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 /**
  * \file bignum.h
  *
  * \brief Multi-precision integer library
  */
 /*
  *  Copyright The Mbed TLS Contributors
  *  SPDX-License-Identifier: Apache-2.0 OR GPL-2.0-or-later
  */
 #ifndef MBEDTLS_BIGNUM_H
 #define MBEDTLS_BIGNUM_H
 #include "mbedtls/private_access.h"
 
 #include "tf-psa-crypto/build_info.h"
 #include "psa/crypto_values.h"
 
 #include <stddef.h>
 #include <stdint.h>
 
 #if defined(MBEDTLS_FS_IO)
 #include <stdio.h>
 #endif
 
 /** An error occurred while reading from or writing to a file. */
 #define MBEDTLS_ERR_MPI_FILE_IO_ERROR                     -0x0002
 /** Bad input parameters to function. */
 #define MBEDTLS_ERR_MPI_BAD_INPUT_DATA                    PSA_ERROR_INVALID_ARGUMENT
 /** There is an invalid character in the digit string. */
 #define MBEDTLS_ERR_MPI_INVALID_CHARACTER                 -0x0006
 /** The buffer is too small to write to. */
 #define MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL                  PSA_ERROR_BUFFER_TOO_SMALL
 /** The input arguments are negative or result in illegal output. */
 #define MBEDTLS_ERR_MPI_NEGATIVE_VALUE                    -0x000A
 /** The input argument for division is zero, which is not allowed. */
 #define MBEDTLS_ERR_MPI_DIVISION_BY_ZERO                  -0x000C
 /** The input arguments are not acceptable. */
 #define MBEDTLS_ERR_MPI_NOT_ACCEPTABLE                    -0x000E
 /** Memory allocation failed. */
 #define MBEDTLS_ERR_MPI_ALLOC_FAILED                      PSA_ERROR_INSUFFICIENT_MEMORY
 
 #define MBEDTLS_MPI_CHK(f)       \
     do                           \
     {                            \
         if ((ret = (f)) != 0) \
         goto cleanup;        \
     } while (0)
 
 /*
  * Maximum size MPIs are allowed to grow to in number of limbs.
  */
 #define MBEDTLS_MPI_MAX_LIMBS                             10000
 
 #if !defined(MBEDTLS_MPI_WINDOW_SIZE)
 /*
  * Maximum window size used for modular exponentiation. Default: 3
  * Minimum value: 1. Maximum value: 6.
  *
  * Result is an array of ( 2 ** MBEDTLS_MPI_WINDOW_SIZE ) MPIs used
  * for the sliding window calculation. (So 8 by default)
  *
  * Reduction in size, reduces speed.
  */
 #define MBEDTLS_MPI_WINDOW_SIZE                           3        /**< Maximum window size used. */
 #endif /* !MBEDTLS_MPI_WINDOW_SIZE */
 
 #if !defined(MBEDTLS_MPI_MAX_SIZE)
 /*
  * Maximum size of MPIs allowed in bits and bytes for user-MPIs.
  * ( Default: 512 bytes => 4096 bits, Maximum tested: 2048 bytes => 16384 bits )
  *
  * Note: Calculations can temporarily result in larger MPIs. So the number
  * of limbs required (MBEDTLS_MPI_MAX_LIMBS) is higher.
  */
 #define MBEDTLS_MPI_MAX_SIZE                              1024     /**< Maximum number of bytes for usable MPIs. */
 #endif /* !MBEDTLS_MPI_MAX_SIZE */
 
 #if defined(MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS)
 #define MBEDTLS_MPI_MAX_BITS                              (8 * MBEDTLS_MPI_MAX_SIZE)      /**< Maximum number of bits for usable MPIs. */
 
 /*
  * When reading from files with mbedtls_mpi_read_file() and writing to files with
  * mbedtls_mpi_write_file() the buffer should have space
  * for a (short) label, the MPI (in the provided radix), the newline
  * characters and the '\0'.
  *
  * By default we assume at least a 10 char label, a minimum radix of 10
  * (decimal) and a maximum of 4096 bit numbers (1234 decimal chars).
  * Autosized at compile time for at least a 10 char label, a minimum radix
  * of 10 (decimal) for a number of MBEDTLS_MPI_MAX_BITS size.
  *
  * This used to be statically sized to 1250 for a maximum of 4096 bit
  * numbers (1234 decimal chars).
  *
  * Calculate using the formula:
  *  MBEDTLS_MPI_RW_BUFFER_SIZE = ceil(MBEDTLS_MPI_MAX_BITS / ln(10) * ln(2)) +
  *                                LabelSize + 6
  */
 #define MBEDTLS_MPI_MAX_BITS_SCALE100          (100 * MBEDTLS_MPI_MAX_BITS)
 #define MBEDTLS_LN_2_DIV_LN_10_SCALE100                 332
 #define MBEDTLS_MPI_RW_BUFFER_SIZE             (((MBEDTLS_MPI_MAX_BITS_SCALE100 + \
                                                   MBEDTLS_LN_2_DIV_LN_10_SCALE100 - 1) / \
                                                  MBEDTLS_LN_2_DIV_LN_10_SCALE100) + 10 + 6)
 #endif /* MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS */
 
