diff --git a/ChangeLog.md b/ChangeLog.md --- a/ChangeLog.md +++ b/ChangeLog.md @@ -1,18 +1,23 @@ # Change log for galois-field +## 0.2.1 +* Add preliminary implementation of BinaryField. +* Add `frob` function for GaloisField. +* Add minor improvements to documentation. + ## 0.2.0 -* Add `deg` for GaloisField -* Add `order` for GaloisField -* Add `pow` for GaloisField -* Add `rnd` for GaloisField +* Add `deg` function for GaloisField. +* Add `order` function for GaloisField. +* Add `pow` function for GaloisField. +* Add `rnd` function for GaloisField. ## 0.1.1 * Add `Arbitrary` instances to PrimeField, PolynomialRing, and ExtensionField. * Add `Bits` instances to PrimeField. * Add `Pretty` instances to PrimeField, PolynomialRing, and ExtensionField. -* Minor optimisations to multiplication and inversion with `INLINE`. +* Add minor optimisations to multiplication and inversion with `INLINE`. ## 0.1.0 diff --git a/README.md b/README.md --- a/README.md +++ b/README.md @@ -1,8 +1,12 @@

- + + Adjoint Logo +

+ [![CircleCI](https://circleci.com/gh/adjoint-io/galois-field.svg?style=svg)](https://circleci.com/gh/adjoint-io/galois-field) +[![Hackage](https://img.shields.io/hackage/v/galois-field.svg)](https://hackage.haskell.org/package/galois-field) # Galois Field @@ -10,11 +14,11 @@ ## Technical background -A **Galois field** GF(pq), for prime p and positive q, is a *field* (GF(pq), +, \*, 0, 1) of finite *order*. Explicitly, -- (GF(pq), +, 0) is an abelian group, -- (GF(pq) \\ \{0\}, \*, 1) is an abelian group, +A **Galois field** GF(p^q), for prime p and positive q, is a *field* (GF(p^q), +, \*, 0, 1) of finite *order*. Explicitly, +- (GF(p^q), +, 0) is an abelian group, +- (GF(p^q) \\ \{0\}, \*, 1) is an abelian group, - \* is distributive over +, and -- \#GF(pq) is finite. +- \#GF(p^q) is finite. ### Prime fields @@ -24,9 +28,9 @@ ### Extension fields -Any Galois field has order a prime power pq for prime p and positive q, and there is a Galois field GF(pq) of any prime power order pq that is *unique up to non-unique isomorphism*. Any Galois field GF(pq) can be constructed as an **extension field** over a smaller Galois subfield GF(pr), through the identification GF(pq) = GF(pr)[X] / \ for an *irreducible monic splitting polynomial* f(X) of degree q - r + 1 in the *polynomial ring* GF(pr)[X]. +Any Galois field has order a prime power p^q for prime p and positive q, and there is a Galois field GF(p^q) of any prime power order p^q that is *unique up to non-unique isomorphism*. Any Galois field GF(p^q) can be constructed as an **extension field** over a smaller Galois subfield GF(p^r), through the identification GF(p^q) = GF(p^r)[X] / \ for an *irreducible monic splitting polynomial* f(X) of degree q - r + 1 in the *polynomial ring* GF(p^r)[X]. -For example, GF(4) has order 22 and can be constructed as an extension field GF(2)[X] / \ where f(X) = X2 + X + 1 is an irreducible monic splitting quadratic polynomial in GF(2)[X]. +For example, GF(4) has order 2^2 and can be constructed as an extension field GF(2)[X] / \ where f(X) = X^2 + X + 1 is an irreducible monic splitting quadratic polynomial in GF(2)[X]. ## Example usage diff --git a/benchmarks/Main.hs b/benchmarks/Main.hs --- a/benchmarks/Main.hs +++ b/benchmarks/Main.hs @@ -16,7 +16,7 @@ data Pu instance IrreducibleMonic Fq Pu