gf2n.h

Go to the documentation of this file.

// gf2n.h - originally written and placed in the public domain by Wei Dai

/// \file gf2n.h
/// \brief Classes and functions for schemes over GF(2^n)

#ifndef CRYPTOPP_GF2N_H
#define CRYPTOPP_GF2N_H

#include "cryptlib.h"
#include "secblock.h"
#include "algebra.h"
#include "misc.h"
#include "asn.h"

#include <iosfwd>

#if CRYPTOPP_MSC_VERSION
# pragma warning(push)
# pragma warning(disable: 4231 4275)
#endif

NAMESPACE_BEGIN(CryptoPP)

/// \brief Polynomial with Coefficients in GF(2)
/*! \nosubgrouping */
class CRYPTOPP_DLL PolynomialMod2
{
public:
    /// \name ENUMS, EXCEPTIONS, and TYPEDEFS
    //@{
        /// \brief Excpetion thrown when divide by zero is encountered
        class DivideByZero : public Exception
        {
        public:
            DivideByZero() : Exception(OTHER_ERROR, "PolynomialMod2: division by zero") {}
        };

        typedef unsigned int RandomizationParameter;
    //@}

    /// \name CREATORS
    //@{
        /// \brief Construct the zero polynomial
        PolynomialMod2();
        /// Copy construct a PolynomialMod2
        PolynomialMod2(const PolynomialMod2& t);

        /// \brief Construct a PolynomialMod2 from a word
        /// \details value should be encoded with the least significant bit as coefficient to x^0
        ///   and most significant bit as coefficient to x^(WORD_BITS-1)
        ///   bitLength denotes how much memory to allocate initially
        PolynomialMod2(word value, size_t bitLength=WORD_BITS);

        /// \brief Construct a PolynomialMod2 from big-endian byte array
        PolynomialMod2(const byte *encodedPoly, size_t byteCount)
            {Decode(encodedPoly, byteCount);}

        /// \brief Construct a PolynomialMod2 from big-endian form stored in a BufferedTransformation
        PolynomialMod2(BufferedTransformation &encodedPoly, size_t byteCount)
            {Decode(encodedPoly, byteCount);}

        /// \brief Create a uniformly distributed random polynomial
        /// \details Create a random polynomial uniformly distributed over all polynomials with degree less than bitcount
        PolynomialMod2(RandomNumberGenerator &rng, size_t bitcount)
            {Randomize(rng, bitcount);}

        /// \brief Provides x^i
        /// \returns x^i
        static PolynomialMod2 CRYPTOPP_API Monomial(size_t i);
        /// \brief Provides x^t0 + x^t1 + x^t2
        /// \returns x^t0 + x^t1 + x^t2
        static PolynomialMod2 CRYPTOPP_API Trinomial(size_t t0, size_t t1, size_t t2);
        /// \brief Provides x^t0 + x^t1 + x^t2 + x^t3 + x^t4
        /// \returns x^t0 + x^t1 + x^t2 + x^t3 + x^t4
        static PolynomialMod2 CRYPTOPP_API Pentanomial(size_t t0, size_t t1, size_t t2, size_t t3, size_t t4);
        /// \brief Provides x^(n-1) + ... + x + 1
        /// \returns x^(n-1) + ... + x + 1
        static PolynomialMod2 CRYPTOPP_API AllOnes(size_t n);

        /// \brief The Zero polinomial
        /// \returns the zero polynomial
        static const PolynomialMod2 & CRYPTOPP_API Zero();
        /// \brief The One polinomial
        /// \returns the one polynomial
        static const PolynomialMod2 & CRYPTOPP_API One();
    //@}

    /// \name ENCODE/DECODE
    //@{
        /// minimum number of bytes to encode this polynomial
        /*! MinEncodedSize of 0 is 1 */
        unsigned int MinEncodedSize() const {return STDMAX(1U, ByteCount());}

