diff --git a/.gitignore b/.gitignore
index 1bbb1b645..075e0c9f2 100644
--- a/.gitignore
+++ b/.gitignore
@@ -13,6 +13,7 @@ cabal-dev
 .cabal-sandbox/
 cabal.sandbox.config
 *.prof
+*.svg
 *.aux
 *.hp
 *.eventlog
diff --git a/symbolic-apps/src/ZkFold/Symbolic/Apps/KYC.hs b/symbolic-apps/src/ZkFold/Symbolic/Apps/KYC.hs
index 61920a3df..03bfcf677 100644
--- a/symbolic-apps/src/ZkFold/Symbolic/Apps/KYC.hs
+++ b/symbolic-apps/src/ZkFold/Symbolic/Apps/KYC.hs
@@ -6,26 +6,27 @@ import           Data.Aeson
 import           Data.Functor                     ((<$>))
 import           Data.Maybe                       (fromJust)
 import           GHC.Generics                     (Generic)
-import           Prelude                          (String, error, ($), (.))
+import           Prelude                          (String, error, fst, snd, ($), (.))
 
 import           ZkFold.Base.Algebra.Basic.Class  (FromConstant (fromConstant))
 import           ZkFold.Base.Algebra.Basic.Field  (Zp)
 import           ZkFold.Base.Algebra.Basic.Number
-import           ZkFold.Base.Data.Vector          (Vector, head, tail, toVector)
+import           ZkFold.Base.Control.HApplicative (HApplicative)
+import           ZkFold.Base.Data.Vector          (Vector, splitAt, toVector)
 import           ZkFold.Symbolic.Class            (Symbolic (BaseField))
 import           ZkFold.Symbolic.Data.Bool        (Bool, not, (&&))
 import           ZkFold.Symbolic.Data.ByteString  (ByteString, Resize (resize), concat, toWords)
-import           ZkFold.Symbolic.Data.Combinators (Ceil, GetRegisterSize, Iso (..), KnownRegisterSize, KnownRegisters,
-                                                   RegisterSize (..))
+import           ZkFold.Symbolic.Data.Class       (SymbolicData)
+import           ZkFold.Symbolic.Data.Combinators (Ceil, GetRegisterSize, Iso (..), KnownRegisterSize,
+                                                   NumberOfRegisters)
 import           ZkFold.Symbolic.Data.Eq          (Eq ((==)), elem)
+import           ZkFold.Symbolic.Data.Input       (SymbolicInput)
 import           ZkFold.Symbolic.Data.Ord         (Ord ((>=)))
 import           ZkFold.Symbolic.Data.UInt        (OrdWord, UInt)
 import           ZkFold.Symbolic.Interpreter      (Interpreter)
 
 type KYCByteString context = ByteString 256 context
 
-type KYCHash context = UInt 256 Auto context
-
 {-
 >>> type Prime256_1 = 28948022309329048855892746252171976963363056481941560715954676764349967630337
 >>> :{
@@ -39,43 +40,68 @@ exKYC = KYCData
 >>> encode exKYC
 "{\"kycHash\":\"bb8\",\"kycID\":4000,\"kycType\":\"3e8\",\"kycValue\":\"7d0\"}"
 -}
-data KYCData n context = KYCData
-    { kycType  :: KYCByteString context
+data KYCData n k r context = KYCData
+    { kycType  :: ByteString k context
+    , kycID    :: UInt k r context
+    , kycHash  :: UInt k r context
     , kycValue :: ByteString n context
-    , kycHash  :: KYCHash context
-    , kycID    :: UInt 64 Auto context
     } deriving Generic
 
 data User r context = User
-    { userAge     :: UInt 64 r context
-    , userCountry :: ByteString 128 context
+    { userAge     :: UInt 8 r context
+    , userCountry :: ByteString 10 context
     } deriving Generic
 
