algebra.h

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// algebra.h - originally written and placed in the public domain by Wei Dai
 
/// \file algebra.h
/// \brief Classes for performing mathematics over different fields
 
#ifndef CRYPTOPP_ALGEBRA_H
#define CRYPTOPP_ALGEBRA_H
 
#include "config.h"
#include "integer.h"
#include "misc.h"
 
NAMESPACE_BEGIN(CryptoPP)
 
class Integer;
 
/// \brief Abstract group
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class CRYPTOPP_NO_VTABLE AbstractGroup
{
public:
    typedef T Element;
 
    virtual ~AbstractGroup() {}
 
    /// \brief Compare two elements for equality
    /// \param a first element
    /// \param b second element
    /// \return true if the elements are equal, false otherwise
    /// \details Equal() tests the elements for equality using <tt>a==b</tt>
    virtual bool Equal(const Element &a, const Element &b) const =0;
 
    /// \brief Provides the Identity element
    /// \return the Identity element
    virtual const Element& Identity() const =0;
 
    /// \brief Adds elements in the group
    /// \param a first element
    /// \param b second element
    /// \return the sum of <tt>a</tt> and <tt>b</tt>
    virtual const Element& Add(const Element &a, const Element &b) const =0;
 
    /// \brief Inverts the element in the group
    /// \param a first element
    /// \return the inverse of the element
    virtual const Element& Inverse(const Element &a) const =0;
 
    /// \brief Determine if inversion is fast
    /// \return true if inversion is fast, false otherwise
    virtual bool InversionIsFast() const {return false;}
 
    /// \brief Doubles an element in the group
    /// \param a the element
    /// \return the element doubled
    virtual const Element& Double(const Element &a) const;
 
    /// \brief Subtracts elements in the group
    /// \param a first element
    /// \param b second element
    /// \return the difference of <tt>a</tt> and <tt>b</tt>. The element <tt>a</tt> must provide a Subtract member function.
    virtual const Element& Subtract(const Element &a, const Element &b) const;
 
    /// \brief TODO
    /// \param a first element
    /// \param b second element
    /// \return TODO
    virtual Element& Accumulate(Element &a, const Element &b) const;
 
    /// \brief Reduces an element in the congruence class
    /// \param a element to reduce
    /// \param b the congruence class
    /// \return the reduced element
    virtual Element& Reduce(Element &a, const Element &b) const;
 
    /// \brief Performs a scalar multiplication
    /// \param a multiplicand
    /// \param e multiplier
    /// \return the product
    virtual Element ScalarMultiply(const Element &a, const Integer &e) const;
 
    /// \brief TODO
    /// \param x first multiplicand
    /// \param e1 the first multiplier
    /// \param y second multiplicand
    /// \param e2 the second multiplier
    /// \return TODO
    virtual Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const;
 
    /// \brief Multiplies a base to multiple exponents in a group
    /// \param results an array of Elements
    /// \param base the base to raise to the exponents
    /// \param exponents an array of exponents
    /// \param exponentsCount the number of exponents in the array
    /// \details SimultaneousMultiply() multiplies the base to each exponent in the exponents array and stores the
    ///   result at the respective position in the results array.
    /// \details SimultaneousMultiply() must be implemented in a derived class.
    /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
    /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
    virtual void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;
};
 
/// \brief Abstract ring
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class CRYPTOPP_NO_VTABLE AbstractRing : public AbstractGroup<T>
{
public:
    typedef T Element;
 
    /// \brief Construct an AbstractRing
    AbstractRing() {m_mg.m_pRing = this;}
 
    /// \brief Copy construct an AbstractRing
    /// \param source other AbstractRing
    AbstractRing(const AbstractRing &source)
        {CRYPTOPP_UNUSED(source); m_mg.m_pRing = this;}
 
    /// \brief Assign an AbstractRing
    /// \param source other AbstractRing
    AbstractRing& operator=(const AbstractRing &source)
        {CRYPTOPP_UNUSED(source); return *this;}
 
    /// \brief Determines whether an element is a unit in the group
    /// \param a the element
    /// \return true if the element is a unit after reduction, false otherwise.
    virtual bool IsUnit(const Element &a) const =0;
 
    /// \brief Retrieves the multiplicative identity
    /// \return the multiplicative identity
    virtual const Element& MultiplicativeIdentity() const =0;
 
    /// \brief Multiplies elements in the group
    /// \param a the multiplicand
    /// \param b the multiplier
    /// \return the product of a and b
    virtual const Element& Multiply(const Element &a, const Element &b) const =0;
 
