finite modulo-N arithmetic. More...

#include <vnl_finite.h>

Public Member Functions
	vnl_finite_int (int x=0)
	Creates a finite int element. More...

	vnl_finite_int (Base const &x)

	~vnl_finite_int ()=default

	operator int () const

	operator int ()

	operator long () const

	operator long ()

	operator short () const

	operator short ()

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

Base &	operator= (int x)

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

bool	operator!= (Base const &x) const

bool	operator== (int x) const

bool	operator!= (int x) const

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

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

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

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

Base &	operator+= (int r)

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

Base &	operator-= (int r)

Base &	operator *= (int r)
	Multiply&assign: replace lhs by lhs * rhs. More...

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

Base	reciproc () const
	Multiplicative inverse. More...

Base &	operator/= (Base const &r)
	Divide&assign. Uses r.reciproc() for efficient computation. More...

Base &	operator++ ()
	Pre-increment (++r). More...

Base &	operator-- ()
	Pre-decrement (–r). More...

Base	operator++ (int)
	Post-increment (r++). More...

Base	operator-- (int)
	Post-decrement (r–). More...

bool	is_unit () const
	Return true only when x is a unit in this ring. More...

bool	is_zero_divisor () const
	Return true only when x is a zero divisor, i.e., is not a unit. More...

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

unsigned int	multiplicative_order () const
	The multiplicative order of x is the smallest r (>0) such that x^r == 1. More...

Base	pow (int r)
	Return the r-th power of this number. More...

unsigned int	log () const
	Return the smallest nonnegative exponent r for which x=g^r, where g is the smallest generator. More...

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

static unsigned int	totient ()
	Return the Euler totient function, i.e., the number of units of this ring. More...

static std::vector< unsigned int >	decompose ()
	Write N as the unique product of prime factors. More...

static bool	is_field ()
	Return true when N is a prime number, i.e., when this ring is a field. More...

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

static Base	exp (int r)
	Return the inverse of log(), i.e., return g^r where g is the smallest generator. More...

static unsigned int	gcd (unsigned int l, unsigned int n)
	Calculate the greatest common divisor of l and N. More...

static unsigned int	gcd (unsigned int l)

Private Types
typedef vnl_finite_int< N >	Base

Private Member Functions
void	set_log (unsigned int r) const
	private function to set cached value of lp1_ when available. More...

Private Attributes
int	val_
	value of this number (smallest nonnegative representation) More...

unsigned int	mo_
	cached value for multiplicative order More...

unsigned int	lp1_
	cached value for 1+log() More...

Related Functions
(Note that these are not member functions.)
template<int N>
std::ostream &	operator<< (std::ostream &s, vnl_finite_int< N > const &r)
	formatted output. More...

template<int N>
std::istream &	operator>> (std::istream &s, vnl_finite_int< N > &r)
	simple input. More...

template<int N>
vnl_finite_int< N >	operator+ (vnl_finite_int< N > const &r1, vnl_finite_int< N > const &r2)
	Returns the sum of two finite int numbers. More...

template<int N>
vnl_finite_int< N >	operator+ (vnl_finite_int< N > const &r1, int r2)
	Returns the sum of two finite int numbers. More...

template<int N>
vnl_finite_int< N >	operator+ (int r2, vnl_finite_int< N > const &r1)
	Returns the sum of two finite int numbers. More...

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

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

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

template<int N>
vnl_finite_int< N >	operator * (vnl_finite_int< N > const &r1, vnl_finite_int< N > const &r2)
	Returns the product of two finite int numbers. More...

template<int N>
vnl_finite_int< N >	operator * (vnl_finite_int< N > const &r1, int r2)
	Returns the product of two finite int numbers. More...

template<int N>
vnl_finite_int< N >	operator * (int r2, vnl_finite_int< N > const &r1)
	Returns the product of two finite int numbers. More...

template<int N>
vnl_finite_int< N >	operator/ (vnl_finite_int< N > const &r1, vnl_finite_int< N > const &r2)
	Returns the quotient of two finite int numbers. More...

