finite modulo-N arithmetic with polynomials of degree < M. More...

#include <vnl_finite.h>

Public Member Functions
	vnl_finite_int_poly (std::vector< Scalar > const &p)
	Creates a general finite_int_poly. More...

	vnl_finite_int_poly (Scalar const &n)
	Creates a degree 0 finite_int_poly. More...

	vnl_finite_int_poly ()
	Default constructor. Creates a degree 0 finite_int_poly equal to 0. More...

	vnl_finite_int_poly (Base const &x)

	~vnl_finite_int_poly ()=default

std::size_t	deg () const
	Formal degree of this polynomial. More...

int	degree () const
	Effective degree of this polynomial; equals -1 when this polynomial is 0. More...

Scalar	operator[] (unsigned int i) const
	Access to individual coefficients. More...

Base &	operator= (Base const &x)
	Assignment. More...

Base &	operator= (Scalar const &n)

bool	operator== (Base const &x) const
	Comparison of finite int polys. More...

bool	operator!= (Base const &x) const

bool	operator== (Scalar const &x) const

bool	operator!= (Scalar const &x) const

Base	operator- () const
	Unary minus - returns the additive inverse. More...

Base	operator+ () const
	Unary plus - returns the current polynomial. More...

bool	operator! () const
	Unary not - returns true if finite int poly is equal to zero. More...

Base &	operator+= (Base const &r)
	Plus&assign: replace lhs by lhs + rhs. More...

Base &	operator-= (Base const &r)
	Minus&assign: replace lhs by lhs - rhs. More...

Base &	operator *= (Scalar const &n)
	Scalar multiple of this. More...

unsigned int	additive_order () const
	The additive order of x is the smallest positive r such that r*x == 0. More...

Base &	operator *= (Base const &r)
	Multiply&assign: replace lhs by lhs * rhs, modulo the "modulo" polynomial. More...

unsigned int	multiplicative_order () const
	Return the multiplicative order of this polynomial. More...

Static Public Member Functions
static unsigned int	cardinality ()
	The number of different finite_int polynomials of this type. More...

static std::vector< Scalar > &	modulo_polynomial (std::vector< Scalar > p=std::vector< Scalar >())
	get/set the (irreducible) modulo polynomial of degree M. More...

static bool	is_field ()
	Return true when this ring is a field. More...

static Base	smallest_generator ()
	Return the smallest multiplicative generator of the units in this ring. More...

Private Types
typedef vnl_finite_int_poly< N, M >	Base

typedef vnl_finite_int< N >	Scalar

Private Member Functions
void	add_modulo_poly (unsigned int m, Scalar const &x)
	Add x to the i-th degree coefficient of val_, possibly reducing modulo the "modulo" poly. More...

Static Private Member Functions
static unsigned int	Ntothe (unsigned int m)

Private Attributes
std::vector< Scalar >	val_
	M-tuple (or degree M-1 polynomial) representing this. More...

Related Functions
(Note that these are not member functions.)
template<int N, int M>
vnl_finite_int_poly< N, M >	operator+ (vnl_finite_int_poly< N, M > const &r1, vnl_finite_int_poly< N, M > const &r2)
	Returns the sum of two finite int polynomials. More...

template<int N, int M>
vnl_finite_int_poly< N, M >	operator- (vnl_finite_int_poly< N, M > const &r1, vnl_finite_int_poly< N, M > const &r2)
	Returns the difference of two finite int polynomials. More...

template<int N, int M>
vnl_finite_int_poly< N, M >	operator * (vnl_finite_int_poly< N, M > const &r1, vnl_finite_int< N > const &r2)
	Returns a scalar multiple of a finite int polynomial. More...

template<int N, int M>
vnl_finite_int_poly< N, M >	operator * (vnl_finite_int< N > const &r2, vnl_finite_int_poly< N, M > const &r1)
	Returns a scalar multiple of a finite int polynomial. More...

template<int N, int M>
vnl_finite_int_poly< N, M >	operator * (vnl_finite_int_poly< N, M > const &r1, vnl_finite_int_poly< N, M > const &r2)
	Multiplies two finite int polynomials. More...

template<int N, int M>
std::ostream &	operator<< (std::ostream &s, vnl_finite_int_poly< N, M > const &r)
	formatted output. More...

