GaloisField

Documentation

class (Arbitrary k, Eq k, Fractional 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, pow, rnd

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.

frob :: k -> k

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

order :: k -> Integer

Order p^q of field.

pow :: k -> Integer -> k

Exponentiation x to the power of y.

rnd :: MonadRandom m => m k

Randomised element x of field.

Instances

KnownNat ib => GaloisField (BinaryField ib)

Defined in BinaryField

Methods

char :: BinaryField ib -> Integer

deg :: BinaryField ib -> Int

frob :: BinaryField ib -> BinaryField ib

order :: BinaryField ib -> Integer

pow :: BinaryField ib -> Integer -> BinaryField ib

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

KnownNat p => GaloisField (PrimeField p)

Defined in PrimeField

Methods

char :: PrimeField p -> Integer

deg :: PrimeField p -> Int

frob :: PrimeField p -> PrimeField p

order :: PrimeField p -> Integer

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

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

(GaloisField k, IrreducibleMonic k im) => GaloisField (ExtensionField k im)

Defined in ExtensionField

Methods

char :: ExtensionField k im -> Integer

deg :: ExtensionField k im -> Int

frob :: ExtensionField k im -> ExtensionField k im

order :: ExtensionField k im -> Integer

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

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