures.
      * - Also, X must have at least as many limbs as N for the calls to the
      *   core functions.
      */
     if (mbedtls_mpi_cmp_mpi(X, N) >= 0) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(X, X, N));
     }
     MBEDTLS_MPI_CHK(mbedtls_mpi_grow(X, N->n));
 
     /*
      * Convert to and from Montgomery around mbedtls_mpi_core_exp_mod().
      */
     {
         mbedtls_mpi_uint mm = mbedtls_mpi_core_montmul_init(N->p);
         mbedtls_mpi_core_to_mont_rep(X->p, X->p, N->p, N->n, mm, RR.p, T);
         if (E_public == MBEDTLS_MPI_IS_PUBLIC) {
             mbedtls_mpi_core_exp_mod_unsafe(X->p, X->p, N->p, N->n, E->p, E->n, RR.p, T);
         } else {
             mbedtls_mpi_core_exp_mod(X->p, X->p, N->p, N->n, E->p, E->n, RR.p, T);
         }
         mbedtls_mpi_core_from_mont_rep(X->p, X->p, N->p, N->n, mm, T);
     }
 
     /*
      * Correct for negative A.
      */
     if (A->s == -1 && (E->p[0] & 1) != 0) {
         mbedtls_ct_condition_t is_x_non_zero = mbedtls_mpi_core_check_zero_ct(X->p, X->n);
         X->s = mbedtls_ct_mpi_sign_if(is_x_non_zero, -1, 1);
 
         MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(X, N, X));
     }
 
 cleanup:
 
     mbedtls_mpi_zeroize_and_free(T, T_limbs);
 
     if (prec_RR == NULL || prec_RR->p == NULL) {
         mbedtls_mpi_free(&RR);
     }
 
     return ret;
 }
 
 int mbedtls_mpi_exp_mod(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *E, const mbedtls_mpi *N,
                         mbedtls_mpi *prec_RR)
 {
     return mbedtls_mpi_exp_mod_optionally_safe(X, A, E, MBEDTLS_MPI_IS_SECRET, N, prec_RR);
 }
 
 int mbedtls_mpi_exp_mod_unsafe(mbedtls_mpi *X, const mbedtls_mpi *A,
                                const mbedtls_mpi *E, const mbedtls_mpi *N,
                                mbedtls_mpi *prec_RR)
 {
     return mbedtls_mpi_exp_mod_optionally_safe(X, A, E, MBEDTLS_MPI_IS_PUBLIC, N, prec_RR);
 }
 
 /*
  * 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;
 
     mbedtls_mpi_init(&TA); mbedtls_mpi_init(&TB);
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TA, A));
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TB, B));
 
     lz = mbedtls_mpi_lsb(&TA);
     lzt = mbedtls_mpi_lsb(&TB);
 
     /* The loop below gives the correct result when A==0 but not when B==0.
      * So have a special case for B==0. Leverage the fact that we just
      * calculated the lsb and lsb(B)==0 iff B is odd or 0 to make the test
      * slightly more efficient than cmp_int(). */
     if (lzt == 0 && mbedtls_mpi_get_bit(&TB, 0) == 0) {
         ret = mbedtls_mpi_copy(G, A);
         goto cleanup;
     }
 
     if (lzt < lz) {
         lz = lzt;
     }
 
     TA.s = TB.s = 1;
 
     /* We mostly follow the procedure described in HAC 14.54, but with some
      * minor differences:
      * - Sequences of multiplications or divisions by 2 are grouped into a
      *   single shift operation.
      * - The procedure in HAC assumes that 0 < TB <= TA.
      *     - The condition TB <= TA is not actually necessary for correctness.
      *       TA and TB have symmetric roles except for the loop termination
      *       condition, and the shifts at the beginning of the loop body
      *       remove any significance from the ordering of TA vs TB before
      *       the shifts.
      *     - If TA = 0, the loop goes through 0 iterations and the result is
      *       correctly TB.
      *     - The case TB = 0 was short-circuited above.
      *
      * For the correctness proof below, decompose the original values of
      * A and B as
      *   A = sa * 2^a * A' with A'=0 or A' odd, and sa = +-1
      *   B = sb * 2^b * B' with B'=0 or B' odd, and sb = +-1
      * Then gcd(A, B) = 2^{min(a,b)} * gcd(A',B'),
      * and gcd(A',B') is odd or 0.
      *
      * At the beginning, we have TA = |A| and TB = |B| so gcd(A,B) = gcd(TA,TB).
      * The code maintains the following invariant:
      *     gcd(A,B) = 2^k * gcd(TA,TB) for some k   (I)
      */
 
