diff --git a/folding-schemes/src/arith/ccs.rs b/folding-schemes/src/arith/ccs.rs
index 2c55fed..80b812f 100644
--- a/folding-schemes/src/arith/ccs.rs
+++ b/folding-schemes/src/arith/ccs.rs
@@ -114,7 +114,6 @@ impl<F: PrimeField> CCS<F> {
 pub mod tests {
     use super::*;
     use crate::arith::r1cs::tests::{get_test_r1cs, get_test_z as r1cs_get_test_z};
-    use ark_ff::PrimeField;
     use ark_pallas::Fr;
 
     pub fn get_test_ccs<F: PrimeField>() -> CCS<F> {
diff --git a/folding-schemes/src/arith/r1cs.rs b/folding-schemes/src/arith/r1cs.rs
index d66754f..ef0c5ba 100644
--- a/folding-schemes/src/arith/r1cs.rs
+++ b/folding-schemes/src/arith/r1cs.rs
@@ -140,7 +140,6 @@ pub mod tests {
     use super::*;
     use crate::utils::vec::tests::{to_F_matrix, to_F_vec};
 
-    use ark_ff::PrimeField;
     use ark_pallas::Fr;
 
     pub fn get_test_r1cs<F: PrimeField>() -> R1CS<F> {
diff --git a/folding-schemes/src/commitment/ipa.rs b/folding-schemes/src/commitment/ipa.rs
index 5645d43..4607834 100644
--- a/folding-schemes/src/commitment/ipa.rs
+++ b/folding-schemes/src/commitment/ipa.rs
@@ -97,7 +97,7 @@ impl<C: CurveGroup, const H: bool> CommitmentScheme<C, H> for IPA<C, H> {
 
     fn prove(
         params: &Self::ProverParams,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         P: &C,                // commitment
         a: &[C::ScalarField], // vector
         blind: &C::ScalarField,
@@ -131,7 +131,7 @@ impl<C: CurveGroup, const H: bool> CommitmentScheme<C, H> for IPA<C, H> {
             r = vec![];
         }
 
-        transcript.absorb_point(P)?;
+        transcript.absorb_nonnative(P);
         let x = transcript.get_challenge(); // challenge value at which we evaluate
         let s = transcript.get_challenge();
         let U = C::generator().mul(s);
@@ -162,8 +162,8 @@ impl<C: CurveGroup, const H: bool> CommitmentScheme<C, H> for IPA<C, H> {
                 R[j] = C::msm_unchecked(&G[..m], &a[m..]) + U.mul(inner_prod(&a[m..], &b[..m])?);
             }
             // get challenge for the j-th round
-            transcript.absorb_point(&L[j])?;
-            transcript.absorb_point(&R[j])?;
+            transcript.absorb_nonnative(&L[j]);
+            transcript.absorb_nonnative(&R[j]);
             u[j] = transcript.get_challenge();
 
             let uj = u[j];
@@ -225,21 +225,21 @@ impl<C: CurveGroup, const H: bool> CommitmentScheme<C, H> for IPA<C, H> {
 
     fn verify(
         params: &Self::VerifierParams,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         P: &C, // commitment
         proof: &Self::Proof,
     ) -> Result<(), Error> {
         let (p, _r) = (proof.0.clone(), proof.1);
         let k = p.L.len();
 
-        transcript.absorb_point(P)?;
+        transcript.absorb_nonnative(P);
         let x = transcript.get_challenge(); // challenge value at which we evaluate
         let s = transcript.get_challenge();
         let U = C::generator().mul(s);
         let mut u: Vec<C::ScalarField> = vec![C::ScalarField::zero(); k];
         for i in (0..k).rev() {
-            transcript.absorb_point(&p.L[i])?;
-            transcript.absorb_point(&p.R[i])?;
+            transcript.absorb_nonnative(&p.L[i]);
+            transcript.absorb_nonnative(&p.R[i]);
             u[i] = transcript.get_challenge();
         }
         let challenge = (x, U, u);
@@ -566,15 +566,15 @@ where
 
 #[cfg(test)]
 mod tests {
+    use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, CryptographicSponge};
     use ark_ec::Group;
     use ark_pallas::{constraints::GVar, Fq, Fr, Projective};
-    use ark_r1cs_std::{alloc::AllocVar, bits::boolean::Boolean, eq::EqGadget};
+    use ark_r1cs_std::eq::EqGadget;
     use ark_relations::r1cs::ConstraintSystem;
-    use ark_std::UniformRand;
     use std::ops::Mul;
 
     use super::*;
-    use crate::transcript::poseidon::{poseidon_canonical_config, PoseidonTranscript};
+    use crate::transcript::poseidon::poseidon_canonical_config;
 
     #[test]
     fn test_ipa() {
@@ -592,9 +592,9 @@ mod tests {
 
         let poseidon_config = poseidon_canonical_config::<Fr>();
         // init Prover's transcript
-        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p = PoseidonSponge::<Fr>::new(&poseidon_config);
         // init Verifier's transcript
-        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         // a is the vector that we're committing
         let a: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(&mut rng))
@@ -636,9 +636,9 @@ mod tests {
 
         let poseidon_config = poseidon_canonical_config::<Fr>();
         // init Prover's transcript
-        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p = PoseidonSponge::<Fr>::new(&poseidon_config);
         // init Verifier's transcript
-        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         let mut a: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(&mut rng))
             .take(d / 2)
@@ -666,15 +666,15 @@ mod tests {
         // circuit
         let cs = ConstraintSystem::<Fq>::new_ref();
 
-        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
-        transcript_v.absorb_point(&cm).unwrap();
+        let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
+        transcript_v.absorb_nonnative(&cm);
         let challenge = transcript_v.get_challenge(); // challenge value at which we evaluate
         let s = transcript_v.get_challenge();
         let U = Projective::generator().mul(s);
         let mut u: Vec<Fr> = vec![Fr::zero(); k];
         for i in (0..k).rev() {
-            transcript_v.absorb_point(&proof.0.L[i]).unwrap();
-            transcript_v.absorb_point(&proof.0.R[i]).unwrap();
+            transcript_v.absorb_nonnative(&proof.0.L[i]);
+            transcript_v.absorb_nonnative(&proof.0.R[i]);
             u[i] = transcript_v.get_challenge();
         }
 
diff --git a/folding-schemes/src/commitment/kzg.rs b/folding-schemes/src/commitment/kzg.rs
index 47a4cbd..d034428 100644
--- a/folding-schemes/src/commitment/kzg.rs
+++ b/folding-schemes/src/commitment/kzg.rs
@@ -156,13 +156,13 @@ where
     /// the Pairing trait.
     fn prove(
         params: &Self::ProverParams,
-        transcript: &mut impl Transcript<E::G1>,
+        transcript: &mut impl Transcript<E::ScalarField>,
         cm: &E::G1,
         v: &[E::ScalarField],
         _blind: &E::ScalarField,
         _rng: Option<&mut dyn RngCore>,
     ) -> Result<Self::Proof, Error> {
-        transcript.absorb_point(cm)?;
+        transcript.absorb_nonnative(cm);
         let challenge = transcript.get_challenge();
         Self::prove_with_challenge(params, challenge, v, _blind, _rng)
     }
@@ -214,11 +214,11 @@ where
 
     fn verify(
         params: &Self::VerifierParams,
-        transcript: &mut impl Transcript<E::G1>,
+        transcript: &mut impl Transcript<E::ScalarField>,
         cm: &E::G1,
         proof: &Self::Proof,
     ) -> Result<(), Error> {
-        transcript.absorb_point(cm)?;
+        transcript.absorb_nonnative(cm);
         let challenge = transcript.get_challenge();
         Self::verify_with_challenge(params, challenge, cm, proof)
     }
@@ -286,17 +286,18 @@ fn convert_to_bigints<F: PrimeField>(p: &[F]) -> Vec<F::BigInt> {
 #[cfg(test)]
 mod tests {
     use ark_bn254::{Bn254, Fr, G1Projective as G1};
+    use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, CryptographicSponge};
     use ark_std::{test_rng, UniformRand};
 
     use super::*;
-    use crate::transcript::poseidon::{poseidon_canonical_config, PoseidonTranscript};
+    use crate::transcript::poseidon::poseidon_canonical_config;
 
     #[test]
     fn test_kzg_commitment_scheme() {
         let mut rng = &mut test_rng();
         let poseidon_config = poseidon_canonical_config::<Fr>();
-        let transcript_p = &mut PoseidonTranscript::<G1>::new(&poseidon_config);
-        let transcript_v = &mut PoseidonTranscript::<G1>::new(&poseidon_config);
+        let transcript_p = &mut PoseidonSponge::<Fr>::new(&poseidon_config);
+        let transcript_v = &mut PoseidonSponge::<Fr>::new(&poseidon_config);
 
         let n = 10;
         let (pk, vk): (ProverKey<G1>, VerifierKey<Bn254>) =
diff --git a/folding-schemes/src/commitment/mod.rs b/folding-schemes/src/commitment/mod.rs
index 1c64457..58ecadd 100644
--- a/folding-schemes/src/commitment/mod.rs
+++ b/folding-schemes/src/commitment/mod.rs
@@ -34,7 +34,7 @@ pub trait CommitmentScheme<C: CurveGroup, const H: bool = false>: Clone + Debug
 
     fn prove(
         params: &Self::ProverParams,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         cm: &C,
         v: &[C::ScalarField],
         blind: &C::ScalarField,
@@ -53,7 +53,7 @@ pub trait CommitmentScheme<C: CurveGroup, const H: bool = false>: Clone + Debug
 
     fn verify(
         params: &Self::VerifierParams,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         cm: &C,
         proof: &Self::Proof,
     ) -> Result<(), Error>;
@@ -72,7 +72,10 @@ pub trait CommitmentScheme<C: CurveGroup, const H: bool = false>: Clone + Debug
 mod tests {
     use super::*;
     use ark_bn254::{Bn254, Fr, G1Projective as G1};
-    use ark_crypto_primitives::sponge::{poseidon::PoseidonConfig, Absorb};
+    use ark_crypto_primitives::sponge::{
+        poseidon::{PoseidonConfig, PoseidonSponge},
+        Absorb, CryptographicSponge,
+    };
     use ark_poly_commit::kzg10::VerifierKey;
     use ark_std::Zero;
     use ark_std::{test_rng, UniformRand};
@@ -80,10 +83,7 @@ mod tests {
     use super::ipa::IPA;
     use super::kzg::{ProverKey, KZG};
     use super::pedersen::Pedersen;
-    use crate::transcript::{
-        poseidon::{poseidon_canonical_config, PoseidonTranscript},
-        Transcript,
-    };
+    use crate::transcript::poseidon::poseidon_canonical_config;
 
     #[test]
     fn test_homomorphic_property_using_Commitment_trait() {
@@ -153,7 +153,7 @@ mod tests {
         let v_3: Vec<C::ScalarField> = v_1.iter().zip(v_2).map(|(a, b)| *a + (r * b)).collect();
 
         // compute the proof of the cm_3
-        let transcript_p = &mut PoseidonTranscript::<C>::new(poseidon_config);
+        let transcript_p = &mut PoseidonSponge::<C::ScalarField>::new(poseidon_config);
         let proof = CS::prove(
             prover_params,
             transcript_p,
@@ -165,7 +165,7 @@ mod tests {
         .unwrap();
 
         // verify the opening proof
-        let transcript_v = &mut PoseidonTranscript::<C>::new(poseidon_config);
+        let transcript_v = &mut PoseidonSponge::<C::ScalarField>::new(poseidon_config);
         CS::verify(verifier_params, transcript_v, &cm_3, &proof).unwrap();
     }
 }
diff --git a/folding-schemes/src/commitment/pedersen.rs b/folding-schemes/src/commitment/pedersen.rs
index a906f21..4d225ce 100644
--- a/folding-schemes/src/commitment/pedersen.rs
+++ b/folding-schemes/src/commitment/pedersen.rs
@@ -81,13 +81,13 @@ impl<C: CurveGroup, const H: bool> CommitmentScheme<C, H> for Pedersen<C, H> {
 
     fn prove(
         params: &Self::ProverParams,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         cm: &C,
         v: &[C::ScalarField],
         r: &C::ScalarField, // blinding factor
         _rng: Option<&mut dyn RngCore>,
     ) -> Result<Self::Proof, Error> {
-        transcript.absorb_point(cm)?;
+        transcript.absorb_nonnative(cm);
         let r1 = transcript.get_challenge();
         let d = transcript.get_challenges(v.len());
 
@@ -98,7 +98,7 @@ impl<C: CurveGroup, const H: bool> CommitmentScheme<C, H> for Pedersen<C, H> {
             R += params.h.mul(r1);
         }
 
-        transcript.absorb_point(&R)?;
+        transcript.absorb_nonnative(&R);
         let e = transcript.get_challenge();
 
         let challenge = (r1, d, R, e);
@@ -133,14 +133,14 @@ impl<C: CurveGroup, const H: bool> CommitmentScheme<C, H> for Pedersen<C, H> {
 
     fn verify(
         params: &Self::VerifierParams,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         cm: &C,
         proof: &Proof<C>,
     ) -> Result<(), Error> {
-        transcript.absorb_point(cm)?;
+        transcript.absorb_nonnative(cm);
         transcript.get_challenge(); // r_1
         transcript.get_challenges(proof.u.len()); // d
-        transcript.absorb_point(&proof.R)?;
+        transcript.absorb_nonnative(&proof.R);
         let e = transcript.get_challenge();
         Self::verify_with_challenge(params, e, cm, proof)
     }
@@ -217,14 +217,14 @@ where
 
 #[cfg(test)]
 mod tests {
+    use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, CryptographicSponge};
     use ark_ff::{BigInteger, PrimeField};
     use ark_pallas::{constraints::GVar, Fq, Fr, Projective};
     use ark_r1cs_std::{alloc::AllocVar, eq::EqGadget};
     use ark_relations::r1cs::ConstraintSystem;
-    use ark_std::UniformRand;
 
     use super::*;
-    use crate::transcript::poseidon::{poseidon_canonical_config, PoseidonTranscript};
+    use crate::transcript::poseidon::poseidon_canonical_config;
 
     #[test]
     fn test_pedersen() {
@@ -240,9 +240,9 @@ mod tests {
         let poseidon_config = poseidon_canonical_config::<Fr>();
 
         // init Prover's transcript
-        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p = PoseidonSponge::<Fr>::new(&poseidon_config);
         // init Verifier's transcript
-        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         let v: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(&mut rng))
             .take(n)
diff --git a/folding-schemes/src/folding/circuits/cyclefold.rs b/folding-schemes/src/folding/circuits/cyclefold.rs
index 3d0e5f8..522971e 100644
--- a/folding-schemes/src/folding/circuits/cyclefold.rs
+++ b/folding-schemes/src/folding/circuits/cyclefold.rs
@@ -1,23 +1,13 @@
 /// Contains [CycleFold](https://eprint.iacr.org/2023/1192.pdf) related circuits and functions that
 /// are shared across the different folding schemes
-use ark_crypto_primitives::{
-    crh::{
-        poseidon::constraints::{CRHGadget, CRHParametersVar},
-        CRHSchemeGadget,
-    },
-    sponge::{
-        constraints::CryptographicSpongeVar,
-        poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
-        Absorb, CryptographicSponge,
-    },
-};
-use ark_ec::{AffineRepr, CurveGroup, Group};
-use ark_ff::{BigInteger, Field, PrimeField, ToConstraintField};
+use ark_crypto_primitives::sponge::{Absorb, CryptographicSponge};
+use ark_ec::{CurveGroup, Group};
+use ark_ff::{BigInteger, PrimeField};
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     boolean::Boolean,
     eq::EqGadget,
-    fields::{fp::FpVar, FieldVar},
+    fields::fp::FpVar,
     groups::GroupOpsBounds,
     prelude::CurveVar,
     ToConstraintFieldGadget,
@@ -26,7 +16,7 @@ use ark_relations::r1cs::{
     ConstraintSynthesizer, ConstraintSystem, ConstraintSystemRef, Namespace, SynthesisError,
 };
 use ark_std::fmt::Debug;
-use ark_std::{One, Zero};
+use ark_std::Zero;
 use core::{borrow::Borrow, marker::PhantomData};
 
 use super::{nonnative::uint::NonNativeUintVar, CF2};
@@ -35,6 +25,7 @@ use crate::commitment::CommitmentScheme;
 use crate::constants::N_BITS_RO;
 use crate::folding::nova::{nifs::NIFS, CommittedInstance, Witness};
 use crate::frontend::FCircuit;
+use crate::transcript::{AbsorbNonNativeGadget, Transcript, TranscriptVar};
 use crate::Error;
 