Detailed Description

template<int N, int M>
class vnl_finite_int_poly< N, M >

finite modulo-N arithmetic with polynomials of degree < M.

This class implements arithmetic with polynomials of degree < M over vnl_finite_int<N>. Multiplication is defined modulo a polynomial of degree M.

For N prime, and when the "modulo" polynomial is irreducible, vnl_finite_int_poly<N,M> implements the finite field GF(N^M).

Addition, subtraction and scalar multiplication are already defined without the presence of a "modulo" polynomial. Restricted to these operations, vnl_finite_int_poly<N,M> forms an M-dimensional vector space over vnl_finite_int<N>. The current implementation does not yet implement anything more than that.

Definition at line 431 of file vnl_finite.h.

Member Typedef Documentation

◆ Base

template<int N, int M>

typedef vnl_finite_int_poly<N,M> vnl_finite_int_poly< N, M >::Base

private

Definition at line 433 of file vnl_finite.h.

◆ Scalar

template<int N, int M>

typedef vnl_finite_int<N> vnl_finite_int_poly< N, M >::Scalar

private

Definition at line 434 of file vnl_finite.h.

Constructor & Destructor Documentation

◆ vnl_finite_int_poly() [1/4]

template<int N, int M>

vnl_finite_int_poly< N, M >::vnl_finite_int_poly ( std::vector< Scalar > const & p )

inline

Creates a general finite_int_poly.

Definition at line 445 of file vnl_finite.h.

◆ vnl_finite_int_poly() [2/4]

template<int N, int M>

vnl_finite_int_poly< N, M >::vnl_finite_int_poly ( Scalar const & n )

inline

Creates a degree 0 finite_int_poly.

Definition at line 447 of file vnl_finite.h.

◆ vnl_finite_int_poly() [3/4]

template<int N, int M>

vnl_finite_int_poly< N, M >::vnl_finite_int_poly ( )

inline

Default constructor. Creates a degree 0 finite_int_poly equal to 0.

Definition at line 449 of file vnl_finite.h.

◆ vnl_finite_int_poly() [4/4]

template<int N, int M>

vnl_finite_int_poly< N, M >::vnl_finite_int_poly ( Base const & x )

inline

Definition at line 451 of file vnl_finite.h.

◆ ~vnl_finite_int_poly()

template<int N, int M>

vnl_finite_int_poly< N, M >::~vnl_finite_int_poly ( )

inlinedefault

Member Function Documentation

◆ add_modulo_poly()

template<int N, int M>

void vnl_finite_int_poly< N, M >::add_modulo_poly	(	unsigned int	m,
		Scalar const &	x
	)

inlineprivate

Add x to the i-th degree coefficient of val_, possibly reducing modulo the "modulo" poly.

Definition at line 578 of file vnl_finite.h.

◆ additive_order()

template<int N, int M>

unsigned int vnl_finite_int_poly< N, M >::additive_order ( ) const

inline

The additive order of x is the smallest positive r such that r*x == 0.

Definition at line 510 of file vnl_finite.h.

◆ cardinality()

template<int N, int M>

static unsigned int vnl_finite_int_poly< N, M >::cardinality ( )

inlinestatic

The number of different finite_int polynomials of this type.

Definition at line 442 of file vnl_finite.h.

◆ deg()

template<int N, int M>

std::size_t vnl_finite_int_poly< N, M >::deg ( ) const

inline

Formal degree of this polynomial.

Definition at line 456 of file vnl_finite.h.

◆ degree()

template<int N, int M>

int vnl_finite_int_poly< N, M >::degree ( ) const

inline

Effective degree of this polynomial; equals -1 when this polynomial is 0.

Definition at line 459 of file vnl_finite.h.

◆ is_field()

template<int N, int M>

static bool vnl_finite_int_poly< N, M >::is_field ( )

inlinestatic

Return true when this ring is a field.

