polynomi.h

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#ifndef CRYPTOPP_POLYNOMI_H
#define CRYPTOPP_POLYNOMI_H

/*! \file */

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

#include <iosfwd>
#include <vector>

NAMESPACE_BEGIN(CryptoPP)

//! represents single-variable polynomials over arbitrary rings
/*!     \nosubgrouping */
template <class T> class PolynomialOver
{
public:
        //! \name ENUMS, EXCEPTIONS, and TYPEDEFS
        //@{
                //! division by zero exception
                class DivideByZero : public Exception 
                {
                public: 
                        DivideByZero() : Exception(OTHER_ERROR, "PolynomialOver<T>: division by zero") {}
                };

                //! specify the distribution for randomization functions
                class RandomizationParameter
                {
                public:
                        RandomizationParameter(unsigned int coefficientCount, const typename T::RandomizationParameter &coefficientParameter )
                                : m_coefficientCount(coefficientCount), m_coefficientParameter(coefficientParameter) {}

                private:
                        unsigned int m_coefficientCount;
                        typename T::RandomizationParameter m_coefficientParameter;
                        friend class PolynomialOver<T>;
                };

                typedef T Ring;
                typedef typename T::Element CoefficientType;
        //@}

        //! \name CREATORS
        //@{
                //! creates the zero polynomial
                PolynomialOver() {}

                //!
                PolynomialOver(const Ring &ring, unsigned int count)
                        : m_coefficients((size_t)count, ring.Identity()) {}

                //! copy constructor
                PolynomialOver(const PolynomialOver<Ring> &t)
                        : m_coefficients(t.m_coefficients.size()) {*this = t;}

                //! construct constant polynomial
                PolynomialOver(const CoefficientType &element)
                        : m_coefficients(1, element) {}

                //! construct polynomial with specified coefficients, starting from coefficient of x^0
                template <typename Iterator> PolynomialOver(Iterator begin, Iterator end)
                        : m_coefficients(begin, end) {}

                //! convert from string
                PolynomialOver(const char *str, const Ring &ring) {FromStr(str, ring);}

                //! convert from big-endian byte array
                PolynomialOver(const byte *encodedPolynomialOver, unsigned int byteCount);

                //! convert from Basic Encoding Rules encoded byte array
                explicit PolynomialOver(const byte *BEREncodedPolynomialOver);

                //! convert from BER encoded byte array stored in a BufferedTransformation object
                explicit PolynomialOver(BufferedTransformation &bt);

                //! create a random PolynomialOver<T>
                PolynomialOver(RandomNumberGenerator &rng, const RandomizationParameter &parameter, const Ring &ring)
                        {Randomize(rng, parameter, ring);}
        //@}

        //! \name ACCESSORS
        //@{
                //! the zero polynomial will return a degree of -1
                int Degree(const Ring &ring) const {return int(CoefficientCount(ring))-1;}
                //!
                unsigned int CoefficientCount(const Ring &ring) const;
                //! return coefficient for x^i
                CoefficientType GetCoefficient(unsigned int i, const Ring &ring) const;
        //@}

        //! \name MANIPULATORS
        //@{
                //!
                PolynomialOver<Ring>&  operator=(const PolynomialOver<Ring>& t);

                //!
                void Randomize(RandomNumberGenerator &rng, const RandomizationParameter &parameter, const Ring &ring);

                //! set the coefficient for x^i to value
                void SetCoefficient(unsigned int i, const CoefficientType &value, const Ring &ring);

                //!
                void Negate(const Ring &ring);

                //!
                void swap(PolynomialOver<Ring> &t);
        //@}


        //! \name BASIC ARITHMETIC ON POLYNOMIALS
        //@{
                bool Equals(const PolynomialOver<Ring> &t, const Ring &ring) const;
                bool IsZero(const Ring &ring) const {return CoefficientCount(ring)==0;}

                PolynomialOver<Ring> Plus(const PolynomialOver<Ring>& t, const Ring &ring) const;
                PolynomialOver<Ring> Minus(const PolynomialOver<Ring>& t, const Ring &ring) const;
                PolynomialOver<Ring> Inverse(const Ring &ring) const;

                PolynomialOver<Ring> Times(const PolynomialOver<Ring>& t, const Ring &ring) const;
                PolynomialOver<Ring> DividedBy(const PolynomialOver<Ring>& t, const Ring &ring) const;
                PolynomialOver<Ring> Modulo(const PolynomialOver<Ring>& t, const Ring &ring) const;
                PolynomialOver<Ring> MultiplicativeInverse(const Ring &ring) const;
                bool IsUnit(const Ring &ring) const;