 /// Public inputs length for the CycleFoldCircuit: |[r, p1.x,y, p2.x,y, p3.x,y]|
@@ -77,7 +68,7 @@ where
     }
 }
 
-impl<C, GC> ToConstraintFieldGadget<CF2<C>> for CycleFoldCommittedInstanceVar<C, GC>
+impl<C, GC> AbsorbNonNativeGadget<C::BaseField> for CycleFoldCommittedInstanceVar<C, GC>
 where
     C: CurveGroup,
     GC: CurveVar<C, CF2<C>> + ToConstraintFieldGadget<CF2<C>>,
@@ -87,25 +78,24 @@ where
     /// 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
     /// concat).
-    fn to_constraint_field(&self) -> Result<Vec<FpVar<CF2<C>>>, SynthesisError> {
+    fn to_native_sponge_field_elements(&self) -> Result<Vec<FpVar<CF2<C>>>, SynthesisError> {
         let mut cmE_elems = self.cmE.to_constraint_field()?;
         let mut cmW_elems = self.cmW.to_constraint_field()?;
 
-        let cmE_is_inf = cmE_elems.pop().unwrap();
-        let cmW_is_inf = cmW_elems.pop().unwrap();
-        // Concatenate `cmE_is_inf` and `cmW_is_inf` to save constraints for CRHGadget::evaluate
-        let is_inf = cmE_is_inf.double()? + cmW_is_inf;
+        // See `transcript/poseidon.rs: TranscriptVar::absorb_point` for details
+        // why the last element is unnecessary.
+        cmE_elems.pop();
+        cmW_elems.pop();
 
         Ok([
-            self.u.to_constraint_field()?,
+            self.u.to_native_sponge_field_elements()?,
             self.x
                 .iter()
-                .map(|i| i.to_constraint_field())
+                .map(|i| i.to_native_sponge_field_elements())
                 .collect::<Result<Vec<_>, _>>()?
                 .concat(),
             cmE_elems,
             cmW_elems,
-            vec![is_inf],
         ]
         .concat())
     }
@@ -124,16 +114,16 @@ where
     /// parameters, so they can be reused in other gadgets avoiding recalculating (reconstraining)
     /// them.
     #[allow(clippy::type_complexity)]
-    pub fn hash(
+    pub fn hash<S: CryptographicSponge, T: TranscriptVar<CF2<C>, S>>(
         self,
-        crh_params: &CRHParametersVar<CF2<C>>,
+        sponge: &T,
         pp_hash: FpVar<CF2<C>>, // public params hash
     ) -> Result<(FpVar<CF2<C>>, Vec<FpVar<CF2<C>>>), SynthesisError> {
-        let U_vec = self.to_constraint_field()?;
-        Ok((
-            CRHGadget::evaluate(crh_params, &[vec![pp_hash], U_vec.clone()].concat())?,
-            U_vec,
-        ))
+        let mut sponge = sponge.clone();
+        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))
     }
 }
 
@@ -254,65 +244,33 @@ where
     <C as CurveGroup>::BaseField: Absorb,
     for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
 {
-    pub fn get_challenge_native(
-        poseidon_config: &PoseidonConfig<C::BaseField>,
+    pub fn get_challenge_native<T: Transcript<C::BaseField>>(
+        transcript: &mut T,
         pp_hash: C::BaseField, // public params hash
         U_i: CommittedInstance<C>,
         u_i: CommittedInstance<C>,
         cmT: C,
-    ) -> Result<Vec<bool>, Error> {
-        let mut sponge = PoseidonSponge::<C::BaseField>::new(poseidon_config);
-
-        let mut U_vec = U_i.to_field_elements().unwrap();
-        let mut u_vec = u_i.to_field_elements().unwrap();
-        let (cmT_x, cmT_y, cmT_is_inf) = match cmT.into_affine().xy() {
-            Some((&x, &y)) => (x, y, C::BaseField::zero()),
-            None => (
-                C::BaseField::zero(),
-                C::BaseField::zero(),
-                C::BaseField::one(),
-            ),
-        };
-
-        let U_cm_is_inf = U_vec.pop().unwrap();
-        let u_cm_is_inf = u_vec.pop().unwrap();
-
-        // Concatenate `U_i.cmE_is_inf`, `U_i.cmW_is_inf`, `u_i.cmE_is_inf`, `u_i.cmW_is_inf`, `cmT_is_inf`
-        // to save constraints for sponge.squeeze_bits in the corresponding circuit
-        let is_inf = U_cm_is_inf * CF2::<C>::from(8u8) + u_cm_is_inf.double() + cmT_is_inf;
-
-        let input = [vec![pp_hash], U_vec, u_vec, vec![cmT_x, cmT_y, is_inf]].concat();
-        sponge.absorb(&input);
-        let bits = sponge.squeeze_bits(N_BITS_RO);
-        Ok(bits)
+    ) -> Vec<bool> {
+        transcript.absorb(&pp_hash);
+        transcript.absorb_nonnative(&U_i);
+        transcript.absorb_nonnative(&u_i);
+        transcript.absorb_point(&cmT);
+        transcript.squeeze_bits(N_BITS_RO)
     }
 
     // compatible with the native get_challenge_native
-    pub fn get_challenge_gadget(
-        cs: ConstraintSystemRef<C::BaseField>,
-        poseidon_config: &PoseidonConfig<C::BaseField>,
+    pub fn get_challenge_gadget<S: CryptographicSponge, T: TranscriptVar<C::BaseField, S>>(
+        transcript: &mut T,
         pp_hash: FpVar<C::BaseField>, // public params hash
-        mut U_i_vec: Vec<FpVar<C::BaseField>>,
+        U_i_vec: Vec<FpVar<C::BaseField>>,
         u_i: CycleFoldCommittedInstanceVar<C, GC>,
         cmT: GC,
     ) -> Result<Vec<Boolean<C::BaseField>>, SynthesisError> {
-        let mut sponge = PoseidonSpongeVar::<C::BaseField>::new(cs, poseidon_config);
-
-        let mut u_i_vec = u_i.to_constraint_field()?;
-        let mut cmT_vec = cmT.to_constraint_field()?;
-
-        let U_cm_is_inf = U_i_vec.pop().unwrap();
-        let u_cm_is_inf = u_i_vec.pop().unwrap();
-        let cmT_is_inf = cmT_vec.pop().unwrap();
-
-        // Concatenate `U_i.cmE_is_inf`, `U_i.cmW_is_inf`, `u_i.cmE_is_inf`, `u_i.cmW_is_inf`, `cmT_is_inf`
-        // to save constraints for sponge.squeeze_bits
-        let is_inf = U_cm_is_inf * CF2::<C>::from(8u8) + u_cm_is_inf.double()? + cmT_is_inf;
-
-        let input = [vec![pp_hash], U_i_vec, u_i_vec, cmT_vec, vec![is_inf]].concat();
-        sponge.absorb(&input)?;
-        let bits = sponge.squeeze_bits(N_BITS_RO)?;
-        Ok(bits)
+        transcript.absorb(&pp_hash)?;
+        transcript.absorb(&U_i_vec)?;
+        transcript.absorb_nonnative(&u_i)?;
+        transcript.absorb_point(&cmT)?;
+        transcript.squeeze_bits(N_BITS_RO)
     }
 }
 
@@ -383,7 +341,7 @@ where
 #[allow(clippy::type_complexity)]
 #[allow(clippy::too_many_arguments)]
 pub fn fold_cyclefold_circuit<C1, GC1, C2, GC2, FC, CS1, CS2>(
-    poseidon_config: &PoseidonConfig<C1::ScalarField>,
+    transcript: &mut impl Transcript<C1::ScalarField>,
     cf_r1cs: R1CS<C2::ScalarField>,
     cf_cs_params: CS2::ProverParams,
     pp_hash: C1::ScalarField,      // public params hash
@@ -445,12 +403,12 @@ where
     )?;
 
     let cf_r_bits = CycleFoldChallengeGadget::<C2, GC2>::get_challenge_native(
-        poseidon_config,
+        transcript,
         pp_hash,
         cf_U_i.clone(),
         cf_u_i.clone(),
         cf_cmT,
-    )?;
+    );
     let cf_r_Fq = C1::BaseField::from_bigint(BigInteger::from_bits_le(&cf_r_bits))
         .expect("cf_r_bits out of bounds");
 
@@ -463,9 +421,11 @@ where
 #[cfg(test)]
 pub mod tests {
     use ark_bn254::{constraints::GVar, Fq, Fr, G1Projective as Projective};
-    use ark_ff::BigInteger;
+    use ark_crypto_primitives::sponge::{
+        constraints::CryptographicSpongeVar,
+        poseidon::{constraints::PoseidonSpongeVar, PoseidonSponge},
+    };
     use ark_r1cs_std::R1CSVar;
-    use ark_relations::r1cs::ConstraintSystem;
     use ark_std::UniformRand;
 
     use super::*;
@@ -583,6 +543,7 @@ pub mod tests {
     fn test_cyclefold_challenge_gadget() {
         let mut rng = ark_std::test_rng();
         let poseidon_config = poseidon_canonical_config::<Fq>();
+        let mut transcript = PoseidonSponge::<Fq>::new(&poseidon_config);
 
         let u_i = CommittedInstance::<Projective> {
             cmE: Projective::zero(), // zero on purpose, so we test also the zero point case
@@ -605,13 +566,12 @@ pub mod tests {
         // compute the challenge natively
         let pp_hash = Fq::from(42u32); // only for test
         let r_bits = CycleFoldChallengeGadget::<Projective, GVar>::get_challenge_native(
-            &poseidon_config,
+            &mut transcript,
             pp_hash,
             U_i.clone(),
             u_i.clone(),
             cmT,
-        )
-        .unwrap();
+        );
 
         let cs = ConstraintSystem::<Fq>::new_ref();
         let u_iVar =
@@ -625,13 +585,14 @@ pub mod tests {
             })
             .unwrap();
         let cmTVar = GVar::new_witness(cs.clone(), || Ok(cmT)).unwrap();
+        let mut transcript_var =
+            PoseidonSpongeVar::<Fq>::new(ConstraintSystem::<Fq>::new_ref(), &poseidon_config);
 
         let pp_hashVar = FpVar::<Fq>::new_witness(cs.clone(), || Ok(pp_hash)).unwrap();
         let r_bitsVar = CycleFoldChallengeGadget::<Projective, GVar>::get_challenge_gadget(
-            cs.clone(),
-            &poseidon_config,
+            &mut transcript_var,
             pp_hashVar,
-            U_iVar.to_constraint_field().unwrap(),
+            U_iVar.to_native_sponge_field_elements().unwrap(),
             u_iVar,
             cmTVar,
         )
@@ -649,6 +610,7 @@ pub mod tests {
     fn test_cyclefold_hash_gadget() {
         let mut rng = ark_std::test_rng();
         let poseidon_config = poseidon_canonical_config::<Fq>();
+        let sponge = PoseidonSponge::<Fq>::new(&poseidon_config);
 
         let U_i = CommittedInstance::<Projective> {
             cmE: Projective::rand(&mut rng),
@@ -659,7 +621,7 @@ pub mod tests {
                 .collect(),
         };
         let pp_hash = Fq::from(42u32); // only for test
-        let h = U_i.hash_cyclefold(&poseidon_config, pp_hash).unwrap();
+        let h = U_i.hash_cyclefold(&sponge, pp_hash);
 
         let cs = ConstraintSystem::<Fq>::new_ref();
         let U_iVar =
@@ -670,7 +632,7 @@ pub mod tests {
         let pp_hashVar = FpVar::<Fq>::new_witness(cs.clone(), || Ok(pp_hash)).unwrap();
         let (hVar, _) = U_iVar
             .hash(
-                &CRHParametersVar::new_constant(cs.clone(), poseidon_config).unwrap(),
+                &PoseidonSpongeVar::new(cs.clone(), &poseidon_config),
                 pp_hashVar,
             )
             .unwrap();
diff --git a/folding-schemes/src/folding/circuits/nonnative/affine.rs b/folding-schemes/src/folding/circuits/nonnative/affine.rs
index 05ee6b1..f4ec661 100644
--- a/folding-schemes/src/folding/circuits/nonnative/affine.rs
+++ b/folding-schemes/src/folding/circuits/nonnative/affine.rs
@@ -1,5 +1,4 @@
 use ark_ec::{AffineRepr, CurveGroup};
-use ark_ff::PrimeField;
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     fields::fp::FpVar,
@@ -9,16 +8,15 @@ use ark_relations::r1cs::{Namespace, SynthesisError};
 use ark_std::Zero;
 use core::borrow::Borrow;
 
+use crate::transcript::{AbsorbNonNative, AbsorbNonNativeGadget};
+
 use super::uint::{nonnative_field_to_field_elements, NonNativeUintVar};
 
 /// NonNativeAffineVar represents an elliptic curve point in Affine representation in the non-native
 /// field, over the constraint field. It is not intended to perform operations, but just to contain
 /// the affine coordinates in order to perform hash operations of the point.
 #[derive(Debug, Clone)]
-pub struct NonNativeAffineVar<C: CurveGroup>
-where
-    <C as CurveGroup>::BaseField: PrimeField,
-{
+pub struct NonNativeAffineVar<C: CurveGroup> {
     pub x: NonNativeUintVar<C::ScalarField>,
     pub y: NonNativeUintVar<C::ScalarField>,
 }
@@ -26,7 +24,6 @@ where
 impl<C> AllocVar<C, C::ScalarField> for NonNativeAffineVar<C>
 where
     C: CurveGroup,
-    <C as CurveGroup>::BaseField: PrimeField,
 {
     fn new_variable<T: Borrow<C>>(
         cs: impl Into<Namespace<C::ScalarField>>,
@@ -37,21 +34,18 @@ where
             let cs = cs.into();
 
             let affine = val.borrow().into_affine();
-            let zero_point = (&C::BaseField::zero(), &C::BaseField::zero());
-            let xy = affine.xy().unwrap_or(zero_point);
+            let zero = (&C::BaseField::zero(), &C::BaseField::zero());
+            let (x, y) = affine.xy().unwrap_or(zero);
 
-            let x = NonNativeUintVar::new_variable(cs.clone(), || Ok(*xy.0), mode)?;
-            let y = NonNativeUintVar::new_variable(cs.clone(), || Ok(*xy.1), mode)?;
+            let x = NonNativeUintVar::new_variable(cs.clone(), || Ok(*x), mode)?;
+            let y = NonNativeUintVar::new_variable(cs.clone(), || Ok(*y), mode)?;
 
             Ok(Self { x, y })
         })
     }
 }
 
-impl<C: CurveGroup> ToConstraintFieldGadget<C::ScalarField> for NonNativeAffineVar<C>
-where
-    <C as CurveGroup>::BaseField: PrimeField,
-{
+impl<C: CurveGroup> ToConstraintFieldGadget<C::ScalarField> for NonNativeAffineVar<C> {
     // Used for converting `NonNativeAffineVar` to a vector of `FpVar` with minimum length in
     // the circuit.
     fn to_constraint_field(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
@@ -63,50 +57,50 @@ where
 