-instance (Symbolic context) => FromJSON (KYCData 256 context)
-instance (Symbolic (Interpreter (Zp p))) => ToJSON (KYCData 256 (Interpreter (Zp p)))
+instance ( Symbolic context
+         , KnownNat n
+         , KnownNat k
+         , KnownRegisterSize r
+         ) => FromJSON (KYCData n k r context)
+
+instance ( Symbolic (Interpreter (Zp p))
+         , KnownNat n
+         , KnownNat k
+         , KnownRegisterSize r
+         ) => ToJSON (KYCData n k r (Interpreter (Zp p)))
+
+instance ( HApplicative context
+         , KnownNat n
+         , KnownNat k
+         , KnownRegisterSize r
+         , KnownNat (NumberOfRegisters (BaseField context) k r)
+         , Symbolic context
+         ) => SymbolicData (KYCData n k r context)
+
+instance (
+  Symbolic context
+  , KnownNat n
+  , KnownNat k
+  , KnownRegisterSize r
+  , KnownNat (NumberOfRegisters (BaseField context) k r)
+  ) => SymbolicInput (KYCData n k r context)
 
 isCitizen :: (Symbolic c) => KYCByteString c -> Vector n (KYCByteString c) -> Bool c
 isCitizen = elem
 
-kycExample :: forall n r rsc context . (
+kycExample :: forall n k r rsc context . (
   Symbolic context
   , KnownNat n
   , KnownNat rsc
-  , Eq (Bool context) (KYCHash context)
   , KnownRegisterSize r
-  , KnownRegisters context 64 r
-  , KnownNat (Ceil (GetRegisterSize (BaseField context) 64 r) OrdWord)
-  ) => KYCData n context -> KYCHash context -> Bool context
+  , KnownNat (NumberOfRegisters (BaseField context) 8 r)
+  , KnownNat (Ceil (GetRegisterSize (BaseField context) 8 r) OrdWord)
+  , KnownNat (NumberOfRegisters (BaseField context) k r)
+  ) => KYCData n k r context -> UInt k r context -> Bool context
 kycExample kycData hash =
   let
-    v :: Vector 3 (ByteString 64 context)
-    v = toWords $ resize $ kycValue kycData
+    v :: (Vector 8 (ByteString 1 context), Vector 10 (ByteString 1 context))
+    v = splitAt @8 @10 $ toWords $ resize $ kycValue kycData
 
     correctHash :: Bool context
     correctHash = hash == kycHash kycData
 
     user :: User r context
-    user = User (from $ head v) (concat $ tail v)
+    user = User (from $ concat $ fst v) (concat $ snd v)
 
     validAge :: Bool context
     validAge = userAge user >= fromConstant (18 :: Natural)
@@ -105,7 +131,7 @@ iso3166 = \case
 restrictedCountries :: forall m context . (
   Symbolic context
   , KnownNat m
-  ) => Vector m (ByteString 128 context)
+  ) => Vector m (ByteString 10 context)
 restrictedCountries =
   fromJust $ toVector $ fromConstant . iso3166 <$>
   [ "FRA"
diff --git a/symbolic-base/src/ZkFold/Base/Algebra/EllipticCurve/Class.hs b/symbolic-base/src/ZkFold/Base/Algebra/EllipticCurve/Class.hs
index 46f47e46a..5822fbf7b 100644
--- a/symbolic-base/src/ZkFold/Base/Algebra/EllipticCurve/Class.hs
+++ b/symbolic-base/src/ZkFold/Base/Algebra/EllipticCurve/Class.hs
@@ -7,6 +7,7 @@
 
 module ZkFold.Base.Algebra.EllipticCurve.Class where
 
+import           Data.Aeson                       (ToJSON)
 import           Data.Functor                     ((<&>))
 import           Data.Kind                        (Type)
 import           Data.String                      (fromString)
@@ -111,6 +112,7 @@ instance
   ) => Show (Point curve) where
     show (Point x y isInf) = if isInf then "Inf" else "(" ++ show x ++ ", " ++ show y ++ ")"
 
+deriving instance (ToJSON (BaseField curve), ToJSON (BooleanOf curve)) => ToJSON (Point curve)
 instance EllipticCurve curve => AdditiveSemigroup (Point curve) where
     (+) = add
 
