integer.h

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// integer.h - originally written and placed in the public domain by Wei Dai

/// \file integer.h
/// \brief Multiple precision integer with arithmetic operations
/// \details The Integer class can represent positive and negative integers
///   with absolute value less than (256**sizeof(word))<sup>(256**sizeof(int))</sup>.
/// \details Internally, the library uses a sign magnitude representation, and the class
///   has two data members. The first is a IntegerSecBlock (a SecBlock<word>) and it is
///   used to hold the representation. The second is a Sign (an enumeration), and it is
///   used to track the sign of the Integer.
/// \details For details on how the Integer class initializes its function pointers using
///   InitializeInteger and how it creates Integer::Zero(), Integer::One(), and
///   Integer::Two(), then see the comments at the top of <tt>integer.cpp</tt>.
/// \since Crypto++ 1.0

#ifndef CRYPTOPP_INTEGER_H
#define CRYPTOPP_INTEGER_H

#include "cryptlib.h"
#include "secblock.h"
#include "stdcpp.h"

#include <iosfwd>

NAMESPACE_BEGIN(CryptoPP)

/// \struct InitializeInteger
/// \brief Performs static initialization of the Integer class
struct InitializeInteger
{
    InitializeInteger();
};

// Always align, http://github.com/weidai11/cryptopp/issues/256
typedef SecBlock<word, AllocatorWithCleanup<word, true> > IntegerSecBlock;

/// \brief Multiple precision integer with arithmetic operations
/// \details The Integer class can represent positive and negative integers
///   with absolute value less than (256**sizeof(word))<sup>(256**sizeof(int))</sup>.
/// \details Internally, the library uses a sign magnitude representation, and the class
///   has two data members. The first is a IntegerSecBlock (a SecBlock<word>) and it is
///   used to hold the representation. The second is a Sign (an enumeration), and it is
///   used to track the sign of the Integer.
/// \details For details on how the Integer class initializes its function pointers using
///   InitializeInteger and how it creates Integer::Zero(), Integer::One(), and
///   Integer::Two(), then see the comments at the top of <tt>integer.cpp</tt>.
/// \since Crypto++ 1.0
/// \nosubgrouping
class CRYPTOPP_DLL Integer : private InitializeInteger, public ASN1Object
{
public:
    /// \name ENUMS, EXCEPTIONS, and TYPEDEFS
    //@{
        /// \brief Exception thrown when division by 0 is encountered
        class DivideByZero : public Exception
        {
        public:
            DivideByZero() : Exception(OTHER_ERROR, "Integer: division by zero") {}
        };

        /// \brief Exception thrown when a random number cannot be found that
        ///   satisfies the condition
        class RandomNumberNotFound : public Exception
        {
        public:
            RandomNumberNotFound() : Exception(OTHER_ERROR, "Integer: no integer satisfies the given parameters") {}
        };

        /// \enum Sign
        /// \brief Used internally to represent the integer
        /// \details Sign is used internally to represent the integer. It is also used in a few API functions.
        /// \sa SetPositive(), SetNegative(), Signedness
        enum Sign {
            /// \brief the value is positive or 0
            POSITIVE=0,
            /// \brief the value is negative
            NEGATIVE=1};

        /// \enum Signedness
        /// \brief Used when importing and exporting integers
        /// \details Signedness is usually used in API functions.
        /// \sa Sign
        enum Signedness {
            /// \brief an unsigned value
            UNSIGNED,
            /// \brief a signed value
            SIGNED};

        /// \enum RandomNumberType
        /// \brief Properties of a random integer
        enum RandomNumberType {
            /// \brief a number with no special properties
            ANY,
            /// \brief a number which is probabilistically prime
            PRIME};
    //@}

    /// \name CREATORS
    //@{
        /// \brief Creates the zero integer
        Integer();

        /// copy constructor
        Integer(const Integer& t);

        /// \brief Convert from signed long
        Integer(signed long value);

        /// \brief Convert from lword
        /// \param sign enumeration indicating Sign
        /// \param value the long word
        Integer(Sign sign, lword value);

        /// \brief Convert from two words
        /// \param sign enumeration indicating Sign
        /// \param highWord the high word
        /// \param lowWord the low word
        Integer(Sign sign, word highWord, word lowWord);

        /// \brief Convert from a C-string
        /// \param str C-string value
        /// \param order the ByteOrder of the string to be processed
        /// \details \p str can be in base 2, 8, 10, or 16. Base is determined by a case
        ///   insensitive suffix of 'h', 'o', or 'b'.  No suffix means base 10.
        /// \details Byte order was added at Crypto++ 5.7 to allow use of little-endian
        ///   integers with curve25519, Poly1305 and Microsoft CAPI.
        explicit Integer(const char *str, ByteOrder order = BIG_ENDIAN_ORDER);

