@@ -78,7 +78,7 @@ }
 }
 
 fn main() {
-    let n_steps = 10;
+    let n_steps = 5;
     // set the initial state
     let z_0 = vec![Fr::from(3_u32)];
 

@@ -28,7 +28,7 @@ transcript::poseidon::poseidon_canonical_config,
     Decider, FoldingScheme,
 };
 use frontends::noir::{load_noir_circuit, NoirFCircuit};
-use std::{env, time::Instant};
+use std::time::Instant;
 
 use solidity_verifiers::{
     evm::{compile_solidity, Evm},
@@ -42,12 +42,7 @@ // set the initial state
     let z_0 = vec![Fr::from(1)];
 
     // initialize the noir fcircuit
-    let cur_path = env::current_dir().unwrap();
-
-    let circuit_path = format!(
-        "{}/frontends/src/noir/test_folder/test_mimc/target/test_mimc.json",
-        cur_path.to_str().unwrap()
-    );
+    let circuit_path = format!("./frontends/src/noir/test_folder/test_mimc/target/test_mimc.json",);
 
     let circuit = load_noir_circuit(circuit_path).unwrap();
     let f_circuit = NoirFCircuit {

@@ -121,8 +121,8 @@ cyclefold::{CycleFoldCommittedInstanceVar, CycleFoldWitnessVar},
             nonnative::uint::NonNativeUintVar,
         },
         nova::{
-            circuits::CommittedInstanceVar, decider_eth_circuit::WitnessVar, CommittedInstance,
-            Witness,
+            decider_eth_circuit::WitnessVar, nifs::nova_circuits::CommittedInstanceVar,
+            CommittedInstance, Witness,
         },
     };
     use crate::frontend::{

@@ -150,7 +150,7 @@ use ark_relations::r1cs::ConstraintSystem;
     use ark_std::UniformRand;
 
     use super::*;
-    use crate::folding::nova::{circuits::CommittedInstanceVar, CommittedInstance};
+    use crate::folding::nova::{nifs::nova_circuits::CommittedInstanceVar, CommittedInstance};
     use crate::transcript::poseidon::poseidon_canonical_config;
 
     // checks that the gadget and native implementations of the challenge computation match

@@ -28,7 +28,7 @@ decider::{EvalGadget, KZGChallengesGadget},
             nonnative::affine::NonNativeAffineVar,
             CF1, CF2,
         },
-        nova::{circuits::CommittedInstanceVar, decider_eth_circuit::WitnessVar},
+        nova::{decider_eth_circuit::WitnessVar, nifs::nova_circuits::CommittedInstanceVar},
         traits::{CommittedInstanceOps, CommittedInstanceVarOps, Dummy, WitnessOps, WitnessVarOps},
     },
 };

@@ -1,25 +1,30 @@
 /// contains [Nova](https://eprint.iacr.org/2021/370.pdf) related circuits
 use ark_crypto_primitives::sponge::{
-    constraints::{AbsorbGadget, CryptographicSpongeVar},
-    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig},
-    Absorb, CryptographicSponge,
+    constraints::CryptographicSpongeVar,
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
+    Absorb,
 };
 use ark_ec::{CurveGroup, Group};
 use ark_ff::PrimeField;
 use ark_r1cs_std::{
-    alloc::{AllocVar, AllocationMode},
+    alloc::AllocVar,
     boolean::Boolean,
     eq::EqGadget,
     fields::{fp::FpVar, FieldVar},
     prelude::CurveVar,
-    uint8::UInt8,
     R1CSVar, ToConstraintFieldGadget,
 };
-use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, Namespace, SynthesisError};
+use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, SynthesisError};
 use ark_std::{fmt::Debug, One, Zero};
-use core::{borrow::Borrow, marker::PhantomData};
+use core::marker::PhantomData;
 
-use super::{CommittedInstance, NovaCycleFoldConfig};
+use super::{
+    nifs::{
+        nova_circuits::{CommittedInstanceVar, NIFSGadget},
+        NIFSGadgetTrait,
+    },
+    CommittedInstance, NovaCycleFoldConfig,
+};
 use crate::folding::circuits::{
     cyclefold::{
         CycleFoldChallengeGadget, CycleFoldCommittedInstance, CycleFoldCommittedInstanceVar,
@@ -28,180 +33,9 @@ },
     nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
     CF1, CF2,
 };
+use crate::folding::traits::{CommittedInstanceVarOps, Dummy};
 use crate::frontend::FCircuit;
-use crate::transcript::{AbsorbNonNativeGadget, Transcript, TranscriptVar};
-use crate::{
-    constants::NOVA_N_BITS_RO,
-    folding::traits::{CommittedInstanceVarOps, Dummy},
-};
-
-/// CommittedInstanceVar contains the u, x, cmE and cmW values which are folded on the main Nova
-/// constraints field (E1::Fr, where E1 is the main curve). The peculiarity is that cmE and cmW are
-/// represented non-natively over the constraint field.
-#[derive(Debug, Clone)]
-pub struct CommittedInstanceVar<C: CurveGroup> {
-    pub u: FpVar<C::ScalarField>,
-    pub x: Vec<FpVar<C::ScalarField>>,
-    pub cmE: NonNativeAffineVar<C>,
-    pub cmW: NonNativeAffineVar<C>,
-}
-
-impl<C> AllocVar<CommittedInstance<C>, CF1<C>> for CommittedInstanceVar<C>
-where
-    C: CurveGroup,
-{
-    fn new_variable<T: Borrow<CommittedInstance<C>>>(
-        cs: impl Into<Namespace<CF1<C>>>,
-        f: impl FnOnce() -> Result<T, SynthesisError>,
-        mode: AllocationMode,
-    ) -> Result<Self, SynthesisError> {
-        f().and_then(|val| {
-            let cs = cs.into();
-
-            let u = FpVar::<C::ScalarField>::new_variable(cs.clone(), || Ok(val.borrow().u), mode)?;
-            let x: Vec<FpVar<C::ScalarField>> =
-                Vec::new_variable(cs.clone(), || Ok(val.borrow().x.clone()), mode)?;
-
-            let cmE =
-                NonNativeAffineVar::<C>::new_variable(cs.clone(), || Ok(val.borrow().cmE), mode)?;
-            let cmW =
-                NonNativeAffineVar::<C>::new_variable(cs.clone(), || Ok(val.borrow().cmW), mode)?;
-
-            Ok(Self { u, x, cmE, cmW })
-        })
-    }
-}
-
-impl<C: CurveGroup> AbsorbGadget<C::ScalarField> for CommittedInstanceVar<C> {
-    fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
-        FpVar::batch_to_sponge_bytes(&self.to_sponge_field_elements()?)
-    }
-
-    fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
-        Ok([
-            vec![self.u.clone()],
-            self.x.clone(),
-            self.cmE.to_constraint_field()?,
-            self.cmW.to_constraint_field()?,
-        ]
-        .concat())
-    }
-}
-
-impl<C: CurveGroup> CommittedInstanceVarOps<C> for CommittedInstanceVar<C> {
-    type PointVar = NonNativeAffineVar<C>;
-
-    fn get_commitments(&self) -> Vec<Self::PointVar> {
-        vec![self.cmW.clone(), self.cmE.clone()]
-    }
-
-    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>] {
-        &self.x
-    }
-
-    fn enforce_incoming(&self) -> Result<(), SynthesisError> {
-        let zero = NonNativeUintVar::new_constant(ConstraintSystemRef::None, CF2::<C>::zero())?;
-        self.cmE.x.enforce_equal_unaligned(&zero)?;
-        self.cmE.y.enforce_equal_unaligned(&zero)?;
-        self.u.enforce_equal(&FpVar::one())
-    }
-
-    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError> {
-        self.u.enforce_equal(&other.u)?;
-        self.x.enforce_equal(&other.x)
-    }
-}
-
-/// Implements the circuit that does the checks of the Non-Interactive Folding Scheme Verifier
-/// described in section 4 of [Nova](https://eprint.iacr.org/2021/370.pdf), where the cmE & cmW checks are
-/// delegated to the NIFSCycleFoldGadget.
-pub struct NIFSGadget<C: CurveGroup> {
-    _c: PhantomData<C>,
-}
-
-impl<C: CurveGroup> NIFSGadget<C> {
-    pub fn fold_committed_instance(
-        r: FpVar<CF1<C>>,
-        ci1: CommittedInstanceVar<C>, // U_i
-        ci2: CommittedInstanceVar<C>, // u_i
-    ) -> Result<CommittedInstanceVar<C>, SynthesisError> {
-        Ok(CommittedInstanceVar {
-            cmE: NonNativeAffineVar::new_constant(ConstraintSystemRef::None, C::zero())?,
-            cmW: NonNativeAffineVar::new_constant(ConstraintSystemRef::None, C::zero())?,
-            // ci3.u = ci1.u + r * ci2.u
-            u: ci1.u + &r * ci2.u,
-            // ci3.x = ci1.x + r * ci2.x
-            x: ci1
-                .x
-                .iter()
-                .zip(ci2.x)
-                .map(|(a, b)| a + &r * &b)
-                .collect::<Vec<FpVar<CF1<C>>>>(),
-        })
-    }
-
-    /// Implements the constraints for NIFS.V for u and x, since cm(E) and cm(W) are delegated to
-    /// the CycleFold circuit.
-    pub fn verify(
-        r: FpVar<CF1<C>>,
-        ci1: CommittedInstanceVar<C>, // U_i
-        ci2: CommittedInstanceVar<C>, // u_i
-        ci3: CommittedInstanceVar<C>, // U_{i+1}
-    ) -> Result<(), SynthesisError> {
-        let ci = Self::fold_committed_instance(r, ci1, ci2)?;
-
-        ci.u.enforce_equal(&ci3.u)?;
-        ci.x.enforce_equal(&ci3.x)?;
-
-        Ok(())
-    }
-}
-
-/// ChallengeGadget computes the RO challenge used for the Nova instances NIFS, it contains a
-/// rust-native and a in-circuit compatible versions.
-pub struct ChallengeGadget<C: CurveGroup, CI: Absorb> {
-    _c: PhantomData<C>,
-    _ci: PhantomData<CI>,
-}
-impl<C: CurveGroup, CI: Absorb> ChallengeGadget<C, CI>
-where
-    <C as Group>::ScalarField: Absorb,
-{
-    pub fn get_challenge_native<T: Transcript<C::ScalarField>>(
-        transcript: &mut T,
-        pp_hash: C::ScalarField, // public params hash
-        U_i: &CI,
-        u_i: &CI,
-        cmT: Option<&C>,
-    ) -> Vec<bool> {
-        transcript.absorb(&pp_hash);
-        transcript.absorb(&U_i);
-        transcript.absorb(&u_i);
-        // in the Nova case we absorb the cmT, in Ova case we don't since it is not used.
-        if let Some(cmT_value) = cmT {
-            transcript.absorb_nonnative(cmT_value);
-        }
-        transcript.squeeze_bits(NOVA_N_BITS_RO)
-    }
-
-    // compatible with the native get_challenge_native
-    pub fn get_challenge_gadget<S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>>(
-        transcript: &mut T,
-        pp_hash: FpVar<CF1<C>>,      // public params hash
-        U_i_vec: Vec<FpVar<CF1<C>>>, // apready processed input, so we don't have to recompute these values
-        u_i: CommittedInstanceVar<C>,
-        cmT: Option<NonNativeAffineVar<C>>,
-    ) -> Result<Vec<Boolean<C::ScalarField>>, SynthesisError> {
-        transcript.absorb(&pp_hash)?;
-        transcript.absorb(&U_i_vec)?;
-        transcript.absorb(&u_i)?;
-        // in the Nova case we absorb the cmT, in Ova case we don't since it is not used.
-        if let Some(cmT_value) = cmT {
-            transcript.absorb_nonnative(&cmT_value)?;
-        }
-        transcript.squeeze_bits(NOVA_N_BITS_RO)
-    }
-}
+use crate::transcript::AbsorbNonNativeGadget;
 