@@ -187,7 +189,7 @@ pointNegate (Point x y isInf) = if isInf then pointInf else pointXY x (negate y)
 pointMul
     :: forall curve s
     .  EllipticCurve curve
-    => BinaryExpansion (s)
+    => BinaryExpansion s
     => Bits s ~ [s]
     => P.Eq s
     => s
diff --git a/symbolic-base/src/ZkFold/Base/Algebra/Polynomials/Univariate.hs b/symbolic-base/src/ZkFold/Base/Algebra/Polynomials/Univariate.hs
index 0b44347b5..1cea3fabb 100644
--- a/symbolic-base/src/ZkFold/Base/Algebra/Polynomials/Univariate.hs
+++ b/symbolic-base/src/ZkFold/Base/Algebra/Polynomials/Univariate.hs
@@ -52,8 +52,8 @@ import           Control.DeepSeq                  (NFData (..))
 import qualified Data.Vector                      as V
 import           GHC.Generics                     (Generic)
 import           GHC.IsList                       (IsList (..))
-import           Prelude                          hiding (Num (..), drop, length, product, replicate, sum, take, (/),
-                                                   (^))
+import           Prelude                          hiding (Num (..), drop, length, product, replicate, sum, take,
+                                                   truncate, (/), (^))
 import qualified Prelude                          as P
 import           Test.QuickCheck                  (Arbitrary (..), chooseInt)
 
@@ -112,7 +112,7 @@ instance (Ring c, Eq c) => AdditiveMonoid (Poly c) where
 instance (Ring c, Eq c) => AdditiveGroup (Poly c) where
     negate (P cs) = P $ fmap negate cs
 
-instance (Field c, Eq c) => MultiplicativeSemigroup (Poly c) where
+instance {-# OVERLAPPABLE #-} (Field c, Eq c) => MultiplicativeSemigroup (Poly c) where
     -- | If it is possible to calculate a primitive root of unity in the field, proceed with FFT multiplication.
     -- Otherwise default to Karatsuba multiplication for polynomials of degree higher than 64 or use naive multiplication otherwise.
     -- 64 is a threshold determined by benchmarking.
@@ -329,7 +329,7 @@ instance (Field c, KnownNat size) => Exponent (PolyVec c size) Natural where
 instance {-# OVERLAPPING #-} (Field c, KnownNat size) => Scale (PolyVec c size) (PolyVec c size)
 
 -- TODO (Issue #18): check for overflow
-instance (Field c, KnownNat size) => MultiplicativeSemigroup (PolyVec c size) where
+instance {-# OVERLAPPABLE #-} (Field c, KnownNat size) => MultiplicativeSemigroup (PolyVec c size) where
     (PV l) * (PV r) = toPolyVec $ mulAdaptive l r
 
 instance (Field c, KnownNat size) => MultiplicativeMonoid (PolyVec c size) where
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/NonInteractiveProof/Internal.hs b/symbolic-base/src/ZkFold/Base/Protocol/NonInteractiveProof/Internal.hs
index 87ef82ed7..707dd6551 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/NonInteractiveProof/Internal.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/NonInteractiveProof/Internal.hs
@@ -11,9 +11,9 @@ import           Data.Word                                  (Word8)
 import           Numeric.Natural                            (Natural)
 import           Prelude                                    hiding (Num ((*)), sum)
 
-import           ZkFold.Base.Algebra.Basic.Class            (Field, MultiplicativeSemigroup ((*)), sum)
+import           ZkFold.Base.Algebra.Basic.Class            (Field, sum, (*))
 import           ZkFold.Base.Algebra.EllipticCurve.Class    (EllipticCurve (..), Point)
-import           ZkFold.Base.Algebra.Polynomials.Univariate (Poly, PolyVec, fromPolyVec)
+import           ZkFold.Base.Algebra.Polynomials.Univariate (Poly, PolyVec, fromPolyVec, qr)
 import           ZkFold.Base.Data.ByteString
 
 class Monoid ts => ToTranscript ts a where
@@ -69,8 +69,13 @@ class (EllipticCurve curve) => CoreFunction curve core where
 
     polyMul :: (f ~ ScalarField curve, Field f, Eq f) => Poly f -> Poly f -> Poly f
 
+    polyQr :: (f ~ ScalarField curve, Field f, Eq f) => Poly f -> Poly f -> (Poly f, Poly f)
+
 data HaskellCore
 
