Onchain decider circuit for Protogalaxy (#145)

* Move r1cs and ccs to standalone folders

* Simplify type bounds of SparseMatrixVar

* Implement `EquivalenceGadget` trait for `FpVar` and `NonNativeUintVar`.

Together with the existing `MatrixGadget` and `VectorGadget`, we can now use the same logic for checking R1CS satisfiability of `R1CSVar` both natively and non-natively.

* Simplify trait bounds

* Implement `ArithGadget` for `R1CSMatricesVar` and `CCSMatricesVar`

* `PedersenGadget::commit` now takes slices as input

* Structs for proofs and auxiliary values in protogalaxy

* `u` in LCCCS should be `z[0]`

* `Inputize` trait

* Generic decider circuits

* Verifier should check the commitments in committed instances

* Update the comments according to the new docs

* Fix examples

* Add `DeciderEnabledNIFS::fold_group_elements_native` to wrap code for folding commitments

* Fix incorrect endian

* Format

* Get rid of `unwrap` when possible

examples/circom_full_flow.rs

@@ -19,9 +19,12 @@ use std::time::Instant;
 
 use folding_schemes::{
     commitment::{kzg::KZG, pedersen::Pedersen},
-    folding::nova::{
-        decider_eth::{prepare_calldata, Decider as DeciderEth},
-        Nova, PreprocessorParam,
+    folding::{
+        nova::{
+            decider_eth::{prepare_calldata, Decider as DeciderEth},
+            Nova, PreprocessorParam,
+        },
+        traits::CommittedInstanceOps,
     },
     frontend::FCircuit,
     transcript::poseidon::poseidon_canonical_config,
@@ -83,7 +86,6 @@ fn main() {
     // prepare the Nova prover & verifier params
     let nova_preprocess_params = PreprocessorParam::new(poseidon_config, f_circuit.clone());
     let nova_params = N::preprocess(&mut rng, &nova_preprocess_params).unwrap();
-    let pp_hash = nova_params.1.pp_hash().unwrap();
 
     // initialize the folding scheme engine, in our case we use Nova
     let mut nova = N::init(&nova_params, f_circuit.clone(), z_0).unwrap();
@@ -117,8 +119,8 @@ fn main() {
         nova.i,
         nova.z_0.clone(),
         nova.z_i.clone(),
-        &nova.U_i,
-        &nova.u_i,
+        &nova.U_i.get_commitments(),
+        &nova.u_i.get_commitments(),
         &proof,
     )
     .unwrap();
@@ -131,7 +133,6 @@ fn main() {
 
     let calldata: Vec<u8> = prepare_calldata(
         function_selector,
-        pp_hash,
         nova.i,
         nova.z_0,
         nova.z_i,

examples/full_flow.rs

@@ -21,9 +21,12 @@ use std::time::Instant;
 
 use folding_schemes::{
     commitment::{kzg::KZG, pedersen::Pedersen},
-    folding::nova::{
-        decider_eth::{prepare_calldata, Decider as DeciderEth},
-        Nova, PreprocessorParam,
+    folding::{
+        nova::{
+            decider_eth::{prepare_calldata, Decider as DeciderEth},
+            Nova, PreprocessorParam,
+        },
+        traits::CommittedInstanceOps,
     },
     frontend::FCircuit,
     transcript::poseidon::poseidon_canonical_config,
@@ -101,7 +104,6 @@ fn main() {
     // prepare the Nova prover & verifier params
     let nova_preprocess_params = PreprocessorParam::new(poseidon_config.clone(), f_circuit);
     let nova_params = N::preprocess(&mut rng, &nova_preprocess_params).unwrap();
-    let pp_hash = nova_params.1.pp_hash().unwrap();
 
     // initialize the folding scheme engine, in our case we use Nova
     let mut nova = N::init(&nova_params, f_circuit, z_0).unwrap();
@@ -125,8 +127,8 @@ fn main() {
         nova.i,
         nova.z_0.clone(),
         nova.z_i.clone(),
-        &nova.U_i,
-        &nova.u_i,
+        &nova.U_i.get_commitments(),
+        &nova.u_i.get_commitments(),
         &proof,
     )
     .unwrap();
@@ -139,7 +141,6 @@ fn main() {
 
     let calldata: Vec<u8> = prepare_calldata(
         function_selector,
-        pp_hash,
         nova.i,
         nova.z_0,
         nova.z_i,

examples/noir_full_flow.rs

@@ -16,9 +16,12 @@ use ark_grumpkin::{constraints::GVar as GVar2, Projective as G2};
 
 use folding_schemes::{
     commitment::{kzg::KZG, pedersen::Pedersen},
-    folding::nova::{
-        decider_eth::{prepare_calldata, Decider as DeciderEth},
-        Nova, PreprocessorParam,
+    folding::{
+        nova::{
+            decider_eth::{prepare_calldata, Decider as DeciderEth},
+            Nova, PreprocessorParam,
+        },
+        traits::CommittedInstanceOps,
     },
     frontend::FCircuit,
     transcript::poseidon::poseidon_canonical_config,
@@ -46,7 +49,7 @@ fn main() {
         cur_path.to_str().unwrap()
     );
 
-    let circuit = load_noir_circuit(circuit_path);
+    let circuit = load_noir_circuit(circuit_path).unwrap();
     let f_circuit = NoirFCircuit {
         circuit,
         state_len: 1,
@@ -72,7 +75,6 @@ fn main() {
     // prepare the Nova prover & verifier params
     let nova_preprocess_params = PreprocessorParam::new(poseidon_config, f_circuit.clone());
     let nova_params = N::preprocess(&mut rng, &nova_preprocess_params).unwrap();
-    let pp_hash = nova_params.1.pp_hash().unwrap();
 
     // initialize the folding scheme engine, in our case we use Nova
     let mut nova = N::init(&nova_params, f_circuit.clone(), z_0).unwrap();
@@ -104,8 +106,8 @@ fn main() {
         nova.i,
         nova.z_0.clone(),
         nova.z_i.clone(),
-        &nova.U_i,
-        &nova.u_i,
+        &nova.U_i.get_commitments(),
+        &nova.u_i.get_commitments(),
         &proof,
     )
     .unwrap();
@@ -118,7 +120,6 @@ fn main() {
 
     let calldata: Vec<u8> = prepare_calldata(
         function_selector,
-        pp_hash,
         nova.i,
         nova.z_0,
         nova.z_i,

examples/noname_full_flow.rs

@@ -17,9 +17,12 @@ use ark_grumpkin::{constraints::GVar as GVar2, Projective as G2};
 
 use folding_schemes::{
     commitment::{kzg::KZG, pedersen::Pedersen},
-    folding::nova::{
-        decider_eth::{prepare_calldata, Decider as DeciderEth},
-        Nova, PreprocessorParam,
+    folding::{
+        nova::{
+            decider_eth::{prepare_calldata, Decider as DeciderEth},
+            Nova, PreprocessorParam,
+        },
+        traits::CommittedInstanceOps,
     },
     frontend::FCircuit,
     transcript::poseidon::poseidon_canonical_config,
@@ -85,7 +88,6 @@ fn main() {
     // prepare the Nova prover & verifier params
     let nova_preprocess_params = PreprocessorParam::new(poseidon_config, f_circuit.clone());
     let nova_params = N::preprocess(&mut rng, &nova_preprocess_params).unwrap();
-    let pp_hash = nova_params.1.pp_hash().unwrap();
 
     // initialize the folding scheme engine, in our case we use Nova
     let mut nova = N::init(&nova_params, f_circuit.clone(), z_0).unwrap();
@@ -119,8 +121,8 @@ fn main() {
         nova.i,
         nova.z_0.clone(),
         nova.z_i.clone(),
-        &nova.U_i,
-        &nova.u_i,
+        &nova.U_i.get_commitments(),
+        &nova.u_i.get_commitments(),
         &proof,
     )
     .unwrap();
@@ -133,7 +135,6 @@ fn main() {
 
     let calldata: Vec<u8> = prepare_calldata(
         function_selector,
-        pp_hash,
         nova.i,
         nova.z_0,
         nova.z_i,

folding-schemes/src/arith/ccs/circuits.rs

@@ -0,0 +1,35 @@
+use super::CCS;
+use crate::utils::gadgets::SparseMatrixVar;
+use ark_ff::PrimeField;
+use ark_r1cs_std::{
+    alloc::{AllocVar, AllocationMode},
+    fields::fp::FpVar,
+};
+use ark_relations::r1cs::{Namespace, SynthesisError};
+use ark_std::borrow::Borrow;
+/// CCSMatricesVar contains the matrices 'M' of the CCS without the rest of CCS parameters.
+///
+#[derive(Debug, Clone)]
+pub struct CCSMatricesVar<F: PrimeField> {
+    // we only need native representation, so the constraint field==F
+    pub M: Vec<SparseMatrixVar<FpVar<F>>>,
+}
+
+impl<F: PrimeField> AllocVar<CCS<F>, F> for CCSMatricesVar<F> {
+    fn new_variable<T: Borrow<CCS<F>>>(
+        cs: impl Into<Namespace<F>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        _mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        f().and_then(|val| {
+            let cs = cs.into();
+            let M: Vec<SparseMatrixVar<FpVar<F>>> = val
+                .borrow()
+                .M
+                .iter()
+                .map(|M| SparseMatrixVar::<FpVar<F>>::new_constant(cs.clone(), M.clone()))
+                .collect::<Result<_, SynthesisError>>()?;
+            Ok(Self { M })
+        })
+    }
+}

folding-schemes/src/arith/ccs.rs → folding-schemes/src/arith/ccs/mod.rs

@@ -9,6 +9,8 @@ use crate::Error;
 use super::ArithSerializer;
 use super::{r1cs::R1CS, Arith};
 
+pub mod circuits;
+
 /// CCS represents the Customizable Constraint Systems structure defined in
 /// the [CCS paper](https://eprint.iacr.org/2023/552)
 #[derive(Debug, Clone, Eq, PartialEq)]

folding-schemes/src/arith/r1cs/circuits.rs

@@ -0,0 +1,301 @@
+use crate::{
+    arith::ArithGadget,
+    utils::gadgets::{EquivalenceGadget, MatrixGadget, SparseMatrixVar, VectorGadget},
+};
+use ark_ff::PrimeField;
+use ark_r1cs_std::alloc::{AllocVar, AllocationMode};
+use ark_relations::r1cs::{Namespace, SynthesisError};
+use ark_std::{borrow::Borrow, marker::PhantomData, One};
+
+use super::R1CS;
+
+/// An in-circuit representation of the `R1CS` struct.
+///
+/// `M` is for the modulo operation involved in the satisfiability check when
+/// the underlying `FVar` is `NonNativeUintVar`.
+#[derive(Debug, Clone)]
+pub struct R1CSMatricesVar<M, FVar> {
+    _m: PhantomData<M>,
+    pub A: SparseMatrixVar<FVar>,
+    pub B: SparseMatrixVar<FVar>,
+    pub C: SparseMatrixVar<FVar>,
+}
+
+impl<F: PrimeField, ConstraintF: PrimeField, FVar: AllocVar<F, ConstraintF>>
+    AllocVar<R1CS<F>, ConstraintF> for R1CSMatricesVar<F, FVar>
+{
+    fn new_variable<T: Borrow<R1CS<F>>>(
+        cs: impl Into<Namespace<ConstraintF>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        _mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        f().and_then(|val| {
+            let cs = cs.into();
+
+            Ok(Self {
+                _m: PhantomData,
+                A: SparseMatrixVar::<FVar>::new_constant(cs.clone(), &val.borrow().A)?,
+                B: SparseMatrixVar::<FVar>::new_constant(cs.clone(), &val.borrow().B)?,
+                C: SparseMatrixVar::<FVar>::new_constant(cs.clone(), &val.borrow().C)?,
+            })
+        })
+    }
+}
+
+impl<M, FVar> R1CSMatricesVar<M, FVar>
+where
+    SparseMatrixVar<FVar>: MatrixGadget<FVar>,
+    [FVar]: VectorGadget<FVar>,
+{
+    pub fn eval_at_z(&self, z: &[FVar]) -> Result<(Vec<FVar>, Vec<FVar>), SynthesisError> {
+        // Multiply Cz by z[0] (u) here, allowing this method to be reused for
+        // both relaxed and unrelaxed R1CS.
+        let Az = self.A.mul_vector(z)?;
+        let Bz = self.B.mul_vector(z)?;
+        let Cz = self.C.mul_vector(z)?;
+        let uCz = Cz.mul_scalar(&z[0])?;
+        let AzBz = Az.hadamard(&Bz)?;
+        Ok((AzBz, uCz))
+    }
+}
+
+impl<M, FVar, WVar: AsRef<[FVar]>, UVar: AsRef<[FVar]>> ArithGadget<WVar, UVar>
+    for R1CSMatricesVar<M, FVar>
+where
+    SparseMatrixVar<FVar>: MatrixGadget<FVar>,
+    [FVar]: VectorGadget<FVar> + EquivalenceGadget<M>,
+    FVar: Clone + One,
+{
+    /// Evaluation is a tuple of two vectors (`AzBz` and `uCz`) instead of a
+    /// single vector `AzBz - uCz`, because subtraction is not supported for
+    /// `FVar = NonNativeUintVar`.
+    type Evaluation = (Vec<FVar>, Vec<FVar>);
+
+    fn eval_relation(&self, w: &WVar, u: &UVar) -> Result<Self::Evaluation, SynthesisError> {
+        self.eval_at_z(&[&[FVar::one()], u.as_ref(), w.as_ref()].concat())
+    }
+
+    fn enforce_evaluation(
+        _w: &WVar,
+        _u: &UVar,
+        (lhs, rhs): Self::Evaluation,
+    ) -> Result<(), SynthesisError> {
+        lhs.enforce_equivalent(&rhs)
+    }
+}
+
+#[cfg(test)]
+pub mod tests {
+    use std::cmp::max;
+
+    use ark_crypto_primitives::crh::{
+        sha256::{
+            constraints::{Sha256Gadget, UnitVar},
+            Sha256,
+        },
+        CRHScheme, CRHSchemeGadget,
+    };
+    use ark_ec::CurveGroup;
+    use ark_ff::BigInteger;
+    use ark_pallas::{Fq, Fr, Projective};
+    use ark_r1cs_std::{bits::uint8::UInt8, eq::EqGadget, fields::fp::FpVar};
+    use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem, ConstraintSystemRef};
+    use ark_std::{
+        rand::{thread_rng, Rng},
+        One, UniformRand,
+    };
+    use ark_vesta::{constraints::GVar as GVar2, Projective as Projective2};
+
+    use super::*;
+    use crate::arith::{
+        r1cs::{
+            extract_r1cs, extract_w_x,
+            tests::{get_test_r1cs, get_test_z},
+        },
+        Arith,
+    };
+    use crate::commitment::{pedersen::Pedersen, CommitmentScheme};
+    use crate::folding::{
+        circuits::{
+            cyclefold::{CycleFoldCommittedInstanceVar, CycleFoldWitnessVar},
+            nonnative::uint::NonNativeUintVar,
+        },
+        nova::{
+            circuits::CommittedInstanceVar, decider_eth_circuit::WitnessVar, CommittedInstance,
+            Witness,
+        },
+    };
+    use crate::frontend::{
+        utils::{CubicFCircuit, CustomFCircuit, WrapperCircuit},
+        FCircuit,
+    };
+
+    fn prepare_instances<C: CurveGroup, CS: CommitmentScheme<C>, R: Rng>(
+        mut rng: R,
+        r1cs: &R1CS<C::ScalarField>,
+        z: &[C::ScalarField],
+    ) -> (Witness<C>, CommittedInstance<C>) {
+        let (w, x) = r1cs.split_z(z);
+
+        let (cs_pp, _) = CS::setup(&mut rng, max(w.len(), r1cs.A.n_rows)).unwrap();
+
+        let mut w = Witness::new::<false>(w, r1cs.A.n_rows, &mut rng);
+        w.E = r1cs.eval_at_z(z).unwrap();
+        let mut u = w.commit::<CS, false>(&cs_pp, x).unwrap();
+        u.u = z[0];
+
+        (w, u)
+    }
+
+    #[test]
+    fn test_relaxed_r1cs_small_gadget_handcrafted() {
+        let rng = &mut thread_rng();
+
+        let r1cs: R1CS<Fr> = get_test_r1cs();
+        let mut z = get_test_z(3);
+        z[0] = Fr::rand(rng);
+        let (w, u) = prepare_instances::<_, Pedersen<Projective>, _>(rng, &r1cs, &z);
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        let wVar = WitnessVar::new_witness(cs.clone(), || Ok(w)).unwrap();
+        let uVar = CommittedInstanceVar::new_witness(cs.clone(), || Ok(u)).unwrap();
+        let r1csVar =
+            R1CSMatricesVar::<Fr, FpVar<Fr>>::new_witness(cs.clone(), || Ok(r1cs)).unwrap();
+
+        r1csVar.enforce_relation(&wVar, &uVar).unwrap();
+        assert!(cs.is_satisfied().unwrap());
+    }
+
+    // gets as input a circuit that implements the ConstraintSynthesizer trait, and that has been
+    // initialized.
+    fn test_relaxed_r1cs_gadget<CS: ConstraintSynthesizer<Fr>>(circuit: CS) {
+        let rng = &mut thread_rng();
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        circuit.generate_constraints(cs.clone()).unwrap();
+        cs.finalize();
+        assert!(cs.is_satisfied().unwrap());
+
+        let cs = cs.into_inner().unwrap();
+
+        let r1cs = extract_r1cs::<Fr>(&cs).unwrap();
+        let (w, x) = extract_w_x::<Fr>(&cs);
+        r1cs.check_relation(&w, &x).unwrap();
+        let mut z = [vec![Fr::one()], x, w].concat();
+        z[0] = Fr::rand(rng);
+
+        let (w, u) = prepare_instances::<_, Pedersen<Projective>, _>(rng, &r1cs, &z);
+        r1cs.check_relation(&w, &u).unwrap();
+
+        // set new CS for the circuit that checks the RelaxedR1CS of our original circuit
+        let cs = ConstraintSystem::<Fr>::new_ref();
+        // prepare the inputs for our circuit
+        let wVar = WitnessVar::new_witness(cs.clone(), || Ok(w)).unwrap();
+        let uVar = CommittedInstanceVar::new_witness(cs.clone(), || Ok(u)).unwrap();
+        let r1csVar =
+            R1CSMatricesVar::<Fr, FpVar<Fr>>::new_witness(cs.clone(), || Ok(r1cs)).unwrap();
+
+        r1csVar.enforce_relation(&wVar, &uVar).unwrap();
+        assert!(cs.is_satisfied().unwrap());
+    }
+
+    #[test]
+    fn test_relaxed_r1cs_small_gadget_arkworks() {
+        let z_i = vec![Fr::from(3_u32)];
+        let cubic_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
+        let circuit = WrapperCircuit::<Fr, CubicFCircuit<Fr>> {
+            FC: cubic_circuit,
+            z_i: Some(z_i.clone()),
+            z_i1: Some(cubic_circuit.step_native(0, z_i, vec![]).unwrap()),
+        };
+
+        test_relaxed_r1cs_gadget(circuit);
+    }
+
+    struct Sha256TestCircuit<F: PrimeField> {
+        _f: PhantomData<F>,
+        pub x: Vec<u8>,
+        pub y: Vec<u8>,
+    }
+    impl<F: PrimeField> ConstraintSynthesizer<F> for Sha256TestCircuit<F> {
+        fn generate_constraints(self, cs: ConstraintSystemRef<F>) -> Result<(), SynthesisError> {
+            let x = Vec::<UInt8<F>>::new_witness(cs.clone(), || Ok(self.x))?;
+            let y = Vec::<UInt8<F>>::new_input(cs.clone(), || Ok(self.y))?;
+
+            let unitVar = UnitVar::default();
+            let comp_y = <Sha256Gadget<F> as CRHSchemeGadget<Sha256, F>>::evaluate(&unitVar, &x)?;
+            comp_y.0.enforce_equal(&y)?;
+            Ok(())
+        }
+    }
+    #[test]
+    fn test_relaxed_r1cs_medium_gadget_arkworks() {
+        let x = Fr::from(5_u32).into_bigint().to_bytes_le();
+        let y = <Sha256 as CRHScheme>::evaluate(&(), x.clone()).unwrap();
+
+        let circuit = Sha256TestCircuit::<Fr> {
+            _f: PhantomData,
+            x,
+            y,
+        };
+        test_relaxed_r1cs_gadget(circuit);
+    }
+
+    #[test]
+    fn test_relaxed_r1cs_custom_circuit() {
+        let n_constraints = 10_000;
+        let custom_circuit = CustomFCircuit::<Fr>::new(n_constraints).unwrap();
+        let z_i = vec![Fr::from(5_u32)];
+        let circuit = WrapperCircuit::<Fr, CustomFCircuit<Fr>> {
+            FC: custom_circuit,
+            z_i: Some(z_i.clone()),
+            z_i1: Some(custom_circuit.step_native(0, z_i, vec![]).unwrap()),
+        };
+        test_relaxed_r1cs_gadget(circuit);
+    }
+
+    #[test]
+    fn test_relaxed_r1cs_nonnative_circuit() {
+        let rng = &mut thread_rng();
+
+        let cs = ConstraintSystem::<Fq>::new_ref();
+        // in practice we would use CycleFoldCircuit, but is a very big circuit (when computed
+        // non-natively inside the RelaxedR1CS circuit), so in order to have a short test we use a
+        // custom circuit.
+        let custom_circuit = CustomFCircuit::<Fq>::new(10).unwrap();
+        let z_i = vec![Fq::from(5_u32)];
+        let circuit = WrapperCircuit::<Fq, CustomFCircuit<Fq>> {
+            FC: custom_circuit,
+            z_i: Some(z_i.clone()),
+            z_i1: Some(custom_circuit.step_native(0, z_i, vec![]).unwrap()),
+        };
+        circuit.generate_constraints(cs.clone()).unwrap();
+        cs.finalize();
+        let cs = cs.into_inner().unwrap();
+        let r1cs = extract_r1cs::<Fq>(&cs).unwrap();
+        let (w, x) = extract_w_x::<Fq>(&cs);
+        let z = [vec![Fq::rand(rng)], x, w].concat();
+
+        let (w, u) = prepare_instances::<_, Pedersen<Projective2>, _>(rng, &r1cs, &z);
+
+        // natively
+        let cs = ConstraintSystem::<Fq>::new_ref();
+        let wVar = WitnessVar::new_witness(cs.clone(), || Ok(&w)).unwrap();
+        let uVar = CommittedInstanceVar::new_witness(cs.clone(), || Ok(&u)).unwrap();
+        let r1csVar =
+            R1CSMatricesVar::<Fq, FpVar<Fq>>::new_witness(cs.clone(), || Ok(&r1cs)).unwrap();
+        r1csVar.enforce_relation(&wVar, &uVar).unwrap();
+
+        // non-natively
+        let cs = ConstraintSystem::<Fr>::new_ref();
+        let wVar = CycleFoldWitnessVar::new_witness(cs.clone(), || Ok(w)).unwrap();
+        let uVar =
+            CycleFoldCommittedInstanceVar::<_, GVar2>::new_witness(cs.clone(), || Ok(u)).unwrap();
+        let r1csVar =
+            R1CSMatricesVar::<Fq, NonNativeUintVar<Fr>>::new_witness(cs.clone(), || Ok(r1cs))
+                .unwrap();
+        r1csVar.enforce_relation(&wVar, &uVar).unwrap();
+    }
+}

folding-schemes/src/arith/r1cs.rs → folding-schemes/src/arith/r1cs/mod.rs

@@ -10,6 +10,8 @@ use crate::utils::vec::{
 };
 use crate::Error;
 
+pub mod circuits;
+
 #[derive(Debug, Clone, Eq, PartialEq, CanonicalSerialize, CanonicalDeserialize)]
 pub struct R1CS<F: PrimeField> {
     pub l: usize, // io len
@@ -121,8 +123,14 @@ impl From> for R1CS {
 
 /// extracts arkworks ConstraintSystem matrices into crate::utils::vec::SparseMatrix format as R1CS
 /// struct.
-pub fn extract_r1cs<F: PrimeField>(cs: &ConstraintSystem<F>) -> R1CS<F> {
-    let m = cs.to_matrices().unwrap();
+pub fn extract_r1cs<F: PrimeField>(cs: &ConstraintSystem<F>) -> Result<R1CS<F>, Error> {
+    let m = cs.to_matrices().ok_or_else(|| {
+        Error::ConversionError(
+            "ConstraintSystem".into(),
+            "ConstraintMatrices".into(),
+            "The matrices have not been generated yet".into(),
+        )
+    })?;
 
     let n_rows = cs.num_constraints;
     let n_cols = cs.num_instance_variables + cs.num_witness_variables; // cs.num_instance_variables already counts the 1
@@ -143,12 +151,12 @@ pub fn extract_r1cs(cs: &ConstraintSystem) -> R1CS {
         coeffs: m.c,
     };
 
-    R1CS::<F> {
+    Ok(R1CS::<F> {
         l: cs.num_instance_variables - 1, // -1 to subtract the first '1'
         A,
         B,
         C,
-    }
+    })
 }
 
 /// extracts the witness and the public inputs from arkworks ConstraintSystem.

folding-schemes/src/commitment/ipa.rs

@@ -13,7 +13,6 @@ use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     boolean::Boolean,
     fields::{nonnative::NonNativeFieldVar, FieldVar},
-    groups::GroupOpsBounds,
     prelude::CurveVar,
     ToBitsGadget,
 };
@@ -492,9 +491,6 @@ impl IPAGadget
 where
     C: CurveGroup,
     GC: CurveVar<C, CF<C>>,
-
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
     /// Verify the IPA opening proof, K=log2(d), where d is the degree of the committed polynomial,
     /// and H indicates if the commitment is in hiding mode and thus uses blinding factors, if not,
@@ -515,7 +511,7 @@ where
             return Err(SynthesisError::Unsatisfiable);
         }
 
-        let P_ = P + U.scalar_mul_le(v.to_bits_le()?.iter())?;
+        let P_ = U.scalar_mul_le(v.to_bits_le()?.iter())? + P;
         let mut q_0 = P_;
         let mut r = r.clone();
 

folding-schemes/src/commitment/pedersen.rs

@@ -1,6 +1,6 @@
 use ark_ec::CurveGroup;
 use ark_ff::Field;
-use ark_r1cs_std::{boolean::Boolean, groups::GroupOpsBounds, prelude::CurveVar};
+use ark_r1cs_std::{boolean::Boolean, prelude::CurveVar};
 use ark_relations::r1cs::SynthesisError;
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
 use ark_std::Zero;
@@ -194,15 +194,12 @@ impl PedersenGadget
 where
     C: CurveGroup,
     GC: CurveVar<C, CF<C>>,
-
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
     pub fn commit(
-        h: GC,
-        g: Vec<GC>,
-        v: Vec<Vec<Boolean<CF<C>>>>,
-        r: Vec<Boolean<CF<C>>>,
+        h: &GC,
+        g: &[GC],
+        v: &[Vec<Boolean<CF<C>>>],
+        r: &[Boolean<CF<C>>],
     ) -> Result<GC, SynthesisError> {
         let mut res = GC::zero();
         if H {
@@ -303,7 +300,7 @@ mod tests {
 
         // use the gadget
         let cmVar =
-            PedersenGadget::<Projective, GVar, hiding>::commit(hVar, gVar, vVar, rVar).unwrap();
+            PedersenGadget::<Projective, GVar, hiding>::commit(&hVar, &gVar, &vVar, &rVar).unwrap();
         cmVar.enforce_equal(&expected_cmVar).unwrap();
     }
 }

folding-schemes/src/folding/circuits/cyclefold.rs

@@ -8,7 +8,6 @@ use ark_r1cs_std::{
     boolean::Boolean,
     eq::EqGadget,
     fields::fp::FpVar,
-    groups::GroupOpsBounds,
     prelude::CurveVar,
     ToConstraintFieldGadget,
 };
@@ -17,16 +16,23 @@ use ark_relations::r1cs::{
 };
 use ark_std::fmt::Debug;
 use ark_std::rand::RngCore;
-use ark_std::Zero;
+use ark_std::{One, Zero};
 use core::{borrow::Borrow, marker::PhantomData};
 
 use super::{nonnative::uint::NonNativeUintVar, CF1, CF2};
-use crate::arith::r1cs::{extract_w_x, R1CS};
 use crate::commitment::CommitmentScheme;
 use crate::constants::NOVA_N_BITS_RO;
 use crate::folding::nova::nifs::{nova::NIFS, NIFSTrait};
 use crate::transcript::{AbsorbNonNative, AbsorbNonNativeGadget, Transcript, TranscriptVar};
+use crate::utils::gadgets::{EquivalenceGadget, VectorGadget};
 use crate::Error;
+use crate::{
+    arith::{
+        r1cs::{circuits::R1CSMatricesVar, extract_w_x, R1CS},
+        ArithGadget,
+    },
+    folding::traits::Inputize,
+};
 use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
 
 /// Re-export the Nova committed instance as `CycleFoldCommittedInstance` and
@@ -35,13 +41,39 @@ pub use crate::folding::nova::{
     CommittedInstance as CycleFoldCommittedInstance, Witness as CycleFoldWitness,
 };
 
+impl<C: CurveGroup, GC: CurveVar<C, CF2<C>>> Inputize<CF2<C>, CycleFoldCommittedInstanceVar<C, GC>>
+    for CycleFoldCommittedInstance<C>
+{
+    fn inputize(&self) -> Vec<CF2<C>> {
+        let zero = (&C::BaseField::zero(), &C::BaseField::zero());
+        let cmE = self.cmE.into_affine();
+        let cmW = self.cmW.into_affine();
+        let (cmE_x, cmE_y) = cmE.xy().unwrap_or(zero);
+        let (cmW_x, cmW_y) = cmW.xy().unwrap_or(zero);
+        self.u
+            .inputize()
+            .into_iter()
+            .chain(self.x.iter().flat_map(|x| x.inputize()))
+            .chain(
+                [
+                    *cmE_x,
+                    *cmE_y,
+                    C::BaseField::one(),
+                    *cmW_x,
+                    *cmW_y,
+                    C::BaseField::one(),
+                ]
+                .into_iter()
+                .flat_map(|x| x.to_base_prime_field_elements()),
+            )
+            .collect()
+    }
+}
+
 /// CycleFoldCommittedInstanceVar is the CycleFold CommittedInstance represented
 /// in folding verifier circuit
 #[derive(Debug, Clone)]
-pub struct CycleFoldCommittedInstanceVar<C: CurveGroup, GC: CurveVar<C, CF2<C>>>
-where
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
-{
+pub struct CycleFoldCommittedInstanceVar<C: CurveGroup, GC: CurveVar<C, CF2<C>>> {
     pub cmE: GC,
     pub u: NonNativeUintVar<CF2<C>>,
     pub cmW: GC,
@@ -51,8 +83,6 @@ impl AllocVar, CF2> for CycleFoldCommitt
 where
     C: CurveGroup,
     GC: CurveVar<C, CF2<C>>,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
     fn new_variable<T: Borrow<CycleFoldCommittedInstance<C>>>(
         cs: impl Into<Namespace<CF2<C>>>,
@@ -101,8 +131,7 @@ impl AbsorbNonNativeGadget for CycleFoldCommittedInstanceVa
 where
     C: CurveGroup,
     GC: CurveVar<C, CF2<C>> + ToConstraintFieldGadget<CF2<C>>,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField + Absorb,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
+    C::BaseField: PrimeField + Absorb,
 {
     /// Extracts the underlying field elements from `CycleFoldCommittedInstanceVar`, in the order
     /// of `u`, `x`, `cmE.x`, `cmE.y`, `cmW.x`, `cmW.y`, `cmE.is_inf || cmW.is_inf` (|| is for
@@ -132,7 +161,7 @@ where
 
 impl<C: CurveGroup> CycleFoldCommittedInstance<C>
 where
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField + Absorb,
+    C::BaseField: PrimeField + Absorb,
 {
     /// hash_cyclefold implements the committed instance hash compatible with the
     /// in-circuit implementation `CycleFoldCommittedInstanceVar::hash`.
@@ -153,8 +182,7 @@ impl CycleFoldCommittedInstanceVar
 where
     C: CurveGroup,
     GC: CurveVar<C, CF2<C>> + ToConstraintFieldGadget<CF2<C>>,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField + Absorb,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
+    C::BaseField: PrimeField + Absorb,
 {
     /// hash implements the committed instance hash compatible with the native
     /// implementation `CycleFoldCommittedInstance::hash_cyclefold`.
@@ -165,7 +193,7 @@ where
     /// (reconstraining) them.
     #[allow(clippy::type_complexity)]
     pub fn hash<S: CryptographicSponge, T: TranscriptVar<CF2<C>, S>>(
-        self,
+        &self,
         sponge: &T,
         pp_hash: FpVar<CF2<C>>, // public params hash
     ) -> Result<(FpVar<CF2<C>>, Vec<FpVar<CF2<C>>>), SynthesisError> {
@@ -173,7 +201,11 @@ where
         let U_vec = self.to_native_sponge_field_elements()?;
         sponge.absorb(&pp_hash)?;
         sponge.absorb(&U_vec)?;
-        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
+        Ok((
+            // `unwrap` is safe because the sponge is guaranteed to return a single element
+            sponge.squeeze_field_elements(1)?.pop().unwrap(),
+            U_vec,
+        ))
     }
 }
 
@@ -182,10 +214,7 @@ where
 /// represented as native points, which are folded on the auxiliary curve constraints field (E2::Fr
 /// = E1::Fq).
 #[derive(Debug, Clone)]
-pub struct CommittedInstanceInCycleFoldVar<C: CurveGroup, GC: CurveVar<C, CF2<C>>>
-where
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
-{
+pub struct CommittedInstanceInCycleFoldVar<C: CurveGroup, GC: CurveVar<C, CF2<C>>> {
     _c: PhantomData<C>,
     pub cmE: GC,
     pub cmW: GC,
@@ -196,7 +225,6 @@ impl AllocVar, CF2>
 where
     C: CurveGroup,
     GC: CurveVar<C, CF2<C>>,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
     fn new_variable<T: Borrow<CycleFoldCommittedInstance<C>>>(
         cs: impl Into<Namespace<CF2<C>>>,
@@ -232,7 +260,7 @@ pub struct CycleFoldWitnessVar {
 impl<C> AllocVar<CycleFoldWitness<C>, CF2<C>> for CycleFoldWitnessVar<C>
 where
     C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: PrimeField,
+    C::BaseField: PrimeField,
 {
     fn new_variable<T: Borrow<CycleFoldWitness<C>>>(
         cs: impl Into<Namespace<CF2<C>>>,
@@ -265,8 +293,7 @@ impl>> NIFSFullGadget
 where
     C: CurveGroup,
     GC: CurveVar<C, CF2<C>>,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
+    C::BaseField: PrimeField,
 {
     pub fn fold_committed_instance(
         r_bits: Vec<Boolean<CF2<C>>>,
@@ -284,13 +311,13 @@ where
         Ok(CycleFoldCommittedInstanceVar {
             cmE: cmT.scalar_mul_le(r_bits.iter())? + ci1.cmE,
             cmW: ci1.cmW + ci2.cmW.scalar_mul_le(r_bits.iter())?,
-            u: ci1.u.add_no_align(&r_nonnat).modulo::<CF1<C>>()?,
+            u: ci1.u.add_no_align(&r_nonnat)?.modulo::<CF1<C>>()?,
             x: ci1
                 .x
                 .iter()
                 .zip(ci2.x)
                 .map(|(a, b)| {
-                    a.add_no_align(&r_nonnat.mul_no_align(&b)?)
+                    a.add_no_align(&r_nonnat.mul_no_align(&b)?)?
                         .modulo::<CF1<C>>()
                 })
                 .collect::<Result<Vec<_>, _>>()?,
@@ -319,6 +346,29 @@ where
     }
 }
 
