nzmath.finitefield.FiniteField Class Reference

Inheritance diagram for nzmath.finitefield.FiniteField:

Collaboration diagram for nzmath.finitefield.FiniteField:

Public Member Functions
def	__init__ (self, characteristic)
def	card (self)
def	getCharacteristic (self)
def	order (self, elem)
def	random_element (self, *args)
def	primitive_element (self)
def	Legendre (self, element)
def	TonelliShanks (self, element)
def	sqrt (self, element)
Public Member Functions inherited from nzmath.ring.Field
def	__init__ (self)
def	createElement (self, *args)
def	isfield (self)
def	gcd (self, a, b)
def	getQuotientField (self)
Public Member Functions inherited from nzmath.ring.CommutativeRing
def	isdomain (self)
def	isnoetherian (self)
def	isufd (self)
def	ispid (self)
def	iseuclidean (self)
def	registerModuleAction (self, action_ring, action)
def	hasaction (self, action_ring)
def	getaction (self, action_ring)
Public Member Functions inherited from nzmath.ring.Ring
def	createElement (self, seed)
def	issubring (self, other)
def	issuperring (self, other)
def	getCommonSuperring (self, other)
def	__eq__ (self, other)
def	__hash__ (self)
def	__ne__ (self, other)

Public Attributes
	char
Public Attributes inherited from nzmath.ring.CommutativeRing
	properties

Private Attributes
	_orderfactor

Detailed Description

The base class for all finite fields.

Definition at line 35 of file finitefield.py.

Constructor & Destructor Documentation

__init__()

def nzmath.finitefield.FiniteField.__init__	(		self,
			characteristic
	)

Reimplemented in nzmath.finitefield.FinitePrimeField.

Definition at line 39 of file finitefield.py.

Member Function Documentation

card()

def nzmath.finitefield.FiniteField.card

(

self

)

Reimplemented in nzmath.finitefield.ExtendedField, and nzmath.finitefield.FinitePrimeField.

Definition at line 47 of file finitefield.py.

Referenced by nzmath.finitefield.FiniteField.Legendre(), nzmath.finitefield.FiniteField.order(), nzmath.finitefield.FiniteField.sqrt(), and nzmath.finitefield.FiniteField.TonelliShanks().

getCharacteristic()

def nzmath.finitefield.FiniteField.getCharacteristic

(

self

)

Return the characteristic of the field.

Reimplemented from nzmath.ring.Ring.

Definition at line 51 of file finitefield.py.

References nzmath.finitefield.FiniteField.char.

Referenced by nzmath.finitefield.ExtendedField.__contains__().

Legendre()

def nzmath.finitefield.FiniteField.Legendre	(		self,
			element
	)

Return generalize Legendre Symbol for FiniteField.

Reimplemented in nzmath.finitefield.FinitePrimeField.

Definition at line 87 of file finitefield.py.

References nzmath.finitefield.FiniteField.card(), nzmath.finitefield.FiniteField.char, nzmath.real.RealField.one, nzmath.poly.ring.PolynomialRing.one, nzmath.imaginary.ComplexField.one, nzmath.intresidue.IntegerResidueClassRing.one, nzmath.algfield.NumberField.one, nzmath.finitefield.FinitePrimeField.one, nzmath.poly.multiutil.PolynomialRingAnonymousVariables.one, nzmath.ring.ResidueClassRing.one, nzmath.poly.ring.RationalFunctionField.one, nzmath.rational.RationalField.one, nzmath.finitefield.ExtendedField.one, nzmath.rational.IntegerRing.one, and nzmath.matrix.MatrixRing.one.

Referenced by nzmath.finitefield.FiniteField.TonelliShanks().

order()

def nzmath.finitefield.FiniteField.order	(		self,
			elem
	)

Find and return the order of the element in the multiplicative
group of the field.

Definition at line 57 of file finitefield.py.