 instance (EllipticCurve curve, f ~ ScalarField curve) => CoreFunction curve HaskellCore where
     msm gs f = sum $ V.zipWith mul (fromPolyVec f) gs
+
     polyMul = (*)
+
+    polyQr = qr
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/Plonk.hs b/symbolic-base/src/ZkFold/Base/Protocol/Plonk.hs
index 507aee7cb..2d3b72f4d 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/Plonk.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/Plonk.hs
@@ -101,4 +101,4 @@ instance forall p i n l c1 c2 (ts :: Type) core .
         in (input, proof)
 
     verify :: SetupVerify (Plonk p i n l c1 c2 ts) -> Input (Plonk p i n l c1 c2 ts) -> Proof (Plonk p i n l c1 c2 ts) -> Bool
-    verify = plonkVerify @p @i @n @l @c1 @c2 @ts
+    verify = plonkVerify @p @i @n @l @c1 @c2 @ts @core
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Prover.hs b/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Prover.hs
index f1fc4fa8b..f7b35994b 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Prover.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Prover.hs
@@ -1,4 +1,5 @@
 {-# LANGUAGE AllowAmbiguousTypes #-}
+{-# LANGUAGE TypeOperators       #-}
 
 module ZkFold.Base.Protocol.Plonk.Prover
     ( plonkProve
@@ -14,7 +15,8 @@ import           Prelude                                             hiding (Num
 import           ZkFold.Base.Algebra.Basic.Class
 import           ZkFold.Base.Algebra.Basic.Number                    (KnownNat, Natural, value)
 import           ZkFold.Base.Algebra.EllipticCurve.Class             (CompressedPoint, EllipticCurve (..), compress)
-import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (qr)
+import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (polyVecDiv, polyVecInLagrangeBasis,
+                                                                      polyVecLagrange, qr)
 import           ZkFold.Base.Data.Vector                             ((!!))
 import           ZkFold.Base.Protocol.NonInteractiveProof
 import           ZkFold.Base.Protocol.Plonkup                        (with4n6)
@@ -50,6 +52,19 @@ plonkProve PlonkupProverSetup {..}
         (@) :: forall size . (KnownNat size) => PolyVec (ScalarField c1) size -> PolyVec (ScalarField c1) size -> PolyVec (ScalarField c1) size
         (@) a b = poly2vec $ polyMul @c1 @core (vec2poly a) (vec2poly b)
 
+        polyVecDiv :: forall c size . (c ~ ScalarField c1, KnownNat size) =>PolyVec c size -> PolyVec c size -> PolyVec c size
+        polyVecDiv l r = poly2vec $ fst $ (polyQr @c1 @core) (vec2poly l) (vec2poly r)
+
+        polyVecLagrange :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            Natural -> c -> PolyVec c size
+        polyVecLagrange i omega' = scalePV (omega'^i // fromConstant (value @m)) $ (polyVecZero @c @m @size - one) `polyVecDiv` polyVecLinear one (negate $ omega'^i)
+
+        polyVecInLagrangeBasis :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            c -> PolyVec c m -> PolyVec c size
+        polyVecInLagrangeBasis omega' cs =
+            let ls = fmap (\i -> polyVecLagrange @c @m @size i omega') (V.generate (V.length (fromPolyVec cs)) (fromIntegral . succ))
+            in sum $ V.zipWith scalePV (fromPolyVec cs) ls
+
         PlonkupCircuitPolynomials {..} = polynomials
         secret i = ps !! (i -! 1)
 
