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