diff --git a/reference/src/Class/Field.hs b/reference/src/Class/Field.hs
new file mode 100644
index 0000000..27fbfe2
--- /dev/null
+++ b/reference/src/Class/Field.hs
@@ -0,0 +1,40 @@
+
+{-# LANGUAGE TypeFamilies #-}
+module Class.Field where
+
+--------------------------------------------------------------------------------
+
+import Data.Kind
+import Data.Proxy
+
+import System.Random
+
+--------------------------------------------------------------------------------
+
+class (Show a, Eq a, Num a) => Ring a where
+  zero      :: a
+  one       :: a
+  isZero    :: a -> Bool
+  isOne     :: a -> Bool
+  square    :: a -> a
+  power     :: a -> Integer -> a
+  power_    :: a -> Int     -> a
+
+class (Ring a, Fractional a) => Field a where
+  inverse   :: a -> a
+  inverse = recip
+
+class Field a => FiniteField a where
+  rndIO     :: IO a
+  fieldSize :: Proxy a -> Integer
+
+-- | Quadratic extensions
+class (Field (BaseField ext), Field ext) => QuadraticExt ext where
+  type BaseField ext :: Type
+  inject  :: BaseField ext -> ext
+  project :: ext -> Maybe (BaseField ext)
+  scale   :: BaseField ext -> ext -> ext
+  quadraticPack   :: (BaseField ext, BaseField ext) -> ext
+  quadraticUnpack :: ext -> (BaseField ext, BaseField ext)
+
+--------------------------------------------------------------------------------
diff --git a/reference/src/Class/Flat.hs b/reference/src/Class/Flat.hs
new file mode 100644
index 0000000..29b640a
--- /dev/null
+++ b/reference/src/Class/Flat.hs
@@ -0,0 +1,7 @@
+
+module Class.Flat 
+  ( module Data.Flat.Class 
+  )
+  where
+
+import Data.Flat.Class
diff --git a/reference/src/Class/NTT.hs b/reference/src/Class/NTT.hs
new file mode 100644
index 0000000..3def71d
--- /dev/null
+++ b/reference/src/Class/NTT.hs
@@ -0,0 +1,40 @@
+
+module NTT.Class where
+
+--------------------------------------------------------------------------------
+
+import Data.Kind
+
+import Class.Field
+import Class.Poly
+
+import NTT.Subgroup
+
+--------------------------------------------------------------------------------
+
+{-
+class NTT ntt where
+  type Field ntt :: Type
+  type Poly  ntt :: Type
+
+  forwardNTT        :: Subgroup F -> Poly F -> FlatArray F
+  inverseNTT        :: Subgroup F -> FlatArray F -> Poly F
+  shiftedForwardNTT :: Subgroup F -> F -> Poly F -> FlatArray F
+  shiftedInverseNTT :: Subgroup F -> F -> FlatArray F -> Poly F
+  asymmForwardNTT   :: Subgroup F -> Poly F -> Subgroup F -> FlatArray F
+-}
+
+{-
+-- | Polynomials which over an FFT-friend field support NTT operations
+class (Univariate p, FFTField (Coeff p)) => UnivariateFFT p where
+  -- | Number-theoretical transform (evaluate on a subgroup)
+  ntt  :: FFTSubgroup (Coeff p) -> p -> FlatArray (Coeff p) 
+  -- | Inverse number-theoretical transform (interpolate on a subgroup)
+  intt :: FFTSubgroup (Coeff p) -> FlatArray (Coeff p) -> p
+  -- | Shifts @f@ by @eta@, evaluating @f(eta*omega^k)@
+  shiftedNTT  :: FFTSubgroup (Coeff p) -> Coeff p -> p -> FlatArray (Coeff p) 
+  -- | Shifts @f@ by @eta^-1@, interpolating @f@ so that @f(eta^-1 * omega^k) = y_k@
+  shiftedINTT :: FFTSubgroup (Coeff p) -> Coeff p -> FlatArray (Coeff p) -> p
+  -- | Evaluate on a larger subgroup than the polynomial is defined on
+  asymmNTT :: FFTSubgroup (Coeff p) -> p -> FFTSubgroup (Coeff p) -> FlatArray (Coeff p) 
+-}
diff --git a/reference/src/Class/Poly.hs b/reference/src/Class/Poly.hs
new file mode 100644
index 0000000..bcacb2f
--- /dev/null
+++ b/reference/src/Class/Poly.hs
@@ -0,0 +1,99 @@
+
+-- | Univariate polynomials
+
+module Class.Poly where
+
+--------------------------------------------------------------------------------
+
+import Data.Kind
+
+import Class.Field
+
+--------------------------------------------------------------------------------
+-- * Dense univariate polynomials over (finite) fields
+
+class (Ring p, Field (Coeff p)) => Univariate p where
+  -- | the type of coefficients
+  type Coeff p :: Type
+  -- | Degree
+  degree :: p -> Int
+  -- | Size (can be larger than degree+1 if the top coefficients are zeor)
+  polySize  :: p -> Int
+  -- | The k-th coefficient
+  kthCoeff :: Int -> p -> Coeff p
+  -- | Evaluation
+  evalAt :: Coeff p -> p -> Coeff p
+  -- | Scaling
+  scale :: Coeff p -> p -> p
+  -- | Create a polynomial from coefficiens
+  mkPoly :: [Coeff p] -> p
+  -- | Coefficients of the polynomial as a list
+  coeffs :: p -> [Coeff p]
+  -- | Coefficients as an Array
+  coeffsArr :: p -> Array Int (Coeff p)
+{-
+  -- | Polynomial long division
+  polyLongDiv :: p -> p -> (p,p)
+  -- | Polynomial quotient
+  polyQuot :: p -> p -> p
+  -- | Polynomial remainder
+  polyRem :: p -> p -> p
+  -- | Divide by the coset vanishing polynomial @(x^n - eta)@
+  divByVanishing :: p -> (Int, Coeff p) -> (p,p)
+  -- | Quotient by the coset vanishing polynomial @(x^n - eta)@
+  quotByVanishing :: p -> (Int, Coeff p) -> Maybe p
+-}
+
+--------------------------------------------------------------------------------
+-- * Some generic functions
+
+-- | Checks whether the input is the constant one polynomial?
+polyIsOne :: Univariate p => p -> Bool
+polyIsOne p = case mbConst p of
+  Nothing -> False
+  Just x  -> ZK.Algebra.Class.Field.isOne x
+
+-- | The constant term of a polynomial
+constTermOf :: Univariate p => p -> Coeff p
+constTermOf = kthCoeff 0
+
+-- | Is this a constant polynomial? 
+--
+-- TODO: this is not efficient for high-degree polynomials, where we could exit early...
+mbConst :: Univariate p => p -> Maybe (Coeff p)
+mbConst p = if degree p <= 0 then Just (constTermOf p) else Nothing
+
+-- | Create a constant polynomial
+constPoly :: Univariate p => Coeff p -> p
+constPoly y = mkPoly [y]
+
+-- | The polynomial @p(x)=x@
+idPoly :: Univariate p => p
+idPoly = mkPoly [0,1]
+
+-- | Create a linear polynomial.
+-- 
+-- > linearPoly a b == a*x + b@
+--
+linearPoly :: Univariate p => Coeff p -> Coeff p -> p
+linearPoly a b = mkPoly [b,a]
+
+showPoly :: Univariate p => p -> String
+showPoly = showPoly' True
+
+showPoly' :: Univariate p => Bool -> p -> String
+showPoly' newlines_flag poly =
+  case newlines_flag of
+    False -> intercalate " +"   terms
+    True  -> intercalate " +\n" terms
+  where
+    pairs  = filter (\kx -> snd kx /= 0) 
+           $ zip [0..] (coeffs poly)
+    terms  = case pairs of 
+               [] -> [" 0"]
+               _  -> map f pairs
+    f (0,x) = ' ' : show x
+    f (1,x) = ' ' : show x ++ " * x"
+    f (k,x) = ' ' : show x ++ " * x^" ++ show k
+
+--------------------------------------------------------------------------------
\ No newline at end of file
diff --git a/reference/src/Class/Vector.hs b/reference/src/Class/Vector.hs
new file mode 100644
index 0000000..5d28d59
--- /dev/null
+++ b/reference/src/Class/Vector.hs
@@ -0,0 +1,112 @@
+
+{-# LANGUAGE TypeFamilies #-}
+module Class.Vector where
+
+--------------------------------------------------------------------------------
+
+import Data.Kind
+
+import Data.Array.IArray
+
+import Class.Field
+import Misc
+
+--------------------------------------------------------------------------------
+
+class Vector v where
+
+  type VecElem v :: Type
+
+  vecLength :: v -> Int
+  vecPeek   :: v -> Int -> VecElem v
+  
+  vecToList    :: v -> [VecElem v]
+  vecFromList  :: [VecElem v] -> v
+
+  vecToArray   :: v -> Array Int (VecElem v)
+  vecFromArray :: Array Int (VecElem v) -> v
+
+  vecTake   :: Int -> v -> v
+  vecDrop   :: Int -> v -> v
+  vecAppend :: v -> v -> v
+  vecConcat :: [v] -> v
+  vecMap    :: (VecElem v -> VecElem v) -> v -> v
+
+vecSize :: Vector v => v -> Int
+vecSize = vecLength
+
+--------------------------------------------------------------------------------
+
+instance Vector [a] where
+  type VecElem [a] = a
+  vecLength    = length
+  vecPeek      = (!!)
+  vecToList    = id
+  vecFromList  = id
+  vecToArray   = listToArray
+  vecFromArray = elems
+  vecTake      = take
+  vecDrop      = drop
+  vecAppend    = (++)
+  vecConcat    = concat
+  vecMap       = map
+
+instance Vector (Array Int a) where
+  type VecElem (Array Int a) = a
+  vecLength    = \arr -> let (0,n1) = bounds arr in n1+1
+  vecPeek      = (!)
+  vecToList    = elems
+  vecFromList  = listToArray
+  vecToArray   = id
+  vecFromArray = id
+  vecTake      = takeArray
+  vecDrop      = dropArray
+  vecAppend    = appendArrays
+  vecConcat    = concatArrays
+  vecMap       = amap
+
+--------------------------------------------------------------------------------
+-- * Pointwise operations
+
+class PointwiseGroup a where
+  -- | Pointwise negation
+  pwNeg :: a -> a 
+  -- | Pointwise addition
+  pwAdd :: a -> a -> a
+  -- | Pointwise subtraction
+  pwSub :: a -> a -> a
+
+class PointwiseGroup a => PointwiseRing a where
+  -- | Pointwise squaring
+  pwSqr :: a -> a
+  -- | Pointwise multiplication
+  pwMul :: a -> a -> a
+  -- | Pointwise @a*b+c@
+  pwMulAdd :: a -> a -> a -> a
+  -- | Pointwise @a*b-c@
+  pwMulSub :: a -> a -> a -> a
+
+class PointwiseRing a => PointwiseField a where
+  -- | Pointwise inversion
+  pwInv :: a -> a
+  -- | Pointwise division
+  pwDiv :: a -> a -> a
+
+--------------------------------------------------------------------------------
+
+-- | Finite dimensional vector spaces
+class (Vector v, Field (VecElem v), PointwiseField v) => VectorSpace v where
+  -- | Scaling by an element
+  vecScale :: VecElem v -> v -> v
+  -- | Dot product
+  dotProd :: v -> v -> VecElem v
+  -- | The array @[ a*b^k | k<-[0..n-1] ]@
+  powers :: VecElem v -> VecElem v -> Int -> v
+  -- | Pointwise multiplication by the array @[ a*b^k | k<-[0..n-1] ]@
+  mulByPowers :: VecElem v -> VecElem v -> v -> v
+  -- | Linear combination @a*x + y@
+  linComb1 :: (VecElem v, v) -> v -> v
+  -- | Linear combination @a*x + b*y@
+  linComb2 :: (VecElem v, v) -> (VecElem v, v) -> v
+
+--------------------------------------------------------------------------------
diff --git a/reference/src/Data/Flat/Array.hs b/reference/src/Data/Flat/Array.hs
index 45ff599..674aa3b 100644
--- a/reference/src/Data/Flat/Array.hs
+++ b/reference/src/Data/Flat/Array.hs
@@ -25,6 +25,7 @@ import System.IO
 import System.IO.Unsafe
 
 import Data.Flat.Class
+import Class.Vector
 import Misc
 
 --------------------------------------------------------------------------------
@@ -33,7 +34,7 @@ import Misc
 -- foreign memory (not managed by the Haskell runtime). 
 --
 -- Note: the @Int@ means the number of objects in the array.
-data FlatArray a 
+data FlatArray (a :: Type) 
   = MkFlatArray !Int !(ForeignPtr Word64)
   deriving Show
 
@@ -128,6 +129,14 @@ dropFlatArrayIO k (MkFlatArray n fptr1) = do
       when (m>0) $ copyBytes ptr2 src (8*sz*m)
   return (MkFlatArray m fptr2)
 
+{-# NOINLINE mapFlatArrayIO #-}
+mapFlatArrayIO :: (a -> a) -> FlatArray a -> IO (FlatArray a)
+mapFlatArrayIO = error "mapFlatArrayIO: not yet implemented"
+
+{-# NOINLINE mapFlatArray #-}
+mapFlatArray :: (a -> a) -> FlatArray a -> FlatArray a
+mapFlatArray f arr = unsafePerformIO (mapFlatArrayIO f arr)
+
 ----------------------------------------
 
 -- | Read a flat array from a raw binary file. The size of the file determines the length of the array.
@@ -214,3 +223,20 @@ unpackFlatArrayToListIO (MkFlatArray len fptr) = do
       makeFlat src
 
 --------------------------------------------------------------------------------
+
+instance Flat a => Vector (FlatArray a) where
+
+  type VecElem (FlatArray a) = a
+
+  vecLength = flatArrayLength
+  vecPeek   = peekFlatArray
+  
+  vecToList    = unpackFlatArrayToList
+  vecFromList  = packFlatArrayFromList
+
+  vecToArray   = unpackFlatArray
+  vecFromArray = packFlatArray 
+
+  vecAppend = error "Vector/vecAppend/FlatArray: not implemented"
+  vecConcat = error "Vector/vecAConcat/FlatArray: not implemented"
+  vecMap    = mapFlatArray
diff --git a/reference/src/Field/Class.hs b/reference/src/Field/Class.hs
deleted file mode 100644
index bade3a7..0000000
--- a/reference/src/Field/Class.hs
+++ /dev/null
@@ -1,77 +0,0 @@
-
-{-# LANGUAGE TypeFamilies #-}
-module Field.Class where
-
---------------------------------------------------------------------------------
-
-import Data.Kind
-import Data.Proxy
-
-import System.Random
-
-import qualified Field.Goldilocks           as Goldi   
-import qualified Field.Goldilocks.Extension as GoldiExt
-
---------------------------------------------------------------------------------
-
-class (Show a, Eq a, Num a, Fractional a) => Field a where
-  fieldSize :: Proxy a -> Integer
-  zero      :: a
-  one       :: a
-  isZero    :: a -> Bool
-  isOne     :: a -> Bool
-  square    :: a -> a
-  power     :: a -> Integer -> a
-  power_    :: a -> Int     -> a
-  rndIO     :: IO a
-
-inverse :: Field a => a -> a
-inverse = recip
-
--- | Quadratic extensions
-class (Field (BaseField ext), Field ext) => QuadraticExt ext where
-  type BaseField ext :: Type
-  inject  :: BaseField ext -> ext
-  project :: ext -> Maybe (BaseField ext)
-  scale   :: BaseField ext -> ext -> ext
-  quadraticPack   :: (BaseField ext, BaseField ext) -> ext
-  quadraticUnpack :: ext -> (BaseField ext, BaseField ext)
-
---------------------------------------------------------------------------------
-
-instance Field Goldi.F where
-  fieldSize _ = Goldi.goldilocksPrime
-  zero        = Goldi.zero
-  one         = Goldi.one
-  isZero      = Goldi.isZero
-  isOne       = Goldi.isOne
-  square      = Goldi.sqr
-  power       = Goldi.pow
-  power_      = Goldi.pow_
-  rndIO       = randomIO
-
---------------------------------------------------------------------------------
-
-instance Field GoldiExt.FExt where
-  fieldSize _ = (Goldi.goldilocksPrime ^ 2)
-  zero        = GoldiExt.zero
-  one         = GoldiExt.one
-  isZero      = GoldiExt.isZero
-  isOne       = GoldiExt.isOne
-  square      = GoldiExt.sqr
-  power       = GoldiExt.pow
-  power_      = GoldiExt.pow_
-  rndIO       = randomIO
-
---------------------------------------------------------------------------------
-
-instance QuadraticExt GoldiExt.FExt where
-  type BaseField GoldiExt.FExt = Goldi.F
-
-  inject  = GoldiExt.inj
-  project = GoldiExt.proj
-  scale   = GoldiExt.scl
-  quadraticPack   = GoldiExt.pack
-  quadraticUnpack = GoldiExt.unpack
-
---------------------------------------------------------------------------------
diff --git a/reference/src/Field/Goldilocks/Extension/BindC.hs b/reference/src/Field/Goldilocks/Extension/BindC.hs
index 88f7ab8..08c07b8 100644
--- a/reference/src/Field/Goldilocks/Extension/BindC.hs
+++ b/reference/src/Field/Goldilocks/Extension/BindC.hs
@@ -5,7 +5,7 @@
 -- with the fast Goldilocks base field operations, but the C versions should be useful 
 -- for the vector operations, and this way we can test them easily.
 
-{-# LANGUAGE ForeignFunctionInterface, BangPatterns, NumericUnderscores #-}
+{-# LANGUAGE ForeignFunctionInterface, BangPatterns, NumericUnderscores, TypeFamilies #-}
 module Field.Goldilocks.Extension.BindC where
 
 --------------------------------------------------------------------------------
@@ -31,10 +31,11 @@ import Data.Binary.Put ( putWord64le )
 
 import Text.Printf
 
-import Field.Goldilocks ( F , Goldilocks(..) )
-import qualified Field.Goldilocks as Goldi
+import Field.Goldilocks.Fast ( F , Goldilocks(..) )
+import qualified Field.Goldilocks.Fast as Goldi
 
 import Data.Flat
+import Class.Field
 
 --------------------------------------------------------------------------------
 
@@ -103,6 +104,29 @@ instance Random F2 where
               in  (F2 x y, g'')
   randomR = error "randomR/F2: doesn't make any sense"
 
+instance Ring FExt where
+  zero        = Field.Goldilocks.Extension.BindC.zero
+  one         = Field.Goldilocks.Extension.BindC.one
+  isZero      = Field.Goldilocks.Extension.BindC.isZero
+  isOne       = Field.Goldilocks.Extension.BindC.isOne
+  square      = Field.Goldilocks.Extension.BindC.sqr
+  power       = Field.Goldilocks.Extension.BindC.pow
+  power_      = Field.Goldilocks.Extension.BindC.pow_
+
+instance Field FExt
+
+instance FiniteField FExt where
+  fieldSize _ = (Goldi.goldilocksPrime ^ 2)
+  rndIO       = randomIO
+
+instance QuadraticExt FExt where
+  type BaseField FExt = Goldi.F
+  inject          = Field.Goldilocks.Extension.BindC.inj
+  project         = Field.Goldilocks.Extension.BindC.proj
+  scale           = Field.Goldilocks.Extension.BindC.scl
+  quadraticPack   = Field.Goldilocks.Extension.BindC.pack
+  quadraticUnpack = Field.Goldilocks.Extension.BindC.unpack
+
 --------------------------------------------------------------------------------
 
 zero, one, two :: F2
diff --git a/reference/src/Field/Goldilocks/Extension/Haskell.hs b/reference/src/Field/Goldilocks/Extension/Haskell.hs
index b182b18..99e0e6f 100644
--- a/reference/src/Field/Goldilocks/Extension/Haskell.hs
+++ b/reference/src/Field/Goldilocks/Extension/Haskell.hs
@@ -3,6 +3,7 @@
 --
 -- We use the irreducble polynomial @x^2 - 7@ to be compatible with Plonky3
 
+{-# LANGUAGE TypeFamilies #-}
 module Field.Goldilocks.Extension.Haskell where
 
 --------------------------------------------------------------------------------
@@ -20,10 +21,11 @@ import Foreign.Marshal
 
 import Data.Binary
 
-import Data.Flat
+import Field.Goldilocks.Slow ( F )
+import qualified Field.Goldilocks.Slow as Goldi
 
-import Field.Goldilocks ( F )
-import qualified Field.Goldilocks as Goldi
+import Data.Flat
+import Class.Field
 
 --------------------------------------------------------------------------------
 
@@ -92,6 +94,29 @@ instance Random F2 where
               in  (F2 x y, g'')
   randomR = error "randomR/F2: doesn't make any sense"
 
+instance Ring FExt where
+  zero        = Field.Goldilocks.Extension.Haskell.zero
+  one         = Field.Goldilocks.Extension.Haskell.one
+  isZero      = Field.Goldilocks.Extension.Haskell.isZero
+  isOne       = Field.Goldilocks.Extension.Haskell.isOne
+  square      = Field.Goldilocks.Extension.Haskell.sqr
+  power       = Field.Goldilocks.Extension.Haskell.pow
+  power_      = Field.Goldilocks.Extension.Haskell.pow_
+
+instance Field FExt 
+
+instance FiniteField FExt where
+  fieldSize _ = (Goldi.goldilocksPrime ^ 2)
+  rndIO       = randomIO
+
+instance QuadraticExt FExt where
+  type BaseField FExt = Goldi.F
+  inject          = Field.Goldilocks.Extension.Haskell.inj
+  project         = Field.Goldilocks.Extension.Haskell.proj
+  scale           = Field.Goldilocks.Extension.Haskell.scl
+  quadraticPack   = Field.Goldilocks.Extension.Haskell.pack
+  quadraticUnpack = Field.Goldilocks.Extension.Haskell.unpack
+
 --------------------------------------------------------------------------------
 
 zero, one, two :: F2
diff --git a/reference/src/Field/Goldilocks/Fast.hs b/reference/src/Field/Goldilocks/Fast.hs
index e04b80a..7cb2578 100644
--- a/reference/src/Field/Goldilocks/Fast.hs
+++ b/reference/src/Field/Goldilocks/Fast.hs
@@ -27,6 +27,7 @@ import Data.Binary.Put ( putWord64le )
 import Text.Printf
 
 import Data.Flat
+import Class.Field 
 
 --------------------------------------------------------------------------------
 
@@ -99,6 +100,21 @@ instance Random Goldilocks where
   random  g = let (x,g') = randomR (0,goldilocksPrimeWord64-1) g in (MkGoldilocks x, g')
   randomR = error "randomR/Goldilocks: doesn't make much sense"
 
+instance Ring F where
+  zero        = Field.Goldilocks.Fast.zero
+  one         = Field.Goldilocks.Fast.one
+  isZero      = Field.Goldilocks.Fast.isZero
+  isOne       = Field.Goldilocks.Fast.isOne
+  square      = Field.Goldilocks.Fast.sqr
+  power       = Field.Goldilocks.Fast.pow
+  power_      = Field.Goldilocks.Fast.pow_
+
+instance Field F 
+
+instance FiniteField F where
+  fieldSize _ = Field.Goldilocks.Fast.goldilocksPrime
+  rndIO       = randomIO
+
 --------------------------------------------------------------------------------
 
 -- | @p = 2^64 - 2^32 + 1@
diff --git a/reference/src/Field/Goldilocks/Slow.hs b/reference/src/Field/Goldilocks/Slow.hs
index ef400a6..556791d 100644
--- a/reference/src/Field/Goldilocks/Slow.hs
+++ b/reference/src/Field/Goldilocks/Slow.hs
@@ -26,6 +26,7 @@ import Data.Binary.Put ( putWord64le )
 import Text.Printf
 
 import Data.Flat
+import Class.Field
 
 --------------------------------------------------------------------------------
 
@@ -94,9 +95,24 @@ instance Fractional Goldilocks where
 
 instance Random Goldilocks where
   -- random :: RandomGen g => g -> (a, g) 
-  random  g = let (x,g') = randomR (0,goldilocksPrime-1) g in (MkGoldilocks x, g')
+  random g = let (x,g') = randomR (0,goldilocksPrime-1) g in (MkGoldilocks x, g')
   randomR = error "randomR/Goldilocks: doesn't make much sense"
 
+instance Ring F where
+  zero        = Field.Goldilocks.Slow.zero
+  one         = Field.Goldilocks.Slow.one
+  isZero      = Field.Goldilocks.Slow.isZero
+  isOne       = Field.Goldilocks.Slow.isOne
+  square      = Field.Goldilocks.Slow.sqr
+  power       = Field.Goldilocks.Slow.pow
+  power_      = Field.Goldilocks.Slow.pow_
+
+instance Field F
+
+instance FiniteField F where
+  fieldSize _ = Field.Goldilocks.Slow.goldilocksPrime
+  rndIO       = randomIO
+
 --------------------------------------------------------------------------------
 
 -- | @p = 2^64 - 2^32 + 1@
@@ -153,4 +169,3 @@ pow x e
       _ -> go (acc*s) (sqr s) (shiftR expo 1)
 
 --------------------------------------------------------------------------------
-
diff --git a/reference/src/Field/Properties.hs b/reference/src/Field/Properties.hs
index 972b9e2..f7188e1 100644
--- a/reference/src/Field/Properties.hs
+++ b/reference/src/Field/Properties.hs
@@ -14,7 +14,7 @@ import Control.Monad
 import System.IO
 import System.Random
 
-import Field.Class
+import Class.Field
 
 --------------------------------------------------------------------------------
 -- compatibility hacks
diff --git a/reference/src/Field/Tests.hs b/reference/src/Field/Tests.hs
index 423ab8a..9f3607d 100644
--- a/reference/src/Field/Tests.hs
+++ b/reference/src/Field/Tests.hs
@@ -8,7 +8,7 @@ import Control.Monad
 import Data.Proxy
 import Data.IORef
 
-import Field.Class
+import Class.Field
 import Field.Properties
 
 import Field.Goldilocks           ( F    )
diff --git a/reference/src/Misc.hs b/reference/src/Misc.hs
index 5cf62dd..1f09e52 100644
--- a/reference/src/Misc.hs
+++ b/reference/src/Misc.hs
@@ -214,6 +214,24 @@ safeIndex def arr j
   where
     (a,b) = bounds arr
 
+takeArray :: Int -> Array Int a -> Array Int a
+takeArray k arr = listArray (0,k-1) $ take k (elems arr)
+
+dropArray :: Int -> Array Int a -> Array Int a
+dropArray k arr = listArray (0,n1-k) [ arr!(k+j) | j<-[0..n1-k] ] where
+  (0,n1) = bounds arr
+
+appendArrays :: Array Int a -> Array Int a -> Array Int a
+appendArrays arr1 arr2 = listArray (0,n1+n2+1) $ (elems arr1 ++ elems arr2) where
+  (0,n1) = bounds arr1
+  (0,n2) = bounds arr2
+
+concatArrays :: [Array Int a] -> Array Int a
+concatArrays []   = error "concatArrays: empty list"
+concatArrays arrs = listArray (0,nn-1) (concatMap elems arrs) where
+  sizes = map arraySize arrs
+  nn    = sum sizes
+
 interleaveArrays' :: Array Int (Array Int a) -> Array Int a
 interleaveArrays' arrs 
   | nubOrd (elems sizes) == [n]   = big
diff --git a/reference/src/NTT/Class.hs b/reference/src/NTT/Class.hs
deleted file mode 100644
index 6541863..0000000
--- a/reference/src/NTT/Class.hs
+++ /dev/null
@@ -1,22 +0,0 @@
-
-module NTT.Class where
-
---------------------------------------------------------------------------------
-
-import Data.Kind
-
-import NTT.Subgroup
-
---------------------------------------------------------------------------------
-
-{-
-class NTT ntt where
-  type Field ntt :: Type
-  type Poly  ntt :: Type
-
-  forwardNTT        :: Subgroup F -> Poly F -> FlatArray F
-  inverseNTT        :: Subgroup F -> FlatArray F -> Poly F
-  shiftedForwardNTT :: Subgroup F -> F -> Poly F -> FlatArray F
-  shiftedInverseNTT :: Subgroup F -> F -> FlatArray F -> Poly F
-  asymmForwardNTT   :: Subgroup F -> Poly F -> Subgroup F -> FlatArray F
--}
diff --git a/reference/src/NTT/FFT/Fast.hs b/reference/src/NTT/FFT/Fast.hs
index 19a13a4..07aee54 100644
--- a/reference/src/NTT/FFT/Fast.hs
+++ b/reference/src/NTT/FFT/Fast.hs
@@ -48,7 +48,7 @@ forwardNTT :: Subgroup F -> Poly F -> FlatArray F
 forwardNTT sg (MkPoly (MkFlatArray n fptr2))
   | subgroupSize sg /= n   = error "forwardNTT: subgroup size differs from the array size"
   | otherwise              = unsafePerformIO $ do
-      fptr3 <- mallocForeignPtrArray n -- (n*4)
+      fptr3 <- mallocForeignPtrArray n 
       let MkGoldilocks gen = subgroupGen sg
       withForeignPtr fptr2 $ \ptr2 -> do
         withForeignPtr fptr3 $ \ptr3 -> do
@@ -60,7 +60,7 @@ inverseNTT :: Subgroup F -> FlatArray F -> Poly F
 inverseNTT sg (MkFlatArray n fptr2)
   | subgroupSize sg /= n   = error "inverseNTT: subgroup size differs from the array size"
   | otherwise              = unsafePerformIO $ do
-      fptr3 <- mallocForeignPtrArray n --(n*4)
+      fptr3 <- mallocForeignPtrArray n 
       let MkGoldilocks gen = subgroupGen sg
       withForeignPtr fptr2 $ \ptr2 -> do
         withForeignPtr fptr3 $ \ptr3 -> do
@@ -73,7 +73,7 @@ shiftedForwardNTT :: Subgroup F -> F -> Poly F -> FlatArray F
 shiftedForwardNTT sg (MkGoldilocks eta) (MkPoly (MkFlatArray n fptr2))
   | subgroupSize sg /= n   = error "shiftedForwardNTT: subgroup size differs from the array size"
   | otherwise              = unsafePerformIO $ do
-      fptr3 <- mallocForeignPtrArray n -- (n*4)
+      fptr3 <- mallocForeignPtrArray n 
       let MkGoldilocks gen = subgroupGen sg
       withForeignPtr fptr2 $ \ptr2 -> do
         withForeignPtr fptr3 $ \ptr3 -> do
@@ -86,7 +86,7 @@ shiftedInverseNTT :: Subgroup F -> F -> FlatArray F -> Poly F
 shiftedInverseNTT sg (MkGoldilocks eta) (MkFlatArray n fptr2)
   | subgroupSize sg /= n   = error "shiftedInverseNTT: subgroup size differs from the array size"
   | otherwise              = unsafePerformIO $ do
-      fptr3 <- mallocForeignPtrArray n -- (n*4)
+      fptr3 <- mallocForeignPtrArray n 
       withForeignPtr fptr2 $ \ptr2 -> do
         withForeignPtr fptr3 $ \ptr3 -> do
           c_ntt_inverse_shifted eta (subgroupCLogSize sg) (subgroupGenAsWord64 sg) ptr2 ptr3
@@ -99,7 +99,7 @@ asymmForwardNTT sg_src (MkPoly (MkFlatArray n fptr2)) sg_tgt
   | subgroupSize sg_src /= n   = error "asymmForwardNTT: subgroup size differs from the array size"
   | m < n                      = error "asymmForwardNTT: target subgroup size should be at least the source subgroup src"
   | otherwise                  = unsafePerformIO $ do
-      fptr3 <- mallocForeignPtrArray m -- (m*4)
+      fptr3 <- mallocForeignPtrArray m 
       let MkGoldilocks sgen1 = subgroupGen sg_src
       let MkGoldilocks sgen2 = subgroupGen sg_tgt
       withForeignPtr fptr2 $ \ptr2 -> do
diff --git a/reference/src/NTT/FFT/Slow.hs b/reference/src/NTT/FFT/Slow.hs
index 0c41c76..3de633a 100644
--- a/reference/src/NTT/FFT/Slow.hs
+++ b/reference/src/NTT/FFT/Slow.hs
@@ -25,6 +25,15 @@ polyEvaluate = subgroupNTT
 polyInterpolate :: Subgroup F -> Array Int F -> Poly F 
 polyInterpolate = subgroupINTT
 
+--------------------------------------------------------------------------------
+-- compatible names
+
+forwardNTT :: Subgroup F -> Poly F -> Array Int F
+forwardNTT = subgroupNTT
+
+inverseNTT :: Subgroup F -> Array Int F -> Poly F
+inverseNTT = subgroupINTT
+
 --------------------------------------------------------------------------------
 
 -- | Evaluates a polynomial on a subgroup /of the same size/
diff --git a/reference/src/NTT/Poly/Flat.hs b/reference/src/NTT/Poly/Flat.hs
index b1179ee..82e3c4c 100644
--- a/reference/src/NTT/Poly/Flat.hs
+++ b/reference/src/NTT/Poly/Flat.hs
@@ -20,8 +20,8 @@ import Foreign.ForeignPtr
 import System.Random
 import System.IO.Unsafe
 
+import Class.Field
 import NTT.Subgroup
-import Field.Class
 import Data.Flat as L
 
 --------------------------------------------------------------------------------
diff --git a/reference/src/NTT/Poly/Naive.hs b/reference/src/NTT/Poly/Naive.hs
index 665d8c6..0e5ada5 100644
--- a/reference/src/NTT/Poly/Naive.hs
+++ b/reference/src/NTT/Poly/Naive.hs
@@ -32,6 +32,9 @@ newtype Poly a
   = Poly (Array Int a) 
   deriving (Show,Functor)
 
+fromPoly :: Poly a -> Array Int a
+fromPoly (Poly arr) = arr
+
 instance Binary a => Binary (Poly a) where
   put (Poly arr) = putSmallArray arr
   get = Poly <$> getSmallArray
diff --git a/reference/src/NTT/Tests.hs b/reference/src/NTT/Tests.hs
index de80263..6c75e19 100644
--- a/reference/src/NTT/Tests.hs
+++ b/reference/src/NTT/Tests.hs
@@ -7,7 +7,7 @@ import Data.Array
 
 import Control.Monad
 
-import Field.Class
+import Class.Field
 import Field.Goldilocks ( F ) 
 import Misc