References nzmath.finitefield.FiniteField._orderfactor, nzmath.finitefield.FiniteField.card(), nzmath.real.RealField.one, nzmath.poly.ring.PolynomialRing.one, nzmath.imaginary.ComplexField.one, nzmath.intresidue.IntegerResidueClassRing.one, nzmath.algfield.NumberField.one, nzmath.finitefield.FinitePrimeField.one, nzmath.poly.multiutil.PolynomialRingAnonymousVariables.one, nzmath.ring.ResidueClassRing.one, nzmath.poly.ring.RationalFunctionField.one, nzmath.rational.RationalField.one, nzmath.finitefield.ExtendedField.one, nzmath.rational.IntegerRing.one, and nzmath.matrix.MatrixRing.one.

Referenced by nzmath.poly.uniutil.DivisionProvider.__divmod__(), nzmath.poly.uniutil.DivisionProvider.__floordiv__(), nzmath.poly.uniutil.DivisionProvider.__mod__(), nzmath.poly.uniutil.DivisionProvider._populate_reduced(), nzmath.poly.uniutil.DivisionProvider._populate_reduced_more(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.distinct_degree_factorization(), nzmath.poly.uniutil.PseudoDivisionProvider.exact_division(), nzmath.poly.uniutil.DivisionProvider.extgcd(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.factor(), nzmath.poly.uniutil.DivisionProvider.gcd(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.isirreducible(), nzmath.poly.multiutil.NestProvider.leading_variable(), nzmath.poly.uniutil.DivisionProvider.mod(), nzmath.poly.uniutil.PseudoDivisionProvider.monic_divmod(), nzmath.poly.uniutil.PseudoDivisionProvider.monic_floordiv(), nzmath.poly.uniutil.PseudoDivisionProvider.monic_mod(), nzmath.poly.uniutil.PseudoDivisionProvider.pseudo_divmod(), nzmath.poly.uniutil.PseudoDivisionProvider.pseudo_floordiv(), nzmath.poly.uniutil.PseudoDivisionProvider.pseudo_mod(), nzmath.poly.uniutil.KaratsubaProvider.ring_mul_karatsuba(), nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.split_same_degrees(), nzmath.poly.uniutil.KaratsubaProvider.square_karatsuba(), and nzmath.poly.uniutil.PrimeCharacteristicFunctionsProvider.squarefree_decomposition().

primitive_element()

def nzmath.finitefield.FiniteField.primitive_element

(

self

)

Return a primitive element of the field, i.e., a generator of
the multiplicative group.

Reimplemented in nzmath.finitefield.ExtendedField, and nzmath.finitefield.FinitePrimeField.

Definition at line 80 of file finitefield.py.

random_element()

def nzmath.finitefield.FiniteField.random_element	(		self,
		*	args
	)

Return a randomly chosen element og the field.

Definition at line 74 of file finitefield.py.

References nzmath.lattice.Lattice.createElement(), nzmath.ring.Ring.createElement(), nzmath.group.Group.createElement(), nzmath.poly.ring.PolynomialRing.createElement(), nzmath.real.RealField.createElement(), nzmath.imaginary.ComplexField.createElement(), nzmath.intresidue.IntegerResidueClassRing.createElement(), nzmath.algfield.NumberField.createElement(), nzmath.finitefield.FinitePrimeField.createElement(), nzmath.poly.multiutil.PolynomialRingAnonymousVariables.createElement(), nzmath.rational.RationalField.createElement(), nzmath.permute.PermGroup.createElement(), nzmath.poly.ring.RationalFunctionField.createElement(), nzmath.finitefield.ExtendedField.createElement(), nzmath.rational.IntegerRing.createElement(), and nzmath.matrix.MatrixRing.createElement().

Referenced by nzmath.finitefield.FiniteField.TonelliShanks().

sqrt()

def nzmath.finitefield.FiniteField.sqrt	(		self,
			element
	)

Return square root if exist.

Definition at line 145 of file finitefield.py.