 /*
  * Define the base integer type, architecture-wise.
  *
  * 32 or 64-bit integer types can be forced regardless of the underlying
  * architecture by defining MBEDTLS_HAVE_INT32 or MBEDTLS_HAVE_INT64
  * respectively and undefining MBEDTLS_HAVE_ASM.
  *
  * Double-width integers (e.g. 128-bit in 64-bit architectures) can be
  * disabled by defining MBEDTLS_NO_UDBL_DIVISION.
  */
 #if !defined(MBEDTLS_HAVE_INT32)
     #if defined(_MSC_VER) && defined(_M_AMD64)
 /* Always choose 64-bit when using MSC */
         #if !defined(MBEDTLS_HAVE_INT64)
             #define MBEDTLS_HAVE_INT64
         #endif /* !MBEDTLS_HAVE_INT64 */
 typedef  int64_t mbedtls_mpi_sint;
 typedef uint64_t mbedtls_mpi_uint;
 #define MBEDTLS_MPI_UINT_MAX                UINT64_MAX
     #elif defined(__GNUC__) && (                         \
     defined(__amd64__) || defined(__x86_64__)     || \
     defined(__ppc64__) || defined(__powerpc64__)  || \
     defined(__ia64__)  || defined(__alpha__)      || \
     (defined(__sparc__) && defined(__arch64__)) || \
     defined(__s390x__) || defined(__mips64)       || \
     defined(__aarch64__))
         #if !defined(MBEDTLS_HAVE_INT64)
             #define MBEDTLS_HAVE_INT64
         #endif /* MBEDTLS_HAVE_INT64 */
 typedef  int64_t mbedtls_mpi_sint;
 typedef uint64_t mbedtls_mpi_uint;
 #define MBEDTLS_MPI_UINT_MAX                UINT64_MAX
         #if !defined(MBEDTLS_NO_UDBL_DIVISION)
 /* mbedtls_t_udbl defined as 128-bit unsigned int */
 typedef unsigned int mbedtls_t_udbl __attribute__((mode(TI)));
             #define MBEDTLS_HAVE_UDBL
         #endif /* !MBEDTLS_NO_UDBL_DIVISION */
     #elif defined(__ARMCC_VERSION) && defined(__aarch64__)
 /*
  * __ARMCC_VERSION is defined for both armcc and armclang and
  * __aarch64__ is only defined by armclang when compiling 64-bit code
  */
         #if !defined(MBEDTLS_HAVE_INT64)
             #define MBEDTLS_HAVE_INT64
         #endif /* !MBEDTLS_HAVE_INT64 */
 typedef  int64_t mbedtls_mpi_sint;
 typedef uint64_t mbedtls_mpi_uint;
 #define MBEDTLS_MPI_UINT_MAX                UINT64_MAX
         #if !defined(MBEDTLS_NO_UDBL_DIVISION)
 /* mbedtls_t_udbl defined as 128-bit unsigned int */
 typedef __uint128_t mbedtls_t_udbl;
             #define MBEDTLS_HAVE_UDBL
         #endif /* !MBEDTLS_NO_UDBL_DIVISION */
     #elif defined(MBEDTLS_HAVE_INT64)
 /* Force 64-bit integers with unknown compiler */
 typedef  int64_t mbedtls_mpi_sint;
 typedef uint64_t mbedtls_mpi_uint;
 #define MBEDTLS_MPI_UINT_MAX                UINT64_MAX
     #endif
 #endif /* !MBEDTLS_HAVE_INT32 */
 
 #if !defined(MBEDTLS_HAVE_INT64)
 /* Default to 32-bit compilation */
     #if !defined(MBEDTLS_HAVE_INT32)
         #define MBEDTLS_HAVE_INT32
     #endif /* !MBEDTLS_HAVE_INT32 */
 typedef  int32_t mbedtls_mpi_sint;
 typedef uint32_t mbedtls_mpi_uint;
 #define MBEDTLS_MPI_UINT_MAX                UINT32_MAX
     #if !defined(MBEDTLS_NO_UDBL_DIVISION)
 typedef uint64_t mbedtls_t_udbl;
         #define MBEDTLS_HAVE_UDBL
     #endif /* !MBEDTLS_NO_UDBL_DIVISION */
 #endif /* !MBEDTLS_HAVE_INT64 */
 
 /*
  * Sanity check that exactly one of MBEDTLS_HAVE_INT32 or MBEDTLS_HAVE_INT64 is defined,
  * so that code elsewhere doesn't have to check.
  */
 #if (!(defined(MBEDTLS_HAVE_INT32) || defined(MBEDTLS_HAVE_INT64))) || \
     (defined(MBEDTLS_HAVE_INT32) && defined(MBEDTLS_HAVE_INT64))
 #error "Only 32-bit or 64-bit limbs are supported in bignum"
 #endif
 
 /** \typedef mbedtls_mpi_uint
  * \brief The type of machine digits in a bignum, called _limbs_.
  *
  * This is always an unsigned integer type with no padding bits. The size
  * is platform-dependent.
  */
 
 /** \typedef mbedtls_mpi_sint
  * \brief The signed type corresponding to #mbedtls_mpi_uint.
  *
  * This is always an signed integer type with no padding bits. The size
  * is platform-dependent.
  */
 
