quadratic extension of the Goldilocks field

GoldilocksExt.hs

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+
+-- | Quadratic extension of the Goldilocks field
+--
+-- The field is defined as @F[X] / (X^2 - 7)@ 
+--
+-- (@X^2 - 7@ is the smallest such irreducible polynomial over Goldilocks)
+--
+
+module GoldilocksExt where
+
+--------------------------------------------------------------------------------
+
+import Data.Bits
+import Data.Ratio
+
+import Goldilocks
+
+--------------------------------------------------------------------------------
+
+type FExt = GoldilocksExt
+
+data GoldilocksExt 
+  = MkExt !Goldilocks !Goldilocks
+  deriving Eq
+
+instance Show GoldilocksExt where
+  show (MkExt real imag) = "(" ++ show real ++ " + X*" ++ show imag ++ ")"
+
+--------------------------------------------------------------------------------
+
+instance Num FExt where
+  fromInteger k = MkExt (fromInteger k) 0
+  negate (MkExt r i) = MkExt (negate r) (negate i)
+  (+) (MkExt r1 i1) (MkExt r2 i2) = MkExt (r1+r2) (i1+i2)
+  (-) (MkExt r1 i1) (MkExt r2 i2) = MkExt (r1-r2) (i1-i2)
+  (*) (MkExt r1 i1) (MkExt r2 i2) = MkExt (r1*r2 + 7*i1*i2) (r1*i2 + r2*i1)
+  signum = const 1
+  abs    = id
+
+instance Fractional FExt where
+  fromRational q = fromInteger (numerator q) / fromInteger (denominator q)
+  recip = invExt
+  (/)   = divExt
+
+--------------------------------------------------------------------------------
+
+sqrExt :: FExt -> FExt
+sqrExt x = x*x
+
+invExt :: FExt -> FExt
+invExt (MkExt a b) = MkExt c d where
+  denom = inv (a*a - 7*b*b)
+  c =        a * denom
+  d = negate b * denom
+
+divExt :: FExt -> FExt -> FExt
+divExt u v = u * invExt v
+
+--------------------------------------------------------------------------------
+
+powExt :: GoldilocksExt -> Integer -> GoldilocksExt
+powExt x e 
+  | e == 0    = 1
+  | e <  0    = powExt (invExt x) (negate e)
+  | otherwise = go 1 x e
+  where
+    go !acc _  0     = acc
+    go !acc !s !expo = case expo .&. 1 of
+      0 -> go acc     (sqrExt s) (shiftR expo 1)
+      _ -> go (acc*s) (sqrExt s) (shiftR expo 1)
+
+--------------------------------------------------------------------------------