Safe Haskell	None
Language	Haskell2010

Math.FiniteField.GaloisField.Zech

Contents

Tables of Zech logarithms
Witness for the existence of the field
Field elements

Description

Small Galois fields via precomputed tables of Zech's logarithms.

https://en.wikipedia.org/wiki/Zech%27s_logarithm

When "creating" a field, we precompute the Zech logarithm table. After that, computations should be fast.

This is practical up to fields of size 10^5 -- 10^6

TODO: also export the tables to C, and write C implementation of of the field operations.

Synopsis

data ZechTable
makeZechTable :: forall p m. WitnessGF p m -> ZechTable
newtype WitnessZech (p :: Nat) (m :: Nat) = WitnessZech {
- fromWitnessZech :: ZechTable
}
data SomeWitnessZech = forall p m. SomeWitnessZech (WitnessZech p m)
mkZechField :: Int -> Int -> Maybe SomeWitnessZech
unsafeZechField :: Int -> Int -> SomeWitnessZech
data Zech (p :: Nat) (m :: Nat)

Tables of Zech logarithms

data ZechTable Source #

Instances

Instances details

Show ZechTable Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods showsPrec :: Int -> ZechTable -> ShowS # show :: ZechTable -> String # showList :: [ZechTable] -> ShowS #

makeZechTable :: forall p m. WitnessGF p m -> ZechTable Source #

Witness for the existence of the field

newtype WitnessZech (p :: Nat) (m :: Nat) Source #

Constructors

WitnessZech
Fields fromWitnessZech :: ZechTable

Instances

Instances details

Show (WitnessZech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods showsPrec :: Int -> WitnessZech p m -> ShowS # show :: WitnessZech p m -> String # showList :: [WitnessZech p m] -> ShowS #

data SomeWitnessZech Source #

Constructors

forall p m. SomeWitnessZech (WitnessZech p m)

Instances

Instances details

Show SomeWitnessZech Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods showsPrec :: Int -> SomeWitnessZech -> ShowS # show :: SomeWitnessZech -> String # showList :: [SomeWitnessZech] -> ShowS #

mkZechField :: Int -> Int -> Maybe SomeWitnessZech Source #

unsafeZechField :: Int -> Int -> SomeWitnessZech Source #

Field elements

data Zech (p :: Nat) (m :: Nat) Source #

Instances

Instances details

Eq (Zech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods (==) :: Zech p m -> Zech p m -> Bool # (/=) :: Zech p m -> Zech p m -> Bool #
Fractional (Zech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods (/) :: Zech p m -> Zech p m -> Zech p m # recip :: Zech p m -> Zech p m # fromRational :: Rational -> Zech p m #
Num (Zech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods (+) :: Zech p m -> Zech p m -> Zech p m # (-) :: Zech p m -> Zech p m -> Zech p m # (*) :: Zech p m -> Zech p m -> Zech p m # negate :: Zech p m -> Zech p m # abs :: Zech p m -> Zech p m # signum :: Zech p m -> Zech p m # fromInteger :: Integer -> Zech p m #
Ord (Zech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods compare :: Zech p m -> Zech p m -> Ordering # (<) :: Zech p m -> Zech p m -> Bool # (<=) :: Zech p m -> Zech p m -> Bool # (>) :: Zech p m -> Zech p m -> Bool # (>=) :: Zech p m -> Zech p m -> Bool # max :: Zech p m -> Zech p m -> Zech p m # min :: Zech p m -> Zech p m -> Zech p m #
Show (Zech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Methods showsPrec :: Int -> Zech p m -> ShowS # show :: Zech p m -> String # showList :: [Zech p m] -> ShowS #
Field (Zech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech Associated Types type Witness (Zech p m) = (r :: Type) Source # Methods characteristic :: Witness (Zech p m) -> Integer Source # dimension :: Witness (Zech p m) -> Integer Source # fieldSize :: Witness (Zech p m) -> Integer Source # zero :: Witness (Zech p m) -> Zech p m Source # one :: Witness (Zech p m) -> Zech p m Source # isZero :: Zech p m -> Bool Source # isOne :: Zech p m -> Bool Source # embed :: Witness (Zech p m) -> Integer -> Zech p m Source # embedSmall :: Witness (Zech p m) -> Int -> Zech p m Source # randomFieldElem :: RandomGen gen => Witness (Zech p m) -> gen -> (Zech p m, gen) Source # randomInvertible :: RandomGen gen => Witness (Zech p m) -> gen -> (Zech p m, gen) Source # primGen :: Witness (Zech p m) -> Zech p m Source # witnessOf :: Zech p m -> Witness (Zech p m) Source # power :: Zech p m -> Integer -> Zech p m Source # powerSmall :: Zech p m -> Int -> Zech p m Source # enumerate :: Witness (Zech p m) -> [Zech p m] Source #
type Witness (Zech p m) Source #
Instance details Defined in Math.FiniteField.GaloisField.Zech type Witness (Zech p m) = WitnessZech p m