 /// The out-circuit counterpart of `NonNativeAffineVar::to_constraint_field`
 #[allow(clippy::type_complexity)]
-pub fn nonnative_affine_to_field_elements<C: CurveGroup>(
+fn nonnative_affine_to_field_elements<C: CurveGroup>(
     p: C,
-) -> Result<(Vec<C::ScalarField>, Vec<C::ScalarField>), SynthesisError>
-where
-    <C as CurveGroup>::BaseField: PrimeField,
-{
+) -> (Vec<C::ScalarField>, Vec<C::ScalarField>) {
     let affine = p.into_affine();
-    if affine.is_zero() {
-        let x = nonnative_field_to_field_elements(&C::BaseField::zero());
-        let y = nonnative_field_to_field_elements(&C::BaseField::zero());
-        return Ok((x, y));
-    }
+    let zero = (&C::BaseField::zero(), &C::BaseField::zero());
+    let (x, y) = affine.xy().unwrap_or(zero);
 
-    let (x, y) = affine.xy().unwrap();
     let x = nonnative_field_to_field_elements(x);
     let y = nonnative_field_to_field_elements(y);
-    Ok((x, y))
+    (x, y)
 }
 
-impl<C: CurveGroup> NonNativeAffineVar<C>
-where
-    <C as CurveGroup>::BaseField: PrimeField,
-{
-    // A wrapper of `point_to_nonnative_limbs_custom_opt` with constraints-focused optimization
-    // type (which is the default optimization type for arkworks' Groth16).
-    // Used for extracting a list of field elements of type `C::ScalarField` from the public input
+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();
-        if affine.is_zero() {
-            let x = NonNativeUintVar::inputize(C::BaseField::zero());
-            let y = NonNativeUintVar::inputize(C::BaseField::zero());
-            return Ok((x, y));
-        }
+        let zero = (&C::BaseField::zero(), &C::BaseField::zero());
+        let (x, y) = affine.xy().unwrap_or(zero);
 
-        let (x, y) = affine.xy().unwrap();
         let x = NonNativeUintVar::inputize(*x);
         let y = NonNativeUintVar::inputize(*y);
         Ok((x, y))
     }
 }
 
+impl<C: CurveGroup> AbsorbNonNative<C::ScalarField> for C {
+    fn to_native_sponge_field_elements(&self, dest: &mut Vec<C::ScalarField>) {
+        let (x, y) = nonnative_affine_to_field_elements(*self);
+        dest.extend(x);
+        dest.extend(y);
+    }
+}
+
+impl<C: CurveGroup> AbsorbNonNativeGadget<C::ScalarField> for NonNativeAffineVar<C> {
+    fn to_native_sponge_field_elements(
+        &self,
+    ) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
+        self.to_constraint_field()
+    }
+}
+
 #[cfg(test)]
 mod tests {
     use super::*;
@@ -132,7 +126,7 @@ 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) = nonnative_affine_to_field_elements(p).unwrap();
+        let (x, y) = nonnative_affine_to_field_elements(p);
         assert_eq!(
             pVar.to_constraint_field().unwrap().value().unwrap(),
             [x, y].concat()
diff --git a/folding-schemes/src/folding/circuits/nonnative/uint.rs b/folding-schemes/src/folding/circuits/nonnative/uint.rs
index 151ed2c..2b32177 100644
--- a/folding-schemes/src/folding/circuits/nonnative/uint.rs
+++ b/folding-schemes/src/folding/circuits/nonnative/uint.rs
@@ -3,7 +3,7 @@ use std::{
     cmp::{max, min},
 };
 
-use ark_ff::{BigInteger, One, PrimeField, Zero};
+use ark_ff::{BigInteger, Field, One, PrimeField, Zero};
 use ark_r1cs_std::{
     alloc::{AllocVar, AllocationMode},
     boolean::Boolean,
@@ -16,7 +16,10 @@ use ark_relations::r1cs::{ConstraintSystemRef, Namespace, SynthesisError};
 use num_bigint::BigUint;
 use num_integer::Integer;
 
-use crate::utils::gadgets::{MatrixGadget, SparseMatrixVar, VectorGadget};
+use crate::{
+    transcript::{AbsorbNonNative, AbsorbNonNativeGadget},
+    utils::gadgets::{MatrixGadget, SparseMatrixVar, VectorGadget},
+};
 
 /// `LimbVar` represents a single limb of a non-native unsigned integer in the
 /// circuit.
@@ -229,7 +232,7 @@ impl<F: PrimeField> AllocVar<BoundedBigUint, F> for NonNativeUintVar<F> {
     }
 }
 
-impl<F: PrimeField, G: PrimeField> AllocVar<G, F> for NonNativeUintVar<F> {
+impl<F: PrimeField, G: Field> AllocVar<G, F> for NonNativeUintVar<F> {
     fn new_variable<T: Borrow<G>>(
         cs: impl Into<Namespace<F>>,
         f: impl FnOnce() -> Result<T, SynthesisError>,
@@ -237,7 +240,8 @@ impl<F: PrimeField, G: PrimeField> AllocVar<G, F> for NonNativeUintVar<F> {
     ) -> Result<Self, SynthesisError> {
         let cs = cs.into().cs();
         let v = f()?;
-        let v = v.borrow();
+        assert_eq!(G::extension_degree(), 1);
+        let v = v.borrow().to_base_prime_field_elements().next().unwrap();
 
         let mut limbs = vec![];
 
@@ -256,8 +260,12 @@ impl<F: PrimeField, G: PrimeField> AllocVar<G, F> for NonNativeUintVar<F> {
 }
 
 impl<F: PrimeField> NonNativeUintVar<F> {
-    pub fn inputize<T: PrimeField>(x: T) -> Vec<F> {
-        x.into_bigint()
+    pub fn inputize<T: Field>(x: T) -> Vec<F> {
+        assert_eq!(T::extension_degree(), 1);
+        x.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())
@@ -802,14 +810,40 @@ impl<F: PrimeField, B: AsRef<[Boolean<F>]>> From<B> for NonNativeUintVar<F> {
     }
 }
 
+// If we impl `AbsorbNonNative` directly for `PrimeField`, rustc will complain
+// that this impl conflicts with the impl for `CurveGroup`.
+// Therefore, we instead impl `AbsorbNonNative` for a slice of `PrimeField` as a
+// workaround.
+impl<TargetField: PrimeField, BaseField: PrimeField> AbsorbNonNative<BaseField>
+    for [TargetField]
+{
+    fn to_native_sponge_field_elements(&self, dest: &mut Vec<BaseField>) {
+        self.iter()
+            .for_each(|x| dest.extend(&nonnative_field_to_field_elements(x)));
+    }
+}
+
+impl<F: PrimeField> AbsorbNonNativeGadget<F> for NonNativeUintVar<F> {
+    fn to_native_sponge_field_elements(&self) -> Result<Vec<FpVar<F>>, SynthesisError> {
+        self.to_constraint_field()
+    }
+}
+
 /// The out-circuit counterpart of `NonNativeUintVar::to_constraint_field`
-pub fn nonnative_field_to_field_elements<TargetField: PrimeField, BaseField: PrimeField>(
+pub(super) fn nonnative_field_to_field_elements<TargetField: Field, BaseField: PrimeField>(
     f: &TargetField,
 ) -> Vec<BaseField> {
-    let bits = f.into_bigint().to_bits_le();
+    assert_eq!(TargetField::extension_degree(), 1);
+    let bits = f
+        .to_base_prime_field_elements()
+        .next()
+        .unwrap()
+        .into_bigint()
+        .to_bits_le();
 
     let bits_per_limb = BaseField::MODULUS_BIT_SIZE as usize - 1;
-    let num_limbs = (TargetField::MODULUS_BIT_SIZE as usize).div_ceil(bits_per_limb);
+    let num_limbs =
+        (TargetField::BasePrimeField::MODULUS_BIT_SIZE as usize).div_ceil(bits_per_limb);
 
     let mut limbs = bits
         .chunks(bits_per_limb)
@@ -897,7 +931,6 @@ mod tests {
     use std::error::Error;
 
     use super::*;
-    use ark_ff::Field;
     use ark_pallas::{Fq, Fr};
     use ark_relations::r1cs::ConstraintSystem;
     use ark_std::{test_rng, UniformRand};
diff --git a/folding-schemes/src/folding/circuits/sum_check.rs b/folding-schemes/src/folding/circuits/sum_check.rs
index f06b4f9..75c7b43 100644
--- a/folding-schemes/src/folding/circuits/sum_check.rs
+++ b/folding-schemes/src/folding/circuits/sum_check.rs
@@ -3,8 +3,7 @@
 /// - Typings to better stick to ark_poly's API
 /// - Uses `folding-schemes`' own `TranscriptVar` trait and `PoseidonTranscriptVar` struct
 /// - API made closer to gadgets found in `folding-schemes`
-use ark_crypto_primitives::sponge::Absorb;
-use ark_ec::{CurveGroup, Group};
+use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, Absorb, CryptographicSponge};
 use ark_ff::PrimeField;
 use ark_poly::{univariate::DensePolynomial, DenseUVPolynomial};
 use ark_r1cs_std::{
@@ -19,7 +18,7 @@ use std::{borrow::Borrow, marker::PhantomData};
 use crate::utils::espresso::sum_check::SumCheck;
 use crate::utils::virtual_polynomial::VPAuxInfo;
 use crate::{
-    transcript::{poseidon::PoseidonTranscript, TranscriptVar},
+    transcript::TranscriptVar,
     utils::sum_check::{structs::IOPProof, IOPSumCheck},
 };
 
@@ -82,35 +81,27 @@ impl<F: PrimeField> DensePolynomialVar<F> {
 }
 
 #[derive(Clone, Debug)]
-pub struct IOPProofVar<C: CurveGroup> {
+pub struct IOPProofVar<F: PrimeField> {
     // We have to be generic over a CurveGroup because instantiating a IOPProofVar will call IOPSumCheck which requires a CurveGroup
-    pub proofs: Vec<DensePolynomialVar<C::ScalarField>>, // = IOPProof.proofs
-    pub claim: FpVar<C::ScalarField>,
+    pub proofs: Vec<DensePolynomialVar<F>>,
+    pub claim: FpVar<F>,
 }
 
-impl<C: CurveGroup> AllocVar<IOPProof<C::ScalarField>, C::ScalarField> for IOPProofVar<C>
-where
-    <C as Group>::ScalarField: Absorb,
-{
-    fn new_variable<T: Borrow<IOPProof<C::ScalarField>>>(
-        cs: impl Into<Namespace<C::ScalarField>>,
+impl<F: PrimeField + Absorb> AllocVar<IOPProof<F>, F> for IOPProofVar<F> {
+    fn new_variable<T: Borrow<IOPProof<F>>>(
+        cs: impl Into<Namespace<F>>,
         f: impl FnOnce() -> Result<T, SynthesisError>,
         mode: AllocationMode,
     ) -> Result<Self, SynthesisError> {
         f().and_then(|c| {
             let cs = cs.into();
-            let cp: &IOPProof<C::ScalarField> = c.borrow();
-            let claim = IOPSumCheck::<C, PoseidonTranscript<C>>::extract_sum(cp);
-            let claim = FpVar::<C::ScalarField>::new_variable(cs.clone(), || Ok(claim), mode)?;
-            let mut proofs =
-                Vec::<DensePolynomialVar<C::ScalarField>>::with_capacity(cp.proofs.len());
+            let cp: &IOPProof<F> = c.borrow();
+            let claim = IOPSumCheck::<F, PoseidonSponge<F>>::extract_sum(cp);
+            let claim = FpVar::<F>::new_variable(cs.clone(), || Ok(claim), mode)?;
+            let mut proofs = Vec::<DensePolynomialVar<F>>::with_capacity(cp.proofs.len());
             for proof in cp.proofs.iter() {
                 let poly = DensePolynomial::from_coefficients_slice(&proof.coeffs);
-                let proof = DensePolynomialVar::<C::ScalarField>::new_variable(
-                    cs.clone(),
-                    || Ok(poly),
-                    mode,
-                )?;
+                let proof = DensePolynomialVar::<F>::new_variable(cs.clone(), || Ok(poly), mode)?;
                 proofs.push(proof);
             }
             Ok(Self { proofs, claim })
@@ -149,28 +140,28 @@ impl<F: PrimeField> AllocVar<VPAuxInfo<F>, F> for VPAuxInfoVar<F> {
 }
 
 #[derive(Debug, Clone)]
-pub struct SumCheckVerifierGadget<C: CurveGroup> {
-    _f: PhantomData<C>,
+pub struct SumCheckVerifierGadget<F: PrimeField> {
+    _f: PhantomData<F>,
 }
 
-impl<C: CurveGroup> SumCheckVerifierGadget<C> {
+impl<F: PrimeField> SumCheckVerifierGadget<F> {
     #[allow(clippy::type_complexity)]
-    pub fn verify(
-        iop_proof_var: &IOPProofVar<C>,
-        poly_aux_info_var: &VPAuxInfoVar<C::ScalarField>,
-        transcript_var: &mut impl TranscriptVar<C::ScalarField>,
-        enabled: Boolean<C::ScalarField>,
-    ) -> Result<(Vec<FpVar<C::ScalarField>>, Vec<FpVar<C::ScalarField>>), SynthesisError> {
+    pub fn verify<S: CryptographicSponge, T: TranscriptVar<F, S>>(
+        iop_proof_var: &IOPProofVar<F>,
+        poly_aux_info_var: &VPAuxInfoVar<F>,
+        transcript_var: &mut T,
+        enabled: Boolean<F>,
+    ) -> Result<(Vec<FpVar<F>>, Vec<FpVar<F>>), SynthesisError> {
         let mut e_vars = vec![iop_proof_var.claim.clone()];
-        let mut r_vars: Vec<FpVar<C::ScalarField>> = Vec::new();
-        transcript_var.absorb(poly_aux_info_var.num_variables.clone())?;
-        transcript_var.absorb(poly_aux_info_var.max_degree.clone())?;
+        let mut r_vars: Vec<FpVar<F>> = Vec::new();
+        transcript_var.absorb(&poly_aux_info_var.num_variables)?;
+        transcript_var.absorb(&poly_aux_info_var.max_degree)?;
 
         for poly_var in iop_proof_var.proofs.iter() {
             let res = poly_var.eval_at_one() + poly_var.eval_at_zero();
             let e_var = e_vars.last().ok_or(SynthesisError::Unsatisfiable)?;
             res.conditional_enforce_equal(e_var, &enabled)?;
-            transcript_var.absorb_vec(&poly_var.coeffs)?;
+            transcript_var.absorb(&poly_var.coeffs)?;
             let r_i_var = transcript_var.get_challenge()?;
             e_vars.push(poly_var.evaluate(&r_i_var));
             r_vars.push(r_i_var);
@@ -182,49 +173,35 @@ impl<C: CurveGroup> SumCheckVerifierGadget<C> {
 
 #[cfg(test)]
 mod tests {
-    use ark_crypto_primitives::sponge::{poseidon::PoseidonConfig, Absorb};
-    use ark_ec::CurveGroup;
-    use ark_ff::Field;
-    use ark_pallas::{Fr, Projective};
-    use ark_poly::{
-        univariate::DensePolynomial, DenseMultilinearExtension, DenseUVPolynomial,
-        MultilinearExtension, Polynomial,
+    use ark_crypto_primitives::sponge::{
+        constraints::CryptographicSpongeVar,
+        poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig},
     };
-    use ark_r1cs_std::{alloc::AllocVar, R1CSVar};
+    use ark_pallas::Fr;
+    use ark_poly::{DenseMultilinearExtension, MultilinearExtension, Polynomial};
+    use ark_r1cs_std::R1CSVar;
     use ark_relations::r1cs::ConstraintSystem;
     use std::sync::Arc;
 
     use super::*;
     use crate::{
-        folding::circuits::sum_check::{IOPProofVar, VPAuxInfoVar},
-        transcript::{
-            poseidon::{poseidon_canonical_config, PoseidonTranscript, PoseidonTranscriptVar},
-            Transcript, TranscriptVar,
-        },
-        utils::{
-            sum_check::{structs::IOPProof, IOPSumCheck, SumCheck},
-            virtual_polynomial::VirtualPolynomial,
-        },
+        transcript::poseidon::poseidon_canonical_config,
+        utils::virtual_polynomial::VirtualPolynomial,
     };
 
     pub type TestSumCheckProof<F> = (VirtualPolynomial<F>, PoseidonConfig<F>, IOPProof<F>);
 