where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type Fq2 = ExtensionField Fq Pu fq2 :: Fq2 @@ -33,7 +33,7 @@ data Pv instance IrreducibleMonic Fq2 Pv where - split _ = x^3 - (9 + t x) + split _ = x ^ (3 :: Int) - (9 + t x) type Fq6 = ExtensionField Fq2 Pv fq6 :: Fq6 @@ -70,7 +70,7 @@ data Pw instance IrreducibleMonic Fq6 Pw where - split _ = x^2 - t x + split _ = x ^ (2 :: Int) - t x type Fq12 = ExtensionField Fq6 Pw fq12 :: Fq12 diff --git a/galois-field.cabal b/galois-field.cabal --- a/galois-field.cabal +++ b/galois-field.cabal @@ -2,12 +2,12 @@ -- -- see: https://github.com/sol/hpack -- --- hash: f66dcf977899a69f9f8785c75595dada58f68cbc1cb35faa4acb24b494fe26fe +-- hash: c0b59111dcbf4f45abd61925f0a42fbe4a93dce27b89db73519e0e36afc5e8a8 name: galois-field -version: 0.2.0 +version: 0.2.1 synopsis: Galois field library -description: Galois field library for cryptography research +description: An efficient implementation of Galois fields used in cryptography research category: Cryptography homepage: https://github.com/adjoint-io/galois-field#readme bug-reports: https://github.com/adjoint-io/galois-field/issues @@ -26,9 +26,10 @@ library exposed-modules: + BinaryField + ExtensionField GaloisField PrimeField - ExtensionField other-modules: PolynomialRing hs-source-dirs: @@ -50,8 +51,8 @@ other-modules: ExtensionFieldTests GaloisFieldTests - PolynomialRingTests PrimeFieldTests + BinaryField ExtensionField GaloisField PolynomialRing @@ -61,14 +62,13 @@ tests src default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses - ghc-options: -O2 -main-is Main + ghc-options: -O2 -Wall -main-is Main build-depends: MonadRandom , base >=4.7 && <5 , integer-gmp , protolude >=0.2 , tasty - , tasty-discover , tasty-quickcheck , wl-pprint-text default-language: Haskell2010 @@ -77,6 +77,7 @@ type: exitcode-stdio-1.0 main-is: Main.hs other-modules: + BinaryField ExtensionField GaloisField PolynomialRing @@ -86,7 +87,7 @@ benchmarks src default-extensions: LambdaCase RecordWildCards OverloadedStrings NoImplicitPrelude FlexibleInstances FlexibleContexts ScopedTypeVariables RankNTypes DataKinds DeriveGeneric GeneralizedNewtypeDeriving KindSignatures MultiParamTypeClasses - ghc-options: -O2 -main-is Main + ghc-options: -O2 -Wall -main-is Main build-depends: MonadRandom , base >=4.7 && <5 diff --git a/src/BinaryField.hs b/src/BinaryField.hs new file mode 100644 --- /dev/null +++ b/src/BinaryField.hs @@ -0,0 +1,108 @@ +module BinaryField + ( BinaryField + ) where + +import Protolude + +import Control.Monad.Random (Random(..), getRandom) +import Test.Tasty.QuickCheck (Arbitrary(..), choose) +import Text.PrettyPrint.Leijen.Text (Pretty(..)) + +import GaloisField (GaloisField(..)) + +-- | Binary fields @GF(2^q)[X]/\@ for @q@ positive and +-- @f(X)@ irreducible monic in @GF(2^q)[X]@ encoded as an integer. +newtype BinaryField (ib :: Nat) = BF Integer + deriving (Eq, Generic, NFData, Show) + +-- Binary fields are arbitrary. +instance KnownNat ib => Arbitrary (BinaryField ib) where + arbitrary = BF <$> choose (0, 2 ^ natVal (witness :: BinaryField ib) - 1) + +-- Binary fields are fields. +instance KnownNat ib => Fractional (BinaryField ib) where + recip y@(BF x) = case inv (natVal y) x of + Just z -> BF z + _ -> panic "no multiplicative inverse." + {-# INLINE recip #-} + fromRational (x:%y) = fromInteger x / fromInteger