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QuotientRing< T > Class Template Reference

Quotient ring. More...

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.

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

AbstractRingoperator= (const AbstractRing &source)
Assign an AbstractRing. 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 InversionIsFast () const
Determine if inversion is fast. 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 Function Documentation

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.

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.

template<class T>
 const Element& QuotientRing< T >::Add ( const Element & a, const Element & b ) const
inlinevirtual

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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