     /// Primarily used for testing the sumcheck gadget
     /// Returns a random virtual polynomial, the poseidon config used and the associated sumcheck proof
-    pub fn get_test_sumcheck_proof<C: CurveGroup>(
+    pub fn get_test_sumcheck_proof<F: PrimeField + Absorb>(
         num_vars: usize,
-    ) -> TestSumCheckProof<C::ScalarField>
-    where
-        <C as ark_ec::Group>::ScalarField: Absorb,
-    {
+    ) -> TestSumCheckProof<F> {
         let mut rng = ark_std::test_rng();
-        let poseidon_config: PoseidonConfig<C::ScalarField> =
-            poseidon_canonical_config::<C::ScalarField>();
-        let mut poseidon_transcript_prove = PoseidonTranscript::<C>::new(&poseidon_config);
+        let poseidon_config: PoseidonConfig<F> = poseidon_canonical_config::<F>();
+        let mut poseidon_transcript_prove = PoseidonSponge::<F>::new(&poseidon_config);
         let poly_mle = DenseMultilinearExtension::rand(num_vars, &mut rng);
-        let virtual_poly =
-            VirtualPolynomial::new_from_mle(&Arc::new(poly_mle), C::ScalarField::ONE);
-        let sum_check: IOPProof<C::ScalarField> = IOPSumCheck::<C, PoseidonTranscript<C>>::prove(
+        let virtual_poly = VirtualPolynomial::new_from_mle(&Arc::new(poly_mle), F::ONE);
+        let sum_check: IOPProof<F> = IOPSumCheck::<F, PoseidonSponge<F>>::prove(
             &virtual_poly,
             &mut poseidon_transcript_prove,
         )
@@ -237,15 +214,15 @@ mod tests {
         for num_vars in 1..15 {
             let cs = ConstraintSystem::<Fr>::new_ref();
             let (virtual_poly, poseidon_config, sum_check) =
-                get_test_sumcheck_proof::<Projective>(num_vars);
-            let mut poseidon_var: PoseidonTranscriptVar<Fr> =
-                PoseidonTranscriptVar::new(cs.clone(), &poseidon_config);
+                get_test_sumcheck_proof::<Fr>(num_vars);
+            let mut poseidon_var: PoseidonSpongeVar<Fr> =
+                PoseidonSpongeVar::new(cs.clone(), &poseidon_config);
             let iop_proof_var =
-                IOPProofVar::<Projective>::new_witness(cs.clone(), || Ok(&sum_check)).unwrap();
+                IOPProofVar::<Fr>::new_witness(cs.clone(), || Ok(&sum_check)).unwrap();
             let poly_aux_info_var =
                 VPAuxInfoVar::<Fr>::new_witness(cs.clone(), || Ok(virtual_poly.aux_info)).unwrap();
             let enabled = Boolean::<Fr>::new_witness(cs.clone(), || Ok(true)).unwrap();
-            let res = SumCheckVerifierGadget::<Projective>::verify(
+            let res = SumCheckVerifierGadget::<Fr>::verify(
                 &iop_proof_var,
                 &poly_aux_info_var,
                 &mut poseidon_var,
@@ -256,8 +233,7 @@ mod tests {
             let (circuit_evals, r_challenges) = res.unwrap();
 
             // 1. assert claim from circuit is equal to the one from the sum-check
-            let claim: Fr =
-                IOPSumCheck::<Projective, PoseidonTranscript<Projective>>::extract_sum(&sum_check);
+            let claim: Fr = IOPSumCheck::<Fr, PoseidonSponge<Fr>>::extract_sum(&sum_check);
             assert_eq!(circuit_evals[0].value().unwrap(), claim);
 
             // 2. assert that all in-circuit evaluations are equal to the ones from the sum-check
diff --git a/folding-schemes/src/folding/hypernova/cccs.rs b/folding-schemes/src/folding/hypernova/cccs.rs
index c81a189..6be8ab6 100644
--- a/folding-schemes/src/folding/hypernova/cccs.rs
+++ b/folding-schemes/src/folding/hypernova/cccs.rs
@@ -1,3 +1,4 @@
+use ark_crypto_primitives::sponge::Absorb;
 use ark_ec::CurveGroup;
 use ark_ff::PrimeField;
 use ark_std::One;
@@ -9,6 +10,7 @@ use ark_std::rand::Rng;
 use super::Witness;
 use crate::arith::{ccs::CCS, Arith};
 use crate::commitment::CommitmentScheme;
+use crate::transcript::AbsorbNonNative;
 use crate::utils::mle::dense_vec_to_dense_mle;
 use crate::utils::vec::mat_vec_mul;
 use crate::utils::virtual_polynomial::{build_eq_x_r_vec, VirtualPolynomial};
@@ -118,6 +120,26 @@ impl<C: CurveGroup> CCCS<C> {
     }
 }
 
+impl<C: CurveGroup> Absorb for CCCS<C>
+where
+    C::ScalarField: Absorb,
+{
+    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
+        // This is never called
+        unimplemented!()
+    }
+
+    fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
+        // We cannot call `to_native_sponge_field_elements(dest)` directly, as
+        // `to_native_sponge_field_elements` needs `F` to be `C::ScalarField`,
+        // but here `F` is a generic `PrimeField`.
+        self.C
+            .to_native_sponge_field_elements_as_vec()
+            .to_sponge_field_elements(dest);
+        self.x.to_sponge_field_elements(dest);
+    }
+}
+
 #[cfg(test)]
 pub mod tests {
     use ark_pallas::Fr;
diff --git a/folding-schemes/src/folding/hypernova/circuits.rs b/folding-schemes/src/folding/hypernova/circuits.rs
index bdc9bd3..17b0442 100644
--- a/folding-schemes/src/folding/hypernova/circuits.rs
+++ b/folding-schemes/src/folding/hypernova/circuits.rs
@@ -1,7 +1,8 @@
 /// Implementation of [HyperNova](https://eprint.iacr.org/2023/573.pdf) circuits
-use ark_crypto_primitives::crh::{
-    poseidon::constraints::{CRHGadget, CRHParametersVar},
-    CRHSchemeGadget,
+use ark_crypto_primitives::sponge::{
+    constraints::CryptographicSpongeVar,
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonSponge},
+    CryptographicSponge,
 };
 use ark_crypto_primitives::sponge::{poseidon::PoseidonConfig, Absorb};
 use ark_ec::{CurveGroup, Group};
@@ -45,10 +46,7 @@ use crate::utils::virtual_polynomial::VPAuxInfo;
 use crate::Error;
 use crate::{
     arith::{ccs::CCS, r1cs::extract_r1cs},
-    transcript::{
-        poseidon::{PoseidonTranscript, PoseidonTranscriptVar},
-        Transcript, TranscriptVar,
-    },
+    transcript::TranscriptVar,
 };
 
 /// Committed CCS instance
@@ -142,12 +140,13 @@ where
     #[allow(clippy::type_complexity)]
     pub fn hash(
         self,
-        crh_params: &CRHParametersVar<CF1<C>>,
+        sponge: &PoseidonSpongeVar<CF1<C>>,
         pp_hash: FpVar<CF1<C>>,
         i: FpVar<CF1<C>>,
         z_0: Vec<FpVar<CF1<C>>>,
         z_i: Vec<FpVar<CF1<C>>>,
     ) -> Result<(FpVar<CF1<C>>, Vec<FpVar<CF1<C>>>), SynthesisError> {
+        let mut sponge = sponge.clone();
         let U_vec = [
             self.C.to_constraint_field()?,
             vec![self.u],
@@ -156,18 +155,19 @@ where
             self.v,
         ]
         .concat();
-        let input = [vec![pp_hash, i], z_0, z_i, U_vec.clone()].concat();
-        Ok((
-            CRHGadget::<C::ScalarField>::evaluate(crh_params, &input)?,
-            U_vec,
-        ))
+        sponge.absorb(&pp_hash)?;
+        sponge.absorb(&i)?;
+        sponge.absorb(&z_0)?;
+        sponge.absorb(&z_i)?;
+        sponge.absorb(&U_vec)?;
+        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
     }
 }
 
 /// ProofVar defines a multifolding proof
 #[derive(Debug)]
 pub struct ProofVar<C: CurveGroup> {
-    pub sc_proof: IOPProofVar<C>,
+    pub sc_proof: IOPProofVar<C::ScalarField>,
     #[allow(clippy::type_complexity)]
     pub sigmas_thetas: (Vec<Vec<FpVar<CF1<C>>>>, Vec<Vec<FpVar<CF1<C>>>>),
 }
@@ -185,7 +185,7 @@ where
         f().and_then(|val| {
             let cs = cs.into();
 
-            let sc_proof = IOPProofVar::<C>::new_variable(
+            let sc_proof = IOPProofVar::<C::ScalarField>::new_variable(
                 cs.clone(),
                 || Ok(val.borrow().sc_proof.clone()),
                 mode,
@@ -223,12 +223,11 @@ where
     /// Runs (in-circuit) the NIMFS.V, which outputs the new folded LCCCS instance together with
     /// the rho_bits, which will be used in other parts of the AugmentedFCircuit
     #[allow(clippy::type_complexity)]
-    pub fn verify(
+    pub fn verify<S: CryptographicSponge, T: TranscriptVar<C::ScalarField, S>>(
         cs: ConstraintSystemRef<CF1<C>>,
         // only used the CCS params, not the matrices
         ccs: &CCS<C::ScalarField>,
-        mut transcript: impl TranscriptVar<C::ScalarField>,
-
+        transcript: &mut T,
         running_instances: &[LCCCSVar<C>],
         new_instances: &[CCCSVar<C>],
         proof: ProofVar<C>,
@@ -244,24 +243,24 @@ where
                 U_i.v.clone(),
             ]
             .concat();
-            transcript.absorb_vec(&v)?;
+            transcript.absorb(&v)?;
         }
         for u_i in new_instances {
             let v = [u_i.C.to_constraint_field()?, u_i.x.clone()].concat();
-            transcript.absorb_vec(&v)?;
+            transcript.absorb(&v)?;
         }
 
         // get the challenges
         let gamma_scalar_raw = C::ScalarField::from_le_bytes_mod_order(b"gamma");
         let gamma_scalar: FpVar<CF1<C>> =
             FpVar::<CF1<C>>::new_constant(cs.clone(), gamma_scalar_raw)?;
-        transcript.absorb(gamma_scalar)?;
+        transcript.absorb(&gamma_scalar)?;
         let gamma: FpVar<CF1<C>> = transcript.get_challenge()?;
 
         let beta_scalar_raw = C::ScalarField::from_le_bytes_mod_order(b"beta");
         let beta_scalar: FpVar<CF1<C>> =
             FpVar::<CF1<C>>::new_constant(cs.clone(), beta_scalar_raw)?;
-        transcript.absorb(beta_scalar)?;
+        transcript.absorb(&beta_scalar)?;
         let beta: Vec<FpVar<CF1<C>>> = transcript.get_challenges(ccs.s)?;
 
         let vp_aux_info_raw = VPAuxInfo::<C::ScalarField> {
@@ -283,10 +282,10 @@ where
         }
 
         // verify the interactive part of the sumcheck
-        let (e_vars, r_vars) = SumCheckVerifierGadget::<C>::verify(
+        let (e_vars, r_vars) = SumCheckVerifierGadget::<C::ScalarField>::verify(
             &proof.sc_proof,
             &vp_aux_info,
-            &mut transcript,
+            transcript,
             enabled.clone(),
         )?;
 
@@ -312,7 +311,7 @@ where
         // get the folding challenge
         let rho_scalar_raw = C::ScalarField::from_le_bytes_mod_order(b"rho");
         let rho_scalar: FpVar<CF1<C>> = FpVar::<CF1<C>>::new_constant(cs.clone(), rho_scalar_raw)?;
-        transcript.absorb(rho_scalar)?;
+        transcript.absorb(&rho_scalar)?;
         let rho_bits: Vec<Boolean<CF1<C>>> = transcript.get_challenge_nbits(N_BITS_RO)?;
         let rho = Boolean::le_bits_to_fp_var(&rho_bits)?;
 
@@ -560,10 +559,10 @@ where
         let w_i = W_i.clone();
         let u_i = CCCS::<C1>::dummy(ccs.l);
 
-        let mut transcript_p: PoseidonTranscript<C1> =
-            PoseidonTranscript::<C1>::new(&self.poseidon_config.clone());
+        let mut transcript_p: PoseidonSponge<C1::ScalarField> =
+            PoseidonSponge::<C1::ScalarField>::new(&self.poseidon_config.clone());
         // since this is only for the number of constraints, no need to absorb the pp_hash here
-        let (nimfs_proof, U_i1, _, _) = NIMFS::<C1, PoseidonTranscript<C1>>::prove(
+        let (nimfs_proof, U_i1, _, _) = NIMFS::<C1, PoseidonSponge<C1::ScalarField>>::prove(
             &mut transcript_p,
             &ccs,
             &[U_i.clone()],
@@ -671,10 +670,7 @@ where
         })?;
         let cf_cmT = GC2::new_witness(cs.clone(), || Ok(self.cf_cmT.unwrap_or_else(C2::zero)))?;
 
-        let crh_params = CRHParametersVar::<C1::ScalarField>::new_constant(
-            cs.clone(),
-            self.poseidon_config.clone(),
-        )?;
+        let sponge = PoseidonSpongeVar::<C1::ScalarField>::new(cs.clone(), &self.poseidon_config);
 
         // get z_{i+1} from the F circuit
         let i_usize = self.i_usize.unwrap_or(0);
@@ -689,14 +685,14 @@ where
         // P.1. Compute u_i.x
         // u_i.x[0] = H(i, z_0, z_i, U_i)
         let (u_i_x, _) = U_i.clone().hash(
-            &crh_params,
+            &sponge,
             pp_hash.clone(),
             i.clone(),
             z_0.clone(),
             z_i.clone(),
         )?;
         // u_i.x[1] = H(cf_U_i)
-        let (cf_u_i_x, cf_U_i_vec) = cf_U_i.clone().hash(&crh_params, pp_hash.clone())?;
+        let (cf_u_i_x, cf_U_i_vec) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
 
         // P.2. Construct u_i
         let u_i = CCCSVar::<C1> {
@@ -712,13 +708,12 @@ where
         // Notice that NIMFSGadget::fold_committed_instance does not fold C. We set `U_i1.C` to
         // unconstrained witnesses `U_i1_C` respectively. Its correctness will be checked on the
         // other curve.
-        let mut transcript =
-            PoseidonTranscriptVar::<C1::ScalarField>::new(cs.clone(), &self.poseidon_config);
-        transcript.absorb(pp_hash.clone())?;
+        let mut transcript = PoseidonSpongeVar::new(cs.clone(), &self.poseidon_config);
+        transcript.absorb(&pp_hash)?;
         let (mut U_i1, rho_bits) = NIMFSGadget::<C1>::verify(
             cs.clone(),
             &self.ccs.clone(),
-            transcript,
+            &mut transcript,
             &[U_i.clone()],
             &[u_i.clone()],
             nimfs_proof,
@@ -728,14 +723,14 @@ where
 