y + {-# INLINABLE fromRational #-} + +-- Binary fields are Galois fields. +instance KnownNat ib => GaloisField (BinaryField ib) where + char = const 2 + {-# INLINE char #-} + deg = bin . natVal + {-# INLINE deg #-} + frob = flip pow 2 + {-# INLINE frob #-} + pow = (^) + {-# INLINE pow #-} + rnd = getRandom + {-# INLINE rnd #-} + +-- Binary fields are fields. +instance KnownNat ib => Num (BinaryField ib) where + BF x + BF y = BF (xor x y) + {-# INLINE (+) #-} + BF x * BF y = fromInteger (mul x y) + {-# INLINE (*) #-} + BF x - BF y = BF (xor x y) + {-# INLINE (-) #-} + negate = identity + {-# INLINE negate #-} + fromInteger = BF . red (natVal (witness :: BinaryField ib)) + {-# INLINABLE fromInteger #-} + abs = panic "not implemented." + signum = panic "not implemented." + +-- Binary fields are pretty. +instance KnownNat ib => Pretty (BinaryField ib) where + pretty (BF x) = pretty x + +-- Binary fields are random. +instance KnownNat ib => Random (BinaryField ib) where + random = first BF . randomR (0, 2 ^ natVal (witness :: BinaryField ib) - 1) + randomR = panic "not implemented." + +-- Binary logarithm. +bin :: Integer -> Int +bin = logP 2 + where + logP :: Integer -> Integer -> Int + logP p x = let l = 2 * logP (p * p) x + in if x < p then 0 else log' l (quot x (p ^ l)) + where + log' :: Int -> Integer -> Int + log' q y = if y < p then q else log' (q + 1) (quot y p) +{-# INLINE bin #-} + +-- Binary multiplication. +mul :: Integer -> Integer -> Integer +mul x y = mul' (bin y) (if testBit y 0 then x else 0) + where + mul' :: Int -> Integer -> Integer + mul' 0 n = n + mul' l n = mul' (l - 1) (if testBit y l then xor n (shift x l) else n) +{-# INLINE mul #-} + +-- Binary reduction. +red :: Integer -> Integer -> Integer +red f = red' + where + red' :: Integer -> Integer + red' x = let n = bin x - bin f + in if n < 0 then x else red' (xor x (shift f n)) +{-# INLINE red #-} + +-- Binary inversion. +inv :: Integer -> Integer -> Maybe Integer +inv f x = case inv' 1 x 0 f of + (y, 1) -> Just y + _ -> Nothing + where + inv' :: Integer -> Integer -> Integer -> Integer -> (Integer, Integer) + inv' t r _ 0 = (t, r) + inv' t r t' r' = let q = max 0 (bin r - bin r') + in inv' t' r' (xor t (shift t' q)) (xor r (shift r' q)) +{-# INLINE inv #-} diff --git a/src/ExtensionField.hs b/src/ExtensionField.hs --- a/src/ExtensionField.hs +++ b/src/ExtensionField.hs @@ -10,57 +10,58 @@ import Protolude import Control.Monad.Random (Random(..), getRandom) -import Test.Tasty.QuickCheck (Arbitrary(..), choose, sized) +import Test.Tasty.QuickCheck (Arbitrary(..), vector) import Text.PrettyPrint.Leijen.Text (Pretty(..)) import GaloisField (GaloisField(..)) import PolynomialRing (Polynomial(..), cut, polyInv, polyMul, polyQR) --- | Irreducible monic splitting polynomial of extension field -class IrreducibleMonic k im where - {-# MINIMAL split #-} - split :: ExtensionField k im -> Polynomial k -- ^ Splitting polynomial - --- | Extension fields @GF(p^q)[X]/@ for @p@ prime, @q@ positive, and --- @f(X)@ irreducible monic in @GF(p^q)[X]@ +-- | Extension fields @GF(p^q)[X]/\@ for @p@ prime, @q@ positive, and +-- @f(X)@ irreducible monic in @GF(p^q)[X]@. newtype ExtensionField k im = EF (Polynomial k) deriving (Eq, Generic, NFData, Show) --- | Extension fields are arbitrary +-- | Irreducible monic splitting polynomial @f(X)@ of extension field. +class IrreducibleMonic k im where + {-# MINIMAL split #-} + -- | Splitting polynomial @f(X)@. + split :: ExtensionField k im -> Polynomial k + +-- Extension fields are arbitrary. instance (Arbitrary k, GaloisField k, IrreducibleMonic k im) => Arbitrary (ExtensionField k im) where - arbitrary = fromList <$> sized (const poly) + arbitrary = fromList <$> vector (length xs - 1) where - poly = choose (1, length xs - 1) >>= mapM (const arbitrary) . enumFromTo 1 - where - X xs = split (witness :: ExtensionField k im) + X xs = split (witness :: ExtensionField k im) --- | Extension fields are fields +-- Extension fields are fields. instance (GaloisField k, IrreducibleMonic k im) => Fractional (ExtensionField k im) where - recip (EF (X ys)) = case polyInv ys xs of + recip y@(EF (X ys)) = case polyInv ys xs of Just zs -> EF (X zs) _ -> panic "no multiplicative inverse." where - X xs = split (witness :: ExtensionField k im) + X xs = split y {-# INLINE recip #-} fromRational (y:%z) = fromInteger y / fromInteger z {-# INLINABLE fromRational #-} --- | Extension fields are Galois fields +-- Extension fields are Galois fields. instance (GaloisField k, IrreducibleMonic k im) => GaloisField (ExtensionField k im) where char = const (char (witness :: k)) {-# INLINE char #-} - deg = const (deg (witness :: k) * length xs - 1) + deg y = deg (witness :: k) * (length xs - 1) where - X xs = split (witness :: ExtensionField k im) + X xs = split y {-# INLINE deg #-} + frob = pow <*> char + {-# INLINE frob #-} pow y@(EF (X ys)) n | n < 0 = pow (recip y) (-n) | otherwise = EF (X (pow' [1] ys n)) where - X xs = split (witness :: ExtensionField k im) + X xs = split y mul = (.) (snd . flip polyQR xs) . polyMul pow' ws zs m | m == 0 = ws @@ -71,30 +72,30 @@ rnd = getRandom {-# INLINE rnd #-} --- | Extension fields are rings +-- Extension fields are rings. instance (GaloisField k, IrreducibleMonic k im) => Num (ExtensionField k im) where - EF y + EF z = EF (y + z) + EF y + EF z = EF (y + z) {-# INLINE (+) #-} - EF (X ys) * EF (X zs) = EF (X (snd (polyQR (polyMul ys zs) xs))) + y@(EF (X ys)) * EF (X zs) = EF (X (snd (polyQR (polyMul ys zs) xs))) where - X xs = split (witness :: ExtensionField k im) + X xs = split y {-# INLINE (*) #-} - EF y - EF z = EF (y - z) + EF y - EF z = EF (y - z) {-# INLINE (-) #-} - negate (EF y) = EF (-y) + negate (EF y) = EF (-y) {-# INLINE negate #-} - fromInteger = EF . fromInteger + fromInteger = EF . fromInteger {-# INLINABLE fromInteger #-} - abs = panic "not implemented." - signum = panic "not implemented." + abs = panic "not implemented." + signum = panic "not implemented." --- | Extension fields are pretty +-- Extension fields are pretty. instance (GaloisField k, IrreducibleMonic k im) => Pretty (ExtensionField k im) where pretty (EF y) = pretty y --- | Extension fields are random +-- Extension fields are random. instance (GaloisField k, IrreducibleMonic k im) => Random (ExtensionField k im) where random = first (EF . X . cut) . unfold (length xs - 1) [] @@ -104,12 +105,12 @@ let (y, g') = random g in unfold (n - 1) (y : ys) g' randomR = panic "not implemented." --- | List from field +-- | Convert from field element to list representation. fromField :: ExtensionField k im -> [k] fromField (EF (X xs)) = xs {-# INLINABLE fromField #-} --- | Field from list +-- | Convert