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)

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

Show fp => Show (GF k fp d)

NFData fp => NFData (GF k fp d)

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

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

C fp => C (GF k fp d)

C fp => C (GF k fp d)

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

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

(GFCtx k fp d, Fact m, Additive (CT m fp)) => C (GF k fp d) (CT m fp)

(GFCtx k fp d, Fact m, Additive (RT m fp)) => C (GF k fp d) (RT m fp)

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

Rp is an F_{p^d}-module when d divides phi(m), by applying d-dimensional Fp-linear transform on d-dim chunks of powerful basis coeffs

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

Rp is an F_{p^d}-module when d divides phi(m), by applying d-dimensional Fp-linear transform on d-dim chunks of powerful basis coeffs

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 fp-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 IrreducibleChar2 and exported from Lol.)

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 GF(p^d)-module structure for tensors over F_p of dimension n, where d | n.

Constructors

Coeffs

Fields

unCoeffs :: [a]

Instances

C a => C (TensorCoeffs a)

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