        /// encode in big-endian format
        /// \details if outputLen < MinEncodedSize, the most significant bytes will be dropped
        ///   if outputLen > MinEncodedSize, the most significant bytes will be padded
        void Encode(byte *output, size_t outputLen) const;
        ///
        void Encode(BufferedTransformation &bt, size_t outputLen) const;

        ///
        void Decode(const byte *input, size_t inputLen);
        ///
        //* Precondition: bt.MaxRetrievable() >= inputLen
        void Decode(BufferedTransformation &bt, size_t inputLen);

        /// encode value as big-endian octet string
        void DEREncodeAsOctetString(BufferedTransformation &bt, size_t length) const;
        /// decode value as big-endian octet string
        void BERDecodeAsOctetString(BufferedTransformation &bt, size_t length);
    //@}

    /// \name ACCESSORS
    //@{
        /// number of significant bits = Degree() + 1
        unsigned int BitCount() const;
        /// number of significant bytes = ceiling(BitCount()/8)
        unsigned int ByteCount() const;
        /// number of significant words = ceiling(ByteCount()/sizeof(word))
        unsigned int WordCount() const;

        /// return the n-th bit, n=0 being the least significant bit
        bool GetBit(size_t n) const {return GetCoefficient(n)!=0;}
        /// return the n-th byte
        byte GetByte(size_t n) const;

        /// the zero polynomial will return a degree of -1
        signed int Degree() const {return (signed int)(BitCount()-1U);}
        /// degree + 1
        unsigned int CoefficientCount() const {return BitCount();}
        /// return coefficient for x^i
        int GetCoefficient(size_t i) const
            {return (i/WORD_BITS < reg.size()) ? int(reg[i/WORD_BITS] >> (i % WORD_BITS)) & 1 : 0;}
        /// return coefficient for x^i
        int operator[](unsigned int i) const {return GetCoefficient(i);}

        ///
        bool IsZero() const {return !*this;}
        ///
        bool Equals(const PolynomialMod2 &rhs) const;
    //@}

    /// \name MANIPULATORS
    //@{
        ///
        PolynomialMod2&  operator=(const PolynomialMod2& t);
        ///
        PolynomialMod2&  operator&=(const PolynomialMod2& t);
        ///
        PolynomialMod2&  operator^=(const PolynomialMod2& t);
        ///
        PolynomialMod2&  operator+=(const PolynomialMod2& t) {return *this ^= t;}
        ///
        PolynomialMod2&  operator-=(const PolynomialMod2& t) {return *this ^= t;}
        ///
        PolynomialMod2&  operator*=(const PolynomialMod2& t);
        ///
        PolynomialMod2&  operator/=(const PolynomialMod2& t);
        ///
        PolynomialMod2&  operator%=(const PolynomialMod2& t);
        ///
        PolynomialMod2&  operator<<=(unsigned int);
        ///
        PolynomialMod2&  operator>>=(unsigned int);

        ///
        void Randomize(RandomNumberGenerator &rng, size_t bitcount);

        ///
        void SetBit(size_t i, int value = 1);
        /// set the n-th byte to value
        void SetByte(size_t n, byte value);

        ///
        void SetCoefficient(size_t i, int value) {SetBit(i, value);}

        ///
        void swap(PolynomialMod2 &a) {reg.swap(a.reg);}
    //@}

    /// \name UNARY OPERATORS
    //@{
        ///
        bool            operator!() const;
        ///
        PolynomialMod2  operator+() const {return *this;}
        ///
        PolynomialMod2  operator-() const {return *this;}
    //@}

    /// \name BINARY OPERATORS
    //@{
        ///
        PolynomialMod2 And(const PolynomialMod2 &b) const;
        ///
        PolynomialMod2 Xor(const PolynomialMod2 &b) const;
        ///
        PolynomialMod2 Plus(const PolynomialMod2 &b) const {return Xor(b);}
        ///
        PolynomialMod2 Minus(const PolynomialMod2 &b) const {return Xor(b);}
        ///
        PolynomialMod2 Times(const PolynomialMod2 &b) const;
        ///
        PolynomialMod2 DividedBy(const PolynomialMod2 &b) const;
        ///
        PolynomialMod2 Modulo(const PolynomialMod2 &b) const;