from list representation to field element. fromList :: forall k im . (GaloisField k, IrreducibleMonic k im) => [k] -> ExtensionField k im fromList = EF . X . snd . flip polyQR xs . cut @@ -117,12 +118,12 @@ X xs = split (witness :: ExtensionField k im) {-# INLINABLE fromList #-} --- | Current indeterminate variable -x :: GaloisField k => Polynomial k -x = X [0, 1] -{-# INLINE x #-} - --- | Descend variable tower +-- | Descend tower of indeterminate variables. t :: Polynomial k -> Polynomial (ExtensionField k im) t = X . return . EF {-# INLINE t #-} + +-- | Current indeterminate variable. +x :: GaloisField k => Polynomial k +x = X [0, 1] +{-# INLINE x #-} diff --git a/src/GaloisField.hs b/src/GaloisField.hs --- a/src/GaloisField.hs +++ b/src/GaloisField.hs @@ -8,22 +8,31 @@ import Test.Tasty.QuickCheck (Arbitrary) import Text.PrettyPrint.Leijen.Text (Pretty) --- | Galois fields @GF(p^q)@ for @p@ prime and @q@ non-negative +-- | Galois fields @GF(p^q)@ for @p@ prime and @q@ non-negative. class (Arbitrary k, Eq k, Fractional k, Pretty k, Random k, Show k) => GaloisField k where - {-# MINIMAL char, deg, pow, rnd #-} + {-# MINIMAL char, deg, frob, pow, rnd #-} -- Characteristics - char :: k -> Integer -- ^ Characteristic @q@ of field - deg :: k -> Int -- ^ Degree @q@ of field + -- | Characteristic @p@ of field and order of prime subfield. + char :: k -> Integer - order :: k -> Integer -- ^ Order @p^q@ of field + -- | Degree @q@ of field as extension field over prime subfield. + deg :: k -> Int + + -- | Frobenius endomorphism @x->x^p@ of prime subfield. + frob :: k -> k + + -- | Order @p^q@ of field. + order :: k -> Integer order = (^) <$> char <*> deg {-# INLINE order #-} -- Functions - pow :: k -> Integer -> k -- @x@ to the power of @y@ - -- Randomisation - rnd :: MonadRandom m => m k -- ^ Random element of field + -- | Exponentiation @x@ to the power of @y@. + pow :: k -> Integer -> k + + -- | Randomised element @x@ of field. + rnd :: MonadRandom m => m k diff --git a/src/PrimeField.hs b/src/PrimeField.hs --- a/src/PrimeField.hs +++ b/src/PrimeField.hs @@ -12,33 +12,35 @@ import GaloisField (GaloisField(..)) --- | Prime fields @GF(p)@ for @p@ prime +-- | Prime fields @GF(p)@ for @p@ prime. newtype PrimeField (p :: Nat) = PF Integer deriving (Bits, Eq, Generic, NFData, Show) --- | Prime fields are arbitrary +-- Prime fields are arbitrary. instance KnownNat p => Arbitrary (PrimeField p) where arbitrary = fromInteger <$> arbitrary --- | Prime fields are fields +-- Prime fields are fields. instance KnownNat p => Fractional (PrimeField p) where recip y@(PF x) = PF (recipModInteger x (natVal y)) {-# INLINE recip #-} fromRational (x:%y) = fromInteger x / fromInteger y {-# INLINABLE fromRational #-} --- | Prime fields are Galois fields +-- Prime fields are Galois fields. instance KnownNat p => GaloisField (PrimeField p) where char = natVal {-# INLINE char #-} deg = const 1 {-# INLINE deg #-} + frob = identity + {-# INLINE frob #-} pow y@(PF x) n = PF (powModInteger x n (natVal y)) {-# INLINE pow #-} rnd = getRandom {-# INLINE rnd #-} --- | Prime fields are rings +-- Prime fields are rings. instance KnownNat p => Num (PrimeField p) where z@(PF x) + PF y = PF (if xyp >= 0 then xyp else xy) where @@ -61,16 +63,16 @@ abs = panic "not implemented." signum = panic "not implemented." --- | Prime fields