diff --git a/src/circuit.rs b/src/circuit.rs
index 7d9ba85..7876e54 100644
--- a/src/circuit.rs
+++ b/src/circuit.rs
@@ -1,8 +1,9 @@
 use std::{rc::{Rc, Weak}, marker::PhantomData, iter::repeat_with, cell::RefCell};
 use elsa::map::FrozenMap;
 use ff::PrimeField;
+use itertools::Itertools;
 
-use crate::{witness::CSWtns, gate::{Gatebb, Gate}, constraint_system::{Variable, ConstraintSystem, CommitKind, Visibility, CS, Constraint}, utils::poly_utils::check_poly, circuit::circuit_operations::{AttachedAdvice, AttachedPolynomialAdvice, AttachedAdvicePub}, external_interface::{RunIndex, RunAllocator} };
+use crate::{witness::{CSWtns, ProtostarWtns, ProtostarLhsWtns}, gate::{Gatebb, Gate}, constraint_system::{Variable, ConstraintSystem, CommitKind, Visibility, CS, Constraint}, utils::poly_utils::check_poly, circuit::circuit_operations::{AttachedAdvice, AttachedPolynomialAdvice, AttachedAdvicePub}, external_interface::{RunIndex, RunAllocator} };
 
 use self::circuit_operations::CircuitOperation;
 
@@ -367,6 +368,35 @@ where
         }
     }
 
+    pub fn end(&self, mut beta: F) -> ProtostarWtns<'constructed, 'circuit, F, G> {
+        let m = self.constructed.circuit.cs.constr_spec().num_nonlinear_constraints;
+        let mut p = 1;
+        let mut protostar_challenges = vec![];
+        while p <= m {
+            protostar_challenges.push(beta);
+            beta = beta * beta;
+            p = p * 2;
+        }
+
+        let mut pubs = vec![];
+        let mut round_wtns = vec![];
+
+        for round in &self.cs.wtns {
+            round_wtns.push(round.privs.iter().map(|x| x.unwrap()).collect_vec());
+            pubs.push(round.pubs.iter().map(|x| x.unwrap()).collect_vec());
+        }
+
+        ProtostarWtns {
+            lhs: ProtostarLhsWtns {
+                round_wtns,
+                pubs,
+                protostar_challenges,
+                circuit: self.constructed,
+            },
+            error: F::ZERO,
+        }
+    }
+
     pub fn set_ext(&mut self, ext: ExternalValue<F>, value: F) -> () {
         self.cs.setext(ext, value);
     }
diff --git a/src/folding/shape.rs b/src/folding/shape.rs
index dd26dfc..b04e9bd 100644
--- a/src/folding/shape.rs
+++ b/src/folding/shape.rs
@@ -4,7 +4,7 @@
 use ff::PrimeField;
 use halo2::halo2curves::CurveAffine;
 use itertools::Itertools;
-use crate::{utils::arith_helper::{log2_ceil, ev}, constraint_system::{WitnessSpec, ConstraintSystem, CS, ConstrSpec}, gate::Gate};
+use crate::{utils::arith_helper::{log2_ceil, ev}, constraint_system::{WitnessSpec, ConstraintSystem, CS, ConstrSpec}, gate::Gate, witness::Module};
 
 /// Encode value as field elements.
 pub trait FEncoding <F: PrimeField> {
@@ -44,20 +44,22 @@ impl<F: PrimeField, C: CurveAffine<ScalarExt = F>> ProtostarLhs<F, C> {
 
         assert_eq!(self.protostar_challenges.len(), log2_ceil(shape.cspec.num_nonlinear_constraints));
     }
+}
 
-    pub fn scale(&mut self, scale: F) {
+impl<F: PrimeField, C: CurveAffine<ScalarExt = F>> Module<F> for ProtostarLhs<F, C> {
+    fn scale(&mut self, scale: F) {
         self.round_commitments.iter_mut().map(|x| *x = (*x * scale).into()).count();
         self.pubs.iter_mut().map(|rpubs| rpubs.iter_mut().map(|x| *x *= scale).count()).count();
         self.protostar_challenges.iter_mut().map(|x| *x *= scale).count();
     }
 
-    pub fn neg(&mut self) {
+    fn neg(&mut self) {
         self.round_commitments.iter_mut().map(|x| *x = -*x).count();
         self.pubs.iter_mut().map(|rpubs| rpubs.iter_mut().map(|x| *x = -*x).count()).count();
         self.protostar_challenges.iter_mut().map(|x| *x = -*x).count();
     }
 
-    pub fn add_assign(&mut self, other: Self) {
+    fn add_assign(&mut self, other: Self) {
         self.round_commitments.iter_mut().zip_eq(other.round_commitments.iter())
             .map(|(a, b)| *a = (*a + *b).into()).count();
         self.pubs.iter_mut().zip_eq(other.pubs.iter())
diff --git a/src/witness.rs b/src/witness.rs
index dd81e7f..63a8fff 100644
--- a/src/witness.rs
+++ b/src/witness.rs
@@ -2,9 +2,10 @@ use std::iter::repeat;
 use std::marker::PhantomData;
 
 use ff::PrimeField;
-use halo2::halo2curves::CurveAffine;
+use halo2::{halo2curves::CurveAffine, arithmetic::best_multiexp};
+use itertools::Itertools;
 