 /// `AugmentedFCircuit` enhances the original step function `F`, so that it can
 /// be used in recursive arguments such as IVC.
@@ -364,31 +198,32 @@ // u_i.x is computed in step 1
             x: vec![u_i_x, cf_u_i_x],
         };
 
-        // P.3. nifs.verify, obtains U_{i+1} by folding u_i & U_i .
-
-        // compute r = H(u_i, U_i, cmT)
-        let r_bits = ChallengeGadget::<C1, CommittedInstance<C1>>::get_challenge_gadget(
+        // P.3. nifs.verify, obtains U_{i+1} by folding u_i & U_i.
+        // Notice that NIFSGadget::verify does not fold cmE & cmW.
+        // We set `U_i1.cmE` and `U_i1.cmW` to unconstrained witnesses `U_i1_cmE` and `U_i1_cmW`
+        // respectively.
+        // The correctness of them will be checked on the other curve.
+        let (mut U_i1, r_bits) = NIFSGadget::<
+            C1,
+            PoseidonSponge<C1::ScalarField>,
+            PoseidonSpongeVar<C1::ScalarField>,
+        >::verify(
             &mut transcript,
             pp_hash.clone(),
+            U_i.clone(),
             U_i_vec,
             u_i.clone(),
             Some(cmT.clone()),
         )?;
-        let r = Boolean::le_bits_to_fp_var(&r_bits)?;
-        // Also convert r_bits to a `NonNativeFieldVar`
+        U_i1.cmE = U_i1_cmE;
+        U_i1.cmW = U_i1_cmW;
+
+        // convert r_bits to a `NonNativeFieldVar`
         let r_nonnat = {
             let mut bits = r_bits;
             bits.resize(C1::BaseField::MODULUS_BIT_SIZE as usize, Boolean::FALSE);
             NonNativeUintVar::from(&bits)
         };
-
-        // Notice that NIFSGadget::fold_committed_instance does not fold cmE & cmW.
-        // We set `U_i1.cmE` and `U_i1.cmW` to unconstrained witnesses `U_i1_cmE` and `U_i1_cmW`
-        // respectively.
-        // The correctness of them will be checked on the other curve.
-        let mut U_i1 = NIFSGadget::<C1>::fold_committed_instance(r, U_i.clone(), u_i.clone())?;
-        U_i1.cmE = U_i1_cmE;
-        U_i1.cmW = U_i1_cmW;
 
         // P.4.a compute and check the first output of F'
 
@@ -507,159 +342,13 @@ #[cfg(test)]
 pub mod tests {
     use super::*;
     use ark_bn254::{Fr, G1Projective as Projective};
-    use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
+    use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, CryptographicSponge};
     use ark_ff::BigInteger;
     use ark_relations::r1cs::ConstraintSystem;
     use ark_std::UniformRand;
 
-    use crate::commitment::pedersen::Pedersen;
-    use crate::folding::nova::nifs::{nova::NIFS, NIFSTrait};
-    use crate::folding::traits::CommittedInstanceOps;
+    use crate::folding::nova::nifs::nova::ChallengeGadget;
     use crate::transcript::poseidon::poseidon_canonical_config;
-
-    #[test]
-    fn test_committed_instance_var() {
-        let mut rng = ark_std::test_rng();
-
-        let ci = CommittedInstance::<Projective> {
-            cmE: Projective::rand(&mut rng),
-            u: Fr::rand(&mut rng),
-            cmW: Projective::rand(&mut rng),
-            x: vec![Fr::rand(&mut rng); 1],
-        };
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-        let ciVar =
-            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
-        assert_eq!(ciVar.u.value().unwrap(), ci.u);
-        assert_eq!(ciVar.x.value().unwrap(), ci.x);
-        // the values cmE and cmW are checked in the CycleFold's circuit
-        // CommittedInstanceInCycleFoldVar in
-        // cyclefold::tests::test_committed_instance_cyclefold_var
-    }
-
-    #[test]
-    fn test_nifs_gadget() {
-        let mut rng = ark_std::test_rng();
-
-        // prepare the committed instances to test in-circuit
-        let ci: Vec<CommittedInstance<Projective>> = (0..2)
-            .into_iter()
-            .map(|_| CommittedInstance::<Projective> {
-                cmE: Projective::rand(&mut rng),
-                u: Fr::rand(&mut rng),
-                cmW: Projective::rand(&mut rng),
-                x: vec![Fr::rand(&mut rng); 1],
-            })
-            .collect();
-        let (ci1, ci2) = (ci[0].clone(), ci[1].clone());
-        let pp_hash = Fr::rand(&mut rng);
-        let cmT = Projective::rand(&mut rng);
-        let poseidon_config = poseidon_canonical_config::<Fr>();
-        let mut transcript = PoseidonSponge::<Fr>::new(&poseidon_config);
-        let (ci3, r_bits) = NIFS::<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>::verify(
-            &mut transcript,
-            pp_hash,
-            &ci1,
-            &ci2,
-            &cmT,
-        )
-        .unwrap();
-        let r_Fr = Fr::from_bigint(BigInteger::from_bits_le(&r_bits)).unwrap();
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-
-        let rVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(r_Fr)).unwrap();
-        let ci1Var =
-            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci1.clone()))
-                .unwrap();
-        let ci2Var =
-            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci2.clone()))
-                .unwrap();
-        let ci3Var =
-            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci3.clone()))
-                .unwrap();
-
-        NIFSGadget::<Projective>::verify(
-            rVar.clone(),
-            ci1Var.clone(),
-            ci2Var.clone(),
-            ci3Var.clone(),
-        )
-        .unwrap();
-        assert!(cs.is_satisfied().unwrap());
-    }
-
-    /// test that checks the native CommittedInstance.to_sponge_{bytes,field_elements}
-    /// vs the R1CS constraints version
-    #[test]
-    pub fn test_committed_instance_to_sponge_preimage() {
-        let mut rng = ark_std::test_rng();
-
-        let ci = CommittedInstance::<Projective> {
-            cmE: Projective::rand(&mut rng),
-            u: Fr::rand(&mut rng),
-            cmW: Projective::rand(&mut rng),
-            x: vec![Fr::rand(&mut rng); 1],
-        };
-
-        let bytes = ci.to_sponge_bytes_as_vec();
-        let field_elements = ci.to_sponge_field_elements_as_vec();
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-
-        let ciVar =
-            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
-        let bytes_var = ciVar.to_sponge_bytes().unwrap();
-        let field_elements_var = ciVar.to_sponge_field_elements().unwrap();
-
-        assert!(cs.is_satisfied().unwrap());
-
-        // check that the natively computed and in-circuit computed hashes match
-        assert_eq!(bytes_var.value().unwrap(), bytes);
-        assert_eq!(field_elements_var.value().unwrap(), field_elements);
-    }
-
-    #[test]
-    fn test_committed_instance_hash() {
-        let mut rng = ark_std::test_rng();
-        let poseidon_config = poseidon_canonical_config::<Fr>();
-        let sponge = PoseidonSponge::<Fr>::new(&poseidon_config);
-        let pp_hash = Fr::from(42u32); // only for test
-
-        let i = Fr::from(3_u32);
-        let z_0 = vec![Fr::from(3_u32)];
-        let z_i = vec![Fr::from(3_u32)];
-        let ci = CommittedInstance::<Projective> {
-            cmE: Projective::rand(&mut rng),
-            u: Fr::rand(&mut rng),
-            cmW: Projective::rand(&mut rng),
-            x: vec![Fr::rand(&mut rng); 1],
-        };
-
-        // compute the CommittedInstance hash natively
-        let h = ci.hash(&sponge, pp_hash, i, &z_0, &z_i);
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-
-        let pp_hashVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(pp_hash)).unwrap();
-        let iVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(i)).unwrap();
-        let z_0Var = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(z_0.clone())).unwrap();
-        let z_iVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(z_i.clone())).unwrap();
-        let ciVar =
-            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
-
-        let sponge = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
-
-        // compute the CommittedInstance hash in-circuit
-        let (hVar, _) = ciVar
-            .hash(&sponge, &pp_hashVar, &iVar, &z_0Var, &z_iVar)
-            .unwrap();
-        assert!(cs.is_satisfied().unwrap());
-
-        // check that the natively computed and in-circuit computed hashes match
-        assert_eq!(hVar.value().unwrap(), h);
-    }
 
     // checks that the gadget and native implementations of the challenge computation match
     #[test]

@@ -288,7 +288,6 @@ &Inputize::<CF2<C2>, CycleFoldCommittedInstanceVar<C2, GC2>>::inputize(&cf_U),
             &proof.cs1_challenges,
             &proof.cs1_proofs.iter().map(|p| p.eval).collect::<Vec<_>>(),
             &proof.cmT.inputize(),
-            &[proof.r],
         ]
         .concat();
 

@@ -55,6 +55,7 @@ const H: bool,
     > TryFrom<Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>> for DeciderCircuit1<C1, C2, GC2>
 where
     CF1<C1>: Absorb,
+    <C1 as CurveGroup>::BaseField: PrimeField,
 {
     type Error = Error;
 

@@ -22,8 +22,7 @@ kzg::{Proof as KZGProof, KZG},
     pedersen::Params as PedersenParams,
     CommitmentScheme,
 };
-use crate::folding::circuits::decider::DeciderEnabledNIFS;
-use crate::folding::circuits::CF2;
+use crate::folding::circuits::{decider::DeciderEnabledNIFS, CF2};
 use crate::folding::traits::{Inputize, WitnessOps};
 use crate::frontend::FCircuit;
 use crate::Error;
@@ -228,7 +227,6 @@ .collect::<Vec<_>>(),
             &proof.kzg_challenges,
             &proof.kzg_proofs.iter().map(|p| p.eval).collect::<Vec<_>>(),
             &proof.cmT.inputize(),
-            &[proof.r],
         ]
         .concat();
 

@@ -11,8 +11,6 @@ use ark_ec::CurveGroup;
 use ark_ff::{BigInteger, PrimeField};
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
-    boolean::Boolean,
-    eq::EqGadget,
     fields::fp::FpVar,
     prelude::CurveVar,
     ToConstraintFieldGadget,
@@ -21,8 +19,8 @@ use ark_relations::r1cs::{Namespace, SynthesisError};
 use ark_std::{borrow::Borrow, marker::PhantomData};
 
 use super::{
-    circuits::{ChallengeGadget, CommittedInstanceVar, NIFSGadget},
-    nifs::{nova::NIFS, NIFSTrait},
+    nifs::nova_circuits::{CommittedInstanceVar, NIFSGadget},
+    nifs::{nova::NIFS, NIFSGadgetTrait, NIFSTrait},
     CommittedInstance, Nova, Witness,
 };
 use crate::commitment::{pedersen::Params as PedersenParams, CommitmentScheme};
@@ -108,6 +106,7 @@ const H: bool,
     > TryFrom<Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>> for DeciderEthCircuit<C1, C2, GC2>
 where
     CF1<C1>: Absorb,
+    <C1 as CurveGroup>::BaseField: PrimeField,
 {
     type Error = Error;
 
@@ -187,21 +186,12 @@ U: CommittedInstanceVar<C>,
         U_vec: Vec<FpVar<CF1<C>>>,
         u: CommittedInstanceVar<C>,
         proof: C,
-        randomness: CF1<C>,
+        _randomness: CF1<C>,
     ) -> Result<CommittedInstanceVar<C>, SynthesisError> {
         let cs = transcript.cs();
         let cmT = NonNativeAffineVar::new_input(cs.clone(), || Ok(proof))?;
-        let r = FpVar::new_input(cs.clone(), || Ok(randomness))?;
-        let r_bits = ChallengeGadget::<C, CommittedInstance<C>>::get_challenge_gadget(
-            transcript,
-            pp_hash,
-            U_vec,
-            u.clone(),
-            Some(cmT),
-        )?;
-        Boolean::le_bits_to_fp_var(&r_bits)?.enforce_equal(&r)?;
-
-        NIFSGadget::<C>::fold_committed_instance(r, U, u)
+        let (new_U, _) = NIFSGadget::verify(transcript, pp_hash, U, U_vec, u, Some(cmT))?;
+        Ok(new_U)
     }
 
     fn fold_group_elements_native(

@@ -17,7 +17,6 @@ use ark_std::fmt::Debug;
 use ark_std::rand::RngCore;
 use ark_std::{One, UniformRand, Zero};
 use core::marker::PhantomData;
-use decider_eth_circuit::WitnessVar;
 
 use crate::folding::circuits::cyclefold::{
     fold_cyclefold_circuit, CycleFoldCircuit, CycleFoldCommittedInstance, CycleFoldConfig,
@@ -38,6 +37,7 @@ constants::NOVA_N_BITS_RO,
     utils::{get_cm_coordinates, pp_hash},
 };
 use crate::{arith::Arith, commitment::CommitmentScheme};
+use decider_eth_circuit::WitnessVar;
 
 pub mod circuits;
 pub mod traits;
@@ -46,8 +46,8 @@
 // NIFS related:
 pub mod nifs;
 
-use circuits::{AugmentedFCircuit, CommittedInstanceVar};
-use nifs::{nova::NIFS, NIFSTrait};
+use circuits::AugmentedFCircuit;
+use nifs::{nova::NIFS, nova_circuits::CommittedInstanceVar, NIFSTrait};
 
 // offchain decider
 pub mod decider;

@@ -1,18 +1,28 @@
-/// This module defines the NIFSTrait, which is set to implement the NIFS (Non-Interactive Folding
-/// Scheme) by the various schemes (Nova, Mova, Ova).
-use ark_crypto_primitives::sponge::Absorb;
+/// This module defines the traits related to the NIFS (Non-Interactive Folding Scheme).
+/// - NIFSTrait, which implements the NIFS interface
+/// - NIFSGadget, which implements the NIFS in-circuit
+/// both traits implemented by the various Nova variants schemes; ie.
+/// [Nova](https://eprint.iacr.org/2021/370.pdf), [Ova](https://hackmd.io/V4838nnlRKal9ZiTHiGYzw),
+/// [Mova](https://eprint.iacr.org/2024/1220.pdf).
+use ark_crypto_primitives::sponge::{constraints::AbsorbGadget, Absorb, CryptographicSponge};
 use ark_ec::CurveGroup;
+use ark_r1cs_std::{alloc::AllocVar, boolean::Boolean, fields::fp::FpVar};
+use ark_relations::r1cs::SynthesisError;
 use ark_std::fmt::Debug;
 use ark_std::rand::RngCore;
 
 use crate::arith::r1cs::R1CS;
 use crate::commitment::CommitmentScheme;
-use crate::transcript::Transcript;
+use crate::folding::circuits::CF1;
+use crate::folding::traits::{CommittedInstanceOps, CommittedInstanceVarOps};
+use crate::transcript::{Transcript, TranscriptVar};
 use crate::Error;
 
 pub mod mova;
 pub mod nova;
+pub mod nova_circuits;
 pub mod ova;
+pub mod ova_circuits;
 pub mod pointvsline;
 
 /// Defines the NIFS (Non-Interactive Folding Scheme) trait, initially defined in
@@ -27,7 +37,7 @@ T: Transcript<C::ScalarField>,
     const H: bool = false,
 >
 {
-    type CommittedInstance: Debug + Clone + Absorb;
+    type CommittedInstance: Debug + Clone + Absorb; // + CommittedInstanceOps<C>;
     type Witness: Debug + Clone;
     type ProverAux: Debug + Clone; // Prover's aux params. eg. in Nova is T
     type Proof: Debug + Clone; // proof. eg. in Nova is cmT
@@ -83,17 +93,56 @@ proof: &Self::Proof,
     ) -> Result<(Self::CommittedInstance, Vec<bool>), Error>;
 }
 
+/// Defines the NIFS (Non-Interactive Folding Scheme) Gadget trait, which specifies the in-circuit
+/// logic of the NIFS.Verify defined in [Nova](https://eprint.iacr.org/2021/370.pdf) and it's
+/// variants [Ova](https://hackmd.io/V4838nnlRKal9ZiTHiGYzw) and
+/// [Mova](https://eprint.iacr.org/2024/1220.pdf).
+pub trait NIFSGadgetTrait<C: CurveGroup, S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>> {
+    type CommittedInstance: Debug + Clone + Absorb + CommittedInstanceOps<C>;
+    type CommittedInstanceVar: Debug
+        + Clone
+        + AbsorbGadget<C::ScalarField>
+        + AllocVar<Self::CommittedInstance, CF1<C>>
+        + CommittedInstanceVarOps<C>;
+    type Proof: Debug + Clone;
+    type ProofVar: Debug + Clone + AllocVar<Self::Proof, CF1<C>>;
+
+    /// Implements the constraints for NIFS.V for u and x, since cm(E) and cm(W) are delegated to
+    /// the CycleFold circuit.
+    #[allow(clippy::type_complexity)]
+    fn verify(
+        transcript: &mut T,
+        pp_hash: FpVar<CF1<C>>,
+        U_i: Self::CommittedInstanceVar,
+        // U_i_vec is passed to reuse the already computed U_i_vec from previous methods
+        U_i_vec: Vec<FpVar<CF1<C>>>,
+        u_i: Self::CommittedInstanceVar,
+        proof: Option<Self::ProofVar>,
+    ) -> Result<(Self::CommittedInstanceVar, Vec<Boolean<CF1<C>>>), SynthesisError>;
+}
+
+/// These tests are the generic tests so that in the tests of Nova, Mova, Ova, we just need to
+/// instantiate these tests to test both the NIFSTrait and NIFSGadgetTrait implementations for each
+/// of the schemes.
 #[cfg(test)]
 pub mod tests {
-    use super::*;
-    use crate::transcript::poseidon::poseidon_canonical_config;
+    use ark_crypto_primitives::sponge::{
+        constraints::{AbsorbGadget, CryptographicSpongeVar},
+        poseidon::constraints::PoseidonSpongeVar,
+        Absorb,
+    };
     use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, CryptographicSponge};
     use ark_pallas::{Fr, Projective};
+    use ark_r1cs_std::{alloc::AllocVar, fields::fp::FpVar, R1CSVar};
+    use ark_relations::r1cs::ConstraintSystem;
     use ark_std::{test_rng, UniformRand};
 
     use super::NIFSTrait;
+    use super::*;
     use crate::arith::r1cs::tests::{get_test_r1cs, get_test_z};
     use crate::commitment::pedersen::Pedersen;
+    use crate::folding::traits::{CommittedInstanceOps, CommittedInstanceVarOps};
+    use crate::transcript::poseidon::poseidon_canonical_config;
 
     /// Test method used to test the different implementations of the NIFSTrait (ie. Nova, Mova,
     /// Ova). Runs a loop using the NIFS trait, and returns the last Witness and CommittedInstance
@@ -101,9 +150,7 @@ /// so that their relation can be checked.
     pub(crate) fn test_nifs_opt<
         N: NIFSTrait<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
     >() -> (N::Witness, N::CommittedInstance) {
-        let r1cs = get_test_r1cs();
-        let z = get_test_z(3);
-        let (w, x) = r1cs.split_z(&z);
+        let r1cs: R1CS<Fr> = get_test_r1cs();
 
         let mut rng = ark_std::test_rng();
         let (pedersen_params, _) = Pedersen::<Projective>::setup(&mut rng, r1cs.A.n_cols).unwrap();
@@ -114,6 +161,8 @@ let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
         let pp_hash = Fr::rand(&mut rng);
 
         // prepare the running instance
+        let z = get_test_z(3);
+        let (w, x) = r1cs.split_z(&z);
         let mut W_i = N::new_witness(w.clone(), r1cs.A.n_rows, test_rng());
         let mut U_i = N::new_instance(&mut rng, &pedersen_params, &W_i, x, vec![]).unwrap();
 
@@ -148,5 +197,124 @@ U_i = folded_committed_instance;
         }
 
         (W_i, U_i)
+    }
+
+    /// Test method used to test the different implementations of the NIFSGadgetTrait (ie. Nova,
+    /// Mova, Ova). It returns the last Witness and CommittedInstance so that it can be checked at
+    /// the parent test that their values match.
+    pub(crate) fn test_nifs_gadget_opt<N, NG>(
+        ci: Vec<NG::CommittedInstance>,
+        proof: NG::Proof,
+    ) -> Result<(NG::CommittedInstance, NG::CommittedInstanceVar), Error>
+    where
+        N: NIFSTrait<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+        NG: NIFSGadgetTrait<
+            Projective,
+            PoseidonSponge<Fr>,
+            PoseidonSpongeVar<Fr>,
+            CommittedInstance = N::CommittedInstance, // constrain that N::CI==NG::CI
+            Proof = N::Proof,                         // constrain that N::Proof==NG::Proof
+        >,
+    {
+        let mut rng = ark_std::test_rng();
+
+        let (U_i, u_i) = (ci[0].clone(), ci[1].clone());
+        let pp_hash = Fr::rand(&mut rng);
+        let poseidon_config = poseidon_canonical_config::<Fr>();
+        let mut transcript = PoseidonSponge::<Fr>::new(&poseidon_config);
+        let (ci3, _) = N::verify(&mut transcript, pp_hash, &U_i, &u_i, &proof)?;
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        let mut transcriptVar = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
+        let pp_hashVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(pp_hash))?;
+        let ci1Var = NG::CommittedInstanceVar::new_witness(cs.clone(), || Ok(U_i.clone()))?;
+        let ci2Var = NG::CommittedInstanceVar::new_witness(cs.clone(), || Ok(u_i.clone()))?;
+        let proofVar = NG::ProofVar::new_witness(cs.clone(), || Ok(proof))?;
+
+        let ci1Var_vec = ci1Var.to_sponge_field_elements()?;
+        let (out, _) = NG::verify(
+            &mut transcriptVar,
+            pp_hashVar,
+            ci1Var.clone(),
+            ci1Var_vec,
+            ci2Var.clone(),
+            Some(proofVar.clone()),
+        )?;
+        assert!(cs.is_satisfied()?);
+
+        // return the NIFS.V and the NIFSGadget.V obtained values, so that they are checked at the
+        // parent test
+        Ok((ci3, out))
+    }
+
+    /// test that checks the native CommittedInstance.to_sponge_{bytes,field_elements}
+    /// vs the R1CS constraints version
+    pub(crate) fn test_committed_instance_to_sponge_preimage_opt<N, NG>(ci: N::CommittedInstance)
+    where
+        N: NIFSTrait<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+        NG: NIFSGadgetTrait<
+            Projective,
+            PoseidonSponge<Fr>,
+            PoseidonSpongeVar<Fr>,
+            CommittedInstance = N::CommittedInstance, // constrain that N::CI==NG::CI
+        >,
+    {
+        let bytes = ci.to_sponge_bytes_as_vec();
+        let field_elements = ci.to_sponge_field_elements_as_vec();
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        let ciVar = NG::CommittedInstanceVar::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
+        let bytes_var = ciVar.to_sponge_bytes().unwrap();
+        let field_elements_var = ciVar.to_sponge_field_elements().unwrap();
+
+        assert!(cs.is_satisfied().unwrap());
+
+        // check that the natively computed and in-circuit computed hashes match
+        assert_eq!(bytes_var.value().unwrap(), bytes);
+        assert_eq!(field_elements_var.value().unwrap(), field_elements);
+    }
+
+    pub(crate) fn test_committed_instance_hash_opt<N, NG>(ci: NG::CommittedInstance)
+    where
+        N: NIFSTrait<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+        NG: NIFSGadgetTrait<
+            Projective,
+            PoseidonSponge<Fr>,
+            PoseidonSpongeVar<Fr>,
+            CommittedInstance = N::CommittedInstance, // constrain that N::CI==NG::CI
+        >,
+        N::CommittedInstance: CommittedInstanceOps<Projective>,
+    {
+        let poseidon_config = poseidon_canonical_config::<Fr>();
+        let sponge = PoseidonSponge::<Fr>::new(&poseidon_config);
+        let pp_hash = Fr::from(42u32); // only for test
+
+        let i = Fr::from(3_u32);
+        let z_0 = vec![Fr::from(3_u32)];
+        let z_i = vec![Fr::from(3_u32)];
+
+        // compute the CommittedInstance hash natively
+        let h = ci.hash(&sponge, pp_hash, i, &z_0, &z_i);
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        let pp_hashVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(pp_hash)).unwrap();
+        let iVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(i)).unwrap();
+        let z_0Var = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(z_0.clone())).unwrap();
+        let z_iVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(z_i.clone())).unwrap();
+        let ciVar = NG::CommittedInstanceVar::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
+
+        let sponge = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
+
+        // compute the CommittedInstance hash in-circuit
+        let (hVar, _) = ciVar
+            .hash(&sponge, &pp_hashVar, &iVar, &z_0Var, &z_iVar)
+            .unwrap();
+        assert!(cs.is_satisfied().unwrap());
+
+        // check that the natively computed and in-circuit computed hashes match
+        assert_eq!(hVar.value().unwrap(), h);
     }
 }

@@ -1,8 +1,10 @@
 /// This module contains the implementation the NIFSTrait for the
 /// [Nova](https://eprint.iacr.org/2021/370.pdf) NIFS (Non-Interactive Folding Scheme).
-use ark_crypto_primitives::sponge::Absorb;
+use ark_crypto_primitives::sponge::{constraints::AbsorbGadget, Absorb, CryptographicSponge};
 use ark_ec::{CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
+use ark_r1cs_std::{boolean::Boolean, fields::fp::FpVar};
+use ark_relations::r1cs::SynthesisError;
 use ark_std::rand::RngCore;
 use ark_std::Zero;
 use std::marker::PhantomData;
@@ -10,12 +12,68 @@
 use super::NIFSTrait;
 use crate::arith::r1cs::R1CS;
 use crate::commitment::CommitmentScheme;
-use crate::folding::circuits::cyclefold::{CycleFoldCommittedInstance, CycleFoldWitness};
-use crate::folding::nova::circuits::ChallengeGadget;
+use crate::constants::NOVA_N_BITS_RO;
+use crate::folding::circuits::{
+    cyclefold::{CycleFoldCommittedInstance, CycleFoldWitness},
+    nonnative::affine::NonNativeAffineVar,
+    CF1,
+};
 use crate::folding::nova::{CommittedInstance, Witness};
-use crate::transcript::Transcript;
+use crate::transcript::{Transcript, TranscriptVar};
 use crate::utils::vec::{hadamard, mat_vec_mul, vec_add, vec_scalar_mul, vec_sub};
 use crate::Error;
+
+/// ChallengeGadget computes the RO challenge used for the Nova instances NIFS, it contains a
+/// rust-native and a in-circuit compatible versions.
+pub struct ChallengeGadget<C: CurveGroup, CI: Absorb> {
+    _c: PhantomData<C>,
+    _ci: PhantomData<CI>,
+}
+impl<C: CurveGroup, CI: Absorb> ChallengeGadget<C, CI>
+where
+    C: CurveGroup,
+    // <C as CurveGroup>::BaseField: PrimeField,
+    <C as Group>::ScalarField: Absorb,
+{
+    pub fn get_challenge_native<T: Transcript<C::ScalarField>>(
+        transcript: &mut T,
+        pp_hash: C::ScalarField, // public params hash
+        U_i: &CI,
+        u_i: &CI,
+        cmT: Option<&C>,
+    ) -> Vec<bool> {
+        transcript.absorb(&pp_hash);
+        transcript.absorb(&U_i);
+        transcript.absorb(&u_i);
+        // in the Nova case we absorb the cmT, in Ova case we don't since it is not used.
+        if let Some(cmT_value) = cmT {
+            transcript.absorb_nonnative(cmT_value);
+        }
+        transcript.squeeze_bits(NOVA_N_BITS_RO)
+    }
+
+    // compatible with the native get_challenge_native
+    pub fn get_challenge_gadget<
+        S: CryptographicSponge,
+        T: TranscriptVar<CF1<C>, S>,
+        CIVar: AbsorbGadget<CF1<C>>,
+    >(
+        transcript: &mut T,
+        pp_hash: FpVar<CF1<C>>,      // public params hash
+        U_i_vec: Vec<FpVar<CF1<C>>>, // apready processed input, so we don't have to recompute these values
+        u_i: CIVar,
+        cmT: Option<NonNativeAffineVar<C>>,
+    ) -> Result<Vec<Boolean<C::ScalarField>>, SynthesisError> {
+        transcript.absorb(&pp_hash)?;
+        transcript.absorb(&U_i_vec)?;
+        transcript.absorb(&u_i)?;
+        // in the Nova case we absorb the cmT, in Ova case we don't since it is not used.
+        if let Some(cmT_value) = cmT {
+            transcript.absorb_nonnative(&cmT_value)?;
+        }
+        transcript.squeeze_bits(NOVA_N_BITS_RO)
+    }
+}
 
 /// Implements the Non-Interactive Folding Scheme described in section 4 of
 /// [Nova](https://eprint.iacr.org/2021/370.pdf).

@@ -0,0 +1,237 @@
+/// contains [Nova](https://eprint.iacr.org/2021/370.pdf) NIFS related circuits
+use ark_crypto_primitives::sponge::{constraints::AbsorbGadget, Absorb, CryptographicSponge};
+use ark_ec::{CurveGroup, Group};
+use ark_r1cs_std::{
+    alloc::{AllocVar, AllocationMode},
+    boolean::Boolean,
+    eq::EqGadget,
+    fields::{fp::FpVar, FieldVar},
+    uint8::UInt8,
+    ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::{ConstraintSystemRef, Namespace, SynthesisError};
+use ark_std::{fmt::Debug, Zero};
+use core::{borrow::Borrow, marker::PhantomData};
+
+use super::NIFSGadgetTrait;
+use crate::folding::circuits::{
+    nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
+    CF1, CF2,
+};
+use crate::folding::nova::CommittedInstance;
+use crate::folding::traits::CommittedInstanceVarOps;
+use crate::transcript::TranscriptVar;
+
+use super::nova::ChallengeGadget;
+
+/// CommittedInstanceVar contains the u, x, cmE and cmW values which are folded on the main Nova
+/// constraints field (E1::Fr, where E1 is the main curve). The peculiarity is that cmE and cmW are
+/// represented non-natively over the constraint field.
+#[derive(Debug, Clone)]
+pub struct CommittedInstanceVar<C: CurveGroup> {
+    pub u: FpVar<C::ScalarField>,
+    pub x: Vec<FpVar<C::ScalarField>>,
+    pub cmE: NonNativeAffineVar<C>,
+    pub cmW: NonNativeAffineVar<C>,
+}
+
+impl<C> AllocVar<CommittedInstance<C>, CF1<C>> for CommittedInstanceVar<C>
+where
+    C: CurveGroup,
+{
+    fn new_variable<T: Borrow<CommittedInstance<C>>>(
+        cs: impl Into<Namespace<CF1<C>>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        f().and_then(|val| {
+            let cs = cs.into();
+
+            let u = FpVar::<C::ScalarField>::new_variable(cs.clone(), || Ok(val.borrow().u), mode)?;
+            let x: Vec<FpVar<C::ScalarField>> =
+                Vec::new_variable(cs.clone(), || Ok(val.borrow().x.clone()), mode)?;
+
+            let cmE =
+                NonNativeAffineVar::<C>::new_variable(cs.clone(), || Ok(val.borrow().cmE), mode)?;
+            let cmW =
+                NonNativeAffineVar::<C>::new_variable(cs.clone(), || Ok(val.borrow().cmW), mode)?;
+
+            Ok(Self { u, x, cmE, cmW })
+        })
+    }
+}
+
+impl<C> AbsorbGadget<C::ScalarField> for CommittedInstanceVar<C>
+where
+    C: CurveGroup,
+{
+    fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
+        FpVar::batch_to_sponge_bytes(&self.to_sponge_field_elements()?)
+    }
+
+    fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
+        Ok([
+            vec![self.u.clone()],
+            self.x.clone(),
+            self.cmE.to_constraint_field()?,
+            self.cmW.to_constraint_field()?,
+        ]
+        .concat())
+    }
+}
+
+impl<C: CurveGroup> CommittedInstanceVarOps<C> for CommittedInstanceVar<C> {
+    type PointVar = NonNativeAffineVar<C>;
+
+    fn get_commitments(&self) -> Vec<Self::PointVar> {
+        vec![self.cmW.clone(), self.cmE.clone()]
+    }
+
+    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>] {
+        &self.x
+    }
+
+    fn enforce_incoming(&self) -> Result<(), SynthesisError> {
+        let zero = NonNativeUintVar::new_constant(ConstraintSystemRef::None, CF2::<C>::zero())?;
+        self.cmE.x.enforce_equal_unaligned(&zero)?;
+        self.cmE.y.enforce_equal_unaligned(&zero)?;
+        self.u.enforce_equal(&FpVar::one())
+    }
+
+    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.u.enforce_equal(&other.u)?;
+        self.x.enforce_equal(&other.x)
+    }
+}
+
+/// Implements the circuit that does the checks of the Non-Interactive Folding Scheme Verifier
+/// described in section 4 of [Nova](https://eprint.iacr.org/2021/370.pdf), where the cmE & cmW checks are
+/// delegated to the NIFSCycleFoldGadget.
+pub struct NIFSGadget<C: CurveGroup, S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>> {
+    _c: PhantomData<C>,
+    _s: PhantomData<S>,
+    _t: PhantomData<T>,
+}
+
+impl<C, S, T> NIFSGadgetTrait<C, S, T> for NIFSGadget<C, S, T>
+where
+    C: CurveGroup,
+    S: CryptographicSponge,
+    T: TranscriptVar<CF1<C>, S>,
+    <C as Group>::ScalarField: Absorb,
+{
+    type CommittedInstance = CommittedInstance<C>;
+    type CommittedInstanceVar = CommittedInstanceVar<C>;
+    type Proof = C;
+    type ProofVar = NonNativeAffineVar<C>;
+
+    fn verify(
+        transcript: &mut T,
+        pp_hash: FpVar<CF1<C>>,
+        U_i: Self::CommittedInstanceVar,
+        // U_i_vec is passed to reuse the already computed U_i_vec from previous methods
+        U_i_vec: Vec<FpVar<CF1<C>>>,
+        u_i: Self::CommittedInstanceVar,
+        cmT: Option<Self::ProofVar>,
+    ) -> Result<(Self::CommittedInstanceVar, Vec<Boolean<CF1<C>>>), SynthesisError> {
+        let r_bits = ChallengeGadget::<C, CommittedInstance<C>>::get_challenge_gadget(
+            transcript,
+            pp_hash.clone(),
+            U_i_vec,
+            u_i.clone(),
+            cmT.clone(),
+        )?;
+        let r = Boolean::le_bits_to_fp_var(&r_bits)?;
+
+        Ok((
+            Self::CommittedInstanceVar {
+                cmE: NonNativeAffineVar::new_constant(ConstraintSystemRef::None, C::zero())?,
+                cmW: NonNativeAffineVar::new_constant(ConstraintSystemRef::None, C::zero())?,
+                // ci3.u = U_i.u + r * u_i.u
+                u: U_i.u + &r * u_i.u,
+                // ci3.x = U_i.x + r * u_i.x
+                x: U_i
+                    .x
+                    .iter()
+                    .zip(u_i.x)
+                    .map(|(a, b)| a + &r * &b)
+                    .collect::<Vec<FpVar<CF1<C>>>>(),
+            },
+            r_bits,
+        ))
+    }
+}
+
+#[cfg(test)]
+pub mod tests {
+    use super::*;
+    use ark_crypto_primitives::sponge::poseidon::constraints::PoseidonSpongeVar;
+    use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
+    use ark_pallas::{Fr, Projective};
+    use ark_r1cs_std::R1CSVar;
+    use ark_std::UniformRand;
+
+    use crate::commitment::pedersen::Pedersen;
+    use crate::folding::nova::nifs::{
+        nova::NIFS,
+        tests::{
+            test_committed_instance_hash_opt, test_committed_instance_to_sponge_preimage_opt,
+            test_nifs_gadget_opt,
+        },
+    };
+
+    #[test]
+    fn test_nifs_gadget() {
+        let mut rng = ark_std::test_rng();
+        // prepare the committed instances to test in-circuit
+        let ci: Vec<CommittedInstance<Projective>> = (0..2)
+            .into_iter()
+            .map(|_| CommittedInstance::<Projective> {
+                cmE: Projective::rand(&mut rng),
+                u: Fr::rand(&mut rng),
+                cmW: Projective::rand(&mut rng),
+                x: vec![Fr::rand(&mut rng); 1],
+            })
+            .collect();
+        let cmT = Projective::rand(&mut rng);
+
+        let (ci_out, ciVar_out) = test_nifs_gadget_opt::<
+            NIFS<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+            NIFSGadget<Projective, PoseidonSponge<Fr>, PoseidonSpongeVar<Fr>>,
+        >(ci, cmT)
+        .unwrap();
+        assert_eq!(ciVar_out.u.value().unwrap(), ci_out.u);
+        assert_eq!(ciVar_out.x.value().unwrap(), ci_out.x);
+    }
+
+    #[test]
+    fn test_committed_instance_to_sponge_preimage() {
+        let mut rng = ark_std::test_rng();
+        let ci = CommittedInstance::<Projective> {
+            cmE: Projective::rand(&mut rng),
+            u: Fr::rand(&mut rng),
+            cmW: Projective::rand(&mut rng),
+            x: vec![Fr::rand(&mut rng); 1],
+        };
+
+        test_committed_instance_to_sponge_preimage_opt::<
+            NIFS<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+            NIFSGadget<Projective, PoseidonSponge<Fr>, PoseidonSpongeVar<Fr>>,
+        >(ci);
+    }
+
+    #[test]
+    fn test_committed_instance_hash() {
+        let mut rng = ark_std::test_rng();
+        let ci = CommittedInstance::<Projective> {
+            cmE: Projective::rand(&mut rng),
+            u: Fr::rand(&mut rng),
+            cmW: Projective::rand(&mut rng),
+            x: vec![Fr::rand(&mut rng); 1],
+        };
+        test_committed_instance_hash_opt::<
+            NIFS<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+            NIFSGadget<Projective, PoseidonSponge<Fr>, PoseidonSpongeVar<Fr>>,
+        >(ci);
+    }
+}

@@ -9,10 +9,12 @@ use ark_std::rand::RngCore;
 use ark_std::{One, UniformRand, Zero};
 use std::marker::PhantomData;
 
+use super::nova::ChallengeGadget;
+use super::ova_circuits::CommittedInstanceVar;
 use super::NIFSTrait;
 use crate::arith::r1cs::R1CS;
 use crate::commitment::CommitmentScheme;
-use crate::folding::nova::circuits::ChallengeGadget;
+use crate::folding::traits::{CommittedInstanceOps, Inputize};
 use crate::folding::{circuits::CF1, traits::Dummy};
 use crate::transcript::{AbsorbNonNative, Transcript};
 use crate::utils::vec::{hadamard, mat_vec_mul, vec_scalar_mul, vec_sub};
@@ -50,6 +52,24 @@ .to_sponge_field_elements(dest);
     }
 }
 
+impl<C: CurveGroup> CommittedInstanceOps<C> for CommittedInstance<C> {
+    type Var = CommittedInstanceVar<C>;
+
+    fn get_commitments(&self) -> Vec<C> {
+        vec![self.cmWE]
+    }
+
+    fn is_incoming(&self) -> bool {
+        self.u == One::one()
+    }
+}
+
+impl<C: CurveGroup> Inputize<C::ScalarField, CommittedInstanceVar<C>> for CommittedInstance<C> {
+    fn inputize(&self) -> Vec<C::ScalarField> {
+        [&[self.u][..], &self.x, &self.cmWE.inputize()].concat()
+    }
+}
+
 /// A Witness in Ova is represented by `w`. It also contains a blinder which can or not be used
 /// when committing to the witness itself.
 #[derive(Debug, Clone, Eq, PartialEq, CanonicalSerialize, CanonicalDeserialize)]
@@ -121,7 +141,9 @@ {
     type CommittedInstance = CommittedInstance<C>;
     type Witness = Witness<C>;
     type ProverAux = ();
-    type Proof = ();
+    // Proof is unused, but set to C::ScalarField so that the NIFSGadgetTrait abstraction can
+    // define the ProofsVar implementing the AllocVar from Proof
+    type Proof = C::ScalarField;
 
     fn new_witness(w: Vec<C::ScalarField>, _e_len: usize, rng: impl RngCore) -> Self::Witness {
         Witness::new::<H>(w, rng)
@@ -182,11 +204,12 @@ .ok_or(Error::OutOfBounds)?;
 
         let w = Self::fold_witness(r_Fr, W_i, w_i, &())?;
 
-        let (ci, _r_bits_v) = Self::verify(&mut transcript_v, pp_hash, U_i, u_i, &())?;
+        let proof = C::ScalarField::zero();
+        let (ci, _r_bits_v) = Self::verify(&mut transcript_v, pp_hash, U_i, u_i, &proof)?;
         #[cfg(test)]
         assert_eq!(_r_bits_v, r_bits);
 
-        Ok((w, ci, (), r_bits))
+        Ok((w, ci, proof, r_bits))
     }
 
     fn verify(
@@ -203,7 +226,7 @@ let r = C::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits))
             .ok_or(Error::OutOfBounds)?;
 
         // recall that r=alpha, and u=mu between Nova and Ova respectively
-        let u = U_i.u + r; // u_i.u is always 1 IN ova as we just can do sequential IVC.
+        let u = U_i.u + r; // u_i.u is always 1 in Ova as we just can do IVC (not PCD).
         let cmWE = U_i.cmWE + u_i.cmWE.mul(r);
         let x = U_i
             .x

@@ -0,0 +1,224 @@
+/// contains [Ova](https://hackmd.io/V4838nnlRKal9ZiTHiGYzw) NIFS related circuits
+use ark_crypto_primitives::sponge::{constraints::AbsorbGadget, Absorb, CryptographicSponge};
+use ark_ec::{CurveGroup, Group};
+use ark_ff::PrimeField;
+use ark_r1cs_std::{
+    alloc::{AllocVar, AllocationMode},
+    boolean::Boolean,
+    eq::EqGadget,
+    fields::{fp::FpVar, FieldVar},
+    uint8::UInt8,
+    ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::{ConstraintSystemRef, Namespace, SynthesisError};
+use ark_std::fmt::Debug;
+use core::{borrow::Borrow, marker::PhantomData};
+
+use super::ova::CommittedInstance;
+use super::NIFSGadgetTrait;
+use crate::folding::circuits::{nonnative::affine::NonNativeAffineVar, CF1};
+use crate::folding::traits::CommittedInstanceVarOps;
+use crate::transcript::TranscriptVar;
+
+use crate::folding::nova::nifs::nova::ChallengeGadget;
+
+#[derive(Debug, Clone)]
+pub struct CommittedInstanceVar<C: CurveGroup> {
+    pub u: FpVar<C::ScalarField>,
+    pub x: Vec<FpVar<C::ScalarField>>,
+    pub cmWE: NonNativeAffineVar<C>,
+}
+
+impl<C> AllocVar<CommittedInstance<C>, CF1<C>> for CommittedInstanceVar<C>
+where
+    C: CurveGroup,
+{
+    fn new_variable<T: Borrow<CommittedInstance<C>>>(
+        cs: impl Into<Namespace<CF1<C>>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        f().and_then(|val| {
+            let cs = cs.into();
+
+            let u = FpVar::<C::ScalarField>::new_variable(cs.clone(), || Ok(val.borrow().u), mode)?;
+            let x: Vec<FpVar<C::ScalarField>> =
+                Vec::new_variable(cs.clone(), || Ok(val.borrow().x.clone()), mode)?;
+
+            let cmWE =
+                NonNativeAffineVar::<C>::new_variable(cs.clone(), || Ok(val.borrow().cmWE), mode)?;
+
+            Ok(Self { u, x, cmWE })
+        })
+    }
+}
+
+impl<C> AbsorbGadget<C::ScalarField> for CommittedInstanceVar<C>
+where
+    C: CurveGroup,
+    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
+{
+    fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
+        FpVar::batch_to_sponge_bytes(&self.to_sponge_field_elements()?)
+    }
+
+    fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
+        Ok([
+            vec![self.u.clone()],
+            self.x.clone(),
+            self.cmWE.to_constraint_field()?,
+        ]
+        .concat())
+    }
+}
+
+impl<C: CurveGroup> CommittedInstanceVarOps<C> for CommittedInstanceVar<C> {
+    type PointVar = NonNativeAffineVar<C>;
+
+    fn get_commitments(&self) -> Vec<Self::PointVar> {
+        vec![self.cmWE.clone()]
+    }
+
+    fn get_public_inputs(&self) -> &[FpVar<CF1<C>>] {
+        &self.x
+    }
+
+    fn enforce_incoming(&self) -> Result<(), SynthesisError> {
+        self.u.enforce_equal(&FpVar::one())
+    }
+
+    fn enforce_partial_equal(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.u.enforce_equal(&other.u)?;
+        self.x.enforce_equal(&other.x)
+    }
+}
+
+/// Implements the circuit that does the checks of the Non-Interactive Folding Scheme Verifier
+/// described of the Ova variant, where the cmWE check is delegated to the NIFSCycleFoldGadget.
+pub struct NIFSGadget<C: CurveGroup, S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>> {
+    _c: PhantomData<C>,
+    _s: PhantomData<S>,
+    _t: PhantomData<T>,
+}
+
+impl<C, S, T> NIFSGadgetTrait<C, S, T> for NIFSGadget<C, S, T>
+where
+    C: CurveGroup,
+    S: CryptographicSponge,
+    T: TranscriptVar<CF1<C>, S>,
+    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
+
+    <C as Group>::ScalarField: Absorb,
+    <C as CurveGroup>::BaseField: PrimeField,
+{
+    type CommittedInstance = CommittedInstance<C>;
+    type CommittedInstanceVar = CommittedInstanceVar<C>;
+    type Proof = C::ScalarField;
+    type ProofVar = FpVar<C::ScalarField>; // unused
+
+    fn verify(
+        transcript: &mut T,
+        pp_hash: FpVar<CF1<C>>,
+        U_i: Self::CommittedInstanceVar,
+        // U_i_vec is passed to reuse the already computed U_i_vec from previous methods
+        U_i_vec: Vec<FpVar<CF1<C>>>,
+        u_i: Self::CommittedInstanceVar,
+        _proof: Option<Self::ProofVar>,
+    ) -> Result<(Self::CommittedInstanceVar, Vec<Boolean<CF1<C>>>), SynthesisError> {
+        let r_bits = ChallengeGadget::<C, CommittedInstance<C>>::get_challenge_gadget(
+            transcript,
+            pp_hash.clone(),
+            U_i_vec,
+            u_i.clone(),
+            None,
+        )?;
+        let r = Boolean::le_bits_to_fp_var(&r_bits)?;
+
+        Ok((
+            Self::CommittedInstanceVar {
+                cmWE: NonNativeAffineVar::new_constant(ConstraintSystemRef::None, C::zero())?,
+                // ci3.u = U_i.u + r * u_i.u (u_i.u is always 1 in Ova)
+                u: U_i.u + &r,
+                // ci3.x = U_i.x + r * u_i.x
+                x: U_i
+                    .x
+                    .iter()
+                    .zip(u_i.x)
+                    .map(|(a, b)| a + &r * &b)
+                    .collect::<Vec<FpVar<CF1<C>>>>(),
+            },
+            r_bits,
+        ))
+    }
+}
+
+#[cfg(test)]
+pub mod tests {
+    use super::*;
+    use ark_crypto_primitives::sponge::poseidon::constraints::PoseidonSpongeVar;
+    use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
+    use ark_pallas::{Fr, Projective};
+    use ark_r1cs_std::R1CSVar;
+    use ark_std::UniformRand;
+    use ark_std::Zero;
+
+    use crate::commitment::pedersen::Pedersen;
+    use crate::folding::nova::nifs::{
+        ova::NIFS,
+        tests::{
+            test_committed_instance_hash_opt, test_committed_instance_to_sponge_preimage_opt,
+            test_nifs_gadget_opt,
+        },
+    };
+
+    #[test]
+    fn test_nifs_gadget() {
+        let mut rng = ark_std::test_rng();
+        // prepare the committed instances to test in-circuit
+        let ci: Vec<CommittedInstance<Projective>> = (0..2)
+            .into_iter()
+            .map(|_| CommittedInstance::<Projective> {
+                u: Fr::rand(&mut rng),
+                x: vec![Fr::rand(&mut rng); 1],
+                cmWE: Projective::rand(&mut rng),
+            })
+            .collect();
+
+        let (ci_out, ciVar_out) = test_nifs_gadget_opt::<
+            NIFS<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+            NIFSGadget<Projective, PoseidonSponge<Fr>, PoseidonSpongeVar<Fr>>,
+        >(ci, Fr::zero())
+        .unwrap();
+        assert_eq!(ciVar_out.u.value().unwrap(), ci_out.u);
+        assert_eq!(ciVar_out.x.value().unwrap(), ci_out.x);
+    }
+
+    #[test]
+    fn test_committed_instance_to_sponge_preimage() {
+        let mut rng = ark_std::test_rng();
+        let ci = CommittedInstance::<Projective> {
+            u: Fr::rand(&mut rng),
+            x: vec![Fr::rand(&mut rng); 1],
+            cmWE: Projective::rand(&mut rng),
+        };
+
+        test_committed_instance_to_sponge_preimage_opt::<
+            NIFS<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+            NIFSGadget<Projective, PoseidonSponge<Fr>, PoseidonSpongeVar<Fr>>,
+        >(ci);
+    }
+
+    #[test]
+    fn test_committed_instance_hash() {
+        let mut rng = ark_std::test_rng();
+        let ci = CommittedInstance::<Projective> {
+            u: Fr::rand(&mut rng),
+            x: vec![Fr::rand(&mut rng); 1],
+            cmWE: Projective::rand(&mut rng),
+        };
+        test_committed_instance_hash_opt::<
+            NIFS<Projective, Pedersen<Projective>, PoseidonSponge<Fr>>,
+            NIFSGadget<Projective, PoseidonSponge<Fr>, PoseidonSpongeVar<Fr>>,
+        >(ci);
+    }
+}

@@ -3,8 +3,8 @@ use ark_r1cs_std::fields::fp::FpVar;
 use ark_relations::r1cs::SynthesisError;
 use ark_std::{rand::RngCore, UniformRand};
 
-use super::circuits::CommittedInstanceVar;
 use super::decider_eth_circuit::WitnessVar;
+use super::nifs::nova_circuits::CommittedInstanceVar;
 use super::{CommittedInstance, Witness};
 use crate::arith::{
     r1cs::{circuits::R1CSMatricesVar, R1CS},

@@ -5,4 +5,4 @@ authors = [""]
 compiler_version = ">=0.30.0"
 
 [dependencies]
-
+mimc = { tag = "v0.1.0", git = "https://github.com/noir-lang/mimc" }

@@ -1,6 +1,4 @@
-use dep::std;
-
 pub fn main(x: pub [Field; 1]) -> pub Field {
-  let hash = std::hash::mimc::mimc_bn254(x);
+  let hash = mimc::mimc_bn254(x);
   hash
 }

@@ -137,9 +137,6 @@ for (uint8 k = 0; k < {{num_limbs}}; k++) {
                 public_inputs[{{ z_len * 2 + 2 + num_limbs * 4 }} + 4 + k] = cmT_x_limbs[k]; 
                 public_inputs[{{ z_len * 2 + 2 + num_limbs * 5 }} + 4 + k] = cmT_y_limbs[k];
             }
-        
-            // last element of the groth16 proof's public inputs is `r`
-            public_inputs[{{ public_inputs_len - 2 }}] = cmT_r[2];
 
             bool success_g16 = this.verifyProof(pA, pB, pC, public_inputs);
             require(success_g16 == true, "Groth16: verifying proof failed");