are pretty +-- Prime fields are pretty. instance KnownNat p => Pretty (PrimeField p) where pretty (PF x) = pretty [x] --- | Prime fields are random +-- Prime fields are random. instance KnownNat p => Random (PrimeField p) where random = first PF . randomR (0, natVal (witness :: PrimeField p) - 1) randomR = panic "not implemented." --- | Embed to integers +-- | Embed field element to integers. toInt :: PrimeField p -> Integer toInt (PF x) = x {-# INLINABLE toInt #-} diff --git a/tests/ExtensionFieldTests.hs b/tests/ExtensionFieldTests.hs --- a/tests/ExtensionFieldTests.hs +++ b/tests/ExtensionFieldTests.hs @@ -3,6 +3,7 @@ import Protolude import ExtensionField +import PolynomialRing import Test.Tasty import GaloisFieldTests @@ -10,111 +11,112 @@ data P11 instance IrreducibleMonic FS2 P11 where - split _ = x^2 + x + 1 + split _ = x ^ (2 :: Int) + x + 1 type FS4 = ExtensionField FS2 P11 -test_S4 :: TestTree -test_S4 = fieldAxioms (Proxy :: Proxy FS4) "FS4" data P110 instance IrreducibleMonic FS2 P110 where - split _ = x^3 + x + 1 + split _ = x ^ (3 :: Int) + x + 1 type FS8 = ExtensionField FS2 P110 -test_S8 :: TestTree -test_S8 = fieldAxioms (Proxy :: Proxy FS8) "FS8" data P101 instance IrreducibleMonic FS2 P101 where - split _ = x^3 + x^2 + 1 + split _ = x ^ (3 :: Int) + x ^ (2 :: Int) + 1 type FS8' = ExtensionField FS2 P101 -test_S8' :: TestTree -test_S8' = fieldAxioms (Proxy :: Proxy FS8') "FS8'" data P10 instance IrreducibleMonic FS3 P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FS9 = ExtensionField FS3 P10 -test_S9 :: TestTree -test_S9 = fieldAxioms (Proxy :: Proxy FS9) "FS9" data P21 instance IrreducibleMonic FS3 P21 where - split _ = x^2 + x - 1 + split _ = x ^ (2 :: Int) + x - 1 type FS9' = ExtensionField FS3 P21 -test_S9' :: TestTree -test_S9' = fieldAxioms (Proxy :: Proxy FS9') "FS9'" data P22 instance IrreducibleMonic FS3 P22 where - split _ = x^2 - x - 1 + split _ = x ^ (2 :: Int) - x - 1 type FS9'' = ExtensionField FS3 P22 -test_S9'' :: TestTree -test_S9'' = fieldAxioms (Proxy :: Proxy FS9'') "FS9''" instance IrreducibleMonic FM0 P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FL0 = ExtensionField FM0 P10 -test_L0 :: TestTree -test_L0 = fieldAxioms (Proxy :: Proxy FL0) "FL0" instance IrreducibleMonic FM1 P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FL1 = ExtensionField FM1 P10 -test_L1 :: TestTree -test_L1 = fieldAxioms (Proxy :: Proxy FL1) "FL1" instance IrreducibleMonic FM2 P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FL2 = ExtensionField FM2 P10 -test_L2 :: TestTree -test_L2 = fieldAxioms (Proxy :: Proxy FL2) "FL2" instance IrreducibleMonic FM3 P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FL3 = ExtensionField FM3 P10 -test_L3 :: TestTree -test_L3 = fieldAxioms (Proxy :: Proxy FL3) "FL3" instance IrreducibleMonic FM4 P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FL4 = ExtensionField FM4 P10 -test_L4 :: TestTree -test_L4 = fieldAxioms (Proxy :: Proxy FL4) "FL4" instance IrreducibleMonic FVL P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FV2 = ExtensionField FVL P10 -test_V2 :: TestTree -test_V2 = fieldAxioms (Proxy :: Proxy FV2) "FV2" instance IrreducibleMonic FXL P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FX2 = ExtensionField FXL P10 -test_X2 :: TestTree -test_X2 = fieldAxioms (Proxy :: Proxy FX2) "FX2" instance