        ///
        PolynomialMod2 operator>>(unsigned int n) const;
        ///
        PolynomialMod2 operator<<(unsigned int n) const;
    //@}

    /// \name OTHER ARITHMETIC FUNCTIONS
    //@{
        /// sum modulo 2 of all coefficients
        unsigned int Parity() const;

        /// check for irreducibility
        bool IsIrreducible() const;

        /// is always zero since we're working modulo 2
        PolynomialMod2 Doubled() const {return Zero();}
        ///
        PolynomialMod2 Squared() const;

        /// only 1 is a unit
        bool IsUnit() const {return Equals(One());}
        /// return inverse if *this is a unit, otherwise return 0
        PolynomialMod2 MultiplicativeInverse() const {return IsUnit() ? One() : Zero();}

        /// greatest common divisor
        static PolynomialMod2 CRYPTOPP_API Gcd(const PolynomialMod2 &a, const PolynomialMod2 &n);
        /// calculate multiplicative inverse of *this mod n
        PolynomialMod2 InverseMod(const PolynomialMod2 &) const;

        /// calculate r and q such that (a == d*q + r) && (deg(r) < deg(d))
        static void CRYPTOPP_API Divide(PolynomialMod2 &r, PolynomialMod2 &q, const PolynomialMod2 &a, const PolynomialMod2 &d);
    //@}

    /// \name INPUT/OUTPUT
    //@{
        ///
        friend std::ostream& operator<<(std::ostream& out, const PolynomialMod2 &a);
    //@}

private:
    friend class GF2NT;
    friend class GF2NT233;

    SecWordBlock reg;
};

///
inline bool operator==(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b)
{return a.Equals(b);}
///
inline bool operator!=(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b)
{return !(a==b);}
/// compares degree
inline bool operator> (const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b)
{return a.Degree() > b.Degree();}
/// compares degree
inline bool operator>=(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b)
{return a.Degree() >= b.Degree();}
/// compares degree
inline bool operator< (const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b)
{return a.Degree() < b.Degree();}
/// compares degree
inline bool operator<=(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b)
{return a.Degree() <= b.Degree();}
///
inline CryptoPP::PolynomialMod2 operator&(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b) {return a.And(b);}
///
inline CryptoPP::PolynomialMod2 operator^(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b) {return a.Xor(b);}
///
inline CryptoPP::PolynomialMod2 operator+(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b) {return a.Plus(b);}
///
inline CryptoPP::PolynomialMod2 operator-(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b) {return a.Minus(b);}
///
inline CryptoPP::PolynomialMod2 operator*(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b) {return a.Times(b);}
///
inline CryptoPP::PolynomialMod2 operator/(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b) {return a.DividedBy(b);}
///
inline CryptoPP::PolynomialMod2 operator%(const CryptoPP::PolynomialMod2 &a, const CryptoPP::PolynomialMod2 &b) {return a.Modulo(b);}

// CodeWarrior 8 workaround: put these template instantiations after overloaded operator declarations,
// but before the use of QuotientRing<EuclideanDomainOf<PolynomialMod2> > for VC .NET 2003
CRYPTOPP_DLL_TEMPLATE_CLASS AbstractGroup<PolynomialMod2>;
CRYPTOPP_DLL_TEMPLATE_CLASS AbstractRing<PolynomialMod2>;
CRYPTOPP_DLL_TEMPLATE_CLASS AbstractEuclideanDomain<PolynomialMod2>;
CRYPTOPP_DLL_TEMPLATE_CLASS EuclideanDomainOf<PolynomialMod2>;
CRYPTOPP_DLL_TEMPLATE_CLASS QuotientRing<EuclideanDomainOf<PolynomialMod2> >;