Note that this requires that N is a prime, but that condition is not sufficient: also the "modulo" polynomial must be irreducible.

Definition at line 557 of file vnl_finite.h.

◆ modulo_polynomial()

template<int N, int M>

static std::vector<Scalar>& vnl_finite_int_poly< N, M >::modulo_polynomial ( std::vector< Scalar > p = std::vector<Scalar>() )

inlinestatic

get/set the (irreducible) modulo polynomial of degree M.

Note that this polynomial has to be set only once, i.e., once set, it applies to all vnl_finite_int_polys with the same N and M.

Definition at line 520 of file vnl_finite.h.

◆ multiplicative_order()

template<int N, int M>

unsigned int vnl_finite_int_poly< N, M >::multiplicative_order ( ) const

inline

Return the multiplicative order of this polynomial.

Definition at line 547 of file vnl_finite.h.

◆ Ntothe()

template<int N, int M>

static unsigned int vnl_finite_int_poly< N, M >::Ntothe ( unsigned int m )

inlinestaticprivate

Definition at line 439 of file vnl_finite.h.

◆ operator *=() [1/2]

template<int N, int M>

Base& vnl_finite_int_poly< N, M >::operator *= ( Scalar const & n )

inline

Scalar multiple of this.

Definition at line 507 of file vnl_finite.h.

◆ operator *=() [2/2]

template<int N, int M>

Base& vnl_finite_int_poly< N, M >::operator *= ( Base const & r )

inline

Multiply&assign: replace lhs by lhs * rhs, modulo the "modulo" polynomial.

Definition at line 537 of file vnl_finite.h.

◆ operator!()

template<int N, int M>

bool vnl_finite_int_poly< N, M >::operator! ( ) const

inline

Unary not - returns true if finite int poly is equal to zero.

Definition at line 489 of file vnl_finite.h.

◆ operator!=() [1/2]

template<int N, int M>

bool vnl_finite_int_poly< N, M >::operator!= ( Base const & x ) const

inline

Definition at line 476 of file vnl_finite.h.

◆ operator!=() [2/2]

template<int N, int M>

bool vnl_finite_int_poly< N, M >::operator!= ( Scalar const & x ) const

inline

Definition at line 482 of file vnl_finite.h.

◆ operator+()

template<int N, int M>

Base vnl_finite_int_poly< N, M >::operator+ ( ) const

inline

Unary plus - returns the current polynomial.

Definition at line 487 of file vnl_finite.h.

◆ operator+=()

template<int N, int M>

Base& vnl_finite_int_poly< N, M >::operator+= ( Base const & r )

inline

Plus&assign: replace lhs by lhs + rhs.

Definition at line 492 of file vnl_finite.h.

◆ operator-()

template<int N, int M>

Base vnl_finite_int_poly< N, M >::operator- ( ) const

inline

Unary minus - returns the additive inverse.

Definition at line 485 of file vnl_finite.h.

◆ operator-=()

template<int N, int M>

Base& vnl_finite_int_poly< N, M >::operator-= ( Base const & r )

inline

Minus&assign: replace lhs by lhs - rhs.

Definition at line 499 of file vnl_finite.h.

◆ operator=() [1/2]

template<int N, int M>

Base& vnl_finite_int_poly< N, M >::operator= ( Base const & x )

inline

Assignment.

Definition at line 465 of file vnl_finite.h.

◆ operator=() [2/2]

template<int N, int M>

Base& vnl_finite_int_poly< N, M >::operator= ( Scalar const & n )

inline

Definition at line 466 of file vnl_finite.h.

◆ operator==() [1/2]

template<int N, int M>

bool vnl_finite_int_poly< N, M >::operator== ( Base const & x ) const

inline

Comparison of finite int polys.

Definition at line 469 of file vnl_finite.h.

◆ operator==() [2/2]

template<int N, int M>

bool vnl_finite_int_poly< N, M >::operator== ( Scalar const & x ) const

inline

Definition at line 477 of file vnl_finite.h.