 #ifdef __cplusplus
 extern "C" {
 #endif
 
 /**
  * \brief          MPI structure
  */
 typedef struct mbedtls_mpi {
     /** Pointer to limbs.
      *
      * This may be \c NULL if \c n is 0.
      */
     mbedtls_mpi_uint *MBEDTLS_PRIVATE(p);
 
     /** Sign: -1 if the mpi is negative, 1 otherwise.
      *
      * The number 0 must be represented with `s = +1`. Although many library
      * functions treat all-limbs-zero as equivalent to a valid representation
      * of 0 regardless of the sign bit, there are exceptions, so bignum
      * functions and external callers must always set \c s to +1 for the
      * number zero.
      *
      * Note that this implies that calloc() or `... = {0}` does not create
      * a valid MPI representation. You must call mbedtls_mpi_init().
      */
     signed short MBEDTLS_PRIVATE(s);
 
     /** Total number of limbs in \c p.  */
     unsigned short MBEDTLS_PRIVATE(n);
     /* Make sure that MBEDTLS_MPI_MAX_LIMBS fits in n.
      * Use the same limit value on all platforms so that we don't have to
      * think about different behavior on the rare platforms where
      * unsigned short can store values larger than the minimum required by
      * the C language, which is 65535.
      */
 #if MBEDTLS_MPI_MAX_LIMBS > 65535
 #error "MBEDTLS_MPI_MAX_LIMBS > 65535 is not supported"
 #endif
 }
 mbedtls_mpi;
 
 #define MBEDTLS_MPI_INIT { 0, 1, 0 }
 
 #if defined(MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS)
 /**
  * \brief           Initialize an MPI context.
  *
  *                  This makes the MPI ready to be set or freed,
  *                  but does not define a value for the MPI.
  *
  * \param X         The MPI context to initialize. This must not be \c NULL.
  */
 void mbedtls_mpi_init(mbedtls_mpi *X);
 
 /**
  * \brief          This function frees the components of an MPI context.
  *
  * \param X        The MPI context to be cleared. This may be \c NULL,
  *                 in which case this function is a no-op. If it is
  *                 not \c NULL, it must point to an initialized MPI.
  */
 void mbedtls_mpi_free(mbedtls_mpi *X);
 
 /**
  * \brief          Enlarge an MPI to the specified number of limbs.
  *
  * \note           This function does nothing if the MPI is
  *                 already large enough.
  *
  * \param X        The MPI to grow. It must be initialized.
  * \param nblimbs  The target number of limbs.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on other kinds of failure.
  */
 int mbedtls_mpi_grow(mbedtls_mpi *X, size_t nblimbs);
 
 /**
  * \brief          This function resizes an MPI downwards, keeping at least the
  *                 specified number of limbs.
  *
  *                 If \c X is smaller than \c nblimbs, it is resized up
  *                 instead.
  *
  * \param X        The MPI to shrink. This must point to an initialized MPI.
  * \param nblimbs  The minimum number of limbs to keep.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed
  *                 (this can only happen when resizing up).
  * \return         Another negative error code on other kinds of failure.
  */
 int mbedtls_mpi_shrink(mbedtls_mpi *X, size_t nblimbs);
 
 /**
  * \brief          Make a copy of an MPI.
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param Y        The source MPI. This must point to an initialized MPI.
  *
  * \note           The limb-buffer in the destination MPI is enlarged
  *                 if necessary to hold the value in the source MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on other kinds of failure.
  */
 int mbedtls_mpi_copy(mbedtls_mpi *X, const mbedtls_mpi *Y);
 
 /**
  * \brief          Swap the contents of two MPIs.
  *
  * \param X        The first MPI. It must be initialized.
  * \param Y        The second MPI. It must be initialized.
  */
 void mbedtls_mpi_swap(mbedtls_mpi *X, mbedtls_mpi *Y);
 
 /**
  * \brief          Perform a safe conditional copy of MPI which doesn't
  *                 reveal whether the condition was true or not.
  *
  * \param X        The MPI to conditionally assign to. This must point
  *                 to an initialized MPI.
  * \param Y        The MPI to be assigned from. This must point to an
  *                 initialized MPI.
  * \param assign   The condition deciding whether to perform the
  *                 assignment or not. Must be either 0 or 1:
  *                 * \c 1: Perform the assignment `X = Y`.
  *                 * \c 0: Keep the original value of \p X.
  *
  * \note           This function is equivalent to
  *                      `if( assign ) mbedtls_mpi_copy( X, Y );`
  *                 except that it avoids leaking any information about whether
  *                 the assignment was done or not (the above code may leak
  *                 information through branch prediction and/or memory access
  *                 patterns analysis).
  *
  * \warning        If \p assign is neither 0 nor 1, the result of this function
  *                 is indeterminate, and the resulting value in \p X might be
  *                 neither its original value nor the value in \p Y.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on other kinds of failure.
  */
 int mbedtls_mpi_safe_cond_assign(mbedtls_mpi *X, const mbedtls_mpi *Y, unsigned char assign);
 