+impl<C: CurveGroup, GC: CurveVar<C, CF2<C>>>
+    ArithGadget<CycleFoldWitnessVar<C>, CycleFoldCommittedInstanceVar<C, GC>>
+    for R1CSMatricesVar<CF1<C>, NonNativeUintVar<CF2<C>>>
+{
+    type Evaluation = (Vec<NonNativeUintVar<CF2<C>>>, Vec<NonNativeUintVar<CF2<C>>>);
+
+    fn eval_relation(
+        &self,
+        w: &CycleFoldWitnessVar<C>,
+        u: &CycleFoldCommittedInstanceVar<C, GC>,
+    ) -> Result<Self::Evaluation, SynthesisError> {
+        self.eval_at_z(&[&[u.u.clone()][..], &u.x, &w.W].concat())
+    }
+
+    fn enforce_evaluation(
+        w: &CycleFoldWitnessVar<C>,
+        _u: &CycleFoldCommittedInstanceVar<C, GC>,
+        (AzBz, uCz): Self::Evaluation,
+    ) -> Result<(), SynthesisError> {
+        EquivalenceGadget::<CF1<C>>::enforce_equivalent(&AzBz[..], &uCz.add(&w.E)?[..])
+    }
+}
+
 /// CycleFoldChallengeGadget computes the RO challenge used for the CycleFold instances NIFS, it contains a
 /// rust-native and a in-circuit compatible versions.
 pub struct CycleFoldChallengeGadget<C: CurveGroup, GC: CurveVar<C, CF2<C>>> {
@@ -329,9 +379,7 @@ impl CycleFoldChallengeGadget
 where
     C: CurveGroup,
     GC: CurveVar<C, CF2<C>> + ToConstraintFieldGadget<CF2<C>>,
-    <C as CurveGroup>::BaseField: PrimeField,
-    <C as CurveGroup>::BaseField: Absorb,
-    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
+    C::BaseField: PrimeField + Absorb,
 {
     pub fn get_challenge_native<T: Transcript<C::BaseField>>(
         transcript: &mut T,
@@ -425,7 +473,6 @@ impl> ConstraintSynthesizer
 where
     GC: ToConstraintFieldGadget<CFG::F>,
     CFG::F: PrimeField,
-    for<'a> &'a GC: GroupOpsBounds<'a, CFG::C, GC>,
 {
     fn generate_constraints(self, cs: ConstraintSystemRef<CFG::F>) -> Result<(), SynthesisError> {
         let r_bits = Vec::<Boolean<CFG::F>>::new_witness(cs.clone(), || {
@@ -510,7 +557,6 @@ pub struct CycleFoldNIFS<
 > where
     <C1 as CurveGroup>::BaseField: PrimeField,
     <C2 as CurveGroup>::BaseField: PrimeField,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     _c1: PhantomData<C1>,
     _c2: PhantomData<C2>,
@@ -526,7 +572,6 @@ where
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
     GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     fn prove(
         cf_r_Fq: C2::ScalarField, // C2::Fr==C1::Fq
@@ -597,8 +642,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     let cs2 = ConstraintSystem::<C1::BaseField>::new_ref();
     cf_circuit.generate_constraints(cs2.clone())?;

folding-schemes/src/folding/circuits/decider/mod.rs

@@ -0,0 +1,211 @@
+use ark_crypto_primitives::sponge::{
+    poseidon::constraints::PoseidonSpongeVar, CryptographicSponge,
+};
+use ark_ec::CurveGroup;
+use ark_ff::PrimeField;
+use ark_poly::Polynomial;
+use ark_r1cs_std::{
+    fields::{fp::FpVar, FieldVar},
+    poly::{domain::Radix2DomainVar, evaluations::univariate::EvaluationsVar},
+    ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::SynthesisError;
+use ark_std::log2;
+
+use crate::folding::traits::{CommittedInstanceOps, CommittedInstanceVarOps, Dummy, WitnessOps};
+use crate::transcript::{Transcript, TranscriptVar};
+use crate::utils::vec::poly_from_vec;
+use crate::Error;
+use crate::{arith::Arith, folding::circuits::CF1};
+
+pub mod off_chain;
+pub mod on_chain;
+
+/// Gadget that computes the KZG challenges.
+/// It also offers the rust native implementation compatible with the gadget.
+pub struct KZGChallengesGadget {}
+
+impl KZGChallengesGadget {
+    pub fn get_challenges_native<
+        C: CurveGroup,
+        T: Transcript<CF1<C>>,
+        U: CommittedInstanceOps<C>,
+    >(
+        transcript: &mut T,
+        U_i: &U,
+    ) -> Vec<CF1<C>> {
+        let mut challenges = vec![];
+        for cm in U_i.get_commitments() {
+            transcript.absorb_nonnative(&cm);
+            challenges.push(transcript.get_challenge());
+        }
+        challenges
+    }
+
+    pub fn get_challenges_gadget<
+        C: CurveGroup,
+        S: CryptographicSponge,
+        T: TranscriptVar<CF1<C>, S>,
+        U: CommittedInstanceVarOps<C>,
+    >(
+        transcript: &mut T,
+        U_i: &U,
+    ) -> Result<Vec<FpVar<CF1<C>>>, SynthesisError> {
+        let mut challenges = vec![];
+        for cm in U_i.get_commitments() {
+            transcript.absorb(&cm.to_constraint_field()?)?;
+            challenges.push(transcript.get_challenge()?);
+        }
+        Ok(challenges)
+    }
+}
+
+/// Gadget that interpolates the polynomial from the given vector and returns
+/// its evaluation at the given point.
+/// It also offers the rust native implementation compatible with the gadget.
+pub struct EvalGadget {}
+
+impl EvalGadget {
+    pub fn evaluate_native<F: PrimeField>(v: &[F], point: F) -> Result<F, Error> {
+        let mut v = v.to_vec();
+        v.resize(v.len().next_power_of_two(), F::zero());
+
+        Ok(poly_from_vec(v)?.evaluate(&point))
+    }
+
+    pub fn evaluate_gadget<F: PrimeField>(
+        v: &[FpVar<F>],
+        point: &FpVar<F>,
+    ) -> Result<FpVar<F>, SynthesisError> {
+        let mut v = v.to_vec();
+        v.resize(v.len().next_power_of_two(), FpVar::zero());
+        let n = v.len() as u64;
+        let gen = F::get_root_of_unity(n).ok_or(SynthesisError::PolynomialDegreeTooLarge)?;
+        // `unwrap` below is safe because `Radix2DomainVar::new` only fails if
+        // `offset.enforce_not_equal(&FpVar::zero())` returns an error.
+        // But in our case, `offset` is `FpVar::one()`, i.e., both operands of
+        // `enforce_not_equal` are constants.
+        // Consequently, `FpVar`'s implementation of `enforce_not_equal` will
+        // always return `Ok(())`.
+        let domain = Radix2DomainVar::new(gen, log2(v.len()) as u64, FpVar::one()).unwrap();
+
+        let evaluations_var = EvaluationsVar::from_vec_and_domain(v, domain, true);
+        evaluations_var.interpolate_and_evaluate(point)
+    }
+}
+
+/// This is a temporary workaround for step 6 (running NIFS.V for group elements
+/// in circuit) in an NIFS-agnostic way, because different folding schemes have
+/// different interfaces of folding verification now.
+///
+/// In the future, we may introduce a better solution that uses a trait for all
+/// folding schemes that specifies their native and in-circuit behaviors.
+pub trait DeciderEnabledNIFS<
+    C: CurveGroup,
+    RU: CommittedInstanceOps<C>, // Running instance
+    IU: CommittedInstanceOps<C>, // Incoming instance
+    W: WitnessOps<CF1<C>>,
+    A: Arith<W, RU>,
+>
+{
+    type ProofDummyCfg;
+    type Proof: Dummy<Self::ProofDummyCfg>;
+    type RandomnessDummyCfg;
+    type Randomness: Dummy<Self::RandomnessDummyCfg>;
+
+    /// Fold the field elements in `U` and `u` inside the circuit.
+    ///
+    /// `U_vec` is `U` expressed as a vector of `FpVar`s, which can be reused
+    /// before or after calling this function to save constraints.
+    #[allow(clippy::too_many_arguments)]
+    fn fold_field_elements_gadget(
+        arith: &A,
+        transcript: &mut PoseidonSpongeVar<CF1<C>>,
+        pp_hash: FpVar<CF1<C>>,
+        U: RU::Var,
+        U_vec: Vec<FpVar<CF1<C>>>,
+        u: IU::Var,
+        proof: Self::Proof,
+        randomness: Self::Randomness,
+    ) -> Result<RU::Var, SynthesisError>;
+
+    /// Fold the group elements (i.e., commitments) in `U` and `u` outside the
+    /// circuit.
+    fn fold_group_elements_native(
+        U_commitments: &[C],
+        u_commitments: &[C],
+        proof: Option<Self::Proof>,
+        randomness: Self::Randomness,
+    ) -> Result<Vec<C>, Error>;
+}
+
+#[cfg(test)]
+pub mod tests {
+    use ark_crypto_primitives::sponge::{
+        constraints::CryptographicSpongeVar, poseidon::PoseidonSponge,
+    };
+    use ark_pallas::{Fr, Projective};
+    use ark_r1cs_std::{alloc::AllocVar, R1CSVar};
+    use ark_relations::r1cs::ConstraintSystem;
+    use ark_std::UniformRand;
+
+    use super::*;
+    use crate::folding::nova::{circuits::CommittedInstanceVar, CommittedInstance};
+    use crate::transcript::poseidon::poseidon_canonical_config;
+
+    // checks that the gadget and native implementations of the challenge computation match
+    #[test]
+    fn test_kzg_challenge_gadget() {
+        let mut rng = ark_std::test_rng();
+        let poseidon_config = poseidon_canonical_config::<Fr>();
+        let mut transcript = PoseidonSponge::<Fr>::new(&poseidon_config);
+
+        let U_i = 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 challenge natively
+        let challenges = KZGChallengesGadget::get_challenges_native(&mut transcript, &U_i);
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+        let U_iVar =
+            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(U_i.clone()))
+                .unwrap();
+        let mut transcript_var = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
+
+        let challenges_var =
+            KZGChallengesGadget::get_challenges_gadget(&mut transcript_var, &U_iVar).unwrap();
+        assert!(cs.is_satisfied().unwrap());
+
+        // check that the natively computed and in-circuit computed hashes match
+        assert_eq!(challenges_var.value().unwrap(), challenges);
+    }
+
+    #[test]
+    fn test_polynomial_interpolation() {
+        let mut rng = ark_std::test_rng();
+        let n = 12;
+        let l = 1 << n;
+
+        let v: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(&mut rng))
+            .take(l)
+            .collect();
+        let challenge = Fr::rand(&mut rng);
+
+        use ark_poly::Polynomial;
+        let polynomial = poly_from_vec(v.to_vec()).unwrap();
+        let eval = polynomial.evaluate(&challenge);
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+        let vVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(v)).unwrap();
+        let challengeVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(challenge)).unwrap();
+
+        let evalVar = EvalGadget::evaluate_gadget(&vVar, &challengeVar).unwrap();
+
+        assert_eq!(evalVar.value().unwrap(), eval);
+        assert!(cs.is_satisfied().unwrap());
+    }
+}

folding-schemes/src/folding/circuits/decider/off_chain.rs

@@ -0,0 +1,316 @@
+/// This file implements a generic offchain decider circuit.
+/// For ethereum use cases, use the `GenericOnchainDeciderCircuit`.
+/// More details can be found at the documentation page:
+/// https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-offchain.html
+use ark_crypto_primitives::sponge::{
+    constraints::{AbsorbGadget, CryptographicSpongeVar},
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig},
+    Absorb,
+};
+use ark_ec::CurveGroup;
+use ark_r1cs_std::{
+    alloc::AllocVar, eq::EqGadget, fields::fp::FpVar, prelude::CurveVar, ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, SynthesisError};
+use ark_std::{marker::PhantomData, Zero};
+
+use crate::{
+    arith::{
+        r1cs::{circuits::R1CSMatricesVar, R1CS},
+        Arith, ArithGadget,
+    },
+    folding::{
+        circuits::{
+            cyclefold::{
+                CycleFoldCommittedInstance, CycleFoldCommittedInstanceVar, CycleFoldWitness,
+            },
+            decider::{EvalGadget, KZGChallengesGadget},
+            nonnative::affine::NonNativeAffineVar,
+            CF1, CF2,
+        },
+        nova::{circuits::CommittedInstanceVar, decider_eth_circuit::WitnessVar},
+        traits::{CommittedInstanceOps, CommittedInstanceVarOps, Dummy, WitnessOps, WitnessVarOps},
+    },
+};
+
+use super::DeciderEnabledNIFS;
+
+/// Circuit that implements part of the in-circuit checks needed for the offchain verification over
+/// the Curve2's BaseField (=Curve1's ScalarField).
+pub struct GenericOffchainDeciderCircuit1<
+    C1: CurveGroup,
+    C2: CurveGroup,
+    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+    RU: CommittedInstanceOps<C1>,       // Running instance
+    IU: CommittedInstanceOps<C1>,       // Incoming instance
+    W: WitnessOps<CF1<C1>>,             // Witness
+    A: Arith<W, RU>,                    // Constraint system
+    AVar: ArithGadget<W::Var, RU::Var>, // In-circuit representation of `A`
+    D: DeciderEnabledNIFS<C1, RU, IU, W, A>,
+> {
+    pub _gc2: PhantomData<GC2>,
+    pub _avar: PhantomData<AVar>,
+    /// Constraint system of the Augmented Function circuit
+    pub arith: A,
+    pub poseidon_config: PoseidonConfig<CF1<C1>>,
+    /// public params hash
+    pub pp_hash: CF1<C1>,
+    pub i: CF1<C1>,
+    /// initial state
+    pub z_0: Vec<CF1<C1>>,
+    /// current i-th state
+    pub z_i: Vec<CF1<C1>>,
+    /// Folding scheme instances
+    pub U_i: RU,
+    pub W_i: W,
+    pub u_i: IU,
+    pub w_i: W,
+    pub U_i1: RU,
+    pub W_i1: W,
+
+    /// Helper for folding verification
+    pub proof: D::Proof,
+    pub randomness: D::Randomness,
+
+    /// CycleFold running instance
+    pub cf_U_i: CycleFoldCommittedInstance<C2>,
+
+    /// KZG challenges
+    pub kzg_challenges: Vec<CF1<C1>>,
+    pub kzg_evaluations: Vec<CF1<C1>>,
+}
+
+impl<
+        C1: CurveGroup,
+        C2: CurveGroup<ScalarField = CF2<C1>, BaseField = CF1<C1>>,
+        GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        RU: CommittedInstanceOps<C1> + for<'a> Dummy<&'a A>,
+        IU: CommittedInstanceOps<C1> + for<'a> Dummy<&'a A>,
+        W: WitnessOps<CF1<C1>> + for<'a> Dummy<&'a A>,
+        A: Arith<W, RU>,
+        AVar: ArithGadget<W::Var, RU::Var> + AllocVar<A, CF1<C1>>,
+        D: DeciderEnabledNIFS<C1, RU, IU, W, A>,
+    >
+    Dummy<(
+        A,
+        &R1CS<CF1<C2>>,
+        PoseidonConfig<CF1<C1>>,
+        D::ProofDummyCfg,
+        D::RandomnessDummyCfg,
+        usize,
+        usize,
+    )> for GenericOffchainDeciderCircuit1<C1, C2, GC2, RU, IU, W, A, AVar, D>
+{
+    fn dummy(
+        (
+            arith,
+            cf_arith,
+            poseidon_config,
+            proof_config,
+            randomness_config,
+            state_len,
+            num_commitments,
+        ): (
+            A,
+            &R1CS<CF1<C2>>,
+            PoseidonConfig<CF1<C1>>,
+            D::ProofDummyCfg,
+            D::RandomnessDummyCfg,
+            usize,
+            usize,
+        ),
+    ) -> Self {
+        Self {
+            _gc2: PhantomData,
+            _avar: PhantomData,
+            poseidon_config,
+            pp_hash: Zero::zero(),
+            i: Zero::zero(),
+            z_0: vec![Zero::zero(); state_len],
+            z_i: vec![Zero::zero(); state_len],
+            U_i: RU::dummy(&arith),
+            W_i: W::dummy(&arith),
+            u_i: IU::dummy(&arith),
+            w_i: W::dummy(&arith),
+            U_i1: RU::dummy(&arith),
+            W_i1: W::dummy(&arith),
+            proof: D::Proof::dummy(proof_config),
+            randomness: D::Randomness::dummy(randomness_config),
+            cf_U_i: CycleFoldCommittedInstance::dummy(cf_arith),
+            kzg_challenges: vec![Zero::zero(); num_commitments],
+            kzg_evaluations: vec![Zero::zero(); num_commitments],
+            arith,
+        }
+    }
+}
+
+impl<
+        C1: CurveGroup,
+        C2: CurveGroup<ScalarField = CF2<C1>, BaseField = CF1<C1>>,
+        GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        RU: CommittedInstanceOps<C1>,
+        IU: CommittedInstanceOps<C1>,
+        W: WitnessOps<CF1<C1>>,
+        A: Arith<W, RU>,
+        AVar: ArithGadget<W::Var, RU::Var> + AllocVar<A, CF1<C1>>,
+        D: DeciderEnabledNIFS<C1, RU, IU, W, A>,
+    > ConstraintSynthesizer<CF1<C1>>
+    for GenericOffchainDeciderCircuit1<C1, C2, GC2, RU, IU, W, A, AVar, D>
+where
+    RU::Var: AbsorbGadget<CF1<C1>> + CommittedInstanceVarOps<C1, PointVar = NonNativeAffineVar<C1>>,
+    CF1<C1>: Absorb,
+{
+    fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
+        let arith = AVar::new_witness(cs.clone(), || Ok(&self.arith))?;
+
+        let pp_hash = FpVar::new_input(cs.clone(), || Ok(self.pp_hash))?;
+        let i = FpVar::new_input(cs.clone(), || Ok(self.i))?;
+        let z_0 = Vec::new_input(cs.clone(), || Ok(self.z_0))?;
+        let z_i = Vec::new_input(cs.clone(), || Ok(self.z_i))?;
+
+        let u_i = IU::Var::new_witness(cs.clone(), || Ok(self.u_i))?;
+        let U_i = RU::Var::new_witness(cs.clone(), || Ok(self.U_i))?;
+        // here (U_i1, W_i1) = NIFS.P( (U_i,W_i), (u_i,w_i))
+        let U_i1_commitments = Vec::<NonNativeAffineVar<C1>>::new_input(cs.clone(), || {
+            Ok(self.U_i1.get_commitments())
+        })?;
+        let U_i1 = RU::Var::new_witness(cs.clone(), || Ok(self.U_i1))?;
+        let W_i1 = W::Var::new_witness(cs.clone(), || Ok(self.W_i1))?;
+        U_i1.get_commitments().enforce_equal(&U_i1_commitments)?;
+
+        let cf_U_i =
+            CycleFoldCommittedInstanceVar::<C2, GC2>::new_input(cs.clone(), || Ok(self.cf_U_i))?;
+
+        // allocate the inputs for the checks 7.1 and 7.2
+        let kzg_challenges = Vec::new_input(cs.clone(), || Ok(self.kzg_challenges))?;
+        let kzg_evaluations = Vec::new_input(cs.clone(), || Ok(self.kzg_evaluations))?;
+
+        // `sponge` is for digest computation.
+        let sponge = PoseidonSpongeVar::new(cs.clone(), &self.poseidon_config);
+        // `transcript` is for challenge generation.
+        let mut transcript = sponge.clone();
+        // notice that the `pp_hash` is absorbed inside the ChallengeGadget::get_challenge_gadget call
+
+        // 1. enforce `U_{i+1}` and `W_{i+1}` satisfy `arith`
+        arith.enforce_relation(&W_i1, &U_i1)?;
+
+        // 2. enforce `u_i` is an incoming instance
+        u_i.enforce_incoming()?;
+
+        // 3. u_i.x[0] == H(i, z_0, z_i, U_i), u_i.x[1] == H(cf_U_i)
+        let (u_i_x, U_i_vec) = U_i.hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
+        let (cf_u_i_x, _) = cf_U_i.hash(&sponge, pp_hash.clone())?;
+        u_i.get_public_inputs().enforce_equal(&[u_i_x, cf_u_i_x])?;
+
+        // 6.1. partially enforce `NIFS.V(U_i, u_i) = U_{i+1}`.
+        D::fold_field_elements_gadget(
+            &self.arith,
+            &mut transcript,
+            pp_hash,
+            U_i,
+            U_i_vec,
+            u_i,
+            self.proof,
+            self.randomness,
+        )?
+        .enforce_partial_equal(&U_i1)?;
+
+        // 7.1. compute and check KZG challenges
+        KZGChallengesGadget::get_challenges_gadget(&mut transcript, &U_i1)?
+            .enforce_equal(&kzg_challenges)?;
+
+        // 7.2. check the claimed evaluations
+        for (((v, _r), c), e) in W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .zip(&kzg_evaluations)
+        {
+            // The randomness `_r` is currently not used.
+            EvalGadget::evaluate_gadget(v, c)?.enforce_equal(e)?;
+        }
+
+        Ok(())
+    }
+}
+
+/// Circuit that implements part of the in-circuit checks needed for the offchain verification over
+/// the Curve1's BaseField (=Curve2's ScalarField).
+pub struct GenericOffchainDeciderCircuit2<C2: CurveGroup> {
+    /// R1CS of the CycleFold circuit
+    pub cf_arith: R1CS<CF1<C2>>,
+    pub poseidon_config: PoseidonConfig<CF1<C2>>,
+    /// public params hash
+    pub pp_hash: CF1<C2>,
+
+    /// CycleFold running instance
+    pub cf_U_i: CycleFoldCommittedInstance<C2>,
+    pub cf_W_i: CycleFoldWitness<C2>,
+
+    /// KZG challenges
+    pub kzg_challenges: Vec<CF1<C2>>,
+    pub kzg_evaluations: Vec<CF1<C2>>,
+}
+
+impl<C2: CurveGroup> Dummy<(R1CS<CF1<C2>>, PoseidonConfig<CF1<C2>>, usize)>
+    for GenericOffchainDeciderCircuit2<C2>
+{
+    fn dummy(
+        (cf_arith, poseidon_config, num_commitments): (
+            R1CS<CF1<C2>>,
+            PoseidonConfig<CF1<C2>>,
+            usize,
+        ),
+    ) -> Self {
+        Self {
+            poseidon_config,
+            pp_hash: Zero::zero(),
+            cf_U_i: CycleFoldCommittedInstance::dummy(&cf_arith),
+            cf_W_i: CycleFoldWitness::dummy(&cf_arith),
+            kzg_challenges: vec![Zero::zero(); num_commitments],
+            kzg_evaluations: vec![Zero::zero(); num_commitments],
+            cf_arith,
+        }
+    }
+}
+
+impl<C2: CurveGroup> ConstraintSynthesizer<CF1<C2>> for GenericOffchainDeciderCircuit2<C2> {
+    fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C2>>) -> Result<(), SynthesisError> {
+        let cf_r1cs = R1CSMatricesVar::<CF1<C2>, FpVar<CF1<C2>>>::new_witness(cs.clone(), || {
+            Ok(self.cf_arith.clone())
+        })?;
+
+        let pp_hash = FpVar::new_input(cs.clone(), || Ok(self.pp_hash))?;
+
+        let cf_U_i = CommittedInstanceVar::new_input(cs.clone(), || Ok(self.cf_U_i))?;
+        let cf_W_i = WitnessVar::new_witness(cs.clone(), || Ok(self.cf_W_i))?;
+
+        // allocate the inputs for the checks 4.1 and 4.2
+        let kzg_challenges = Vec::new_input(cs.clone(), || Ok(self.kzg_challenges))?;
+        let kzg_evaluations = Vec::new_input(cs.clone(), || Ok(self.kzg_evaluations))?;
+
+        // `transcript` is for challenge generation.
+        let mut transcript = PoseidonSpongeVar::new(cs.clone(), &self.poseidon_config);
+        transcript.absorb(&pp_hash)?;
+
+        // 5. enforce `cf_U_i` and `cf_W_i` satisfy `cf_r1cs`
+        cf_r1cs.enforce_relation(&cf_W_i, &cf_U_i)?;
+
+        // 4.1. compute and check KZG challenges
+        KZGChallengesGadget::get_challenges_gadget(&mut transcript, &cf_U_i)?
+            .enforce_equal(&kzg_challenges)?;
+
+        // 4.2. check the claimed evaluations
+        for (((v, _r), c), e) in cf_W_i
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .zip(&kzg_evaluations)
+        {
+            // The randomness `_r` is currently not used.
+            EvalGadget::evaluate_gadget(v, c)?.enforce_equal(e)?;
+        }
+
+        Ok(())
+    }
+}

folding-schemes/src/folding/circuits/decider/on_chain.rs

@@ -0,0 +1,325 @@
+/// This file implements the onchain (Ethereum's EVM) decider circuit. For non-ethereum use cases,
+/// other more efficient approaches can be used.
+use ark_crypto_primitives::sponge::{
+    constraints::{AbsorbGadget, CryptographicSpongeVar},
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig},
+    Absorb,
+};
+use ark_ec::CurveGroup;
+use ark_r1cs_std::{
+    alloc::AllocVar, eq::EqGadget, fields::fp::FpVar, prelude::CurveVar, ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, SynthesisError};
+use ark_std::{marker::PhantomData, Zero};
+
+use crate::{
+    arith::{r1cs::R1CS, Arith, ArithGadget},
+    commitment::pedersen::Params as PedersenParams,
+    folding::{
+        circuits::{
+            cyclefold::{
+                CycleFoldCommittedInstance, CycleFoldCommittedInstanceVar, CycleFoldWitness,
+            },
+            decider::{EvalGadget, KZGChallengesGadget},
+            nonnative::affine::NonNativeAffineVar,
+            CF1, CF2,
+        },
+        traits::{CommittedInstanceOps, CommittedInstanceVarOps, Dummy, WitnessOps, WitnessVarOps},
+    },
+};
+
+use super::DeciderEnabledNIFS;
+
+/// A generic circuit tailored for the onchain (Ethereum's EVM) verification of
+/// IVC proofs, where we support IVC built upon any folding scheme.
+///
+/// Specifically, `GenericDeciderEthCircuit` implements the in-circuit version
+/// of the IVC verification algorithm, which essentially checks the following:
+/// - `R_arith(W_i, U_i)`:
+///   The running instance `U_i` and witness `W_i` satisfy `arith`,
+///   and the commitments in `U_i` open to the values in `W_i`.
+/// - `R_arith(w_i, u_i)`:
+///   The incoming instance `u_i` and witness `w_i` satisfy `arith`,
+///   and the commitments in `u_i` open to the values in `w_i`.
+/// - `R_cf_arith(cf_W_i, cf_U_i)`:
+///   The CycleFold instance `cf_U_i` and witness `cf_W_i` satisfy `cf_arith`,
+///   and the commitments in `cf_U_i` open to the values in `cf_W_i`.
+/// - `u_i` contains the correct hash of the initial and final states.
+///
+/// To reduce the number of relation checks, the prover, before invoking the
+/// circuit, further folds `U_i, u_i` into `U_{i+1}`, and `W_i, w_i` into
+/// `W_{i+1}`.
+/// Now, the circuit only needs to perform two relation checks, i.e.,
+/// `R_arith(W_{i+1}, U_{i+1})` and `R_cf_arith(cf_W_i, cf_U_i)`, plus a few
+/// constraints for enforcing the correct hash in `u_i` and the correct folding
+/// from `U_i, u_i` to `U_{i+1}`.
+///
+/// We further reduce the circuit size by avoiding the non-native commitment
+/// checks involved in `R_arith(W_{i+1}, U_{i+1})`.
+/// Now, we now only check the satisfiability of the constraint system `arith`
+/// with the witness `W_{i+1}` and instance `U_{i+1}` in the circuit, but the
+/// actual commitment checks are done with the help of KZG.
+///
+/// For more details, see [https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-onchain.html].
+pub struct GenericOnchainDeciderCircuit<
+    C1: CurveGroup,
+    C2: CurveGroup,
+    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+    RU: CommittedInstanceOps<C1>,       // Running instance
+    IU: CommittedInstanceOps<C1>,       // Incoming instance
+    W: WitnessOps<CF1<C1>>,             // Witness
+    A: Arith<W, RU>,                    // Constraint system
+    AVar: ArithGadget<W::Var, RU::Var>, // In-circuit representation of `A`
+    D: DeciderEnabledNIFS<C1, RU, IU, W, A>,
+> {
+    pub _gc2: PhantomData<GC2>,
+    pub _avar: PhantomData<AVar>,
+    /// Constraint system of the Augmented Function circuit
+    pub arith: A,
+    /// R1CS of the CycleFold circuit
+    pub cf_arith: R1CS<CF1<C2>>,
+    /// CycleFold PedersenParams over C2
+    pub cf_pedersen_params: PedersenParams<C2>,
+    pub poseidon_config: PoseidonConfig<CF1<C1>>,
+    /// public params hash
+    pub pp_hash: CF1<C1>,
+    pub i: CF1<C1>,
+    /// initial state
+    pub z_0: Vec<CF1<C1>>,
+    /// current i-th state
+    pub z_i: Vec<CF1<C1>>,
+    /// Folding scheme instances
+    pub U_i: RU,
+    pub W_i: W,
+    pub u_i: IU,
+    pub w_i: W,
+    pub U_i1: RU,
+    pub W_i1: W,
+
+    /// Helper for folding verification
+    pub proof: D::Proof,
+    pub randomness: D::Randomness,
+
+    /// CycleFold running instance
+    pub cf_U_i: CycleFoldCommittedInstance<C2>,
+    pub cf_W_i: CycleFoldWitness<C2>,
+
+    /// KZG challenges
+    pub kzg_challenges: Vec<CF1<C1>>,
+    pub kzg_evaluations: Vec<CF1<C1>>,
+}
+
+impl<
+        C1: CurveGroup,
+        C2: CurveGroup<ScalarField = CF2<C1>, BaseField = CF1<C1>>,
+        GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        RU: CommittedInstanceOps<C1> + for<'a> Dummy<&'a A>,
+        IU: CommittedInstanceOps<C1> + for<'a> Dummy<&'a A>,
+        W: WitnessOps<CF1<C1>> + for<'a> Dummy<&'a A>,
+        A: Arith<W, RU>,
+        AVar: ArithGadget<W::Var, RU::Var> + AllocVar<A, CF1<C1>>,
+        D: DeciderEnabledNIFS<C1, RU, IU, W, A>,
+    >
+    Dummy<(
+        A,
+        R1CS<CF1<C2>>,
+        PedersenParams<C2>,
+        PoseidonConfig<CF1<C1>>,
+        D::ProofDummyCfg,
+        D::RandomnessDummyCfg,
+        usize,
+        usize,
+    )> for GenericOnchainDeciderCircuit<C1, C2, GC2, RU, IU, W, A, AVar, D>
+{
+    fn dummy(
+        (
+            arith,
+            cf_arith,
+            cf_pedersen_params,
+            poseidon_config,
+            proof_config,
+            randomness_config,
+            state_len,
+            num_commitments,
+        ): (
+            A,
+            R1CS<CF1<C2>>,
+            PedersenParams<C2>,
+            PoseidonConfig<CF1<C1>>,
+            D::ProofDummyCfg,
+            D::RandomnessDummyCfg,
+            usize,
+            usize,
+        ),
+    ) -> Self {
+        Self {
+            _gc2: PhantomData,
+            _avar: PhantomData,
+            cf_pedersen_params,
+            poseidon_config,
+            pp_hash: Zero::zero(),
+            i: Zero::zero(),
+            z_0: vec![Zero::zero(); state_len],
+            z_i: vec![Zero::zero(); state_len],
+            U_i: RU::dummy(&arith),
+            W_i: W::dummy(&arith),
+            u_i: IU::dummy(&arith),
+            w_i: W::dummy(&arith),
+            U_i1: RU::dummy(&arith),
+            W_i1: W::dummy(&arith),
+            proof: D::Proof::dummy(proof_config),
+            randomness: D::Randomness::dummy(randomness_config),
+            cf_U_i: CycleFoldCommittedInstance::dummy(&cf_arith),
+            cf_W_i: CycleFoldWitness::dummy(&cf_arith),
+            kzg_challenges: vec![Zero::zero(); num_commitments],
+            kzg_evaluations: vec![Zero::zero(); num_commitments],
+            arith,
+            cf_arith,
+        }
+    }
+}
+
+impl<
+        C1: CurveGroup,
+        C2: CurveGroup<ScalarField = CF2<C1>, BaseField = CF1<C1>>,
+        GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        RU: CommittedInstanceOps<C1>,
+        IU: CommittedInstanceOps<C1>,
+        W: WitnessOps<CF1<C1>>,
+        A: Arith<W, RU>,
+        AVar: ArithGadget<W::Var, RU::Var> + AllocVar<A, CF1<C1>>,
+        D: DeciderEnabledNIFS<C1, RU, IU, W, A>,
+    > ConstraintSynthesizer<CF1<C1>>
+    for GenericOnchainDeciderCircuit<C1, C2, GC2, RU, IU, W, A, AVar, D>
+where
+    RU::Var: AbsorbGadget<CF1<C1>> + CommittedInstanceVarOps<C1, PointVar = NonNativeAffineVar<C1>>,
+    CF1<C1>: Absorb,
+{
+    fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
+        let arith = AVar::new_witness(cs.clone(), || Ok(&self.arith))?;
+
+        let pp_hash = FpVar::new_input(cs.clone(), || Ok(self.pp_hash))?;
+        let i = FpVar::new_input(cs.clone(), || Ok(self.i))?;
+        let z_0 = Vec::new_input(cs.clone(), || Ok(self.z_0))?;
+        let z_i = Vec::new_input(cs.clone(), || Ok(self.z_i))?;
+
+        let u_i = IU::Var::new_witness(cs.clone(), || Ok(self.u_i))?;
+        let U_i = RU::Var::new_witness(cs.clone(), || Ok(self.U_i))?;
+        // here (U_i1, W_i1) = NIFS.P( (U_i,W_i), (u_i,w_i))
+        let U_i1_commitments = Vec::<NonNativeAffineVar<C1>>::new_input(cs.clone(), || {
+            Ok(self.U_i1.get_commitments())
+        })?;
+        let U_i1 = RU::Var::new_witness(cs.clone(), || Ok(self.U_i1))?;
+        let W_i1 = W::Var::new_witness(cs.clone(), || Ok(self.W_i1))?;
+        U_i1.get_commitments().enforce_equal(&U_i1_commitments)?;
+
+        let cf_U_i =
+            CycleFoldCommittedInstanceVar::<C2, GC2>::new_witness(cs.clone(), || Ok(self.cf_U_i))?;
+
+        // allocate the inputs for the check 7.1 and 7.2
+        let kzg_challenges = Vec::new_input(cs.clone(), || Ok(self.kzg_challenges))?;
+        let kzg_evaluations = Vec::new_input(cs.clone(), || Ok(self.kzg_evaluations))?;
+
+        // `sponge` is for digest computation.
+        let sponge = PoseidonSpongeVar::new(cs.clone(), &self.poseidon_config);
+        // `transcript` is for challenge generation.
+        let mut transcript = sponge.clone();
+
+        // NOTE: we use the same enumeration as in
+        // https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-onchain.html
+        // in order to make it easier to reason about.
+
+        // 1. enforce `U_{i+1}` and `W_{i+1}` satisfy `arith`
+        arith.enforce_relation(&W_i1, &U_i1)?;
+
+        // 2. enforce `u_i` is an incoming instance
+        u_i.enforce_incoming()?;
+
+        // 3. u_i.x[0] == H(i, z_0, z_i, U_i), u_i.x[1] == H(cf_U_i)
+        let (u_i_x, U_i_vec) = U_i.hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
+        let (cf_u_i_x, _) = cf_U_i.hash(&sponge, pp_hash.clone())?;
+        u_i.get_public_inputs().enforce_equal(&[u_i_x, cf_u_i_x])?;
+
+        #[cfg(feature = "light-test")]
+        log::warn!("[WARNING]: Running with the 'light-test' feature, skipping the big part of the DeciderEthCircuit.\n           Only for testing purposes.");
+
+        // The following two checks (and their respective allocations) are disabled for normal
+        // tests since they take several millions of constraints and would take several minutes
+        // (and RAM) to run the test. It is active by default, and not active only when
+        // 'light-test' feature is used.
+        #[cfg(not(feature = "light-test"))]
+        {
+            // imports here instead of at the top of the file, so we avoid having multiple
+            // `#[cfg(not(test))]`
+            use crate::{
+                arith::r1cs::circuits::R1CSMatricesVar,
+                commitment::pedersen::PedersenGadget,
+                folding::circuits::{
+                    cyclefold::CycleFoldWitnessVar, nonnative::uint::NonNativeUintVar,
+                },
+            };
+            use ark_r1cs_std::ToBitsGadget;
+            let cf_W_i = CycleFoldWitnessVar::<C2>::new_witness(cs.clone(), || Ok(self.cf_W_i))?;
+            // 4. check Pedersen commitments of cf_U_i.{cmE, cmW}
+            let H = GC2::constant(self.cf_pedersen_params.h);
+            let G = self
+                .cf_pedersen_params
+                .generators
+                .iter()
+                .map(|&g| GC2::constant(g.into()))
+                .collect::<Vec<_>>();
+            let cf_W_i_E_bits = cf_W_i
+                .E
+                .iter()
+                .map(|E_i| E_i.to_bits_le())
+                .collect::<Result<Vec<_>, _>>()?;
+            let cf_W_i_W_bits = cf_W_i
+                .W
+                .iter()
+                .map(|W_i| W_i.to_bits_le())
+                .collect::<Result<Vec<_>, _>>()?;
+            PedersenGadget::<C2, GC2>::commit(&H, &G, &cf_W_i_E_bits, &cf_W_i.rE.to_bits_le()?)?
+                .enforce_equal(&cf_U_i.cmE)?;
+            PedersenGadget::<C2, GC2>::commit(&H, &G, &cf_W_i_W_bits, &cf_W_i.rW.to_bits_le()?)?
+                .enforce_equal(&cf_U_i.cmW)?;
+
+            let cf_r1cs = R1CSMatricesVar::<CF1<C2>, NonNativeUintVar<CF2<C2>>>::new_constant(
+                ConstraintSystemRef::None,
+                self.cf_arith,
+            )?;
+
+            // 5. enforce `cf_U_i` and `cf_W_i` satisfy `cf_r1cs`
+            cf_r1cs.enforce_relation(&cf_W_i, &cf_U_i)?;
+        }
+
+        // 6.1. partially enforce `NIFS.V(U_i, u_i) = U_{i+1}`.
+        D::fold_field_elements_gadget(
+            &self.arith,
+            &mut transcript,
+            pp_hash,
+            U_i,
+            U_i_vec,
+            u_i,
+            self.proof,
+            self.randomness,
+        )?
+        .enforce_partial_equal(&U_i1)?;
+
+        // 7.1. compute and check KZG challenges
+        KZGChallengesGadget::get_challenges_gadget(&mut transcript, &U_i1)?
+            .enforce_equal(&kzg_challenges)?;
+
+        // 7.2. check the claimed evaluations
+        for (((v, _r), c), e) in W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .zip(&kzg_evaluations)
+        {
+            // The randomness `_r` is currently not used.
+            EvalGadget::evaluate_gadget(v, c)?.enforce_equal(e)?;
+        }
+
+        Ok(())
+    }
+}

folding-schemes/src/folding/circuits/mod.rs

@@ -1,8 +1,9 @@
 /// Circuits and gadgets shared across the different folding schemes.
-use ark_ec::{AffineRepr, CurveGroup};
+use ark_ec::{CurveGroup, Group};
 use ark_ff::Field;
 
 pub mod cyclefold;
+pub mod decider;
 pub mod nonnative;
 pub mod sum_check;
 pub mod utils;
@@ -10,7 +11,7 @@ pub mod utils;
 /// CF1 uses the ScalarField of the given C. CF1 represents the ConstraintField used for the main
 /// folding circuit which is over E1::Fr, where E1 is the main curve where we do the folding.
 /// In CF1, the points of C can not be natively represented.
-pub type CF1<C> = <<C as CurveGroup>::Affine as AffineRepr>::ScalarField;
+pub type CF1<C> = <C as Group>::ScalarField;
 /// CF2 uses the BaseField of the given C. CF2 represents the ConstraintField used for the
 /// CycleFold circuit which is over E2::Fr=E1::Fq, where E2 is the auxiliary curve (from
 /// [CycleFold](https://eprint.iacr.org/2023/1192.pdf) approach) where we check the folding of the

folding-schemes/src/folding/circuits/nonnative/affine.rs

@@ -2,7 +2,9 @@ use ark_ec::{short_weierstrass::SWFlags, AffineRepr, CurveGroup};
 use ark_ff::{Field, PrimeField};
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
+    eq::EqGadget,
     fields::fp::FpVar,
+    prelude::Boolean,
     R1CSVar, ToConstraintFieldGadget,
 };
 use ark_relations::r1cs::{ConstraintSystemRef, Namespace, SynthesisError};
@@ -10,7 +12,10 @@ use ark_serialize::{CanonicalSerialize, CanonicalSerializeWithFlags};
 use ark_std::Zero;
 use core::borrow::Borrow;
 
-use crate::transcript::{AbsorbNonNative, AbsorbNonNativeGadget};
+use crate::{
+    folding::traits::Inputize,
+    transcript::{AbsorbNonNative, AbsorbNonNativeGadget},
+};
 
 use super::uint::{nonnative_field_to_field_elements, NonNativeUintVar};
 
@@ -63,8 +68,15 @@ impl R1CSVar for NonNativeAffineVar {
         let y = <C::BaseField as Field>::BasePrimeField::from_le_bytes_mod_order(
             &self.y.value()?.to_bytes_le(),
         );
+        // Below is a workaround to convert the `x` and `y` coordinates to a
+        // point. This is because the `CurveGroup` trait does not provide a
+        // method to construct a point from `BaseField` elements.
         let mut bytes = vec![];
+        // `unwrap` below is safe because serialization of a `PrimeField` value
+        // only fails if the serialization flag has more than 8 bits, but here
+        // we call `serialize_uncompressed` which uses an empty flag.
         x.serialize_uncompressed(&mut bytes).unwrap();
+        // `unwrap` below is also safe, because the bit size of `SWFlags` is 2.
         y.serialize_with_flags(
             &mut bytes,
             if x.is_zero() && y.is_zero() {
@@ -76,6 +88,9 @@ impl R1CSVar for NonNativeAffineVar {
             },
         )
         .unwrap();
+        // `unwrap` below is safe because `bytes` is constructed from the `x`
+        // and `y` coordinates of a valid point, and these coordinates are
+        // serialized in the same way as the `CurveGroup` implementation.
         Ok(C::deserialize_uncompressed_unchecked(&bytes[..]).unwrap())
     }
 }
@@ -90,6 +105,53 @@ impl ToConstraintFieldGadget for NonNativeAffineV
     }
 }
 
+impl<C: CurveGroup> EqGadget<C::ScalarField> for NonNativeAffineVar<C> {
+    fn is_eq(&self, other: &Self) -> Result<Boolean<C::ScalarField>, SynthesisError> {
+        let mut result = Boolean::TRUE;
+        if self.x.0.len() != other.x.0.len() {
+            return Err(SynthesisError::Unsatisfiable);
+        }
+        if self.y.0.len() != other.y.0.len() {
+            return Err(SynthesisError::Unsatisfiable);
+        }
+        for (l, r) in self
+            .x
+            .0
+            .iter()
+            .chain(&self.y.0)
+            .zip(other.x.0.iter().chain(&other.y.0))
+        {
+            if l.ub != r.ub {
+                return Err(SynthesisError::Unsatisfiable);
+            }
+            result = result.and(&l.v.is_eq(&r.v)?)?;
+        }
+        Ok(result)
+    }
+
+    fn enforce_equal(&self, other: &Self) -> Result<(), SynthesisError> {
+        if self.x.0.len() != other.x.0.len() {
+            return Err(SynthesisError::Unsatisfiable);
+        }
+        if self.y.0.len() != other.y.0.len() {
+            return Err(SynthesisError::Unsatisfiable);
+        }
+        for (l, r) in self
+            .x
+            .0
+            .iter()
+            .chain(&self.y.0)
+            .zip(other.x.0.iter().chain(&other.y.0))
+        {
+            if l.ub != r.ub {
+                return Err(SynthesisError::Unsatisfiable);
+            }
+            l.v.enforce_equal(&r.v)?;
+        }
+        Ok(())
+    }
+}
+
 /// The out-circuit counterpart of `NonNativeAffineVar::to_constraint_field`
 #[allow(clippy::type_complexity)]
 pub(crate) fn nonnative_affine_to_field_elements<C: CurveGroup>(
@@ -104,22 +166,22 @@ pub(crate) fn nonnative_affine_to_field_elements(
     (x, y)
 }
 
-impl<C: CurveGroup> NonNativeAffineVar<C> {
-    // Extracts a list of field elements of type `C::ScalarField` from the public input
-    // `p`, in exactly the same way as how `NonNativeAffineVar` is represented as limbs of type
-    // `FpVar` in-circuit.
-    #[allow(clippy::type_complexity)]
-    pub fn inputize(p: C) -> Result<(Vec<C::ScalarField>, Vec<C::ScalarField>), SynthesisError> {
-        let affine = p.into_affine();
+impl<C: CurveGroup> Inputize<C::ScalarField, NonNativeAffineVar<C>> for C {
+    fn inputize(&self) -> Vec<C::ScalarField> {
+        let affine = self.into_affine();
         let zero = (&C::BaseField::zero(), &C::BaseField::zero());
         let (x, y) = affine.xy().unwrap_or(zero);
 
-        let x = NonNativeUintVar::inputize(*x);
-        let y = NonNativeUintVar::inputize(*y);
-        Ok((x, y))
+        let x = x.inputize();
+        let y = y.inputize();
+        [x, y].concat()
     }
+}
 
+impl<C: CurveGroup> NonNativeAffineVar<C> {
     pub fn zero() -> Self {
+        // `unwrap` below is safe because we are allocating a constant value,
+        // which is guaranteed to succeed.
         Self::new_constant(ConstraintSystemRef::None, C::zero()).unwrap()
     }
 }
@@ -179,9 +241,11 @@ mod tests {
         let mut rng = ark_std::test_rng();
         let p = Projective::rand(&mut rng);
         let pVar = NonNativeAffineVar::<Projective>::new_witness(cs.clone(), || Ok(p)).unwrap();
-        let (x, y) = NonNativeAffineVar::inputize(p).unwrap();
+        let xy = p.inputize();
 
-        assert_eq!(pVar.x.0.value().unwrap(), x);
-        assert_eq!(pVar.y.0.value().unwrap(), y);
+        assert_eq!(
+            [pVar.x.0.value().unwrap(), pVar.y.0.value().unwrap()].concat(),
+            xy
+        );
     }
 }

folding-schemes/src/folding/circuits/nonnative/uint.rs

@@ -17,8 +17,9 @@ use num_bigint::BigUint;
 use num_integer::Integer;
 
 use crate::{
+    folding::traits::Inputize,
     transcript::{AbsorbNonNative, AbsorbNonNativeGadget},
-    utils::gadgets::{MatrixGadget, SparseMatrixVar, VectorGadget},
+    utils::gadgets::{EquivalenceGadget, MatrixGadget, SparseMatrixVar, VectorGadget},
 };
 
 /// `LimbVar` represents a single limb of a non-native unsigned integer in the
@@ -36,6 +37,11 @@ pub struct LimbVar {
 impl<F: PrimeField, B: AsRef<[Boolean<F>]>> From<B> for LimbVar<F> {
     fn from(bits: B) -> Self {
         Self {
+            // `Boolean::le_bits_to_fp_var` will return an error if the internal
+            // invocation of `Boolean::enforce_in_field_le` fails.
+            // However, this method is only called when the length of `bits` is
+            // greater than `F::MODULUS_BIT_SIZE`, which should not happen in
+            // our case where `bits` is guaranteed to be short.
             v: Boolean::le_bits_to_fp_var(bits.as_ref()).unwrap(),
             ub: (BigUint::one() << bits.as_ref().len()) - BigUint::one(),
         }
@@ -219,7 +225,7 @@ impl AllocVar for NonNativeUintVar {
             .collect::<Vec<_>>()
             .chunks(Self::bits_per_limb())
         {
-            let limb = F::from_bigint(F::BigInt::from_bits_le(chunk)).unwrap();
+            let limb = F::from(F::BigInt::from_bits_le(chunk));
             let limb = FpVar::new_variable(cs.clone(), || Ok(limb), mode)?;
             Self::enforce_bit_length(&limb, chunk.len())?;
             limbs.push(LimbVar {
@@ -241,12 +247,15 @@ impl AllocVar for NonNativeUintVar {
         let cs = cs.into().cs();
         let v = f()?;
         assert_eq!(G::extension_degree(), 1);
+        // `unwrap` is safe because `G` is a field with extension degree 1, and
+        // thus `G::to_base_prime_field_elements` should return an iterator with
+        // exactly one element.
         let v = v.borrow().to_base_prime_field_elements().next().unwrap();
 
         let mut limbs = vec![];
 
         for chunk in v.into_bigint().to_bits_le().chunks(Self::bits_per_limb()) {
-            let limb = F::from_bigint(F::BigInt::from_bits_le(chunk)).unwrap();
+            let limb = F::from(F::BigInt::from_bits_le(chunk));
             let limb = FpVar::new_variable(cs.clone(), || Ok(limb), mode)?;
             Self::enforce_bit_length(&limb, chunk.len())?;
             limbs.push(LimbVar {
@@ -259,16 +268,19 @@ impl AllocVar for NonNativeUintVar {
     }
 }
 
-impl<F: PrimeField> NonNativeUintVar<F> {
-    pub fn inputize<T: Field>(x: T) -> Vec<F> {
+impl<F: PrimeField, T: Field> Inputize<F, NonNativeUintVar<F>> for T {
+    fn inputize(&self) -> Vec<F> {
         assert_eq!(T::extension_degree(), 1);
-        x.to_base_prime_field_elements()
+        // `unwrap` is safe because `T` is a field with extension degree 1, and
+        // thus `T::to_base_prime_field_elements` should return an iterator with
+        // exactly one element.
+        self.to_base_prime_field_elements()
             .next()
             .unwrap()
             .into_bigint()
             .to_bits_le()
-            .chunks(Self::bits_per_limb())
-            .map(|chunk| F::from_bigint(F::BigInt::from_bits_le(chunk)).unwrap())
+            .chunks(NonNativeUintVar::<F>::bits_per_limb())
+            .map(|chunk| F::from(F::BigInt::from_bits_le(chunk)))
             .collect()
     }
 }
@@ -450,7 +462,7 @@ impl NonNativeUintVar {
                 // (i.e., all of them are "non-negative"), implying that all
                 // limbs should be zero to make the sum zero.
                 LimbVar::add_many(&remaining_limbs[1..])
-                    .unwrap()
+                    .ok_or(SynthesisError::Unsatisfiable)?
                     .v
                     .enforce_equal(&FpVar::zero())?;
                 remaining_limbs[0].v.clone()
@@ -621,15 +633,15 @@ impl NonNativeUintVar {
     }
 
     /// Compute `self + other`, without aligning the limbs.
-    pub fn add_no_align(&self, other: &Self) -> Self {
+    pub fn add_no_align(&self, other: &Self) -> Result<Self, SynthesisError> {
         let mut z = vec![LimbVar::zero(); max(self.0.len(), other.0.len())];
         for (i, v) in self.0.iter().enumerate() {
-            z[i] = z[i].add(v).unwrap();
+            z[i] = z[i].add(v).ok_or(SynthesisError::Unsatisfiable)?;
         }
         for (i, v) in other.0.iter().enumerate() {
-            z[i] = z[i].add(v).unwrap();
+            z[i] = z[i].add(v).ok_or(SynthesisError::Unsatisfiable)?;
         }
-        Self(z)
+        Ok(Self(z))
     }
 
     /// Compute `self * other`, without aligning the limbs.
@@ -650,7 +662,7 @@ impl NonNativeUintVar {
                     )
                 })
                 .collect::<Option<Vec<_>>>()
-                .unwrap();
+                .ok_or(SynthesisError::Unsatisfiable)?;
             return Ok(Self(z));
         }
         let cs = self.cs().or(other.cs());
@@ -753,7 +765,7 @@ impl NonNativeUintVar {
         let m = Self::new_constant(cs.clone(), BoundedBigUint(m, M::MODULUS_BIT_SIZE as usize))?;
         // Enforce `self = q * m + r`
         q.mul_no_align(&m)?
-            .add_no_align(&r)
+            .add_no_align(&r)?
             .enforce_equal_unaligned(self)?;
         // Enforce `r < m` (and `r >= 0` already holds)
         r.enforce_lt(&m)?;
@@ -789,8 +801,8 @@ impl NonNativeUintVar {
 
         let zero = Self::new_constant(cs.clone(), BoundedBigUint(BigUint::zero(), bits))?;
         let m = Self::new_constant(cs.clone(), BoundedBigUint(m, M::MODULUS_BIT_SIZE as usize))?;
-        let l = self.add_no_align(&is_ge.select(&zero, &q)?.mul_no_align(&m)?);
-        let r = other.add_no_align(&is_ge.select(&q, &zero)?.mul_no_align(&m)?);
+        let l = self.add_no_align(&is_ge.select(&zero, &q)?.mul_no_align(&m)?)?;
+        let r = other.add_no_align(&is_ge.select(&q, &zero)?.mul_no_align(&m)?)?;
         // If `self >= other`, enforce `self = other + q * m`
         // Otherwise, enforce `self + q * m = other`
         // Soundness holds because if `self` and `other` are not congruent, then
@@ -799,6 +811,12 @@ impl NonNativeUintVar {
     }
 }
 
+impl<F: PrimeField, M: PrimeField> EquivalenceGadget<M> for NonNativeUintVar<F> {
+    fn enforce_equivalent(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.enforce_congruent::<M>(other)
+    }
+}
+
 impl<F: PrimeField, B: AsRef<[Boolean<F>]>> From<B> for NonNativeUintVar<F> {
     fn from(bits: B) -> Self {
         Self(
@@ -834,6 +852,9 @@ pub(super) fn nonnative_field_to_field_elements
     f: &TargetField,
 ) -> Vec<BaseField> {
     assert_eq!(TargetField::extension_degree(), 1);
+    // `unwrap` is safe because `TargetField` is a field with extension degree
+    // 1, and thus `TargetField::to_base_prime_field_elements` should return an
+    // iterator with exactly one element.
     let bits = f
         .to_base_prime_field_elements()
         .next()
@@ -864,11 +885,10 @@ pub(super) fn nonnative_field_to_field_elements
 
 impl<F: PrimeField> VectorGadget<NonNativeUintVar<F>> for [NonNativeUintVar<F>] {
     fn add(&self, other: &Self) -> Result<Vec<NonNativeUintVar<F>>, SynthesisError> {
-        Ok(self
-            .iter()
+        self.iter()
             .zip(other.iter())
             .map(|(x, y)| x.add_no_align(y))
-            .collect())
+            .collect()
     }
 
     fn hadamard(&self, other: &Self) -> Result<Vec<NonNativeUintVar<F>>, SynthesisError> {
@@ -886,15 +906,12 @@ impl VectorGadget> for [NonNativeUintVar]
     }
 }
 
-impl<F: PrimeField, CF: PrimeField> MatrixGadget<NonNativeUintVar<CF>>
-    for SparseMatrixVar<F, CF, NonNativeUintVar<CF>>
-{
+impl<CF: PrimeField> MatrixGadget<NonNativeUintVar<CF>> for SparseMatrixVar<NonNativeUintVar<CF>> {
     fn mul_vector(
         &self,
         v: &[NonNativeUintVar<CF>],
     ) -> Result<Vec<NonNativeUintVar<CF>>, SynthesisError> {
-        Ok(self
-            .coeffs
+        self.coeffs
             .iter()
             .map(|row| {
                 let len = row
@@ -906,7 +923,7 @@ impl MatrixGadget>
                 // that results in more flattened `LinearCombination`s.
                 // Consequently, `ConstraintSystem::inline_all_lcs` costs less
                 // time, thus making trusted setup and proof generation faster.
-                let limbs = (0..len)
+                (0..len)
                     .map(|i| {
                         LimbVar::add_many(
                             &row.iter()
@@ -919,10 +936,10 @@ impl MatrixGadget>
                         )
                     })
                     .collect::<Option<Vec<_>>>()
-                    .unwrap();
-                NonNativeUintVar(limbs)
+                    .ok_or(SynthesisError::Unsatisfiable)
+                    .map(NonNativeUintVar)
             })
-            .collect())
+            .collect::<Result<Vec<_>, _>>()
     }
 }
 
@@ -1041,7 +1058,7 @@ mod tests {
         let mut r_var =
             NonNativeUintVar::new_constant(cs.clone(), BoundedBigUint(BigUint::zero(), 0))?;
         for (a, b) in a_var.into_iter().zip(b_var.into_iter()) {
-            r_var = r_var.add_no_align(&a.mul_no_align(&b)?);
+            r_var = r_var.add_no_align(&a.mul_no_align(&b)?)?;
         }
         r_var.enforce_congruent::<Fq>(&c_var)?;
 

folding-schemes/src/folding/hypernova/cccs.rs

@@ -10,6 +10,7 @@ use super::Witness;
 use crate::arith::{ccs::CCS, Arith};
 use crate::commitment::CommitmentScheme;
 use crate::folding::circuits::CF1;
+use crate::folding::traits::Inputize;
 use crate::folding::traits::{CommittedInstanceOps, Dummy};
 use crate::transcript::AbsorbNonNative;
 use crate::utils::mle::dense_vec_to_dense_mle;
@@ -152,6 +153,12 @@ impl CommittedInstanceOps for CCCS {
     }
 }
 
+impl<C: CurveGroup> Inputize<C::ScalarField, CCCSVar<C>> for CCCS<C> {
+    fn inputize(&self) -> Vec<C::ScalarField> {
+        [&self.C.inputize()[..], &self.x].concat()
+    }
+}
+
 #[cfg(test)]
 pub mod tests {
     use ark_pallas::Fr;

folding-schemes/src/folding/hypernova/circuits.rs

@@ -12,7 +12,6 @@ use ark_r1cs_std::{
     boolean::Boolean,
     eq::EqGadget,
     fields::{fp::FpVar, FieldVar},
-    groups::GroupOpsBounds,
     prelude::CurveVar,
     uint8::UInt8,
     R1CSVar, ToConstraintFieldGadget,
@@ -195,7 +194,6 @@ pub struct ProofVar {
 impl<C> AllocVar<NIMFSProof<C>, CF1<C>> for ProofVar<C>
 where
     C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: PrimeField,
     <C as Group>::ScalarField: Absorb,
 {
     fn new_variable<T: Borrow<NIMFSProof<C>>>(
@@ -237,10 +235,7 @@ where
 pub struct NIMFSGadget<C: CurveGroup> {
     _c: PhantomData<C>,
 }
-impl<C: CurveGroup> NIMFSGadget<C>
-where
-    <C as CurveGroup>::BaseField: PrimeField,
-{
+impl<C: CurveGroup> NIMFSGadget<C> {
     /// Runs (in-circuit) the NIMFS.V, which outputs the new folded LCCCS instance together with
     /// the rho_powers, which will be used in other parts of the AugmentedFCircuit
     #[allow(clippy::type_complexity)]
@@ -492,9 +487,7 @@ pub struct AugmentedFCircuit<
     FC: FCircuit<CF1<C1>>,
     const MU: usize,
     const NU: usize,
-> where
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
-{
+> {
     pub(super) _c2: PhantomData<C2>,
     pub(super) _gc2: PhantomData<GC2>,
     pub(super) poseidon_config: PoseidonConfig<CF1<C1>>,
@@ -532,7 +525,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     pub fn default(
         poseidon_config: &PoseidonConfig<CF1<C1>>,
@@ -588,11 +580,9 @@ where
             M: vec![],
         };
         let mut augmented_f_circuit = Self::default(poseidon_config, F, initial_ccs)?;
-        if ccs.is_some() {
-            augmented_f_circuit.ccs = ccs.unwrap();
-        } else {
-            augmented_f_circuit.ccs = augmented_f_circuit.upper_bound_ccs()?;
-        }
+        augmented_f_circuit.ccs = ccs
+            .ok_or(())
+            .or_else(|_| augmented_f_circuit.upper_bound_ccs())?;
         Ok(augmented_f_circuit)
     }
 
@@ -601,7 +591,7 @@ where
     /// For a stable FCircuit circuit, the CCS parameters can be computed in advance and can be
     /// feed in as parameter for the AugmentedFCircuit::empty method to avoid computing them there.
     pub fn upper_bound_ccs(&self) -> Result<CCS<C1::ScalarField>, Error> {
-        let r1cs = get_r1cs_from_cs::<CF1<C1>>(self.clone()).unwrap();
+        let r1cs = get_r1cs_from_cs::<CF1<C1>>(self.clone())?;
         let mut ccs = CCS::from(r1cs);
 
         let z_0 = vec![C1::ScalarField::zero(); self.F.state_len()];
@@ -690,7 +680,7 @@ where
         self.clone().generate_constraints(cs.clone())?;
         cs.finalize();
         let cs = cs.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let r1cs = extract_r1cs::<C1::ScalarField>(&cs);
+        let r1cs = extract_r1cs::<C1::ScalarField>(&cs)?;
         let ccs = CCS::from(r1cs);
 
         Ok((cs, ccs))
@@ -709,7 +699,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
         let pp_hash = FpVar::<CF1<C1>>::new_witness(cs.clone(), || {
@@ -883,7 +872,7 @@ where
         // Non-base case: u_{i+1}.x[1] == H(cf_U_{i+1})
         let (cf_u_i1_x, _) = cf_U_i1.clone().hash(&sponge, pp_hash.clone())?;
         let (cf_u_i1_x_base, _) =
-            CycleFoldCommittedInstanceVar::new_constant(cs.clone(), cf_u_dummy)?
+            CycleFoldCommittedInstanceVar::<C2, GC2>::new_constant(cs.clone(), cf_u_dummy)?
                 .hash(&sponge, pp_hash)?;
         let cf_x = FpVar::new_input(cs.clone(), || {
             Ok(self.cf_x.unwrap_or(cf_u_i1_x_base.value()?))
@@ -1232,7 +1221,7 @@ mod tests {
             .into_inner()
             .ok_or(Error::NoInnerConstraintSystem)
             .unwrap();
-        let cf_r1cs = extract_r1cs::<Fq>(&cs2);
+        let cf_r1cs = extract_r1cs::<Fq>(&cs2).unwrap();
         println!("CF m x n: {} x {}", cf_r1cs.A.n_rows, cf_r1cs.A.n_cols);
 
         let (pedersen_params, _) =

folding-schemes/src/folding/hypernova/decider_eth.rs

@@ -2,7 +2,7 @@
 use ark_crypto_primitives::sponge::Absorb;
 use ark_ec::{CurveGroup, Group};
 use ark_ff::PrimeField;
-use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar, ToConstraintFieldGadget};
+use ark_r1cs_std::{prelude::CurveVar, ToConstraintFieldGadget};
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
 use ark_snark::SNARK;
 use ark_std::rand::{CryptoRng, RngCore};
@@ -10,12 +10,15 @@ use ark_std::{One, Zero};
 use core::marker::PhantomData;
 
 pub use super::decider_eth_circuit::DeciderEthCircuit;
-use super::{lcccs::LCCCS, HyperNova};
+use super::decider_eth_circuit::DeciderHyperNovaGadget;
+use super::HyperNova;
 use crate::commitment::{
     kzg::Proof as KZGProof, pedersen::Params as PedersenParams, CommitmentScheme,
 };
-use crate::folding::circuits::{nonnative::affine::NonNativeAffineVar, CF2};
+use crate::folding::circuits::decider::DeciderEnabledNIFS;
+use crate::folding::circuits::CF2;
 use crate::folding::nova::decider_eth::VerifierParam;
+use crate::folding::traits::{Inputize, WitnessOps};
 use crate::frontend::FCircuit;
 use crate::Error;
 use crate::{Decider as DeciderTrait, FoldingScheme};
@@ -31,7 +34,6 @@ where
     kzg_proof: CS1::Proof,
     // rho used at the last fold, U_{i+1}=NIMFS.V(rho, U_i, u_i), it is checked in-circuit
     rho: C1::ScalarField,
-    U_i1: LCCCS<C1>, // U_{i+1}, which is checked in-circuit
     // the KZG challenge is provided by the prover, but in-circuit it is checked to match
     // the in-circuit computed computed one.
     kzg_challenge: C1::ScalarField,
@@ -75,8 +77,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'b> &'b GC1: GroupOpsBounds<'b, C1, GC1>,
-    for<'b> &'b GC2: GroupOpsBounds<'b, C2, GC2>,
     // constrain FS into HyperNova, since this is a Decider specifically for HyperNova
     HyperNova<C1, GC1, C2, GC2, FC, CS1, CS2, MU, NU, false>: From<FS>,
     crate::folding::hypernova::ProverParams<C1, C2, CS1, CS2, false>:
@@ -89,21 +89,18 @@ where
     type Proof = Proof<C1, CS1, S>;
     type VerifierParam = VerifierParam<C1, CS1::VerifierParams, S::VerifyingKey>;
     type PublicInput = Vec<C1::ScalarField>;
-    type CommittedInstance = ();
+    type CommittedInstance = Vec<C1>;
 
     fn preprocess(
         mut rng: impl RngCore + CryptoRng,
         prep_param: Self::PreprocessorParam,
         fs: FS,
     ) -> Result<(Self::ProverParam, Self::VerifierParam), Error> {
-        let circuit =
-            DeciderEthCircuit::<C1, GC1, C2, GC2, CS1, CS2>::from_hypernova::<FC, MU, NU>(
-                fs.into(),
-            )
-            .unwrap();
+        let circuit = DeciderEthCircuit::<C1, C2, GC2>::try_from(HyperNova::from(fs))?;
 
         // get the Groth16 specific setup for the circuit
-        let (g16_pk, g16_vk) = S::circuit_specific_setup(circuit, &mut rng).unwrap();
+        let (g16_pk, g16_vk) = S::circuit_specific_setup(circuit, &mut rng)
+            .map_err(|e| Error::SNARKSetupFail(e.to_string()))?;
 
         // get the FoldingScheme prover & verifier params from HyperNova
         #[allow(clippy::type_complexity)]
@@ -122,7 +119,7 @@ where
 
         let pp = (g16_pk, hypernova_pp.cs_pp);
 
-        let vp = VerifierParam {
+        let vp = Self::VerifierParam {
             pp_hash,
             snark_vp: g16_vk,
             cs_vp: hypernova_vp.cs_vp,
@@ -137,40 +134,37 @@ where
     ) -> Result<Self::Proof, Error> {
         let (snark_pk, cs_pk): (S::ProvingKey, CS1::ProverParams) = pp;
 
-        let circuit = DeciderEthCircuit::<C1, GC1, C2, GC2, CS1, CS2>::from_hypernova::<FC, MU, NU>(
-            folding_scheme.into(),
-        )?;
+        let circuit = DeciderEthCircuit::<C1, C2, GC2>::try_from(HyperNova::from(folding_scheme))?;
 
-        let snark_proof = S::prove(&snark_pk, circuit.clone(), &mut rng)
-            .map_err(|e| Error::Other(e.to_string()))?;
-
-        // Notice that since the `circuit` has been constructed at the `from_hypernova` call, which
-        // in case of failure it would have returned an error there, the next two unwraps should
-        // never reach an error.
-        let rho_Fr = circuit.rho.ok_or(Error::Empty)?;
-        let W_i1 = circuit.W_i1.ok_or(Error::Empty)?;
+        let rho = circuit.randomness;
 
         // get the challenges that have been already computed when preparing the circuit inputs in
-        // the above `from_hypernova` call
-        let challenge_W = circuit
-            .kzg_challenge
-            .ok_or(Error::MissingValue("kzg_challenge".to_string()))?;
+        // the above `try_from` call
+        let kzg_challenges = circuit.kzg_challenges.clone();
 
         // generate KZG proofs
-        let U_cmW_proof = CS1::prove_with_challenge(
-            &cs_pk,
-            challenge_W,
-            &W_i1.w,
-            &C1::ScalarField::zero(),
-            None,
-        )?;
+        let kzg_proofs = circuit
+            .W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .map(|((v, _), &c)| {
+                CS1::prove_with_challenge(&cs_pk, c, v, &C1::ScalarField::zero(), None)
+            })
+            .collect::<Result<Vec<_>, _>>()?;
+
+        let snark_proof =
+            S::prove(&snark_pk, circuit, &mut rng).map_err(|e| Error::Other(e.to_string()))?;
 
         Ok(Self::Proof {
             snark_proof,
-            kzg_proof: U_cmW_proof,
-            rho: rho_Fr,
-            U_i1: circuit.U_i1.ok_or(Error::Empty)?,
-            kzg_challenge: challenge_W,
+            rho,
+            kzg_proof: (kzg_proofs.len() == 1)
+                .then(|| kzg_proofs[0].clone())
+                .ok_or(Error::NotExpectedLength(kzg_proofs.len(), 1))?,
+            kzg_challenge: (kzg_challenges.len() == 1)
+                .then(|| kzg_challenges[0])
+                .ok_or(Error::NotExpectedLength(kzg_challenges.len(), 1))?,
         })
     }
 
@@ -180,45 +174,48 @@ where
         z_0: Vec<C1::ScalarField>,
         z_i: Vec<C1::ScalarField>,
         // we don't use the instances at the verifier level, since we check them in-circuit
-        _running_instance: &Self::CommittedInstance,
-        _incoming_instance: &Self::CommittedInstance,
+        running_commitments: &Self::CommittedInstance,
+        incoming_commitments: &Self::CommittedInstance,
         proof: &Self::Proof,
     ) -> Result<bool, Error> {
         if i <= C1::ScalarField::one() {
             return Err(Error::NotEnoughSteps);
         }
 
-        let (pp_hash, snark_vk, cs_vk): (C1::ScalarField, S::VerifyingKey, CS1::VerifierParams) =
-            (vp.pp_hash, vp.snark_vp, vp.cs_vp);
+        let Self::VerifierParam {
+            pp_hash,
+            snark_vp,
+            cs_vp,
+        } = vp;
+
+        // 6.2. Fold the commitments
+        let C = DeciderHyperNovaGadget::fold_group_elements_native(
+            running_commitments,
+            incoming_commitments,
+            None,
+            proof.rho,
+        )?[0];
 
         // Note: the NIMFS proof is checked inside the DeciderEthCircuit, which ensures that the
         // 'proof.U_i1' is correctly computed
-
-        let (cmC_x, cmC_y) = NonNativeAffineVar::inputize(proof.U_i1.C)?;
-
         let public_input: Vec<C1::ScalarField> = [
-            vec![pp_hash, i],
-            z_0,
-            z_i,
-            // U_i+1:
-            cmC_x,
-            cmC_y,
-            vec![proof.U_i1.u],
-            proof.U_i1.x.clone(),
-            proof.U_i1.r_x.clone(),
-            proof.U_i1.v.clone(),
-            vec![proof.kzg_challenge, proof.kzg_proof.eval, proof.rho],
+            &[pp_hash, i][..],
+            &z_0,
+            &z_i,
+            &C.inputize(),
+            &[proof.kzg_challenge, proof.kzg_proof.eval, proof.rho],
         ]
         .concat();
 
-        let snark_v = S::verify(&snark_vk, &public_input, &proof.snark_proof)
+        let snark_v = S::verify(&snark_vp, &public_input, &proof.snark_proof)
             .map_err(|e| Error::Other(e.to_string()))?;
         if !snark_v {
             return Err(Error::SNARKVerificationFail);
         }
 
+        // 7.3. Verify the KZG proof
         // we're at the Ethereum EVM case, so the CS1 is KZG commitments
-        CS1::verify_with_challenge(&cs_vk, proof.kzg_challenge, &proof.U_i1.C, &proof.kzg_proof)?;
+        CS1::verify_with_challenge(&cs_vp, proof.kzg_challenge, &C, &proof.kzg_proof)?;
 
         Ok(true)
     }
@@ -229,13 +226,14 @@ pub mod tests {
     use ark_bn254::{constraints::GVar, Bn254, Fr, G1Projective as Projective};
     use ark_groth16::Groth16;
     use ark_grumpkin::{constraints::GVar as GVar2, Projective as Projective2};
-    use ark_serialize::{Compress, Validate};
+    use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Compress, Validate};
 
     use super::*;
     use crate::commitment::{kzg::KZG, pedersen::Pedersen};
     use crate::folding::hypernova::cccs::CCCS;
+    use crate::folding::hypernova::lcccs::LCCCS;
     use crate::folding::hypernova::PreprocessorParam;
-    use crate::folding::nova::decider_eth::VerifierParam;
+    use crate::folding::traits::CommittedInstanceOps;
     use crate::frontend::utils::CubicFCircuit;
     use crate::transcript::poseidon::poseidon_canonical_config;
 
@@ -300,8 +298,8 @@ pub mod tests {
             hypernova.i,
             hypernova.z_0,
             hypernova.z_i,
-            &(),
-            &(),
+            &hypernova.U_i.get_commitments(),
+            &hypernova.u_i.get_commitments(),
             &proof,
         )
         .unwrap();
@@ -403,8 +401,8 @@ pub mod tests {
             hypernova.i,
             hypernova.z_0.clone(),
             hypernova.z_i.clone(),
-            &(),
-            &(),
+            &hypernova.U_i.get_commitments(),
+            &hypernova.u_i.get_commitments(),
             &proof,
         )
         .unwrap();
@@ -470,8 +468,8 @@ pub mod tests {
             i_deserialized,
             z_0_deserialized.clone(),
             z_i_deserialized.clone(),
-            &(),
-            &(),
+            &hypernova.U_i.get_commitments(),
+            &hypernova.u_i.get_commitments(),
             &proof_deserialized,
         )
         .unwrap();

folding-schemes/src/folding/hypernova/decider_eth_circuit.rs

@@ -2,50 +2,73 @@
 /// other more efficient approaches can be used.
 use ark_crypto_primitives::sponge::{
     constraints::CryptographicSpongeVar,
-    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonSponge},
     Absorb, CryptographicSponge,
 };
-use ark_ec::{CurveGroup, Group};
+use ark_ec::CurveGroup;
 use ark_ff::PrimeField;
-use ark_poly::Polynomial;
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     boolean::Boolean,
     eq::EqGadget,
     fields::fp::FpVar,
-    groups::GroupOpsBounds,
     prelude::CurveVar,
     ToConstraintFieldGadget,
 };
-use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, Namespace, SynthesisError};
-use ark_std::Zero;
-use core::{borrow::Borrow, marker::PhantomData};
+use ark_relations::r1cs::{Namespace, SynthesisError};
+use ark_std::{borrow::Borrow, log2, marker::PhantomData};
 
 use super::{
     circuits::{CCCSVar, LCCCSVar, NIMFSGadget, ProofVar as NIMFSProofVar},
     nimfs::{NIMFSProof, NIMFS},
     HyperNova, Witness, CCCS, LCCCS,
 };
-use crate::folding::circuits::{
-    cyclefold::{CycleFoldCommittedInstance, CycleFoldWitness},
-    CF1, CF2,
-};
+use crate::folding::circuits::{decider::on_chain::GenericOnchainDeciderCircuit, CF1, CF2};
+use crate::folding::traits::{WitnessOps, WitnessVarOps};
 use crate::frontend::FCircuit;
-use crate::transcript::{Transcript, TranscriptVar};
-use crate::utils::{
-    gadgets::{eval_mle, MatrixGadget, SparseMatrixVar},
-    vec::poly_from_vec,
-};
+use crate::utils::gadgets::{eval_mle, MatrixGadget};
 use crate::Error;
 use crate::{
-    arith::{ccs::CCS, r1cs::R1CS},
-    folding::traits::{CommittedInstanceVarOps, Dummy, WitnessVarOps},
+    arith::{
+        ccs::{circuits::CCSMatricesVar, CCS},
+        ArithGadget,
+    },
+    folding::circuits::decider::{EvalGadget, KZGChallengesGadget},
 };
 use crate::{
     commitment::{pedersen::Params as PedersenParams, CommitmentScheme},
-    folding::nova::decider_eth_circuit::evaluate_gadget,
+    folding::circuits::decider::DeciderEnabledNIFS,
 };
 
+impl<C: CurveGroup> ArithGadget<WitnessVar<CF1<C>>, LCCCSVar<C>> for CCSMatricesVar<CF1<C>> {
+    type Evaluation = Vec<FpVar<CF1<C>>>;
+
+    fn eval_relation(
+        &self,
+        w: &WitnessVar<CF1<C>>,
+        u: &LCCCSVar<C>,
+    ) -> Result<Self::Evaluation, SynthesisError> {
+        let z = [&[u.u.clone()][..], &u.x, &w.w].concat();
+
+        self.M
+            .iter()
+            .map(|M_j| {
+                let s = log2(M_j.n_rows) as usize;
+                let Mz = M_j.mul_vector(&z)?;
+                Ok(eval_mle(s, Mz, u.r_x.clone()))
+            })
+            .collect()
+    }
+
+    fn enforce_evaluation(
+        _w: &WitnessVar<CF1<C>>,
+        u: &LCCCSVar<C>,
+        v: Self::Evaluation,
+    ) -> Result<(), SynthesisError> {
+        v.enforce_equal(&u.v)
+    }
+}
+
 /// In-circuit representation of the Witness associated to the CommittedInstance.
 #[derive(Debug, Clone)]
 pub struct WitnessVar<F: PrimeField> {
@@ -77,135 +100,38 @@ impl WitnessVarOps for WitnessVar {
     }
 }
 
-/// CCSMatricesVar contains the matrices 'M' of the CCS without the rest of CCS parameters.
-#[derive(Debug, Clone)]
-pub struct CCSMatricesVar<F: PrimeField> {
-    // we only need native representation, so the constraint field==F
-    pub M: Vec<SparseMatrixVar<F, F, FpVar<F>>>,
-}
-
-impl<F> AllocVar<CCS<F>, F> for CCSMatricesVar<F>
-where
-    F: PrimeField,
-{
-    fn new_variable<T: Borrow<CCS<F>>>(
-        cs: impl Into<Namespace<F>>,
-        f: impl FnOnce() -> Result<T, SynthesisError>,
-        _mode: AllocationMode,
-    ) -> Result<Self, SynthesisError> {
-        f().and_then(|val| {
-            let cs = cs.into();
-
-            let M: Vec<SparseMatrixVar<F, F, FpVar<F>>> = val
-                .borrow()
-                .M
-                .iter()
-                .map(|M| SparseMatrixVar::<F, F, FpVar<F>>::new_constant(cs.clone(), M.clone()))
-                .collect::<Result<_, SynthesisError>>()?;
-
-            Ok(Self { M })
-        })
-    }
-}
-
-/// Gadget to check the LCCCS relation both over the native constraint field and over the
-/// non-native constraint field.
-#[derive(Debug, Clone)]
-pub struct LCCCSCheckerGadget {}
-impl LCCCSCheckerGadget {
-    /// performs in-circuit the RelaxedR1CS check for native variables (Az∘Bz==uCz+E)
-    pub fn check<F: PrimeField>(
-        s: usize,
-        ccs_mat: CCSMatricesVar<F>,
-        z: Vec<FpVar<F>>,
-        // LCCCS values
-        r_x: Vec<FpVar<F>>,
-        v: Vec<FpVar<F>>,
-    ) -> Result<(), SynthesisError> {
-        let computed_v: Vec<FpVar<F>> = ccs_mat
-            .M
-            .iter()
-            .map(|M_j| {
-                let Mz = M_j.mul_vector(&z)?;
-                Ok(eval_mle(s, Mz, r_x.clone()))
-            })
-            .collect::<Result<Vec<FpVar<F>>, SynthesisError>>()?;
-
-        (computed_v).enforce_equal(&v)?;
-        Ok(())
-    }
-}
-
-/// Circuit that implements the in-circuit checks needed for the HyperNova's onchain (Ethereum's
-/// EVM) verification.
-#[derive(Clone, Debug)]
-pub struct DeciderEthCircuit<C1, GC1, C2, GC2, CS1, CS2, const H: bool = false>
+pub type DeciderEthCircuit<C1, C2, GC2> = GenericOnchainDeciderCircuit<
+    C1,
+    C2,
+    GC2,
+    LCCCS<C1>,
+    CCCS<C1>,
+    Witness<CF1<C1>>,
+    CCS<CF1<C1>>,
+    CCSMatricesVar<CF1<C1>>,
+    DeciderHyperNovaGadget,
+>;
+
+impl<
+        C1: CurveGroup,
+        GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
+        C2: CurveGroup,
+        GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        FC: FCircuit<C1::ScalarField>,
+        CS1: CommitmentScheme<C1, H>,
+        // enforce that the CS2 is Pedersen commitment scheme, since we're at Ethereum's EVM decider
+        CS2: CommitmentScheme<C2, H, ProverParams = PedersenParams<C2>>,
+        const MU: usize,
+        const NU: usize,
+        const H: bool,
+    > TryFrom<HyperNova<C1, GC1, C2, GC2, FC, CS1, CS2, MU, NU, H>>
+    for DeciderEthCircuit<C1, C2, GC2>
 where
-    C1: CurveGroup,
-    GC1: CurveVar<C1, CF2<C1>>,
-    C2: CurveGroup,
-    GC2: CurveVar<C2, CF2<C2>>,
-    CS1: CommitmentScheme<C1, H>,
-    CS2: CommitmentScheme<C2, H>,
+    CF1<C1>: Absorb,
 {
-    _c1: PhantomData<C1>,
-    _gc1: PhantomData<GC1>,
-    _c2: PhantomData<C2>,
-    _gc2: PhantomData<GC2>,
-    _cs1: PhantomData<CS1>,
-    _cs2: PhantomData<CS2>,
-
-    /// E vector's length of the CycleFold instance witness
-    pub cf_E_len: usize,
-    /// CCS of the Augmented Function circuit
-    pub ccs: CCS<C1::ScalarField>,
-    /// R1CS of the CycleFold circuit
-    pub cf_r1cs: R1CS<C2::ScalarField>,
-    /// CycleFold PedersenParams over C2
-    pub cf_pedersen_params: PedersenParams<C2>,
-    pub poseidon_config: PoseidonConfig<CF1<C1>>,
-    /// public params hash
-    pub pp_hash: Option<C1::ScalarField>,
-    pub i: Option<CF1<C1>>,
-    /// initial state
-    pub z_0: Option<Vec<C1::ScalarField>>,
-    /// current i-th state
-    pub z_i: Option<Vec<C1::ScalarField>>,
-    /// Nova instances
-    pub U_i: Option<LCCCS<C1>>,
-    pub W_i: Option<Witness<C1::ScalarField>>,
-    pub u_i: Option<CCCS<C1>>,
-    pub w_i: Option<Witness<C1::ScalarField>>,
-    pub U_i1: Option<LCCCS<C1>>,
-    pub W_i1: Option<Witness<C1::ScalarField>>,
-    pub nimfs_proof: Option<NIMFSProof<C1>>,
-    // rho is the 'random' value used for the fold of the last 2 instances
-    pub rho: Option<C1::ScalarField>,
-    /// CycleFold running instance
-    pub cf_U_i: Option<CycleFoldCommittedInstance<C2>>,
-    pub cf_W_i: Option<CycleFoldWitness<C2>>,
-
-    /// KZG challenge & eval
-    pub kzg_challenge: Option<C1::ScalarField>,
-    pub eval_W: Option<C1::ScalarField>,
-}
+    type Error = Error;
 