        /// \brief Convert from a wide C-string
        /// \param str wide C-string value
        /// \param order the ByteOrder of the string to be processed
        /// \details \p str can be in base 2, 8, 10, or 16. Base is determined by a case
        ///   insensitive suffix of 'h', 'o', or 'b'.  No suffix means base 10.
        /// \details Byte order was added at Crypto++ 5.7 to allow use of little-endian
        ///   integers with curve25519, Poly1305 and Microsoft CAPI.
        explicit Integer(const wchar_t *str, ByteOrder order = BIG_ENDIAN_ORDER);

        /// \brief Convert from a big-endian byte array
        /// \param encodedInteger big-endian byte array
        /// \param byteCount length of the byte array
        /// \param sign enumeration indicating Signedness
        /// \param order the ByteOrder of the array to be processed
        /// \details Byte order was added at Crypto++ 5.7 to allow use of little-endian
        ///   integers with curve25519, Poly1305 and Microsoft CAPI.
        Integer(const byte *encodedInteger, size_t byteCount, Signedness sign=UNSIGNED, ByteOrder order = BIG_ENDIAN_ORDER);

        /// \brief Convert from a big-endian array
        /// \param bt BufferedTransformation object with big-endian byte array
        /// \param byteCount length of the byte array
        /// \param sign enumeration indicating Signedness
        /// \param order the ByteOrder of the data to be processed
        /// \details Byte order was added at Crypto++ 5.7 to allow use of little-endian
        ///   integers with curve25519, Poly1305 and Microsoft CAPI.
        Integer(BufferedTransformation &bt, size_t byteCount, Signedness sign=UNSIGNED, ByteOrder order = BIG_ENDIAN_ORDER);

        /// \brief Convert from a BER encoded byte array
        /// \param bt BufferedTransformation object with BER encoded byte array
        explicit Integer(BufferedTransformation &bt);

        /// \brief Create a random integer
        /// \param rng RandomNumberGenerator used to generate material
        /// \param bitCount the number of bits in the resulting integer
        /// \details The random integer created is uniformly distributed over <tt>[0, 2<sup>bitCount</sup>]</tt>.
        Integer(RandomNumberGenerator &rng, size_t bitCount);

        /// \brief Integer representing 0
        /// \return an Integer representing 0
        /// \details Zero() avoids calling constructors for frequently used integers
        static const Integer & CRYPTOPP_API Zero();
        /// \brief Integer representing 1
        /// \return an Integer representing 1
        /// \details One() avoids calling constructors for frequently used integers
        static const Integer & CRYPTOPP_API One();
        /// \brief Integer representing 2
        /// \return an Integer representing 2
        /// \details Two() avoids calling constructors for frequently used integers
        static const Integer & CRYPTOPP_API Two();

        /// \brief Create a random integer of special form
        /// \param rng RandomNumberGenerator used to generate material
        /// \param min the minimum value
        /// \param max the maximum value
        /// \param rnType RandomNumberType to specify the type
        /// \param equiv the equivalence class based on the parameter \p mod
        /// \param mod the modulus used to reduce the equivalence class
        /// \throw RandomNumberNotFound if the set is empty.
        /// \details Ideally, the random integer created should be uniformly distributed
        ///   over <tt>{x | min <= x <= max</tt> and \p x is of rnType and <tt>x \% mod == equiv}</tt>.
        ///   However the actual distribution may not be uniform because sequential
        ///   search is used to find an appropriate number from a random starting
        ///   point.
        /// \details May return (with very small probability) a pseudoprime when a prime
        ///   is requested and <tt>max > lastSmallPrime*lastSmallPrime</tt>. \p lastSmallPrime
        ///   is declared in nbtheory.h.
        Integer(RandomNumberGenerator &rng, const Integer &min, const Integer &max, RandomNumberType rnType=ANY, const Integer &equiv=Zero(), const Integer &mod=One());

        /// \brief Exponentiates to a power of 2
        /// \return the Integer 2<sup>e</sup>
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        static Integer CRYPTOPP_API Power2(size_t e);
    //@}

    /// \name ENCODE/DECODE
    //@{
        /// \brief Minimum number of bytes to encode this integer
        /// \param sign enumeration indicating Signedness
        /// \note The MinEncodedSize() of 0 is 1.
        size_t MinEncodedSize(Signedness sign=UNSIGNED) const;