 /**
  * \brief          Perform a safe conditional swap which doesn't
  *                 reveal whether the condition was true or not.
  *
  * \param X        The first MPI. This must be initialized.
  * \param Y        The second MPI. This must be initialized.
  * \param swap     The condition deciding whether to perform
  *                 the swap or not. Must be either 0 or 1:
  *                 * \c 1: Swap the values of \p X and \p Y.
  *                 * \c 0: Keep the original values of \p X and \p Y.
  *
  * \note           This function is equivalent to
  *                      if( swap ) mbedtls_mpi_swap( X, Y );
  *                 except that it avoids leaking any information about whether
  *                 the swap was done or not (the above code may leak
  *                 information through branch prediction and/or memory access
  *                 patterns analysis).
  *
  * \warning        If \p swap is neither 0 nor 1, the result of this function
  *                 is indeterminate, and both \p X and \p Y might end up with
  *                 values different to either of the original ones.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on other kinds of failure.
  *
  */
 int mbedtls_mpi_safe_cond_swap(mbedtls_mpi *X, mbedtls_mpi *Y, unsigned char swap);
 
 /**
  * \brief          Store integer value in MPI.
  *
  * \param X        The MPI to set. This must be initialized.
  * \param z        The value to use.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on other kinds of failure.
  */
 int mbedtls_mpi_lset(mbedtls_mpi *X, mbedtls_mpi_sint z);
 
 /**
  * \brief          Get a specific bit from an MPI.
  *
  * \param X        The MPI to query. This must be initialized.
  * \param pos      Zero-based index of the bit to query.
  *
  * \return         \c 0 or \c 1 on success, depending on whether bit \c pos
  *                 of \c X is unset or set.
  * \return         A negative error code on failure.
  */
 int mbedtls_mpi_get_bit(const mbedtls_mpi *X, size_t pos);
 
 /**
  * \brief          Modify a specific bit in an MPI.
  *
  * \note           This function will grow the target MPI if necessary to set a
  *                 bit to \c 1 in a not yet existing limb. It will not grow if
  *                 the bit should be set to \c 0.
  *
  * \param X        The MPI to modify. This must be initialized.
  * \param pos      Zero-based index of the bit to modify.
  * \param val      The desired value of bit \c pos: \c 0 or \c 1.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on other kinds of failure.
  */
 int mbedtls_mpi_set_bit(mbedtls_mpi *X, size_t pos, unsigned char val);
 
 /**
  * \brief          Return the number of bits of value \c 0 before the
  *                 least significant bit of value \c 1.
  *
  * \note           This is the same as the zero-based index of
  *                 the least significant bit of value \c 1.
  *
  * \param X        The MPI to query.
  *
  * \return         The number of bits of value \c 0 before the least significant
  *                 bit of value \c 1 in \p X.
  */
 size_t mbedtls_mpi_lsb(const mbedtls_mpi *X);
 
 /**
  * \brief          Return the number of bits up to and including the most
  *                 significant bit of value \c 1.
  *
  * * \note         This is same as the one-based index of the most
  *                 significant bit of value \c 1.
  *
  * \param X        The MPI to query. This must point to an initialized MPI.
  *
  * \return         The number of bits up to and including the most
  *                 significant bit of value \c 1.
  */
 size_t mbedtls_mpi_bitlen(const mbedtls_mpi *X);
 
 /**
  * \brief          Return the total size of an MPI value in bytes.
  *
  * \param X        The MPI to use. This must point to an initialized MPI.
  *
  * \note           The value returned by this function may be less than
  *                 the number of bytes used to store \p X internally.
  *                 This happens if and only if there are trailing bytes
  *                 of value zero.
  *
  * \return         The least number of bytes capable of storing
  *                 the absolute value of \p X.
  */
 size_t mbedtls_mpi_size(const mbedtls_mpi *X);
 
 /**
  * \brief          Import an MPI from an ASCII string.
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param radix    The numeric base of the input string.
  * \param s        Null-terminated string buffer.
  *
  * \return         \c 0 if successful.
  * \return         A negative error code on failure.
  */
 int mbedtls_mpi_read_string(mbedtls_mpi *X, int radix, const char *s);
 
 /**
  * \brief          Export an MPI to an ASCII string.
  *
  * \param X        The source MPI. This must point to an initialized MPI.
  * \param radix    The numeric base of the output string.
  * \param buf      The buffer to write the string to. This must be writable
  *                 buffer of length \p buflen Bytes.
  * \param buflen   The available size in Bytes of \p buf.
  * \param olen     The address at which to store the length of the string
  *                 written, including the  final \c NULL byte. This must
  *                 not be \c NULL.
  *
  * \note           You can call this function with `buflen == 0` to obtain the
  *                 minimum required buffer size in `*olen`.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if the target buffer \p buf
  *                 is too small to hold the value of \p X in the desired base.
  *                 In this case, `*olen` is nonetheless updated to contain the
  *                 size of \p buf required for a successful call.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_write_string(const mbedtls_mpi *X, int radix,
                              char *buf, size_t buflen, size_t *olen);
 