         // P.4.a compute and check the first output of F'
         let (u_i1_x, _) = U_i1.clone().hash(
-            &crh_params,
+            &sponge,
             pp_hash.clone(),
             i + FpVar::<CF1<C1>>::one(),
             z_0.clone(),
             z_i1.clone(),
         )?;
         let (u_i1_x_base, _) = LCCCSVar::new_constant(cs.clone(), U_dummy)?.hash(
-            &crh_params,
+            &sponge,
             pp_hash.clone(),
             FpVar::<CF1<C1>>::one(),
             z_0.clone(),
@@ -776,8 +771,7 @@ where
         // compute cf_r = H(cf_u_i, cf_U_i, cf_cmT)
         // cf_r_bits is denoted by rho* in the paper.
         let cf_r_bits = CycleFoldChallengeGadget::<C2, GC2>::get_challenge_gadget(
-            cs.clone(),
-            &self.poseidon_config,
+            &mut transcript,
             pp_hash.clone(),
             cf_U_i_vec,
             cf_u_i.clone(),
@@ -802,10 +796,10 @@ where
         // P.4.b compute and check the second output of F'
         // Base case: u_{i+1}.x[1] == H(cf_U_{\bot})
         // Non-base case: u_{i+1}.x[1] == H(cf_U_{i+1})
-        let (cf_u_i1_x, _) = cf_U_i1.clone().hash(&crh_params, pp_hash.clone())?;
+        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)?
-                .hash(&crh_params, pp_hash)?;
+                .hash(&sponge, pp_hash)?;
         let cf_x = FpVar::new_input(cs.clone(), || {
             Ok(self.cf_x.unwrap_or(cf_u_i1_x_base.value()?))
         })?;
@@ -820,34 +814,23 @@ mod tests {
     use ark_bn254::{constraints::GVar, Fq, Fr, G1Projective as Projective};
     use ark_ff::BigInteger;
     use ark_grumpkin::{constraints::GVar as GVar2, Projective as Projective2};
-    use ark_r1cs_std::{alloc::AllocVar, fields::fp::FpVar, R1CSVar};
     use ark_std::{test_rng, UniformRand};
     use std::time::Instant;
 
     use super::*;
     use crate::{
         arith::{
-            ccs::{
-                tests::{get_test_ccs, get_test_z},
-                CCS,
-            },
+            ccs::tests::{get_test_ccs, get_test_z},
             r1cs::extract_w_x,
         },
         commitment::{pedersen::Pedersen, CommitmentScheme},
         folding::{
             circuits::cyclefold::{fold_cyclefold_circuit, CycleFoldCircuit},
-            hypernova::{
-                nimfs::NIMFS,
-                utils::{compute_c, compute_sigmas_thetas},
-                Witness,
-            },
-            nova::{traits::NovaR1CS, CommittedInstance, Witness as NovaWitness},
+            hypernova::utils::{compute_c, compute_sigmas_thetas},
+            nova::{traits::NovaR1CS, Witness as NovaWitness},
         },
         frontend::tests::CubicFCircuit,
-        transcript::{
-            poseidon::{poseidon_canonical_config, PoseidonTranscript, PoseidonTranscriptVar},
-            Transcript,
-        },
+        transcript::poseidon::poseidon_canonical_config,
         utils::get_cm_coordinates,
     };
 
@@ -996,12 +979,11 @@ mod tests {
 
         // Prover's transcript
         let poseidon_config = poseidon_canonical_config::<Fr>();
-        let mut transcript_p: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         // Run the prover side of the multifolding
         let (proof, folded_lcccs, folded_witness, _) =
-            NIMFS::<Projective, PoseidonTranscript<Projective>>::prove(
+            NIMFS::<Projective, PoseidonSponge<Fr>>::prove(
                 &mut transcript_p,
                 &ccs,
                 &lcccs_instances,
@@ -1012,11 +994,10 @@ mod tests {
             .unwrap();
 
         // Verifier's transcript
-        let mut transcript_v: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         // Run the verifier side of the multifolding
-        let folded_lcccs_v = NIMFS::<Projective, PoseidonTranscript<Projective>>::verify(
+        let folded_lcccs_v = NIMFS::<Projective, PoseidonSponge<Fr>>::verify(
             &mut transcript_v,
             &ccs,
             &lcccs_instances,
@@ -1039,13 +1020,13 @@ mod tests {
                 .unwrap();
         let proofVar =
             ProofVar::<Projective>::new_witness(cs.clone(), || Ok(proof.clone())).unwrap();
-        let transcriptVar = PoseidonTranscriptVar::<Fr>::new(cs.clone(), &poseidon_config);
+        let mut transcriptVar = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
 
         let enabled = Boolean::<Fr>::new_witness(cs.clone(), || Ok(true)).unwrap();
         let (folded_lcccsVar, _) = NIMFSGadget::<Projective>::verify(
             cs.clone(),
             &ccs,
-            transcriptVar,
+            &mut transcriptVar,
             &lcccs_instancesVar,
             &cccs_instancesVar,
             proofVar,
@@ -1061,6 +1042,7 @@ mod tests {
     pub fn test_lcccs_hash() {
         let mut rng = test_rng();
         let poseidon_config = poseidon_canonical_config::<Fr>();
+        let sponge = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         let ccs = get_test_ccs();
         let z1 = get_test_z::<Fr>(3);
@@ -1077,12 +1059,11 @@ mod tests {
             .unwrap();
         let h = lcccs
             .clone()
-            .hash(&poseidon_config, pp_hash, i, z_0.clone(), z_i.clone())
-            .unwrap();
+            .hash(&sponge, pp_hash, i, z_0.clone(), z_i.clone());
 
         let cs = ConstraintSystem::<Fr>::new_ref();
 
-        let crh_params = CRHParametersVar::<Fr>::new_constant(cs.clone(), poseidon_config).unwrap();
+        let spongeVar = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
         let pp_hashVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(pp_hash)).unwrap();
         let iVar = FpVar::<Fr>::new_witness(cs.clone(), || Ok(i)).unwrap();
         let z_0Var = Vec::<FpVar<Fr>>::new_witness(cs.clone(), || Ok(z_0.clone())).unwrap();
@@ -1091,7 +1072,7 @@ mod tests {
         let (hVar, _) = lcccsVar
             .clone()
             .hash(
-                &crh_params,
+                &spongeVar,
                 pp_hashVar,
                 iVar.clone(),
                 z_0Var.clone(),
@@ -1108,6 +1089,7 @@ mod tests {
     pub fn test_augmented_f_circuit() {
         let mut rng = test_rng();
         let poseidon_config = poseidon_canonical_config::<Fr>();
+        let sponge = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         let start = Instant::now();
         let F_circuit = CubicFCircuit::<Fr>::new(()).unwrap();
@@ -1161,15 +1143,8 @@ mod tests {
         let mut cf_W_i = cf_W_dummy.clone();
         let mut cf_U_i = cf_U_dummy.clone();
         u_i.x = vec![
-            U_i.hash(
-                &poseidon_config,
-                pp_hash,
-                Fr::zero(),
-                z_0.clone(),
-                z_i.clone(),
-            )
-            .unwrap(),
-            cf_U_i.hash_cyclefold(&poseidon_config, pp_hash).unwrap(),
+            U_i.hash(&sponge, pp_hash, Fr::zero(), z_0.clone(), z_i.clone()),
+            cf_U_i.hash_cyclefold(&sponge, pp_hash),
         ];
 
         let n_steps: usize = 4;
@@ -1185,19 +1160,11 @@ mod tests {
                 W_i1 = Witness::<Fr>::dummy(&ccs);
                 U_i1 = LCCCS::dummy(ccs.l, ccs.t, ccs.s);
 
-                let u_i1_x = U_i1
-                    .hash(
-                        &poseidon_config,
-                        pp_hash,
-                        Fr::one(),
-                        z_0.clone(),
-                        z_i1.clone(),
-                    )
-                    .unwrap();
+                let u_i1_x = U_i1.hash(&sponge, pp_hash, Fr::one(), z_0.clone(), z_i1.clone());
 
                 // hash the initial (dummy) CycleFold instance, which is used as the 2nd public
                 // input in the AugmentedFCircuit
-                let cf_u_i1_x = cf_U_i.hash_cyclefold(&poseidon_config, pp_hash).unwrap();
+                let cf_u_i1_x = cf_U_i.hash_cyclefold(&sponge, pp_hash);
 
                 augmented_f_circuit =
                     AugmentedFCircuit::<Projective, Projective2, GVar2, CubicFCircuit<Fr>> {
@@ -1225,12 +1192,12 @@ mod tests {
                         cf_cmT: None,
                     };
             } else {
-                let mut transcript_p: PoseidonTranscript<Projective> =
-                    PoseidonTranscript::<Projective>::new(&poseidon_config.clone());
+                let mut transcript_p: PoseidonSponge<Fr> =
+                    PoseidonSponge::<Fr>::new(&poseidon_config.clone());
                 transcript_p.absorb(&pp_hash);
                 let (rho_bits, nimfs_proof);
                 (nimfs_proof, U_i1, W_i1, rho_bits) =
-                    NIMFS::<Projective, PoseidonTranscript<Projective>>::prove(
+                    NIMFS::<Projective, PoseidonSponge<Fr>>::prove(
                         &mut transcript_p,
                         &ccs,
                         &[U_i.clone()],
@@ -1243,15 +1210,8 @@ mod tests {
                 // sanity check: check the folded instance relation
                 U_i1.check_relation(&ccs, &W_i1).unwrap();
 
-                let u_i1_x = U_i1
-                    .hash(
-                        &poseidon_config,
-                        pp_hash,
-                        iFr + Fr::one(),
-                        z_0.clone(),
-                        z_i1.clone(),
-                    )
-                    .unwrap();
+                let u_i1_x =
+                    U_i1.hash(&sponge, pp_hash, iFr + Fr::one(), z_0.clone(), z_i1.clone());
 
                 let rho_Fq = Fq::from_bigint(BigInteger::from_bits_le(&rho_bits)).unwrap();
                 // CycleFold part:
@@ -1282,7 +1242,7 @@ mod tests {
                     Pedersen<Projective>,
                     Pedersen<Projective2>,
                 >(
-                    &poseidon_config,
+                    &mut transcript_p,
                     cf_r1cs.clone(),
                     cf_pedersen_params.clone(),
                     pp_hash,
@@ -1295,7 +1255,7 @@ mod tests {
 
                 // hash the CycleFold folded instance, which is used as the 2nd public input in the
                 // AugmentedFCircuit
-                let cf_u_i1_x = cf_U_i1.hash_cyclefold(&poseidon_config, pp_hash).unwrap();
+                let cf_u_i1_x = cf_U_i1.hash_cyclefold(&sponge, pp_hash);
 
                 augmented_f_circuit =
                     AugmentedFCircuit::<Projective, Projective2, GVar2, CubicFCircuit<Fr>> {
@@ -1346,16 +1306,9 @@ mod tests {
             assert_eq!(u_i.x, r1cs_x_i1);
             assert_eq!(u_i.x[0], augmented_f_circuit.x.unwrap());
             assert_eq!(u_i.x[1], augmented_f_circuit.cf_x.unwrap());
-            let expected_u_i1_x = U_i1
-                .hash(
-                    &poseidon_config,
-                    pp_hash,
-                    iFr + Fr::one(),
-                    z_0.clone(),
-                    z_i1.clone(),
-                )
-                .unwrap();
-            let expected_cf_U_i1_x = cf_U_i.hash_cyclefold(&poseidon_config, pp_hash).unwrap();
+            let expected_u_i1_x =
+                U_i1.hash(&sponge, pp_hash, iFr + Fr::one(), z_0.clone(), z_i1.clone());
+            let expected_cf_U_i1_x = cf_U_i.hash_cyclefold(&sponge, pp_hash);
             // u_i is already u_i1 at this point, check that has the expected value at x[0]
             assert_eq!(u_i.x[0], expected_u_i1_x);
             assert_eq!(u_i.x[1], expected_cf_U_i1_x);
diff --git a/folding-schemes/src/folding/hypernova/lcccs.rs b/folding-schemes/src/folding/hypernova/lcccs.rs
index 95fd937..801ec4e 100644
--- a/folding-schemes/src/folding/hypernova/lcccs.rs
+++ b/folding-schemes/src/folding/hypernova/lcccs.rs
@@ -1,7 +1,4 @@
-use ark_crypto_primitives::{
-    crh::{poseidon::CRH, CRHScheme},
-    sponge::{poseidon::PoseidonConfig, Absorb},
-};
+use ark_crypto_primitives::sponge::Absorb;
 use ark_ec::{CurveGroup, Group};
 use ark_ff::PrimeField;
 use ark_poly::DenseMultilinearExtension;
@@ -12,7 +9,7 @@ use ark_std::Zero;
 use super::Witness;
 use crate::arith::ccs::CCS;
 use crate::commitment::CommitmentScheme;
-use crate::folding::circuits::nonnative::affine::nonnative_affine_to_field_elements;
+use crate::transcript::{AbsorbNonNative, Transcript};
 use crate::utils::mle::dense_vec_to_dense_mle;
 use crate::utils::vec::mat_vec_mul;
 use crate::Error;
@@ -118,6 +115,29 @@ impl<C: CurveGroup> LCCCS<C> {
     }
 }
 
+impl<C: CurveGroup> Absorb for LCCCS<C>
+where
+    C::ScalarField: Absorb,
+{
+    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
+        // This is never called
+        unimplemented!()
+    }
+
+    fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
+        // We cannot call `to_native_sponge_field_elements(dest)` directly, as
+        // `to_native_sponge_field_elements` needs `F` to be `C::ScalarField`,
+        // but here `F` is a generic `PrimeField`.
+        self.C
+            .to_native_sponge_field_elements_as_vec()
+            .to_sponge_field_elements(dest);
+        self.u.to_sponge_field_elements(dest);
+        self.x.to_sponge_field_elements(dest);
+        self.r_x.to_sponge_field_elements(dest);
+        self.v.to_sponge_field_elements(dest);
+    }
+}
+
 impl<C: CurveGroup> LCCCS<C>
 where
     <C as Group>::ScalarField: Absorb,
@@ -126,32 +146,21 @@ where
     /// [`LCCCS`].hash implements the committed instance hash compatible with the gadget
     /// implemented in nova/circuits.rs::CommittedInstanceVar.hash.
     /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U_i` is the LCCCS.
-    pub fn hash(
+    pub fn hash<T: Transcript<C::ScalarField>>(
         &self,
-        poseidon_config: &PoseidonConfig<C::ScalarField>,
+        sponge: &T,
         pp_hash: C::ScalarField,
         i: C::ScalarField,
         z_0: Vec<C::ScalarField>,
         z_i: Vec<C::ScalarField>,
-    ) -> Result<C::ScalarField, Error> {
-        let (C_x, C_y) = nonnative_affine_to_field_elements::<C>(self.C)?;
-
-        CRH::<C::ScalarField>::evaluate(
-            poseidon_config,
-            vec![
-                vec![pp_hash, i],
-                z_0,
-                z_i,
-                C_x,
-                C_y,
-                vec![self.u],
-                self.x.clone(),
-                self.r_x.clone(),
-                self.v.clone(),
-            ]
-            .concat(),
-        )
-        .map_err(|e| Error::Other(e.to_string()))
+    ) -> C::ScalarField {
+        let mut sponge = sponge.clone();
+        sponge.absorb(&pp_hash);
+        sponge.absorb(&i);
+        sponge.absorb(&z_0);
+        sponge.absorb(&z_i);
+        sponge.absorb(&self);
+        sponge.squeeze_field_elements(1)[0]
     }
 }
 
@@ -161,7 +170,6 @@ pub mod tests {
     use ark_std::test_rng;
     use ark_std::One;
     use ark_std::UniformRand;
-    use ark_std::Zero;
     use std::sync::Arc;
 
     use super::*;
diff --git a/folding-schemes/src/folding/hypernova/mod.rs b/folding-schemes/src/folding/hypernova/mod.rs
index c1b5939..0a3db8c 100644
--- a/folding-schemes/src/folding/hypernova/mod.rs
+++ b/folding-schemes/src/folding/hypernova/mod.rs
@@ -1,5 +1,8 @@
 /// Implements the scheme described in [HyperNova](https://eprint.iacr.org/2023/573.pdf)
-use ark_crypto_primitives::sponge::{poseidon::PoseidonConfig, Absorb};
+use ark_crypto_primitives::sponge::{
+    poseidon::{PoseidonConfig, PoseidonSponge},
+    Absorb, CryptographicSponge,
+};
 use ark_ec::{CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
 use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar, ToConstraintFieldGadget};
@@ -18,6 +21,10 @@ use cccs::CCCS;
 use lcccs::LCCCS;
 use nimfs::NIMFS;
 