        /// \brief Encode in big-endian format
        /// \param output big-endian byte array
        /// \param outputLen length of the byte array
        /// \param sign enumeration indicating Signedness
        /// \details Unsigned means encode absolute value, signed means encode two's complement if negative.
        /// \details outputLen can be used to ensure an Integer is encoded to an exact size (rather than a
        ///   minimum size). An exact size is useful, for example, when encoding to a field element size.
        void Encode(byte *output, size_t outputLen, Signedness sign=UNSIGNED) const;

        /// \brief Encode in big-endian format
        /// \param bt BufferedTransformation object
        /// \param outputLen length of the encoding
        /// \param sign enumeration indicating Signedness
        /// \details Unsigned means encode absolute value, signed means encode two's complement if negative.
        /// \details outputLen can be used to ensure an Integer is encoded to an exact size (rather than a
        ///   minimum size). An exact size is useful, for example, when encoding to a field element size.
        void Encode(BufferedTransformation &bt, size_t outputLen, Signedness sign=UNSIGNED) const;

        /// \brief Encode in DER format
        /// \param bt BufferedTransformation object
        /// \details Encodes the Integer using Distinguished Encoding Rules
        ///   The result is placed into a BufferedTransformation object
        void DEREncode(BufferedTransformation &bt) const;

        /// \brief Encode absolute value as big-endian octet string
        /// \param bt BufferedTransformation object
        /// \param length the number of mytes to decode
        void DEREncodeAsOctetString(BufferedTransformation &bt, size_t length) const;

        /// \brief Encode absolute value in OpenPGP format
        /// \param output big-endian byte array
        /// \param bufferSize length of the byte array
        /// \return length of the output
        /// \details OpenPGPEncode places result into the buffer and returns the
        ///   number of bytes used for the encoding
        size_t OpenPGPEncode(byte *output, size_t bufferSize) const;

        /// \brief Encode absolute value in OpenPGP format
        /// \param bt BufferedTransformation object
        /// \return length of the output
        /// \details OpenPGPEncode places result into a BufferedTransformation object and returns the
        ///   number of bytes used for the encoding
        size_t OpenPGPEncode(BufferedTransformation &bt) const;

        /// \brief Decode from big-endian byte array
        /// \param input big-endian byte array
        /// \param inputLen length of the byte array
        /// \param sign enumeration indicating Signedness
        void Decode(const byte *input, size_t inputLen, Signedness sign=UNSIGNED);

        /// \brief Decode nonnegative value from big-endian byte array
        /// \param bt BufferedTransformation object
        /// \param inputLen length of the byte array
        /// \param sign enumeration indicating Signedness
        /// \note <tt>bt.MaxRetrievable() >= inputLen</tt>.
        void Decode(BufferedTransformation &bt, size_t inputLen, Signedness sign=UNSIGNED);

        /// \brief Decode from BER format
        /// \param input big-endian byte array
        /// \param inputLen length of the byte array
        void BERDecode(const byte *input, size_t inputLen);

        /// \brief Decode from BER format
        /// \param bt BufferedTransformation object
        void BERDecode(BufferedTransformation &bt);

        /// \brief Decode nonnegative value from big-endian octet string
        /// \param bt BufferedTransformation object
        /// \param length length of the byte array
        void BERDecodeAsOctetString(BufferedTransformation &bt, size_t length);

        /// \brief Exception thrown when an error is encountered decoding an OpenPGP integer
        class OpenPGPDecodeErr : public Exception
        {
        public:
            OpenPGPDecodeErr() : Exception(INVALID_DATA_FORMAT, "OpenPGP decode error") {}
        };

        /// \brief Decode from OpenPGP format
        /// \param input big-endian byte array
        /// \param inputLen length of the byte array
        void OpenPGPDecode(const byte *input, size_t inputLen);
        /// \brief Decode from OpenPGP format
        /// \param bt BufferedTransformation object
        void OpenPGPDecode(BufferedTransformation &bt);
    //@}

    /// \name ACCESSORS
    //@{
        /// \brief Determines if the Integer is convertable to Long
        /// \return true if *this can be represented as a signed long
        /// \sa ConvertToLong()
        bool IsConvertableToLong() const;
        /// \brief Convert the Integer to Long
        /// \return equivalent signed long if possible, otherwise undefined
        /// \sa IsConvertableToLong()
        signed long ConvertToLong() const;

        /// \brief Determines the number of bits required to represent the Integer
        /// \return number of significant bits
        /// \details BitCount is calculated as <tt>floor(log2(abs(*this))) + 1</tt>.
        unsigned int BitCount() const;
        /// \brief Determines the number of bytes required to represent the Integer
        /// \return number of significant bytes
        /// \details ByteCount is calculated as <tt>ceiling(BitCount()/8)</tt>.
        unsigned int ByteCount() const;
        /// \brief Determines the number of words required to represent the Integer
        /// \return number of significant words
        /// \details WordCount is calculated as <tt>ceiling(ByteCount()/sizeof(word))</tt>.
        unsigned int WordCount() const;