 #if defined(MBEDTLS_FS_IO)
 /**
  * \brief          Read an MPI from a line in an opened file.
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param radix    The numeric base of the string representation used
  *                 in the source line.
  * \param fin      The input file handle to use. This must not be \c NULL.
  *
  * \note           On success, this function advances the file stream
  *                 to the end of the current line or to EOF.
  *
  *                 The function returns \c 0 on an empty line.
  *
  *                 Leading whitespaces are ignored, as is a
  *                 '0x' prefix for radix \c 16.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if the file read buffer
  *                 is too small.
  * \return         Another negative error code on failure.
  */
 int mbedtls_mpi_read_file(mbedtls_mpi *X, int radix, FILE *fin);
 
 /**
  * \brief          Export an MPI into an opened file.
  *
  * \param p        A string prefix to emit prior to the MPI data.
  *                 For example, this might be a label, or "0x" when
  *                 printing in base \c 16. This may be \c NULL if no prefix
  *                 is needed.
  * \param X        The source MPI. This must point to an initialized MPI.
  * \param radix    The numeric base to be used in the emitted string.
  * \param fout     The output file handle. This may be \c NULL, in which case
  *                 the output is written to \c stdout.
  *
  * \return         \c 0 if successful.
  * \return         A negative error code on failure.
  */
 int mbedtls_mpi_write_file(const char *p, const mbedtls_mpi *X,
                            int radix, FILE *fout);
 #endif /* MBEDTLS_FS_IO */
 
 /**
  * \brief          Import an MPI from unsigned big endian binary data.
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param buf      The input buffer. This must be a readable buffer of length
  *                 \p buflen Bytes.
  * \param buflen   The length of the input buffer \p buf in Bytes.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_read_binary(mbedtls_mpi *X, const unsigned char *buf,
                             size_t buflen);
 
 /**
  * \brief          Import X from unsigned binary data, little endian
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param buf      The input buffer. This must be a readable buffer of length
  *                 \p buflen Bytes.
  * \param buflen   The length of the input buffer \p buf in Bytes.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_read_binary_le(mbedtls_mpi *X,
                                const unsigned char *buf, size_t buflen);
 
 /**
  * \brief          Export X into unsigned binary data, big endian.
  *                 Always fills the whole buffer, which will start with zeros
  *                 if the number is smaller.
  *
  * \param X        The source MPI. This must point to an initialized MPI.
  * \param buf      The output buffer. This must be a writable buffer of length
  *                 \p buflen Bytes.
  * \param buflen   The size of the output buffer \p buf in Bytes.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p buf isn't
  *                 large enough to hold the value of \p X.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_write_binary(const mbedtls_mpi *X, unsigned char *buf,
                              size_t buflen);
 
 /**
  * \brief          Export X into unsigned binary data, little endian.
  *                 Always fills the whole buffer, which will end with zeros
  *                 if the number is smaller.
  *
  * \param X        The source MPI. This must point to an initialized MPI.
  * \param buf      The output buffer. This must be a writable buffer of length
  *                 \p buflen Bytes.
  * \param buflen   The size of the output buffer \p buf in Bytes.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p buf isn't
  *                 large enough to hold the value of \p X.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_write_binary_le(const mbedtls_mpi *X,
                                 unsigned char *buf, size_t buflen);
 
 /**
  * \brief          Perform a left-shift on an MPI: X <<= count
  *
  * \param X        The MPI to shift. This must point to an initialized MPI.
  *                 The MPI pointed by \p X may be resized to fit
  *                 the resulting number.
  * \param count    The number of bits to shift by.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_shift_l(mbedtls_mpi *X, size_t count);
 
 /**
  * \brief          Perform a right-shift on an MPI: X >>= count
  *
  * \param X        The MPI to shift. This must point to an initialized MPI.
  * \param count    The number of bits to shift by.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_shift_r(mbedtls_mpi *X, size_t count);
 
 /**
  * \brief          Compare the absolute values of two MPIs.
  *
  * \param X        The left-hand MPI. This must point to an initialized MPI.
  * \param Y        The right-hand MPI. This must point to an initialized MPI.
  *
  * \return         \c 1 if `|X|` is greater than `|Y|`.
  * \return         \c -1 if `|X|` is lesser than `|Y|`.
  * \return         \c 0 if `|X|` is equal to `|Y|`.
  */
 int mbedtls_mpi_cmp_abs(const mbedtls_mpi *X, const mbedtls_mpi *Y);
 
 /**
  * \brief          Compare two MPIs.
  *
  * \param X        The left-hand MPI. This must point to an initialized MPI.
  * \param Y        The right-hand MPI. This must point to an initialized MPI.
  *
  * \return         \c 1 if \p X is greater than \p Y.
  * \return         \c -1 if \p X is lesser than \p Y.
  * \return         \c 0 if \p X is equal to \p Y.
  */
 int mbedtls_mpi_cmp_mpi(const mbedtls_mpi *X, const mbedtls_mpi *Y);
 
 /**
  * \brief          Check if an MPI is less than the other in constant time.
  *
  * \param X        The left-hand MPI. This must point to an initialized MPI
  *                 with the same allocated length as Y.
  * \param Y        The right-hand MPI. This must point to an initialized MPI
  *                 with the same allocated length as X.
  * \param ret      The result of the comparison:
  *                 \c 1 if \p X is less than \p Y.
  *                 \c 0 if \p X is greater than or equal to \p Y.
  *
  * \return         0 on success.
  * \return         MBEDTLS_ERR_MPI_BAD_INPUT_DATA if the allocated length of
  *                 the two input MPIs is not the same.
  */
 int mbedtls_mpi_lt_mpi_ct(const mbedtls_mpi *X, const mbedtls_mpi *Y,
                           unsigned *ret);
 