-use crate::{gate::Gate, constraint_system::{ConstraintSystem, Variable, CS, Visibility, WitnessSpec}, commitment::{CommitmentKey, CkWtns, CtRound, ErrGroup, CkRelaxed}, circuit::ExternalValue};
+use crate::{gate::Gate, constraint_system::{ConstraintSystem, Variable, CS, Visibility, WitnessSpec}, commitment::{CommitmentKey, CkWtns, CtRound, ErrGroup, CkRelaxed}, circuit::{ExternalValue, ConstructedCircuit, PolyOp}, utils::field_precomp::FieldUtils, folding::shape::{ProtostarLhs, ProtostarInstance}};
 
 #[derive(Clone)]
 pub struct RoundWtns<F: PrimeField> {
@@ -138,3 +139,95 @@ impl<'c, F: PrimeField, T: Gate<'c, F>, G:CurveAffine<ScalarExt=F>> CSSystemComm
         (ck.0.commit(&self.cs.wtns),  ck.1.commit(&self.err))
     }
 }
+
+pub trait Module<F> {
+    fn add_assign(&mut self, other: Self) -> ();
+    fn neg(&mut self) -> ();
+    fn scale(&mut self, scale: F) -> ();
+}
+
+pub struct ProtostarLhsWtns<'constructed, 'circuit, F: PrimeField, G: Gate<'circuit, F> + From<PolyOp<'circuit, F>>> {
+    pub round_wtns: Vec<Vec<F>>,
+    pub pubs: Vec<Vec<F>>,
+    pub protostar_challenges: Vec<F>,
+    pub circuit: &'constructed ConstructedCircuit<'circuit, F, G>,
+}
+
+impl<'constructed, 'circuit, F: PrimeField, G: Gate<'circuit, F> + From<PolyOp<'circuit, F>>> ProtostarLhsWtns<'constructed, 'circuit, F, G> {
+    pub fn commit<C: CurveAffine<ScalarExt=F>> (&self, commitment_key: Vec<Vec<C>>) -> ProtostarLhs<F, C> {
+        ProtostarLhs { 
+            round_commitments: self.round_wtns.iter().zip_eq(commitment_key).map(|(wtns, ck)| best_multiexp(&wtns, &ck).into()).collect_vec(),
+            pubs: self.pubs.clone(),
+            protostar_challenges: self.protostar_challenges.clone(),
+        }
+    }
+}
+
+impl<'constructed, 'circuit, F: PrimeField, G: Gate<'circuit, F> + From<PolyOp<'circuit, F>>> Module<F> for ProtostarLhsWtns<'constructed, 'circuit, F, G> {
+    fn add_assign(&mut self, other: Self) -> () {
+        self.round_wtns.iter_mut().zip_eq(other.round_wtns.iter()).map(|(s, o)| {
+            s.iter_mut().zip_eq(o.iter()).map(|(s, o)| *s = *s + o)
+        }).last();
+        self.pubs.iter_mut().zip_eq(other.pubs.iter()).map(|(s, o)| {
+            s.iter_mut().zip_eq(o.iter()).map(|(s, o)| *s = *s + o)
+        }).last();
+        self.protostar_challenges.iter_mut().zip_eq(other.protostar_challenges.iter()).map(|(s, o)| {
+            *s = *s + o
+        }).last();
+    }
+
+    fn neg(&mut self) -> () {
+        self.round_wtns.iter_mut().map(|s| {
+            s.iter_mut().map(|s| *s = -*s)
+        }).last();
+        self.pubs.iter_mut().map(|s| {
+            s.iter_mut().map(|s| *s = -*s)
+        }).last();
+        self.protostar_challenges.iter_mut().map(|s| {
+            *s = -*s
+        }).last();
+    }
+
+    fn scale(&mut self, scale: F) -> () {
+        self.round_wtns.iter_mut().map(|s| {
+            s.iter_mut().map(|s| *s = *s * scale)
+        }).last();
+        self.pubs.iter_mut().map(|s| {
+            s.iter_mut().map(|s| *s = *s * scale)
+        }).last();
+        self.protostar_challenges.iter_mut().map(|s| {
+            *s = *s * scale
+        }).last();
+    }
+}
+
+pub struct ProtostarWtns<'constructed, 'circuit, F: PrimeField, G: Gate<'circuit, F> + From<PolyOp<'circuit, F>>> {
+    pub lhs: ProtostarLhsWtns<'constructed, 'circuit, F, G>,
+    pub error: F
+}
+
+impl<'constructed, 'circuit, F: PrimeField, G: Gate<'circuit, F> + From<PolyOp<'circuit, F>>> Module<F> for ProtostarWtns<'constructed, 'circuit, F, G> {
+    fn add_assign(&mut self, other: Self) -> () {
+        self.error += other.error;
+        self.lhs.add_assign(other.lhs);
+    }
+
+    fn neg(&mut self) -> () {
+        self.error = -self.error;
+        self.lhs.neg();
+    }
+
+    fn scale(&mut self, scale: F) -> () {
+        self.error *= scale;
+        self.lhs.scale(scale);
+    }
+}
+
+impl<'constructed, 'circuit, F: PrimeField, G: Gate<'circuit, F> + From<PolyOp<'circuit, F>>> ProtostarWtns<'constructed, 'circuit, F, G> {
+    pub fn commit<C: CurveAffine<ScalarExt=F>> (&self, commitment_key: Vec<Vec<C>>) -> ProtostarInstance<F, C> {
+        ProtostarInstance {
+            lhs: self.lhs.commit(commitment_key),
+            error: self.error,
+        }
+    }
+}
\ No newline at end of file