                PolynomialOver<Ring>& Accumulate(const PolynomialOver<Ring>& t, const Ring &ring);
                PolynomialOver<Ring>& Reduce(const PolynomialOver<Ring>& t, const Ring &ring);

                //!
                PolynomialOver<Ring> Doubled(const Ring &ring) const {return Plus(*this, ring);}
                //!
                PolynomialOver<Ring> Squared(const Ring &ring) const {return Times(*this, ring);}

                CoefficientType EvaluateAt(const CoefficientType &x, const Ring &ring) const;

                PolynomialOver<Ring>& ShiftLeft(unsigned int n, const Ring &ring);
                PolynomialOver<Ring>& ShiftRight(unsigned int n, const Ring &ring);

                //! calculate r and q such that (a == d*q + r) && (0 <= degree of r < degree of d)
                static void Divide(PolynomialOver<Ring> &r, PolynomialOver<Ring> &q, const PolynomialOver<Ring> &a, const PolynomialOver<Ring> &d, const Ring &ring);
        //@}

        //! \name INPUT/OUTPUT
        //@{
                std::istream& Input(std::istream &in, const Ring &ring);
                std::ostream& Output(std::ostream &out, const Ring &ring) const;
        //@}

private:
        void FromStr(const char *str, const Ring &ring);

        std::vector<CoefficientType> m_coefficients;
};

//! Polynomials over a fixed ring
/*! Having a fixed ring allows overloaded operators */
template <class T, int instance> class PolynomialOverFixedRing : private PolynomialOver<T>
{
        typedef PolynomialOver<T> B;
        typedef PolynomialOverFixedRing<T, instance> ThisType;

public:
        typedef T Ring;
        typedef typename T::Element CoefficientType;
        typedef typename B::DivideByZero DivideByZero;
        typedef typename B::RandomizationParameter RandomizationParameter;

        //! \name CREATORS
        //@{
                //! creates the zero polynomial
                PolynomialOverFixedRing(unsigned int count = 0) : B(ms_fixedRing, count) {}

                //! copy constructor
                PolynomialOverFixedRing(const ThisType &t) : B(t) {}

                explicit PolynomialOverFixedRing(const B &t) : B(t) {}

                //! construct constant polynomial
                PolynomialOverFixedRing(const CoefficientType &element) : B(element) {}

                //! construct polynomial with specified coefficients, starting from coefficient of x^0
                template <typename Iterator> PolynomialOverFixedRing(Iterator first, Iterator last)
                        : B(first, last) {}

                //! convert from string
                explicit PolynomialOverFixedRing(const char *str) : B(str, ms_fixedRing) {}

                //! convert from big-endian byte array
                PolynomialOverFixedRing(const byte *encodedPoly, unsigned int byteCount) : B(encodedPoly, byteCount) {}

                //! convert from Basic Encoding Rules encoded byte array
                explicit PolynomialOverFixedRing(const byte *BEREncodedPoly) : B(BEREncodedPoly) {}

                //! convert from BER encoded byte array stored in a BufferedTransformation object
                explicit PolynomialOverFixedRing(BufferedTransformation &bt) : B(bt) {}

                //! create a random PolynomialOverFixedRing
                PolynomialOverFixedRing(RandomNumberGenerator &rng, const RandomizationParameter &parameter) : B(rng, parameter, ms_fixedRing) {}

                static const ThisType &Zero();
                static const ThisType &One();
        //@}

        //! \name ACCESSORS
        //@{
                //! the zero polynomial will return a degree of -1
                int Degree() const {return B::Degree(ms_fixedRing);}
                //! degree + 1
                unsigned int CoefficientCount() const {return B::CoefficientCount(ms_fixedRing);}
                //! return coefficient for x^i
                CoefficientType GetCoefficient(unsigned int i) const {return B::GetCoefficient(i, ms_fixedRing);}
                //! return coefficient for x^i
                CoefficientType operator[](unsigned int i) const {return B::GetCoefficient(i, ms_fixedRing);}
        //@}

        //! \name MANIPULATORS
        //@{
                //!
                ThisType&  operator=(const ThisType& t) {B::operator=(t); return *this;}
                //!
                ThisType&  operator+=(const ThisType& t) {Accumulate(t, ms_fixedRing); return *this;}
                //!
                ThisType&  operator-=(const ThisType& t) {Reduce(t, ms_fixedRing); return *this;}
                //!
                ThisType&  operator*=(const ThisType& t) {return *this = *this*t;}
                //!
                ThisType&  operator/=(const ThisType& t) {return *this = *this/t;}
                //!
                ThisType&  operator%=(const ThisType& t) {return *this = *this%t;}