 /**
  * \brief          Compare an MPI with an integer.
  *
  * \param X        The left-hand MPI. This must point to an initialized MPI.
  * \param z        The integer value to compare \p X to.
  *
  * \return         \c 1 if \p X is greater than \p z.
  * \return         \c -1 if \p X is lesser than \p z.
  * \return         \c 0 if \p X is equal to \p z.
  */
 int mbedtls_mpi_cmp_int(const mbedtls_mpi *X, mbedtls_mpi_sint z);
 
 /**
  * \brief          Perform an unsigned addition of MPIs: X = |A| + |B|
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The first summand. This must point to an initialized MPI.
  * \param B        The second summand. This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_add_abs(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *B);
 
 /**
  * \brief          Perform an unsigned subtraction of MPIs: X = |A| - |B|
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The minuend. This must point to an initialized MPI.
  * \param B        The subtrahend. This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_NEGATIVE_VALUE if \p B is greater than \p A.
  * \return         Another negative error code on different kinds of failure.
  *
  */
 int mbedtls_mpi_sub_abs(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *B);
 
 /**
  * \brief          Perform a signed addition of MPIs: X = A + B
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The first summand. This must point to an initialized MPI.
  * \param B        The second summand. This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_add_mpi(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *B);
 
 /**
  * \brief          Perform a signed subtraction of MPIs: X = A - B
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The minuend. This must point to an initialized MPI.
  * \param B        The subtrahend. This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_sub_mpi(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *B);
 
 /**
  * \brief          Perform a signed addition of an MPI and an integer: X = A + b
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The first summand. This must point to an initialized MPI.
  * \param b        The second summand.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_add_int(mbedtls_mpi *X, const mbedtls_mpi *A,
                         mbedtls_mpi_sint b);
 
 /**
  * \brief          Perform a signed subtraction of an MPI and an integer:
  *                 X = A - b
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The minuend. This must point to an initialized MPI.
  * \param b        The subtrahend.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_sub_int(mbedtls_mpi *X, const mbedtls_mpi *A,
                         mbedtls_mpi_sint b);
 
 /**
  * \brief          Perform a multiplication of two MPIs: X = A * B
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The first factor. This must point to an initialized MPI.
  * \param B        The second factor. This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  *
  */
 int mbedtls_mpi_mul_mpi(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *B);
 
 /**
  * \brief          Perform a multiplication of an MPI with an unsigned integer:
  *                 X = A * b
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param A        The first factor. This must point to an initialized MPI.
  * \param b        The second factor.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  *
  */
 int mbedtls_mpi_mul_int(mbedtls_mpi *X, const mbedtls_mpi *A,
                         mbedtls_mpi_uint b);
 
 /**
  * \brief          Perform a division with remainder of two MPIs:
  *                 A = Q * B + R
  *
  * \param Q        The destination MPI for the quotient.
  *                 This may be \c NULL if the value of the
  *                 quotient is not needed. This must not alias A or B.
  * \param R        The destination MPI for the remainder value.
  *                 This may be \c NULL if the value of the
  *                 remainder is not needed. This must not alias A or B.
  * \param A        The dividend. This must point to an initialized MPI.
  * \param B        The divisor. This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_DIVISION_BY_ZERO if \p B equals zero.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_div_mpi(mbedtls_mpi *Q, mbedtls_mpi *R, const mbedtls_mpi *A,
                         const mbedtls_mpi *B);
 
 /**
  * \brief          Perform a division with remainder of an MPI by an integer:
  *                 A = Q * b + R
  *
  * \param Q        The destination MPI for the quotient.
  *                 This may be \c NULL if the value of the
  *                 quotient is not needed.  This must not alias A.
  * \param R        The destination MPI for the remainder value.
  *                 This may be \c NULL if the value of the
  *                 remainder is not needed.  This must not alias A.
  * \param A        The dividend. This must point to an initialized MPi.
  * \param b        The divisor.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_DIVISION_BY_ZERO if \p b equals zero.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_div_int(mbedtls_mpi *Q, mbedtls_mpi *R, const mbedtls_mpi *A,
                         mbedtls_mpi_sint b);
 
 /**
  * \brief          Perform a modular reduction. R = A mod B
  *
  * \param R        The destination MPI for the residue value.
  *                 This must point to an initialized MPI.
  * \param A        The MPI to compute the residue of.
  *                 This must point to an initialized MPI.
  * \param B        The base of the modular reduction.
  *                 This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_DIVISION_BY_ZERO if \p B equals zero.
  * \return         #MBEDTLS_ERR_MPI_NEGATIVE_VALUE if \p B is negative.
  * \return         Another negative error code on different kinds of failure.
  *
  */
 int mbedtls_mpi_mod_mpi(mbedtls_mpi *R, const mbedtls_mpi *A,
                         const mbedtls_mpi *B);
 
