GaloisField

Documentation

class (Euclidean k, Ring k) => Field k where

Fields.

Minimal complete definition

Nothing

Methods

divide :: k -> k -> k

Division.

invert :: k -> k

Inversion.

minus :: k -> k -> k

Subtraction.

Instances

KnownNat im => Field (BinaryField im)

Defined in BinaryField

Methods

divide :: BinaryField im -> BinaryField im -> BinaryField im

invert :: BinaryField im -> BinaryField im

minus :: BinaryField im -> BinaryField im -> BinaryField im

KnownNat p => Field (PrimeField p)

Defined in PrimeField

Methods

divide :: PrimeField p -> PrimeField p -> PrimeField p

invert :: PrimeField p -> PrimeField p

minus :: PrimeField p -> PrimeField p -> PrimeField p

IrreducibleMonic k im => Field (ExtensionField k im)

Defined in ExtensionField

Methods

divide :: ExtensionField k im -> ExtensionField k im -> ExtensionField k im

invert :: ExtensionField k im -> ExtensionField k im

minus :: ExtensionField k im -> ExtensionField k im -> ExtensionField k im

class (Arbitrary k, Field k, Fractional k, Generic k, Ord k, Pretty k, Random k, Show k) => GaloisField k where

Galois fields GF(p^q) for p prime and q non-negative.

Minimal complete definition

char, deg, frob

Methods

char :: k -> Integer

Characteristic p of field and order of prime subfield.

deg :: k -> Int

Degree q of field as extension field over prime subfield.

order :: k -> Integer

Order p^q of field.

frob :: k -> k

Frobenius endomorphism x -> x^p of prime subfield.

pow :: k -> Integer -> k

Exponentiation of field element to integer.

qnr :: k

Get randomised quadratic nonresidue.

qr :: k -> Bool

Check if quadratic residue.

quad :: k -> k -> k -> Maybe k

Solve quadratic ax^2 + bx + c = 0 over field.

rnd :: MonadRandom m => m k

Randomised field element.

sr :: k -> Maybe k

Square root of field element.

Instances

KnownNat im => GaloisField (BinaryField im)

Defined in BinaryField

Methods

char :: BinaryField im -> Integer

deg :: BinaryField im -> Int

order :: BinaryField im -> Integer

frob :: BinaryField im -> BinaryField im

pow :: BinaryField im -> Integer -> BinaryField im

qnr :: BinaryField im

qr :: BinaryField im -> Bool

quad :: BinaryField im -> BinaryField im -> BinaryField im -> Maybe (BinaryField im)

rnd :: MonadRandom m => m (BinaryField im)

sr :: BinaryField im -> Maybe (BinaryField im)

KnownNat p => GaloisField (PrimeField p)

Defined in PrimeField

Methods

char :: PrimeField p -> Integer

deg :: PrimeField p -> Int

order :: PrimeField p -> Integer

frob :: PrimeField p -> PrimeField p

pow :: PrimeField p -> Integer -> PrimeField p

qnr :: PrimeField p

qr :: PrimeField p -> Bool

quad :: PrimeField p -> PrimeField p -> PrimeField p -> Maybe (PrimeField p)

rnd :: MonadRandom m => m (PrimeField p)

sr :: PrimeField p -> Maybe (PrimeField p)

IrreducibleMonic k im => GaloisField (ExtensionField k im)

Defined in ExtensionField

Methods

char :: ExtensionField k im -> Integer

deg :: ExtensionField k im -> Int

order :: ExtensionField k im -> Integer

frob :: ExtensionField k im -> ExtensionField k im

pow :: ExtensionField k im -> Integer -> ExtensionField k im

qnr :: ExtensionField k im

qr :: ExtensionField k im -> Bool

quad :: ExtensionField k im -> ExtensionField k im -> ExtensionField k im -> Maybe (ExtensionField k im)

rnd :: MonadRandom m => m (ExtensionField k im)

sr :: ExtensionField k im -> Maybe (ExtensionField k im)