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Verifier.hs b/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Verifier.hs
index 777690346..063c9f895 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Verifier.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/Plonk/Verifier.hs
@@ -1,10 +1,12 @@
 {-# LANGUAGE AllowAmbiguousTypes  #-}
+{-# LANGUAGE TypeOperators        #-}
 {-# LANGUAGE UndecidableInstances #-}
 
 module ZkFold.Base.Protocol.Plonk.Verifier
     ( plonkVerify
     ) where
 
+import qualified Data.Vector                                         as V
 import           Data.Word                                           (Word8)
 import           GHC.IsList                                          (IsList (..))
 import           Prelude                                             hiding (Num (..), Ord, drop, length, sum, take,
@@ -13,7 +15,8 @@ import           Prelude                                             hiding (Num
 import           ZkFold.Base.Algebra.Basic.Class
 import           ZkFold.Base.Algebra.Basic.Number                    (KnownNat, Natural, value)
 import           ZkFold.Base.Algebra.EllipticCurve.Class
-import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (qr)
+import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (polyVecDiv, polyVecInLagrangeBasis,
+                                                                      polyVecLagrange, qr)
 import           ZkFold.Base.Protocol.NonInteractiveProof            hiding (verify)
 import           ZkFold.Base.Protocol.Plonkup.Input
 import           ZkFold.Base.Protocol.Plonkup.Internal
@@ -23,7 +26,7 @@ import           ZkFold.Base.Protocol.Plonkup.Verifier.Setup
 import           ZkFold.Symbolic.Compiler.ArithmeticCircuit.Internal
 import           ZkFold.Symbolic.Data.Ord
 
-plonkVerify :: forall p i n l c1 c2 ts .
+plonkVerify :: forall p i n l c1 c2 ts core .
     ( KnownNat n
     , Foldable l
     , Pairing c1 c2
@@ -34,6 +37,7 @@ plonkVerify :: forall p i n l c1 c2 ts .
     , ToTranscript ts (ScalarField c1)
     , ToTranscript ts (CompressedPoint c1)
     , FromTranscript ts (ScalarField c1)
+    , CoreFunction c1 core
     ) => PlonkupVerifierSetup p i n l c1 c2 -> PlonkupInput l c1 -> PlonkupProof c1 -> Bool
 plonkVerify
     PlonkupVerifierSetup {..}
@@ -150,3 +154,16 @@ plonkVerify
         -- Step 13: Compute the pairing
         p1 = pairing (proof1 + eta `mul` proof2) h1
         p2 = pairing (xi `mul` proof1 + (eta * xi * omega) `mul` proof2 + f - e) (pointGen @c2)
+
+        polyVecDiv :: forall c size . (c ~ ScalarField c1, KnownNat size) =>PolyVec c size -> PolyVec c size -> PolyVec c size
+        polyVecDiv l r = poly2vec $ fst $ (polyQr @c1 @core) (vec2poly l) (vec2poly r)
+
+        polyVecLagrange :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            Natural -> c -> PolyVec c size
+        polyVecLagrange i omega' = scalePV (omega'^i // fromConstant (value @m)) $ (polyVecZero @c @m @size - one) `polyVecDiv` polyVecLinear one (negate $ omega'^i)
+
+        polyVecInLagrangeBasis :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            c -> PolyVec c m -> PolyVec c size
+        polyVecInLagrangeBasis omega' cs =
+            let ls = fmap (\i -> polyVecLagrange @c @m @size i omega') (V.generate (V.length (fromPolyVec cs)) (fromIntegral . succ))
+            in sum $ V.zipWith scalePV (fromPolyVec cs) ls
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup.hs b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup.hs
index e6dca4e05..11f62ad6f 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup.hs
@@ -71,5 +71,5 @@ instance forall p i n l c1 c2 ts core.
         in (input, proof)
 
     verify :: SetupVerify (Plonkup p i n l c1 c2 ts) -> Input (Plonkup p i n l c1 c2 ts) -> Proof (Plonkup p i n l c1 c2 ts) -> Bool
-    verify = with4n6 @n $ plonkupVerify @p @i @n @l @c1 @c2 @ts
+    verify = with4n6 @n $ plonkupVerify @p @i @n @l @c1 @c2 @ts @core
 
