InnerPrimeField pΒΆ

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InnerPrimeField(p) implements the field with p elements by using IntegerMod p. Note: argument p MUST be a prime (this domain does not check). See PrimeField for a domain that does check. In addition to the inherited operations of IntegerMod p, the domain provides exploits the structure of the cyclic group of its invertible elements. It stores a primitive element w, i.a. generator of this group and it stores a logarithm table for w as soon as this is required. sqrt was added in 2018.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, Fraction Integer) -> %

from RightModule Fraction Integer

*: (Fraction Integer, %) -> %

from LeftModule Fraction Integer

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, %) -> %

from Field

=: (%, %) -> Boolean

from BasicType

^: (%, Integer) -> %

from DivisionRing

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

algebraic?: % -> Boolean

from ExtensionField %

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %

from NonAssociativeSemiRng

associates?: (%, %) -> Boolean

from EntireRing

associator: (%, %, %) -> %

from NonAssociativeRng

basis: () -> Vector %

from FramedModule %

basis: PositiveInteger -> Vector %

from FiniteAlgebraicExtensionField %

characteristic: () -> NonNegativeInteger

from NonAssociativeRing

characteristicPolynomial: % -> SparseUnivariatePolynomial %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

charthRoot: % -> %

from FiniteFieldCategory

charthRoot: % -> Union(%, failed)

from PolynomialFactorizationExplicit

coerce: % -> %

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> %

from Algebra Fraction Integer

coerce: Integer -> %

from NonAssociativeRing

commutator: (%, %) -> %

from NonAssociativeRng

conditionP: Matrix % -> Union(Vector %, failed)

from PolynomialFactorizationExplicit

convert: % -> InputForm

from ConvertibleTo InputForm

convert: % -> Integer

from ConvertibleTo Integer

convert: % -> Vector %

from FramedModule %

convert: Vector % -> %

from FramedModule %

coordinates: % -> Vector %

from FramedModule %

coordinates: (%, Vector %) -> Vector %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

coordinates: (Vector %, Vector %) -> Matrix %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

coordinates: Vector % -> Matrix %

from FramedModule %

createNormalElement: () -> %

from FiniteAlgebraicExtensionField %

createPrimitiveElement: () -> %

from FiniteFieldCategory

D: % -> %

from DifferentialRing

D: (%, NonNegativeInteger) -> %

from DifferentialRing

definingPolynomial: () -> SparseUnivariatePolynomial %

from FiniteAlgebraicExtensionField %

degree: % -> OnePointCompletion PositiveInteger

from ExtensionField %

degree: % -> PositiveInteger

from FiniteAlgebraicExtensionField %

differentiate: % -> %

from DifferentialRing

differentiate: (%, NonNegativeInteger) -> %

from DifferentialRing

discreteLog: % -> NonNegativeInteger

from FiniteFieldCategory

discreteLog: (%, %) -> Union(NonNegativeInteger, failed)

from FieldOfPrimeCharacteristic

discriminant: () -> %

from FramedAlgebra(%, SparseUnivariatePolynomial %)

discriminant: Vector % -> %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

divide: (%, %) -> Record(quotient: %, remainder: %)

from EuclideanDomain

enumerate: () -> List %

from Finite

euclideanSize: % -> NonNegativeInteger

from EuclideanDomain

expressIdealMember: (List %, %) -> Union(List %, failed)

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed)

from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %)

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed)

from EuclideanDomain

extensionDegree: () -> OnePointCompletion PositiveInteger

from ExtensionField %

extensionDegree: () -> PositiveInteger

from FiniteAlgebraicExtensionField %

factor: % -> Factored %

from UniqueFactorizationDomain

factorPolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

factorsOfCyclicGroupSize: () -> List Record(factor: Integer, exponent: NonNegativeInteger)

from FiniteFieldCategory

factorSquareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

Frobenius: % -> %

from ExtensionField %

Frobenius: (%, NonNegativeInteger) -> %

from ExtensionField %

gcd: (%, %) -> %

from GcdDomain

gcd: List % -> %

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

generator: () -> %

from FiniteAlgebraicExtensionField %

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

index: PositiveInteger -> %

from Finite

inGroundField?: % -> Boolean

from ExtensionField %

init: %

from StepThrough

inv: % -> %

from DivisionRing

latex: % -> String

from SetCategory

lcm: (%, %) -> %

from GcdDomain

lcm: List % -> %

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %)