        /// \brief Provides the i-th bit of the Integer
        /// \return the i-th bit, i=0 being the least significant bit
        bool GetBit(size_t i) const;
        /// \brief Provides the i-th byte of the Integer
        /// \return the i-th byte
        byte GetByte(size_t i) const;
        /// \brief Provides the low order bits of the Integer
        /// \return n lowest bits of *this >> i
        lword GetBits(size_t i, size_t n) const;

        /// \brief Determines if the Integer is 0
        /// \return true if the Integer is 0, false otherwise
        bool IsZero() const {return !*this;}
        /// \brief Determines if the Integer is non-0
        /// \return true if the Integer is non-0, false otherwise
        bool NotZero() const {return !IsZero();}
        /// \brief Determines if the Integer is negative
        /// \return true if the Integer is negative, false otherwise
        bool IsNegative() const {return sign == NEGATIVE;}
        /// \brief Determines if the Integer is non-negative
        /// \return true if the Integer is non-negative, false otherwise
        bool NotNegative() const {return !IsNegative();}
        /// \brief Determines if the Integer is positive
        /// \return true if the Integer is positive, false otherwise
        bool IsPositive() const {return NotNegative() && NotZero();}
        /// \brief Determines if the Integer is non-positive
        /// \return true if the Integer is non-positive, false otherwise
        bool NotPositive() const {return !IsPositive();}
        /// \brief Determines if the Integer is even parity
        /// \return true if the Integer is even, false otherwise
        bool IsEven() const {return GetBit(0) == 0;}
        /// \brief Determines if the Integer is odd parity
        /// \return true if the Integer is odd, false otherwise
        bool IsOdd() const  {return GetBit(0) == 1;}
    //@}

    /// \name MANIPULATORS
    //@{
        /// \brief Assignment
        /// \param t the other Integer
        /// \return the result of assignment
        Integer&  operator=(const Integer& t);
        /// \brief Addition Assignment
        /// \param t the other Integer
        /// \return the result of <tt>*this + t</tt>
        Integer&  operator+=(const Integer& t);
        /// \brief Subtraction Assignment
        /// \param t the other Integer
        /// \return the result of <tt>*this - t</tt>
        Integer&  operator-=(const Integer& t);
        /// \brief Multiplication Assignment
        /// \param t the other Integer
        /// \return the result of <tt>*this * t</tt>
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        Integer&  operator*=(const Integer& t)  {return *this = Times(t);}
        /// \brief Division Assignment
        /// \param t the other Integer
        /// \return the result of <tt>*this / t</tt>
        Integer&  operator/=(const Integer& t)  {return *this = DividedBy(t);}
        /// \brief Remainder Assignment
        /// \param t the other Integer
        /// \return the result of <tt>*this % t</tt>
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        Integer&  operator%=(const Integer& t)  {return *this = Modulo(t);}
        /// \brief Division Assignment
        /// \param t the other word
        /// \return the result of <tt>*this / t</tt>
        Integer&  operator/=(word t)  {return *this = DividedBy(t);}
        /// \brief Remainder Assignment
        /// \param t the other word
        /// \return the result of <tt>*this % t</tt>
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        Integer&  operator%=(word t)  {return *this = Integer(POSITIVE, 0, Modulo(t));}

        /// \brief Left-shift Assignment
        /// \param n number of bits to shift
        /// \return reference to this Integer
        Integer&  operator<<=(size_t n);
        /// \brief Right-shift Assignment
        /// \param n number of bits to shift
        /// \return reference to this Integer
        Integer&  operator>>=(size_t n);

        /// \brief Bitwise AND Assignment
        /// \param t the other Integer
        /// \return the result of *this & t
        /// \details operator&=() performs a bitwise AND on *this. Missing bits are truncated
        ///   at the most significant bit positions, so the result is as small as the
        ///   smaller of the operands.
        /// \details Internally, Crypto++ uses a sign-magnitude representation. The library
        ///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
        ///   the integer should be converted to a 2's compliment representation before performing
        ///   the operation.
        /// \since Crypto++ 6.0
        Integer& operator&=(const Integer& t);
        /// \brief Bitwise OR Assignment
        /// \param t the second Integer
        /// \return the result of *this | t
        /// \details operator|=() performs a bitwise OR on *this. Missing bits are shifted in
        ///   at the most significant bit positions, so the result is as large as the
        ///   larger of the operands.
        /// \details Internally, Crypto++ uses a sign-magnitude representation. The library
        ///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
        ///   the integer should be converted to a 2's compliment representation before performing
        ///   the operation.
        /// \since Crypto++ 6.0
        Integer& operator|=(const Integer& t);
        /// \brief Bitwise XOR Assignment
        /// \param t the other Integer
        /// \return the result of *this ^ t
        /// \details operator^=() performs a bitwise XOR on *this. Missing bits are shifted
        ///   in at the most significant bit positions, so the result is as large as the
        ///   larger of the operands.
        /// \details Internally, Crypto++ uses a sign-magnitude representation. The library
        ///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
        ///   the integer should be converted to a 2's compliment representation before performing
        ///   the operation.
        /// \since Crypto++ 6.0
        Integer& operator^=(const Integer& t);

        /// \brief Set this Integer to random integer
        /// \param rng RandomNumberGenerator used to generate material
        /// \param bitCount the number of bits in the resulting integer
        /// \details The random integer created is uniformly distributed over <tt>[0, 2<sup>bitCount</sup>]</tt>.
        void Randomize(RandomNumberGenerator &rng, size_t bitCount);

        /// \brief Set this Integer to random integer
        /// \param rng RandomNumberGenerator used to generate material
        /// \param min the minimum value
        /// \param max the maximum value
        /// \details The random integer created is uniformly distributed over <tt>[min, max]</tt>.
        void Randomize(RandomNumberGenerator &rng, const Integer &min, const Integer &max);

        /// \brief Set this Integer to random integer of special form
        /// \param rng RandomNumberGenerator used to generate material
        /// \param min the minimum value
        /// \param max the maximum value
        /// \param rnType RandomNumberType to specify the type
        /// \param equiv the equivalence class based on the parameter \p mod
        /// \param mod the modulus used to reduce the equivalence class
        /// \throw RandomNumberNotFound if the set is empty.
        /// \details Ideally, the random integer created should be uniformly distributed
        ///   over <tt>{x | min <= x <= max</tt> and \p x is of rnType and <tt>x \% mod == equiv}</tt>.
        ///   However the actual distribution may not be uniform because sequential
        ///   search is used to find an appropriate number from a random starting
        ///   point.
        /// \details May return (with very small probability) a pseudoprime when a prime
        ///   is requested and <tt>max > lastSmallPrime*lastSmallPrime</tt>. \p lastSmallPrime
        ///   is declared in nbtheory.h.
        bool Randomize(RandomNumberGenerator &rng, const Integer &min, const Integer &max, RandomNumberType rnType, const Integer &equiv=Zero(), const Integer &mod=One());

        /// \brief Generate a random number
        /// \param rng RandomNumberGenerator used to generate material
        /// \param params additional parameters that cannot be passed directly to the function
        /// \return true if a random number was generated, false otherwise
        /// \details GenerateRandomNoThrow attempts to generate a random number according to the
        ///   parameters specified in params. The function does not throw RandomNumberNotFound.
        /// \details The example below generates a prime number using NameValuePairs that Integer
        ///   class recognizes. The names are not provided in argnames.h.
        /// <pre>
        ///     AutoSeededRandomPool prng;
        ///     AlgorithmParameters params = MakeParameters("BitLength", 2048)
        ///                                                ("RandomNumberType", Integer::PRIME);
        ///     Integer x;
        ///     if (x.GenerateRandomNoThrow(prng, params) == false)
        ///         throw std::runtime_error("Failed to generate prime number");
        /// </pre>
        bool GenerateRandomNoThrow(RandomNumberGenerator &rng, const NameValuePairs &params = g_nullNameValuePairs);

        /// \brief Generate a random number
        /// \param rng RandomNumberGenerator used to generate material
        /// \param params additional parameters that cannot be passed directly to the function
        /// \throw RandomNumberNotFound if a random number is not found
        /// \details GenerateRandom attempts to generate a random number according to the
        ///   parameters specified in params.
        /// \details The example below generates a prime number using NameValuePairs that Integer
        ///   class recognizes. The names are not provided in argnames.h.
        /// <pre>
        ///     AutoSeededRandomPool prng;
        ///     AlgorithmParameters params = MakeParameters("BitLength", 2048)
        ///                                                ("RandomNumberType", Integer::PRIME);
        ///     Integer x;
        ///     try { x.GenerateRandom(prng, params); }
        ///     catch (RandomNumberNotFound&) { x = -1; }
        /// </pre>
        void GenerateRandom(RandomNumberGenerator &rng, const NameValuePairs &params = g_nullNameValuePairs)
        {
            if (!GenerateRandomNoThrow(rng, params))
                throw RandomNumberNotFound();
        }