     /* Proof that the loop terminates:
      * At each iteration, either the right-shift by 1 is made on a nonzero
      * value and the nonnegative integer bitlen(TA) + bitlen(TB) decreases
      * by at least 1, or the right-shift by 1 is made on zero and then
      * TA becomes 0 which ends the loop (TB cannot be 0 if it is right-shifted
      * since in that case TB is calculated from TB-TA with the condition TB>TA).
      */
     while (mbedtls_mpi_cmp_int(&TA, 0) != 0) {
         /* Divisions by 2 preserve the invariant (I). */
         MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TA, mbedtls_mpi_lsb(&TA)));
         MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TB, mbedtls_mpi_lsb(&TB)));
 
         /* Set either TA or TB to |TA-TB|/2. Since TA and TB are both odd,
          * TA-TB is even so the division by 2 has an integer result.
          * Invariant (I) is preserved since any odd divisor of both TA and TB
          * also divides |TA-TB|/2, and any odd divisor of both TA and |TA-TB|/2
          * also divides TB, and any odd divisor of both TB and |TA-TB|/2 also
          * divides TA.
          */
         if (mbedtls_mpi_cmp_mpi(&TA, &TB) >= 0) {
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(&TA, &TA, &TB));
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TA, 1));
         } else {
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(&TB, &TB, &TA));
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TB, 1));
         }
         /* Note that one of TA or TB is still odd. */
     }
 
     /* By invariant (I), gcd(A,B) = 2^k * gcd(TA,TB) for some k.
      * At the loop exit, TA = 0, so gcd(TA,TB) = TB.
      * - If there was at least one loop iteration, then one of TA or TB is odd,
      *   and TA = 0, so TB is odd and gcd(TA,TB) = gcd(A',B'). In this case,
      *   lz = min(a,b) so gcd(A,B) = 2^lz * TB.
      * - If there was no loop iteration, then A was 0, and gcd(A,B) = B.
      *   In this case, lz = 0 and B = TB so gcd(A,B) = B = 2^lz * TB as well.
      */
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&TB, lz));
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(G, &TB));
 
 cleanup:
 
     mbedtls_mpi_free(&TA); mbedtls_mpi_free(&TB);
 
     return ret;
 }
 
 /*
  * Fill X with size bytes of random.
  * The bytes returned from the RNG are used in a specific order which
  * is suitable for deterministic ECDSA (see the specification of
  * mbedtls_mpi_random() and the implementation in mbedtls_mpi_fill_random()).
  */
 int mbedtls_mpi_fill_random(mbedtls_mpi *X, size_t size,
                             int (*f_rng)(void *, unsigned char *, size_t),
                             void *p_rng)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     const size_t limbs = CHARS_TO_LIMBS(size);
 
     /* Ensure that target MPI has exactly the necessary number of limbs */
     MBEDTLS_MPI_CHK(mbedtls_mpi_resize_clear(X, limbs));
     if (size == 0) {
         return 0;
     }
 
     ret = mbedtls_mpi_core_fill_random(X->p, X->n, size, f_rng, p_rng);
 
 cleanup:
     return ret;
 }
 
 int mbedtls_mpi_random(mbedtls_mpi *X,
                        mbedtls_mpi_sint min,
                        const mbedtls_mpi *N,
                        int (*f_rng)(void *, unsigned char *, size_t),
                        void *p_rng)
 {
     if (min < 0) {
         return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
     }
     if (mbedtls_mpi_cmp_int(N, min) <= 0) {
         return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
     }
 