-impl<C1, GC1, C2, GC2, CS1, CS2, const H: bool> DeciderEthCircuit<C1, GC1, C2, GC2, CS1, CS2, H>
-where
-    C1: CurveGroup,
-    C2: CurveGroup,
-    GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
-    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
-    CS1: CommitmentScheme<C1, H>,
-    // enforce that the CS2 is Pedersen commitment scheme, since we're at Ethereum's EVM decider
-    CS2: CommitmentScheme<C2, H, ProverParams = PedersenParams<C2>>,
-    <C1 as Group>::ScalarField: Absorb,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-{
-    /// returns an instance of the DeciderEthCircuit from the given HyperNova struct
-    pub fn from_hypernova<FC: FCircuit<C1::ScalarField>, const MU: usize, const NU: usize>(
-        hn: HyperNova<C1, GC1, C2, GC2, FC, CS1, CS2, MU, NU, H>,
-    ) -> Result<Self, Error> {
+    fn try_from(hn: HyperNova<C1, GC1, C2, GC2, FC, CS1, CS2, MU, NU, H>) -> Result<Self, Error> {
         // compute the U_{i+1}, W_{i+1}, by folding the last running & incoming instances
         let mut transcript = PoseidonSponge::<C1::ScalarField>::new(&hn.poseidon_config);
         transcript.absorb(&hn.pp_hash);
@@ -219,288 +145,92 @@ where
         )?;
 
         // compute the KZG challenges used as inputs in the circuit
-        let kzg_challenge =
-            KZGChallengeGadget::<C1>::get_challenge_native(&mut transcript, U_i1.clone())?;
+        let kzg_challenges = KZGChallengesGadget::get_challenges_native(&mut transcript, &U_i1);
 
         // get KZG evals
-        let mut W = W_i1.w.clone();
-        W.extend(
-            std::iter::repeat(C1::ScalarField::zero())
-                .take(W_i1.w.len().next_power_of_two() - W_i1.w.len()),
-        );
-        let p_W = poly_from_vec(W.to_vec())?;
-        let eval_W = p_W.evaluate(&kzg_challenge);
+        let kzg_evaluations = W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .map(|((v, _), &c)| EvalGadget::evaluate_native(v, c))
+            .collect::<Result<Vec<_>, _>>()?;
 
         Ok(Self {
-            _c1: PhantomData,
-            _gc1: PhantomData,
-            _c2: PhantomData,
             _gc2: PhantomData,
-            _cs1: PhantomData,
-            _cs2: PhantomData,
-
-            cf_E_len: hn.cf_W_i.E.len(),
-            ccs: hn.ccs,
-            cf_r1cs: hn.cf_r1cs,
+            _avar: PhantomData,
+            arith: hn.ccs,
+            cf_arith: hn.cf_r1cs,
             cf_pedersen_params: hn.cf_cs_pp,
             poseidon_config: hn.poseidon_config,
-            pp_hash: Some(hn.pp_hash),
-            i: Some(hn.i),
-            z_0: Some(hn.z_0),
-            z_i: Some(hn.z_i),
-            U_i: Some(hn.U_i),
-            W_i: Some(hn.W_i),
-            u_i: Some(hn.u_i),
-            w_i: Some(hn.w_i),
-            U_i1: Some(U_i1),
-            W_i1: Some(W_i1),
-            nimfs_proof: Some(nimfs_proof),
-            rho: Some(rho),
-            cf_U_i: Some(hn.cf_U_i),
-            cf_W_i: Some(hn.cf_W_i),
-            kzg_challenge: Some(kzg_challenge),
-            eval_W: Some(eval_W),
+            pp_hash: hn.pp_hash,
+            i: hn.i,
+            z_0: hn.z_0,
+            z_i: hn.z_i,
+            U_i: hn.U_i,
+            W_i: hn.W_i,
+            u_i: hn.u_i,
+            w_i: hn.w_i,
+            U_i1,
+            W_i1,
+            proof: nimfs_proof,
+            randomness: rho,
+            cf_U_i: hn.cf_U_i,
+            cf_W_i: hn.cf_W_i,
+            kzg_challenges,
+            kzg_evaluations,
         })
     }
 }
 
-impl<C1, GC1, C2, GC2, CS1, CS2> ConstraintSynthesizer<CF1<C1>>
-    for DeciderEthCircuit<C1, GC1, C2, GC2, CS1, CS2>
+pub struct DeciderHyperNovaGadget;
+
+impl<C: CurveGroup> DeciderEnabledNIFS<C, LCCCS<C>, CCCS<C>, Witness<C::ScalarField>, CCS<CF1<C>>>
+    for DeciderHyperNovaGadget
 where
-    C1: CurveGroup,
-    C2: CurveGroup,
-    GC1: CurveVar<C1, CF2<C1>>,
-    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
-    CS1: CommitmentScheme<C1>,
-    CS2: CommitmentScheme<C2>,
-    C1::ScalarField: PrimeField,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-    <C2 as CurveGroup>::BaseField: PrimeField,
-    <C1 as Group>::ScalarField: Absorb,
-    <C2 as Group>::ScalarField: Absorb,
-    C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'b> &'b GC2: GroupOpsBounds<'b, C2, GC2>,
+    CF1<C>: Absorb,
 {
-    fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
-        let ccs_matrices =
-            CCSMatricesVar::<C1::ScalarField>::new_witness(cs.clone(), || Ok(self.ccs.clone()))?;
-
-        let pp_hash = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.pp_hash.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let i =
-            FpVar::<CF1<C1>>::new_input(cs.clone(), || Ok(self.i.unwrap_or_else(CF1::<C1>::zero)))?;
-        let z_0 = Vec::<FpVar<CF1<C1>>>::new_input(cs.clone(), || {
-            Ok(self.z_0.unwrap_or(vec![CF1::<C1>::zero()]))
-        })?;
-        let z_i = Vec::<FpVar<CF1<C1>>>::new_input(cs.clone(), || {
-            Ok(self.z_i.unwrap_or(vec![CF1::<C1>::zero()]))
-        })?;
-
-        let U_dummy_native = LCCCS::<C1>::dummy(&self.ccs);
-        let u_dummy_native = CCCS::<C1>::dummy(&self.ccs);
-        let w_dummy_native = Witness::<C1::ScalarField>::new(
-            vec![C1::ScalarField::zero(); self.ccs.n - 3 /* (3=2+1, since u_i.x.len=2) */],
-        );
-
-        let U_i = LCCCSVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.U_i.unwrap_or(U_dummy_native.clone()))
-        })?;
-        let u_i = CCCSVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.u_i.unwrap_or(u_dummy_native.clone()))
-        })?;
-        // here (U_i1, W_i1) = NIFS.P( (U_i,W_i), (u_i,w_i))
-        let U_i1 = LCCCSVar::<C1>::new_input(cs.clone(), || {
-            Ok(self.U_i1.unwrap_or(U_dummy_native.clone()))
-        })?;
-        let W_i1 = WitnessVar::<C1::ScalarField>::new_witness(cs.clone(), || {
-            Ok(self.W_i1.unwrap_or(w_dummy_native.clone()))
-        })?;
-        let nimfs_proof_dummy = NIMFSProof::<C1>::dummy((&self.ccs, 1, 1)); // mu=1 & nu=1 because the last fold is 2-to-1
-        let nimfs_proof = NIMFSProofVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.nimfs_proof.unwrap_or(nimfs_proof_dummy))
-        })?;
-
-        // allocate the inputs for the check 6
-        let kzg_challenge = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.kzg_challenge.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let eval_W = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.eval_W.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-
-        // `sponge` is for digest computation.
-        let sponge = PoseidonSpongeVar::<C1::ScalarField>::new(cs.clone(), &self.poseidon_config);
-        // `transcript` is for challenge generation.
-        let mut transcript = sponge.clone();
+    type ProofDummyCfg = (usize, usize, usize, usize);
+    type Proof = NIMFSProof<C>;
+    type Randomness = CF1<C>;
+    type RandomnessDummyCfg = ();
+
+    fn fold_field_elements_gadget(
+        arith: &CCS<CF1<C>>,
+        transcript: &mut PoseidonSpongeVar<CF1<C>>,
+        pp_hash: FpVar<CF1<C>>,
+        U: LCCCSVar<C>,
+        _U_vec: Vec<FpVar<CF1<C>>>,
+        u: CCCSVar<C>,
+        proof: Self::Proof,
+        randomness: Self::Randomness,
+    ) -> Result<LCCCSVar<C>, SynthesisError> {
+        let cs = transcript.cs();
         transcript.absorb(&pp_hash)?;
-
-        // NOTE: we use the same enumeration as in Nova's DeciderEthCircuit described at
-        // https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-onchain.html
-        // in order to make it easier to reason about.
-
-        // 1. check LCCCS relation of U_{i+1}
-        let z_U1: Vec<FpVar<CF1<C1>>> =
-            [vec![U_i1.u.clone()], U_i1.x.to_vec(), W_i1.w.to_vec()].concat();
-        LCCCSCheckerGadget::check(
-            self.ccs.s,
-            ccs_matrices,
-            z_U1,
-            U_i1.r_x.clone(),
-            U_i1.v.clone(),
-        )?;
-
-        // 3.a u_i.x[0] == H(i, z_0, z_i, U_i)
-        let (u_i_x, _) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
-        (u_i.x[0]).enforce_equal(&u_i_x)?;
-
-        #[cfg(feature = "light-test")]
-        log::warn!("[WARNING]: Running with the 'light-test' feature, skipping the big part of the DeciderEthCircuit.\n           Only for testing purposes.");
-
-        // The following two checks (and their respective allocations) are disabled for normal
-        // tests since they take several millions of constraints and would take several minutes
-        // (and RAM) to run the test. It is active by default, and not active only when
-        // 'light-test' feature is used.
-        #[cfg(not(feature = "light-test"))]
-        {
-            // imports here instead of at the top of the file, so we avoid having multiple
-            // `#[cfg(not(test))]`
-            use crate::commitment::pedersen::PedersenGadget;
-            use crate::folding::circuits::nonnative::uint::NonNativeUintVar;
-            use crate::folding::nova::decider_eth_circuit::{R1CSVar, RelaxedR1CSGadget};
-            use crate::folding::{
-                circuits::cyclefold::{
-                    CycleFoldCommittedInstanceVar, CycleFoldConfig, CycleFoldWitnessVar,
-                },
-                nova::NovaCycleFoldConfig,
-            };
-            use ark_r1cs_std::ToBitsGadget;
-
-            let cf_u_dummy_native =
-                CycleFoldCommittedInstance::<C2>::dummy(NovaCycleFoldConfig::<C1>::IO_LEN);
-            let cf_w_dummy_native = CycleFoldWitness::<C2>::dummy(&self.cf_r1cs);
-            let cf_U_i = CycleFoldCommittedInstanceVar::<C2, GC2>::new_witness(cs.clone(), || {
-                Ok(self.cf_U_i.unwrap_or_else(|| cf_u_dummy_native.clone()))
-            })?;
-            let cf_W_i = CycleFoldWitnessVar::<C2>::new_witness(cs.clone(), || {
-                Ok(self.cf_W_i.unwrap_or(cf_w_dummy_native.clone()))
-            })?;
-
-            // 3.b u_i.x[1] == H(cf_U_i)
-            let (cf_u_i_x, _) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
-            (u_i.x[1]).enforce_equal(&cf_u_i_x)?;
-
-            // 4. check Pedersen commitments of cf_U_i.{cmE, cmW}
-            let H = GC2::new_constant(cs.clone(), self.cf_pedersen_params.h)?;
-            let G = Vec::<GC2>::new_constant(cs.clone(), self.cf_pedersen_params.generators)?;
-            let cf_W_i_E_bits: Result<Vec<Vec<Boolean<CF1<C1>>>>, SynthesisError> =
-                cf_W_i.E.iter().map(|E_i| E_i.to_bits_le()).collect();
-            let cf_W_i_W_bits: Result<Vec<Vec<Boolean<CF1<C1>>>>, SynthesisError> =
-                cf_W_i.W.iter().map(|W_i| W_i.to_bits_le()).collect();
-
-            let computed_cmE = PedersenGadget::<C2, GC2>::commit(
-                H.clone(),
-                G.clone(),
-                cf_W_i_E_bits?,
-                cf_W_i.rE.to_bits_le()?,
-            )?;
-            cf_U_i.cmE.enforce_equal(&computed_cmE)?;
-            let computed_cmW =
-                PedersenGadget::<C2, GC2>::commit(H, G, cf_W_i_W_bits?, cf_W_i.rW.to_bits_le()?)?;
-            cf_U_i.cmW.enforce_equal(&computed_cmW)?;
-
-            let cf_r1cs =
-                R1CSVar::<C1::BaseField, CF1<C1>, NonNativeUintVar<CF1<C1>>>::new_witness(
-                    cs.clone(),
-                    || Ok(self.cf_r1cs.clone()),
-                )?;
-
-            // 5. check RelaxedR1CS of cf_U_i
-            let cf_z_U = [vec![cf_U_i.u.clone()], cf_U_i.x.to_vec(), cf_W_i.W.to_vec()].concat();
-            RelaxedR1CSGadget::check_nonnative(cf_r1cs, cf_W_i.E, cf_U_i.u.clone(), cf_z_U)?;
-        }
-
-        // The following steps are in non-increasing order because the `computed_U_i1` is computed
-        // at step 8, and later used at step 6. Notice that in Nova, we compute U_i1 outside of the
-        // circuit, in the smart contract, but here we're computing it in-circuit, and we reuse the
-        // `rho_bits` computed along the way of computing `computed_U_i1` for the later `rho_powers`
-        // check (6.b).
-
-        // 8.a verify the NIMFS.V of the final fold, and check that the obtained rho_powers from the
-        // transcript match the one from the public input (so we avoid the onchain logic of the
-        // verifier computing it).
-        // Notice that the NIMFSGadget performs all the logic except of checking the fold of the
-        // instances C parameter, which would require non-native arithmetic, henceforth we perform
-        // that check outside the circuit.
-        let (computed_U_i1, rho_bits) = NIMFSGadget::<C1>::verify(
+        let nimfs_proof = NIMFSProofVar::<C>::new_witness(cs.clone(), || Ok(proof))?;
+        let rho = FpVar::<CF1<C>>::new_input(cs.clone(), || Ok(randomness))?;
+        let (computed_U_i1, rho_bits) = NIMFSGadget::<C>::verify(
             cs.clone(),
-            &self.ccs.clone(),
-            &mut transcript,
-            &[U_i],
-            &[u_i],
+            arith,
+            transcript,
+            &[U],
+            &[u],
             nimfs_proof,
             Boolean::TRUE, // enabled
         )?;
-
-        // 6.a check KZG challenges
-        // Notice that this step is done after the NIMFS.V, to follow the transcript absorbs order
-        // done outside the circuit, where to compute the challenge it needs first to compute the
-        // U_{i+1} through the NIMFS.V
-        let incircuit_challenge =
-            KZGChallengeGadget::<C1>::get_challenge_gadget(&mut transcript, U_i1.clone())?;
-        incircuit_challenge.enforce_equal(&kzg_challenge)?;
-
-        // 6.b check that the obtained U_{i+1} from the NIMFS.V matches the U_{i+1} from the input,
-        // except for the C parameter, which to compute its folding would require non-native logic
-        // in-circuit, and we defer it to outside the circuit.
-        computed_U_i1.u.enforce_equal(&U_i1.u)?;
-        computed_U_i1.r_x.enforce_equal(&U_i1.r_x)?;
-        computed_U_i1.v.enforce_equal(&U_i1.v)?;
-
-        // 7. check eval_W==p_W(c_W)
-        let incircuit_eval_W = evaluate_gadget::<CF1<C1>>(W_i1.w, incircuit_challenge)?;
-        incircuit_eval_W.enforce_equal(&eval_W)?;
-
-        // 8.b check that the in-circuit computed r is equal to the inputted r.
-
-        let rho = Boolean::le_bits_to_fp_var(&rho_bits)?;
-        let external_rho =
-            FpVar::<CF1<C1>>::new_input(cs.clone(), || Ok(self.rho.unwrap_or(CF1::<C1>::zero())))?;
-        rho.enforce_equal(&external_rho)?;
-
-        Ok(())
+        Boolean::le_bits_to_fp_var(&rho_bits)?.enforce_equal(&rho)?;
+        Ok(computed_U_i1)
     }
-}
 
-/// Gadget that computes the KZG challenges, also offers the rust native implementation compatible
-/// with the gadget.
-pub struct KZGChallengeGadget<C: CurveGroup> {
-    _c: PhantomData<C>,
-}
-#[allow(clippy::type_complexity)]
-impl<C> KZGChallengeGadget<C>
-where
-    C: CurveGroup,
-    C::ScalarField: PrimeField,
-    <C as CurveGroup>::BaseField: PrimeField,
-    C::ScalarField: Absorb,
-{
-    pub fn get_challenge_native<T: Transcript<C::ScalarField>>(
-        transcript: &mut T,
-        U_i: LCCCS<C>,
-    ) -> Result<C::ScalarField, Error> {
-        // compute the KZG challenges, which are computed in-circuit and checked that it matches
-        // the inputted one
-        transcript.absorb_nonnative(&U_i.C);
-        Ok(transcript.get_challenge())
-    }
-    // compatible with the native get_challenge_native
-    pub fn get_challenge_gadget<S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>>(
-        transcript: &mut T,
-        U_i: LCCCSVar<C>,
-    ) -> Result<FpVar<C::ScalarField>, SynthesisError> {
-        transcript.absorb(&U_i.C.to_constraint_field()?)?;
-        transcript.get_challenge()
+    fn fold_group_elements_native(
+        U_commitments: &[C],
+        u_commitments: &[C],
+        _: Option<Self::Proof>,
+        r: Self::Randomness,
+    ) -> Result<Vec<C>, Error> {
+        let U_C = U_commitments[0];
+        let u_C = u_commitments[0];
+        let C = U_C + u_C.mul(r);
+        Ok(vec![C])
     }
 }
 
@@ -508,10 +238,11 @@ where
 pub mod tests {
     use ark_bn254::{constraints::GVar, Fr, G1Projective as Projective};
     use ark_grumpkin::{constraints::GVar as GVar2, Projective as Projective2};
-    use ark_relations::r1cs::ConstraintSystem;
+    use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
     use ark_std::{test_rng, UniformRand};
 
     use super::*;
+    use crate::arith::r1cs::R1CS;
     use crate::commitment::pedersen::Pedersen;
     use crate::folding::nova::PreprocessorParam;
     use crate::frontend::utils::CubicFCircuit;
@@ -530,7 +261,7 @@ pub mod tests {
         let (pedersen_params, _) =
             Pedersen::<Projective>::setup(&mut rng, ccs.n - ccs.l - 1).unwrap();
 
-        let (lcccs, _) = ccs
+        let (lcccs, w) = ccs
             .to_lcccs::<_, Projective, Pedersen<Projective>, false>(&mut rng, &pedersen_params, &z)
             .unwrap();
 
@@ -538,11 +269,10 @@ pub mod tests {
 
         // CCS's (sparse) matrices are constants in the circuit
         let ccs_mat = CCSMatricesVar::<Fr>::new_constant(cs.clone(), ccs.clone()).unwrap();
-        let zVar = Vec::<FpVar<Fr>>::new_input(cs.clone(), || Ok(z)).unwrap();
-        let r_xVar = Vec::<FpVar<Fr>>::new_input(cs.clone(), || Ok(lcccs.r_x)).unwrap();
-        let vVar = Vec::<FpVar<Fr>>::new_input(cs.clone(), || Ok(lcccs.v)).unwrap();
+        let w_var = WitnessVar::new_witness(cs.clone(), || Ok(w)).unwrap();
+        let lcccs_var = LCCCSVar::new_input(cs.clone(), || Ok(lcccs)).unwrap();
 
-        LCCCSCheckerGadget::check(ccs.s, ccs_mat, zVar, r_xVar, vVar).unwrap();
+        ccs_mat.enforce_relation(&w_var, &lcccs_var).unwrap();
 
         assert!(cs.is_satisfied().unwrap());
     }
@@ -588,15 +318,8 @@ pub mod tests {
         HN::verify(hn_params.1, ivc_proof).unwrap();
 
         // load the DeciderEthCircuit from the generated Nova instance
-        let decider_circuit = DeciderEthCircuit::<
-            Projective,
-            GVar,
-            Projective2,
-            GVar2,
-            Pedersen<Projective>,
-            Pedersen<Projective2>,
-        >::from_hypernova(hypernova)
-        .unwrap();
+        let decider_circuit =
+            DeciderEthCircuit::<Projective, Projective2, GVar2>::try_from(hypernova).unwrap();
 
         let cs = ConstraintSystem::<Fr>::new_ref();
 

folding-schemes/src/folding/hypernova/lcccs.rs

@@ -14,6 +14,7 @@ use crate::arith::ccs::CCS;
 use crate::arith::Arith;
 use crate::commitment::CommitmentScheme;
 use crate::folding::circuits::CF1;
+use crate::folding::traits::Inputize;
 use crate::folding::traits::{CommittedInstanceOps, Dummy};
 use crate::transcript::AbsorbNonNative;
 use crate::utils::mle::dense_vec_to_dense_mle;
@@ -72,7 +73,7 @@ impl CCS {
         Ok((
             LCCCS::<C> {
                 C,
-                u: C::ScalarField::one(),
+                u: z[0],
                 x: z[1..(1 + self.l)].to_vec(),
                 r_x,
                 v,
@@ -154,6 +155,19 @@ impl CommittedInstanceOps for LCCCS {
     }
 }
 
+impl<C: CurveGroup> Inputize<C::ScalarField, LCCCSVar<C>> for LCCCS<C> {
+    fn inputize(&self) -> Vec<C::ScalarField> {
+        [
+            &self.C.inputize(),
+            &[self.u][..],
+            &self.x,
+            &self.r_x,
+            &self.v,
+        ]
+        .concat()
+    }
+}
+
 #[cfg(test)]
 pub mod tests {
     use ark_pallas::{Fr, Projective};

folding-schemes/src/folding/hypernova/mod.rs

@@ -5,7 +5,7 @@ use ark_crypto_primitives::sponge::{
 };
 use ark_ec::{CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
-use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar, ToConstraintFieldGadget};
+use ark_r1cs_std::{prelude::CurveVar, ToConstraintFieldGadget};
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Compress, SerializationError};
 use ark_std::{fmt::Debug, marker::PhantomData, rand::RngCore, One, Zero};
 
@@ -293,8 +293,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     type RunningInstance = (LCCCS<C1>, Witness<C1::ScalarField>);
     type IncomingInstance = (CCCS<C1>, Witness<C1::ScalarField>);
@@ -360,8 +358,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     /// internal helper for new_running_instance & new_incoming_instance methods, returns the R1CS
     /// z=[u,x,w] vector to be used to create the LCCCS & CCCS fresh instances.
@@ -467,8 +463,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     /// Reuse Nova's PreprocessorParam.
     type PreprocessorParam = PreprocessorParam<C1, C2, FC, CS1, CS2, H>;
@@ -567,23 +561,14 @@ where
         let cf_r1cs = get_r1cs_from_cs::<C2::ScalarField>(cf_circuit)?;
 
         // if cs params exist, use them, if not, generate new ones
-        let cs_pp: CS1::ProverParams;
-        let cs_vp: CS1::VerifierParams;
-        let cf_cs_pp: CS2::ProverParams;
-        let cf_cs_vp: CS2::VerifierParams;
-        if prep_param.cs_pp.is_some()
-            && prep_param.cf_cs_pp.is_some()
-            && prep_param.cs_vp.is_some()
-            && prep_param.cf_cs_vp.is_some()
-        {
-            cs_pp = prep_param.clone().cs_pp.unwrap();
-            cs_vp = prep_param.clone().cs_vp.unwrap();
-            cf_cs_pp = prep_param.clone().cf_cs_pp.unwrap();
-            cf_cs_vp = prep_param.clone().cf_cs_vp.unwrap();
-        } else {
-            (cs_pp, cs_vp) = CS1::setup(&mut rng, ccs.n - ccs.l - 1)?;
-            (cf_cs_pp, cf_cs_vp) = CS2::setup(&mut rng, cf_r1cs.A.n_cols - cf_r1cs.l - 1)?;
-        }
+        let (cs_pp, cs_vp) = match (&prep_param.cs_pp, &prep_param.cs_vp) {
+            (Some(cs_pp), Some(cs_vp)) => (cs_pp.clone(), cs_vp.clone()),
+            _ => CS1::setup(&mut rng, ccs.n - ccs.l - 1)?,
+        };
+        let (cf_cs_pp, cf_cs_vp) = match (&prep_param.cf_cs_pp, &prep_param.cf_cs_vp) {
+            (Some(cf_cs_pp), Some(cf_cs_vp)) => (cf_cs_pp.clone(), cf_cs_vp.clone()),
+            _ => CS2::setup(&mut rng, cf_r1cs.A.n_cols - cf_r1cs.l - 1)?,
+        };
 
         let pp = ProverParams::<C1, C2, CS1, CS2, H> {
             poseidon_config: prep_param.poseidon_config.clone(),
@@ -726,9 +711,9 @@ where
             let (Us, Ws): (Vec<LCCCS<C1>>, Vec<Witness<C1::ScalarField>>) =
                 lcccs.into_iter().unzip();
             let (us, ws): (Vec<CCCS<C1>>, Vec<Witness<C1::ScalarField>>) = cccs.into_iter().unzip();
-            (Some(Us), Some(Ws), Some(us), Some(ws))
+            (Us, Ws, us, ws)
         } else {
-            (None, None, None, None)
+            (vec![], vec![], vec![], vec![])
         };
 
         let augmented_f_circuit: AugmentedFCircuit<C1, C2, GC2, FC, MU, NU>;
@@ -807,9 +792,9 @@ where
                 z_i: Some(self.z_i.clone()),
                 external_inputs: Some(external_inputs.clone()),
                 U_i: Some(self.U_i.clone()),
-                Us: Us.clone(),
+                Us: Some(Us),
                 u_i_C: Some(self.u_i.C),
-                us: us.clone(),
+                us: Some(us),
                 U_i1_C: Some(U_i1.C),
                 F: self.F.clone(),
                 x: Some(u_i1_x),
@@ -826,21 +811,12 @@ where
                 PoseidonSponge::<C1::ScalarField>::new(&self.poseidon_config);
             transcript_p.absorb(&self.pp_hash);
 
-            let (all_Us, all_us, all_Ws, all_ws) = if MU > 1 || NU > 1 {
-                (
-                    [vec![self.U_i.clone()], Us.clone().unwrap()].concat(),
-                    [vec![self.u_i.clone()], us.clone().unwrap()].concat(),
-                    [vec![self.W_i.clone()], Ws.unwrap()].concat(),
-                    [vec![self.w_i.clone()], ws.unwrap()].concat(),
-                )
-            } else {
-                (
-                    vec![self.U_i.clone()],
-                    vec![self.u_i.clone()],
-                    vec![self.W_i.clone()],
-                    vec![self.w_i.clone()],
-                )
-            };
+            let (all_Us, all_us, all_Ws, all_ws) = (
+                [&[self.U_i.clone()][..], &Us].concat(),
+                [&[self.u_i.clone()][..], &us].concat(),
+                [vec![self.W_i.clone()], Ws].concat(),
+                [vec![self.w_i.clone()], ws].concat(),
+            );
 
             let (rho, nimfs_proof);
             (nimfs_proof, U_i1, W_i1, rho) = NIMFS::<C1, PoseidonSponge<C1::ScalarField>>::prove(
@@ -865,7 +841,9 @@ where
             );
 
             let rho_bits = rho.into_bigint().to_bits_le()[..NOVA_N_BITS_RO].to_vec();
-            let rho_Fq = C1::BaseField::from_bigint(BigInteger::from_bits_le(&rho_bits)).unwrap();
+            let rho_Fq = C1::BaseField::from(<C1::BaseField as PrimeField>::BigInt::from_bits_le(
+                &rho_bits,
+            ));
 
             // CycleFold part:
             // get the vector used as public inputs 'x' in the CycleFold circuit.
@@ -935,9 +913,9 @@ where
                 z_i: Some(self.z_i.clone()),
                 external_inputs: Some(external_inputs),
                 U_i: Some(self.U_i.clone()),
-                Us: Us.clone(),
+                Us: Some(Us),
                 u_i_C: Some(self.u_i.C),
-                us: us.clone(),
+                us: Some(us),
                 U_i1_C: Some(U_i1.C),
                 F: self.F.clone(),
                 x: Some(u_i1_x),
@@ -1030,7 +1008,7 @@ where
         } = ivc_proof;
         let (pp, vp) = params;
 
-        let f_circuit = FC::new(fcircuit_params).unwrap();
+        let f_circuit = FC::new(fcircuit_params)?;
         let augmented_f_circuit = AugmentedFCircuit::<C1, C2, GC2, FC, MU, NU>::empty(
             &pp.poseidon_config,
             f_circuit.clone(),

folding-schemes/src/folding/hypernova/nimfs.rs

@@ -31,28 +31,34 @@ pub struct NIMFSProof {
     pub sigmas_thetas: SigmasThetas<C::ScalarField>,
 }
 
-impl<C: CurveGroup> Dummy<(&CCS<CF1<C>>, usize, usize)> for NIMFSProof<C> {
-    fn dummy((ccs, mu, nu): (&CCS<CF1<C>>, usize, usize)) -> Self {
+impl<C: CurveGroup> Dummy<(usize, usize, usize, usize)> for NIMFSProof<C> {
+    fn dummy((s, t, mu, nu): (usize, usize, usize, usize)) -> Self {
         // use 'C::ScalarField::one()' instead of 'zero()' to enforce the NIMFSProof to have the
         // same in-circuit representation to match the number of constraints of an actual proof.
         NIMFSProof::<C> {
             sc_proof: SumCheckProof::<C::ScalarField> {
-                point: vec![C::ScalarField::one(); ccs.s],
+                point: vec![C::ScalarField::one(); s],
                 proofs: vec![
                     IOPProverMessage {
-                        coeffs: vec![C::ScalarField::one(); ccs.t + 1]
+                        coeffs: vec![C::ScalarField::one(); t + 1]
                     };
-                    ccs.s
+                    s
                 ],
             },
             sigmas_thetas: SigmasThetas(
-                vec![vec![C::ScalarField::one(); ccs.t]; mu],
-                vec![vec![C::ScalarField::one(); ccs.t]; nu],
+                vec![vec![C::ScalarField::one(); t]; mu],
+                vec![vec![C::ScalarField::one(); t]; nu],
             ),
         }
     }
 }
 
+impl<C: CurveGroup> Dummy<(&CCS<CF1<C>>, usize, usize)> for NIMFSProof<C> {
+    fn dummy((ccs, mu, nu): (&CCS<CF1<C>>, usize, usize)) -> Self {
+        NIMFSProof::dummy((ccs.s, ccs.t, mu, nu))
+    }
+}
+
 #[derive(Clone, Debug, Eq, PartialEq)]
 pub struct SigmasThetas<F: PrimeField>(pub Vec<Vec<F>>, pub Vec<Vec<F>>);
 
@@ -67,7 +73,6 @@ pub struct NIMFS> {
 impl<C: CurveGroup, T: Transcript<C::ScalarField>> NIMFS<C, T>
 where
     <C as Group>::ScalarField: Absorb,
-    C::BaseField: PrimeField,
 {
     pub fn fold(
         lcccs: &[LCCCS<C>],
@@ -258,8 +263,9 @@ where
         let rho_scalar = C::ScalarField::from_le_bytes_mod_order(b"rho");
         transcript.absorb(&rho_scalar);
         let rho_bits: Vec<bool> = transcript.get_challenge_nbits(NOVA_N_BITS_RO);
-        let rho: C::ScalarField =
-            C::ScalarField::from_bigint(BigInteger::from_bits_le(&rho_bits)).unwrap();
+        let rho: C::ScalarField = C::ScalarField::from(
+            <C::ScalarField as PrimeField>::BigInt::from_bits_le(&rho_bits),
+        );
 
         // Step 7: Create the folded instance
         let folded_lcccs = Self::fold(
@@ -377,8 +383,9 @@ where
         let rho_scalar = C::ScalarField::from_le_bytes_mod_order(b"rho");
         transcript.absorb(&rho_scalar);
         let rho_bits: Vec<bool> = transcript.get_challenge_nbits(NOVA_N_BITS_RO);
-        let rho: C::ScalarField =
-            C::ScalarField::from_bigint(BigInteger::from_bits_le(&rho_bits)).unwrap();
+        let rho: C::ScalarField = C::ScalarField::from(
+            <C::ScalarField as PrimeField>::BigInt::from_bits_le(&rho_bits),
+        );
 
         // Step 7: Compute the folded instance
         Ok(Self::fold(

folding-schemes/src/folding/nova/circuits.rs

@@ -11,7 +11,6 @@ use ark_r1cs_std::{
     boolean::Boolean,
     eq::EqGadget,
     fields::{fp::FpVar, FieldVar},
-    groups::GroupOpsBounds,
     prelude::CurveVar,
     uint8::UInt8,
     R1CSVar, ToConstraintFieldGadget,
@@ -73,11 +72,7 @@ where
     }
 }
 
-impl<C> AbsorbGadget<C::ScalarField> for CommittedInstanceVar<C>
-where
-    C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-{
+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()?)
     }
@@ -124,11 +119,7 @@ pub struct NIFSGadget {
     _c: PhantomData<C>,
 }
 
-impl<C: CurveGroup> NIFSGadget<C>
-where
-    C: CurveGroup,
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
-{
+impl<C: CurveGroup> NIFSGadget<C> {
     pub fn fold_committed_instance(
         r: FpVar<CF1<C>>,
         ci1: CommittedInstanceVar<C>, // U_i
@@ -174,8 +165,6 @@ pub struct ChallengeGadget {
 }
 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>>(
@@ -232,9 +221,7 @@ pub struct AugmentedFCircuit<
     C2: CurveGroup,
     GC2: CurveVar<C2, CF2<C2>>,
     FC: FCircuit<CF1<C1>>,
-> where
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
-{
+> {
     pub(super) _gc2: PhantomData<GC2>,
     pub(super) poseidon_config: PoseidonConfig<CF1<C1>>,
     pub(super) pp_hash: Option<CF1<C1>>,
@@ -264,8 +251,6 @@ pub struct AugmentedFCircuit<
 
 impl<C1: CurveGroup, C2: CurveGroup, GC2: CurveVar<C2, CF2<C2>>, FC: FCircuit<CF1<C1>>>
     AugmentedFCircuit<C1, C2, GC2, FC>
-where
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     pub fn empty(poseidon_config: &PoseidonConfig<CF1<C1>>, F_circuit: FC) -> Self {
         Self {
@@ -306,7 +291,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
         let pp_hash = FpVar::<CF1<C1>>::new_witness(cs.clone(), || {
@@ -508,7 +492,7 @@ where
         // Non-base case: u_{i+1}.x[1] == H(cf_U_{i+1})
         let (cf_u_i1_x, _) = cf_U_i1.clone().hash(&sponge, pp_hash.clone())?;
         let (cf_u_i1_x_base, _) =
-            CycleFoldCommittedInstanceVar::new_constant(cs.clone(), cf_u_dummy)?
+            CycleFoldCommittedInstanceVar::<C2, GC2>::new_constant(cs.clone(), cf_u_dummy)?
                 .hash(&sponge, pp_hash)?;
         let cf_x = FpVar::new_input(cs.clone(), || {
             Ok(self.cf_x.unwrap_or(cf_u_i1_x_base.value()?))

folding-schemes/src/folding/nova/decider.rs

@@ -2,8 +2,8 @@
 /// DeciderEth from decider_eth.rs file.
 /// More details can be found at the documentation page:
 /// https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-offchain.html
-use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, Absorb, CryptographicSponge};
-use ark_ec::{AffineRepr, CurveGroup, Group};
+use ark_crypto_primitives::sponge::Absorb;
+use ark_ec::{CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
 use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar, ToConstraintFieldGadget};
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
@@ -13,18 +13,16 @@ use ark_std::{One, Zero};
 use core::marker::PhantomData;
 
 use super::decider_circuits::{DeciderCircuit1, DeciderCircuit2};
-use super::{
-    nifs::{nova::NIFS, NIFSTrait},
-    CommittedInstance, Nova,
-};
+use super::decider_eth_circuit::DeciderNovaGadget;
+use super::Nova;
 use crate::commitment::CommitmentScheme;
+use crate::folding::circuits::decider::DeciderEnabledNIFS;
 use crate::folding::circuits::{
-    cyclefold::CycleFoldCommittedInstance,
-    nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
+    cyclefold::{CycleFoldCommittedInstance, CycleFoldCommittedInstanceVar},
     CF2,
 };
+use crate::folding::traits::{CommittedInstanceOps, Inputize, WitnessOps};
 use crate::frontend::FCircuit;
-use crate::transcript::poseidon::poseidon_canonical_config;
 use crate::Error;
 use crate::{Decider as DeciderTrait, FoldingScheme};
 
@@ -45,8 +43,9 @@ where
     // cmT and r are values for the last fold, U_{i+1}=NIFS.V(r, U_i, u_i, cmT), and they are
     // checked in-circuit
     cmT: C1,
+    r: C1::ScalarField,
     // cyclefold committed instance
-    cf_U_i: CycleFoldCommittedInstance<C2>,
+    cf_U_final: CycleFoldCommittedInstance<C2>,
     // the CS challenges are provided by the prover, but in-circuit they are checked to match the
     // in-circuit computed computed ones.
     cs1_challenges: [C1::ScalarField; 2],
@@ -147,22 +146,21 @@ where
         S2::VerifyingKey,
     >;
     type PublicInput = Vec<C1::ScalarField>;
-    type CommittedInstance = CommittedInstance<C1>;
+    type CommittedInstance = Vec<C1>;
 
     fn preprocess(
         mut rng: impl RngCore + CryptoRng,
         prep_param: Self::PreprocessorParam,
         fs: FS,
     ) -> Result<(Self::ProverParam, Self::VerifierParam), Error> {
-        let circuit1 = DeciderCircuit1::<C1, C2, GC2>::from_nova::<GC1, CS1, CS2, false, FC>(
-            fs.clone().into(),
-        )?;
-        let circuit2 =
-            DeciderCircuit2::<C1, GC1, C2>::from_nova::<GC2, CS1, CS2, false, FC>(fs.into())?;
+        let circuit1 = DeciderCircuit1::<C1, C2, GC2>::try_from(Nova::from(fs.clone()))?;
+        let circuit2 = DeciderCircuit2::<C2>::try_from(Nova::from(fs))?;
 
         // get the Groth16 specific setup for the circuits
-        let (c1_g16_pk, c1_g16_vk) = S1::circuit_specific_setup(circuit1, &mut rng).unwrap();
-        let (c2_g16_pk, c2_g16_vk) = S2::circuit_specific_setup(circuit2, &mut rng).unwrap();
+        let (c1_g16_pk, c1_g16_vk) = S1::circuit_specific_setup(circuit1, &mut rng)
+            .map_err(|e| Error::SNARKSetupFail(e.to_string()))?;
+        let (c2_g16_pk, c2_g16_vk) = S2::circuit_specific_setup(circuit2, &mut rng)
+            .map_err(|e| Error::SNARKSetupFail(e.to_string()))?;
 
         // get the FoldingScheme prover & verifier params from Nova
         #[allow(clippy::type_complexity)]
@@ -200,76 +198,57 @@ where
         pp: Self::ProverParam,
         fs: FS,
     ) -> Result<Self::Proof, Error> {
-        let circuit1 = DeciderCircuit1::<C1, C2, GC2>::from_nova::<GC1, CS1, CS2, false, FC>(
-            fs.clone().into(),
-        )?;
-        let circuit2 =
-            DeciderCircuit2::<C1, GC1, C2>::from_nova::<GC2, CS1, CS2, false, FC>(fs.into())?;
-
-        let c1_snark_proof = S1::prove(&pp.c1_snark_pp, circuit1.clone(), &mut rng)
+        let circuit1 = DeciderCircuit1::<C1, C2, GC2>::try_from(Nova::from(fs.clone()))?;
+        let circuit2 = DeciderCircuit2::<C2>::try_from(Nova::from(fs))?;
+
+        let cmT = circuit1.proof;
+        let r = circuit1.randomness;
+        let cf_U_final = circuit1.cf_U_i.clone();
+
+        let c1_kzg_challenges = circuit1.kzg_challenges.clone();
+        let c1_kzg_proofs = circuit1
+            .W_i1
+            .get_openings()
+            .iter()
+            .zip(&c1_kzg_challenges)
+            .map(|((v, _), &c)| {
+                CS1::prove_with_challenge(&pp.c1_cs_pp, c, v, &C1::ScalarField::zero(), None)
+            })
+            .collect::<Result<Vec<_>, _>>()?;
+        let c2_kzg_challenges = circuit2.kzg_challenges.clone();
+        let c2_kzg_proofs = circuit2
+            .cf_W_i
+            .get_openings()
+            .iter()
+            .zip(&c2_kzg_challenges)
+            .map(|((v, _), &c)| {
+                CS2::prove_with_challenge(&pp.c2_cs_pp, c, v, &C2::ScalarField::zero(), None)
+            })
+            .collect::<Result<Vec<_>, _>>()?;
+
+        let c1_snark_proof = S1::prove(&pp.c1_snark_pp, circuit1, &mut rng)
             .map_err(|e| Error::Other(e.to_string()))?;
-        let c2_snark_proof = S2::prove(&pp.c2_snark_pp, circuit2.clone(), &mut rng)
+        let c2_snark_proof = S2::prove(&pp.c2_snark_pp, circuit2, &mut rng)
             .map_err(|e| Error::Other(e.to_string()))?;
 
-        let cmT = circuit1.cmT.unwrap();
-        let W_i1 = circuit1.W_i1.unwrap();
-        let cf_W_i = circuit2.cf_W_i.unwrap();
-
-        // get the challenges that have been already computed when preparing the circuits inputs in
-        // the above `from_nova` calls
-        let challenge_W = circuit1
-            .cs_c_W
-            .ok_or(Error::MissingValue("cs_c_W".to_string()))?;
-        let challenge_E = circuit1
-            .cs_c_E
-            .ok_or(Error::MissingValue("cs_c_E".to_string()))?;
-        let c2_challenge_W = circuit2
-            .cs_c_W
-            .ok_or(Error::MissingValue("c2's cs_c_W".to_string()))?;
-        let c2_challenge_E = circuit2
-            .cs_c_E
-            .ok_or(Error::MissingValue("c2's cs_c_E".to_string()))?;
-
-        // generate CommitmentScheme proofs for the main instance
-        let U_cmW_proof = CS1::prove_with_challenge(
-            &pp.c1_cs_pp,
-            challenge_W,
-            &W_i1.W,
-            &C1::ScalarField::zero(),
-            None,
-        )?;
-        let U_cmE_proof = CS1::prove_with_challenge(
-            &pp.c1_cs_pp,
-            challenge_E,
-            &W_i1.E,
-            &C1::ScalarField::zero(),
-            None,
-        )?;
-        // CS proofs for the CycleFold instance
-        let cf_cmW_proof = CS2::prove_with_challenge(
-            &pp.c2_cs_pp,
-            c2_challenge_W,
-            &cf_W_i.W,
-            &C2::ScalarField::zero(),
-            None,
-        )?;
-        let cf_cmE_proof = CS2::prove_with_challenge(
-            &pp.c2_cs_pp,
-            c2_challenge_E,
-            &cf_W_i.E,
-            &C2::ScalarField::zero(),
-            None,
-        )?;
-
         Ok(Self::Proof {
             c1_snark_proof,
             c2_snark_proof,
-            cs1_proofs: [U_cmW_proof, U_cmE_proof],
-            cs2_proofs: [cf_cmW_proof, cf_cmE_proof],
+            cs1_proofs: c1_kzg_proofs
+                .try_into()
+                .map_err(|e: Vec<_>| Error::NotExpectedLength(e.len(), 2))?,
+            cs2_proofs: c2_kzg_proofs
+                .try_into()
+                .map_err(|e: Vec<_>| Error::NotExpectedLength(e.len(), 2))?,
             cmT,
-            cf_U_i: circuit1.cf_U_i.unwrap(),
-            cs1_challenges: [challenge_W, challenge_E],
-            cs2_challenges: [c2_challenge_W, c2_challenge_E],
+            r,
+            cf_U_final,
+            cs1_challenges: c1_kzg_challenges
+                .try_into()
+                .map_err(|e: Vec<_>| Error::NotExpectedLength(e.len(), 2))?,
+            cs2_challenges: c2_kzg_challenges
+                .try_into()
+                .map_err(|e: Vec<_>| Error::NotExpectedLength(e.len(), 2))?,
         })
     }
 
@@ -278,72 +257,38 @@ where
         i: C1::ScalarField,
         z_0: Vec<C1::ScalarField>,
         z_i: Vec<C1::ScalarField>,
-        running_instance: &Self::CommittedInstance,
-        incoming_instance: &Self::CommittedInstance,
+        // we don't use the instances at the verifier level, since we check them in-circuit
+        running_commitments: &Self::CommittedInstance,
+        incoming_commitments: &Self::CommittedInstance,
         proof: &Self::Proof,
     ) -> Result<bool, Error> {
         if i <= C1::ScalarField::one() {
             return Err(Error::NotEnoughSteps);
         }
 
-        // compute U = U_{d+1}= NIFS.V(U_d, u_d, cmT)
-        let poseidon_config = poseidon_canonical_config::<C1::ScalarField>();
-        let mut transcript = PoseidonSponge::<C1::ScalarField>::new(&poseidon_config);
-        let (U, r_bits) = NIFS::<C1, CS1, PoseidonSponge<C1::ScalarField>>::verify(
-            &mut transcript,
-            vp.pp_hash,
-            running_instance,
-            incoming_instance,
-            &proof.cmT,
+        // 6.2. Fold the commitments
+        let U_final_commitments = DeciderNovaGadget::fold_group_elements_native(
+            running_commitments,
+            incoming_commitments,
+            Some(proof.cmT),
+            proof.r,
         )?;
-        let r = C1::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits))
-            .ok_or(Error::OutOfBounds)?;
-
-        let (cmE_x, cmE_y) = NonNativeAffineVar::inputize(U.cmE)?;
-        let (cmW_x, cmW_y) = NonNativeAffineVar::inputize(U.cmW)?;
-        let (cmT_x, cmT_y) = NonNativeAffineVar::inputize(proof.cmT)?;
-
-        let zero = (&C2::BaseField::zero(), &C2::BaseField::zero());
-        let cmE_affine = proof.cf_U_i.cmE.into_affine();
-        let cmW_affine = proof.cf_U_i.cmW.into_affine();
-        let (cf_cmE_x, cf_cmE_y) = cmE_affine.xy().unwrap_or(zero);
-        let cf_cmE_z = C1::ScalarField::one();
-        let (cf_cmW_x, cf_cmW_y) = cmW_affine.xy().unwrap_or(zero);
-        let cf_cmW_z = C1::ScalarField::one();
+        let cf_U = proof.cf_U_final.clone();
 
         // snark proof 1
-        let c1_public_input: Vec<C1::ScalarField> = [
-            vec![vp.pp_hash, i],
-            z_0,
-            z_i,
-            // U_{i+1} values:
-            vec![U.u],
-            U.x.clone(),
-            cmE_x,
-            cmE_y,
-            cmW_x,
-            cmW_y,
-            // CS1 values:
-            proof.cs1_challenges.to_vec(), // c_W, c_E
-            vec![
-                proof.cs1_proofs[0].eval, // eval_W
-                proof.cs1_proofs[1].eval, // eval_E
-            ],
-            // cf_U_i values
-            NonNativeUintVar::<CF2<C2>>::inputize(proof.cf_U_i.u),
-            proof
-                .cf_U_i
-                .x
+        let c1_public_input = [
+            &[vp.pp_hash, i][..],
+            &z_0,
+            &z_i,
+            &U_final_commitments
                 .iter()
-                .flat_map(|&x_i| NonNativeUintVar::<CF2<C2>>::inputize(x_i))
-                .collect::<Vec<C1::ScalarField>>(),
-            vec![
-                *cf_cmE_x, *cf_cmE_y, cf_cmE_z, *cf_cmW_x, *cf_cmW_y, cf_cmW_z,
-            ],
-            // NIFS values:
-            cmT_x,
-            cmT_y,
-            vec![r],
+                .flat_map(|c| c.inputize())
+                .collect::<Vec<_>>(),
+            &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();
 
@@ -353,26 +298,15 @@ where
             return Err(Error::SNARKVerificationFail);
         }
 
-        let (cf2_cmE_x, cf2_cmE_y) = NonNativeAffineVar::inputize(proof.cf_U_i.cmE)?;
-        let (cf2_cmW_x, cf2_cmW_y) = NonNativeAffineVar::inputize(proof.cf_U_i.cmW)?;
-
         // snark proof 2
         // migrate pp_hash from C1::Fr to C1::Fq
         let pp_hash_Fq =
             C2::ScalarField::from_le_bytes_mod_order(&vp.pp_hash.into_bigint().to_bytes_le());
         let c2_public_input: Vec<C2::ScalarField> = [
-            vec![pp_hash_Fq],
-            vec![proof.cf_U_i.u],
-            proof.cf_U_i.x.clone(),
-            cf2_cmE_x,
-            cf2_cmE_y,
-            cf2_cmW_x,
-            cf2_cmW_y,
-            proof.cs2_challenges.to_vec(),
-            vec![
-                proof.cs2_proofs[0].eval, // eval_W
-                proof.cs2_proofs[1].eval, // eval_E
-            ],
+            &[pp_hash_Fq][..],
+            &cf_U.inputize(),
+            &proof.cs2_challenges,
+            &proof.cs2_proofs.iter().map(|p| p.eval).collect::<Vec<_>>(),
         ]
         .concat();
 
@@ -382,33 +316,24 @@ where
             return Err(Error::SNARKVerificationFail);
         }
 
-        // check C1 commitments (main instance commitments)
-        CS1::verify_with_challenge(
-            &vp.c1_cs_vp,
-            proof.cs1_challenges[0],
-            &U.cmW,
-            &proof.cs1_proofs[0],
-        )?;
-        CS1::verify_with_challenge(
-            &vp.c1_cs_vp,
-            proof.cs1_challenges[1],
-            &U.cmE,
-            &proof.cs1_proofs[1],
-        )?;
+        // 7.3. check C1 commitments (main instance commitments)
+        for ((cm, &c), pi) in U_final_commitments
+            .iter()
+            .zip(&proof.cs1_challenges)
+            .zip(&proof.cs1_proofs)
+        {
+            CS1::verify_with_challenge(&vp.c1_cs_vp, c, cm, pi)?;
+        }
 
-        // check C2 commitments (CycleFold instance commitments)
-        CS2::verify_with_challenge(
-            &vp.c2_cs_vp,
-            proof.cs2_challenges[0],
-            &proof.cf_U_i.cmW,
-            &proof.cs2_proofs[0],
-        )?;
-        CS2::verify_with_challenge(
-            &vp.c2_cs_vp,
-            proof.cs2_challenges[1],
-            &proof.cf_U_i.cmE,
-            &proof.cs2_proofs[1],
-        )?;
+        // 4.3. check C2 commitments (CycleFold instance commitments)
+        for ((cm, &c), pi) in cf_U
+            .get_commitments()
+            .iter()
+            .zip(&proof.cs2_challenges)
+            .zip(&proof.cs2_proofs)
+        {
+            CS2::verify_with_challenge(&vp.c2_cs_vp, c, cm, pi)?;
+        }
 
         Ok(true)
     }
@@ -494,7 +419,13 @@ pub mod tests {
         // decider proof verification
         let start = Instant::now();
         let verified = D::verify(
-            decider_vp, nova.i, nova.z_0, nova.z_i, &nova.U_i, &nova.u_i, &proof,
+            decider_vp,
+            nova.i,
+            nova.z_0,
+            nova.z_i,
+            &nova.U_i.get_commitments(),
+            &nova.u_i.get_commitments(),
+            &proof,
         )
         .unwrap();
         assert!(verified);

folding-schemes/src/folding/nova/decider_circuits.rs

@@ -2,116 +2,63 @@
 /// DeciderEthCircuit.
 /// More details can be found at the documentation page:
 /// https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-offchain.html
-use ark_crypto_primitives::sponge::{
-    constraints::CryptographicSpongeVar,
-    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
-    Absorb, CryptographicSponge,
-};
-use ark_ec::{CurveGroup, Group};
+use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, Absorb, CryptographicSponge};
+use ark_ec::CurveGroup;
 use ark_ff::{BigInteger, PrimeField};
-use ark_poly::Polynomial;
-use ark_r1cs_std::{
-    alloc::AllocVar,
-    boolean::Boolean,
-    eq::EqGadget,
-    fields::{fp::FpVar, FieldVar},
-    groups::GroupOpsBounds,
-    prelude::CurveVar,
-    ToConstraintFieldGadget,
-};
-use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, SynthesisError};
-use ark_std::Zero;
+use ark_r1cs_std::{fields::fp::FpVar, prelude::CurveVar, ToConstraintFieldGadget};
 use core::marker::PhantomData;
 
 use super::{
-    circuits::{ChallengeGadget, CommittedInstanceVar},
-    decider_eth_circuit::{
-        evaluate_gadget, KZGChallengesGadget, R1CSVar, RelaxedR1CSGadget, WitnessVar,
-    },
+    decider_eth_circuit::DeciderNovaGadget,
     nifs::{nova::NIFS, NIFSTrait},
     CommittedInstance, Nova, Witness,
 };
-use crate::arith::r1cs::R1CS;
-use crate::commitment::CommitmentScheme;
-use crate::folding::circuits::{
-    cyclefold::{
-        CycleFoldCommittedInstance, CycleFoldCommittedInstanceVar, CycleFoldConfig,
-        CycleFoldWitness,
-    },
-    nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
-    CF1, CF2,
+use crate::folding::{
+    circuits::{CF1, CF2},
+    traits::WitnessOps,
 };
-use crate::folding::nova::NovaCycleFoldConfig;
-use crate::folding::traits::{CommittedInstanceVarOps, Dummy};
 use crate::frontend::FCircuit;
-use crate::utils::vec::poly_from_vec;
 use crate::Error;
+use crate::{
+    arith::r1cs::{circuits::R1CSMatricesVar, R1CS},
+    folding::circuits::decider::{
+        off_chain::{GenericOffchainDeciderCircuit1, GenericOffchainDeciderCircuit2},
+        EvalGadget, KZGChallengesGadget,
+    },
+};
+use crate::{commitment::CommitmentScheme, transcript::poseidon::poseidon_canonical_config};
 
 /// Circuit that implements part of the in-circuit checks needed for the offchain verification over
 /// the Curve2's BaseField (=Curve1's ScalarField).
-#[derive(Clone, Debug)]
-pub struct DeciderCircuit1<C1, C2, GC2>
-where
-    C1: CurveGroup,
-    C2: CurveGroup,
-    GC2: CurveVar<C2, CF2<C2>>,
-{
-    _c1: PhantomData<C1>,
-    _c2: PhantomData<C2>,
-    _gc2: PhantomData<GC2>,
-
-    /// E vector's length of the Nova instance witness
-    pub E_len: usize,
-    /// E vector's length of the CycleFold instance witness
-    pub cf_E_len: usize,
-    /// R1CS of the Augmented Function circuit
-    pub r1cs: R1CS<C1::ScalarField>,
-    pub poseidon_config: PoseidonConfig<CF1<C1>>,
-    /// public params hash
-    pub pp_hash: Option<C1::ScalarField>,
-    pub i: Option<CF1<C1>>,
-    /// initial state
-    pub z_0: Option<Vec<C1::ScalarField>>,
-    /// current i-th state
-    pub z_i: Option<Vec<C1::ScalarField>>,
-    /// Nova instances
-    pub u_i: Option<CommittedInstance<C1>>,
-    pub w_i: Option<Witness<C1>>,
-    pub U_i: Option<CommittedInstance<C1>>,
-    pub W_i: Option<Witness<C1>>,
-    pub U_i1: Option<CommittedInstance<C1>>,
-    pub W_i1: Option<Witness<C1>>,
-    pub cmT: Option<C1>,
-    pub r: Option<C1::ScalarField>,
-    /// CycleFold running instance
-    pub cf_U_i: Option<CycleFoldCommittedInstance<C2>>,
 
-    /// Commitment Scheme challenges
-    pub cs_c_W: Option<C1::ScalarField>,
-    pub cs_c_E: Option<C1::ScalarField>,
-    /// Evaluations of the committed polynomials at the challenge
-    pub eval_W: Option<C1::ScalarField>,
-    pub eval_E: Option<C1::ScalarField>,
-}
-impl<C1, C2, GC2> DeciderCircuit1<C1, C2, GC2>
-where
-    C1: CurveGroup,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-    <C1 as Group>::ScalarField: Absorb,
-    C2: CurveGroup,
-    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
-    for<'b> &'b GC2: GroupOpsBounds<'b, C2, GC2>,
-{
-    pub fn from_nova<GC1, CS1, CS2, const H: bool, FC: FCircuit<C1::ScalarField>>(
-        nova: Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>,
-    ) -> Result<Self, Error>
-    where
-        C2: CurveGroup,
+pub type DeciderCircuit1<C1, C2, GC2> = GenericOffchainDeciderCircuit1<
+    C1,
+    C2,
+    GC2,
+    CommittedInstance<C1>,
+    CommittedInstance<C1>,
+    Witness<C1>,
+    R1CS<CF1<C1>>,
+    R1CSMatricesVar<CF1<C1>, FpVar<CF1<C1>>>,
+    DeciderNovaGadget,
+>;
+
+impl<
+        C1: CurveGroup,
         GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
+        C2: CurveGroup,
         GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        FC: FCircuit<C1::ScalarField>,
         CS1: CommitmentScheme<C1, H>,
         CS2: CommitmentScheme<C2, H>,
-    {
+        const H: bool,
+    > TryFrom<Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>> for DeciderCircuit1<C1, C2, GC2>
+where
+    CF1<C1>: Absorb,
+{
+    type Error = Error;
+
+    fn try_from(nova: Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>) -> Result<Self, Error> {
         let mut transcript = PoseidonSponge::<C1::ScalarField>::new(&nova.poseidon_config);
         // pp_hash is absorbed to transcript at the NIFS::prove call
 
@@ -129,360 +76,98 @@ where
         let r_Fr = C1::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits))
             .ok_or(Error::OutOfBounds)?;
 
-        // compute the commitment scheme challenges used as inputs in the circuit
-        let (cs_challenge_W, cs_challenge_E) =
-            KZGChallengesGadget::<C1>::get_challenges_native(&mut transcript, U_i1.clone());
+        // compute the KZG challenges used as inputs in the circuit
+        let kzg_challenges = KZGChallengesGadget::get_challenges_native(&mut transcript, &U_i1);
 
-        // get evals of the committed polys at the challenges
-        let mut W = W_i1.W.clone();
-        W.extend(
-            std::iter::repeat(C1::ScalarField::zero())
-                .take(W_i1.W.len().next_power_of_two() - W_i1.W.len()),
-        );
-        let mut E = W_i1.E.clone();
-        E.extend(
-            std::iter::repeat(C1::ScalarField::zero())
-                .take(W_i1.E.len().next_power_of_two() - W_i1.E.len()),
-        );
-        let p_W = poly_from_vec(W.to_vec())?;
-        let eval_W = p_W.evaluate(&cs_challenge_W);
-        let p_E = poly_from_vec(E.to_vec())?;
-        let eval_E = p_E.evaluate(&cs_challenge_E);
+        // get KZG evals
+        let kzg_evaluations = W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .map(|((v, _), &c)| EvalGadget::evaluate_native(v, c))
+            .collect::<Result<Vec<_>, _>>()?;
 
         Ok(Self {
-            _c1: PhantomData,
-            _c2: PhantomData,
             _gc2: PhantomData,
-
-            E_len: nova.W_i.E.len(),
-            cf_E_len: nova.cf_W_i.E.len(),
-            r1cs: nova.r1cs,
+            _avar: PhantomData,
+            arith: nova.r1cs,
             poseidon_config: nova.poseidon_config,
-            pp_hash: Some(nova.pp_hash),
-            i: Some(nova.i),
-            z_0: Some(nova.z_0),
-            z_i: Some(nova.z_i),
-            u_i: Some(nova.u_i),
-            w_i: Some(nova.w_i),
-            U_i: Some(nova.U_i),
-            W_i: Some(nova.W_i),
-            U_i1: Some(U_i1),
-            W_i1: Some(W_i1),
-            cmT: Some(cmT),
-            r: Some(r_Fr),
-            cf_U_i: Some(nova.cf_U_i),
-            cs_c_W: Some(cs_challenge_W),
-            cs_c_E: Some(cs_challenge_E),
-            eval_W: Some(eval_W),
-            eval_E: Some(eval_E),
+            pp_hash: nova.pp_hash,
+            i: nova.i,
+            z_0: nova.z_0,
+            z_i: nova.z_i,
+            U_i: nova.U_i,
+            W_i: nova.W_i,
+            u_i: nova.u_i,
+            w_i: nova.w_i,
+            U_i1,
+            W_i1,
+            proof: cmT,
+            randomness: r_Fr,
+            cf_U_i: nova.cf_U_i,
+            kzg_challenges,
+            kzg_evaluations,
         })
     }
 }
 
-impl<C1, C2, GC2> ConstraintSynthesizer<CF1<C1>> for DeciderCircuit1<C1, C2, GC2>
-where
-    C1: CurveGroup,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-    <C1 as Group>::ScalarField: Absorb,
-    C2: CurveGroup,
-    <C2 as CurveGroup>::BaseField: PrimeField,
-    <C2 as Group>::ScalarField: Absorb,
-    C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
-    for<'b> &'b GC2: GroupOpsBounds<'b, C2, GC2>,
-{
-    fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
-        let r1cs =
-            R1CSVar::<C1::ScalarField, CF1<C1>, FpVar<CF1<C1>>>::new_witness(cs.clone(), || {
-                Ok(self.r1cs.clone())
-            })?;
-
-        let pp_hash = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.pp_hash.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let i =
-            FpVar::<CF1<C1>>::new_input(cs.clone(), || Ok(self.i.unwrap_or_else(CF1::<C1>::zero)))?;
-        let z_0 = Vec::<FpVar<CF1<C1>>>::new_input(cs.clone(), || {
-            Ok(self.z_0.unwrap_or(vec![CF1::<C1>::zero()]))
-        })?;
-        let z_i = Vec::<FpVar<CF1<C1>>>::new_input(cs.clone(), || {
-            Ok(self.z_i.unwrap_or(vec![CF1::<C1>::zero()]))
-        })?;
-
-        let u_dummy_native = CommittedInstance::<C1>::dummy(&self.r1cs);
-        let w_dummy_native = Witness::<C1>::dummy(&self.r1cs);
-
-        let u_i = CommittedInstanceVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.u_i.unwrap_or(u_dummy_native.clone()))
-        })?;
-        let U_i = CommittedInstanceVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.U_i.unwrap_or(u_dummy_native.clone()))
-        })?;
-        // here (U_i1, W_i1) = NIFS.P( (U_i,W_i), (u_i,w_i))
-        let U_i1 = CommittedInstanceVar::<C1>::new_input(cs.clone(), || {
-            Ok(self.U_i1.unwrap_or(u_dummy_native.clone()))
-        })?;
-        let W_i1 = WitnessVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.W_i1.unwrap_or(w_dummy_native.clone()))
-        })?;
-
-        // allocate the inputs for the check 6
-        let cs_c_W = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.cs_c_W.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let cs_c_E = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.cs_c_E.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let eval_W = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.eval_W.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let eval_E = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.eval_E.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-
-        // `sponge` is for digest computation.
-        let sponge = PoseidonSpongeVar::<C1::ScalarField>::new(cs.clone(), &self.poseidon_config);
-        // `transcript` is for challenge generation.
-        let mut transcript = sponge.clone();
-        // notice that the `pp_hash` is absorbed inside the ChallengeGadget::get_challenge_gadget call
-
-        // 2. u_i.cmE==cm(0), u_i.u==1
-        // Here zero is the x & y coordinates of the zero point affine representation.
-        let zero = NonNativeUintVar::new_constant(cs.clone(), C1::BaseField::zero())?;
-        u_i.cmE.x.enforce_equal_unaligned(&zero)?;
-        u_i.cmE.y.enforce_equal_unaligned(&zero)?;
-        (u_i.u.is_one()?).enforce_equal(&Boolean::TRUE)?;
-
-        // 3.a u_i.x[0] == H(i, z_0, z_i, U_i)
-        let (u_i_x, U_i_vec) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
-        (u_i.x[0]).enforce_equal(&u_i_x)?;
-
-        // 3.b u_i.x[1] == H(cf_U_i)
-        let cf_u_dummy_native =
-            CycleFoldCommittedInstance::<C2>::dummy(NovaCycleFoldConfig::<C1>::IO_LEN);
-        let cf_U_i = CycleFoldCommittedInstanceVar::<C2, GC2>::new_input(cs.clone(), || {
-            Ok(self.cf_U_i.unwrap_or_else(|| cf_u_dummy_native.clone()))
-        })?;
-        let (cf_u_i_x, _) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
-        (u_i.x[1]).enforce_equal(&cf_u_i_x)?;
-
-        // 4. check RelaxedR1CS of U_{i+1}
-        let z_U1: Vec<FpVar<CF1<C1>>> =
-            [vec![U_i1.u.clone()], U_i1.x.to_vec(), W_i1.W.to_vec()].concat();
-        RelaxedR1CSGadget::check_native(r1cs, W_i1.E.clone(), U_i1.u.clone(), z_U1)?;
-
-        // 1.1.a, 5.1 compute NIFS.V and Commitment Scheme challenges.
-        // We need to ensure the order of challenge generation is the same as
-        // the native counterpart, so we first compute the challenges here and
-        // do the actual checks later.
-        let cmT =
-            NonNativeAffineVar::new_input(cs.clone(), || Ok(self.cmT.unwrap_or_else(C1::zero)))?;
-        let r_bits = ChallengeGadget::<C1, CommittedInstance<C1>>::get_challenge_gadget(
-            &mut transcript,
-            pp_hash,
-            U_i_vec,
-            u_i.clone(),
-            Some(cmT.clone()),
-        )?;
-        // 5.1.
-        let (incircuit_c_W, incircuit_c_E) =
-            KZGChallengesGadget::<C1>::get_challenges_gadget(&mut transcript, U_i1.clone())?;
-        incircuit_c_W.enforce_equal(&cs_c_W)?;
-        incircuit_c_E.enforce_equal(&cs_c_E)?;
-
-        // 5.2. check eval_W==p_W(c_W) and eval_E==p_E(c_E)
-        let incircuit_eval_W = evaluate_gadget::<CF1<C1>>(W_i1.W, incircuit_c_W)?;
-        let incircuit_eval_E = evaluate_gadget::<CF1<C1>>(W_i1.E, incircuit_c_E)?;
-        incircuit_eval_W.enforce_equal(&eval_W)?;
-        incircuit_eval_E.enforce_equal(&eval_E)?;
-
-        // 1.1.b check that the NIFS.V challenge matches the one from the public input (so we avoid
-        //   the verifier computing it)
-        let r_Fr = Boolean::le_bits_to_fp_var(&r_bits)?;
-        // check that the in-circuit computed r is equal to the inputted r
-        let r =
-            FpVar::<CF1<C1>>::new_input(cs.clone(), || Ok(self.r.unwrap_or_else(CF1::<C1>::zero)))?;
-        r_Fr.enforce_equal(&r)?;
-
-        Ok(())
-    }
-}
-
 /// Circuit that implements part of the in-circuit checks needed for the offchain verification over
 /// the Curve1's BaseField (=Curve2's ScalarField).
-#[derive(Clone, Debug)]
-pub struct DeciderCircuit2<C1, GC1, C2>
-where
-    C1: CurveGroup,
-    C2: CurveGroup,
-{
-    _c1: PhantomData<C1>,
-    _gc1: PhantomData<GC1>,
-    _c2: PhantomData<C2>,
-
-    /// E vector's length of the CycleFold instance witness
-    pub cf_E_len: usize,
-    /// R1CS of the CycleFold circuit
-    pub cf_r1cs: R1CS<C2::ScalarField>,
-    pub poseidon_config: PoseidonConfig<CF1<C2>>,
-    /// public params hash
-    pub pp_hash: Option<C2::ScalarField>,
+pub type DeciderCircuit2<C2> = GenericOffchainDeciderCircuit2<C2>;
 
-    /// CycleFold running instance. Notice that here we use Nova's CommittedInstance (instead of
-    /// CycleFoldCommittedInstance), since we are over C2::Fr, so that the CycleFold instances can
-    /// be computed natively
-    pub cf_U_i: Option<CommittedInstance<C2>>,
-    pub cf_W_i: Option<CycleFoldWitness<C2>>,
-    /// Commitment Scheme challenges
-    pub cs_c_W: Option<C2::ScalarField>,
-    pub cs_c_E: Option<C2::ScalarField>,
-    /// Evaluations of the committed polynomials at the challenge
-    pub eval_W: Option<C2::ScalarField>,
-    pub eval_E: Option<C2::ScalarField>,
-}
-impl<C1, GC1, C2> DeciderCircuit2<C1, GC1, C2>
-where
-    C1: CurveGroup,
-    C2: CurveGroup,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-    <C1 as Group>::ScalarField: Absorb,
-    <C2 as CurveGroup>::BaseField: PrimeField,
-    <C2 as Group>::ScalarField: Absorb,
-    GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
-{
-    pub fn from_nova<GC2, CS1, CS2, const H: bool, FC: FCircuit<C1::ScalarField>>(
-        nova: Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>,
-    ) -> Result<Self, Error>
-    where
+impl<
+        C1: CurveGroup,
+        GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
+        C2: CurveGroup,
         GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        FC: FCircuit<C1::ScalarField>,
         CS1: CommitmentScheme<C1, H>,
         CS2: CommitmentScheme<C2, H>,
-    {
+        const H: bool,
+    > TryFrom<Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>> for DeciderCircuit2<C2>
+where
+    CF1<C2>: Absorb,
+{
+    type Error = Error;
+
+    fn try_from(nova: Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>) -> Result<Self, Error> {
         // compute the Commitment Scheme challenges of the CycleFold instance commitments, used as
         // inputs in the circuit
-        let poseidon_config =
-            crate::transcript::poseidon::poseidon_canonical_config::<C2::ScalarField>();
+        let poseidon_config = poseidon_canonical_config::<C2::ScalarField>();
         let mut transcript = PoseidonSponge::<C2::ScalarField>::new(&poseidon_config);
         let pp_hash_Fq =
             C2::ScalarField::from_le_bytes_mod_order(&nova.pp_hash.into_bigint().to_bytes_le());
         transcript.absorb(&pp_hash_Fq);
 
-        let (cs_challenge_W, cs_challenge_E) =
-            KZGChallengesGadget::<C2>::get_challenges_native(&mut transcript, nova.cf_U_i.clone());
+        // compute the KZG challenges used as inputs in the circuit
+        let kzg_challenges =
+            KZGChallengesGadget::get_challenges_native(&mut transcript, &nova.cf_U_i);
 
-        // get evals of the committed polynomials at the challenge
-        let mut W = nova.cf_W_i.W.clone();
-        W.extend(
-            std::iter::repeat(C2::ScalarField::zero())
-                .take(nova.cf_W_i.W.len().next_power_of_two() - nova.cf_W_i.W.len()),
-        );
-        let mut E = nova.cf_W_i.E.clone();
-        E.extend(
-            std::iter::repeat(C2::ScalarField::zero())
-                .take(nova.cf_W_i.E.len().next_power_of_two() - nova.cf_W_i.E.len()),
-        );
-        let p_W = poly_from_vec(W.to_vec())?;
-        let eval_W = p_W.evaluate(&cs_challenge_W);
-        let p_E = poly_from_vec(E.to_vec())?;
-        let eval_E = p_E.evaluate(&cs_challenge_E);
+        // get KZG evals
+        let kzg_evaluations = nova
+            .cf_W_i
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .map(|((v, _), &c)| EvalGadget::evaluate_native(v, c))
+            .collect::<Result<Vec<_>, _>>()?;
 
         Ok(Self {
-            _c1: PhantomData,
-            _gc1: PhantomData,
-            _c2: PhantomData,
-
-            cf_E_len: nova.cf_W_i.E.len(),
-            cf_r1cs: nova.cf_r1cs,
+            cf_arith: nova.cf_r1cs,
             poseidon_config,
-            pp_hash: Some(pp_hash_Fq),
-
-            cf_U_i: Some(nova.cf_U_i),
-            cf_W_i: Some(nova.cf_W_i),
-
-            // CycleFold instance commitments challenges
-            cs_c_W: Some(cs_challenge_W),
-            cs_c_E: Some(cs_challenge_E),
-            eval_W: Some(eval_W),
-            eval_E: Some(eval_E),
+            pp_hash: pp_hash_Fq,
+            cf_U_i: nova.cf_U_i,
+            cf_W_i: nova.cf_W_i,
+            kzg_challenges,
+            kzg_evaluations,
         })
     }
 }
 
-impl<C1, GC1, C2> ConstraintSynthesizer<CF1<C2>> for DeciderCircuit2<C1, GC1, C2>
-where
-    C1: CurveGroup,
-    C2: CurveGroup,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-    <C2 as CurveGroup>::BaseField: PrimeField,
-    <C1 as Group>::ScalarField: Absorb,
-    <C2 as Group>::ScalarField: Absorb,
-    C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-{
-    fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C2>>) -> Result<(), SynthesisError> {
-        let pp_hash = FpVar::<CF1<C2>>::new_input(cs.clone(), || {
-            Ok(self.pp_hash.unwrap_or_else(CF1::<C2>::zero))
-        })?;
-
-        let cf_u_dummy_native = CommittedInstance::<C2>::dummy(&self.cf_r1cs);
-        let w_dummy_native = Witness::<C2>::dummy(&self.cf_r1cs);
-        let cf_U_i = CommittedInstanceVar::<C2>::new_input(cs.clone(), || {
-            Ok(self.cf_U_i.unwrap_or_else(|| cf_u_dummy_native.clone()))
-        })?;
-        let cf_W_i = WitnessVar::<C2>::new_witness(cs.clone(), || {
-            Ok(self.cf_W_i.unwrap_or(w_dummy_native.clone()))
-        })?;
-
-        let cf_r1cs =
-            R1CSVar::<C2::ScalarField, CF1<C2>, FpVar<CF1<C2>>>::new_witness(cs.clone(), || {
-                Ok(self.cf_r1cs.clone())
-            })?;
-
-        // 6. check RelaxedR1CS of cf_U_i
-        let cf_z_U = [vec![cf_U_i.u.clone()], cf_U_i.x.to_vec(), cf_W_i.W.to_vec()].concat();
-        RelaxedR1CSGadget::check_native(cf_r1cs, cf_W_i.E.clone(), cf_U_i.u.clone(), cf_z_U)?;
-
-        // `transcript` is for challenge generation.
-        let mut transcript =
-            PoseidonSpongeVar::<C2::ScalarField>::new(cs.clone(), &self.poseidon_config);
-        transcript.absorb(&pp_hash)?;
-
-        // allocate the inputs for the check 7.1
-        let cs_c_W = FpVar::<CF1<C2>>::new_input(cs.clone(), || {
-            Ok(self.cs_c_W.unwrap_or_else(CF1::<C2>::zero))
-        })?;
-        let cs_c_E = FpVar::<CF1<C2>>::new_input(cs.clone(), || {
-            Ok(self.cs_c_E.unwrap_or_else(CF1::<C2>::zero))
-        })?;
-        // allocate the inputs for the check 7.2
-        let eval_W = FpVar::<CF1<C2>>::new_input(cs.clone(), || {
-            Ok(self.eval_W.unwrap_or_else(CF1::<C2>::zero))
-        })?;
-        let eval_E = FpVar::<CF1<C2>>::new_input(cs.clone(), || {
-            Ok(self.eval_E.unwrap_or_else(CF1::<C2>::zero))
-        })?;
-
-        // 7.1. check the commitment scheme challenges correct computation
-        let (incircuit_c_W, incircuit_c_E) =
-            KZGChallengesGadget::<C2>::get_challenges_gadget(&mut transcript, cf_U_i.clone())?;
-        incircuit_c_W.enforce_equal(&cs_c_W)?;
-        incircuit_c_E.enforce_equal(&cs_c_E)?;
-
-        // 7.2. check eval_W==p_W(c_W) and eval_E==p_E(c_E)
-        let incircuit_eval_W = evaluate_gadget::<CF1<C2>>(cf_W_i.W, incircuit_c_W)?;
-        let incircuit_eval_E = evaluate_gadget::<CF1<C2>>(cf_W_i.E, incircuit_c_E)?;
-        incircuit_eval_W.enforce_equal(&eval_W)?;
-        incircuit_eval_E.enforce_equal(&eval_E)?;
-
-        Ok(())
-    }
-}
-
 #[cfg(test)]
 pub mod tests {
     use ark_pallas::{constraints::GVar, Fq, Fr, Projective};
-    use ark_relations::r1cs::ConstraintSystem;
+    use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
     use ark_vesta::{constraints::GVar as GVar2, Projective as Projective2};
 
     use super::*;
@@ -530,9 +215,8 @@ pub mod tests {
 
         // load the DeciderCircuit 1 & 2 from the Nova instance
         let decider_circuit1 =
-            DeciderCircuit1::<Projective, Projective2, GVar2>::from_nova(nova.clone()).unwrap();
-        let decider_circuit2 =
-            DeciderCircuit2::<Projective, GVar, Projective2>::from_nova(nova).unwrap();
+            DeciderCircuit1::<Projective, Projective2, GVar2>::try_from(nova.clone()).unwrap();
+        let decider_circuit2 = DeciderCircuit2::<Projective2>::try_from(nova).unwrap();
 
         // generate the constraints of both circuits and check that are satisfied by the inputs
         let cs1 = ConstraintSystem::<Fr>::new_ref();

folding-schemes/src/folding/nova/decider_eth.rs

@@ -3,11 +3,11 @@
 /// More details can be found at the documentation page:
 /// https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-onchain.html
 use ark_bn254::Bn254;
-use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, Absorb, CryptographicSponge};
+use ark_crypto_primitives::sponge::Absorb;
 use ark_ec::{AffineRepr, CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
 use ark_groth16::Groth16;
-use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar, ToConstraintFieldGadget};
+use ark_r1cs_std::{prelude::CurveVar, ToConstraintFieldGadget};
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
 use ark_snark::SNARK;
 use ark_std::rand::{CryptoRng, RngCore};
@@ -15,37 +15,36 @@ use ark_std::{One, Zero};
 use core::marker::PhantomData;
 
 pub use super::decider_eth_circuit::DeciderEthCircuit;
-use super::{
-    nifs::{nova::NIFS, NIFSTrait},
-    CommittedInstance, Nova,
-};
+use super::decider_eth_circuit::DeciderNovaGadget;
+use super::{CommittedInstance, Nova};
 use crate::commitment::{
     kzg::{Proof as KZGProof, KZG},
     pedersen::Params as PedersenParams,
     CommitmentScheme,
 };
-use crate::folding::circuits::{nonnative::affine::NonNativeAffineVar, CF2};
-use crate::folding::nova::circuits::ChallengeGadget;
+use crate::folding::circuits::decider::DeciderEnabledNIFS;
+use crate::folding::circuits::CF2;
+use crate::folding::traits::{Inputize, WitnessOps};
 use crate::frontend::FCircuit;
-use crate::transcript::poseidon::poseidon_canonical_config;
 use crate::Error;
 use crate::{Decider as DeciderTrait, FoldingScheme};
 
 #[derive(Debug, Clone, Eq, PartialEq, CanonicalSerialize, CanonicalDeserialize)]
-pub struct Proof<C1, CS1, S>
+pub struct Proof<C, CS, S>
 where
-    C1: CurveGroup,
-    CS1: CommitmentScheme<C1, ProverChallenge = C1::ScalarField, Challenge = C1::ScalarField>,
-    S: SNARK<C1::ScalarField>,
+    C: CurveGroup,
+    CS: CommitmentScheme<C, ProverChallenge = C::ScalarField, Challenge = C::ScalarField>,
+    S: SNARK<C::ScalarField>,
 {
     snark_proof: S::Proof,
-    kzg_proofs: [CS1::Proof; 2],
+    kzg_proofs: [CS::Proof; 2],
     // cmT and r are values for the last fold, U_{i+1}=NIFS.V(r, U_i, u_i, cmT), and they are
     // checked in-circuit
-    cmT: C1,
+    cmT: C,
+    r: C::ScalarField,
     // the KZG challenges are provided by the prover, but in-circuit they are checked to match
     // the in-circuit computed computed ones.
-    kzg_challenges: [C1::ScalarField; 2],
+    kzg_challenges: [C::ScalarField; 2],
 }
 
 #[derive(Debug, Clone, Eq, PartialEq, CanonicalSerialize, CanonicalDeserialize)]
@@ -98,8 +97,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'b> &'b GC1: GroupOpsBounds<'b, C1, GC1>,
-    for<'b> &'b GC2: GroupOpsBounds<'b, C2, GC2>,
     // constrain FS into Nova, since this is a Decider specifically for Nova
     Nova<C1, GC1, C2, GC2, FC, CS1, CS2, false>: From<FS>,
     crate::folding::nova::ProverParams<C1, C2, CS1, CS2, false>:
@@ -112,18 +109,18 @@ where
     type Proof = Proof<C1, CS1, S>;
     type VerifierParam = VerifierParam<C1, CS1::VerifierParams, S::VerifyingKey>;
     type PublicInput = Vec<C1::ScalarField>;
-    type CommittedInstance = CommittedInstance<C1>;
+    type CommittedInstance = Vec<C1>;
 
     fn preprocess(
         mut rng: impl RngCore + CryptoRng,
         prep_param: Self::PreprocessorParam,
         fs: FS,
     ) -> Result<(Self::ProverParam, Self::VerifierParam), Error> {
-        let circuit =
-            DeciderEthCircuit::<C1, GC1, C2, GC2, CS1, CS2>::from_nova::<FC>(fs.into()).unwrap();
+        let circuit = DeciderEthCircuit::<C1, C2, GC2>::try_from(Nova::from(fs))?;
 
         // get the Groth16 specific setup for the circuit
-        let (g16_pk, g16_vk) = S::circuit_specific_setup(circuit, &mut rng).unwrap();
+        let (g16_pk, g16_vk) = S::circuit_specific_setup(circuit, &mut rng)
+            .map_err(|e| Error::SNARKSetupFail(e.to_string()))?;
 
         // get the FoldingScheme prover & verifier params from Nova
         #[allow(clippy::type_complexity)]
@@ -156,46 +153,39 @@ where
     ) -> Result<Self::Proof, Error> {
         let (snark_pk, cs_pk): (S::ProvingKey, CS1::ProverParams) = pp;
 
-        let circuit = DeciderEthCircuit::<C1, GC1, C2, GC2, CS1, CS2>::from_nova::<FC>(
-            folding_scheme.into(),
-        )?;
+        let circuit = DeciderEthCircuit::<C1, C2, GC2>::try_from(Nova::from(folding_scheme))?;
 
-        let snark_proof = S::prove(&snark_pk, circuit.clone(), &mut rng)
-            .map_err(|e| Error::Other(e.to_string()))?;
-
-        let cmT = circuit.cmT.unwrap();
-        let W_i1 = circuit.W_i1.unwrap();
+        let cmT = circuit.proof;
+        let r = circuit.randomness;
 
         // get the challenges that have been already computed when preparing the circuit inputs in
-        // the above `from_nova` call
-        let challenge_W = circuit
-            .kzg_c_W
-            .ok_or(Error::MissingValue("kzg_c_W".to_string()))?;
-        let challenge_E = circuit
-            .kzg_c_E
-            .ok_or(Error::MissingValue("kzg_c_E".to_string()))?;
+        // the above `try_from` call
+        let kzg_challenges = circuit.kzg_challenges.clone();
 
         // generate KZG proofs
-        let U_cmW_proof = CS1::prove_with_challenge(
-            &cs_pk,
-            challenge_W,
-            &W_i1.W,
-            &C1::ScalarField::zero(),
-            None,
-        )?;
-        let U_cmE_proof = CS1::prove_with_challenge(
-            &cs_pk,
-            challenge_E,
-            &W_i1.E,
-            &C1::ScalarField::zero(),
-            None,
-        )?;
+        let kzg_proofs = circuit
+            .W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .map(|((v, _), &c)| {
+                CS1::prove_with_challenge(&cs_pk, c, v, &C1::ScalarField::zero(), None)
+            })
+            .collect::<Result<Vec<_>, _>>()?;
+
+        let snark_proof =
+            S::prove(&snark_pk, circuit, &mut rng).map_err(|e| Error::Other(e.to_string()))?;
 
         Ok(Self::Proof {
             snark_proof,
-            kzg_proofs: [U_cmW_proof, U_cmE_proof],
             cmT,
-            kzg_challenges: [challenge_W, challenge_E],
+            r,
+            kzg_proofs: kzg_proofs
+                .try_into()
+                .map_err(|e: Vec<_>| Error::NotExpectedLength(e.len(), 2))?,
+            kzg_challenges: kzg_challenges
+                .try_into()
+                .map_err(|e: Vec<_>| Error::NotExpectedLength(e.len(), 2))?,
         })
     }
 
@@ -204,71 +194,59 @@ where
         i: C1::ScalarField,
         z_0: Vec<C1::ScalarField>,
         z_i: Vec<C1::ScalarField>,
-        running_instance: &Self::CommittedInstance,
-        incoming_instance: &Self::CommittedInstance,
+        // we don't use the instances at the verifier level, since we check them in-circuit
+        running_commitments: &Self::CommittedInstance,
+        incoming_commitments: &Self::CommittedInstance,
         proof: &Self::Proof,
     ) -> Result<bool, Error> {
         if i <= C1::ScalarField::one() {
             return Err(Error::NotEnoughSteps);
         }
 
-        // compute U = U_{d+1}= NIFS.V(U_d, u_d, cmT)
-        let poseidon_config = poseidon_canonical_config::<C1::ScalarField>();
-        let mut transcript = PoseidonSponge::<C1::ScalarField>::new(&poseidon_config);
-        let (U, r_bits) = NIFS::<C1, CS1, PoseidonSponge<C1::ScalarField>>::verify(
-            &mut transcript,
-            vp.pp_hash,
-            running_instance,
-            incoming_instance,
-            &proof.cmT,
+        let Self::VerifierParam {
+            pp_hash,
+            snark_vp,
+            cs_vp,
+        } = vp;
+
+        // 6.2. Fold the commitments
+        let U_final_commitments = DeciderNovaGadget::fold_group_elements_native(
+            running_commitments,
+            incoming_commitments,
+            Some(proof.cmT),
+            proof.r,
         )?;
-        let r = C1::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits))
-            .ok_or(Error::OutOfBounds)?;
-
-        let (cmE_x, cmE_y) = NonNativeAffineVar::inputize(U.cmE)?;
-        let (cmW_x, cmW_y) = NonNativeAffineVar::inputize(U.cmW)?;
-        let (cmT_x, cmT_y) = NonNativeAffineVar::inputize(proof.cmT)?;
-
-        let public_input: Vec<C1::ScalarField> = [
-            vec![vp.pp_hash, i],
-            z_0,
-            z_i,
-            vec![U.u],
-            U.x.clone(),
-            cmE_x,
-            cmE_y,
-            cmW_x,
-            cmW_y,
-            proof.kzg_challenges.to_vec(),
-            vec![
-                proof.kzg_proofs[0].eval, // eval_W
-                proof.kzg_proofs[1].eval, // eval_E
-            ],
-            cmT_x,
-            cmT_y,
-            vec![r],
+
+        let public_input = [
+            &[pp_hash, i][..],
+            &z_0,
+            &z_i,
+            &U_final_commitments
+                .iter()
+                .flat_map(|c| c.inputize())
+                .collect::<Vec<_>>(),
+            &proof.kzg_challenges,
+            &proof.kzg_proofs.iter().map(|p| p.eval).collect::<Vec<_>>(),
+            &proof.cmT.inputize(),
+            &[proof.r],
         ]
         .concat();
 
-        let snark_v = S::verify(&vp.snark_vp, &public_input, &proof.snark_proof)
+        let snark_v = S::verify(&snark_vp, &public_input, &proof.snark_proof)
             .map_err(|e| Error::Other(e.to_string()))?;
         if !snark_v {
             return Err(Error::SNARKVerificationFail);
         }
 
-        // we're at the Ethereum EVM case, so the CS1 is KZG commitments
-        CS1::verify_with_challenge(
-            &vp.cs_vp,
-            proof.kzg_challenges[0],
-            &U.cmW,
-            &proof.kzg_proofs[0],
-        )?;
-        CS1::verify_with_challenge(
-            &vp.cs_vp,
-            proof.kzg_challenges[1],
-            &U.cmE,
-            &proof.kzg_proofs[1],
-        )?;
+        // 7.3. Verify the KZG proofs
+        for ((cm, &c), pi) in U_final_commitments
+            .iter()
+            .zip(&proof.kzg_challenges)
+            .zip(&proof.kzg_proofs)
+        {
+            // we're at the Ethereum EVM case, so the CS1 is KZG commitments
+            CS1::verify_with_challenge(&cs_vp, c, cm, pi)?;
+        }
 
         Ok(true)
     }
@@ -278,7 +256,6 @@ where
 #[allow(clippy::too_many_arguments)]
 pub fn prepare_calldata(
     function_signature_check: [u8; 4],
-    pp_hash: ark_bn254::Fr,
     i: ark_bn254::Fr,
     z_0: Vec<ark_bn254::Fr>,
     z_i: Vec<ark_bn254::Fr>,
@@ -286,22 +263,6 @@ pub fn prepare_calldata(
     incoming_instance: &CommittedInstance<ark_bn254::G1Projective>,
     proof: Proof<ark_bn254::G1Projective, KZG<'static, Bn254>, Groth16<Bn254>>,
 ) -> Result<Vec<u8>, Error> {
-    // compute the challenge r
-    let poseidon_config = poseidon_canonical_config::<ark_bn254::Fr>();
-    let mut transcript = PoseidonSponge::<ark_bn254::Fr>::new(&poseidon_config);
-    let r_bits = ChallengeGadget::<
-        ark_bn254::G1Projective,
-        CommittedInstance<ark_bn254::G1Projective>,
-    >::get_challenge_native(
-        &mut transcript,
-        pp_hash,
-        running_instance,
-        incoming_instance,
-        Some(&proof.cmT),
-    );
-    let r =
-        ark_bn254::Fr::from_bigint(BigInteger::from_bits_le(&r_bits)).ok_or(Error::OutOfBounds)?;
-
     Ok(vec![
         function_signature_check.to_vec(),
         i.into_bigint().to_bytes_be(), // i
@@ -311,26 +272,14 @@ pub fn prepare_calldata(
         z_i.iter()
             .flat_map(|v| v.into_bigint().to_bytes_be())
             .collect::<Vec<u8>>(), // z_i
-        point_to_eth_format(running_instance.cmW.into_affine())?, // U_i_cmW
-        point_to_eth_format(running_instance.cmE.into_affine())?, // U_i_cmE
-        running_instance.u.into_bigint().to_bytes_be(), // U_i_u
-        incoming_instance.u.into_bigint().to_bytes_be(), // u_i_u
-        r.into_bigint().to_bytes_be(), // r
-        running_instance
-            .x
-            .iter()
-            .flat_map(|v| v.into_bigint().to_bytes_be())
-            .collect::<Vec<u8>>(), // U_i_x
-        point_to_eth_format(incoming_instance.cmW.into_affine())?, // u_i_cmW
-        incoming_instance
-            .x
-            .iter()
-            .flat_map(|v| v.into_bigint().to_bytes_be())
-            .collect::<Vec<u8>>(), // u_i_x
+        point_to_eth_format(running_instance.cmW.into_affine())?,
+        point_to_eth_format(running_instance.cmE.into_affine())?,
+        point_to_eth_format(incoming_instance.cmW.into_affine())?,
         point_to_eth_format(proof.cmT.into_affine())?, // cmT
-        point_to_eth_format(proof.snark_proof.a)?, // pA
-        point2_to_eth_format(proof.snark_proof.b)?, // pB
-        point_to_eth_format(proof.snark_proof.c)?, // pC
+        proof.r.into_bigint().to_bytes_be(),           // r
+        point_to_eth_format(proof.snark_proof.a)?,     // pA
+        point2_to_eth_format(proof.snark_proof.b)?,    // pB
+        point_to_eth_format(proof.snark_proof.c)?,     // pC
         proof.kzg_challenges[0].into_bigint().to_bytes_be(), // challenge_W
         proof.kzg_challenges[1].into_bigint().to_bytes_be(), // challenge_E
         proof.kzg_proofs[0].eval.into_bigint().to_bytes_be(), // eval W
@@ -373,6 +322,7 @@ pub mod tests {
     use super::*;
     use crate::commitment::pedersen::Pedersen;
     use crate::folding::nova::{PreprocessorParam, ProverParams as NovaProverParams};
+    use crate::folding::traits::CommittedInstanceOps;
     use crate::frontend::utils::CubicFCircuit;
     use crate::transcript::poseidon::poseidon_canonical_config;
 
@@ -434,8 +384,8 @@ pub mod tests {
             nova.i,
             nova.z_0.clone(),
             nova.z_i.clone(),
-            &nova.U_i,
-            &nova.u_i,
+            &nova.U_i.get_commitments(),
+            &nova.u_i.get_commitments(),
             &proof,
         )
         .unwrap();
@@ -444,7 +394,13 @@ pub mod tests {
 
         // decider proof verification using the deserialized data
         let verified = D::verify(
-            decider_vp, nova.i, nova.z_0, nova.z_i, &nova.U_i, &nova.u_i, &proof,
+            decider_vp,
+            nova.i,
+            nova.z_0,
+            nova.z_i,
+            &nova.U_i.get_commitments(),
+            &nova.u_i.get_commitments(),
+            &proof,
         )
         .unwrap();
         assert!(verified);
@@ -550,8 +506,8 @@ pub mod tests {
             nova.i,
             nova.z_0.clone(),
             nova.z_i.clone(),
-            &nova.U_i,
-            &nova.u_i,
+            &nova.U_i.get_commitments(),
+            &nova.u_i.get_commitments(),
             &proof,
         )
         .unwrap();
@@ -579,12 +535,6 @@ pub mod tests {
         nova.z_i
             .serialize_compressed(&mut public_inputs_serialized)
             .unwrap();
-        nova.U_i
-            .serialize_compressed(&mut public_inputs_serialized)
-            .unwrap();
-        nova.u_i
-            .serialize_compressed(&mut public_inputs_serialized)
-            .unwrap();
 
         // deserialize back the verifier_params, proof and public inputs
         let decider_vp_deserialized =
@@ -605,10 +555,6 @@ pub mod tests {
         let i_deserialized = Fr::deserialize_compressed(&mut reader).unwrap();
         let z_0_deserialized = Vec::<Fr>::deserialize_compressed(&mut reader).unwrap();
         let z_i_deserialized = Vec::<Fr>::deserialize_compressed(&mut reader).unwrap();
-        let U_i_deserialized =
-            CommittedInstance::<Projective>::deserialize_compressed(&mut reader).unwrap();
-        let u_i_deserialized =
-            CommittedInstance::<Projective>::deserialize_compressed(&mut reader).unwrap();
 
         // decider proof verification using the deserialized data
         let verified = D::verify(
@@ -616,8 +562,8 @@ pub mod tests {
             i_deserialized,
             z_0_deserialized,
             z_i_deserialized,
-            &U_i_deserialized,
-            &u_i_deserialized,
+            &nova.U_i.get_commitments(),
+            &nova.u_i.get_commitments(),
             &proof_deserialized,
         )
         .unwrap();

folding-schemes/src/folding/nova/decider_eth_circuit.rs

@@ -4,130 +4,42 @@
 /// https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-onchain.html
 use ark_crypto_primitives::sponge::{
     constraints::CryptographicSpongeVar,
-    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonSponge},
     Absorb, CryptographicSponge,
 };
-use ark_ec::{CurveGroup, Group};
+use ark_ec::CurveGroup;
 use ark_ff::{BigInteger, PrimeField};
-use ark_poly::Polynomial;
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     boolean::Boolean,
     eq::EqGadget,
-    fields::{fp::FpVar, FieldVar},
-    groups::GroupOpsBounds,
-    poly::{domain::Radix2DomainVar, evaluations::univariate::EvaluationsVar},
+    fields::fp::FpVar,
     prelude::CurveVar,
     ToConstraintFieldGadget,
 };
-use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, Namespace, SynthesisError};
-use ark_std::{log2, Zero};
-use core::{borrow::Borrow, marker::PhantomData};
+use ark_relations::r1cs::{Namespace, SynthesisError};
+use ark_std::{borrow::Borrow, marker::PhantomData};
 
 use super::{
-    circuits::{ChallengeGadget, CommittedInstanceVar},
+    circuits::{ChallengeGadget, CommittedInstanceVar, NIFSGadget},
     nifs::{nova::NIFS, NIFSTrait},
     CommittedInstance, Nova, Witness,
 };
 use crate::commitment::{pedersen::Params as PedersenParams, CommitmentScheme};
-use crate::folding::circuits::{
-    cyclefold::{CycleFoldCommittedInstance, CycleFoldWitness},
-    nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
-    CF1, CF2,
+use crate::folding::{
+    circuits::{
+        decider::on_chain::GenericOnchainDeciderCircuit, nonnative::affine::NonNativeAffineVar,
+        CF1, CF2,
+    },
+    traits::{WitnessOps, WitnessVarOps},
 };
 use crate::frontend::FCircuit;
-use crate::transcript::{Transcript, TranscriptVar};
-use crate::utils::{
-    gadgets::{MatrixGadget, SparseMatrixVar, VectorGadget},
-    vec::poly_from_vec,
-};
 use crate::Error;
 use crate::{
-    arith::r1cs::R1CS,
-    folding::traits::{CommittedInstanceVarOps, Dummy, WitnessVarOps},
+    arith::r1cs::{circuits::R1CSMatricesVar, R1CS},
+    folding::circuits::decider::{DeciderEnabledNIFS, EvalGadget, KZGChallengesGadget},
 };
 
-#[derive(Debug, Clone)]
-pub struct RelaxedR1CSGadget {}
-impl RelaxedR1CSGadget {
-    /// performs the RelaxedR1CS check for native variables (Az∘Bz==uCz+E)
-    pub fn check_native<F: PrimeField>(
-        r1cs: R1CSVar<F, F, FpVar<F>>,
-        E: Vec<FpVar<F>>,
-        u: FpVar<F>,
-        z: Vec<FpVar<F>>,
-    ) -> Result<(), SynthesisError> {
-        let Az = r1cs.A.mul_vector(&z)?;
-        let Bz = r1cs.B.mul_vector(&z)?;
-        let Cz = r1cs.C.mul_vector(&z)?;
-        let uCzE = Cz.mul_scalar(&u)?.add(&E)?;
-        let AzBz = Az.hadamard(&Bz)?;
-        AzBz.enforce_equal(&uCzE)?;
-        Ok(())
-    }
-
-    /// performs the RelaxedR1CS check for non-native variables (Az∘Bz==uCz+E)
-    pub fn check_nonnative<F: PrimeField, CF: PrimeField>(
-        r1cs: R1CSVar<F, CF, NonNativeUintVar<CF>>,
-        E: Vec<NonNativeUintVar<CF>>,
-        u: NonNativeUintVar<CF>,
-        z: Vec<NonNativeUintVar<CF>>,
-    ) -> Result<(), SynthesisError> {
-        // First we do addition and multiplication without mod F's order
-        let Az = r1cs.A.mul_vector(&z)?;
-        let Bz = r1cs.B.mul_vector(&z)?;
-        let Cz = r1cs.C.mul_vector(&z)?;
-        let uCzE = Cz.mul_scalar(&u)?.add(&E)?;
-        let AzBz = Az.hadamard(&Bz)?;
-
-        // Then we compare the results by checking if they are congruent
-        // modulo the field order
-        AzBz.into_iter()
-            .zip(uCzE)
-            .try_for_each(|(a, b)| a.enforce_congruent::<F>(&b))
-    }
-}
-
-#[derive(Debug, Clone)]
-pub struct R1CSVar<F: PrimeField, CF: PrimeField, FV: AllocVar<F, CF>> {
-    _f: PhantomData<F>,
-    _cf: PhantomData<CF>,
-    _fv: PhantomData<FV>,
-    pub A: SparseMatrixVar<F, CF, FV>,
-    pub B: SparseMatrixVar<F, CF, FV>,
-    pub C: SparseMatrixVar<F, CF, FV>,
-}
-
-impl<F, CF, FV> AllocVar<R1CS<F>, CF> for R1CSVar<F, CF, FV>
-where
-    F: PrimeField,
-    CF: PrimeField,
-    FV: AllocVar<F, CF>,
-{
-    fn new_variable<T: Borrow<R1CS<F>>>(
-        cs: impl Into<Namespace<CF>>,
-        f: impl FnOnce() -> Result<T, SynthesisError>,
-        _mode: AllocationMode,
-    ) -> Result<Self, SynthesisError> {
-        f().and_then(|val| {
-            let cs = cs.into();
-
-            let A = SparseMatrixVar::<F, CF, FV>::new_constant(cs.clone(), &val.borrow().A)?;
-            let B = SparseMatrixVar::<F, CF, FV>::new_constant(cs.clone(), &val.borrow().B)?;
-            let C = SparseMatrixVar::<F, CF, FV>::new_constant(cs.clone(), &val.borrow().C)?;
-
-            Ok(Self {
-                _f: PhantomData,
-                _cf: PhantomData,
-                _fv: PhantomData,
-                A,
-                B,
-                C,
-            })
-        })
-    }
-}
-
 /// In-circuit representation of the Witness associated to the CommittedInstance.
 #[derive(Debug, Clone)]
 pub struct WitnessVar<C: CurveGroup> {
@@ -166,82 +78,40 @@ where
 
 impl<C: CurveGroup> WitnessVarOps<C::ScalarField> for WitnessVar<C> {
     fn get_openings(&self) -> Vec<(&[FpVar<C::ScalarField>], FpVar<C::ScalarField>)> {
-        vec![(&self.E, self.rE.clone()), (&self.W, self.rW.clone())]
+        vec![(&self.W, self.rW.clone()), (&self.E, self.rE.clone())]
     }
 }
 
-/// Circuit that implements the in-circuit checks needed for the onchain (Ethereum's EVM)
-/// verification.
-#[derive(Clone, Debug)]
-pub struct DeciderEthCircuit<C1, GC1, C2, GC2, CS1, CS2, const H: bool = false>
+pub type DeciderEthCircuit<C1, C2, GC2> = GenericOnchainDeciderCircuit<
+    C1,
+    C2,
+    GC2,
+    CommittedInstance<C1>,
+    CommittedInstance<C1>,
+    Witness<C1>,
+    R1CS<CF1<C1>>,
+    R1CSMatricesVar<CF1<C1>, FpVar<CF1<C1>>>,
+    DeciderNovaGadget,
+>;
+
+/// returns an instance of the DeciderEthCircuit from the given Nova struct
+impl<
+        C1: CurveGroup,
+        GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
+        C2: CurveGroup,
+        GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        FC: FCircuit<C1::ScalarField>,
+        CS1: CommitmentScheme<C1, H>,
+        // enforce that the CS2 is Pedersen commitment scheme, since we're at Ethereum's EVM decider
+        CS2: CommitmentScheme<C2, H, ProverParams = PedersenParams<C2>>,
+        const H: bool,
+    > TryFrom<Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>> for DeciderEthCircuit<C1, C2, GC2>
 where
-    C1: CurveGroup,
-    GC1: CurveVar<C1, CF2<C1>>,
-    C2: CurveGroup,
-    GC2: CurveVar<C2, CF2<C2>>,
-    CS1: CommitmentScheme<C1, H>,
-    CS2: CommitmentScheme<C2, H>,
+    CF1<C1>: Absorb,
 {
-    _c1: PhantomData<C1>,
-    _gc1: PhantomData<GC1>,
-    _c2: PhantomData<C2>,
-    _gc2: PhantomData<GC2>,
-    _cs1: PhantomData<CS1>,
-    _cs2: PhantomData<CS2>,
-
-    /// E vector's length of the Nova instance witness
-    pub E_len: usize,
-    /// E vector's length of the CycleFold instance witness
-    pub cf_E_len: usize,
-    /// R1CS of the Augmented Function circuit
-    pub r1cs: R1CS<C1::ScalarField>,
-    /// R1CS of the CycleFold circuit
-    pub cf_r1cs: R1CS<C2::ScalarField>,
-    /// CycleFold PedersenParams over C2
-    pub cf_pedersen_params: PedersenParams<C2>,
-    pub poseidon_config: PoseidonConfig<CF1<C1>>,
-    /// public params hash
-    pub pp_hash: Option<C1::ScalarField>,
-    pub i: Option<CF1<C1>>,
-    /// initial state
-    pub z_0: Option<Vec<C1::ScalarField>>,
-    /// current i-th state
-    pub z_i: Option<Vec<C1::ScalarField>>,
-    /// Nova instances
-    pub u_i: Option<CommittedInstance<C1>>,
-    pub w_i: Option<Witness<C1>>,
-    pub U_i: Option<CommittedInstance<C1>>,
-    pub W_i: Option<Witness<C1>>,
-    pub U_i1: Option<CommittedInstance<C1>>,
-    pub W_i1: Option<Witness<C1>>,
-    pub cmT: Option<C1>,
-    pub r: Option<C1::ScalarField>,
-    /// CycleFold running instance
-    pub cf_U_i: Option<CycleFoldCommittedInstance<C2>>,
-    pub cf_W_i: Option<CycleFoldWitness<C2>>,
+    type Error = Error;
 
-    /// KZG challenges
-    pub kzg_c_W: Option<C1::ScalarField>,
-    pub kzg_c_E: Option<C1::ScalarField>,
-    pub eval_W: Option<C1::ScalarField>,
-    pub eval_E: Option<C1::ScalarField>,
-}
-impl<C1, GC1, C2, GC2, CS1, CS2, const H: bool> DeciderEthCircuit<C1, GC1, C2, GC2, CS1, CS2, H>
-where
-    C1: CurveGroup,
-    C2: CurveGroup,
-    GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
-    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
-    CS1: CommitmentScheme<C1, H>,
-    // enforce that the CS2 is Pedersen commitment scheme, since we're at Ethereum's EVM decider
-    CS2: CommitmentScheme<C2, H, ProverParams = PedersenParams<C2>>,
-    <C1 as Group>::ScalarField: Absorb,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-{
-    /// returns an instance of the DeciderEthCircuit from the given Nova struct
-    pub fn from_nova<FC: FCircuit<C1::ScalarField>>(
-        nova: Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>,
-    ) -> Result<Self, Error> {
+    fn try_from(nova: Nova<C1, GC1, C2, GC2, FC, CS1, CS2, H>) -> Result<Self, Error> {
         let mut transcript = PoseidonSponge::<C1::ScalarField>::new(&nova.poseidon_config);
 
         // compute the U_{i+1}, W_{i+1}
@@ -259,517 +129,116 @@ where
             .ok_or(Error::OutOfBounds)?;
 
         // compute the KZG challenges used as inputs in the circuit
-        let (kzg_challenge_W, kzg_challenge_E) =
-            KZGChallengesGadget::<C1>::get_challenges_native(&mut transcript, U_i1.clone());
+        let kzg_challenges = KZGChallengesGadget::get_challenges_native(&mut transcript, &U_i1);
 
         // get KZG evals
-        let mut W = W_i1.W.clone();
-        W.extend(
-            std::iter::repeat(C1::ScalarField::zero())
-                .take(W_i1.W.len().next_power_of_two() - W_i1.W.len()),
-        );
-        let mut E = W_i1.E.clone();
-        E.extend(
-            std::iter::repeat(C1::ScalarField::zero())
-                .take(W_i1.E.len().next_power_of_two() - W_i1.E.len()),
-        );
-        let p_W = poly_from_vec(W.to_vec())?;
-        let eval_W = p_W.evaluate(&kzg_challenge_W);
-        let p_E = poly_from_vec(E.to_vec())?;
-        let eval_E = p_E.evaluate(&kzg_challenge_E);
+        let kzg_evaluations = W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .map(|((v, _), &c)| EvalGadget::evaluate_native(v, c))
+            .collect::<Result<Vec<_>, _>>()?;
 
         Ok(Self {
-            _c1: PhantomData,
-            _gc1: PhantomData,
-            _c2: PhantomData,
             _gc2: PhantomData,
-            _cs1: PhantomData,
-            _cs2: PhantomData,
-
-            E_len: nova.W_i.E.len(),
-            cf_E_len: nova.cf_W_i.E.len(),
-            r1cs: nova.r1cs,
-            cf_r1cs: nova.cf_r1cs,
+            _avar: PhantomData,
+            arith: nova.r1cs,
+            cf_arith: nova.cf_r1cs,
             cf_pedersen_params: nova.cf_cs_pp,
             poseidon_config: nova.poseidon_config,
-            pp_hash: Some(nova.pp_hash),
-            i: Some(nova.i),
-            z_0: Some(nova.z_0),
-            z_i: Some(nova.z_i),
-            u_i: Some(nova.u_i),
-            w_i: Some(nova.w_i),
-            U_i: Some(nova.U_i),
-            W_i: Some(nova.W_i),
-            U_i1: Some(U_i1),
-            W_i1: Some(W_i1),
-            cmT: Some(cmT),
-            r: Some(r_Fr),
-            cf_U_i: Some(nova.cf_U_i),
-            cf_W_i: Some(nova.cf_W_i),
-            kzg_c_W: Some(kzg_challenge_W),
-            kzg_c_E: Some(kzg_challenge_E),
-            eval_W: Some(eval_W),
-            eval_E: Some(eval_E),
+            pp_hash: nova.pp_hash,
+            i: nova.i,
+            z_0: nova.z_0,
+            z_i: nova.z_i,
+            U_i: nova.U_i,
+            W_i: nova.W_i,
+            u_i: nova.u_i,
+            w_i: nova.w_i,
+            U_i1,
+            W_i1,
+            proof: cmT,
+            randomness: r_Fr,
+            cf_U_i: nova.cf_U_i,
+            cf_W_i: nova.cf_W_i,
+            kzg_challenges,
+            kzg_evaluations,
         })
     }
 }
 
-impl<C1, GC1, C2, GC2, CS1, CS2> ConstraintSynthesizer<CF1<C1>>
-    for DeciderEthCircuit<C1, GC1, C2, GC2, CS1, CS2>
+pub struct DeciderNovaGadget;
+
+impl<C: CurveGroup>
+    DeciderEnabledNIFS<C, CommittedInstance<C>, CommittedInstance<C>, Witness<C>, R1CS<CF1<C>>>
+    for DeciderNovaGadget
 where
-    C1: CurveGroup,
-    C2: CurveGroup,
-    GC1: CurveVar<C1, CF2<C1>>,
-    GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
-    CS1: CommitmentScheme<C1>,
-    CS2: CommitmentScheme<C2>,
-    <C1 as CurveGroup>::BaseField: PrimeField,
-    <C2 as CurveGroup>::BaseField: PrimeField,
-    <C1 as Group>::ScalarField: Absorb,
-    <C2 as Group>::ScalarField: Absorb,
-    C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'b> &'b GC2: GroupOpsBounds<'b, C2, GC2>,
+    CF1<C>: Absorb,
 {
-    fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
-        let r1cs =
-            R1CSVar::<C1::ScalarField, CF1<C1>, FpVar<CF1<C1>>>::new_witness(cs.clone(), || {
-                Ok(self.r1cs.clone())
-            })?;
-
-        let pp_hash = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.pp_hash.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let i =
-            FpVar::<CF1<C1>>::new_input(cs.clone(), || Ok(self.i.unwrap_or_else(CF1::<C1>::zero)))?;
-        let z_0 = Vec::<FpVar<CF1<C1>>>::new_input(cs.clone(), || {
-            Ok(self.z_0.unwrap_or(vec![CF1::<C1>::zero()]))
-        })?;
-        let z_i = Vec::<FpVar<CF1<C1>>>::new_input(cs.clone(), || {
-            Ok(self.z_i.unwrap_or(vec![CF1::<C1>::zero()]))
-        })?;
-
-        let u_dummy_native = CommittedInstance::<C1>::dummy(&self.r1cs);
-        let w_dummy_native = Witness::<C1>::dummy(&self.r1cs);
-
-        let u_i = CommittedInstanceVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.u_i.unwrap_or(u_dummy_native.clone()))
-        })?;
-        let U_i = CommittedInstanceVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.U_i.unwrap_or(u_dummy_native.clone()))
-        })?;
-        // here (U_i1, W_i1) = NIFS.P( (U_i,W_i), (u_i,w_i))
-        let U_i1 = CommittedInstanceVar::<C1>::new_input(cs.clone(), || {
-            Ok(self.U_i1.unwrap_or(u_dummy_native.clone()))
-        })?;
-        let W_i1 = WitnessVar::<C1>::new_witness(cs.clone(), || {
-            Ok(self.W_i1.unwrap_or(w_dummy_native.clone()))
-        })?;
-
-        // allocate the inputs for the check 5.1
-        let kzg_c_W = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.kzg_c_W.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let kzg_c_E = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.kzg_c_E.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        // allocate the inputs for the check 5.2
-        let eval_W = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.eval_W.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-        let eval_E = FpVar::<CF1<C1>>::new_input(cs.clone(), || {
-            Ok(self.eval_E.unwrap_or_else(CF1::<C1>::zero))
-        })?;
-
-        // `sponge` is for digest computation.
-        let sponge = PoseidonSpongeVar::<C1::ScalarField>::new(cs.clone(), &self.poseidon_config);
-        // `transcript` is for challenge generation.
-        let mut transcript = sponge.clone();
-
-        // The following enumeration of the steps matches the one used at the documentation page
-        // https://privacy-scaling-explorations.github.io/sonobe-docs/design/nova-decider-onchain.html
-
-        // 2. u_i.cmE==cm(0), u_i.u==1
-        // Here zero is the x & y coordinates of the zero point affine representation.
-        let zero = NonNativeUintVar::new_constant(cs.clone(), C1::BaseField::zero())?;
-        u_i.cmE.x.enforce_equal_unaligned(&zero)?;
-        u_i.cmE.y.enforce_equal_unaligned(&zero)?;
-        (u_i.u.is_one()?).enforce_equal(&Boolean::TRUE)?;
-
-        // 3.a u_i.x[0] == H(i, z_0, z_i, U_i)
-        let (u_i_x, U_i_vec) = U_i.clone().hash(&sponge, &pp_hash, &i, &z_0, &z_i)?;
-        (u_i.x[0]).enforce_equal(&u_i_x)?;
-
-        // 4. check RelaxedR1CS of U_{i+1}
-        let z_U1: Vec<FpVar<CF1<C1>>> =
-            [vec![U_i1.u.clone()], U_i1.x.to_vec(), W_i1.W.to_vec()].concat();
-        RelaxedR1CSGadget::check_native(r1cs, W_i1.E.clone(), U_i1.u.clone(), z_U1)?;
-
-        #[cfg(feature = "light-test")]
-        log::warn!("[WARNING]: Running with the 'light-test' feature, skipping the big part of the DeciderEthCircuit.\n           Only for testing purposes.");
-
-        // The following two checks (and their respective allocations) are disabled for normal
-        // tests since they take several millions of constraints and would take several minutes
-        // (and RAM) to run the test. It is active by default, and not active only when
-        // 'light-test' feature is used.
-        #[cfg(not(feature = "light-test"))]
-        {
-            // imports here instead of at the top of the file, so we avoid having multiple
-            // `#[cfg(not(test))]`
-            use crate::commitment::pedersen::PedersenGadget;
-            use crate::folding::{
-                circuits::cyclefold::{
-                    CycleFoldCommittedInstanceVar, CycleFoldConfig, CycleFoldWitnessVar,
-                },
-                nova::NovaCycleFoldConfig,
-            };
-            use ark_r1cs_std::ToBitsGadget;
-
-            let cf_u_dummy_native =
-                CycleFoldCommittedInstance::<C2>::dummy(NovaCycleFoldConfig::<C1>::IO_LEN);
-            let w_dummy_native = CycleFoldWitness::<C2>::dummy(&self.cf_r1cs);
-            let cf_U_i = CycleFoldCommittedInstanceVar::<C2, GC2>::new_witness(cs.clone(), || {
-                Ok(self.cf_U_i.unwrap_or_else(|| cf_u_dummy_native.clone()))
-            })?;
-            let cf_W_i = CycleFoldWitnessVar::<C2>::new_witness(cs.clone(), || {
-                Ok(self.cf_W_i.unwrap_or(w_dummy_native.clone()))
-            })?;
-
-            // 3.b u_i.x[1] == H(cf_U_i)
-            let (cf_u_i_x, _) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
-            (u_i.x[1]).enforce_equal(&cf_u_i_x)?;
-
-            // 7. check Pedersen commitments of cf_U_i.{cmE, cmW}
-            let H = GC2::new_constant(cs.clone(), self.cf_pedersen_params.h)?;
-            let G = Vec::<GC2>::new_constant(cs.clone(), self.cf_pedersen_params.generators)?;
-            let cf_W_i_E_bits: Result<Vec<Vec<Boolean<CF1<C1>>>>, SynthesisError> =
-                cf_W_i.E.iter().map(|E_i| E_i.to_bits_le()).collect();
-            let cf_W_i_W_bits: Result<Vec<Vec<Boolean<CF1<C1>>>>, SynthesisError> =
-                cf_W_i.W.iter().map(|W_i| W_i.to_bits_le()).collect();
-
-            let computed_cmE = PedersenGadget::<C2, GC2>::commit(
-                H.clone(),
-                G.clone(),
-                cf_W_i_E_bits?,
-                cf_W_i.rE.to_bits_le()?,
-            )?;
-            cf_U_i.cmE.enforce_equal(&computed_cmE)?;
-            let computed_cmW =
-                PedersenGadget::<C2, GC2>::commit(H, G, cf_W_i_W_bits?, cf_W_i.rW.to_bits_le()?)?;
-            cf_U_i.cmW.enforce_equal(&computed_cmW)?;
-
-            let cf_r1cs =
-                R1CSVar::<C1::BaseField, CF1<C1>, NonNativeUintVar<CF1<C1>>>::new_witness(
-                    cs.clone(),
-                    || Ok(self.cf_r1cs.clone()),
-                )?;
-
-            // 6. check RelaxedR1CS of cf_U_i (CycleFold instance)
-            let cf_z_U = [vec![cf_U_i.u.clone()], cf_U_i.x.to_vec(), cf_W_i.W.to_vec()].concat();
-            RelaxedR1CSGadget::check_nonnative(cf_r1cs, cf_W_i.E, cf_U_i.u.clone(), cf_z_U)?;
-        }
-
-        // 1.1.a, 5.1. compute NIFS.V and KZG challenges.
-        // We need to ensure the order of challenge generation is the same as
-        // the native counterpart, so we first compute the challenges here and
-        // do the actual checks later.
-        let cmT =
-            NonNativeAffineVar::new_input(cs.clone(), || Ok(self.cmT.unwrap_or_else(C1::zero)))?;
-        // 1.1.a
-        let r_bits = ChallengeGadget::<C1, CommittedInstance<C1>>::get_challenge_gadget(
-            &mut transcript,
+    type ProofDummyCfg = ();
+    type Proof = C;
+    type RandomnessDummyCfg = ();
+    type Randomness = CF1<C>;
+
+    fn fold_field_elements_gadget(
+        _arith: &R1CS<CF1<C>>,
+        transcript: &mut PoseidonSpongeVar<CF1<C>>,
+        pp_hash: FpVar<CF1<C>>,
+        U: CommittedInstanceVar<C>,
+        U_vec: Vec<FpVar<CF1<C>>>,
+        u: CommittedInstanceVar<C>,
+        proof: 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_i_vec,
-            u_i.clone(),
+            U_vec,
+            u.clone(),
             Some(cmT),
         )?;
-        // 5.1.
-        let (incircuit_c_W, incircuit_c_E) =
-            KZGChallengesGadget::<C1>::get_challenges_gadget(&mut transcript, U_i1.clone())?;
-        incircuit_c_W.enforce_equal(&kzg_c_W)?;
-        incircuit_c_E.enforce_equal(&kzg_c_E)?;
-
-        // 5.2. check eval_W==p_W(c_W) and eval_E==p_E(c_E)
-        let incircuit_eval_W = evaluate_gadget::<CF1<C1>>(W_i1.W, incircuit_c_W)?;
-        let incircuit_eval_E = evaluate_gadget::<CF1<C1>>(W_i1.E, incircuit_c_E)?;
-        incircuit_eval_W.enforce_equal(&eval_W)?;
-        incircuit_eval_E.enforce_equal(&eval_E)?;
-
-        // 1.1.b check that the NIFS.V challenge matches the one from the public input (so we avoid
-        //   the verifier computing it)
-        let r_Fr = Boolean::le_bits_to_fp_var(&r_bits)?;
-        // check that the in-circuit computed r is equal to the inputted r
-        let r =
-            FpVar::<CF1<C1>>::new_input(cs.clone(), || Ok(self.r.unwrap_or_else(CF1::<C1>::zero)))?;
-        r_Fr.enforce_equal(&r)?;
-
-        Ok(())
-    }
-}
-
-/// Interpolates the polynomial from the given vector, and then returns it's evaluation at the
-/// given point.
-#[allow(unused)] // unused while check 7 is disabled
-pub fn evaluate_gadget<F: PrimeField>(
-    mut v: Vec<FpVar<F>>,
-    point: FpVar<F>,
-) -> Result<FpVar<F>, SynthesisError> {
-    v.resize(v.len().next_power_of_two(), FpVar::zero());
-    let n = v.len() as u64;
-    let gen = F::get_root_of_unity(n).unwrap();
-    let domain = Radix2DomainVar::new(gen, log2(v.len()) as u64, FpVar::one()).unwrap();
+        Boolean::le_bits_to_fp_var(&r_bits)?.enforce_equal(&r)?;
 
-    let evaluations_var = EvaluationsVar::from_vec_and_domain(v, domain, true);
-    evaluations_var.interpolate_and_evaluate(&point)
-}
-
-/// Gadget that computes the KZG challenges, also offers the rust native implementation compatible
-/// with the gadget.
-pub struct KZGChallengesGadget<C: CurveGroup> {
-    _c: PhantomData<C>,
-}
-#[allow(clippy::type_complexity)]
-impl<C> KZGChallengesGadget<C>
-where
-    C: CurveGroup,
-    C::ScalarField: PrimeField,
-    <C as CurveGroup>::BaseField: PrimeField,
-    C::ScalarField: Absorb,
-{
-    pub fn get_challenges_native<T: Transcript<C::ScalarField>>(
-        transcript: &mut T,
-        U_i: CommittedInstance<C>,
-    ) -> (C::ScalarField, C::ScalarField) {
-        // compute the KZG challenges, which are computed in-circuit and checked that it matches
-        // the inputted one
-        transcript.absorb_nonnative(&U_i.cmW);
-        let challenge_W = transcript.get_challenge();
-        transcript.absorb_nonnative(&U_i.cmE);
-        let challenge_E = transcript.get_challenge();
-
-        (challenge_W, challenge_E)
+        NIFSGadget::<C>::fold_committed_instance(r, U, u)
     }
-    // compatible with the native get_challenges_native
-    pub fn get_challenges_gadget<S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>>(
-        transcript: &mut T,
-        U_i: CommittedInstanceVar<C>,
-    ) -> Result<(FpVar<C::ScalarField>, FpVar<C::ScalarField>), SynthesisError> {
-        transcript.absorb(&U_i.cmW.to_constraint_field()?)?;
-        let challenge_W = transcript.get_challenge()?;
 
-        transcript.absorb(&U_i.cmE.to_constraint_field()?)?;
-        let challenge_E = transcript.get_challenge()?;
-
-        Ok((challenge_W, challenge_E))
+    fn fold_group_elements_native(
+        U_commitments: &[C],
+        u_commitments: &[C],
+        cmT: Option<Self::Proof>,
+        r: Self::Randomness,
+    ) -> Result<Vec<C>, Error> {
+        let cmT = cmT.ok_or(Error::Empty)?;
+        let U_cmW = U_commitments[0];
+        let U_cmE = U_commitments[1];
+        let u_cmW = u_commitments[0];
+        let u_cmE = u_commitments[1];
+        if !u_cmE.is_zero() {
+            return Err(Error::NotIncomingCommittedInstance);
+        }
+        let cmW = U_cmW + u_cmW.mul(r);
+        let cmE = U_cmE + cmT.mul(r);
+        Ok(vec![cmW, cmE])
     }
 }
 
 #[cfg(test)]
 pub mod tests {
-    use std::cmp::max;
-
-    use ark_crypto_primitives::crh::{
-        sha256::{
-            constraints::{Sha256Gadget, UnitVar},
-            Sha256,
-        },
-        CRHScheme, CRHSchemeGadget,
-    };
-    use ark_pallas::{constraints::GVar, Fq, Fr, Projective};
-    use ark_r1cs_std::bits::uint8::UInt8;
-    use ark_relations::r1cs::ConstraintSystem;
-    use ark_std::{
-        rand::{thread_rng, Rng},
-        UniformRand,
-    };
+    use ark_pallas::{constraints::GVar, Fr, Projective};
+    use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
     use ark_vesta::{constraints::GVar as GVar2, Projective as Projective2};
 
     use super::*;
-    use crate::arith::{
-        r1cs::{
-            extract_r1cs, extract_w_x,
-            tests::{get_test_r1cs, get_test_z},
-        },
-        Arith,
-    };
     use crate::commitment::pedersen::Pedersen;
     use crate::folding::nova::PreprocessorParam;
-    use crate::frontend::utils::{CubicFCircuit, CustomFCircuit, WrapperCircuit};
+    use crate::frontend::utils::CubicFCircuit;
     use crate::transcript::poseidon::poseidon_canonical_config;
     use crate::FoldingScheme;
 
-    // Convert `z` to a witness-instance pair for the relaxed R1CS
-    fn prepare_relaxed_witness_instance<C: CurveGroup, CS: CommitmentScheme<C>, R: Rng>(
-        mut rng: R,
-        r1cs: &R1CS<C::ScalarField>,
-        z: &[C::ScalarField],
-    ) -> (Witness<C>, CommittedInstance<C>) {
-        let (w, x) = r1cs.split_z(z);
-
-        let (cs_pp, _) = CS::setup(&mut rng, max(w.len(), r1cs.A.n_rows)).unwrap();
-
-        let mut w = Witness::new::<false>(w, r1cs.A.n_rows, &mut rng);
-        w.E = r1cs.eval_at_z(z).unwrap();
-        let mut u = w.commit::<CS, false>(&cs_pp, x).unwrap();
-        u.u = z[0];
-
-        (w, u)
-    }
-
     #[test]
-    fn test_relaxed_r1cs_small_gadget_handcrafted() {
-        let rng = &mut thread_rng();
-
-        let r1cs: R1CS<Fr> = get_test_r1cs();
-
-        let mut z = get_test_z(3);
-        z[0] = Fr::rand(rng); // Randomize `z[0]` (i.e. `u.u`) to test the relaxed R1CS
-        let (w, u) = prepare_relaxed_witness_instance::<_, Pedersen<Projective>, _>(rng, &r1cs, &z);
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-
-        let zVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(z)).unwrap();
-        let EVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(w.E)).unwrap();
-        let uVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(u.u)).unwrap();
-        let r1csVar = R1CSVar::<Fr, Fr, FpVar<Fr>>::new_witness(cs.clone(), || Ok(r1cs)).unwrap();
-
-        RelaxedR1CSGadget::check_native(r1csVar, EVar, uVar, zVar).unwrap();
-        assert!(cs.is_satisfied().unwrap());
-    }
-
-    // gets as input a circuit that implements the ConstraintSynthesizer trait, and that has been
-    // initialized.
-    fn test_relaxed_r1cs_gadget<CS: ConstraintSynthesizer<Fr>>(circuit: CS) {
-        let rng = &mut thread_rng();
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-
-        circuit.generate_constraints(cs.clone()).unwrap();
-        cs.finalize();
-        assert!(cs.is_satisfied().unwrap());
-
-        let cs = cs.into_inner().unwrap();
-
-        let r1cs = extract_r1cs::<Fr>(&cs);
-        let (w, x) = extract_w_x::<Fr>(&cs);
-        r1cs.check_relation(&w, &x).unwrap();
-
-        let z = [vec![Fr::rand(rng)], x, w].concat();
-        let (w, u) = prepare_relaxed_witness_instance::<_, Pedersen<Projective>, _>(rng, &r1cs, &z);
-        r1cs.check_relation(&w, &u).unwrap();
-
-        // set new CS for the circuit that checks the RelaxedR1CS of our original circuit
-        let cs = ConstraintSystem::<Fr>::new_ref();
-        // prepare the inputs for our circuit
-        let zVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(z)).unwrap();
-        let EVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(w.E)).unwrap();
-        let uVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(u.u)).unwrap();
-        let r1csVar = R1CSVar::<Fr, Fr, FpVar<Fr>>::new_witness(cs.clone(), || Ok(r1cs)).unwrap();
-
-        RelaxedR1CSGadget::check_native(r1csVar, EVar, uVar, zVar).unwrap();
-        assert!(cs.is_satisfied().unwrap());
-    }
-
-    #[test]
-    fn test_relaxed_r1cs_small_gadget_arkworks() {
-        let z_i = vec![Fr::from(3_u32)];
-        let cubic_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
-        let circuit = WrapperCircuit::<Fr, CubicFCircuit<Fr>> {
-            FC: cubic_circuit,
-            z_i: Some(z_i.clone()),
-            z_i1: Some(cubic_circuit.step_native(0, z_i, vec![]).unwrap()),
-        };
-
-        test_relaxed_r1cs_gadget(circuit);
-    }
-
-    struct Sha256TestCircuit<F: PrimeField> {
-        _f: PhantomData<F>,
-        pub x: Vec<u8>,
-        pub y: Vec<u8>,
-    }
-    impl<F: PrimeField> ConstraintSynthesizer<F> for Sha256TestCircuit<F> {
-        fn generate_constraints(self, cs: ConstraintSystemRef<F>) -> Result<(), SynthesisError> {
-            let x = Vec::<UInt8<F>>::new_witness(cs.clone(), || Ok(self.x))?;
-            let y = Vec::<UInt8<F>>::new_input(cs.clone(), || Ok(self.y))?;
-
-            let unitVar = UnitVar::default();
-            let comp_y = <Sha256Gadget<F> as CRHSchemeGadget<Sha256, F>>::evaluate(&unitVar, &x)?;
-            comp_y.0.enforce_equal(&y)?;
-            Ok(())
-        }
-    }
-    #[test]
-    fn test_relaxed_r1cs_medium_gadget_arkworks() {
-        let x = Fr::from(5_u32).into_bigint().to_bytes_le();
-        let y = <Sha256 as CRHScheme>::evaluate(&(), x.clone()).unwrap();
-
-        let circuit = Sha256TestCircuit::<Fr> {
-            _f: PhantomData,
-            x,
-            y,
-        };
-        test_relaxed_r1cs_gadget(circuit);
-    }
-
-    #[test]
-    fn test_relaxed_r1cs_custom_circuit() {
-        let n_constraints = 10_000;
-        let custom_circuit = CustomFCircuit::<Fr>::new(n_constraints).unwrap();
-        let z_i = vec![Fr::from(5_u32)];
-        let circuit = WrapperCircuit::<Fr, CustomFCircuit<Fr>> {
-            FC: custom_circuit,
-            z_i: Some(z_i.clone()),
-            z_i1: Some(custom_circuit.step_native(0, z_i, vec![]).unwrap()),
-        };
-        test_relaxed_r1cs_gadget(circuit);
-    }
-
-    #[test]
-    fn test_relaxed_r1cs_nonnative_circuit() {
-        let rng = &mut thread_rng();
-
-        let cs = ConstraintSystem::<Fq>::new_ref();
-        // in practice we would use CycleFoldCircuit, but is a very big circuit (when computed
-        // non-natively inside the RelaxedR1CS circuit), so in order to have a short test we use a
-        // custom circuit.
-        let custom_circuit = CustomFCircuit::<Fq>::new(10).unwrap();
-        let z_i = vec![Fq::from(5_u32)];
-        let circuit = WrapperCircuit::<Fq, CustomFCircuit<Fq>> {
-            FC: custom_circuit,
-            z_i: Some(z_i.clone()),
-            z_i1: Some(custom_circuit.step_native(0, z_i, vec![]).unwrap()),
-        };
-        circuit.generate_constraints(cs.clone()).unwrap();
-        cs.finalize();
-        let cs = cs.into_inner().unwrap();
-        let r1cs = extract_r1cs::<Fq>(&cs);
-        let (w, x) = extract_w_x::<Fq>(&cs);
-
-        let z = [vec![Fq::rand(rng)], x, w].concat();
-        let (w, u) =
-            prepare_relaxed_witness_instance::<_, Pedersen<Projective2>, _>(rng, &r1cs, &z);
-
-        // natively
-        let cs = ConstraintSystem::<Fq>::new_ref();
-        let zVar = Vec::<FpVar<Fq>>::new_witness(cs.clone(), || Ok(z.clone())).unwrap();
-        let EVar = Vec::<FpVar<Fq>>::new_witness(cs.clone(), || Ok(w.E.clone())).unwrap();
-        let uVar = FpVar::<Fq>::new_witness(cs.clone(), || Ok(u.u)).unwrap();
-        let r1csVar =
-            R1CSVar::<Fq, Fq, FpVar<Fq>>::new_witness(cs.clone(), || Ok(r1cs.clone())).unwrap();
-        RelaxedR1CSGadget::check_native(r1csVar, EVar, uVar, zVar).unwrap();
-
-        // non-natively
-        let cs = ConstraintSystem::<Fr>::new_ref();
-        let zVar = Vec::new_witness(cs.clone(), || Ok(z)).unwrap();
-        let EVar = Vec::new_witness(cs.clone(), || Ok(w.E)).unwrap();
-        let uVar = NonNativeUintVar::<Fr>::new_witness(cs.clone(), || Ok(u.u)).unwrap();
-        let r1csVar =
-            R1CSVar::<Fq, Fr, NonNativeUintVar<Fr>>::new_witness(cs.clone(), || Ok(r1cs)).unwrap();
-        RelaxedR1CSGadget::check_nonnative(r1csVar, EVar, uVar, zVar).unwrap();
-    }
-
-    #[test]
-    fn test_decider_eth_circuit() {
+    fn test_decider_circuit() {
         let mut rng = ark_std::test_rng();
         let poseidon_config = poseidon_canonical_config::<Fr>();
 
@@ -803,84 +272,14 @@ pub mod tests {
         let ivc_proof = nova.ivc_proof();
         N::verify(nova_params.1, ivc_proof).unwrap();
 
-        // load the DeciderEthCircuit from the Nova instance
-        let decider_eth_circuit = DeciderEthCircuit::<
-            Projective,
-            GVar,
-            Projective2,
-            GVar2,
-            Pedersen<Projective>,
-            Pedersen<Projective2>,
-        >::from_nova(nova)
-        .unwrap();
+        // load the DeciderEthCircuit from the generated Nova instance
+        let decider_circuit =
+            DeciderEthCircuit::<Projective, Projective2, GVar2>::try_from(nova).unwrap();
 
         let cs = ConstraintSystem::<Fr>::new_ref();
 
         // generate the constraints and check that are satisfied by the inputs
-        decider_eth_circuit
-            .generate_constraints(cs.clone())
-            .unwrap();
-        assert!(cs.is_satisfied().unwrap());
-    }
-
-    // checks that the gadget and native implementations of the challenge computation match
-    #[test]
-    fn test_kzg_challenge_gadget() {
-        let mut rng = ark_std::test_rng();
-        let poseidon_config = poseidon_canonical_config::<Fr>();
-        let mut transcript = PoseidonSponge::<Fr>::new(&poseidon_config);
-
-        let U_i = 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 challenge natively
-        let (challenge_W, challenge_E) =
-            KZGChallengesGadget::<Projective>::get_challenges_native(&mut transcript, U_i.clone());
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-        let U_iVar =
-            CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(U_i.clone()))
-                .unwrap();
-        let mut transcript_var = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
-
-        let (challenge_W_Var, challenge_E_Var) =
-            KZGChallengesGadget::<Projective>::get_challenges_gadget(&mut transcript_var, U_iVar)
-                .unwrap();
-        assert!(cs.is_satisfied().unwrap());
-
-        // check that the natively computed and in-circuit computed hashes match
-        use ark_r1cs_std::R1CSVar;
-        assert_eq!(challenge_W_Var.value().unwrap(), challenge_W);
-        assert_eq!(challenge_E_Var.value().unwrap(), challenge_E);
-    }
-
-    #[test]
-    fn test_polynomial_interpolation() {
-        let mut rng = ark_std::test_rng();
-        let n = 12;
-        let l = 1 << n;
-
-        let v: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(&mut rng))
-            .take(l)
-            .collect();
-        let challenge = Fr::rand(&mut rng);
-
-        use ark_poly::Polynomial;
-        let polynomial = poly_from_vec(v.to_vec()).unwrap();
-        let eval = polynomial.evaluate(&challenge);
-
-        let cs = ConstraintSystem::<Fr>::new_ref();
-        let vVar = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(v)).unwrap();
-        let challengeVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(challenge)).unwrap();
-
-        let evalVar = evaluate_gadget::<Fr>(vVar, challengeVar).unwrap();
-
-        use ark_r1cs_std::R1CSVar;
-        assert_eq!(evalVar.value().unwrap(), eval);
+        decider_circuit.generate_constraints(cs.clone()).unwrap();
         assert!(cs.is_satisfied().unwrap());
     }
 }

folding-schemes/src/folding/nova/mod.rs

@@ -10,7 +10,7 @@ use ark_crypto_primitives::sponge::{
 };
 use ark_ec::{CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
-use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar, ToConstraintFieldGadget};
+use ark_r1cs_std::{prelude::CurveVar, ToConstraintFieldGadget};
 use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize, Valid};
 use ark_std::fmt::Debug;
@@ -56,7 +56,7 @@ pub mod decider_circuits;
 pub mod decider_eth;
 pub mod decider_eth_circuit;
 
-use super::traits::{CommittedInstanceOps, WitnessOps};
+use super::traits::{CommittedInstanceOps, Inputize, WitnessOps};
 
 /// Configuration for Nova's CycleFold circuit
 pub struct NovaCycleFoldConfig<C: CurveGroup> {
@@ -137,6 +137,18 @@ impl CommittedInstanceOps for CommittedInstance {
     }
 }
 
+impl<C: CurveGroup> Inputize<C::ScalarField, CommittedInstanceVar<C>> for CommittedInstance<C> {
+    fn inputize(&self) -> Vec<C::ScalarField> {
+        [
+            &[self.u][..],
+            &self.x,
+            &self.cmE.inputize(),
+            &self.cmW.inputize(),
+        ]
+        .concat()
+    }
+}
+
 #[derive(Debug, Clone, Eq, PartialEq, CanonicalSerialize, CanonicalDeserialize)]
 pub struct Witness<C: CurveGroup> {
     pub E: Vec<C::ScalarField>,
@@ -477,8 +489,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     type PreprocessorParam = PreprocessorParam<C1, C2, FC, CS1, CS2, H>;
     type ProverParam = ProverParams<C1, C2, CS1, CS2, H>;
@@ -518,7 +528,7 @@ where
         augmented_F_circuit.generate_constraints(cs.clone())?;
         cs.finalize();
         let cs = cs.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let r1cs = extract_r1cs::<C1::ScalarField>(&cs);
+        let r1cs = extract_r1cs::<C1::ScalarField>(&cs)?;
 
         // CycleFold circuit R1CS
         let cs2 = ConstraintSystem::<C1::BaseField>::new_ref();
@@ -526,7 +536,7 @@ where
         cf_circuit.generate_constraints(cs2.clone())?;
         cs2.finalize();
         let cs2 = cs2.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2);
+        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2)?;
 
         let cs_vp = CS1::VerifierParams::deserialize_with_mode(&mut reader, compress, validate)?;
         let cf_cs_vp = CS2::VerifierParams::deserialize_with_mode(&mut reader, compress, validate)?;
@@ -548,23 +558,14 @@ where
             get_r1cs::<C1, GC1, C2, GC2, FC>(&prep_param.poseidon_config, prep_param.F.clone())?;
 
         // if cs params exist, use them, if not, generate new ones
-        let cs_pp: CS1::ProverParams;
-        let cs_vp: CS1::VerifierParams;
-        let cf_cs_pp: CS2::ProverParams;
-        let cf_cs_vp: CS2::VerifierParams;
-        if prep_param.cs_pp.is_some()
-            && prep_param.cf_cs_pp.is_some()
-            && prep_param.cs_vp.is_some()
-            && prep_param.cf_cs_vp.is_some()
-        {
-            cs_pp = prep_param.clone().cs_pp.unwrap();
-            cs_vp = prep_param.clone().cs_vp.unwrap();
-            cf_cs_pp = prep_param.clone().cf_cs_pp.unwrap();
-            cf_cs_vp = prep_param.clone().cf_cs_vp.unwrap();
-        } else {
-            (cs_pp, cs_vp) = CS1::setup(&mut rng, r1cs.A.n_rows)?;
-            (cf_cs_pp, cf_cs_vp) = CS2::setup(&mut rng, cf_r1cs.A.n_rows)?;
-        }
+        let (cs_pp, cs_vp) = match (&prep_param.cs_pp, &prep_param.cs_vp) {
+            (Some(cs_pp), Some(cs_vp)) => (cs_pp.clone(), cs_vp.clone()),
+            _ => CS1::setup(&mut rng, r1cs.A.n_rows)?,
+        };
+        let (cf_cs_pp, cf_cs_vp) = match (&prep_param.cf_cs_pp, &prep_param.cf_cs_vp) {
+            (Some(cf_cs_pp), Some(cf_cs_vp)) => (cf_cs_pp.clone(), cf_cs_vp.clone()),
+            _ => CS2::setup(&mut rng, cf_r1cs.A.n_rows)?,
+        };
 
         let prover_params = ProverParams::<C1, C2, CS1, CS2, H> {
             poseidon_config: prep_param.poseidon_config.clone(),
@@ -601,12 +602,12 @@ where
         augmented_F_circuit.generate_constraints(cs.clone())?;
         cs.finalize();
         let cs = cs.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let r1cs = extract_r1cs::<C1::ScalarField>(&cs);
+        let r1cs = extract_r1cs::<C1::ScalarField>(&cs)?;
 
         cf_circuit.generate_constraints(cs2.clone())?;
         cs2.finalize();
         let cs2 = cs2.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2);
+        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2)?;
 
         // compute the public params hash
         let pp_hash = vp.pp_hash()?;
@@ -946,7 +947,7 @@ where
         } = ivc_proof;
         let (pp, vp) = params;
 
-        let f_circuit = FC::new(fcircuit_params).unwrap();
+        let f_circuit = FC::new(fcircuit_params)?;
         let cs = ConstraintSystem::<C1::ScalarField>::new_ref();
         let cs2 = ConstraintSystem::<C1::BaseField>::new_ref();
         let augmented_F_circuit =
@@ -956,12 +957,12 @@ where
         augmented_F_circuit.generate_constraints(cs.clone())?;
         cs.finalize();
         let cs = cs.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let r1cs = extract_r1cs::<C1::ScalarField>(&cs);
+        let r1cs = extract_r1cs::<C1::ScalarField>(&cs)?;
 
         cf_circuit.generate_constraints(cs2.clone())?;
         cs2.finalize();
         let cs2 = cs2.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2);
+        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2)?;
 
         Ok(Self {
             _gc1: PhantomData,
@@ -1056,8 +1057,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     // folds the given cyclefold circuit and its instances
     #[allow(clippy::type_complexity)]
@@ -1102,7 +1101,7 @@ pub fn get_r1cs_from_cs(
     circuit.generate_constraints(cs.clone())?;
     cs.finalize();
     let cs = cs.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-    let r1cs = extract_r1cs::<F>(&cs);
+    let r1cs = extract_r1cs::<F>(&cs)?;
     Ok(r1cs)
 }
 
@@ -1123,8 +1122,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     let augmented_F_circuit =
         AugmentedFCircuit::<C1, C2, GC2, FC>::empty(poseidon_config, F_circuit);
@@ -1151,8 +1148,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     let (r1cs, cf_r1cs) = get_r1cs::<C1, GC1, C2, GC2, FC>(poseidon_config, F_circuit)?;
     Ok((r1cs.A.n_rows, cf_r1cs.A.n_rows))

folding-schemes/src/folding/nova/nifs/nova.rs

@@ -35,8 +35,6 @@ impl, T: Transcript, c
     NIFSTrait<C, CS, T, H> for NIFS<C, CS, T, H>
 where
     <C as Group>::ScalarField: Absorb,
-    <C as CurveGroup>::BaseField: PrimeField,
-    <C as Group>::ScalarField: PrimeField,
 {
     type CommittedInstance = CommittedInstance<C>;
     type Witness = Witness<C>;
@@ -150,7 +148,6 @@ impl, T: Transcript, c
     NIFS<C, CS, T, H>
 where
     <C as Group>::ScalarField: Absorb,
-    <C as CurveGroup>::BaseField: PrimeField,
 {
     /// compute_T: compute cross-terms T
     pub fn compute_T(

folding-schemes/src/folding/nova/traits.rs

@@ -1,11 +1,18 @@
 use ark_ec::CurveGroup;
+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::{CommittedInstance, Witness};
-use crate::arith::ArithSampler;
-use crate::arith::{r1cs::R1CS, Arith};
+use crate::arith::{
+    r1cs::{circuits::R1CSMatricesVar, R1CS},
+    Arith, ArithGadget, ArithSampler,
+};
 use crate::commitment::CommitmentScheme;
 use crate::folding::circuits::CF1;
+use crate::utils::gadgets::{EquivalenceGadget, VectorGadget};
 use crate::Error;
 
 /// Implements `Arith` for R1CS, where the witness is of type [`Witness`], and
@@ -95,3 +102,25 @@ impl ArithSampler, CommittedInstance> for R1CS
         Ok((witness, cm_witness))
     }
 }
+
+impl<C: CurveGroup> ArithGadget<WitnessVar<C>, CommittedInstanceVar<C>>
+    for R1CSMatricesVar<C::ScalarField, FpVar<C::ScalarField>>
+{
+    type Evaluation = (Vec<FpVar<C::ScalarField>>, Vec<FpVar<C::ScalarField>>);
+
+    fn eval_relation(
+        &self,
+        w: &WitnessVar<C>,
+        u: &CommittedInstanceVar<C>,
+    ) -> Result<Self::Evaluation, SynthesisError> {
+        self.eval_at_z(&[&[u.u.clone()][..], &u.x, &w.W].concat())
+    }
+
+    fn enforce_evaluation(
+        w: &WitnessVar<C>,
+        _u: &CommittedInstanceVar<C>,
+        (AzBz, uCz): Self::Evaluation,
+    ) -> Result<(), SynthesisError> {
+        EquivalenceGadget::<C::ScalarField>::enforce_equivalent(&AzBz[..], &uCz.add(&w.E)?[..])
+    }
+}

folding-schemes/src/folding/nova/zk.rs

@@ -43,10 +43,7 @@ use ark_crypto_primitives::sponge::{
     Absorb, CryptographicSponge,
 };
 use ark_ec::{CurveGroup, Group};
-use ark_r1cs_std::{
-    groups::{CurveVar, GroupOpsBounds},
-    ToConstraintFieldGadget,
-};
+use ark_r1cs_std::{groups::CurveVar, ToConstraintFieldGadget};
 
 use crate::{commitment::CommitmentScheme, folding::circuits::CF2, frontend::FCircuit, Error};
 
@@ -90,7 +87,6 @@ where
         <C2 as Group>::ScalarField: PrimeField,
         <C2 as CurveGroup>::BaseField: PrimeField,
         <C2 as CurveGroup>::BaseField: Absorb,
-        for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
         GC2: ToConstraintFieldGadget<<C2 as CurveGroup>::BaseField>,
         C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
     {
@@ -161,7 +157,6 @@ where
         <C2 as Group>::ScalarField: Absorb,
         <C2 as CurveGroup>::BaseField: PrimeField,
         <C2 as CurveGroup>::BaseField: Absorb,
-        for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
         GC2: ToConstraintFieldGadget<<C2 as CurveGroup>::BaseField>,
         C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
     {

folding-schemes/src/folding/protogalaxy/circuits.rs

@@ -11,7 +11,7 @@ use ark_r1cs_std::{
     boolean::Boolean,
     eq::EqGadget,
     fields::{fp::FpVar, FieldVar},
-    groups::{CurveVar, GroupOpsBounds},
+    groups::CurveVar,
     poly::polynomial::univariate::dense::DensePolynomialVar,
     R1CSVar, ToBitsGadget, ToConstraintFieldGadget,
 };
@@ -77,7 +77,8 @@ impl FoldingGadget {
         let betas_star = betas_star_var(&instance.betas, &deltas, &alpha);
 
         let k = vec_instances.len();
-        let H = GeneralEvaluationDomain::new(k + 1).unwrap();
+        let H =
+            GeneralEvaluationDomain::new(k + 1).ok_or(SynthesisError::PolynomialDegreeTooLarge)?;
         let L_X = lagrange_polys(H)
             .into_iter()
             .map(|poly| {
@@ -183,7 +184,6 @@ impl AugmentationGadget {
     ) -> Result<CycleFoldCommittedInstanceVar<C2, GC2>, SynthesisError>
     where
         C2::BaseField: PrimeField + Absorb,
-        for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
     {
         assert_eq!(cf_u_cmWs.len(), cf_u_xs.len());
         assert_eq!(cf_u_xs.len(), cf_cmTs.len());
@@ -268,8 +268,6 @@ pub struct AugmentedFCircuit<
 
 impl<C1: CurveGroup, C2: CurveGroup, GC2: CurveVar<C2, CF2<C2>>, FC: FCircuit<CF1<C1>>>
     AugmentedFCircuit<C1, C2, GC2, FC>
-where
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     pub fn empty(
         poseidon_config: &PoseidonConfig<CF1<C1>>,
@@ -317,7 +315,6 @@ where
     GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
     FC: FCircuit<CF1<C1>>,
     C2::BaseField: PrimeField + Absorb,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     fn generate_constraints(self, cs: ConstraintSystemRef<CF1<C1>>) -> Result<(), SynthesisError> {
         let pp_hash = FpVar::<CF1<C1>>::new_witness(cs.clone(), || Ok(self.pp_hash))?;
@@ -498,7 +495,7 @@ mod tests {
         let mut transcript_p = PoseidonSponge::new(&poseidon_config);
         let mut transcript_v = PoseidonSponge::new(&poseidon_config);
 
-        let (_, _, F_coeffs, K_coeffs, _, _) = Folding::<Projective>::prove(
+        let (_, _, proof, _) = Folding::<Projective>::prove(
             &mut transcript_p,
             &r1cs,
             &instance,
@@ -507,20 +504,15 @@ mod tests {
             &witnesses,
         )?;
 
-        let folded_instance = Folding::<Projective>::verify(
-            &mut transcript_v,
-            &instance,
-            &instances,
-            F_coeffs.clone(),
-            K_coeffs.clone(),
-        )?;
+        let folded_instance =
+            Folding::<Projective>::verify(&mut transcript_v, &instance, &instances, proof.clone())?;
 
         let cs = ConstraintSystem::new_ref();
         let mut transcript_var = PoseidonSpongeVar::new(cs.clone(), &poseidon_config);
         let instance_var = CommittedInstanceVar::new_witness(cs.clone(), || Ok(instance))?;
         let instances_var = Vec::new_witness(cs.clone(), || Ok(instances))?;
-        let F_coeffs_var = Vec::new_witness(cs.clone(), || Ok(F_coeffs))?;
-        let K_coeffs_var = Vec::new_witness(cs.clone(), || Ok(K_coeffs))?;
+        let F_coeffs_var = Vec::new_witness(cs.clone(), || Ok(proof.F_coeffs))?;
+        let K_coeffs_var = Vec::new_witness(cs.clone(), || Ok(proof.K_coeffs))?;
 
         let (folded_instance_var, _) = FoldingGadget::fold_committed_instance(
             &mut transcript_var,

folding-schemes/src/folding/protogalaxy/decider_eth_circuit.rs

@@ -0,0 +1,252 @@
+/// This file implements the onchain (Ethereum's EVM) decider circuit. For non-ethereum use cases,
+/// other more efficient approaches can be used.
+use ark_crypto_primitives::sponge::{
+    constraints::CryptographicSpongeVar,
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonSponge},
+    Absorb, CryptographicSponge,
+};
+use ark_ec::CurveGroup;
+use ark_ff::PrimeField;
+use ark_r1cs_std::{
+    alloc::{AllocVar, AllocationMode},
+    eq::EqGadget,
+    fields::fp::FpVar,
+    groups::CurveVar,
+    ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::{Namespace, SynthesisError};
+use ark_std::{borrow::Borrow, marker::PhantomData};
+
+use crate::{
+    arith::r1cs::{circuits::R1CSMatricesVar, R1CS},
+    commitment::{pedersen::Params as PedersenParams, CommitmentScheme},
+    folding::{
+        circuits::{
+            decider::{
+                on_chain::GenericOnchainDeciderCircuit, DeciderEnabledNIFS, EvalGadget,
+                KZGChallengesGadget,
+            },
+            CF1, CF2,
+        },
+        traits::{WitnessOps, WitnessVarOps},
+    },
+    frontend::FCircuit,
+    Error,
+};
+
+use super::{
+    circuits::FoldingGadget,
+    constants::{INCOMING, RUNNING},
+    folding::{Folding, ProtoGalaxyProof},
+    CommittedInstance, CommittedInstanceVar, ProtoGalaxy, Witness,
+};
+
+/// In-circuit representation of the Witness associated to the CommittedInstance.
+#[derive(Debug, Clone)]
+pub struct WitnessVar<F: PrimeField> {
+    pub W: Vec<FpVar<F>>,
+    pub rW: FpVar<F>,
+}
+
+impl<F: PrimeField> AllocVar<Witness<F>, F> for WitnessVar<F> {
+    fn new_variable<T: Borrow<Witness<F>>>(
+        cs: impl Into<Namespace<F>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        f().and_then(|val| {
+            let cs = cs.into();
+
+            let W = Vec::new_variable(cs.clone(), || Ok(val.borrow().w.to_vec()), mode)?;
+            let rW = FpVar::new_variable(cs.clone(), || Ok(val.borrow().r_w), mode)?;
+
+            Ok(Self { W, rW })
+        })
+    }
+}
+
+impl<F: PrimeField> WitnessVarOps<F> for WitnessVar<F> {
+    fn get_openings(&self) -> Vec<(&[FpVar<F>], FpVar<F>)> {
+        vec![(&self.W, self.rW.clone())]
+    }
+}
+
+pub type DeciderEthCircuit<C1, C2, GC2> = GenericOnchainDeciderCircuit<
+    C1,
+    C2,
+    GC2,
+    CommittedInstance<C1, RUNNING>,
+    CommittedInstance<C1, INCOMING>,
+    Witness<CF1<C1>>,
+    R1CS<CF1<C1>>,
+    R1CSMatricesVar<CF1<C1>, FpVar<CF1<C1>>>,
+    DeciderProtoGalaxyGadget,
+>;
+
+/// returns an instance of the DeciderEthCircuit from the given ProtoGalaxy struct
+impl<
+        C1: CurveGroup,
+        GC1: CurveVar<C1, CF2<C1>> + ToConstraintFieldGadget<CF2<C1>>,
+        C2: CurveGroup,
+        GC2: CurveVar<C2, CF2<C2>> + ToConstraintFieldGadget<CF2<C2>>,
+        FC: FCircuit<C1::ScalarField>,
+        CS1: CommitmentScheme<C1, false>,
+        // enforce that the CS2 is Pedersen commitment scheme, since we're at Ethereum's EVM decider
+        CS2: CommitmentScheme<C2, false, ProverParams = PedersenParams<C2>>,
+    > TryFrom<ProtoGalaxy<C1, GC1, C2, GC2, FC, CS1, CS2>> for DeciderEthCircuit<C1, C2, GC2>
+where
+    CF1<C1>: Absorb,
+{
+    type Error = Error;
+
+    fn try_from(protogalaxy: ProtoGalaxy<C1, GC1, C2, GC2, FC, CS1, CS2>) -> Result<Self, Error> {
+        let mut transcript = PoseidonSponge::<C1::ScalarField>::new(&protogalaxy.poseidon_config);
+
+        let (U_i1, W_i1, proof, aux) = Folding::prove(
+            &mut transcript,
+            &protogalaxy.r1cs,
+            &protogalaxy.U_i,
+            &protogalaxy.W_i,
+            &[protogalaxy.u_i.clone()],
+            &[protogalaxy.w_i.clone()],
+        )?;
+
+        // compute the KZG challenges used as inputs in the circuit
+        let kzg_challenges = KZGChallengesGadget::get_challenges_native(&mut transcript, &U_i1);
+
+        // get KZG evals
+        let kzg_evaluations = W_i1
+            .get_openings()
+            .iter()
+            .zip(&kzg_challenges)
+            .map(|((v, _), &c)| EvalGadget::evaluate_native(v, c))
+            .collect::<Result<Vec<_>, _>>()?;
+
+        Ok(Self {
+            _gc2: PhantomData,
+            _avar: PhantomData,
+            arith: protogalaxy.r1cs,
+            cf_arith: protogalaxy.cf_r1cs,
+            cf_pedersen_params: protogalaxy.cf_cs_params,
+            poseidon_config: protogalaxy.poseidon_config,
+            pp_hash: protogalaxy.pp_hash,
+            i: protogalaxy.i,
+            z_0: protogalaxy.z_0,
+            z_i: protogalaxy.z_i,
+            U_i: protogalaxy.U_i,
+            W_i: protogalaxy.W_i,
+            u_i: protogalaxy.u_i,
+            w_i: protogalaxy.w_i,
+            U_i1,
+            W_i1,
+            proof,
+            randomness: aux.L_X_evals,
+            cf_U_i: protogalaxy.cf_U_i,
+            cf_W_i: protogalaxy.cf_W_i,
+            kzg_challenges,
+            kzg_evaluations,
+        })
+    }
+}
+
+pub struct DeciderProtoGalaxyGadget;
+
+impl<C: CurveGroup>
+    DeciderEnabledNIFS<
+        C,
+        CommittedInstance<C, RUNNING>,
+        CommittedInstance<C, INCOMING>,
+        Witness<CF1<C>>,
+        R1CS<CF1<C>>,
+    > for DeciderProtoGalaxyGadget
+{
+    type Proof = ProtoGalaxyProof<CF1<C>>;
+    type ProofDummyCfg = (usize, usize, usize);
+    type Randomness = Vec<CF1<C>>;
+    type RandomnessDummyCfg = usize;
+
+    fn fold_field_elements_gadget(
+        _arith: &R1CS<CF1<C>>,
+        transcript: &mut PoseidonSpongeVar<CF1<C>>,
+        _pp_hash: FpVar<CF1<C>>,
+        U: CommittedInstanceVar<C, RUNNING>,
+        _U_vec: Vec<FpVar<CF1<C>>>,
+        u: CommittedInstanceVar<C, INCOMING>,
+        proof: Self::Proof,
+        randomness: Self::Randomness,
+    ) -> Result<CommittedInstanceVar<C, RUNNING>, SynthesisError> {
+        let cs = transcript.cs();
+        let F_coeffs = Vec::new_witness(cs.clone(), || Ok(&proof.F_coeffs[..]))?;
+        let K_coeffs = Vec::new_witness(cs.clone(), || Ok(&proof.K_coeffs[..]))?;
+        let randomness = Vec::new_input(cs.clone(), || Ok(randomness))?;
+
+        let (U_next, L_X_evals) =
+            FoldingGadget::fold_committed_instance(transcript, &U, &[u], F_coeffs, K_coeffs)?;
+        L_X_evals.enforce_equal(&randomness)?;
+
+        Ok(U_next)
+    }
+
+    fn fold_group_elements_native(
+        U_commitments: &[C],
+        u_commitments: &[C],
+        _: Option<Self::Proof>,
+        L_X_evals: Self::Randomness,
+    ) -> Result<Vec<C>, Error> {
+        let U_phi = U_commitments[0];
+        let u_phi = u_commitments[0];
+        Ok(vec![U_phi * L_X_evals[0] + u_phi * L_X_evals[1]])
+    }
+}
+
+#[cfg(test)]
+pub mod tests {
+    use ark_bn254::{constraints::GVar, Fr, G1Projective as Projective};
+    use ark_grumpkin::{constraints::GVar as GVar2, Projective as Projective2};
+    use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
+
+    use super::*;
+    use crate::commitment::pedersen::Pedersen;
+    use crate::folding::protogalaxy::ProtoGalaxy;
+    use crate::frontend::{utils::CubicFCircuit, FCircuit};
+    use crate::transcript::poseidon::poseidon_canonical_config;
+    use crate::FoldingScheme;
+
+    #[test]
+    fn test_decider_circuit() {
+        let mut rng = ark_std::test_rng();
+        let poseidon_config = poseidon_canonical_config::<Fr>();
+
+        let F_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
+        let z_0 = vec![Fr::from(3_u32)];
+
+        type PG = ProtoGalaxy<
+            Projective,
+            GVar,
+            Projective2,
+            GVar2,
+            CubicFCircuit<Fr>,
+            Pedersen<Projective>,
+            Pedersen<Projective2>,
+        >;
+        let pg_params = PG::preprocess(&mut rng, &(poseidon_config, F_circuit)).unwrap();
+
+        // generate a Nova instance and do a step of it
+        let mut protogalaxy = PG::init(&pg_params, F_circuit, z_0.clone()).unwrap();
+        protogalaxy.prove_step(&mut rng, vec![], None).unwrap();
+
+        let ivc_proof = protogalaxy.ivc_proof();
+        PG::verify(pg_params.1, ivc_proof).unwrap();
+
+        // load the DeciderEthCircuit from the generated Nova instance
+        let decider_circuit =
+            DeciderEthCircuit::<Projective, Projective2, GVar2>::try_from(protogalaxy).unwrap();
+
+        let cs = ConstraintSystem::<Fr>::new_ref();
+
+        // generate the constraints and check that are satisfied by the inputs
+        decider_circuit.generate_constraints(cs.clone()).unwrap();
+        assert!(cs.is_satisfied().unwrap());
+        dbg!(cs.num_constraints());
+    }
+}

folding-schemes/src/folding/protogalaxy/folding.rs

@@ -15,10 +15,32 @@ use super::ProtoGalaxyError;
 use super::{CommittedInstance, Witness};
 
 use crate::arith::r1cs::R1CS;
+use crate::folding::traits::Dummy;
 use crate::transcript::Transcript;
 use crate::utils::vec::*;
 use crate::Error;
 
