Crypto.Lol.Types.FiniteField

Basic (unoptimized) finite field arithmetic.

Documentation

data GF fp d

A finite field of given degree over \(\F_p\).

Instances

GFCtx k fp d => CRTrans Maybe (GF k fp d)

Methods

crtInfo :: Reflects k1 m Int => TaggedT * k1 m Maybe (CRTInfo (GF k fp d))

(C fp, Eq fp) => Eq (GF k fp d)

Methods

(==) :: GF k fp d -> GF k fp d -> Bool

(/=) :: GF k fp d -> GF k fp d -> Bool

Show fp => Show (GF k fp d)

Methods

showsPrec :: Int -> GF k fp d -> ShowS

show :: GF k fp d -> String

showList :: [GF k fp d] -> ShowS

(Random fp, Reflects k d Int) => Random (GF k fp d)

Methods

randomR :: RandomGen g => (GF k fp d, GF k fp d) -> g -> (GF k fp d, g)

random :: RandomGen g => g -> (GF k fp d, g)

randomRs :: RandomGen g => (GF k fp d, GF k fp d) -> g -> [GF k fp d]

randoms :: RandomGen g => g -> [GF k fp d]

randomRIO :: (GF k fp d, GF k fp d) -> IO (GF k fp d)

randomIO :: IO (GF k fp d)

NFData fp => NFData (GF k fp d)

Methods

rnf :: GF k fp d -> ()

GFCtx k fp d => C (GF k fp d)

Methods

(/) :: GF k fp d -> GF k fp d -> GF k fp d

recip :: GF k fp d -> GF k fp d

fromRational' :: Rational -> GF k fp d

(^-) :: GF k fp d -> Integer -> GF k fp d

C fp => C (GF k fp d)

Methods

isZero :: GF k fp d -> Bool

GFCtx k fp d => C (GF k fp d)

Methods

(*) :: GF k fp d -> GF k fp d -> GF k fp d

one :: GF k fp d

fromInteger :: Integer -> GF k fp d

(^) :: GF k fp d -> Integer -> GF k fp d

C fp => C (GF k fp d)

Methods

zero :: GF k fp d

(+) :: GF k fp d -> GF k fp d -> GF k fp d

(-) :: GF k fp d -> GF k fp d -> GF k fp d

negate :: GF k fp d -> GF k fp d

GFCtx k fp d => Enumerable (GF k fp d)

Methods

values :: [GF k fp d]

(Additive fp, Ring (GF k fp d), Reflects k d Int) => C (GF k fp d) (TensorCoeffs fp)

Methods

(*>) :: GF k fp d -> TensorCoeffs fp -> TensorCoeffs fp

(GFCtx k fp d, Fact m, CElt t fp) => C (GF k fp d) (Cyc t m fp)

\(R_p\) is an \(\F_{p^d}\)-module when \(d\) divides \(\varphi(m)\), by applying \(d\)-dimensional \(\F_p\)-linear transform on \(d\)-dim chunks of powerful basis coeffs.

Methods

(*>) :: GF k fp d -> Cyc t m fp -> Cyc t m fp

(GFCtx k fp d, Fact m, Tensor t, TElt t fp) => C (GF k fp d) (UCyc t m P fp)

\(R_p\) is an \(\F_{p^d}\)-module when \(d\) divides \(\varphi(m)\), by applying \(d\)-dimensional \(\F_p\)-linear transform on \(d\)-dim chunks of powerful basis coeffs.

Methods

(*>) :: GF k fp d -> UCyc t m P fp -> UCyc t m P fp

type PrimeField fp = (Enumerable fp, Field fp, Eq fp, ZeroTestable fp, Prime (CharOf fp), IrreduciblePoly fp)

Constraint synonym for a prime field.

type GFCtx fp d = (PrimeField fp, Reflects d Int)

Constraint synonym for a finite field.

size :: GFCtx fp d => Tagged (GF fp d) Int

The order of the field: size (GF fp d) = \( p^d \)

trace :: forall fp d. GFCtx fp d => GF fp d -> fp

Trace into the prime subfield.

toList :: forall fp d. (Reflects d Int, Additive fp) => GF fp d -> [fp]

Yield a list of length exactly \(d\) (i.e., including trailing zeros) of the \(\F_p\)-coefficients with respect to the power basis.

fromList :: forall fp d. Reflects d Int => [fp] -> GF fp d

Yield a field element given up to \(d\) coefficients with respect to the power basis.

class Field fp => IrreduciblePoly fp where

Represents fields over which we can get irreducible polynomials of desired degrees. (An instance of this class is defined in Crypto.Lol.Types.IrreducibleChar2 and exported from Crypto.Lol.Types.)

Minimal complete definition

irreduciblePoly

Methods

irreduciblePoly :: Reflects d Int => Tagged d (Polynomial fp)

data X

Convenience data type for writing IrreduciblePoly instances.

Constructors

X

(^^) :: Ring a => X -> Int -> Polynomial a

Convenience function for writing IrreduciblePoly instances.

newtype TensorCoeffs a

This wrapper for a list of coefficients is used to define a \(\F_{p^d}\)-module structure for tensors over \(\F_p\) of dimension \(n\), where \(d \mid n\).

Constructors

Coeffs

Fields

• unCoeffs :: [a]

Instances

C a => C (TensorCoeffs a)

Methods

zero :: TensorCoeffs a

(+) :: TensorCoeffs a -> TensorCoeffs a -> TensorCoeffs a

(-) :: TensorCoeffs a -> TensorCoeffs a -> TensorCoeffs a

negate :: TensorCoeffs a -> TensorCoeffs a

(Additive fp, Ring (GF k fp d), Reflects k d Int) => C (GF k fp d) (TensorCoeffs fp)

Methods

(*>) :: GF k fp d -> TensorCoeffs fp -> TensorCoeffs fp