        /// \brief Set the n-th bit to value
        /// \details 0-based numbering.
        void SetBit(size_t n, bool value=1);

        /// \brief Set the n-th byte to value
        /// \details 0-based numbering.
        void SetByte(size_t n, byte value);

        /// \brief Reverse the Sign of the Integer
        void Negate();

        /// \brief Sets the Integer to positive
        void SetPositive() {sign = POSITIVE;}

        /// \brief Sets the Integer to negative
        void SetNegative() {if (!!(*this)) sign = NEGATIVE;}

        /// \brief Swaps this Integer with another Integer
        void swap(Integer &a);
    //@}

    /// \name UNARY OPERATORS
    //@{
        /// \brief Negation
        bool        operator!() const;
        /// \brief Addition
        Integer     operator+() const {return *this;}
        /// \brief Subtraction
        Integer     operator-() const;
        /// \brief Pre-increment
        Integer&    operator++();
        /// \brief Pre-decrement
        Integer&    operator--();
        /// \brief Post-increment
        Integer     operator++(int) {Integer temp = *this; ++*this; return temp;}
        /// \brief Post-decrement
        Integer     operator--(int) {Integer temp = *this; --*this; return temp;}
    //@}

    /// \name BINARY OPERATORS
    //@{
        /// \brief Perform signed comparison
        /// \param a the Integer to comapre
        ///   \retval -1 if <tt>*this < a</tt>
        ///   \retval  0 if <tt>*this = a</tt>
        ///   \retval  1 if <tt>*this > a</tt>
        int Compare(const Integer& a) const;

        /// \brief Addition
        Integer Plus(const Integer &b) const;
        /// \brief Subtraction
        Integer Minus(const Integer &b) const;
        /// \brief Multiplication
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        Integer Times(const Integer &b) const;
        /// \brief Division
        Integer DividedBy(const Integer &b) const;
        /// \brief Remainder
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        Integer Modulo(const Integer &b) const;
        /// \brief Division
        Integer DividedBy(word b) const;
        /// \brief Remainder
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        word Modulo(word b) const;

        /// \brief Bitwise AND
        /// \param t the other Integer
        /// \return the result of <tt>*this & t</tt>
        /// \details And() performs a bitwise AND on the operands. Missing bits are truncated
        ///   at the most significant bit positions, so the result is as small as the
        ///   smaller of the operands.
        /// \details Internally, Crypto++ uses a sign-magnitude representation. The library
        ///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
        ///   the integer should be converted to a 2's compliment representation before performing
        ///   the operation.
        /// \since Crypto++ 6.0
        Integer And(const Integer& t) const;

        /// \brief Bitwise OR
        /// \param t the other Integer
        /// \return the result of <tt>*this | t</tt>
        /// \details Or() performs a bitwise OR on the operands. Missing bits are shifted in
        ///   at the most significant bit positions, so the result is as large as the
        ///   larger of the operands.
        /// \details Internally, Crypto++ uses a sign-magnitude representation. The library
        ///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
        ///   the integer should be converted to a 2's compliment representation before performing
        ///   the operation.
        /// \since Crypto++ 6.0
        Integer Or(const Integer& t) const;

        /// \brief Bitwise XOR
        /// \param t the other Integer
        /// \return the result of <tt>*this ^ t</tt>
        /// \details Xor() performs a bitwise XOR on the operands. Missing bits are shifted in
        ///   at the most significant bit positions, so the result is as large as the
        ///   larger of the operands.
        /// \details Internally, Crypto++ uses a sign-magnitude representation. The library
        ///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
        ///   the integer should be converted to a 2's compliment representation before performing
        ///   the operation.
        /// \since Crypto++ 6.0
        Integer Xor(const Integer& t) const;

        /// \brief Right-shift
        Integer operator>>(size_t n) const  {return Integer(*this)>>=n;}
        /// \brief Left-shift
        Integer operator<<(size_t n) const  {return Integer(*this)<<=n;}
    //@}

    /// \name OTHER ARITHMETIC FUNCTIONS
    //@{
        /// \brief Retrieve the absolute value of this integer
        Integer AbsoluteValue() const;
        /// \brief Add this integer to itself
        Integer Doubled() const {return Plus(*this);}
        /// \brief Multiply this integer by itself
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        Integer Squared() const {return Times(*this);}
        /// \brief Extract square root
        /// \details if negative return 0, else return floor of square root
        Integer SquareRoot() const;
        /// \brief Determine whether this integer is a perfect square
        bool IsSquare() const;