template<int N>
vnl_finite_int< N >	operator/ (vnl_finite_int< N > const &r1, int r2)
	Returns the quotient of two finite int numbers. More...

template<int N>
vnl_finite_int< N >	operator/ (int r1, vnl_finite_int< N > const &r2)
	Returns the quotient of two finite int numbers. More...

template<int N>
bool	operator== (int r1, vnl_finite_int< N > const &r2)

template<int N>
bool	operator!= (int r1, vnl_finite_int< N > const &r2)

template<int N>
vnl_finite_int< N >	squared_magnitude (vnl_finite_int< N > const &x)

template<int N>
vnl_finite_int< N >	sqr (vnl_finite_int< N > const &x)

template<int N>
bool	isnan (vnl_finite_int< N > const &)

template<int N>
bool	isfinite (vnl_finite_int< N > const &)

Detailed Description

template<int N>
class vnl_finite_int< N >

finite modulo-N arithmetic.

The templated vnl_finite_int<N> provides arithmetic "modulo N", i.e., arithmetic in the finite (Galois) field GF(N) in case N is a prime or just in the finite ring (or semi-field) of integers modulo N otherwise. In that case division makes no sense (unless no zero divisor is involved), but all other operations remain valid.

Definition at line 52 of file vnl_finite.h.

Member Typedef Documentation

◆ Base

template<int N>

typedef vnl_finite_int<N> vnl_finite_int< N >::Base

private

Definition at line 57 of file vnl_finite.h.

Constructor & Destructor Documentation

◆ vnl_finite_int() [1/2]

template<int N>

vnl_finite_int< N >::vnl_finite_int ( int x = 0 )

inline

Creates a finite int element.

Default constructor gives 0. Also serves as automatic cast from int to vnl_finite_int.

Definition at line 65 of file vnl_finite.h.

◆ vnl_finite_int() [2/2]

template<int N>

vnl_finite_int< N >::vnl_finite_int ( Base const & x )

inline

Definition at line 67 of file vnl_finite.h.

◆ ~vnl_finite_int()

template<int N>

vnl_finite_int< N >::~vnl_finite_int ( )

inlinedefault

Member Function Documentation

◆ additive_order()

template<int N>

unsigned int vnl_finite_int< N >::additive_order ( ) const

inline

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

Definition at line 191 of file vnl_finite.h.

◆ cardinality()

template<int N>

static unsigned int vnl_finite_int< N >::cardinality ( )

inlinestatic

The number of different finite_int numbers of this type.

Definition at line 60 of file vnl_finite.h.

◆ decompose()

template<int N>

static std::vector<unsigned int> vnl_finite_int< N >::decompose ( )

inlinestatic

Write N as the unique product of prime factors.

Definition at line 166 of file vnl_finite.h.

◆ exp()

template<int N>

static Base vnl_finite_int< N >::exp ( int r )

inlinestatic

Return the inverse of log(), i.e., return g^r where g is the smallest generator.

Definition at line 247 of file vnl_finite.h.

◆ gcd() [1/2]

template<int N>

static unsigned int vnl_finite_int< N >::gcd	(	unsigned int	l,
		unsigned int	n
	)

inlinestatic

Calculate the greatest common divisor of l and N.

Definition at line 254 of file vnl_finite.h.

◆ gcd() [2/2]

template<int N>

static unsigned int vnl_finite_int< N >::gcd ( unsigned int l )

inlinestatic

Definition at line 259 of file vnl_finite.h.

◆ is_field()

template<int N>

static bool vnl_finite_int< N >::is_field ( )

inlinestatic

Return true when N is a prime number, i.e., when this ring is a field.

Definition at line 177 of file vnl_finite.h.

◆ is_unit()

template<int N>

bool vnl_finite_int< N >::is_unit ( ) const

inline

Return true only when x is a unit in this ring.

In a field, all numbers except 0 are units. The total number of units is given by the Euler totient function.

Definition at line 185 of file vnl_finite.h.

◆ is_zero_divisor()

template<int N>

bool vnl_finite_int< N >::is_zero_divisor ( ) const

inline

Return true only when x is a zero divisor, i.e., is not a unit.