+use crate::arith::{
+    ccs::CCS,
+    r1cs::{extract_w_x, R1CS},
+};
 use crate::commitment::CommitmentScheme;
 use crate::folding::circuits::{
     cyclefold::{fold_cyclefold_circuit, CycleFoldCircuit},
@@ -31,13 +38,6 @@ use crate::frontend::FCircuit;
 use crate::utils::{get_cm_coordinates, pp_hash};
 use crate::Error;
 use crate::FoldingScheme;
-use crate::{
-    arith::{
-        ccs::CCS,
-        r1cs::{extract_w_x, R1CS},
-    },
-    transcript::{poseidon::PoseidonTranscript, Transcript},
-};
 
 /// Witness for the LCCCS & CCCS, containing the w vector, and the r_w used as randomness in the Pedersen commitment.
 #[derive(Debug, Clone, Eq, PartialEq)]
@@ -233,6 +233,8 @@ where
         z_0: Vec<C1::ScalarField>,
     ) -> Result<Self, Error> {
         let (pp, vp) = params;
+        // `sponge` is for digest computation.
+        let sponge = PoseidonSponge::<C1::ScalarField>::new(&pp.poseidon_config);
 
         // prepare the HyperNova's AugmentedFCircuit and CycleFold's circuits and obtain its CCS
         // and R1CS respectively
@@ -258,13 +260,13 @@ where
             cf_r1cs.dummy_instance();
         u_dummy.x = vec![
             U_dummy.hash(
-                &pp.poseidon_config,
+                &sponge,
                 pp_hash,
                 C1::ScalarField::zero(),
                 z_0.clone(),
                 z_0.clone(),
-            )?,
-            cf_U_dummy.hash_cyclefold(&pp.poseidon_config, pp_hash)?,
+            ),
+            cf_U_dummy.hash_cyclefold(&sponge, pp_hash),
         ];
 
         // W_dummy=W_0 is a 'dummy witness', all zeroes, but with the size corresponding to the
@@ -299,6 +301,9 @@ where
         mut rng: impl RngCore,
         external_inputs: Vec<C1::ScalarField>,
     ) -> Result<(), Error> {
+        // `sponge` is for digest computation.
+        let sponge = PoseidonSponge::<C1::ScalarField>::new(&self.poseidon_config);
+
         let augmented_f_circuit: AugmentedFCircuit<C1, C2, GC2, FC>;
 
         if self.z_i.len() != self.F.state_len() {
@@ -339,18 +344,16 @@ where
             U_i1 = LCCCS::dummy(self.ccs.l, self.ccs.t, self.ccs.s);
 
             let u_i1_x = U_i1.hash(
-                &self.poseidon_config,
+                &sponge,
                 self.pp_hash,
                 C1::ScalarField::one(),
                 self.z_0.clone(),
                 z_i1.clone(),
-            )?;
+            );
 
             // hash the initial (dummy) CycleFold instance, which is used as the 2nd public
             // input in the AugmentedFCircuit
-            cf_u_i1_x = self
-                .cf_U_i
-                .hash_cyclefold(&self.poseidon_config, self.pp_hash)?;
+            cf_u_i1_x = self.cf_U_i.hash_cyclefold(&sponge, self.pp_hash);
 
             augmented_f_circuit = AugmentedFCircuit::<C1, C2, GC2, FC> {
                 _c2: PhantomData,
@@ -377,30 +380,31 @@ where
                 cf_cmT: None,
             };
         } else {
-            let mut transcript_p: PoseidonTranscript<C1> =
-                PoseidonTranscript::<C1>::new(&self.poseidon_config);
+            let mut transcript_p: PoseidonSponge<C1::ScalarField> =
+                PoseidonSponge::<C1::ScalarField>::new(&self.poseidon_config);
             transcript_p.absorb(&self.pp_hash);
             let (rho_bits, nimfs_proof);
-            (nimfs_proof, U_i1, W_i1, rho_bits) = NIMFS::<C1, PoseidonTranscript<C1>>::prove(
-                &mut transcript_p,
-                &self.ccs,
-                &[self.U_i.clone()],
-                &[self.u_i.clone()],
-                &[self.W_i.clone()],
-                &[self.w_i.clone()],
-            )?;
+            (nimfs_proof, U_i1, W_i1, rho_bits) =
+                NIMFS::<C1, PoseidonSponge<C1::ScalarField>>::prove(
+                    &mut transcript_p,
+                    &self.ccs,
+                    &[self.U_i.clone()],
+                    &[self.u_i.clone()],
+                    &[self.W_i.clone()],
+                    &[self.w_i.clone()],
+                )?;
 
             // sanity check: check the folded instance relation
             #[cfg(test)]
             U_i1.check_relation(&self.ccs, &W_i1)?;
 
             let u_i1_x = U_i1.hash(
-                &self.poseidon_config,
+                &sponge,
                 self.pp_hash,
                 self.i + C1::ScalarField::one(),
                 self.z_0.clone(),
                 z_i1.clone(),
-            )?;
+            );
 
             let rho_Fq = C2::ScalarField::from_bigint(BigInteger::from_bits_le(&rho_bits))
                 .ok_or(Error::OutOfBounds)?;
@@ -425,7 +429,7 @@ where
 
             let (_cf_w_i, cf_u_i, cf_W_i1, cf_U_i1, cf_cmT, _) =
                 fold_cyclefold_circuit::<C1, GC1, C2, GC2, FC, CS1, CS2>(
-                    &self.poseidon_config,
+                    &mut transcript_p,
                     self.cf_r1cs.clone(),
                     self.cf_cs_params.clone(),
                     self.pp_hash,
@@ -435,7 +439,7 @@ where
                     cf_circuit,
                 )?;
 
-            cf_u_i1_x = cf_U_i1.hash_cyclefold(&self.poseidon_config, self.pp_hash)?;
+            cf_u_i1_x = cf_U_i1.hash_cyclefold(&sponge, self.pp_hash);
 
             augmented_f_circuit = AugmentedFCircuit::<C1, C2, GC2, FC> {
                 _c2: PhantomData,
@@ -541,6 +545,8 @@ where
             }
             return Ok(());
         }
+        // `sponge` is for digest computation.
+        let sponge = PoseidonSponge::<C1::ScalarField>::new(&vp.poseidon_config);
 
         let (U_i, W_i) = running_instance;
         let (u_i, w_i) = incoming_instance;
@@ -553,12 +559,12 @@ where
 
         // check that u_i's output points to the running instance
         // u_i.X[0] == H(i, z_0, z_i, U_i)
-        let expected_u_i_x = U_i.hash(&vp.poseidon_config, pp_hash, num_steps, z_0, z_i.clone())?;
+        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, z_0, z_i.clone());
         if expected_u_i_x != u_i.x[0] {
             return Err(Error::IVCVerificationFail);
         }
         // u_i.X[1] == H(cf_U_i)
-        let expected_cf_u_i_x = cf_U_i.hash_cyclefold(&vp.poseidon_config, pp_hash)?;
+        let expected_cf_u_i_x = cf_U_i.hash_cyclefold(&sponge, pp_hash);
         if expected_cf_u_i_x != u_i.x[1] {
             return Err(Error::IVCVerificationFail);
         }
diff --git a/folding-schemes/src/folding/hypernova/nimfs.rs b/folding-schemes/src/folding/hypernova/nimfs.rs
index 5e32d52..ab3bb52 100644
--- a/folding-schemes/src/folding/hypernova/nimfs.rs
+++ b/folding-schemes/src/folding/hypernova/nimfs.rs
@@ -13,7 +13,6 @@ use super::{
 };
 use crate::arith::ccs::CCS;
 use crate::constants::N_BITS_RO;
-use crate::folding::circuits::nonnative::affine::nonnative_affine_to_field_elements;
 use crate::transcript::Transcript;
 use crate::utils::sum_check::structs::{IOPProof as SumCheckProof, IOPProverMessage};
 use crate::utils::sum_check::{IOPSumCheck, SumCheck};
@@ -58,12 +57,12 @@ pub struct SigmasThetas<F: PrimeField>(pub Vec<Vec<F>>, pub Vec<Vec<F>>);
 #[derive(Debug)]
 /// Implements the Non-Interactive Multi Folding Scheme described in section 5 of
 /// [HyperNova](https://eprint.iacr.org/2023/573.pdf)
-pub struct NIMFS<C: CurveGroup, T: Transcript<C>> {
+pub struct NIMFS<C: CurveGroup, T: Transcript<C::ScalarField>> {
     pub _c: PhantomData<C>,
     pub _t: PhantomData<T>,
 }
 
-impl<C: CurveGroup, T: Transcript<C>> NIMFS<C, T>
+impl<C: CurveGroup, T: Transcript<C::ScalarField>> NIMFS<C, T>
 where
     <C as Group>::ScalarField: Absorb,
     C::BaseField: PrimeField,
@@ -178,7 +177,7 @@ where
     /// contains the sumcheck proof and the helper sumcheck claim sigmas and thetas.
     #[allow(clippy::type_complexity)]
     pub fn prove(
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         ccs: &CCS<C::ScalarField>,
         running_instances: &[LCCCS<C>],
         new_instances: &[CCCS<C>],
@@ -186,24 +185,8 @@ where
         w_cccs: &[Witness<C::ScalarField>],
     ) -> Result<(NIMFSProof<C>, LCCCS<C>, Witness<C::ScalarField>, Vec<bool>), Error> {
         // absorb instances to transcript
-        for U_i in running_instances {
-            let (C_x, C_y) = nonnative_affine_to_field_elements::<C>(U_i.C)?;
-            let v = [
-                C_x,
-                C_y,
-                vec![U_i.u],
-                U_i.x.clone(),
-                U_i.r_x.clone(),
-                U_i.v.clone(),
-            ]
-            .concat();
-            transcript.absorb_vec(&v);
-        }
-        for u_i in new_instances {
-            let (C_x, C_y) = nonnative_affine_to_field_elements::<C>(u_i.C)?;
-            let v = [C_x, C_y, u_i.x.clone()].concat();
-            transcript.absorb_vec(&v);
-        }
+        transcript.absorb(&running_instances);
+        transcript.absorb(&new_instances);
 
         if running_instances.is_empty() {
             return Err(Error::Empty);
@@ -247,7 +230,7 @@ where
         let g = compute_g(ccs, running_instances, &z_lcccs, &z_cccs, gamma, &beta)?;
 
         // Step 3: Run the sumcheck prover
-        let sumcheck_proof = IOPSumCheck::<C, T>::prove(&g, transcript)
+        let sumcheck_proof = IOPSumCheck::<C::ScalarField, T>::prove(&g, transcript)
             .map_err(|err| Error::SumCheckProveError(err.to_string()))?;
 
         // Step 2: dig into the sumcheck and extract r_x_prime
@@ -290,31 +273,15 @@ where
     /// into a single LCCCS instance.
     /// Returns the folded LCCCS instance.
     pub fn verify(
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<C::ScalarField>,
         ccs: &CCS<C::ScalarField>,
         running_instances: &[LCCCS<C>],
         new_instances: &[CCCS<C>],
         proof: NIMFSProof<C>,
     ) -> Result<LCCCS<C>, Error> {
         // absorb instances to transcript
-        for U_i in running_instances {
-            let (C_x, C_y) = nonnative_affine_to_field_elements::<C>(U_i.C)?;
-            let v = [
-                C_x,
-                C_y,
-                vec![U_i.u],
-                U_i.x.clone(),
-                U_i.r_x.clone(),
-                U_i.v.clone(),
-            ]
-            .concat();
-            transcript.absorb_vec(&v);
-        }
-        for u_i in new_instances {
-            let (C_x, C_y) = nonnative_affine_to_field_elements::<C>(u_i.C)?;
-            let v = [C_x, C_y, u_i.x.clone()].concat();
-            transcript.absorb_vec(&v);
-        }
+        transcript.absorb(&running_instances);
+        transcript.absorb(&new_instances);
 
         if running_instances.is_empty() {
             return Err(Error::Empty);
@@ -349,9 +316,13 @@ where
         }
 
         // Verify the interactive part of the sumcheck
-        let sumcheck_subclaim =
-            IOPSumCheck::<C, T>::verify(sum_v_j_gamma, &proof.sc_proof, &vp_aux_info, transcript)
-                .map_err(|err| Error::SumCheckVerifyError(err.to_string()))?;
+        let sumcheck_subclaim = IOPSumCheck::<C::ScalarField, T>::verify(
+            sum_v_j_gamma,
+            &proof.sc_proof,
+            &vp_aux_info,
+            transcript,
+        )
+        .map_err(|err| Error::SumCheckVerifyError(err.to_string()))?;
 
         // Step 2: Dig into the sumcheck claim and extract the randomness used
         let r_x_prime = sumcheck_subclaim.point.clone();
@@ -413,7 +384,8 @@ pub mod tests {
         Arith,
     };
     use crate::transcript::poseidon::poseidon_canonical_config;
-    use crate::transcript::poseidon::PoseidonTranscript;
+    use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
+    use ark_crypto_primitives::sponge::CryptographicSponge;
     use ark_std::test_rng;
     use ark_std::UniformRand;
 
@@ -450,7 +422,7 @@ pub mod tests {
         let mut rng = test_rng();
         let rho = Fr::rand(&mut rng);
 
-        let folded = NIMFS::<Projective, PoseidonTranscript<Projective>>::fold(
+        let folded = NIMFS::<Projective, PoseidonSponge<Fr>>::fold(
             &[lcccs],
             &[cccs],
             &sigmas_thetas,
@@ -458,8 +430,7 @@ pub mod tests {
             rho,
         );
 
-        let w_folded =
-            NIMFS::<Projective, PoseidonTranscript<Projective>>::fold_witness(&[w1], &[w2], rho);
+        let w_folded = NIMFS::<Projective, PoseidonSponge<Fr>>::fold_witness(&[w1], &[w2], rho);
 
         // check lcccs relation
         folded.check_relation(&ccs, &w_folded).unwrap();
@@ -491,13 +462,12 @@ pub mod tests {
 
         // Prover's transcript
         let poseidon_config = poseidon_canonical_config::<Fr>();
-        let mut transcript_p: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_p.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
         // Run the prover side of the multifolding
         let (proof, folded_lcccs, folded_witness, _) =
-            NIMFS::<Projective, PoseidonTranscript<Projective>>::prove(
+            NIMFS::<Projective, PoseidonSponge<Fr>>::prove(
                 &mut transcript_p,
                 &ccs,
                 &[running_instance.clone()],
@@ -508,12 +478,11 @@ pub mod tests {
             .unwrap();
 
         // Verifier's transcript
-        let mut transcript_v: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_v.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
         // Run the verifier side of the multifolding
-        let folded_lcccs_v = NIMFS::<Projective, PoseidonTranscript<Projective>>::verify(
+        let folded_lcccs_v = NIMFS::<Projective, PoseidonSponge<Fr>>::verify(
             &mut transcript_v,
             &ccs,
             &[running_instance.clone()],
@@ -545,12 +514,10 @@ pub mod tests {
 
         let poseidon_config = poseidon_canonical_config::<Fr>();
 
-        let mut transcript_p: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_p.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
-        let mut transcript_v: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_v.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
         let n: usize = 10;
@@ -564,7 +531,7 @@ pub mod tests {
 
             // run the prover side of the multifolding
             let (proof, folded_lcccs, folded_witness, _) =
-                NIMFS::<Projective, PoseidonTranscript<Projective>>::prove(
+                NIMFS::<Projective, PoseidonSponge<Fr>>::prove(
                     &mut transcript_p,
                     &ccs,
                     &[running_instance.clone()],
@@ -575,7 +542,7 @@ pub mod tests {
                 .unwrap();
 
             // run the verifier side of the multifolding
-            let folded_lcccs_v = NIMFS::<Projective, PoseidonTranscript<Projective>>::verify(
+            let folded_lcccs_v = NIMFS::<Projective, PoseidonSponge<Fr>>::verify(
                 &mut transcript_v,
                 &ccs,
                 &[running_instance.clone()],
@@ -641,13 +608,12 @@ pub mod tests {
 
         // Prover's transcript
         let poseidon_config = poseidon_canonical_config::<Fr>();
-        let mut transcript_p: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_p.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
         // Run the prover side of the multifolding
         let (proof, folded_lcccs, folded_witness, _) =
-            NIMFS::<Projective, PoseidonTranscript<Projective>>::prove(
+            NIMFS::<Projective, PoseidonSponge<Fr>>::prove(
                 &mut transcript_p,
                 &ccs,
                 &lcccs_instances,
@@ -658,12 +624,11 @@ pub mod tests {
             .unwrap();
 
         // Verifier's transcript
-        let mut transcript_v: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_v.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
         // Run the verifier side of the multifolding
-        let folded_lcccs_v = NIMFS::<Projective, PoseidonTranscript<Projective>>::verify(
+        let folded_lcccs_v = NIMFS::<Projective, PoseidonSponge<Fr>>::verify(
             &mut transcript_v,
             &ccs,
             &lcccs_instances,
@@ -690,13 +655,11 @@ pub mod tests {
 
         let poseidon_config = poseidon_canonical_config::<Fr>();
         // Prover's transcript
-        let mut transcript_p: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_p.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
         // Verifier's transcript
-        let mut transcript_v: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v: PoseidonSponge<Fr> = PoseidonSponge::<Fr>::new(&poseidon_config);
         transcript_v.absorb(&Fr::from_le_bytes_mod_order(b"init init"));
 
         let n_steps = 3;
@@ -741,7 +704,7 @@ pub mod tests {
 
             // Run the prover side of the multifolding
             let (proof, folded_lcccs, folded_witness, _) =
-                NIMFS::<Projective, PoseidonTranscript<Projective>>::prove(
+                NIMFS::<Projective, PoseidonSponge<Fr>>::prove(
                     &mut transcript_p,
                     &ccs,
                     &lcccs_instances,
@@ -752,7 +715,7 @@ pub mod tests {
                 .unwrap();
 
             // Run the verifier side of the multifolding
-            let folded_lcccs_v = NIMFS::<Projective, PoseidonTranscript<Projective>>::verify(
+            let folded_lcccs_v = NIMFS::<Projective, PoseidonSponge<Fr>>::verify(
                 &mut transcript_v,
                 &ccs,
                 &lcccs_instances,
diff --git a/folding-schemes/src/folding/hypernova/utils.rs b/folding-schemes/src/folding/hypernova/utils.rs
index 8b247c9..160ae5a 100644
--- a/folding-schemes/src/folding/hypernova/utils.rs
+++ b/folding-schemes/src/folding/hypernova/utils.rs
@@ -176,7 +176,6 @@ pub mod tests {
     use crate::utils::hypercube::BooleanHypercube;
     use crate::utils::mle::matrix_to_dense_mle;
     use crate::utils::multilinear_polynomial::tests::fix_last_variables;
-    use crate::utils::virtual_polynomial::eq_eval;
 
     /// Given M(x,y) matrix and a random field element `r`, test that ~M(r,y) is is an s'-variable polynomial which
     /// compresses every column j of the M(x,y) matrix by performing a random linear combination between the elements
diff --git a/folding-schemes/src/folding/nova/circuits.rs b/folding-schemes/src/folding/nova/circuits.rs
index 28721c6..3ad2269 100644
--- a/folding-schemes/src/folding/nova/circuits.rs
+++ b/folding-schemes/src/folding/nova/circuits.rs
@@ -1,11 +1,7 @@
 /// contains [Nova](https://eprint.iacr.org/2021/370.pdf) related circuits
-use ark_crypto_primitives::crh::{
-    poseidon::constraints::{CRHGadget, CRHParametersVar},
-    CRHSchemeGadget,
-};
 use ark_crypto_primitives::sponge::{
-    constraints::CryptographicSpongeVar,
-    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
+    constraints::{AbsorbGadget, CryptographicSpongeVar},
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig},
     Absorb, CryptographicSponge,
 };
 use ark_ec::{CurveGroup, Group};
@@ -17,6 +13,7 @@ use ark_r1cs_std::{
     fields::{fp::FpVar, FieldVar},
     groups::GroupOpsBounds,
     prelude::CurveVar,
+    uint8::UInt8,
     R1CSVar, ToConstraintFieldGadget,
 };
 use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystemRef, Namespace, SynthesisError};
@@ -29,13 +26,11 @@ use crate::folding::circuits::{
     cyclefold::{
         CycleFoldChallengeGadget, CycleFoldCommittedInstanceVar, NIFSFullGadget, CF_IO_LEN,
     },
-    nonnative::{
-        affine::{nonnative_affine_to_field_elements, NonNativeAffineVar},
-        uint::NonNativeUintVar,
-    },
+    nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
     CF1, CF2,
 };
 use crate::frontend::FCircuit;
+use crate::transcript::{AbsorbNonNativeGadget, Transcript, TranscriptVar};
 
 /// CommittedInstanceVar contains the u, x, cmE and cmW values which are folded on the main Nova
 /// constraints field (E1::Fr, where E1 is the main curve). The peculiarity is that cmE and cmW are
@@ -78,6 +73,26 @@ where
     }
 }
 
+impl<C> AbsorbGadget<C::ScalarField> for CommittedInstanceVar<C>
+where
+    C: CurveGroup,
+    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
+{
+    fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
+        unimplemented!()
+    }
+
+    fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
+        Ok([
+            vec![self.u.clone()],
+            self.x.clone(),
+            self.cmE.to_constraint_field()?,
+            self.cmW.to_constraint_field()?,
+        ]
+        .concat())
+    }
+}
+
 impl<C> CommittedInstanceVar<C>
 where
     C: CurveGroup,
@@ -91,26 +106,22 @@ where
     /// Additionally it returns the vector of the field elements from the self parameters, so they
     /// can be reused in other gadgets avoiding recalculating (reconstraining) them.
     #[allow(clippy::type_complexity)]
-    pub fn hash(
+    pub fn hash<S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>>(
         self,
-        crh_params: &CRHParametersVar<CF1<C>>,
+        sponge: &T,
         pp_hash: FpVar<CF1<C>>,
         i: FpVar<CF1<C>>,
         z_0: Vec<FpVar<CF1<C>>>,
         z_i: Vec<FpVar<CF1<C>>>,
     ) -> Result<(FpVar<CF1<C>>, Vec<FpVar<CF1<C>>>), SynthesisError> {
-        let U_vec = [
-            vec![self.u],
-            self.x,
-            self.cmE.to_constraint_field()?,
-            self.cmW.to_constraint_field()?,
-        ]
-        .concat();
-        let input = [vec![pp_hash, i], z_0, z_i, U_vec.clone()].concat();
-        Ok((
-            CRHGadget::<C::ScalarField>::evaluate(crh_params, &input)?,
-            U_vec,
-        ))
+        let mut sponge = sponge.clone();
+        let U_vec = self.to_sponge_field_elements()?;
+        sponge.absorb(&pp_hash)?;
+        sponge.absorb(&i)?;
+        sponge.absorb(&z_0)?;
+        sponge.absorb(&z_i)?;
+        sponge.absorb(&U_vec)?;
+        Ok((sponge.squeeze_field_elements(1)?.pop().unwrap(), U_vec))
     }
 }
 
@@ -174,67 +185,33 @@ where
     <C as CurveGroup>::BaseField: PrimeField,
     <C as Group>::ScalarField: Absorb,
 {
-    pub fn get_challenge_native(
-        poseidon_config: &PoseidonConfig<C::ScalarField>,
+    pub fn get_challenge_native<T: Transcript<C::ScalarField>>(
+        transcript: &mut T,
         pp_hash: C::ScalarField, // public params hash
         U_i: CommittedInstance<C>,
         u_i: CommittedInstance<C>,
         cmT: C,
-    ) -> Result<Vec<bool>, SynthesisError> {
-        let (U_cmE_x, U_cmE_y) = nonnative_affine_to_field_elements::<C>(U_i.cmE)?;
-        let (U_cmW_x, U_cmW_y) = nonnative_affine_to_field_elements::<C>(U_i.cmW)?;
-        let (u_cmE_x, u_cmE_y) = nonnative_affine_to_field_elements::<C>(u_i.cmE)?;
-        let (u_cmW_x, u_cmW_y) = nonnative_affine_to_field_elements::<C>(u_i.cmW)?;
-        let (cmT_x, cmT_y) = nonnative_affine_to_field_elements::<C>(cmT)?;
-
-        let mut sponge = PoseidonSponge::<C::ScalarField>::new(poseidon_config);
-        let input = vec![
-            vec![pp_hash],
-            vec![U_i.u],
-            U_i.x.clone(),
-            U_cmE_x,
-            U_cmE_y,
-            U_cmW_x,
-            U_cmW_y,
-            vec![u_i.u],
-            u_i.x.clone(),
-            u_cmE_x,
-            u_cmE_y,
-            u_cmW_x,
-            u_cmW_y,
-            cmT_x,
-            cmT_y,
-        ]
-        .concat();
-        sponge.absorb(&input);
-        let bits = sponge.squeeze_bits(N_BITS_RO);
-        Ok(bits)
+    ) -> Vec<bool> {
+        transcript.absorb(&pp_hash);
+        transcript.absorb(&U_i);
+        transcript.absorb(&u_i);
+        transcript.absorb_nonnative(&cmT);
+        transcript.squeeze_bits(N_BITS_RO)
     }
 
     // compatible with the native get_challenge_native
-    pub fn get_challenge_gadget(
-        cs: ConstraintSystemRef<C::ScalarField>,
-        poseidon_config: &PoseidonConfig<C::ScalarField>,
+    pub fn get_challenge_gadget<S: CryptographicSponge, T: TranscriptVar<CF1<C>, S>>(
+        transcript: &mut T,
         pp_hash: FpVar<CF1<C>>,      // public params hash
         U_i_vec: Vec<FpVar<CF1<C>>>, // apready processed input, so we don't have to recompute these values
         u_i: CommittedInstanceVar<C>,
         cmT: NonNativeAffineVar<C>,
     ) -> Result<Vec<Boolean<C::ScalarField>>, SynthesisError> {
-        let mut sponge = PoseidonSpongeVar::<C::ScalarField>::new(cs, poseidon_config);
-
-        let input: Vec<FpVar<C::ScalarField>> = [
-            vec![pp_hash],
-            U_i_vec,
-            vec![u_i.u.clone()],
-            u_i.x.clone(),
-            u_i.cmE.to_constraint_field()?,
-            u_i.cmW.to_constraint_field()?,
-            cmT.to_constraint_field()?,
-        ]
-        .concat();
-        sponge.absorb(&input)?;
-        let bits = sponge.squeeze_bits(N_BITS_RO)?;
-        Ok(bits)
+        transcript.absorb(&pp_hash)?;
+        transcript.absorb(&U_i_vec)?;
+        transcript.absorb(&u_i)?;
+        transcript.absorb_nonnative(&cmT)?;
+        transcript.squeeze_bits(N_BITS_RO)
     }
 }
 
@@ -367,10 +344,10 @@ where
         let cf1_cmT = GC2::new_witness(cs.clone(), || Ok(self.cf1_cmT.unwrap_or_else(C2::zero)))?;
         let cf2_cmT = GC2::new_witness(cs.clone(), || Ok(self.cf2_cmT.unwrap_or_else(C2::zero)))?;
 
-        let crh_params = CRHParametersVar::<C1::ScalarField>::new_constant(
-            cs.clone(),
-            self.poseidon_config.clone(),
-        )?;
+        // `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();
 
         // get z_{i+1} from the F circuit
         let i_usize = self.i_usize.unwrap_or(0);
@@ -384,14 +361,14 @@ where
         // P.1. Compute u_i.x
         // u_i.x[0] = H(i, z_0, z_i, U_i)
         let (u_i_x, U_i_vec) = U_i.clone().hash(
-            &crh_params,
+            &sponge,
             pp_hash.clone(),
             i.clone(),
             z_0.clone(),
             z_i.clone(),
         )?;
         // u_i.x[1] = H(cf_U_i)
-        let (cf_u_i_x, cf_U_i_vec) = cf_U_i.clone().hash(&crh_params, pp_hash.clone())?;
+        let (cf_u_i_x, cf_U_i_vec) = cf_U_i.clone().hash(&sponge, pp_hash.clone())?;
 
         // P.2. Construct u_i
         let u_i = CommittedInstanceVar {
@@ -411,8 +388,7 @@ where
 
         // compute r = H(u_i, U_i, cmT)
         let r_bits = ChallengeGadget::<C1>::get_challenge_gadget(
-            cs.clone(),
-            &self.poseidon_config,
+            &mut transcript,
             pp_hash.clone(),
             U_i_vec,
             u_i.clone(),
@@ -438,14 +414,14 @@ where
         // Base case: u_{i+1}.x[0] == H((i+1, z_0, z_{i+1}, U_{\bot})
         // Non-base case: u_{i+1}.x[0] == H((i+1, z_0, z_{i+1}, U_{i+1})
         let (u_i1_x, _) = U_i1.clone().hash(
-            &crh_params,
+            &sponge,
             pp_hash.clone(),
             i + FpVar::<CF1<C1>>::one(),
             z_0.clone(),
             z_i1.clone(),
         )?;
         let (u_i1_x_base, _) = CommittedInstanceVar::new_constant(cs.clone(), u_dummy)?.hash(
-            &crh_params,
+            &sponge,
             pp_hash.clone(),
             FpVar::<CF1<C1>>::one(),
             z_0.clone(),
@@ -499,8 +475,7 @@ where
         // compute cf1_r = H(cf1_u_i, cf_U_i, cf1_cmT)
         // cf_r_bits is denoted by rho* in the paper.
         let cf1_r_bits = CycleFoldChallengeGadget::<C2, GC2>::get_challenge_gadget(
-            cs.clone(),
-            &self.poseidon_config,
+            &mut transcript,
             pp_hash.clone(),
             cf_U_i_vec,
             cf1_u_i.clone(),
@@ -523,10 +498,9 @@ where
 
         // same for cf2_r:
         let cf2_r_bits = CycleFoldChallengeGadget::<C2, GC2>::get_challenge_gadget(
-            cs.clone(),
-            &self.poseidon_config,
+            &mut transcript,
             pp_hash.clone(),
-            cf1_U_i1.to_constraint_field()?,
+            cf1_U_i1.to_native_sponge_field_elements()?,
             cf2_u_i.clone(),
             cf2_cmT.clone(),
         )?;
@@ -547,10 +521,10 @@ where
         // P.4.b compute and check the second output of F'
         // Base case: u_{i+1}.x[1] == H(cf_U_{\bot})
         // Non-base case: u_{i+1}.x[1] == H(cf_U_{i+1})
-        let (cf_u_i1_x, _) = cf_U_i1.clone().hash(&crh_params, pp_hash.clone())?;
+        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)?
-                .hash(&crh_params, pp_hash)?;
+                .hash(&sponge, pp_hash)?;
         let cf_x = FpVar::new_input(cs.clone(), || {
             Ok(self.cf_x.unwrap_or(cf_u_i1_x_base.value()?))
         })?;
@@ -564,6 +538,7 @@ where
 pub mod tests {
     use super::*;
     use ark_bn254::{Fr, G1Projective as Projective};
+    use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
     use ark_ff::BigInteger;
     use ark_relations::r1cs::ConstraintSystem;
     use ark_std::UniformRand;
@@ -628,6 +603,7 @@ pub mod tests {
     fn test_committed_instance_hash() {
         let mut rng = ark_std::test_rng();
         let poseidon_config = poseidon_canonical_config::<Fr>();
+        let sponge = PoseidonSponge::<Fr>::new(&poseidon_config);
         let pp_hash = Fr::from(42u32); // only for test
 
         let i = Fr::from(3_u32);
@@ -641,9 +617,7 @@ pub mod tests {
         };
 
         // compute the CommittedInstance hash natively
-        let h = ci
-            .hash(&poseidon_config, pp_hash, i, z_0.clone(), z_i.clone())
-            .unwrap();
+        let h = ci.hash(&sponge, pp_hash, i, z_0.clone(), z_i.clone());
 
         let cs = ConstraintSystem::<Fr>::new_ref();
 