                //!
                ThisType&  operator<<=(unsigned int n) {ShiftLeft(n, ms_fixedRing); return *this;}
                //!
                ThisType&  operator>>=(unsigned int n) {ShiftRight(n, ms_fixedRing); return *this;}

                //! set the coefficient for x^i to value
                void SetCoefficient(unsigned int i, const CoefficientType &value) {B::SetCoefficient(i, value, ms_fixedRing);}

                //!
                void Randomize(RandomNumberGenerator &rng, const RandomizationParameter &parameter) {B::Randomize(rng, parameter, ms_fixedRing);}

                //!
                void Negate() {B::Negate(ms_fixedRing);}

                void swap(ThisType &t) {B::swap(t);}
        //@}

        //! \name UNARY OPERATORS
        //@{
                //!
                bool operator!() const {return CoefficientCount()==0;}
                //!
                ThisType operator+() const {return *this;}
                //!
                ThisType operator-() const {return ThisType(Inverse(ms_fixedRing));}
        //@}

        //! \name BINARY OPERATORS
        //@{
                //!
                friend ThisType operator>>(ThisType a, unsigned int n)  {return ThisType(a>>=n);}
                //!
                friend ThisType operator<<(ThisType a, unsigned int n)  {return ThisType(a<<=n);}
        //@}

        //! \name OTHER ARITHMETIC FUNCTIONS
        //@{
                //!
                ThisType MultiplicativeInverse() const {return ThisType(B::MultiplicativeInverse(ms_fixedRing));}
                //!
                bool IsUnit() const {return B::IsUnit(ms_fixedRing);}

                //!
                ThisType Doubled() const {return ThisType(B::Doubled(ms_fixedRing));}
                //!
                ThisType Squared() const {return ThisType(B::Squared(ms_fixedRing));}

                CoefficientType EvaluateAt(const CoefficientType &x) const {return B::EvaluateAt(x, ms_fixedRing);}

                //! calculate r and q such that (a == d*q + r) && (0 <= r < abs(d))
                static void Divide(ThisType &r, ThisType &q, const ThisType &a, const ThisType &d)
                        {B::Divide(r, q, a, d, ms_fixedRing);}
        //@}

        //! \name INPUT/OUTPUT
        //@{
                //!
                friend std::istream& operator>>(std::istream& in, ThisType &a)
                        {return a.Input(in, ms_fixedRing);}
                //!
                friend std::ostream& operator<<(std::ostream& out, const ThisType &a)
                        {return a.Output(out, ms_fixedRing);}
        //@}

private:
        struct NewOnePolynomial
        {
                ThisType * operator()() const
                {
                        return new ThisType(ms_fixedRing.MultiplicativeIdentity());
                }
        };

        static const Ring ms_fixedRing;
};

//! Ring of polynomials over another ring
template <class T> class RingOfPolynomialsOver : public AbstractEuclideanDomain<PolynomialOver<T> >
{
public:
        typedef T CoefficientRing;
        typedef PolynomialOver<T> Element;
        typedef typename Element::CoefficientType CoefficientType;
        typedef typename Element::RandomizationParameter RandomizationParameter;

        RingOfPolynomialsOver(const CoefficientRing &ring) : m_ring(ring) {}

        Element RandomElement(RandomNumberGenerator &rng, const RandomizationParameter &parameter)
                {return Element(rng, parameter, m_ring);}

        bool Equal(const Element &a, const Element &b) const
                {return a.Equals(b, m_ring);}

        const Element& Identity() const
                {return this->result = m_ring.Identity();}

        const Element& Add(const Element &a, const Element &b) const
                {return this->result = a.Plus(b, m_ring);}

        Element& Accumulate(Element &a, const Element &b) const
                {a.Accumulate(b, m_ring); return a;}

        const Element& Inverse(const Element &a) const
                {return this->result = a.Inverse(m_ring);}

        const Element& Subtract(const Element &a, const Element &b) const
                {return this->result = a.Minus(b, m_ring);}