    /// \brief Calculate the multiplicative inverse of an element in the group
    /// \param a the element
    virtual const Element& MultiplicativeInverse(const Element &a) const =0;
 
    /// \brief Square an element in the group
    /// \param a the element
    /// \return the element squared
    virtual const Element& Square(const Element &a) const;
 
    /// \brief Divides elements in the group
    /// \param a the dividend
    /// \param b the divisor
    /// \return the quotient
    virtual const Element& Divide(const Element &a, const Element &b) const;
 
    /// \brief Raises a base to an exponent in the group
    /// \param a the base
    /// \param e the exponent
    /// \return the exponentiation
    virtual Element Exponentiate(const Element &a, const Integer &e) const;
 
    /// \brief TODO
    /// \param x first element
    /// \param e1 first exponent
    /// \param y second element
    /// \param e2 second exponent
    /// \return TODO
    virtual Element CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const;
 
    /// \brief Exponentiates a base to multiple exponents in the Ring
    /// \param results an array of Elements
    /// \param base the base to raise to the exponents
    /// \param exponents an array of exponents
    /// \param exponentsCount the number of exponents in the array
    /// \details SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the
    ///   result at the respective position in the results array.
    /// \details SimultaneousExponentiate() must be implemented in a derived class.
    /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
    /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
    virtual void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;
 
    /// \brief Retrieves the multiplicative group
    /// \return the multiplicative group
    virtual const AbstractGroup<T>& MultiplicativeGroup() const
        {return m_mg;}
 
private:
    class MultiplicativeGroupT : public AbstractGroup<T>
    {
    public:
        const AbstractRing<T>& GetRing() const
            {return *m_pRing;}
 
        bool Equal(const Element &a, const Element &b) const
            {return GetRing().Equal(a, b);}
 
        const Element& Identity() const
            {return GetRing().MultiplicativeIdentity();}
 
        const Element& Add(const Element &a, const Element &b) const
            {return GetRing().Multiply(a, b);}
 
        Element& Accumulate(Element &a, const Element &b) const
            {return a = GetRing().Multiply(a, b);}
 
        const Element& Inverse(const Element &a) const
            {return GetRing().MultiplicativeInverse(a);}
 
        const Element& Subtract(const Element &a, const Element &b) const
            {return GetRing().Divide(a, b);}
 
        Element& Reduce(Element &a, const Element &b) const
            {return a = GetRing().Divide(a, b);}
 
        const Element& Double(const Element &a) const
            {return GetRing().Square(a);}
 
        Element ScalarMultiply(const Element &a, const Integer &e) const
            {return GetRing().Exponentiate(a, e);}
 
        Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
            {return GetRing().CascadeExponentiate(x, e1, y, e2);}
 
        void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
            {GetRing().SimultaneousExponentiate(results, base, exponents, exponentsCount);}
 
        const AbstractRing<T> *m_pRing;
    };
 
    MultiplicativeGroupT m_mg;
};
 
// ********************************************************
 
/// \brief Base and exponent
/// \tparam T base class or type
/// \tparam E exponent class or type
template <class T, class E = Integer>
struct BaseAndExponent
{
public:
    BaseAndExponent() {}
    BaseAndExponent(const T &base, const E &exponent) : base(base), exponent(exponent) {}
    bool operator<(const BaseAndExponent<T, E> &rhs) const {return exponent < rhs.exponent;}
    T base;
    E exponent;
};
 
// VC60 workaround: incomplete member template support
template <class Element, class Iterator>
    Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end);
template <class Element, class Iterator>
    Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end);
 
// ********************************************************
 
/// \brief Abstract Euclidean domain
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class CRYPTOPP_NO_VTABLE AbstractEuclideanDomain : public AbstractRing<T>
{
public:
    typedef T Element;
 
    /// \brief Performs the division algorithm on two elements in the ring
    /// \param r the remainder
    /// \param q the quotient
    /// \param a the dividend
    /// \param d the divisor
    virtual void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const =0;
 
    /// \brief Performs a modular reduction in the ring
    /// \param a the element
    /// \param b the modulus
    /// \return the result of <tt>a%b</tt>.
    virtual const Element& Mod(const Element &a, const Element &b) const =0;
 
    /// \brief Calculates the greatest common denominator in the ring
    /// \param a the first element
    /// \param b the second element
    /// \return the greatest common denominator of a and b.
    virtual const Element& Gcd(const Element &a, const Element &b) const;
 
protected:
    mutable Element result;
};
 