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

linearAssociatedExp: (%, SparseUnivariatePolynomial %) -> %

from FiniteAlgebraicExtensionField %

linearAssociatedLog: % -> SparseUnivariatePolynomial %

from FiniteAlgebraicExtensionField %

linearAssociatedLog: (%, %) -> Union(SparseUnivariatePolynomial %, failed)

from FiniteAlgebraicExtensionField %

linearAssociatedOrder: % -> SparseUnivariatePolynomial %

from FiniteAlgebraicExtensionField %

lookup: % -> PositiveInteger

from Finite

minimalPolynomial: % -> SparseUnivariatePolynomial %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

minimalPolynomial: (%, PositiveInteger) -> SparseUnivariatePolynomial %

from FiniteAlgebraicExtensionField %

multiEuclidean: (List %, %) -> Union(List %, failed)

from EuclideanDomain

nextItem: % -> Union(%, failed)

from StepThrough

norm: % -> %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

norm: (%, PositiveInteger) -> %

from FiniteAlgebraicExtensionField %

normal?: % -> Boolean

from FiniteAlgebraicExtensionField %

normalElement: () -> %

from FiniteAlgebraicExtensionField %

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

order: % -> OnePointCompletion PositiveInteger

from FieldOfPrimeCharacteristic

order: % -> PositiveInteger

from FiniteFieldCategory

prime?: % -> Boolean

from UniqueFactorizationDomain

primeFrobenius: % -> %

from FieldOfPrimeCharacteristic

primeFrobenius: (%, NonNegativeInteger) -> %

from FieldOfPrimeCharacteristic

primitive?: % -> Boolean

from FiniteFieldCategory

primitiveElement: () -> %

from FiniteFieldCategory

principalIdeal: List % -> Record(coef: List %, generator: %)

from PrincipalIdealDomain

quadraticNonResidue: () -> %

quadraticNonResidue() computes the smallest non negative integer, which represents a quadratic non residue.

quo: (%, %) -> %

from EuclideanDomain

random: () -> %

from Finite

rank: () -> PositiveInteger

from FramedModule %

recip: % -> Union(%, failed)

from MagmaWithUnit

regularRepresentation: % -> Matrix %

from FramedAlgebra(%, SparseUnivariatePolynomial %)

regularRepresentation: (%, Vector %) -> Matrix %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

rem: (%, %) -> %

from EuclideanDomain

representationType: () -> Union(prime, polynomial, normal, cyclic)

from FiniteFieldCategory

represents: (Vector %, Vector %) -> %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

represents: Vector % -> %

from FramedModule %

retract: % -> %

from RetractableTo %

retractIfCan: % -> Union(%, failed)

from RetractableTo %

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

size: () -> NonNegativeInteger

from Finite

sizeLess?: (%, %) -> Boolean

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

solveLinearPolynomialEquation: (List SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> Union(List SparseUnivariatePolynomial %, failed)

from PolynomialFactorizationExplicit

sqrt: % -> %

sqrt(x) computes one y such that y^2 = x, error if there is no square root, i.e. jacobi(x,p) = -1. Implementation according to http://www.staff.uni-mainz.de/pommeren/Cryptology/Asymmetric/5_NTh/

squareFree: % -> Factored %

from UniqueFactorizationDomain

squareFreePart: % -> %

from UniqueFactorizationDomain

squareFreePolynomial: SparseUnivariatePolynomial % -> Factored SparseUnivariatePolynomial %

from PolynomialFactorizationExplicit

subtractIfCan: (%, %) -> Union(%, failed)

from CancellationAbelianMonoid

tableForDiscreteLogarithm: Integer -> Table(PositiveInteger, NonNegativeInteger)

from FiniteFieldCategory

trace: % -> %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

trace: (%, PositiveInteger) -> %

from FiniteAlgebraicExtensionField %

traceMatrix: () -> Matrix %

from FramedAlgebra(%, SparseUnivariatePolynomial %)

traceMatrix: Vector % -> Matrix %

from FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

transcendenceDegree: () -> NonNegativeInteger

from ExtensionField %

transcendent?: % -> Boolean

from ExtensionField %

unit?: % -> Boolean

from EntireRing

unitCanonical: % -> %

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %)

from EntireRing

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(Fraction Integer, Fraction Integer)

CancellationAbelianMonoid

canonicalsClosed

canonicalUnitNormal

CharacteristicNonZero

CoercibleFrom %

CoercibleTo OutputForm

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo InputForm

ConvertibleTo Integer

DifferentialRing

DivisionRing

EntireRing

EuclideanDomain

ExtensionField %

Field

FieldOfPrimeCharacteristic

Finite

FiniteAlgebraicExtensionField %

FiniteFieldCategory

FiniteRankAlgebra(%, SparseUnivariatePolynomial %)

FramedAlgebra(%, SparseUnivariatePolynomial %)

FramedModule %

GcdDomain

IntegralDomain

LeftModule %

LeftModule Fraction Integer

LeftOreRing

Magma

MagmaWithUnit

Module %

Module Fraction Integer

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

PolynomialFactorizationExplicit

PrincipalIdealDomain

RetractableTo %

RightModule %

RightModule Fraction Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown