Quotient ring. More...

#include <algebra.h>

Inheritance diagram for QuotientRing< T >:

Public Types
typedef T	EuclideanDomain

typedef T::Element	Element

Public Types inherited from AbstractRing< T::Element >
typedef T	Element

Public Types inherited from AbstractGroup< T >
typedef T	Element

Public Member Functions
	QuotientRing (const EuclideanDomain &domain, const Element &modulus)

const EuclideanDomain &	GetDomain () const

const Element &	GetModulus () const

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Detailed Description

template<class T>
class QuotientRing< T >

Quotient ring.

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 386 of file algebra.h.

Member Typedef Documentation

◆ EuclideanDomain

template<class T >

typedef T QuotientRing< T >::EuclideanDomain

Definition at line 389 of file algebra.h.

◆ Element

template<class T >

typedef T::Element QuotientRing< T >::Element

Definition at line 390 of file algebra.h.

Constructor & Destructor Documentation

◆ QuotientRing()

template<class T >

QuotientRing< T >::QuotientRing	(	const EuclideanDomain &	domain,
		const Element &	modulus
	)

inline

Definition at line 392 of file algebra.h.

Member Function Documentation

◆ GetDomain()

template<class T >

const EuclideanDomain & QuotientRing< T >::GetDomain ( ) const

inline

Definition at line 395 of file algebra.h.

◆ GetModulus()

template<class T >

const Element & QuotientRing< T >::GetModulus ( ) const

inline

Definition at line 398 of file algebra.h.

◆ Equal()

template<class T >

bool QuotientRing< 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 401 of file algebra.h.

◆ Identity()

template<class T >

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

inlinevirtual

Provides the Identity element.

Returns: the Identity element

Implements AbstractGroup< T >.

Definition at line 404 of file algebra.h.

◆ Add()

template<class T >

const Element & QuotientRing< 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 407 of file algebra.h.

◆ Accumulate()

template<class T >

Element & QuotientRing< 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 410 of file algebra.h.

◆ Inverse()

template<class T >

const Element & QuotientRing< 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 413 of file algebra.h.

◆ Subtract()

template<class T >

const Element & QuotientRing< 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 416 of file algebra.h.

◆ Reduce()

template<class T >

Element & QuotientRing< 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 419 of file algebra.h.

◆ Double()

template<class T >

const Element & QuotientRing< 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 422 of file algebra.h.

◆ IsUnit()

template<class T >

bool QuotientRing< 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::Element >.

Definition at line 425 of file algebra.h.

◆ MultiplicativeIdentity()

template<class T >

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

inlinevirtual

Retrieves the multiplicative identity.

Returns: the multiplicative identity

Implements AbstractRing< T::Element >.

Definition at line 428 of file algebra.h.

◆ Multiply()

template<class T >

const Element & QuotientRing< 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::Element >.

Definition at line 431 of file algebra.h.

◆ Square()

template<class T >

const Element & QuotientRing< 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::Element >.

Definition at line 434 of file algebra.h.

◆ MultiplicativeInverse()

template<class T >

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

virtual

Calculate the multiplicative inverse of an element in the group.

Parameters

a	the element

Implements AbstractRing< T::Element >.

Definition at line 70 of file algebra.cpp.

◆ operator==()

template<class T >

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

inline

Definition at line 439 of file algebra.h.

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

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ EuclideanDomain

◆ Element

Constructor & Destructor Documentation

◆ QuotientRing()

Member Function Documentation

◆ GetDomain()

◆ GetModulus()

◆ Equal()

◆ Identity()

◆ Add()

◆ Accumulate()

◆ Inverse()

◆ Subtract()

◆ Reduce()

◆ Double()

◆ IsUnit()

◆ MultiplicativeIdentity()

◆ Multiply()

◆ Square()

◆ MultiplicativeInverse()

◆ operator==()