Copyright	(c) Eric Crockett 2011-2017 Chris Peikert 2011-2017
License	GPL-3
Maintainer	ecrockett0@email.com
Stability	experimental
Portability	POSIX \( \def\F{\mathbb{F}} \)
Safe Haskell	None
Language	Haskell2010

Crypto.Lol.Types.FiniteField

Description

Basic (unoptimized) finite field arithmetic.

Synopsis

data GF fp d
type PrimeField fp = (Enumerable fp, Field fp, Eq fp, ZeroTestable fp, Prime (CharOf fp), IrreduciblePoly fp)
type GFCtx fp d = (PrimeField fp, Reflects d Int)
size :: GFCtx fp d => Tagged (GF fp d) Int
trace :: forall fp d. GFCtx fp d => GF fp d -> fp
toList :: forall fp d. (Reflects d Int, Additive fp) => GF fp d -> [fp]
fromList :: forall fp d. Reflects d Int => [fp] -> GF fp d
class Field fp => IrreduciblePoly fp where
- irreduciblePoly :: Reflects d Int => Tagged d (Polynomial fp)
data X = X
(^^) :: Ring a => X -> Int -> Polynomial a
newtype TensorCoeffs a = Coeffs {
- unCoeffs :: [a]
}

Documentation

data GF fp d Source #

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

Instances

(GFCtx fp d, NFData fp) => CRTrans Maybe (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods crtInfo :: Reflects m Int => TaggedT m Maybe (CRTInfo (GF fp d)) Source #
(Eq fp, C fp) => Eq (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods (==) :: GF fp d -> GF fp d -> Bool # (/=) :: GF fp d -> GF fp d -> Bool #
Show fp => Show (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods showsPrec :: Int -> GF fp d -> ShowS # show :: GF fp d -> String # showList :: [GF fp d] -> ShowS #
(Random fp, Reflects d Int) => Random (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods randomR :: RandomGen g => (GF fp d, GF fp d) -> g -> (GF fp d, g) # random :: RandomGen g => g -> (GF fp d, g) # randomRs :: RandomGen g => (GF fp d, GF fp d) -> g -> [GF fp d] # randoms :: RandomGen g => g -> [GF fp d] # randomRIO :: (GF fp d, GF fp d) -> IO (GF fp d) # randomIO :: IO (GF fp d) #
NFData fp => NFData (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods rnf :: GF fp d -> () #
GFCtx fp d => C (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods (/) :: GF fp d -> GF fp d -> GF fp d # recip :: GF fp d -> GF fp d # fromRational' :: Rational -> GF fp d # (^-) :: GF fp d -> Integer -> GF fp d #
GFCtx fp d => C (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods (*) :: GF fp d -> GF fp d -> GF fp d # one :: GF fp d # fromInteger :: Integer -> GF fp d # (^) :: GF fp d -> Integer -> GF fp d #
C fp => C (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods isZero :: GF fp d -> Bool #
C fp => C (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods zero :: GF fp d # (+) :: GF fp d -> GF fp d -> GF fp d # (-) :: GF fp d -> GF fp d -> GF fp d # negate :: GF fp d -> GF fp d #
GFCtx fp d => Enumerable (GF fp d) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods values :: [GF fp d] Source #
(Additive fp, Ring (GF fp d), Reflects d Int) => C (GF fp d) (TensorCoeffs fp) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods (*>) :: GF fp d -> TensorCoeffs fp -> TensorCoeffs fp #
(Ring (GF (ZqBasic q z) d), Module (GF (ZqBasic q z) d) (CycG t m (ZqBasic q z))) => C (GF (ZqBasic q z) d) (Cyc t m (ZqBasic q z)) Source #
Instance details Defined in Crypto.Lol.Cyclotomic.Cyc Methods (*>) :: GF (ZqBasic q z) d -> Cyc t m (ZqBasic q z) -> Cyc t m (ZqBasic q z) #
(GFCtx fp d, Fact m, TensorPowDec t fp, Module (GF fp d) (t m fp)) => C (GF fp d) (CycRep t P m fp) Source #	\(R_p\) 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.
Instance details Defined in Crypto.Lol.Cyclotomic.CycRep Methods (*>) :: GF fp d -> CycRep t P m fp -> CycRep t P m fp #

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

Constraint synonym for a prime field.

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

Constraint synonym for a finite field.

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

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

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

Trace into the prime subfield.

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

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 Source #

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

class Field fp => IrreduciblePoly fp where Source #

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.)

Methods

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

Instances

((CharOf a :: PrimeBin) ~ Prime2, Field a) => IrreduciblePoly a Source #
Instance details Defined in Crypto.Lol.Types.IrreducibleChar2 Methods irreduciblePoly :: Reflects d Int => Tagged d (Polynomial a) Source #

data X Source #

Convenience data type for writing IrreduciblePoly instances.

Constructors

X

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

Convenience function for writing IrreduciblePoly instances.

newtype TensorCoeffs a Source #

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) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField 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 fp d), Reflects d Int) => C (GF fp d) (TensorCoeffs fp) Source #
Instance details Defined in Crypto.Lol.Types.FiniteField Methods (*>) :: GF fp d -> TensorCoeffs fp -> TensorCoeffs fp #