Abstract ring. More...

#include <algebra.h>

Inheritance diagram for AbstractRing< T >:

Public Types
typedef T	Element

Public Types inherited from AbstractGroup< T >
typedef T	Element

Public Member Functions
	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 AbstractRing< T >

Abstract 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 118 of file algebra.h.

Member Typedef Documentation

◆ Element

template<class T >

typedef T AbstractRing< T >::Element

Definition at line 121 of file algebra.h.

Constructor & Destructor Documentation

◆ AbstractRing() [1/2]

template<class T >

AbstractRing< T >::AbstractRing ( )

inline

Construct an AbstractRing.

Definition at line 124 of file algebra.h.

◆ AbstractRing() [2/2]

template<class T >

AbstractRing< T >::AbstractRing ( const AbstractRing< T > & source )

inline

Copy construct an AbstractRing.

Parameters

source other AbstractRing

Definition at line 128 of file algebra.h.

Member Function Documentation

◆ operator=()

template<class T >

AbstractRing & AbstractRing< T >::operator= ( const AbstractRing< T > & source )

inline

Assign an AbstractRing.

Parameters

source other AbstractRing

Definition at line 133 of file algebra.h.

◆ IsUnit()

template<class T >

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

pure virtual

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.

Implemented in EuclideanDomainOf< T >, GFP2_ONB< F >, and ModularArithmetic.

◆ MultiplicativeIdentity()

template<class T >

virtual const Element & AbstractRing< T >::MultiplicativeIdentity ( ) const

pure virtual

Retrieves the multiplicative identity.

Returns: the multiplicative identity

Implemented in EuclideanDomainOf< T >, EuclideanDomainOf< PolynomialMod2 >, QuotientRing< T >, QuotientRing< EuclideanDomainOf< PolynomialMod2 > >, ModularArithmetic, MontgomeryRepresentation, RingOfPolynomialsOver< T >, and GFP2_ONB< F >.

◆ Multiply()

template<class T >

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

pure virtual

Multiplies elements in the group.

Parameters

a	the multiplicand
b	the multiplier

Returns: the product of a and b

Implemented in EuclideanDomainOf< T >, GFP2_ONB< F >, ModularArithmetic, and MontgomeryRepresentation.

◆ MultiplicativeInverse()

template<class T >

virtual const Element & AbstractRing< T >::MultiplicativeInverse ( const Element & a ) const

pure virtual

Calculate the multiplicative inverse of an element in the group.

Parameters

a	the element

Implemented in EuclideanDomainOf< T >, GFP2_ONB< F >, ModularArithmetic, and MontgomeryRepresentation.

◆ Square()

template<class T >

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

virtual

Square an element in the group.

Parameters

a	the element

Returns: the element squared

Reimplemented in EuclideanDomainOf< T >, GFP2_ONB< F >, ModularArithmetic, and MontgomeryRepresentation.

Definition at line 37 of file algebra.cpp.

◆ Divide()

template<class T >

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

virtual

Divides elements in the group.

Parameters

a	the dividend
b	the divisor

Returns: the quotient

Reimplemented in EuclideanDomainOf< T >, and ModularArithmetic.

Definition at line 42 of file algebra.cpp.

◆ Exponentiate()

template<class T >

T AbstractRing< T >::Exponentiate	(	const Element &	a,
		const Integer &	e
	)		const

virtual

Raises a base to an exponent in the group.

Parameters

a	the base
e	the exponent

Returns: the exponentiation

Reimplemented in GFP2_ONB< F >.

Definition at line 316 of file algebra.cpp.

◆ CascadeExponentiate()

template<class T >

T AbstractRing< T >::CascadeExponentiate	(	const Element &	x,
		const Integer &	e1,
		const Element &	y,
		const Integer &	e2
	)		const

virtual

TODO.

Parameters

x	first element
e1	first exponent
y	second element
e2	second exponent

Returns: TODO

Reimplemented in ModularArithmetic, and MontgomeryRepresentation.

Definition at line 323 of file algebra.cpp.

◆ SimultaneousExponentiate()

template<class T >

void AbstractRing< T >::SimultaneousExponentiate	(	Element *	results,
		const Element &	base,
		const Integer *	exponents,
		unsigned int	exponentsCount
	)		const

virtual

Exponentiates a base to multiple exponents in the Ring.

Parameters

results	an array of Elements
base	the base to raise to the exponents
exponents	an array of exponents
exponentsCount	the number of exponents in the array

SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the result at the respective position in the results array.

SimultaneousExponentiate() must be implemented in a derived class.

Precondition: COUNTOF(results) == exponentsCount; COUNTOF(exponents) == exponentsCount

Reimplemented in ModularArithmetic, and MontgomeryRepresentation.

Definition at line 334 of file algebra.cpp.

◆ MultiplicativeGroup()

template<class T >

virtual const AbstractGroup< T > & AbstractRing< T >::MultiplicativeGroup ( ) const

inlinevirtual

Retrieves the multiplicative group.

Returns: the multiplicative group

Definition at line 194 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

◆ Element

Constructor & Destructor Documentation

◆ AbstractRing() [1/2]

◆ AbstractRing() [2/2]

Member Function Documentation

◆ operator=()

◆ IsUnit()

◆ MultiplicativeIdentity()

◆ Multiply()

◆ MultiplicativeInverse()

◆ Square()

◆ Divide()

◆ Exponentiate()

◆ CascadeExponentiate()

◆ SimultaneousExponentiate()

◆ MultiplicativeGroup()