References nzmath.finitefield.FiniteField.card(), nzmath.finitefield.FiniteField.char, nzmath.real.RealField.one, nzmath.poly.ring.PolynomialRing.one, nzmath.imaginary.ComplexField.one, nzmath.intresidue.IntegerResidueClassRing.one, nzmath.algfield.NumberField.one, nzmath.finitefield.FinitePrimeField.one, nzmath.poly.multiutil.PolynomialRingAnonymousVariables.one, nzmath.ring.ResidueClassRing.one, nzmath.poly.ring.RationalFunctionField.one, nzmath.rational.RationalField.one, nzmath.finitefield.ExtendedField.one, nzmath.rational.IntegerRing.one, nzmath.matrix.MatrixRing.one, and nzmath.finitefield.FiniteField.TonelliShanks().

Referenced by nzmath.finitefield.FiniteField.TonelliShanks().

TonelliShanks()

def nzmath.finitefield.FiniteField.TonelliShanks	(		self,
			element
	)

Return square root of element if exist.
assume that characteristic have to be more than three.

Definition at line 106 of file finitefield.py.

References nzmath.finitefield.FiniteField.card(), nzmath.finitefield.FiniteField.char, nzmath.lattice.Lattice.createElement(), nzmath.ring.Ring.createElement(), nzmath.group.Group.createElement(), nzmath.poly.ring.PolynomialRing.createElement(), nzmath.real.RealField.createElement(), nzmath.imaginary.ComplexField.createElement(), nzmath.intresidue.IntegerResidueClassRing.createElement(), nzmath.algfield.NumberField.createElement(), nzmath.finitefield.FinitePrimeField.createElement(), nzmath.poly.multiutil.PolynomialRingAnonymousVariables.createElement(), nzmath.rational.RationalField.createElement(), nzmath.permute.PermGroup.createElement(), nzmath.poly.ring.RationalFunctionField.createElement(), nzmath.finitefield.ExtendedField.createElement(), nzmath.rational.IntegerRing.createElement(), nzmath.matrix.MatrixRing.createElement(), nzmath.finitefield.FiniteField.Legendre(), nzmath.real.RealField.one, nzmath.poly.ring.PolynomialRing.one, nzmath.imaginary.ComplexField.one, nzmath.intresidue.IntegerResidueClassRing.one, nzmath.algfield.NumberField.one, nzmath.finitefield.FinitePrimeField.one, nzmath.poly.multiutil.PolynomialRingAnonymousVariables.one, nzmath.ring.ResidueClassRing.one, nzmath.poly.ring.RationalFunctionField.one, nzmath.rational.RationalField.one, nzmath.finitefield.ExtendedField.one, nzmath.rational.IntegerRing.one, nzmath.matrix.MatrixRing.one, nzmath.finitefield.FiniteField.random_element(), and nzmath.finitefield.FiniteField.sqrt().

Referenced by nzmath.finitefield.FiniteField.sqrt().

Member Data Documentation

_orderfactor

nzmath.finitefield.FiniteField._orderfactor

private

Definition at line 45 of file finitefield.py.

Referenced by nzmath.finitefield.FiniteField.order(), nzmath.finitefield.FinitePrimeField.primitive_element(), and nzmath.finitefield.ExtendedField.primitive_element().

char

nzmath.finitefield.FiniteField.char

Definition at line 44 of file finitefield.py.

Referenced by nzmath.finitefield.FinitePrimeField.__contains__(), nzmath.finitefield.ExtendedField.__eq__(), nzmath.finitefield.FinitePrimeField.__hash__(), nzmath.finitefield.ExtendedField.__hash__(), nzmath.finitefield.FinitePrimeField.__repr__(), nzmath.finitefield.ExtendedField.__repr__(), nzmath.finitefield.FinitePrimeField.__str__(), nzmath.finitefield.FinitePrimeField.card(), nzmath.finitefield.ExtendedField.card(), nzmath.finitefield.FinitePrimeField.createElement(), nzmath.finitefield.FiniteField.getCharacteristic(), nzmath.finitefield.FinitePrimeField.issubring(), nzmath.finitefield.ExtendedField.issubring(), nzmath.finitefield.FiniteField.Legendre(), nzmath.finitefield.FinitePrimeField.primitive_element(), nzmath.finitefield.FiniteField.sqrt(), and nzmath.finitefield.FiniteField.TonelliShanks().

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

• nzmath/finitefield.py