     /* Ensure that target MPI has exactly the same number of limbs
      * as the upper bound, even if the upper bound has leading zeros.
      * This is necessary for mbedtls_mpi_core_random. */
     int ret = mbedtls_mpi_resize_clear(X, N->n);
     if (ret != 0) {
         return ret;
     }
 
     return mbedtls_mpi_core_random(X->p, min, N->p, X->n, f_rng, p_rng);
 }
 
 /*
  * Modular inverse: X = A^-1 mod N  (HAC 14.61 / 14.64)
  */
 int mbedtls_mpi_inv_mod(mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *N)
 {
     int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
     mbedtls_mpi G, TA, TU, U1, U2, TB, TV, V1, V2;
 
     if (mbedtls_mpi_cmp_int(N, 1) <= 0) {
         return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;
     }
 
     mbedtls_mpi_init(&TA); mbedtls_mpi_init(&TU); mbedtls_mpi_init(&U1); mbedtls_mpi_init(&U2);
     mbedtls_mpi_init(&G); mbedtls_mpi_init(&TB); mbedtls_mpi_init(&TV);
     mbedtls_mpi_init(&V1); mbedtls_mpi_init(&V2);
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_gcd(&G, A, N));
 
     if (mbedtls_mpi_cmp_int(&G, 1) != 0) {
         ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
         goto cleanup;
     }
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_mod_mpi(&TA, A, N));
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TU, &TA));
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TB, N));
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&TV, N));
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&U1, 1));
     MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&U2, 0));
     MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&V1, 0));
     MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&V2, 1));
 
     do {
         while ((TU.p[0] & 1) == 0) {
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TU, 1));
 
             if ((U1.p[0] & 1) != 0 || (U2.p[0] & 1) != 0) {
                 MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&U1, &U1, &TB));
                 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U2, &U2, &TA));
             }
 
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&U1, 1));
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&U2, 1));
         }
 
         while ((TV.p[0] & 1) == 0) {
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&TV, 1));
 
             if ((V1.p[0] & 1) != 0 || (V2.p[0] & 1) != 0) {
                 MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&V1, &V1, &TB));
                 MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V2, &V2, &TA));
             }
 
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&V1, 1));
             MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&V2, 1));
         }
 
         if (mbedtls_mpi_cmp_mpi(&TU, &TV) >= 0) {
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&TU, &TU, &TV));
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U1, &U1, &V1));
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U2, &U2, &V2));
         } else {
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&TV, &TV, &TU));
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V1, &V1, &U1));
             MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V2, &V2, &U2));
         }
     } while (mbedtls_mpi_cmp_int(&TU, 0) != 0);
 
     while (mbedtls_mpi_cmp_int(&V1, 0) < 0) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&V1, &V1, N));
     }
 
     while (mbedtls_mpi_cmp_mpi(&V1, N) >= 0) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&V1, &V1, N));
     }
 
     MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, &V1));
 
 cleanup:
 
     mbedtls_mpi_free(&TA); mbedtls_mpi_free(&TU); mbedtls_mpi_free(&U1); mbedtls_mpi_free(&U2);
     mbedtls_mpi_free(&G); mbedtls_mpi_free(&TB); mbedtls_mpi_free(&TV);
     mbedtls_mpi_free(&V1); mbedtls_mpi_free(&V2);
 
     return ret;
 }
 
 #if defined(MBEDTLS_GENPRIME)
 