Definition at line 188 of file vnl_finite.h.

◆ log()

template<int N>

unsigned int vnl_finite_int< N >::log ( ) const

inline

Return the smallest nonnegative exponent r for which x=g^r, where g is the smallest generator.

Definition at line 233 of file vnl_finite.h.

◆ multiplicative_order()

template<int N>

unsigned int vnl_finite_int< N >::multiplicative_order ( ) const

inline

The multiplicative order of x is the smallest r (>0) such that x^r == 1.

Definition at line 194 of file vnl_finite.h.

◆ operator *=() [1/2]

template<int N>

Base& vnl_finite_int< N >::operator *= ( int r )

inline

Multiply&assign: replace lhs by lhs * rhs.

Definition at line 103 of file vnl_finite.h.

◆ operator *=() [2/2]

template<int N>

Base& vnl_finite_int< N >::operator *= ( Base const & r )

inline

Multiply&assign: replace lhs by lhs * rhs.

Definition at line 110 of file vnl_finite.h.

◆ operator int() [1/2]

template<int N>

vnl_finite_int< N >::operator int ( ) const

inline

Definition at line 71 of file vnl_finite.h.

◆ operator int() [2/2]

template<int N>

vnl_finite_int< N >::operator int ( )

inline

Definition at line 72 of file vnl_finite.h.

◆ operator long() [1/2]

template<int N>

vnl_finite_int< N >::operator long ( ) const

inline

Definition at line 73 of file vnl_finite.h.

◆ operator long() [2/2]

template<int N>

vnl_finite_int< N >::operator long ( )

inline

Definition at line 74 of file vnl_finite.h.

◆ operator short() [1/2]

template<int N>

vnl_finite_int< N >::operator short ( ) const

inline

Definition at line 75 of file vnl_finite.h.

◆ operator short() [2/2]

template<int N>

vnl_finite_int< N >::operator short ( )

inline

Definition at line 76 of file vnl_finite.h.

◆ operator!()

template<int N>

bool vnl_finite_int< N >::operator! ( ) const

inline

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

Definition at line 94 of file vnl_finite.h.

◆ operator!=() [1/2]

template<int N>

bool vnl_finite_int< N >::operator!= ( Base const & x ) const

inline

Definition at line 85 of file vnl_finite.h.

◆ operator!=() [2/2]

template<int N>

bool vnl_finite_int< N >::operator!= ( int x ) const

inline

Definition at line 87 of file vnl_finite.h.

◆ operator+()

template<int N>

Base vnl_finite_int< N >::operator+ ( ) const

inline

Unary plus - returns the current number.

Definition at line 92 of file vnl_finite.h.

◆ operator++() [1/2]

template<int N>

Base& vnl_finite_int< N >::operator++ ( )

inline

Pre-increment (++r).

Definition at line 157 of file vnl_finite.h.

◆ operator++() [2/2]

template<int N>

Base vnl_finite_int< N >::operator++ ( int )

inline

Post-increment (r++).

Definition at line 161 of file vnl_finite.h.

◆ operator+=() [1/2]

template<int N>

Base& vnl_finite_int< N >::operator+= ( Base const & r )

inline

Plus&assign: replace lhs by lhs + rhs.

Definition at line 97 of file vnl_finite.h.

◆ operator+=() [2/2]

template<int N>

Base& vnl_finite_int< N >::operator+= ( int r )

inline

Definition at line 98 of file vnl_finite.h.

◆ operator-()

template<int N>

Base vnl_finite_int< N >::operator- ( ) const

inline

Unary minus - returns the additive inverse.

Definition at line 90 of file vnl_finite.h.

◆ operator--() [1/2]

template<int N>

Base& vnl_finite_int< N >::operator-- ( )

inline

Pre-decrement (–r).

Definition at line 159 of file vnl_finite.h.

◆ operator--() [2/2]

template<int N>

Base vnl_finite_int< N >::operator-- ( int )

inline

Post-decrement (r–).

Definition at line 163 of file vnl_finite.h.

