Euclidean domain. More...

#include <algebra.h>

Inheritance diagram for EuclideanDomainOf< T >:

Public Types
typedef T	Element

Public Types inherited from AbstractEuclideanDomain< T >
typedef T	Element

Public Types inherited from AbstractRing< T >
typedef T	Element

Public Types inherited from AbstractGroup< T >
typedef T	Element

Public Member Functions
bool	Equal (const Element &a, const Element &b) const
	Compare two elements for equality.

const Element &	Identity () const
	Provides the Identity element.

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

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

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

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

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

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

const Element &	MultiplicativeIdentity () const
	Retrieves the multiplicative identity.

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

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

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

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

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

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

void	DivisionAlgorithm (Element &r, Element &q, const Element &a, const Element &d) const
	Performs the division algorithm on two elements in the ring.

bool	operator== (const EuclideanDomainOf< T > &rhs) const

Public Member Functions inherited from AbstractEuclideanDomain< T >
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 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	InversionIsFast () const
	Determine if inversion is fast.

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 EuclideanDomainOf< T >

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 315 of file algebra.h.

Member Typedef Documentation

◆ Element

template<class T >

typedef T EuclideanDomainOf< T >::Element

Definition at line 318 of file algebra.h.

Constructor & Destructor Documentation

◆ EuclideanDomainOf()

template<class T >

EuclideanDomainOf< T >::EuclideanDomainOf ( )

inline

Definition at line 320 of file algebra.h.

Member Function Documentation

◆ Equal()

template<class T >

bool EuclideanDomainOf< T >::Equal	(	const Element &	a,
		const Element &	b ) const

inlinevirtual

Compare two elements for equality.

Parameters

a	first element
b	second element

Returns: true if the elements are equal, false otherwise

Equal() tests the elements for equality using a==b

Implements AbstractGroup< T >.

Definition at line 322 of file algebra.h.

◆ Identity()

template<class T >

const Element & EuclideanDomainOf< T >::Identity ( ) const

inlinevirtual

Provides the Identity element.

Returns: the Identity element

Implements AbstractGroup< T >.

Definition at line 325 of file algebra.h.

◆ Add()

template<class T >

const Element & EuclideanDomainOf< T >::Add	(	const Element &	a,
		const Element &	b ) const

inlinevirtual

Adds elements in the group.

Parameters

a	first element
b	second element

Returns: the sum of a and b

Implements AbstractGroup< T >.

Definition at line 328 of file algebra.h.

◆ Accumulate()

template<class T >

Element & EuclideanDomainOf< T >::Accumulate	(	Element &	a,
		const Element &	b ) const

inlinevirtual

TODO.

Parameters

a	first element
b	second element

Returns: TODO

Reimplemented from AbstractGroup< T >.

Definition at line 331 of file algebra.h.

◆ Inverse()

template<class T >

const Element & EuclideanDomainOf< T >::Inverse ( const Element & a ) const

inlinevirtual

Inverts the element in the group.

Parameters

a	first element

Returns: the inverse of the element

Implements AbstractGroup< T >.

Definition at line 334 of file algebra.h.

◆ Subtract()

template<class T >

const Element & EuclideanDomainOf< T >::Subtract	(	const Element &	a,
		const Element &	b ) const

inlinevirtual

Subtracts elements in the group.

Parameters

a	first element
b	second element

Returns: the difference of a and b. The element a must provide a Subtract member function.

Reimplemented from AbstractGroup< T >.

Definition at line 337 of file algebra.h.

◆ Reduce()

template<class T >

Element & EuclideanDomainOf< T >::Reduce	(	Element &	a,
		const Element &	b ) const

inlinevirtual

Reduces an element in the congruence class.

Parameters

a	element to reduce
b	the congruence class

Returns: the reduced element

Reimplemented from AbstractGroup< T >.

Definition at line 340 of file algebra.h.

◆ Double()

template<class T >

const Element & EuclideanDomainOf< T >::Double ( const Element & a ) const

inlinevirtual

Doubles an element in the group.

Parameters

a	the element

Returns: the element doubled

Reimplemented from AbstractGroup< T >.

Definition at line 343 of file algebra.h.

◆ MultiplicativeIdentity()

template<class T >

const Element & EuclideanDomainOf< T >::MultiplicativeIdentity ( ) const

inlinevirtual

Retrieves the multiplicative identity.

Returns: the multiplicative identity

Implements AbstractRing< T >.

Definition at line 346 of file algebra.h.

◆ Multiply()

template<class T >

const Element & EuclideanDomainOf< T >::Multiply	(	const Element &	a,
		const Element &	b ) const

inlinevirtual

Multiplies elements in the group.

Parameters

a	the multiplicand
b	the multiplier

Returns: the product of a and b

Implements AbstractRing< T >.

Definition at line 349 of file algebra.h.

◆ Square()

template<class T >

const Element & EuclideanDomainOf< T >::Square ( const Element & a ) const

inlinevirtual

Square an element in the group.

Parameters

a	the element

Returns: the element squared

Reimplemented from AbstractRing< T >.

Definition at line 352 of file algebra.h.

◆ IsUnit()

template<class T >

bool EuclideanDomainOf< T >::IsUnit ( const Element & a ) const

inlinevirtual

Determines whether an element is a unit in the group.

Parameters

a	the element

Returns: true if the element is a unit after reduction, false otherwise.

Implements AbstractRing< T >.

Definition at line 355 of file algebra.h.

◆ MultiplicativeInverse()

template<class T >

const Element & EuclideanDomainOf< T >::MultiplicativeInverse ( const Element & a ) const

inlinevirtual

Calculate the multiplicative inverse of an element in the group.

Parameters

a	the element

Implements AbstractRing< T >.

Definition at line 358 of file algebra.h.

◆ Divide()

template<class T >

const Element & EuclideanDomainOf< T >::Divide	(	const Element &	a,
		const Element &	b ) const

inlinevirtual

Divides elements in the group.

Parameters

a	the dividend
b	the divisor

Returns: the quotient

Reimplemented from AbstractRing< T >.

Definition at line 361 of file algebra.h.

◆ Mod()

template<class T >

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

inlinevirtual

Performs a modular reduction in the ring.

Parameters

a	the element
b	the modulus

Returns: the result of ab.

Implements AbstractEuclideanDomain< T >.

Definition at line 364 of file algebra.h.

◆ DivisionAlgorithm()

template<class T >

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

inlinevirtual

Performs the division algorithm on two elements in the ring.

Parameters

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

Implements AbstractEuclideanDomain< T >.

Definition at line 367 of file algebra.h.

◆ operator==()

template<class T >

bool EuclideanDomainOf< T >::operator== ( const EuclideanDomainOf< T > & rhs ) const

inline

Definition at line 370 of file algebra.h.

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

algebra.h

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ Element

Constructor & Destructor Documentation

◆ EuclideanDomainOf()

Member Function Documentation

◆ Equal()

◆ Identity()

◆ Add()

◆ Accumulate()

◆ Inverse()

◆ Subtract()

◆ Reduce()

◆ Double()

◆ MultiplicativeIdentity()

◆ Multiply()

◆ Square()

◆ IsUnit()

◆ MultiplicativeInverse()

◆ Divide()

◆ Mod()

◆ DivisionAlgorithm()

◆ operator==()