/// \brief GF(2^n) with Polynomial Basis
class CRYPTOPP_DLL GF2NP : public QuotientRing<EuclideanDomainOf<PolynomialMod2> >
{
public:
    GF2NP(const PolynomialMod2 &modulus);

    virtual GF2NP * Clone() const {return new GF2NP(*this);}
    virtual void DEREncode(BufferedTransformation &bt) const
        {CRYPTOPP_UNUSED(bt); CRYPTOPP_ASSERT(false);}  // no ASN.1 syntax yet for general polynomial basis

    void DEREncodeElement(BufferedTransformation &out, const Element &a) const;
    void BERDecodeElement(BufferedTransformation &in, Element &a) const;

    bool Equal(const Element &a, const Element &b) const
        {CRYPTOPP_ASSERT(a.Degree() < m_modulus.Degree() && b.Degree() < m_modulus.Degree()); return a.Equals(b);}

    bool IsUnit(const Element &a) const
        {CRYPTOPP_ASSERT(a.Degree() < m_modulus.Degree()); return !!a;}

    unsigned int MaxElementBitLength() const
        {return m;}

    unsigned int MaxElementByteLength() const
        {return (unsigned int)BitsToBytes(MaxElementBitLength());}

    Element SquareRoot(const Element &a) const;

    Element HalfTrace(const Element &a) const;

    // returns z such that z^2 + z == a
    Element SolveQuadraticEquation(const Element &a) const;

protected:
    unsigned int m;
};

/// \brief GF(2^n) with Trinomial Basis
class CRYPTOPP_DLL GF2NT : public GF2NP
{
public:
    // polynomial modulus = x^t0 + x^t1 + x^t2, t0 > t1 > t2
    GF2NT(unsigned int t0, unsigned int t1, unsigned int t2);

    GF2NP * Clone() const {return new GF2NT(*this);}
    void DEREncode(BufferedTransformation &bt) const;

    const Element& Multiply(const Element &a, const Element &b) const;

    const Element& Square(const Element &a) const
        {return Reduced(a.Squared());}

    const Element& MultiplicativeInverse(const Element &a) const;

protected:
    const Element& Reduced(const Element &a) const;

    unsigned int t0, t1;
    mutable PolynomialMod2 result;
};

/// \brief GF(2^n) for b233 and k233
/// \details GF2NT233 is a specialization of GF2NT that provides Multiply()
///   and Square() operations when carryless multiplies is available.
class CRYPTOPP_DLL GF2NT233 : public GF2NT
{
public:
    // polynomial modulus = x^t0 + x^t1 + x^t2, t0 > t1 > t2
    GF2NT233(unsigned int t0, unsigned int t1, unsigned int t2);

    GF2NP * Clone() const {return new GF2NT233(*this);}

    const Element& Multiply(const Element &a, const Element &b) const;

    const Element& Square(const Element &a) const;
};

/// \brief GF(2^n) with Pentanomial Basis
class CRYPTOPP_DLL GF2NPP : public GF2NP
{
public:
    // polynomial modulus = x^t0 + x^t1 + x^t2 + x^t3 + x^t4, t0 > t1 > t2 > t3 > t4
    GF2NPP(unsigned int t0, unsigned int t1, unsigned int t2, unsigned int t3, unsigned int t4)
        : GF2NP(PolynomialMod2::Pentanomial(t0, t1, t2, t3, t4)), t1(t1), t2(t2), t3(t3) {}

    GF2NP * Clone() const {return new GF2NPP(*this);}
    void DEREncode(BufferedTransformation &bt) const;

private:
    unsigned int t1, t2, t3;
};

// construct new GF2NP from the ASN.1 sequence Characteristic-two
CRYPTOPP_DLL GF2NP * CRYPTOPP_API BERDecodeGF2NP(BufferedTransformation &bt);

NAMESPACE_END

#ifndef __BORLANDC__
NAMESPACE_BEGIN(std)
template<> inline void swap(CryptoPP::PolynomialMod2 &a, CryptoPP::PolynomialMod2 &b)
{
    a.swap(b);
}
NAMESPACE_END
#endif

#if CRYPTOPP_MSC_VERSION
# pragma warning(pop)
#endif

#endif