        /// \brief Determine if 1 or -1
        /// \return true if this integer is 1 or -1, false otherwise
        bool IsUnit() const;
        /// \brief Calculate multiplicative inverse
        /// \return MultiplicativeInverse inverse if 1 or -1, otherwise return 0.
        Integer MultiplicativeInverse() const;

        /// \brief Extended Division
        /// \param r a reference for the remainder
        /// \param q a reference for the quotient
        /// \param a a reference to the dividend
        /// \param d a reference to the divisor
        /// \details Divide calculates r and q such that (a == d*q + r) && (0 <= r < abs(d)).
        static void CRYPTOPP_API Divide(Integer &r, Integer &q, const Integer &a, const Integer &d);

        /// \brief Extended Division
        /// \param r a reference for the remainder
        /// \param q a reference for the quotient
        /// \param a a reference to the dividend
        /// \param d a reference to the divisor
        /// \details Divide calculates r and q such that (a == d*q + r) && (0 <= r < abs(d)).
        ///   This overload uses a faster division algorithm because the divisor is short.
        static void CRYPTOPP_API Divide(word &r, Integer &q, const Integer &a, word d);

        /// \brief Extended Division
        /// \param r a reference for the remainder
        /// \param q a reference for the quotient
        /// \param a a reference to the dividend
        /// \param n a reference to the divisor
        /// \details DivideByPowerOf2 calculates r and q such that (a == d*q + r) && (0 <= r < abs(d)).
        ///   It returns same result as Divide(r, q, a, Power2(n)), but faster.
        ///   This overload uses a faster division algorithm because the divisor is a power of 2.
        static void CRYPTOPP_API DivideByPowerOf2(Integer &r, Integer &q, const Integer &a, unsigned int n);

        /// \brief Calculate greatest common divisor
        /// \param a a reference to the first number
        /// \param n a reference to the secind number
        /// \return the greatest common divisor <tt>a</tt> and <tt>n</tt>.
        static Integer CRYPTOPP_API Gcd(const Integer &a, const Integer &n);

        /// \brief Calculate multiplicative inverse
        /// \param n a reference to the modulus
        /// \return an Integer <tt>*this % n</tt>.
        /// \details InverseMod returns the multiplicative inverse of the Integer <tt>*this</tt>
        ///  modulo the Integer <tt>n</tt>. If no Integer exists then Integer 0 is returned.
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        Integer InverseMod(const Integer &n) const;

        /// \brief Calculate multiplicative inverse
        /// \param n the modulus
        /// \return a word <tt>*this % n</tt>.
        /// \details InverseMod returns the multiplicative inverse of the Integer <tt>*this</tt>
        ///  modulo the word <tt>n</tt>. If no Integer exists then word 0 is returned.
        /// \sa a_times_b_mod_c() and a_exp_b_mod_c()
        word InverseMod(word n) const;
    //@}

    /// \name INPUT/OUTPUT
    //@{
        /// \brief Extraction operator
        /// \param in a reference to a std::istream
        /// \param a a reference to an Integer
        /// \return a reference to a std::istream reference
        friend CRYPTOPP_DLL std::istream& CRYPTOPP_API operator>>(std::istream& in, Integer &a);

        /// \brief Insertion operator
        /// \param out a reference to a std::ostream
        /// \param a a constant reference to an Integer
        /// \return a reference to a std::ostream reference
        /// \details The output integer responds to std::hex, std::oct, std::hex, std::upper and
        ///   std::lower. The output includes the suffix \a h (for hex), \a . (\a dot, for dec)
        ///   and \a o (for octal). There is currently no way to suppress the suffix.
        /// \details If you want to print an Integer without the suffix or using an arbitrary base, then
        ///   use IntToString<Integer>().
        /// \sa IntToString<Integer>
        friend CRYPTOPP_DLL std::ostream& CRYPTOPP_API operator<<(std::ostream& out, const Integer &a);
    //@}

    /// \brief Modular multiplication
    /// \param x a reference to the first term
    /// \param y a reference to the second term
    /// \param m a reference to the modulus
    /// \return an Integer <tt>(a * b) % m</tt>.
    CRYPTOPP_DLL friend Integer CRYPTOPP_API a_times_b_mod_c(const Integer &x, const Integer& y, const Integer& m);
    /// \brief Modular exponentiation
    /// \param x a reference to the base
    /// \param e a reference to the exponent
    /// \param m a reference to the modulus
    /// \return an Integer <tt>(a ^ b) % m</tt>.
    CRYPTOPP_DLL friend Integer CRYPTOPP_API a_exp_b_mod_c(const Integer &x, const Integer& e, const Integer& m);

protected:

    // http://github.com/weidai11/cryptopp/issues/602
    Integer InverseModNext(const Integer &n) const;

private:

    Integer(word value, size_t length);
    int PositiveCompare(const Integer &t) const;

    IntegerSecBlock reg;
    Sign sign;

#ifndef CRYPTOPP_DOXYGEN_PROCESSING
    friend class ModularArithmetic;
    friend class MontgomeryRepresentation;
    friend class HalfMontgomeryRepresentation;

    friend void PositiveAdd(Integer &sum, const Integer &a, const Integer &b);
    friend void PositiveSubtract(Integer &diff, const Integer &a, const Integer &b);
    friend void PositiveMultiply(Integer &product, const Integer &a, const Integer &b);
    friend void PositiveDivide(Integer &remainder, Integer &quotient, const Integer &dividend, const Integer &divisor);
#endif
};

/// \brief Comparison
inline bool operator==(const CryptoPP::Integer& a, const CryptoPP::Integer& b) {return a.Compare(b)==0;}
/// \brief Comparison
inline bool operator!=(const CryptoPP::Integer& a, const CryptoPP::Integer& b) {return a.Compare(b)!=0;}
/// \brief Comparison
inline bool operator> (const CryptoPP::Integer& a, const CryptoPP::Integer& b) {return a.Compare(b)> 0;}
/// \brief Comparison
inline bool operator>=(const CryptoPP::Integer& a, const CryptoPP::Integer& b) {return a.Compare(b)>=0;}
/// \brief Comparison
inline bool operator< (const CryptoPP::Integer& a, const CryptoPP::Integer& b) {return a.Compare(b)< 0;}
/// \brief Comparison
inline bool operator<=(const CryptoPP::Integer& a, const CryptoPP::Integer& b) {return a.Compare(b)<=0;}
/// \brief Addition
inline CryptoPP::Integer operator+(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.Plus(b);}
/// \brief Subtraction
inline CryptoPP::Integer operator-(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.Minus(b);}
/// \brief Multiplication
/// \sa a_times_b_mod_c() and a_exp_b_mod_c()
inline CryptoPP::Integer operator*(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.Times(b);}
/// \brief Division
inline CryptoPP::Integer operator/(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.DividedBy(b);}
/// \brief Remainder
/// \sa a_times_b_mod_c() and a_exp_b_mod_c()
inline CryptoPP::Integer operator%(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.Modulo(b);}
/// \brief Division
inline CryptoPP::Integer operator/(const CryptoPP::Integer &a, CryptoPP::word b) {return a.DividedBy(b);}
/// \brief Remainder
/// \sa a_times_b_mod_c() and a_exp_b_mod_c()
inline CryptoPP::word    operator%(const CryptoPP::Integer &a, CryptoPP::word b) {return a.Modulo(b);}

/// \brief Bitwise AND
/// \param a the first Integer
/// \param b the second Integer
/// \return the result of a & b
/// \details operator&() performs a bitwise AND on the operands. Missing bits are truncated
///   at the most significant bit positions, so the result is as small as the
///   smaller of the operands.
/// \details Internally, Crypto++ uses a sign-magnitude representation. The library
///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
///   the integer should be converted to a 2's compliment representation before performing
///   the operation.
/// \since Crypto++ 6.0
inline CryptoPP::Integer operator&(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.And(b);}

/// \brief Bitwise OR
/// \param a the first Integer
/// \param b the second Integer
/// \return the result of a | b
/// \details operator|() performs a bitwise OR on the operands. Missing bits are shifted in
///   at the most significant bit positions, so the result is as large as the
///   larger of the operands.
/// \details Internally, Crypto++ uses a sign-magnitude representation. The library
///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
///   the integer should be converted to a 2's compliment representation before performing
///   the operation.
/// \since Crypto++ 6.0
inline CryptoPP::Integer operator|(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.Or(b);}

/// \brief Bitwise XOR
/// \param a the first Integer
/// \param b the second Integer
/// \return the result of a ^ b
/// \details operator^() performs a bitwise XOR on the operands. Missing bits are shifted
///   in at the most significant bit positions, so the result is as large as the
///   larger of the operands.
/// \details Internally, Crypto++ uses a sign-magnitude representation. The library
///   does not attempt to interpret bits, and the result is always POSITIVE. If needed,
///   the integer should be converted to a 2's compliment representation before performing
///   the operation.
/// \since Crypto++ 6.0
inline CryptoPP::Integer operator^(const CryptoPP::Integer &a, const CryptoPP::Integer &b) {return a.Xor(b);}

NAMESPACE_END

#ifndef __BORLANDC__
NAMESPACE_BEGIN(std)
inline void swap(CryptoPP::Integer &a, CryptoPP::Integer &b)
{
    a.swap(b);
}
NAMESPACE_END
#endif

#endif