Abstract Euclidean domain. More...

#include <algebra.h>

Inheritance diagram for AbstractEuclideanDomain< T >:

Public Types
typedef T	Element

Public Types inherited from AbstractRing< T >
typedef T	Element

Public Types inherited from AbstractGroup< T >
typedef T	Element

Public Member Functions
virtual void	DivisionAlgorithm (Element &r, Element &q, const Element &a, const Element &d) const =0
	Performs the division algorithm on two elements in the ring.

virtual const Element &	Mod (const Element &a, const Element &b) const =0
	Performs a modular reduction in the ring.

virtual const Element &	Gcd (const Element &a, const Element &b) const
	Calculates the greatest common denominator in the ring.

Public Member Functions inherited from AbstractRing< T >
	AbstractRing ()
	Construct an AbstractRing.

	AbstractRing (const AbstractRing &source)
	Copy construct an AbstractRing.

AbstractRing &	operator= (const AbstractRing &source)
	Assign an AbstractRing.

virtual bool	IsUnit (const Element &a) const =0
	Determines whether an element is a unit in the group.

virtual const Element &	MultiplicativeIdentity () const =0
	Retrieves the multiplicative identity.

virtual const Element &	Multiply (const Element &a, const Element &b) const =0
	Multiplies elements in the group.

virtual const Element &	MultiplicativeInverse (const Element &a) const =0
	Calculate the multiplicative inverse of an element in the group.

virtual const Element &	Square (const Element &a) const
	Square an element in the group.

virtual const Element &	Divide (const Element &a, const Element &b) const
	Divides elements in the group.

virtual Element	Exponentiate (const Element &a, const Integer &e) const
	Raises a base to an exponent in the group.

virtual Element	CascadeExponentiate (const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
	TODO.

virtual void	SimultaneousExponentiate (Element results, const Element &base, const Integer exponents, unsigned int exponentsCount) const
	Exponentiates a base to multiple exponents in the Ring.

virtual const AbstractGroup< T > &	MultiplicativeGroup () const
	Retrieves the multiplicative group.

Public Member Functions inherited from AbstractGroup< T >
virtual bool	Equal (const Element &a, const Element &b) const =0
	Compare two elements for equality.

virtual const Element &	Identity () const =0
	Provides the Identity element.

virtual const Element &	Add (const Element &a, const Element &b) const =0
	Adds elements in the group.

virtual const Element &	Inverse (const Element &a) const =0
	Inverts the element in the group.

virtual bool	InversionIsFast () const
	Determine if inversion is fast.

virtual const Element &	Double (const Element &a) const
	Doubles an element in the group.

virtual const Element &	Subtract (const Element &a, const Element &b) const
	Subtracts elements in the group.

virtual Element &	Accumulate (Element &a, const Element &b) const
	TODO.

virtual Element &	Reduce (Element &a, const Element &b) const
	Reduces an element in the congruence class.

virtual Element	ScalarMultiply (const Element &a, const Integer &e) const
	Performs a scalar multiplication.

virtual Element	CascadeScalarMultiply (const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
	TODO.

virtual void	SimultaneousMultiply (Element results, const Element &base, const Integer exponents, unsigned int exponentsCount) const
	Multiplies a base to multiple exponents in a group.

Detailed Description

template<class T>
class AbstractEuclideanDomain< T >

Abstract Euclidean domain.

Template Parameters

T	element class or type

const Element& 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:

    abcd = group.Add(group.Add(a,b), group.Add(c,d));

But this should be fine:

    abcd = group.Add(a, group.Add(b, group.Add(c,d));

Definition at line 276 of file algebra.h.

Member Typedef Documentation

◆ Element

template<class T >

typedef T AbstractEuclideanDomain< T >::Element

Definition at line 279 of file algebra.h.

Member Function Documentation

◆ DivisionAlgorithm()

template<class T >

virtual void AbstractEuclideanDomain< T >::DivisionAlgorithm	(	Element &	r,
		Element &	q,
		const Element &	a,
		const Element &	d
	)		const

pure virtual

Performs the division algorithm on two elements in the ring.

Parameters

r	the remainder
q	the quotient
a	the dividend
d	the divisor

Implemented in EuclideanDomainOf< T >, and RingOfPolynomialsOver< T >.

◆ Mod()

template<class T >

const T & AbstractEuclideanDomain< T >::Mod	(	const Element &	a,
		const Element &	b
	)		const

pure virtual

Performs a modular reduction in the ring.

Parameters

a	the element
b	the modulus

Returns: the result of ab.

Implemented in EuclideanDomainOf< T >, and RingOfPolynomialsOver< T >.

Definition at line 49 of file algebra.cpp.

◆ Gcd()

template<class T >

const T & AbstractEuclideanDomain< T >::Gcd	(	const Element &	a,
		const Element &	b
	)		const

virtual

Calculates the greatest common denominator in the ring.

Parameters

a	the first element
b	the second element

Returns: the greatest common denominator of a and b.

Definition at line 56 of file algebra.cpp.

The documentation for this class was generated from the following files:

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ Element

Member Function Documentation

◆ DivisionAlgorithm()

◆ Mod()

◆ Gcd()