        Element& Reduce(Element &a, const Element &b) const
                {return a.Reduce(b, m_ring);}

        const Element& Double(const Element &a) const
                {return this->result = a.Doubled(m_ring);}

        const Element& MultiplicativeIdentity() const
                {return this->result = m_ring.MultiplicativeIdentity();}

        const Element& Multiply(const Element &a, const Element &b) const
                {return this->result = a.Times(b, m_ring);}

        const Element& Square(const Element &a) const
                {return this->result = a.Squared(m_ring);}

        bool IsUnit(const Element &a) const
                {return a.IsUnit(m_ring);}

        const Element& MultiplicativeInverse(const Element &a) const
                {return this->result = a.MultiplicativeInverse(m_ring);}

        const Element& Divide(const Element &a, const Element &b) const
                {return this->result = a.DividedBy(b, m_ring);}

        const Element& Mod(const Element &a, const Element &b) const
                {return this->result = a.Modulo(b, m_ring);}

        void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const
                {Element::Divide(r, q, a, d, m_ring);}

        class InterpolationFailed : public Exception
        {
        public:
                InterpolationFailed() : Exception(OTHER_ERROR, "RingOfPolynomialsOver<T>: interpolation failed") {}
        };

        Element Interpolate(const CoefficientType x[], const CoefficientType y[], unsigned int n) const;

        // a faster version of Interpolate(x, y, n).EvaluateAt(position)
        CoefficientType InterpolateAt(const CoefficientType &position, const CoefficientType x[], const CoefficientType y[], unsigned int n) const;
/*
        void PrepareBulkInterpolation(CoefficientType *w, const CoefficientType x[], unsigned int n) const;
        void PrepareBulkInterpolationAt(CoefficientType *v, const CoefficientType &position, const CoefficientType x[], const CoefficientType w[], unsigned int n) const;
        CoefficientType BulkInterpolateAt(const CoefficientType y[], const CoefficientType v[], unsigned int n) const;
*/
protected:
        void CalculateAlpha(std::vector<CoefficientType> &alpha, const CoefficientType x[], const CoefficientType y[], unsigned int n) const;

        CoefficientRing m_ring;
};

template <class Ring, class Element>
void PrepareBulkPolynomialInterpolation(const Ring &ring, Element *w, const Element x[], unsigned int n);
template <class Ring, class Element>
void PrepareBulkPolynomialInterpolationAt(const Ring &ring, Element *v, const Element &position, const Element x[], const Element w[], unsigned int n);
template <class Ring, class Element>
Element BulkPolynomialInterpolateAt(const Ring &ring, const Element y[], const Element v[], unsigned int n);

//!
template <class T, int instance>
inline bool operator==(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return a.Equals(b, a.ms_fixedRing);}
//!
template <class T, int instance>
inline bool operator!=(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return !(a==b);}

//!
template <class T, int instance>
inline bool operator> (const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return a.Degree() > b.Degree();}
//!
template <class T, int instance>
inline bool operator>=(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return a.Degree() >= b.Degree();}
//!
template <class T, int instance>
inline bool operator< (const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return a.Degree() < b.Degree();}
//!
template <class T, int instance>
inline bool operator<=(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return a.Degree() <= b.Degree();}

//!
template <class T, int instance>
inline CryptoPP::PolynomialOverFixedRing<T, instance> operator+(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return CryptoPP::PolynomialOverFixedRing<T, instance>(a.Plus(b, a.ms_fixedRing));}
//!
template <class T, int instance>
inline CryptoPP::PolynomialOverFixedRing<T, instance> operator-(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return CryptoPP::PolynomialOverFixedRing<T, instance>(a.Minus(b, a.ms_fixedRing));}
//!
template <class T, int instance>
inline CryptoPP::PolynomialOverFixedRing<T, instance> operator*(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return CryptoPP::PolynomialOverFixedRing<T, instance>(a.Times(b, a.ms_fixedRing));}
//!
template <class T, int instance>
inline CryptoPP::PolynomialOverFixedRing<T, instance> operator/(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return CryptoPP::PolynomialOverFixedRing<T, instance>(a.DividedBy(b, a.ms_fixedRing));}
//!
template <class T, int instance>
inline CryptoPP::PolynomialOverFixedRing<T, instance> operator%(const CryptoPP::PolynomialOverFixedRing<T, instance> &a, const CryptoPP::PolynomialOverFixedRing<T, instance> &b)
        {return CryptoPP::PolynomialOverFixedRing<T, instance>(a.Modulo(b, a.ms_fixedRing));}

NAMESPACE_END

NAMESPACE_BEGIN(std)
template<class T> inline void swap(CryptoPP::PolynomialOver<T> &a, CryptoPP::PolynomialOver<T> &b)
{
        a.swap(b);
}
template<class T, int i> inline void swap(CryptoPP::PolynomialOverFixedRing<T,i> &a, CryptoPP::PolynomialOverFixedRing<T,i> &b)
{
        a.swap(b);
}
NAMESPACE_END

#endif

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