◆ operator-=() [1/2]

template<int N>

Base& vnl_finite_int< N >::operator-= ( Base const & r )

inline

Minus&assign: replace lhs by lhs - rhs.

Definition at line 100 of file vnl_finite.h.

◆ operator-=() [2/2]

template<int N>

Base& vnl_finite_int< N >::operator-= ( int r )

inline

Definition at line 101 of file vnl_finite.h.

◆ operator/=()

template<int N>

Base& vnl_finite_int< N >::operator/= ( Base const & r )

inline

Divide&assign. Uses r.reciproc() for efficient computation.

Definition at line 151 of file vnl_finite.h.

◆ operator=() [1/2]

template<int N>

Base& vnl_finite_int< N >::operator= ( Base const & x )

inline

Assignment.

Definition at line 79 of file vnl_finite.h.

◆ operator=() [2/2]

template<int N>

Base& vnl_finite_int< N >::operator= ( int x )

inline

Definition at line 80 of file vnl_finite.h.

◆ operator==() [1/2]

template<int N>

bool vnl_finite_int< N >::operator== ( Base const & x ) const

inline

Comparison of finite int numbers.

Note that finite ints have no order, so < and > make no sense.

Definition at line 84 of file vnl_finite.h.

◆ operator==() [2/2]

template<int N>

bool vnl_finite_int< N >::operator== ( int x ) const

inline

Definition at line 86 of file vnl_finite.h.

◆ pow()

template<int N>

Base vnl_finite_int< N >::pow ( int r )

inline

Return the r-th power of this number.

Definition at line 224 of file vnl_finite.h.

◆ reciproc()

template<int N>

Base vnl_finite_int< N >::reciproc ( ) const

inline

Multiplicative inverse.

Uses exp() and log() for efficient computation, unless log() is not defined.

Definition at line 138 of file vnl_finite.h.

◆ set_log()

template<int N>

void vnl_finite_int< N >::set_log ( unsigned int r ) const

inlineprivate

private function to set cached value of lp1_ when available.

Definition at line 263 of file vnl_finite.h.

◆ smallest_generator()

template<int N>

static Base vnl_finite_int< N >::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 206 of file vnl_finite.h.

◆ totient()

template<int N>

static unsigned int vnl_finite_int< N >::totient ( )

inlinestatic

Return the Euler totient function, i.e., the number of units of this ring.

This number also equals the periodicity of the exponent: every unit, when raised to this power, yields 1.

Definition at line 122 of file vnl_finite.h.

Friends And Related Function Documentation

◆ isfinite()

template<int N>

bool isfinite ( vnl_finite_int< N > const & )

◆ isnan()

template<int N>

bool isnan ( vnl_finite_int< N > const & )

◆ operator *() [1/3]

template<int N>

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

Definition at line 336 of file vnl_finite.h.

◆ operator *() [2/3]

template<int N>

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

Definition at line 344 of file vnl_finite.h.

◆ operator *() [3/3]

template<int N>

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

Definition at line 352 of file vnl_finite.h.

◆ operator!=()

template<int N>

bool operator!=	(	int	r1,
		vnl_finite_int< N > const &	r2
	)

◆ operator+() [1/3]

template<int N>

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

Definition at line 288 of file vnl_finite.h.

◆ operator+() [2/3]

template<int N>

vnl_finite_int< N > operator+	(	vnl_finite_int< N > const &	r1,
		int	r2
	)

Definition at line 296 of file vnl_finite.h.

◆ operator+() [3/3]

template<int N>

vnl_finite_int< N > operator+	(	int	r2,
		vnl_finite_int< N > const &	r1
	)

Definition at line 304 of file vnl_finite.h.

◆ operator-() [1/3]

template<int N>

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

Definition at line 312 of file vnl_finite.h.

◆ operator-() [2/3]

template<int N>

vnl_finite_int< N > operator-	(	vnl_finite_int< N > const &	r1,
		int	r2
	)

Definition at line 320 of file vnl_finite.h.

◆ operator-() [3/3]

template<int N>

vnl_finite_int< N > operator-	(	int	r2,
		vnl_finite_int< N > const &	r1
	)

Definition at line 328 of file vnl_finite.h.

