| Safe Haskell | None |
|---|---|
| Language | Haskell2010 |
Math.FiniteField.GaloisField.Small
Description
Small Galois fields via a precomputed table of Conway polynomials.
This covers:
- all fields with order <= 2^30
- all fields with characteristic < 2^16 and order < 2^64 (?)
- higher powers for very small prime characteristic
- some more
To look up Conway polynomials, see the module Math.FiniteField.Conway.
Synopsis
- data WitnessGF (p :: Nat) (m :: Nat) where
- WitnessFp :: IsSmallPrime p -> WitnessGF p 1
- WitnessFq :: HasConwayPoly p m -> WitnessGF p m
- data SomeWitnessGF = forall p m. SomeWitnessGF (WitnessGF p m)
- mkGaloisField :: Int -> Int -> Maybe SomeWitnessGF
- unsafeGaloisField :: Int -> Int -> SomeWitnessGF
- data GF (p :: Nat) (m :: Nat)
Witness for the existence of the field
data WitnessGF (p :: Nat) (m :: Nat) where Source #
We need either a Conway polynomial, or in the m=1 case, a proof that p is prime
Constructors
| WitnessFp :: IsSmallPrime p -> WitnessGF p 1 | |
| WitnessFq :: HasConwayPoly p m -> WitnessGF p m |
data SomeWitnessGF Source #
Constructors
| forall p m. SomeWitnessGF (WitnessGF p m) |
Instances
| Show SomeWitnessGF Source # | |
Defined in Math.FiniteField.GaloisField.Small Methods showsPrec :: Int -> SomeWitnessGF -> ShowS # show :: SomeWitnessGF -> String # showList :: [SomeWitnessGF] -> ShowS # | |
mkGaloisField :: Int -> Int -> Maybe SomeWitnessGF Source #
Usage:
mkGaloisField p m
to construct the field with q = p^m elements
Implementation note: For m=1 we may do a primality test, which is very
slow at the moment. You can use unsafeGaloisField below to avoid this.
unsafeGaloisField :: Int -> Int -> SomeWitnessGF Source #
In the case of m=1 you are responsible for guaranteeing that p is a prime
(for m>1 we have to look up a Conway polynomial anyway).
Field elements
data GF (p :: Nat) (m :: Nat) Source #
An element of the Galois field of order q = p^m