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Prover.hs b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Prover.hs
index 3fdb24c9d..d3e7e0538 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Prover.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Prover.hs
@@ -1,4 +1,5 @@
 {-# LANGUAGE AllowAmbiguousTypes #-}
+{-# LANGUAGE TypeOperators       #-}
 
 module ZkFold.Base.Protocol.Plonkup.Prover
     ( module ZkFold.Base.Protocol.Plonkup.Prover.Polynomials
@@ -17,7 +18,8 @@ import           Prelude                                             hiding (Num
 import           ZkFold.Base.Algebra.Basic.Class
 import           ZkFold.Base.Algebra.Basic.Number                    (KnownNat, Natural, value)
 import           ZkFold.Base.Algebra.EllipticCurve.Class             (CompressedPoint, EllipticCurve (..), compress)
-import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (qr)
+import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (polyVecDiv, polyVecInLagrangeBasis,
+                                                                      polyVecLagrange, qr)
 import           ZkFold.Base.Data.Vector                             ((!!))
 import           ZkFold.Base.Protocol.NonInteractiveProof
 import           ZkFold.Base.Protocol.Plonkup.Input
@@ -53,6 +55,19 @@ plonkupProve PlonkupProverSetup {..}
         (@) :: forall size . (KnownNat size) => PolyVec (ScalarField c1) size -> PolyVec (ScalarField c1) size -> PolyVec (ScalarField c1) size
         (@) a b = poly2vec $ polyMul @c1 @core (vec2poly a) (vec2poly b)
 
+        polyVecDiv :: forall c size . (c ~ ScalarField c1, KnownNat size) =>PolyVec c size -> PolyVec c size -> PolyVec c size
+        polyVecDiv l r = poly2vec $ fst $ (polyQr @c1 @core) (vec2poly l) (vec2poly r)
+
+        polyVecLagrange :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            Natural -> c -> PolyVec c size
+        polyVecLagrange i omega' = scalePV (omega'^i // fromConstant (value @m)) $ (polyVecZero @c @m @size - one) `polyVecDiv` polyVecLinear one (negate $ omega'^i)
+
+        polyVecInLagrangeBasis :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            c -> PolyVec c m -> PolyVec c size
+        polyVecInLagrangeBasis omega' cs =
+            let ls = fmap (\i -> polyVecLagrange @c @m @size i omega') (V.generate (V.length (fromPolyVec cs)) (fromIntegral . succ))
+            in sum $ V.zipWith scalePV (fromPolyVec cs) ls
+
         PlonkupCircuitPolynomials {..} = polynomials
         secret i = ps !! (i -! 1)
 
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Setup.hs b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Setup.hs
index 9a9316fea..8b672c8a0 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Setup.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Setup.hs
@@ -16,7 +16,8 @@ import           ZkFold.Base.Algebra.Basic.Class
 import           ZkFold.Base.Algebra.Basic.Number
 import           ZkFold.Base.Algebra.Basic.Permutations              (fromPermutation)
 import           ZkFold.Base.Algebra.EllipticCurve.Class             (EllipticCurve (..), Pairing, Point)
-import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (qr)
+import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (polyVecDiv, polyVecInLagrangeBasis,
+                                                                      polyVecLagrange, qr)
 import           ZkFold.Base.Protocol.NonInteractiveProof            (CoreFunction (..))
 import           ZkFold.Base.Protocol.Plonkup.Internal
 import           ZkFold.Base.Protocol.Plonkup.Prover
@@ -119,3 +120,17 @@ plonkupSetup Plonkup {..} =
         commitments = PlonkupCircuitCommitments {..}
 
     in PlonkupSetup {..}
+    where
+        polyVecDiv :: forall c size . (c ~ ScalarField c1, KnownNat size) =>PolyVec c size -> PolyVec c size -> PolyVec c size
+        polyVecDiv l r = poly2vec $ fst $ (polyQr @c1 @core) (vec2poly l) (vec2poly r)
+
+        polyVecLagrange :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            Natural -> c -> PolyVec c size
+        polyVecLagrange i omega' = scalePV (omega'^i // fromConstant (value @m)) $ (polyVecZero @c @m @size - one) `polyVecDiv` polyVecLinear one (negate $ omega'^i)
+
+        polyVecInLagrangeBasis :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            c -> PolyVec c m -> PolyVec c size
+        polyVecInLagrangeBasis omega' cs =
+            let ls = fmap (\i -> polyVecLagrange @c @m @size i omega') (V.generate (V.length (fromPolyVec cs)) (fromIntegral . succ))
+            in sum $ V.zipWith scalePV (fromPolyVec cs) ls
+
diff --git a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Verifier.hs b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Verifier.hs
index aa1d7df99..324bee4aa 100644
--- a/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Verifier.hs
+++ b/symbolic-base/src/ZkFold/Base/Protocol/Plonkup/Verifier.hs
@@ -1,4 +1,5 @@
 {-# LANGUAGE AllowAmbiguousTypes  #-}
+{-# LANGUAGE TypeOperators        #-}
 {-# LANGUAGE UndecidableInstances #-}
 
