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EuclideanDomainOf< T > Class Template Reference

Euclidean domain. More...

+ 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. 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...
 
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...
 
bool IsUnit (const Element &a) const
 Determines whether an element is a unit in the group. More...
 
const Element & MultiplicativeInverse (const Element &a) const
 Calculate the multiplicative inverse of an element in the group. More...
 
const Element & Divide (const Element &a, const Element &b) const
 Divides elements in the group. More...
 
const Element & Mod (const Element &a, const Element &b) const
 Performs a modular reduction in the ring. More...
 
void DivisionAlgorithm (Element &r, Element &q, const Element &a, const Element &d) const
 Performs the division algorithm on two elements in the ring. More...
 
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. More...
 
- Public Member Functions inherited from AbstractRing< T >
 AbstractRing ()
 Construct an AbstractRing.
 
 AbstractRing (const AbstractRing &source)
 Copy construct an AbstractRing. More...
 
AbstractRingoperator= (const AbstractRing &source)
 Assign an AbstractRing. 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 EuclideanDomainOf< T >

Euclidean domain.

Template Parameters
Telement 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 Function Documentation

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

Compare two elements for equality.

Parameters
afirst element
bsecond 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.

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.

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

Adds elements in the group.

Parameters
afirst element
bsecond element
Returns
the sum of a and b

Implements AbstractGroup< T >.

Definition at line 328 of file algebra.h.

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

TODO.

Parameters
afirst element
bsecond element
Returns
TODO

Reimplemented from AbstractGroup< T >.

Definition at line 331 of file algebra.h.

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

Inverts the element in the group.

Parameters
afirst element
Returns
the inverse of the element

Implements AbstractGroup< T >.

Definition at line 334 of file algebra.h.

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

Subtracts elements in the group.

Parameters
afirst element
bsecond 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.

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

Reduces an element in the congruence class.

Parameters
aelement to reduce
bthe congruence class
Returns
the reduced element

Reimplemented from AbstractGroup< T >.

Definition at line 340 of file algebra.h.

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

Doubles an element in the group.

Parameters
athe element
Returns
the element doubled

Reimplemented from AbstractGroup< T >.

Definition at line 343 of file algebra.h.

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.

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

Multiplies elements in the group.

Parameters
athe multiplicand
bthe multiplier
Returns
the product of a and b

Implements AbstractRing< T >.

Definition at line 349 of file algebra.h.

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

Square an element in the group.

Parameters
athe element
Returns
the element squared

Reimplemented from AbstractRing< T >.

Definition at line 352 of file algebra.h.

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

Determines whether an element is a unit in the group.

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

Implements AbstractRing< T >.

Definition at line 355 of file algebra.h.

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

Calculate the multiplicative inverse of an element in the group.

Parameters
athe element

Implements AbstractRing< T >.

Definition at line 358 of file algebra.h.

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

Divides elements in the group.

Parameters
athe dividend
bthe divisor
Returns
the quotient

Reimplemented from AbstractRing< T >.

Definition at line 361 of file algebra.h.

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

Performs a modular reduction in the ring.

Parameters
athe element
bthe modulus
Returns
the result of ab.

Implements AbstractEuclideanDomain< T >.

Definition at line 364 of file algebra.h.

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
rthe remainder
qthe quotient
athe dividend
dthe divisor

Implements AbstractEuclideanDomain< T >.

Definition at line 367 of file algebra.h.


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