IrreducibleMonic FZL P10 where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type FZ2 = ExtensionField FZL P10 -test_Z2 :: TestTree -test_Z2 = fieldAxioms (Proxy :: Proxy FZ2) "FZ2" data Pu instance IrreducibleMonic Fq Pu where - split _ = x^2 + 1 + split _ = x ^ (2 :: Int) + 1 type Fq2 = ExtensionField Fq Pu -test_Fq2 :: TestTree -test_Fq2 = fieldAxioms (Proxy :: Proxy Fq2) "Fq2" data Pv instance IrreducibleMonic Fq2 Pv where - split _ = x^3 - (9 + t x) + split _ = x ^ (3 :: Int) - (9 + t x) type Fq6 = ExtensionField Fq2 Pv -test_Fq6 :: TestTree -test_Fq6 = fieldAxioms (Proxy :: Proxy Fq6) "Fq6" data Pw instance IrreducibleMonic Fq6 Pw where - split _ = x^2 - t x + split _ = x ^ (2 :: Int) - t x type Fq12 = ExtensionField Fq6 Pw -test_Fq12 :: TestTree -test_Fq12 = fieldAxioms (Proxy :: Proxy Fq12) "Fq12" + +testExtensionField :: TestTree +testExtensionField = testGroup "Extension fields" + [ testGroup "Polynomial rings" + [ ringAxioms "FS2[X]" (witness :: Polynomial FS2) + , ringAxioms "FS3[X]" (witness :: Polynomial FS3) + , ringAxioms "FS5[X]" (witness :: Polynomial FS5) + , ringAxioms "FS7[X]" (witness :: Polynomial FS7) + , ringAxioms "FM0[X]" (witness :: Polynomial FM0) + , ringAxioms "FM1[X]" (witness :: Polynomial FM1) + , ringAxioms "FM2[X]" (witness :: Polynomial FM2) + , ringAxioms "FM3[X]" (witness :: Polynomial FM3) + , ringAxioms "FM4[X]" (witness :: Polynomial FM4) + , ringAxioms "FVL[X]" (witness :: Polynomial FVL) + , ringAxioms "FXL[X]" (witness :: Polynomial FXL) + , ringAxioms "FZL[X]" (witness :: Polynomial FZL) + ] + , fieldAxioms "FS4" (witness :: FS4 ) + , fieldAxioms "FS8" (witness :: FS8 ) + , fieldAxioms "FS8'" (witness :: FS8' ) + , fieldAxioms "FS9" (witness :: FS9 ) + , fieldAxioms "FS9'" (witness :: FS9' ) + , fieldAxioms "FS9''" (witness :: FS9'') + , fieldAxioms "FL0" (witness :: FL0 ) + , fieldAxioms "FL1" (witness :: FL1 ) + , fieldAxioms "FL2" (witness :: FL2 ) + , fieldAxioms "FL3" (witness :: FL3 ) + , fieldAxioms "FL4" (witness :: FL4 ) + , fieldAxioms "FV2" (witness :: FV2 ) + , fieldAxioms "FX2" (witness :: FX2 ) + , fieldAxioms "FZ2" (witness :: FZ2 ) + , fieldAxioms "Fq2" (witness :: Fq2 ) + , fieldAxioms "Fq6" (witness :: Fq6 ) + , fieldAxioms "Fq12" (witness :: Fq12 ) + ] diff --git a/tests/GaloisFieldTests.hs b/tests/GaloisFieldTests.hs --- a/tests/GaloisFieldTests.hs +++ b/tests/GaloisFieldTests.hs @@ -22,8 +22,8 @@ inverses op inv e x = op x (inv x) == e && op (inv x) x == e ringAxioms :: forall r . (Arbitrary r, Eq r, Num r, Show r) - => Proxy r -> TestName -> TestTree -ringAxioms _ str = testGroup ("Test ring axioms of " <> str) + => TestName -> r -> TestTree +ringAxioms s _ = testGroup ("Ring axioms of " <> s) [ testProperty "commutativity of addition" $ commutativity ((+) :: r -> r -> r) , testProperty "commutativity of multiplication" @@ -43,9 +43,9 @@ ] fieldAxioms :: forall k . (Arbitrary k, Eq k, Fractional k, Show k) - => Proxy k -> TestName -> TestTree -fieldAxioms p str = testGroup ("Test field axioms of " <> str) - [ ringAxioms p str + => TestName -> k -> TestTree +fieldAxioms s k = testGroup ("Field axioms of " <> s) + [ ringAxioms s k , testProperty "multiplicative inverses" $ \n -> n /= 0 ==> inverses ((*) :: k -> k -> k) recip 1 n ] diff --git a/tests/Main.hs b/tests/Main.hs --- a/tests/Main.hs +++ b/tests/Main.hs @@ -1,1 +1,11 @@ -{-# OPTIONS_GHC -F -pgmF tasty-discover -optF --tree-display #-} +module Main where + +import Protolude + +import Test.Tasty + +import ExtensionFieldTests +import PrimeFieldTests + +main :: IO () +main = defaultMain $ testGroup "Tests" [testPrimeField, testExtensionField] diff --git a/tests/PolynomialRingTests.hs b/tests/PolynomialRingTests.hs deleted file mode 100644 --- a/tests/PolynomialRingTests.hs +++ /dev/null @@ -1,45 +0,0 @@ -module PolynomialRingTests where - -import Protolude - -import PolynomialRing -import Test.Tasty - -import PrimeFieldTests -import GaloisFieldTests - -test_S2X :: TestTree -test_S2X = ringAxioms (Proxy :: Proxy (Polynomial FS2)) "FS2[X]" - -test_S3X :: TestTree -test_S3X = ringAxioms (Proxy :: Proxy (Polynomial FS3)) "FS3[X]" - -test_S5X :: TestTree -test_S5X = ringAxioms (Proxy :: Proxy (Polynomial FS5)) "FS5[X]" - -test_S7X :: TestTree -test_S7X = ringAxioms (Proxy :: Proxy (Polynomial FS7)) "FS7[X]" - -test_M0X :: TestTree -test_M0X = ringAxioms (Proxy :: Proxy (Polynomial FM0)) "FM0[X]" - -test_M1X :: TestTree -test_M1X = ringAxioms (Proxy :: Proxy (Polynomial FM1)) "FM1[X]" - -test_M2X :: TestTree -test_M2X = ringAxioms (Proxy :: Proxy (Polynomial FM2)) "FM2[X]" - -test_M3X :: TestTree -test_M3X = ringAxioms (Proxy :: Proxy (Polynomial FM3)) "FM3[X]" - -test_M4X :: TestTree -test_M4X = ringAxioms (Proxy :: Proxy (Polynomial FM4)) "FM4[X]" - -test_VLX :: TestTree -test_VLX = ringAxioms (Proxy :: Proxy (Polynomial FVL)) "FVL[X]" - -test_XLX :: TestTree -test_XLX = ringAxioms (Proxy :: Proxy (Polynomial FXL)) "FXL[X]" - -test_ZLX :: TestTree -test_ZLX = ringAxioms (Proxy :: Proxy (Polynomial FZL)) "FZL[X]" diff --git a/tests/PrimeFieldTests.hs b/tests/PrimeFieldTests.hs --- a/tests/PrimeFieldTests.hs +++ b/tests/PrimeFieldTests.hs @@ -24,39 +24,18 @@ type Fq = PrimeField 21888242871839275222246405745257275088696311157297823662689037894645226208583 - -test_S2 :: TestTree -test_S2 = fieldAxioms (Proxy :: Proxy FS2) "FS2" - -test_S3 :: TestTree -test_S3 = fieldAxioms (Proxy :: Proxy FS3) "FS3" - -test_S5 :: TestTree -test_S5 = fieldAxioms (Proxy :: Proxy FS5) "FS5" - -test_S7 :: TestTree -test_S7 = fieldAxioms (Proxy :: Proxy FS7) "FS7" - -test_M0 :: TestTree -test_M0 = fieldAxioms (Proxy :: Proxy FM0) "FM0" - -test_M1 :: TestTree -test_M1 = fieldAxioms (Proxy :: Proxy FM1) "FM1" - -test_M2 :: TestTree -test_M2 = fieldAxioms (Proxy :: Proxy FM2) "FM2" - -test_M3 :: TestTree -test_M3 = fieldAxioms (Proxy :: Proxy FM3) "FM3" - -test_M4 :: TestTree -test_M4 = fieldAxioms (Proxy :: Proxy FM4) "FM4" - -test_VL :: TestTree -test_VL = fieldAxioms (Proxy :: Proxy FVL) "FVL" - -test_XL :: TestTree -test_XL = fieldAxioms (Proxy :: Proxy FXL) "FXL" - -test_ZL :: TestTree -test_ZL = fieldAxioms (Proxy :: Proxy FZL) "FZL" +testPrimeField :: TestTree +testPrimeField = testGroup "Prime fields" + [ fieldAxioms "FS2" (witness :: FS2) + , fieldAxioms "FS3" (witness :: FS3) + , fieldAxioms "FS5" (witness :: FS5) + , fieldAxioms "FS7" (witness :: FS7) + , fieldAxioms "FM0" (witness :: FM0) + , fieldAxioms "FM1" (witness :: FM1) + , fieldAxioms "FM2" (witness :: FM2) + , fieldAxioms "FM3" (witness :: FM3) + , fieldAxioms "FM4" (witness :: FM4) + , fieldAxioms "FVL" (witness :: FVL) + , fieldAxioms "FXL" (witness :: FXL) + , fieldAxioms "FZL" (witness :: FZL) + ]