 /**
  * \brief          Perform a modular reduction with respect to an integer.
  *                 r = A mod b
  *
  * \param r        The address at which to store the residue.
  *                 This must not be \c NULL.
  * \param A        The MPI to compute the residue of.
  *                 This must point to an initialized MPi.
  * \param b        The integer base of the modular reduction.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_DIVISION_BY_ZERO if \p b equals zero.
  * \return         #MBEDTLS_ERR_MPI_NEGATIVE_VALUE if \p b is negative.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_mod_int(mbedtls_mpi_uint *r, const mbedtls_mpi *A,
                         mbedtls_mpi_sint b);
 
 /**
  * \brief          Perform a modular exponentiation: X = A^E mod N
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  *                 This must not alias E or N.
  * \param A        The base of the exponentiation.
  *                 This must point to an initialized MPI.
  * \param E        The exponent MPI. This must point to an initialized MPI.
  * \param N        The base for the modular reduction. This must point to an
  *                 initialized MPI.
  * \param prec_RR  A helper MPI depending solely on \p N which can be used to
  *                 speed-up multiple modular exponentiations for the same value
  *                 of \p N. This may be \c NULL. If it is not \c NULL, it must
  *                 point to an initialized MPI. If it hasn't been used after
  *                 the call to mbedtls_mpi_init(), this function will compute
  *                 the helper value and store it in \p prec_RR for reuse on
  *                 subsequent calls to this function. Otherwise, the function
  *                 will assume that \p prec_RR holds the helper value set by a
  *                 previous call to mbedtls_mpi_exp_mod(), and reuse it.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \c N is negative or
  *                 even, or if \c E is negative.
  * \return         Another negative error code on different kinds of failures.
  *
  */
 int mbedtls_mpi_exp_mod(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *E, const mbedtls_mpi *N,
                         mbedtls_mpi *prec_RR);
 
 /**
  * \brief          Fill an MPI with a number of random bytes.
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param size     The number of random bytes to generate.
  * \param f_rng    The RNG function to use. This must not be \c NULL.
  * \param p_rng    The RNG parameter to be passed to \p f_rng. This may be
  *                 \c NULL if \p f_rng doesn't need a context argument.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on failure.
  *
  * \note           The bytes obtained from the RNG are interpreted
  *                 as a big-endian representation of an MPI; this can
  *                 be relevant in applications like deterministic ECDSA.
  */
 int mbedtls_mpi_fill_random(mbedtls_mpi *X, size_t size,
                             int (*f_rng)(void *, unsigned char *, size_t),
                             void *p_rng);
 
 /** Generate a random number uniformly in a range.
  *
  * This function generates a random number between \p min inclusive and
  * \p N exclusive.
  *
  * The procedure complies with RFC 6979 §3.3 (deterministic ECDSA)
  * when the RNG is a suitably parametrized instance of HMAC_DRBG
  * and \p min is \c 1.
  *
  * \note           There are `N - min` possible outputs. The lower bound
  *                 \p min can be reached, but the upper bound \p N cannot.
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  * \param min      The minimum value to return.
  *                 It must be nonnegative.
  * \param N        The upper bound of the range, exclusive.
  *                 In other words, this is one plus the maximum value to return.
  *                 \p N must be strictly larger than \p min.
  * \param f_rng    The RNG function to use. This must not be \c NULL.
  * \param p_rng    The RNG parameter to be passed to \p f_rng.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p min or \p N is invalid
  *                 or if they are incompatible.
  * \return         #MBEDTLS_ERR_MPI_NOT_ACCEPTABLE if the implementation was
  *                 unable to find a suitable value within a limited number
  *                 of attempts. This has a negligible probability if \p N
  *                 is significantly larger than \p min, which is the case
  *                 for all usual cryptographic applications.
  * \return         Another negative error code on failure.
  */
 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);
 
 /**
  * \brief          Compute the greatest common divisor: G = gcd(A, B)
  *
  * \param G        The destination MPI. This must point to an initialized MPI.
  *                 This will always be positive or 0.
  * \param A        The first operand. This must point to an initialized MPI.
  * \param B        The second operand. This must point to an initialized MPI.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         Another negative error code on different kinds of failure.
  */
 int mbedtls_mpi_gcd(mbedtls_mpi *G, const mbedtls_mpi *A,
                     const mbedtls_mpi *B);
 
 /**
  * \brief          Compute the modular inverse: X = A^-1 mod N
  *
  * \param X        The destination MPI. This must point to an initialized MPI.
  *                 The value returned on success will be between [1, N-1].
  * \param A        The MPI to calculate the modular inverse of. This must point
  *                 to an initialized MPI. This value can be negative, in which
  *                 case a positive answer will still be returned in \p X.
  * \param N        The base of the modular inversion. This must point to an
  *                 initialized MPI and be greater than one.
  *
  * \return         \c 0 if successful.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p N is less than
  *                 or equal to one.
  * \return         #MBEDTLS_ERR_MPI_NOT_ACCEPTABLE if \p A has no modular
  *                 inverse with respect to \p N.
  */
 int mbedtls_mpi_inv_mod(mbedtls_mpi *X, const mbedtls_mpi *A,
                         const mbedtls_mpi *N);
 