 /* Gaps between primes, starting at 3. https://oeis.org/A001223 */
 static const unsigned char small_prime_gaps[] = {
     2, 2, 4, 2, 4, 2, 4, 6,
     2, 6, 4, 2, 4, 6, 6, 2,
     6, 4, 2, 6, 4, 6, 8, 4,
     2, 4, 2, 4, 14, 4, 6, 2,
     10, 2, 6, 6, 4, 6, 6, 2,
     10, 2, 4, 2, 12, 12, 4, 2,
     4, 6, 2, 10, 6, 6, 6, 2,
     6, 4, 2, 10, 14, 4, 2, 4,
     14, 6, 10, 2, 4, 6, 8, 6,
     6, 4, 6, 8, 4, 8, 10, 2,
     10, 2, 6, 4, 6, 8, 4, 2,
     4, 12, 8, 4, 8, 4, 6, 12,
     2, 18, 6, 10, 6, 6, 2, 6,
     10, 6, 6, 2, 6, 6, 4, 2,
     12, 10, 2, 4, 6, 6, 2, 12,
     4, 6, 8, 10, 8, 10, 8, 6,
     6, 4, 8, 6, 4, 8, 4, 14,
     10, 12, 2, 10, 2, 4, 2, 10,
     14, 4, 2, 4, 14, 4, 2, 4,
     20, 4, 8, 10, 8, 4, 6, 6,
     14, 4, 6, 6, 8, 6, /*reaches 997*/
     0 /* the last entry is effectively unused */
 };
 
 /*
  * Small divisors test (X must be positive)
  *
  * Return values:
  * 0: no small factor (possible prime, more tests needed)
  * 1: certain prime
  * MBEDTLS_ERR_MPI_NOT_ACCEPTABLE: certain non-prime
  * other negative: error
  */
 static int mpi_check_small_factors(const mbedtls_mpi *X)
 {
     int ret = 0;
     size_t i;
     mbedtls_mpi_uint r;
     unsigned p = 3; /* The first odd prime */
 
     if ((X->p[0] & 1) == 0) {
         return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
     }
 
     for (i = 0; i < sizeof(small_prime_gaps); p += small_prime_gaps[i], i++) {
         MBEDTLS_MPI_CHK(mbedtls_mpi_mod_int(&r, X, p));
         if (r == 0) {
             if (mbedtls_mpi_cmp_int(X, p) == 0) {
                 return 1;
             } else {
                 return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
             }
         }
     }
 
 cleanup:
     return ret;
 }
 
 /*
  * Miller-Rabin pseudo-primality test  (HAC 4.24)
  */
 static int mpi_miller_rabin(const mbedtls_mpi *X, size_t rounds,
                             int (*f_rng)(void *, unsigned char *, size_t),
                             void *p_rng)
 {
     int ret, count;
     size_t i, j, k, s;
     mbedtls_mpi W, R, T, A, RR;
 
     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;
 
     XX.s = 1;
     XX.n = X->n;
     XX.p = X->p;
 
     if (mbedtls_mpi_cmp_int(&XX, 0) == 0 ||
         mbedtls_mpi_cmp_int(&XX, 1) == 0) {
         return MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
     }
 
     if (mbedtls_mpi_cmp_int(&XX, 2) == 0) {
         return 0;
     }
 
     if ((ret = mpi_check_small_factors(&XX)) != 0) {
         if (ret == 1) {
             return 0;
         }
 
         return ret;
     }
 
     return mpi_miller_rabin(&XX, rounds, f_rng, p_rng);
 }
 
 /*
  * Prime number generation
  *
  * To generate an RSA key in a way recommended by FIPS 186-4, both primes must
  * be either 1024 bits or 1536 bits long, and flags must contain
  * MBEDTLS_MPI_GEN_PRIME_FLAG_LOW_ERR.
  */
 int mbedtls_mpi_gen_prime(mbedtls_mpi *X, size_t nbits, int flags,
                           int (*f_rng)(void *, unsigned char *, size_t),
                           void *p_rng)
 {
 #ifdef MBEDTLS_HAVE_INT64
 // ceil(2^63.5)
 #define CEIL_MAXUINT_DIV_SQRT2 0xb504f333f9de6485ULL
 #else
 // ceil(2^31.5)
 #define CEIL_MAXUINT_DIV_SQRT2 0xb504f334U
 #endif
     int ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE;
     size_t k, n;
     int rounds;
     mbedtls_mpi_uint r;
     mbedtls_mpi Y;
 
     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 */