◆ operator/() [1/3]

template<int N>

vnl_finite_int< N > operator/	(	vnl_finite_int< N > const &	r1,
		vnl_finite_int< N > const &	r2
	)

Uses r2.reciproc() for efficient computation.

Definition at line 361 of file vnl_finite.h.

◆ operator/() [2/3]

template<int N>

vnl_finite_int< N > operator/	(	vnl_finite_int< N > const &	r1,
		int	r2
	)

Definition at line 369 of file vnl_finite.h.

◆ operator/() [3/3]

template<int N>

vnl_finite_int< N > operator/	(	int	r1,
		vnl_finite_int< N > const &	r2
	)

Definition at line 377 of file vnl_finite.h.

◆ operator<<()

template<int N>

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

Definition at line 272 of file vnl_finite.h.

◆ operator==()

template<int N>

bool operator==	(	int	r1,
		vnl_finite_int< N > const &	r2
	)

◆ operator>>()

template<int N>

std::istream & operator>>	(	std::istream &	s,
		vnl_finite_int< N > &	r
	)

Definition at line 280 of file vnl_finite.h.

◆ sqr()

template<int N>

vnl_finite_int< N > sqr ( vnl_finite_int< N > const & x )

◆ squared_magnitude()

template<int N>

vnl_finite_int< N > squared_magnitude ( vnl_finite_int< N > const & x )

Member Data Documentation

◆ lp1_

template<int N>

unsigned int vnl_finite_int< N >::lp1_

mutableprivate

cached value for 1+log()

Definition at line 266 of file vnl_finite.h.

◆ mo_

template<int N>

unsigned int vnl_finite_int< N >::mo_

mutableprivate

cached value for multiplicative order

Definition at line 265 of file vnl_finite.h.

◆ val_

template<int N>

int vnl_finite_int< N >::val_

private

value of this number (smallest nonnegative representation)

Definition at line 55 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

Private Attributes

Related Functions

Detailed Description

template<int N> class vnl_finite_int< N >

Member Typedef Documentation

◆ Base

Constructor & Destructor Documentation

◆ vnl_finite_int() [1/2]

◆ vnl_finite_int() [2/2]

◆ ~vnl_finite_int()

Member Function Documentation

◆ additive_order()

◆ cardinality()

◆ decompose()

◆ exp()

◆ gcd() [1/2]

◆ gcd() [2/2]

◆ is_field()

◆ is_unit()

◆ is_zero_divisor()

◆ log()

◆ multiplicative_order()

◆ operator *=() [1/2]

◆ operator *=() [2/2]

◆ operator int() [1/2]

◆ operator int() [2/2]

◆ operator long() [1/2]

◆ operator long() [2/2]

◆ operator short() [1/2]

◆ operator short() [2/2]

◆ operator!()

◆ operator!=() [1/2]

◆ operator!=() [2/2]

◆ operator+()

◆ operator++() [1/2]

◆ operator++() [2/2]

◆ operator+=() [1/2]

◆ operator+=() [2/2]

◆ operator-()

◆ operator--() [1/2]

◆ operator--() [2/2]

◆ operator-=() [1/2]

◆ operator-=() [2/2]

◆ operator/=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==() [1/2]

◆ operator==() [2/2]

◆ pow()

◆ reciproc()

◆ set_log()

◆ smallest_generator()

◆ totient()

Friends And Related Function Documentation

◆ isfinite()

◆ isnan()

◆ operator *() [1/3]

◆ operator *() [2/3]

◆ operator *() [3/3]

◆ operator!=()

◆ operator+() [1/3]

◆ operator+() [2/3]

◆ operator+() [3/3]

◆ operator-() [1/3]

◆ operator-() [2/3]

◆ operator-() [3/3]

◆ operator/() [1/3]

◆ operator/() [2/3]

◆ operator/() [3/3]

◆ operator<<()

◆ operator==()

◆ operator>>()

◆ sqr()

◆ squared_magnitude()

Member Data Documentation

◆ lp1_

template<int N>
class vnl_finite_int< N >