 module ZkFold.Base.Protocol.Plonkup.Verifier
@@ -7,6 +8,7 @@ module ZkFold.Base.Protocol.Plonkup.Verifier
     , plonkupVerify
     ) where
 
+import qualified Data.Vector                                         as V
 import           Data.Word                                           (Word8)
 import           GHC.IsList                                          (IsList (..))
 import           Prelude                                             hiding (Num (..), Ord (..), drop, length, sum,
@@ -15,7 +17,8 @@ import           Prelude                                             hiding (Num
 import           ZkFold.Base.Algebra.Basic.Class
 import           ZkFold.Base.Algebra.Basic.Number                    (KnownNat, Natural, value)
 import           ZkFold.Base.Algebra.EllipticCurve.Class
-import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (qr)
+import           ZkFold.Base.Algebra.Polynomials.Univariate          hiding (polyVecDiv, polyVecInLagrangeBasis,
+                                                                      polyVecLagrange, qr)
 import           ZkFold.Base.Protocol.NonInteractiveProof            hiding (verify)
 import           ZkFold.Base.Protocol.Plonkup.Input
 import           ZkFold.Base.Protocol.Plonkup.Internal
@@ -25,7 +28,7 @@ import           ZkFold.Base.Protocol.Plonkup.Verifier.Setup
 import           ZkFold.Symbolic.Compiler.ArithmeticCircuit.Internal
 import           ZkFold.Symbolic.Data.Ord
 
-plonkupVerify :: forall p i n l c1 c2 ts .
+plonkupVerify :: forall p i n l c1 c2 ts core .
     ( KnownNat n
     , KnownNat (PlonkupPolyExtendedLength n)
     , Foldable l
@@ -37,6 +40,7 @@ plonkupVerify :: forall p i n l c1 c2 ts .
     , ToTranscript ts (ScalarField c1)
     , ToTranscript ts (CompressedPoint c1)
     , FromTranscript ts (ScalarField c1)
+    , CoreFunction c1 core
     ) => PlonkupVerifierSetup p i n l c1 c2 -> PlonkupInput l c1 -> PlonkupProof c1 -> Bool
 plonkupVerify
     PlonkupVerifierSetup {..}
@@ -154,3 +158,16 @@ plonkupVerify
         -- Step 13: Compute the pairing
         p1 = pairing (proof1 + eta `mul` proof2) h1
         p2 = pairing (xi `mul` proof1 + (eta * xi * omega) `mul` proof2 + f - e) (pointGen @c2)
+
+        polyVecDiv :: forall c size . (c ~ ScalarField c1, KnownNat size) =>PolyVec c size -> PolyVec c size -> PolyVec c size
+        polyVecDiv l r = poly2vec $ fst $ (polyQr @c1 @core) (vec2poly l) (vec2poly r)
+
+        polyVecLagrange :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            Natural -> c -> PolyVec c size
+        polyVecLagrange i omega' = scalePV (omega'^i // fromConstant (value @m)) $ (polyVecZero @c @m @size - one) `polyVecDiv` polyVecLinear one (negate $ omega'^i)
+
+        polyVecInLagrangeBasis :: forall c m size . (c ~ ScalarField c1, KnownNat m, KnownNat size) =>
+            c -> PolyVec c m -> PolyVec c size
+        polyVecInLagrangeBasis omega' cs =
+            let ls = fmap (\i -> polyVecLagrange @c @m @size i omega') (V.generate (V.length (fromPolyVec cs)) (fromIntegral . succ))
+            in sum $ V.zipWith scalePV (fromPolyVec cs) ls