◆ operator[]()

template<int N, int M>

Scalar vnl_finite_int_poly< N, M >::operator[] ( unsigned int i ) const

inline

Access to individual coefficients.

Definition at line 462 of file vnl_finite.h.

◆ smallest_generator()

template<int N, int M>

static Base vnl_finite_int_poly< N, M >::smallest_generator ( )

inlinestatic

Return the smallest multiplicative generator of the units in this ring.

This is only possible if the units form a cyclic group for multiplication. If not, smallest_generator() returns 1 to indicate this fact. Note that the multiplication group of a finite field is always cyclic.

Definition at line 569 of file vnl_finite.h.

Friends And Related Function Documentation

◆ operator *() [1/3]

template<int N, int M>

vnl_finite_int_poly< N, M > operator *	(	vnl_finite_int_poly< N, M > const &	r1,
		vnl_finite_int< N > const &	r2
	)

Definition at line 608 of file vnl_finite.h.

◆ operator *() [2/3]

template<int N, int M>

vnl_finite_int_poly< N, M > operator *	(	vnl_finite_int< N > const &	r2,
		vnl_finite_int_poly< N, M > const &	r1
	)

Definition at line 617 of file vnl_finite.h.

◆ operator *() [3/3]

template<int N, int M>

vnl_finite_int_poly< N, M > operator *	(	vnl_finite_int_poly< N, M > const &	r1,
		vnl_finite_int_poly< N, M > const &	r2
	)

NOTE: this requires the "modulo" polynomial to be set. Do this by calling modulo_polynomial(p), where p is a vector of length M+1.

Definition at line 627 of file vnl_finite.h.

◆ operator+()

template<int N, int M>

vnl_finite_int_poly< N, M > operator+	(	vnl_finite_int_poly< N, M > const &	r1,
		vnl_finite_int_poly< N, M > const &	r2
	)

Definition at line 591 of file vnl_finite.h.

◆ operator-()

template<int N, int M>

vnl_finite_int_poly< N, M > operator-	(	vnl_finite_int_poly< N, M > const &	r1,
		vnl_finite_int_poly< N, M > const &	r2
	)

Definition at line 599 of file vnl_finite.h.

◆ operator<<()

template<int N, int M>

std::ostream & operator<<	(	std::ostream &	s,
		vnl_finite_int_poly< N, M > const &	r
	)

Definition at line 635 of file vnl_finite.h.

Member Data Documentation

◆ val_

template<int N, int M>

std::vector<Scalar> vnl_finite_int_poly< N, M >::val_

private

M-tuple (or degree M-1 polynomial) representing this.

Definition at line 436 of file vnl_finite.h.

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

core/vnl/vnl_finite.h

Public Member Functions

Static Public Member Functions

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Related Functions

Detailed Description

template<int N, int M> class vnl_finite_int_poly< N, M >

Member Typedef Documentation

◆ Base

◆ Scalar

Constructor & Destructor Documentation

◆ vnl_finite_int_poly() [1/4]

◆ vnl_finite_int_poly() [2/4]

◆ vnl_finite_int_poly() [3/4]

◆ vnl_finite_int_poly() [4/4]

◆ ~vnl_finite_int_poly()

Member Function Documentation

◆ add_modulo_poly()

◆ additive_order()

◆ cardinality()

◆ deg()

◆ degree()

◆ is_field()

◆ modulo_polynomial()

◆ multiplicative_order()

◆ Ntothe()

◆ operator *=() [1/2]

◆ operator *=() [2/2]

◆ operator!()

◆ operator!=() [1/2]

◆ operator!=() [2/2]

◆ operator+()

◆ operator+=()

◆ operator-()

◆ operator-=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==() [1/2]

◆ operator==() [2/2]

◆ operator[]()

◆ smallest_generator()

Friends And Related Function Documentation

◆ operator *() [1/3]

◆ operator *() [2/3]

◆ operator *() [3/3]

◆ operator+()

◆ operator-()

◆ operator<<()

Member Data Documentation

◆ val_

template<int N, int M>
class vnl_finite_int_poly< N, M >