 /**
  * \brief          Miller-Rabin primality test.
  *
  * \warning        If \p X is potentially generated by an adversary, for example
  *                 when validating cryptographic parameters that you didn't
  *                 generate yourself and that are supposed to be prime, then
  *                 \p rounds should be at least the half of the security
  *                 strength of the cryptographic algorithm. On the other hand,
  *                 if \p X is chosen uniformly or non-adversarially (as is the
  *                 case when mbedtls_mpi_gen_prime calls this function), then
  *                 \p rounds can be much lower.
  *
  * \param X        The MPI to check for primality.
  *                 This must point to an initialized MPI.
  * \param rounds   The number of bases to perform the Miller-Rabin primality
  *                 test for. The probability of returning 0 on a composite is
  *                 at most 2<sup>-2*\p rounds </sup>.
  * \param f_rng    The RNG function to use. This must not be \c NULL.
  * \param p_rng    The RNG parameter to be passed to \p f_rng.
  *                 This may be \c NULL if \p f_rng doesn't use
  *                 a context parameter.
  *
  * \return         \c 0 if successful, i.e. \p X is probably prime.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_NOT_ACCEPTABLE if \p X is not prime.
  * \return         Another negative error code on other kinds of failure.
  */
 int mbedtls_mpi_is_prime_ext(const mbedtls_mpi *X, int rounds,
                              int (*f_rng)(void *, unsigned char *, size_t),
                              void *p_rng);
 /**
  * \brief Flags for mbedtls_mpi_gen_prime()
  *
  * Each of these flags is a constraint on the result X returned by
  * mbedtls_mpi_gen_prime().
  */
 typedef enum {
     MBEDTLS_MPI_GEN_PRIME_FLAG_DH =      0x0001, /**< (X-1)/2 is prime too */
     MBEDTLS_MPI_GEN_PRIME_FLAG_LOW_ERR = 0x0002, /**< lower error rate from 2<sup>-80</sup> to 2<sup>-128</sup> */
 } mbedtls_mpi_gen_prime_flag_t;
 
 /**
  * \brief          Generate a prime number.
  *
  * \param X        The destination MPI to store the generated prime in.
  *                 This must point to an initialized MPi.
  * \param nbits    The required size of the destination MPI in bits.
  *                 This must be between \c 3 and #MBEDTLS_MPI_MAX_BITS.
  * \param flags    A mask of flags of type #mbedtls_mpi_gen_prime_flag_t.
  * \param f_rng    The RNG function to use. This must not be \c NULL.
  * \param p_rng    The RNG parameter to be passed to \p f_rng.
  *                 This may be \c NULL if \p f_rng doesn't use
  *                 a context parameter.
  *
  * \return         \c 0 if successful, in which case \p X holds a
  *                 probably prime number.
  * \return         #MBEDTLS_ERR_MPI_ALLOC_FAILED if a memory allocation failed.
  * \return         #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if `nbits` is not between
  *                 \c 3 and #MBEDTLS_MPI_MAX_BITS.
  */
 int mbedtls_mpi_gen_prime(mbedtls_mpi *X, size_t nbits, int flags,
                           int (*f_rng)(void *, unsigned char *, size_t),
                           void *p_rng);
 
 /**
  * \brief       Retrieve an integer ASN.1 tag and its value.
  *              Updates the pointer to immediately behind the full tag.
  *              Legacy function, which is now for internal use only.
  *              Please use mbedtls_asn1_get_integer() instead.
  *
  * \param p     On entry, \c *p points to the start of the ASN.1 element.
  *              On successful completion, \c *p points to the first byte
  *              beyond the ASN.1 element.
  *              On error, the value of \c *p is undefined.
  * \param end   End of data.
  * \param X     On success, the parsed value.
  *
  * \return      0 if successful.
  * \return      An ASN.1 error code if the input does not start with
  *              a valid ASN.1 INTEGER.
  * \return      #MBEDTLS_ERR_ASN1_INVALID_LENGTH if the parsed value does
  *              not fit in an \c int.
  * \return      An MPI error code if the parsed value is too large.
  */
 int mbedtls_asn1_get_mpi(unsigned char **p,
                          const unsigned char *end,
                          mbedtls_mpi *X);
 
 #if defined(MBEDTLS_SELF_TEST)
 
 /**
  * \brief          Checkup routine
  *
  * \return         0 if successful, or 1 if the test failed
  */
 int mbedtls_mpi_self_test(int verbose);
 
 #endif /* MBEDTLS_SELF_TEST */
 
 #endif /* MBEDTLS_DECLARE_PRIVATE_IDENTIFIERS */
 /**
  * \brief           Write an arbitrary-precision number (#MBEDTLS_ASN1_INTEGER)
  *                  in ASN.1 format.
  *
  * \note            This function works backwards in data buffer.
  *
  * \param p         The reference to the current position pointer.
  * \param start     The start of the buffer, for bounds-checking.
  * \param X         The MPI to write.
  *                  It must be non-negative.
  *
  * \return          The number of bytes written to \p p on success.
  * \return          A negative \c MBEDTLS_ERR_ASN1_XXX error code on failure.
  */
 int mbedtls_asn1_write_mpi(unsigned char **p, const unsigned char *start,
                            const mbedtls_mpi *X);
 
 #ifdef __cplusplus
 }
 #endif
 
 #endif /* bignum.h */

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