+#[derive(Debug, Clone)]
+pub struct ProtoGalaxyProof<F: PrimeField> {
+    pub F_coeffs: Vec<F>,
+    pub K_coeffs: Vec<F>,
+}
+
+impl<F: PrimeField> Dummy<(usize, usize, usize)> for ProtoGalaxyProof<F> {
+    fn dummy((t, d, k): (usize, usize, usize)) -> Self {
+        Self {
+            F_coeffs: vec![F::zero(); t],
+            K_coeffs: vec![F::zero(); d * k + 1],
+        }
+    }
+}
+
+#[derive(Debug, Clone)]
+pub struct ProtoGalaxyAux<C: CurveGroup> {
+    pub L_X_evals: Vec<C::ScalarField>,
+    pub phi_stars: Vec<C>,
+}
+
 #[derive(Clone, Debug)]
 /// Implements the protocol described in section 4 of
 /// [ProtoGalaxy](https://eprint.iacr.org/2023/1106.pdf)
@@ -28,7 +50,6 @@ pub struct Folding {
 impl<C: CurveGroup> Folding<C>
 where
     <C as Group>::ScalarField: Absorb,
-    <C as CurveGroup>::BaseField: Absorb,
 {
     #![allow(clippy::type_complexity)]
     /// implements the non-interactive Prover from the folding scheme described in section 4
@@ -45,10 +66,8 @@ where
         (
             CommittedInstance<C, true>,
             Witness<C::ScalarField>,
-            Vec<C::ScalarField>, // F_X coeffs
-            Vec<C::ScalarField>, // K_X coeffs
-            Vec<C::ScalarField>, // L_X evals
-            Vec<C>,              // phi_stars
+            ProtoGalaxyProof<C::ScalarField>,
+            ProtoGalaxyAux<C>,
         ),
         Error,
     > {
@@ -243,10 +262,11 @@ where
                 w: w_star,
                 r_w: r_w_star,
             },
-            F_coeffs,
-            K_coeffs,
-            L_X_evals,
-            phi_stars,
+            ProtoGalaxyProof { F_coeffs, K_coeffs },
+            ProtoGalaxyAux {
+                L_X_evals,
+                phi_stars,
+            },
         ))
     }
 
@@ -258,8 +278,7 @@ where
         // incoming instances
         vec_instances: &[CommittedInstance<C, false>],
         // polys from P
-        F_coeffs: Vec<C::ScalarField>,
-        K_coeffs: Vec<C::ScalarField>,
+        proof: ProtoGalaxyProof<C::ScalarField>,
     ) -> Result<CommittedInstance<C, true>, Error> {
         let t = instance.betas.len();
 
@@ -270,20 +289,20 @@ where
         let delta = transcript.get_challenge();
         let deltas = exponential_powers(delta, t);
 
-        transcript.absorb(&F_coeffs);
+        transcript.absorb(&proof.F_coeffs);
 
         let alpha = transcript.get_challenge();
         let alphas = all_powers(alpha, t);
 
         // F(alpha) = e + \sum_t F_i * alpha^i
         let mut F_alpha = instance.e;
-        for (i, F_i) in F_coeffs.iter().skip(1).enumerate() {
+        for (i, F_i) in proof.F_coeffs.iter().skip(1).enumerate() {
             F_alpha += *F_i * alphas[i + 1];
         }
 
         let betas_star = betas_star(&instance.betas, &deltas, alpha);
 
-        transcript.absorb(&K_coeffs);
+        transcript.absorb(&proof.K_coeffs);
 
         let k = vec_instances.len();
         let H =
@@ -291,7 +310,7 @@ where
         let L_X: Vec<DensePolynomial<C::ScalarField>> = lagrange_polys(H);
         let Z_X: DensePolynomial<C::ScalarField> = H.vanishing_polynomial().into();
         let K_X: DensePolynomial<C::ScalarField> =
-            DensePolynomial::<C::ScalarField>::from_coefficients_vec(K_coeffs);
+            DensePolynomial::<C::ScalarField>::from_coefficients_vec(proof.K_coeffs);
 
         let gamma = transcript.get_challenge();
 
@@ -483,27 +502,20 @@ pub mod tests {
         let mut transcript_p = PoseidonSponge::<Fr>::new(&poseidon_config);
         let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
 
-        let (folded_instance, folded_witness, F_coeffs, K_coeffs, _, _) =
-            Folding::<Projective>::prove(
-                &mut transcript_p,
-                &r1cs,
-                &instance,
-                &witness,
-                &instances,
-                &witnesses,
-            )
-            .unwrap();
-
-        // verifier
-        let folded_instance_v = Folding::<Projective>::verify(
-            &mut transcript_v,
+        let (folded_instance, folded_witness, proof, _) = Folding::<Projective>::prove(
+            &mut transcript_p,
+            &r1cs,
             &instance,
+            &witness,
             &instances,
-            F_coeffs,
-            K_coeffs,
+            &witnesses,
         )
         .unwrap();
 
+        // verifier
+        let folded_instance_v =
+            Folding::<Projective>::verify(&mut transcript_v, &instance, &instances, proof).unwrap();
+
         // check that prover & verifier folded instances are the same values
         assert_eq!(folded_instance.phi, folded_instance_v.phi);
         assert_eq!(folded_instance.betas, folded_instance_v.betas);
@@ -533,24 +545,22 @@ pub mod tests {
             // generate the instances to be fold
             let (_, _, witnesses, instances) = prepare_inputs(k);
 
-            let (folded_instance, folded_witness, F_coeffs, K_coeffs, _, _) =
-                Folding::<Projective>::prove(
-                    &mut transcript_p,
-                    &r1cs,
-                    &running_instance,
-                    &running_witness,
-                    &instances,
-                    &witnesses,
-                )
-                .unwrap();
+            let (folded_instance, folded_witness, proof, _) = Folding::<Projective>::prove(
+                &mut transcript_p,
+                &r1cs,
+                &running_instance,
+                &running_witness,
+                &instances,
+                &witnesses,
+            )
+            .unwrap();
 
             // verifier
             let folded_instance_v = Folding::<Projective>::verify(
                 &mut transcript_v,
                 &running_instance,
                 &instances,
-                F_coeffs,
-                K_coeffs,
+                proof,
             )
             .unwrap();
 

folding-schemes/src/folding/protogalaxy/mod.rs

@@ -9,7 +9,7 @@ use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     eq::EqGadget,
     fields::{fp::FpVar, FieldVar},
-    groups::{CurveVar, GroupOpsBounds},
+    groups::CurveVar,
     R1CSVar, ToConstraintFieldGadget,
 };
 use ark_relations::r1cs::{
@@ -44,6 +44,7 @@ use crate::{
 
 pub mod circuits;
 pub mod constants;
+pub mod decider_eth_circuit;
 pub mod folding;
 pub mod traits;
 pub(crate) mod utils;
@@ -52,7 +53,7 @@ use circuits::AugmentedFCircuit;
 use folding::Folding;
 
 use super::traits::{
-    CommittedInstanceOps, CommittedInstanceVarOps, Dummy, WitnessOps, WitnessVarOps,
+    CommittedInstanceOps, CommittedInstanceVarOps, Dummy, Inputize, WitnessOps, WitnessVarOps,
 };
 
 /// Configuration for ProtoGalaxy's CycleFold circuit
@@ -121,6 +122,14 @@ impl CommittedInstanceOps for CommittedInsta
     }
 }
 
+impl<C: CurveGroup, const TYPE: bool> Inputize<C::ScalarField, CommittedInstanceVar<C, TYPE>>
+    for CommittedInstance<C, TYPE>
+{
+    fn inputize(&self) -> Vec<C::ScalarField> {
+        [&self.phi.inputize(), &self.betas, &[self.e][..], &self.x].concat()
+    }
+}
+
 #[derive(Clone, Debug)]
 pub struct CommittedInstanceVar<C: CurveGroup, const TYPE: bool> {
     phi: NonNativeAffineVar<C>,
@@ -534,8 +543,6 @@ where
     C1::ScalarField: Absorb,
     C2::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     /// This method computes the parameter `t` in ProtoGalaxy for folding `F'`,
     /// the augmented circuit of `F`
@@ -544,7 +551,7 @@ where
         F: &FC,
         d: usize,
         k: usize,
-    ) -> Result<usize, SynthesisError> {
+    ) -> Result<usize, Error> {
         // In ProtoGalaxy, prover and verifier are parameterized by `t = log(n)`
         // where `n` is the number of constraints in the circuit (known as the
         // mapping `f` in the paper).
@@ -581,7 +588,7 @@ where
         // Create a dummy circuit with the same state length and external inputs
         // length as `F`, which replaces `F` in the augmented circuit `F'`.
         let dummy_circuit: DummyCircuit =
-            FCircuit::<C1::ScalarField>::new((state_len, external_inputs_len)).unwrap();
+            FCircuit::<C1::ScalarField>::new((state_len, external_inputs_len))?;
 
         // Compute `augmentation_constraints`, the size of `F'` without `F`.
         let cs = ConstraintSystem::<C1::ScalarField>::new_ref();
@@ -640,8 +647,6 @@ where
     C1::ScalarField: Absorb,
     C2::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     type PreprocessorParam = (PoseidonConfig<CF1<C1>>, FC);
     type ProverParam = ProverParams<C1, C2, CS1, CS2>;
@@ -692,7 +697,7 @@ where
         augmented_F_circuit.generate_constraints(cs.clone())?;
         cs.finalize();
         let cs = cs.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let r1cs = extract_r1cs::<C1::ScalarField>(&cs);
+        let r1cs = extract_r1cs::<C1::ScalarField>(&cs)?;
 
         // CycleFold circuit R1CS
         let cs2 = ConstraintSystem::<C1::BaseField>::new_ref();
@@ -700,7 +705,7 @@ where
         cf_circuit.generate_constraints(cs2.clone())?;
         cs2.finalize();
         let cs2 = cs2.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2);
+        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2)?;
 
         let cs_vp = CS1::VerifierParams::deserialize_with_mode(&mut reader, compress, validate)?;
         let cf_cs_vp = CS2::VerifierParams::deserialize_with_mode(&mut reader, compress, validate)?;
@@ -740,12 +745,12 @@ where
         augmented_F_circuit.generate_constraints(cs.clone())?;
         cs.finalize();
         let cs = cs.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let r1cs = extract_r1cs::<C1::ScalarField>(&cs);
+        let r1cs = extract_r1cs::<C1::ScalarField>(&cs)?;
 
         cf_circuit.generate_constraints(cs2.clone())?;
         cs2.finalize();
         let cs2 = cs2.into_inner().ok_or(Error::NoInnerConstraintSystem)?;
-        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2);
+        let cf_r1cs = extract_r1cs::<C1::BaseField>(&cs2)?;
 
         let (cs_pp, cs_vp) = CS1::setup(&mut rng, r1cs.A.n_rows)?;
         let (cf_cs_pp, cf_cs_vp) = CS2::setup(&mut rng, max(cf_r1cs.A.n_rows, cf_r1cs.A.n_cols))?;
@@ -898,7 +903,7 @@ where
         } else {
             // Primary part:
             // Compute `U_{i+1}` by folding `u_i` into `U_i`.
-            let (U_i1, W_i1, F_coeffs, K_coeffs, L_evals, phi_stars) = Folding::prove(
+            let (U_i1, W_i1, proof, aux) = Folding::prove(
                 &mut transcript_prover,
                 &self.r1cs,
                 &self.U_i,
@@ -909,8 +914,8 @@ where
 
             // CycleFold part:
             // get the vector used as public inputs 'x' in the CycleFold circuit
-            let mut r0_bits = L_evals[0].into_bigint().to_bits_le();
-            let mut r1_bits = L_evals[1].into_bigint().to_bits_le();
+            let mut r0_bits = aux.L_X_evals[0].into_bigint().to_bits_le();
+            let mut r1_bits = aux.L_X_evals[1].into_bigint().to_bits_le();
             r0_bits.resize(C1::ScalarField::MODULUS_BIT_SIZE as usize, false);
             r1_bits.resize(C1::ScalarField::MODULUS_BIT_SIZE as usize, false);
 
@@ -919,13 +924,12 @@ where
             let cf1_u_i_x = [
                 r0_bits
                     .chunks(C1::BaseField::MODULUS_BIT_SIZE as usize - 1)
-                    .map(BigInteger::from_bits_le)
-                    .map(C1::BaseField::from_bigint)
-                    .collect::<Option<Vec<_>>>()
-                    .unwrap(),
+                    .map(<C1::BaseField as PrimeField>::BigInt::from_bits_le)
+                    .map(C1::BaseField::from)
+                    .collect::<Vec<_>>(),
                 get_cm_coordinates(&C1::zero()),
                 get_cm_coordinates(&self.U_i.phi),
-                get_cm_coordinates(&phi_stars[0]),
+                get_cm_coordinates(&aux.phi_stars[0]),
             ]
             .concat();
             let cf1_circuit = ProtoGalaxyCycleFoldCircuit::<C1, GC1> {
@@ -941,11 +945,10 @@ where
             let cf2_u_i_x = [
                 r1_bits
                     .chunks(C1::BaseField::MODULUS_BIT_SIZE as usize - 1)
-                    .map(BigInteger::from_bits_le)
-                    .map(C1::BaseField::from_bigint)
-                    .collect::<Option<Vec<_>>>()
-                    .unwrap(),
-                get_cm_coordinates(&phi_stars[0]),
+                    .map(<C1::BaseField as PrimeField>::BigInt::from_bits_le)
+                    .map(C1::BaseField::from)
+                    .collect::<Vec<_>>(),
+                get_cm_coordinates(&aux.phi_stars[0]),
                 get_cm_coordinates(&self.u_i.phi),
                 get_cm_coordinates(&U_i1.phi),
             ]
@@ -953,7 +956,7 @@ where
             let cf2_circuit = ProtoGalaxyCycleFoldCircuit::<C1, GC1> {
                 _gc: PhantomData,
                 r_bits: Some(r1_bits),
-                points: Some(vec![phi_stars[0], self.u_i.phi]),
+                points: Some(vec![aux.phi_stars[0], self.u_i.phi]),
                 x: Some(cf2_u_i_x.clone()),
             };
 
@@ -998,9 +1001,9 @@ where
                 u_i_phi: self.u_i.phi,
                 U_i: self.U_i.clone(),
                 U_i1_phi: U_i1.phi,
-                F_coeffs: F_coeffs.clone(),
-                K_coeffs: K_coeffs.clone(),
-                phi_stars,
+                F_coeffs: proof.F_coeffs.clone(),
+                K_coeffs: proof.K_coeffs.clone(),
+                phi_stars: aux.phi_stars,
                 F: self.F.clone(),
                 x: Some(u_i1_x),
                 // cyclefold values
@@ -1020,8 +1023,7 @@ where
                         &mut transcript_verifier,
                         &self.U_i,
                         &[self.u_i.clone()],
-                        F_coeffs,
-                        K_coeffs
+                        proof
                     )?,
                     U_i1
                 );
@@ -1108,7 +1110,7 @@ where
         } = ivc_proof;
         let (pp, vp) = params;
 
-        let f_circuit = FC::new(fcircuit_params).unwrap();
+        let f_circuit = FC::new(fcircuit_params)?;
 
         Ok(Self {
             _gc1: PhantomData,
@@ -1195,8 +1197,6 @@ where
     <C1 as Group>::ScalarField: Absorb,
     <C2 as Group>::ScalarField: Absorb,
     C1: CurveGroup<BaseField = C2::ScalarField, ScalarField = C2::BaseField>,
-    for<'a> &'a GC1: GroupOpsBounds<'a, C1, GC1>,
-    for<'a> &'a GC2: GroupOpsBounds<'a, C2, GC2>,
 {
     // folds the given cyclefold circuit and its instances
     #[allow(clippy::type_complexity)]

folding-schemes/src/folding/protogalaxy/traits.rs

@@ -1,14 +1,26 @@
 use ark_crypto_primitives::sponge::{constraints::AbsorbGadget, Absorb};
 use ark_ec::CurveGroup;
 use ark_ff::PrimeField;
-use ark_r1cs_std::{fields::fp::FpVar, uint8::UInt8, ToConstraintFieldGadget};
+use ark_r1cs_std::{
+    eq::EqGadget,
+    fields::{fp::FpVar, FieldVar},
+    uint8::UInt8,
+    ToConstraintFieldGadget,
+};
 use ark_relations::r1cs::SynthesisError;
 use ark_std::{cfg_into_iter, log2, One};
 use rayon::prelude::*;
 
-use super::{constants::RUNNING, utils::pow_i, CommittedInstance, CommittedInstanceVar, Witness};
+use super::{
+    constants::RUNNING,
+    utils::{pow_i, pow_i_var},
+    CommittedInstance, CommittedInstanceVar, Witness, WitnessVar,
+};
 use crate::{
-    arith::{r1cs::R1CS, Arith},
+    arith::{
+        r1cs::{circuits::R1CSMatricesVar, R1CS},
+        Arith, ArithGadget,
+    },
     folding::circuits::CF1,
     transcript::AbsorbNonNative,
     utils::vec::is_zero_vec,
@@ -99,6 +111,37 @@ impl Arith>, CommittedInstance
     }
 }
 
+/// Unlike its native counterpart, we only need to support running instances in
+/// circuit, as the decider circuit only checks running instance satisfiability.
+impl<C: CurveGroup> ArithGadget<WitnessVar<CF1<C>>, CommittedInstanceVar<C, RUNNING>>
+    for R1CSMatricesVar<CF1<C>, FpVar<CF1<C>>>
+{
+    type Evaluation = (Vec<FpVar<CF1<C>>>, Vec<FpVar<CF1<C>>>);
+
+    fn eval_relation(
+        &self,
+        w: &WitnessVar<CF1<C>>,
+        u: &CommittedInstanceVar<C, RUNNING>,
+    ) -> Result<Self::Evaluation, SynthesisError> {
+        self.eval_at_z(&[&[FpVar::one()][..], &u.x, &w.W].concat())
+    }
+
+    fn enforce_evaluation(
+        _w: &WitnessVar<C::ScalarField>,
+        u: &CommittedInstanceVar<C, RUNNING>,
+        (AzBz, uCz): Self::Evaluation,
+    ) -> Result<(), SynthesisError> {
+        let mut e = vec![];
+        for (i, (l, r)) in AzBz.iter().zip(uCz).enumerate() {
+            e.push(pow_i_var(i, &u.betas) * (l - r));
+        }
+        // Call `sum` on a vector instead of computing the sum in the above loop
+        // to avoid stack overflow (the cause of this is similar to issue #80
+        // https://github.com/privacy-scaling-explorations/sonobe/issues/80)
+        e.iter().sum::<FpVar<_>>().enforce_equal(&u.e)
+    }
+}
+
 #[cfg(test)]
 pub mod tests {
     use super::*;

folding-schemes/src/folding/traits.rs

@@ -12,7 +12,7 @@ use crate::{transcript::Transcript, Error};
 
 use super::circuits::CF1;
 
-pub trait CommittedInstanceOps<C: CurveGroup> {
+pub trait CommittedInstanceOps<C: CurveGroup>: Inputize<CF1<C>, Self::Var> {
     /// The in-circuit representation of the committed instance.
     type Var: AllocVar<Self, CF1<C>> + CommittedInstanceVarOps<C>;
     /// `hash` implements the committed instance hash compatible with the
@@ -85,7 +85,11 @@ pub trait CommittedInstanceVarOps {
         sponge.absorb(&z_0)?;
         sponge.absorb(&z_i)?;
         sponge.absorb(&U_vec)?;
-        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
+        Ok((
+            // `unwrap` is safe because the sponge is guaranteed to return a single element
+            sponge.squeeze_field_elements(1)?.pop().unwrap(),
+            U_vec,
+        ))
     }
 
     /// Returns the commitments contained in the committed instance.
@@ -129,3 +133,16 @@ impl Dummy for Vec {
         vec![Default::default(); cfg]
     }
 }
+
+impl<T: Default> Dummy<()> for T {
+    fn dummy(_: ()) -> Self {
+        Default::default()
+    }
+}
+
+/// Converts a value `self` into a vector of field elements, ordered in the same
+/// way as how a variable of type `Var` would be represented in the circuit.
+/// This is useful for the verifier to compute the public inputs.
+pub trait Inputize<F, Var> {
+    fn inputize(&self) -> Vec<F>;
+}

folding-schemes/src/lib.rs

@@ -38,6 +38,8 @@ pub enum Error {
     // Relation errors
     #[error("Relation not satisfied")]
     NotSatisfied,
+    #[error("SNARK setup failed: {0}")]
+    SNARKSetupFail(String),
     #[error("SNARK verification failed")]
     SNARKVerificationFail,
     #[error("IVC verification failed")]
@@ -102,8 +104,8 @@ pub enum Error {
     MaxStep,
     #[error("Witness calculation error: {0}")]
     WitnessCalculationError(String),
-    #[error("BigInt to PrimeField conversion error: {0}")]
-    BigIntConversionError(String),
+    #[error("Failed to convert {0} into {1}: {2}")]
+    ConversionError(String, String, String),
     #[error("Failed to serde: {0}")]
     JSONSerdeError(String),
     #[error("Multi instances folding not supported in this scheme")]

folding-schemes/src/utils/gadgets.rs

@@ -1,14 +1,36 @@
 use ark_ff::PrimeField;
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
+    eq::EqGadget,
     fields::{fp::FpVar, FieldVar},
     R1CSVar,
 };
 use ark_relations::r1cs::{Namespace, SynthesisError};
-use core::{borrow::Borrow, marker::PhantomData};
+use core::borrow::Borrow;
 
 use crate::utils::vec::SparseMatrix;
 
+/// `EquivalenceGadget` enforces that two in-circuit variables are equivalent,
+/// where the equivalence relation is parameterized by `M`:
+/// - For `FpVar`, it is simply a equality relation, and `M` is unused.
+/// - For `NonNativeUintVar`, we consider equivalence as a congruence relation,
+///   in terms of modular arithmetic, so `M` specifies the modulus.
+pub trait EquivalenceGadget<M> {
+    fn enforce_equivalent(&self, other: &Self) -> Result<(), SynthesisError>;
+}
+impl<M, F: PrimeField> EquivalenceGadget<M> for FpVar<F> {
+    fn enforce_equivalent(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.enforce_equal(other)
+    }
+}
+impl<M, T: EquivalenceGadget<M>> EquivalenceGadget<M> for [T] {
+    fn enforce_equivalent(&self, other: &Self) -> Result<(), SynthesisError> {
+        self.iter()
+            .zip(other)
+            .try_for_each(|(a, b)| a.enforce_equivalent(b))
+    }
+}
+
 pub trait MatrixGadget<FV> {
     fn mul_vector(&self, v: &[FV]) -> Result<Vec<FV>, SynthesisError>;
 }
@@ -42,17 +64,14 @@ impl VectorGadget> for [FpVar] {
 }
 
 #[derive(Debug, Clone)]
-pub struct SparseMatrixVar<F: PrimeField, CF: PrimeField, FV: AllocVar<F, CF>> {
-    _f: PhantomData<F>,
-    _cf: PhantomData<CF>,
-    _fv: PhantomData<FV>,
+pub struct SparseMatrixVar<FV> {
     pub n_rows: usize,
     pub n_cols: usize,
     // same format as the native SparseMatrix (which follows ark_relations::r1cs::Matrix format
     pub coeffs: Vec<Vec<(FV, usize)>>,
 }
 
-impl<F, CF, FV> AllocVar<SparseMatrix<F>, CF> for SparseMatrixVar<F, CF, FV>
+impl<F, CF, FV> AllocVar<SparseMatrix<F>, CF> for SparseMatrixVar<FV>
 where
     F: PrimeField,
     CF: PrimeField,
@@ -77,9 +96,6 @@ where
             }
 
             Ok(Self {
-                _f: PhantomData,
-                _cf: PhantomData,
-                _fv: PhantomData,
                 n_rows: val.borrow().n_rows,
                 n_cols: val.borrow().n_cols,
                 coeffs,
@@ -88,7 +104,7 @@ where
     }
 }
 
-impl<F: PrimeField> MatrixGadget<FpVar<F>> for SparseMatrixVar<F, F, FpVar<F>> {
+impl<F: PrimeField> MatrixGadget<FpVar<F>> for SparseMatrixVar<FpVar<F>> {
     fn mul_vector(&self, v: &[FpVar<F>]) -> Result<Vec<FpVar<F>>, SynthesisError> {
         Ok(self
             .coeffs

frontends/src/circom/utils.rs

@@ -108,11 +108,19 @@ impl CircomWrapper {
 
     // Converts a num_bigint::BigInt to a PrimeField::BigInt.
     pub fn num_bigint_to_ark_bigint(&self, value: &BigInt) -> Result<F::BigInt, Error> {
-        let big_uint = value
-            .to_biguint()
-            .ok_or_else(|| Error::BigIntConversionError("BigInt is negative".to_string()))?;
+        let big_uint = value.to_biguint().ok_or_else(|| {
+            Error::ConversionError(
+                "BigInt".into(),
+                "BigUint".into(),
+                "BigInt is negative".into(),
+            )
+        })?;
         F::BigInt::try_from(big_uint).map_err(|_| {
-            Error::BigIntConversionError("Failed to convert to PrimeField::BigInt".to_string())
+            Error::ConversionError(
+                "BigUint".into(),
+                "PrimeField::BigInt".into(),
+                "BigUint is too large to fit into PrimeField::BigInt".into(),
+            )
         })
     }
 

frontends/src/noir/mod.rs

@@ -16,6 +16,7 @@ use ark_relations::r1cs::{ConstraintSystemRef, SynthesisError};
 use folding_schemes::{frontend::FCircuit, utils::PathOrBin, Error};
 use noir_arkworks_backend::{
     read_program_from_binary, read_program_from_file, sonobe_bridge::AcirCircuitSonobe,
+    FilesystemError,
 };
 
 #[derive(Clone, Debug)]
@@ -216,10 +217,12 @@ impl FCircuit for NoirFCircuit {
     }
 }
 
-pub fn load_noir_circuit<F: PrimeField>(path: String) -> Circuit<GenericFieldElement<F>> {
-    let program: Program<GenericFieldElement<F>> = read_program_from_file(path).unwrap();
+pub fn load_noir_circuit<F: PrimeField>(
+    path: String,
+) -> Result<Circuit<GenericFieldElement<F>>, FilesystemError> {
+    let program: Program<GenericFieldElement<F>> = read_program_from_file(path)?;
     let circuit: Circuit<GenericFieldElement<F>> = program.functions[0].clone();
-    circuit
+    Ok(circuit)
 }
 
 #[cfg(test)]
@@ -241,7 +244,7 @@ mod tests {
             "{}/src/noir/test_folder/test_circuit/target/test_circuit.json",
             cur_path.to_str().unwrap()
         );
-        let circuit = load_noir_circuit(circuit_path);
+        let circuit = load_noir_circuit(circuit_path).unwrap();
         let noirfcircuit = NoirFCircuit {
             circuit,
             state_len: 2,
@@ -261,7 +264,7 @@ mod tests {
             "{}/src/noir/test_folder/test_circuit/target/test_circuit.json",
             cur_path.to_str().unwrap()
         );
-        let circuit = load_noir_circuit(circuit_path);
+        let circuit = load_noir_circuit(circuit_path).unwrap();
         let noirfcircuit = NoirFCircuit {
             circuit,
             state_len: 2,
@@ -285,7 +288,7 @@ mod tests {
             "{}/src/noir/test_folder/test_no_external_inputs/target/test_no_external_inputs.json",
             cur_path.to_str().unwrap()
         );
-        let circuit = load_noir_circuit(circuit_path);
+        let circuit = load_noir_circuit(circuit_path).unwrap();
         let noirfcircuit = NoirFCircuit {
             circuit,
             state_len: 2,

solidity-verifiers/src/utils/mod.rs

@@ -22,7 +22,7 @@ pub fn get_function_selector_for_nova_cyclefold_verifier(
     first_param_array_length: usize,
 ) -> [u8; 4] {
     let mut hasher = Sha3::keccak256();
-    let fn_sig = format!("verifyNovaProof(uint256[{}],uint256[4],uint256[3],uint256[4],uint256[4],uint256[2],uint256[2][2],uint256[2],uint256[4],uint256[2][2])", first_param_array_length);
+    let fn_sig = format!("verifyNovaProof(uint256[{}],uint256[4],uint256[2],uint256[3],uint256[2],uint256[2][2],uint256[2],uint256[4],uint256[2][2])", first_param_array_length);
     hasher.input_str(&fn_sig);
     let hash = &mut [0u8; 32];
     hasher.result(hash);

solidity-verifiers/src/verifiers/nova_cyclefold.rs

@@ -153,9 +153,12 @@ mod tests {
 
     use folding_schemes::{
         commitment::{kzg::KZG, pedersen::Pedersen},
-        folding::nova::{
-            decider_eth::{prepare_calldata, Decider as DeciderEth},
-            Nova, PreprocessorParam,
+        folding::{
+            nova::{
+                decider_eth::{prepare_calldata, Decider as DeciderEth},
+                Nova, PreprocessorParam,
+            },
+            traits::CommittedInstanceOps,
         },
         frontend::FCircuit,
         transcript::poseidon::poseidon_canonical_config,
@@ -366,7 +369,6 @@ mod tests {
         n_steps: usize,
     ) {
         let (decider_pp, decider_vp) = decider_params;
-        let pp_hash = fs_params.1.pp_hash().unwrap();
 
         let f_circuit = FC::new(()).unwrap();
 
@@ -389,8 +391,8 @@ mod tests {
             nova.i,
             nova.z_0.clone(),
             nova.z_i.clone(),
-            &nova.U_i,
-            &nova.u_i,
+            &nova.U_i.get_commitments(),
+            &nova.u_i.get_commitments(),
             &proof,
         )
         .unwrap();
@@ -401,7 +403,6 @@ mod tests {
 
         let calldata: Vec<u8> = prepare_calldata(
             function_selector,
-            pp_hash,
             nova.i,
             nova.z_0,
             nova.z_i,

solidity-verifiers/templates/nova_cyclefold_decider.askama.sol

@@ -63,9 +63,8 @@ contract NovaDecider is Groth16Verifier, KZG10Verifier {
         // inputs are grouped to prevent errors due stack too deep
         uint256[{{ 1 + z_len * 2 }}] calldata i_z0_zi, // [i, z0, zi] where |z0| == |zi|
         uint256[4] calldata U_i_cmW_U_i_cmE, // [U_i_cmW[2], U_i_cmE[2]]
-        uint256[3] calldata U_i_u_u_i_u_r, // [U_i_u, u_i_u, r]
-        uint256[4] calldata U_i_x_u_i_cmW, // [U_i_x[2], u_i_cmW[2]]
-        uint256[4] calldata u_i_x_cmT, // [u_i_x[2], cmT[2]]
+        uint256[2] calldata u_i_cmW, // [u_i_cmW[2]]
+        uint256[3] calldata cmT_r, // [cmT[2], r]
         uint256[2] calldata pA, // groth16 
         uint256[2][2] calldata pB, // groth16
         uint256[2] calldata pC, // groth16
@@ -86,20 +85,26 @@ contract NovaDecider is Groth16Verifier, KZG10Verifier {
         }
 
         {
-            // U_i.u + r * u_i.u
-            uint256 u = rlc(U_i_u_u_i_u_r[0], U_i_u_u_i_u_r[2], U_i_u_u_i_u_r[1]);
-            // U_i.x + r * u_i.x
-            uint256 x0 = rlc(U_i_x_u_i_cmW[0], U_i_u_u_i_u_r[2], u_i_x_cmT[0]);
-            uint256 x1 = rlc(U_i_x_u_i_cmW[1], U_i_u_u_i_u_r[2], u_i_x_cmT[1]);
-
-            public_inputs[{{ z_len * 2 + 2 }}] = u;
-            public_inputs[{{ z_len * 2 + 3 }}] = x0;
-            public_inputs[{{ z_len * 2 + 4 }}] = x1;
+            // U_i.cmW + r * u_i.cmW
+            uint256[2] memory mulScalarPoint = super.mulScalar([u_i_cmW[0], u_i_cmW[1]], cmT_r[2]);
+            uint256[2] memory cmW = super.add([U_i_cmW_U_i_cmE[0], U_i_cmW_U_i_cmE[1]], mulScalarPoint);
+
+            {
+                uint256[{{num_limbs}}] memory cmW_x_limbs = LimbsDecomposition.decompose(cmW[0]);
+                uint256[{{num_limbs}}] memory cmW_y_limbs = LimbsDecomposition.decompose(cmW[1]);
+        
+                for (uint8 k = 0; k < {{num_limbs}}; k++) {
+                    public_inputs[{{ z_len * 2 + 2 }} + k] = cmW_x_limbs[k];
+                    public_inputs[{{ z_len * 2 + 2 + num_limbs }} + k] = cmW_y_limbs[k];
+                }
+            }
+        
+            require(this.check(cmW, kzg_proof[0], challenge_W_challenge_E_kzg_evals[0], challenge_W_challenge_E_kzg_evals[2]), "KZG: verifying proof for challenge W failed");
         }
 
         {
-            // U_i.cmE + r * u_i.cmT
-            uint256[2] memory mulScalarPoint = super.mulScalar([u_i_x_cmT[2], u_i_x_cmT[3]], U_i_u_u_i_u_r[2]);
+            // U_i.cmE + r * cmT
+            uint256[2] memory mulScalarPoint = super.mulScalar([cmT_r[0], cmT_r[1]], cmT_r[2]);
             uint256[2] memory cmE = super.add([U_i_cmW_U_i_cmE[2], U_i_cmW_U_i_cmE[3]], mulScalarPoint);
 
             {
@@ -107,53 +112,35 @@ contract NovaDecider is Groth16Verifier, KZG10Verifier {
                 uint256[{{num_limbs}}] memory cmE_y_limbs = LimbsDecomposition.decompose(cmE[1]);
             
                 for (uint8 k = 0; k < {{num_limbs}}; k++) {
-                    public_inputs[{{ z_len * 2 + 5 }} + k] = cmE_x_limbs[k];
-                    public_inputs[{{ z_len * 2 + 5 + num_limbs }} + k] = cmE_y_limbs[k];
+                    public_inputs[{{ z_len * 2 + 2 + num_limbs * 2 }} + k] = cmE_x_limbs[k];
+                    public_inputs[{{ z_len * 2 + 2 + num_limbs * 3 }} + k] = cmE_y_limbs[k];
                 }
             }
 
             require(this.check(cmE, kzg_proof[1], challenge_W_challenge_E_kzg_evals[1], challenge_W_challenge_E_kzg_evals[3]), "KZG: verifying proof for challenge E failed");
         }
 
-        {
-            // U_i.cmW + r * u_i.cmW
-            uint256[2] memory mulScalarPoint = super.mulScalar([U_i_x_u_i_cmW[2], U_i_x_u_i_cmW[3]], U_i_u_u_i_u_r[2]);
-            uint256[2] memory cmW = super.add([U_i_cmW_U_i_cmE[0], U_i_cmW_U_i_cmE[1]], mulScalarPoint);
-        
-            {
-                uint256[{{num_limbs}}] memory cmW_x_limbs = LimbsDecomposition.decompose(cmW[0]);
-                uint256[{{num_limbs}}] memory cmW_y_limbs = LimbsDecomposition.decompose(cmW[1]);
-        
-                for (uint8 k = 0; k < {{num_limbs}}; k++) {
-                    public_inputs[{{ z_len * 2 + 5 + num_limbs * 2 }} + k] = cmW_x_limbs[k];
-                    public_inputs[{{ z_len * 2 + 5 + num_limbs * 3 }} + k] = cmW_y_limbs[k];
-                }
-            }
-        
-            require(this.check(cmW, kzg_proof[0], challenge_W_challenge_E_kzg_evals[0], challenge_W_challenge_E_kzg_evals[2]), "KZG: verifying proof for challenge W failed");
-        }
-
         {
             // add challenges
-            public_inputs[{{ z_len * 2 + 5 + num_limbs * 4 }}] = challenge_W_challenge_E_kzg_evals[0];
-            public_inputs[{{ z_len * 2 + 5 + num_limbs * 4 + 1 }}] = challenge_W_challenge_E_kzg_evals[1];
-            public_inputs[{{ z_len * 2 + 5 + num_limbs * 4 + 2 }}] = challenge_W_challenge_E_kzg_evals[2];
-            public_inputs[{{ z_len * 2 + 5 + num_limbs * 4 + 3 }}] = challenge_W_challenge_E_kzg_evals[3];
-        
+            public_inputs[{{ z_len * 2 + 2 + num_limbs * 4 }}] = challenge_W_challenge_E_kzg_evals[0];
+            public_inputs[{{ z_len * 2 + 2 + num_limbs * 4 + 1 }}] = challenge_W_challenge_E_kzg_evals[1];
+            public_inputs[{{ z_len * 2 + 2 + num_limbs * 4 + 2 }}] = challenge_W_challenge_E_kzg_evals[2];
+            public_inputs[{{ z_len * 2 + 2 + num_limbs * 4 + 3 }}] = challenge_W_challenge_E_kzg_evals[3];
+
             uint256[{{num_limbs}}] memory cmT_x_limbs;
             uint256[{{num_limbs}}] memory cmT_y_limbs;
         
-            cmT_x_limbs = LimbsDecomposition.decompose(u_i_x_cmT[2]);
-            cmT_y_limbs = LimbsDecomposition.decompose(u_i_x_cmT[3]);
+            cmT_x_limbs = LimbsDecomposition.decompose(cmT_r[0]);
+            cmT_y_limbs = LimbsDecomposition.decompose(cmT_r[1]);
         
             for (uint8 k = 0; k < {{num_limbs}}; k++) {
-                public_inputs[{{ z_len * 2 + 5 + num_limbs * 4 }} + 4 + k] = cmT_x_limbs[k]; 
-                public_inputs[{{ z_len * 2 + 5 + num_limbs * 5}} + 4 + k] = cmT_y_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 }}] = U_i_u_u_i_u_r[2];
-            
+            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");
         }