// ********************************************************
 
/// \brief Euclidean domain
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class EuclideanDomainOf : public AbstractEuclideanDomain<T>
{
public:
    typedef T Element;
 
    EuclideanDomainOf() {}
 
    bool Equal(const Element &a, const Element &b) const
        {return a==b;}
 
    const Element& Identity() const
        {return Element::Zero();}
 
    const Element& Add(const Element &a, const Element &b) const
        {return result = a+b;}
 
    Element& Accumulate(Element &a, const Element &b) const
        {return a+=b;}
 
    const Element& Inverse(const Element &a) const
        {return result = -a;}
 
    const Element& Subtract(const Element &a, const Element &b) const
        {return result = a-b;}
 
    Element& Reduce(Element &a, const Element &b) const
        {return a-=b;}
 
    const Element& Double(const Element &a) const
        {return result = a.Doubled();}
 
    const Element& MultiplicativeIdentity() const
        {return Element::One();}
 
    const Element& Multiply(const Element &a, const Element &b) const
        {return result = a*b;}
 
    const Element& Square(const Element &a) const
        {return result = a.Squared();}
 
    bool IsUnit(const Element &a) const
        {return a.IsUnit();}
 
    const Element& MultiplicativeInverse(const Element &a) const
        {return result = a.MultiplicativeInverse();}
 
    const Element& Divide(const Element &a, const Element &b) const
        {return result = a/b;}
 
    const Element& Mod(const Element &a, const Element &b) const
        {return result = a%b;}
 
    void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const
        {Element::Divide(r, q, a, d);}
 
    bool operator==(const EuclideanDomainOf<T> &rhs) const
        {CRYPTOPP_UNUSED(rhs); return true;}
 
private:
    mutable Element result;
};
 
/// \brief Quotient ring
/// \tparam T element class or type
/// \details <tt>const Element&</tt> returned by member functions are references
///   to internal data members. Since each object may have only
///   one such data member for holding results, the following code
///   will produce incorrect results:
///   <pre>    abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
///   But this should be fine:
///   <pre>    abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
template <class T> class QuotientRing : public AbstractRing<typename T::Element>
{
public:
    typedef T EuclideanDomain;
    typedef typename T::Element Element;
 
    QuotientRing(const EuclideanDomain &domain, const Element &modulus)
        : m_domain(domain), m_modulus(modulus) {}
 
    const EuclideanDomain & GetDomain() const
        {return m_domain;}
 
    const Element& GetModulus() const
        {return m_modulus;}
 
    bool Equal(const Element &a, const Element &b) const
        {return m_domain.Equal(m_domain.Mod(m_domain.Subtract(a, b), m_modulus), m_domain.Identity());}
 
    const Element& Identity() const
        {return m_domain.Identity();}
 
    const Element& Add(const Element &a, const Element &b) const
        {return m_domain.Add(a, b);}
 
    Element& Accumulate(Element &a, const Element &b) const
        {return m_domain.Accumulate(a, b);}
 
    const Element& Inverse(const Element &a) const
        {return m_domain.Inverse(a);}
 
    const Element& Subtract(const Element &a, const Element &b) const
        {return m_domain.Subtract(a, b);}
 
    Element& Reduce(Element &a, const Element &b) const
        {return m_domain.Reduce(a, b);}
 
    const Element& Double(const Element &a) const
        {return m_domain.Double(a);}
 
    bool IsUnit(const Element &a) const
        {return m_domain.IsUnit(m_domain.Gcd(a, m_modulus));}
 
    const Element& MultiplicativeIdentity() const
        {return m_domain.MultiplicativeIdentity();}
 
    const Element& Multiply(const Element &a, const Element &b) const
        {return m_domain.Mod(m_domain.Multiply(a, b), m_modulus);}
 
    const Element& Square(const Element &a) const
        {return m_domain.Mod(m_domain.Square(a), m_modulus);}
 
    const Element& MultiplicativeInverse(const Element &a) const;
 
    bool operator==(const QuotientRing<T> &rhs) const
        {return m_domain == rhs.m_domain && m_modulus == rhs.m_modulus;}
 
protected:
    EuclideanDomain m_domain;
    Element m_modulus;
};
 
NAMESPACE_END
 
#ifdef CRYPTOPP_MANUALLY_INSTANTIATE_TEMPLATES
#include "algebra.cpp"
#endif
 
#endif