@@ -654,11 +628,11 @@ pub mod tests {
         let ciVar =
             CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(ci.clone())).unwrap();
 
-        let crh_params = CRHParametersVar::<Fr>::new_constant(cs.clone(), poseidon_config).unwrap();
+        let sponge = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
 
         // compute the CommittedInstance hash in-circuit
         let (hVar, _) = ciVar
-            .hash(&crh_params, pp_hashVar, iVar, z_0Var, z_iVar)
+            .hash(&sponge, pp_hashVar, iVar, z_0Var, z_iVar)
             .unwrap();
         assert!(cs.is_satisfied().unwrap());
 
@@ -671,6 +645,7 @@ pub mod tests {
     fn test_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),
@@ -690,13 +665,12 @@ pub mod tests {
 
         // compute the challenge natively
         let r_bits = ChallengeGadget::<Projective>::get_challenge_native(
-            &poseidon_config,
+            &mut transcript,
             pp_hash,
             U_i.clone(),
             u_i.clone(),
             cmT,
-        )
-        .unwrap();
+        );
         let r = Fr::from_bigint(BigInteger::from_bits_le(&r_bits)).unwrap();
 
         let cs = ConstraintSystem::<Fr>::new_ref();
@@ -708,6 +682,7 @@ pub mod tests {
             CommittedInstanceVar::<Projective>::new_witness(cs.clone(), || Ok(U_i.clone()))
                 .unwrap();
         let cmTVar = NonNativeAffineVar::<Projective>::new_witness(cs.clone(), || Ok(cmT)).unwrap();
+        let mut transcriptVar = PoseidonSpongeVar::<Fr>::new(cs.clone(), &poseidon_config);
 
         // compute the challenge in-circuit
         let U_iVar_vec = [
@@ -718,8 +693,7 @@ pub mod tests {
         ]
         .concat();
         let r_bitsVar = ChallengeGadget::<Projective>::get_challenge_gadget(
-            cs.clone(),
-            &poseidon_config,
+            &mut transcriptVar,
             pp_hashVar,
             U_iVar_vec,
             u_iVar,
diff --git a/folding-schemes/src/folding/nova/decider_eth.rs b/folding-schemes/src/folding/nova/decider_eth.rs
index c0727d4..4b011e8 100644
--- a/folding-schemes/src/folding/nova/decider_eth.rs
+++ b/folding-schemes/src/folding/nova/decider_eth.rs
@@ -317,13 +317,11 @@ fn point2_to_eth_format(p: ark_bn254::G2Affine) -> Result<Vec<u8>, Error> {
 
 #[cfg(test)]
 pub mod tests {
-    use ark_bn254::{constraints::GVar, Bn254, Fr, G1Projective as Projective};
-    use ark_groth16::Groth16;
+    use ark_bn254::{constraints::GVar, Fr, G1Projective as Projective};
     use ark_grumpkin::{constraints::GVar as GVar2, Projective as Projective2};
     use std::time::Instant;
 
     use super::*;
-    use crate::commitment::kzg::KZG;
     use crate::commitment::pedersen::Pedersen;
     use crate::folding::nova::PreprocessorParam;
     use crate::frontend::tests::CubicFCircuit;
diff --git a/folding-schemes/src/folding/nova/decider_eth_circuit.rs b/folding-schemes/src/folding/nova/decider_eth_circuit.rs
index 8e399c9..2a0ab0b 100644
--- a/folding-schemes/src/folding/nova/decider_eth_circuit.rs
+++ b/folding-schemes/src/folding/nova/decider_eth_circuit.rs
@@ -1,7 +1,10 @@
 /// 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::crh::poseidon::constraints::CRHParametersVar;
-use ark_crypto_primitives::sponge::{poseidon::PoseidonConfig, Absorb};
+use ark_crypto_primitives::sponge::{
+    constraints::CryptographicSpongeVar,
+    poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
+    Absorb, CryptographicSponge,
+};
 use ark_ec::{CurveGroup, Group};
 use ark_ff::{BigInteger, PrimeField};
 use ark_poly::Polynomial;
@@ -23,18 +26,12 @@ use super::{circuits::ChallengeGadget, nifs::NIFS};
 use crate::arith::r1cs::R1CS;
 use crate::commitment::{pedersen::Params as PedersenParams, CommitmentScheme};
 use crate::folding::circuits::{
-    nonnative::{
-        affine::{nonnative_affine_to_field_elements, NonNativeAffineVar},
-        uint::NonNativeUintVar,
-    },
+    nonnative::{affine::NonNativeAffineVar, uint::NonNativeUintVar},
     CF1, CF2,
 };
 use crate::folding::nova::{circuits::CommittedInstanceVar, CommittedInstance, Nova, Witness};
 use crate::frontend::FCircuit;
-use crate::transcript::{
-    poseidon::{PoseidonTranscript, PoseidonTranscriptVar},
-    Transcript, TranscriptVar,
-};
+use crate::transcript::{Transcript, TranscriptVar};
 use crate::utils::{
     gadgets::{MatrixGadget, SparseMatrixVar, VectorGadget},
     vec::poly_from_vec,
@@ -264,6 +261,8 @@ where
     pub fn from_nova<FC: FCircuit<C1::ScalarField>>(
         nova: Nova<C1, GC1, C2, GC2, FC, CS1, CS2>,
     ) -> Result<Self, Error> {
+        let mut transcript = PoseidonSponge::<C1::ScalarField>::new(&nova.poseidon_config);
+
         // compute the U_{i+1}, W_{i+1}
         let (T, cmT) = NIFS::<C1, CS1>::compute_cmT(
             &nova.cs_pp,
@@ -274,12 +273,12 @@ where
             &nova.U_i.clone(),
         )?;
         let r_bits = ChallengeGadget::<C1>::get_challenge_native(
-            &nova.poseidon_config,
+            &mut transcript,
             nova.pp_hash,
             nova.U_i.clone(),
             nova.u_i.clone(),
             cmT,
-        )?;
+        );
         let r_Fr = C1::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits))
             .ok_or(Error::OutOfBounds)?;
         let (W_i1, U_i1) = NIFS::<C1, CS1>::fold_instances(
@@ -288,7 +287,7 @@ where
 
         // compute the KZG challenges used as inputs in the circuit
         let (kzg_challenge_W, kzg_challenge_E) =
-            KZGChallengesGadget::<C1>::get_challenges_native(&nova.poseidon_config, U_i1.clone())?;
+            KZGChallengesGadget::<C1>::get_challenges_native(&mut transcript, U_i1.clone());
 
         // get KZG evals
         let mut W = W_i1.W.clone();
@@ -410,10 +409,10 @@ where
             Ok(self.eval_E.unwrap_or_else(CF1::<C1>::zero))
         })?;
 
-        let crh_params = CRHParametersVar::<C1::ScalarField>::new_constant(
-            cs.clone(),
-            self.poseidon_config.clone(),
-        )?;
+        // `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();
 
         // 1. check RelaxedR1CS of U_{i+1}
         let z_U1: Vec<FpVar<CF1<C1>>> =
@@ -429,7 +428,7 @@ where
 
         // 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(
-            &crh_params,
+            &sponge,
             pp_hash.clone(),
             i.clone(),
             z_0.clone(),
@@ -465,7 +464,7 @@ where
             })?;
 
             // 3.b u_i.x[1] == H(cf_U_i)
-            let (cf_u_i_x, _) = cf_U_i.clone().hash(&crh_params, pp_hash.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 Pedersen commitments of cf_U_i.{cmE, cmW}
@@ -498,12 +497,23 @@ where
             RelaxedR1CSGadget::check_nonnative(cf_r1cs, cf_W_i.E, cf_U_i.u.clone(), cf_z_U)?;
         }
 
-        // 6. check KZG challenges
-        let (incircuit_c_W, incircuit_c_E) = KZGChallengesGadget::<C1>::get_challenges_gadget(
-            cs.clone(),
-            &self.poseidon_config,
-            U_i1.clone(),
+        // 8.a, 6.a 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)))?;
+        let r_bits = ChallengeGadget::<C1>::get_challenge_gadget(
+            &mut transcript,
+            pp_hash,
+            U_i_vec,
+            u_i.clone(),
+            cmT.clone(),
         )?;
+        let (incircuit_c_W, incircuit_c_E) =
+            KZGChallengesGadget::<C1>::get_challenges_gadget(&mut transcript, U_i1.clone())?;
+
+        // 6.b check KZG challenges
         incircuit_c_W.enforce_equal(&kzg_c_W)?;
         incircuit_c_E.enforce_equal(&kzg_c_E)?;
 
@@ -516,18 +526,8 @@ where
         // incircuit_eval_W.enforce_equal(&eval_W)?;
         // incircuit_eval_E.enforce_equal(&eval_E)?;
 
-        // 8. compute the NIFS.V challenge and check that matches the one from the public input (so we
+        // 8.b check the NIFS.V challenge matches the one from the public input (so we
         // avoid the verifier computing it)
-        let cmT =
-            NonNativeAffineVar::new_input(cs.clone(), || Ok(self.cmT.unwrap_or_else(C1::zero)))?;
-        let r_bits = ChallengeGadget::<C1>::get_challenge_gadget(
-            cs.clone(),
-            &self.poseidon_config,
-            pp_hash,
-            U_i_vec,
-            u_i.clone(),
-            cmT.clone(),
-        )?;
         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 =
@@ -569,38 +569,28 @@ where
     <C as CurveGroup>::BaseField: PrimeField,
     C::ScalarField: Absorb,
 {
-    pub fn get_challenges_native(
-        poseidon_config: &PoseidonConfig<C::ScalarField>,
+    pub fn get_challenges_native<T: Transcript<C::ScalarField>>(
+        transcript: &mut T,
         U_i: CommittedInstance<C>,
-    ) -> Result<(C::ScalarField, C::ScalarField), Error> {
-        let (cmE_x_limbs, cmE_y_limbs) = nonnative_affine_to_field_elements(U_i.cmE)?;
-        let (cmW_x_limbs, cmW_y_limbs) = nonnative_affine_to_field_elements(U_i.cmW)?;
-
-        let transcript = &mut PoseidonTranscript::<C>::new(poseidon_config);
+    ) -> (C::ScalarField, C::ScalarField) {
         // compute the KZG challenges, which are computed in-circuit and checked that it matches
         // the inputted one
-        transcript.absorb_vec(&cmW_x_limbs);
-        transcript.absorb_vec(&cmW_y_limbs);
+        transcript.absorb_nonnative(&U_i.cmW);
         let challenge_W = transcript.get_challenge();
-        transcript.absorb_vec(&cmE_x_limbs);
-        transcript.absorb_vec(&cmE_y_limbs);
+        transcript.absorb_nonnative(&U_i.cmE);
         let challenge_E = transcript.get_challenge();
 
-        Ok((challenge_W, challenge_E))
+        (challenge_W, challenge_E)
     }
     // compatible with the native get_challenges_native
-    pub fn get_challenges_gadget(
-        cs: ConstraintSystemRef<C::ScalarField>,
-        poseidon_config: &PoseidonConfig<C::ScalarField>,
+    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> {
-        let mut transcript =
-            PoseidonTranscriptVar::<CF1<C>>::new(cs.clone(), &poseidon_config.clone());
-
-        transcript.absorb_vec(&U_i.cmW.to_constraint_field()?[..])?;
+        transcript.absorb(&U_i.cmW.to_constraint_field()?)?;
         let challenge_W = transcript.get_challenge()?;
 
-        transcript.absorb_vec(&U_i.cmE.to_constraint_field()?[..])?;
+        transcript.absorb(&U_i.cmE.to_constraint_field()?)?;
         let challenge_E = transcript.get_challenge()?;
 
         Ok((challenge_W, challenge_E))
@@ -852,6 +842,7 @@ pub mod tests {
     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),
@@ -862,21 +853,17 @@ pub mod tests {
 
         // compute the challenge natively
         let (challenge_W, challenge_E) =
-            KZGChallengesGadget::<Projective>::get_challenges_native(&poseidon_config, U_i.clone())
-                .unwrap();
+            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(
-                cs.clone(),
-                &poseidon_config,
-                U_iVar,
-            )
-            .unwrap();
+            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
diff --git a/folding-schemes/src/folding/nova/mod.rs b/folding-schemes/src/folding/nova/mod.rs
index 071959c..4847314 100644
--- a/folding-schemes/src/folding/nova/mod.rs
+++ b/folding-schemes/src/folding/nova/mod.rs
@@ -1,11 +1,11 @@
 /// Implements the scheme described in [Nova](https://eprint.iacr.org/2021/370.pdf) and
 /// [CycleFold](https://eprint.iacr.org/2023/1192.pdf).
-use ark_crypto_primitives::{
-    crh::{poseidon::CRH, CRHScheme},
-    sponge::{poseidon::PoseidonConfig, Absorb},
+use ark_crypto_primitives::sponge::{
+    poseidon::{PoseidonConfig, PoseidonSponge},
+    Absorb, CryptographicSponge,
 };
 use ark_ec::{AffineRepr, CurveGroup, Group};
-use ark_ff::{BigInteger, Field, PrimeField, ToConstraintField};
+use ark_ff::{BigInteger, PrimeField};
 use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar, ToConstraintFieldGadget};
 use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
 use ark_serialize::{CanonicalDeserialize, CanonicalSerialize};
@@ -14,19 +14,18 @@ use ark_std::rand::RngCore;
 use ark_std::{One, Zero};
 use core::marker::PhantomData;
 
-use crate::arith::r1cs::{extract_r1cs, extract_w_x, R1CS};
 use crate::commitment::CommitmentScheme;
-use crate::folding::circuits::{
-    cyclefold::{fold_cyclefold_circuit, CycleFoldCircuit},
-    nonnative::{
-        affine::nonnative_affine_to_field_elements, uint::nonnative_field_to_field_elements,
-    },
-    CF2,
-};
+use crate::folding::circuits::cyclefold::{fold_cyclefold_circuit, CycleFoldCircuit};
+use crate::folding::circuits::CF2;
 use crate::frontend::FCircuit;
-use crate::utils::{get_cm_coordinates, pp_hash, vec::is_zero_vec};
+use crate::transcript::{AbsorbNonNative, Transcript};
+use crate::utils::vec::is_zero_vec;
 use crate::Error;
 use crate::FoldingScheme;
+use crate::{
+    arith::r1cs::{extract_r1cs, extract_w_x, R1CS},
+    utils::{get_cm_coordinates, pp_hash},
+};
 
 pub mod circuits;
 pub mod decider_eth;
@@ -57,6 +56,54 @@ impl<C: CurveGroup> CommittedInstance<C> {
     }
 }
 
+impl<C: CurveGroup> Absorb for CommittedInstance<C>
+where
+    C::ScalarField: Absorb,
+{
+    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
+        // This is never called
+        unimplemented!()
+    }
+
+    fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
+        self.u.to_sponge_field_elements(dest);
+        self.x.to_sponge_field_elements(dest);
+        // We cannot call `to_native_sponge_field_elements(dest)` directly, as
+        // `to_native_sponge_field_elements` needs `F` to be `C::ScalarField`,
+        // but here `F` is a generic `PrimeField`.
+        self.cmE
+            .to_native_sponge_field_elements_as_vec()
+            .to_sponge_field_elements(dest);
+        self.cmW
+            .to_native_sponge_field_elements_as_vec()
+            .to_sponge_field_elements(dest);
+    }
+}
+
+impl<C: CurveGroup> AbsorbNonNative<C::BaseField> for CommittedInstance<C>
+where
+    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField + Absorb,
+{
+    // Compatible with the in-circuit `CycleFoldCommittedInstanceVar::to_native_sponge_field_elements`
+    // in `cyclefold.rs`.
+    fn to_native_sponge_field_elements(&self, dest: &mut Vec<C::BaseField>) {
+        [self.u].to_native_sponge_field_elements(dest);
+        self.x.to_native_sponge_field_elements(dest);
+        let (cmE_x, cmE_y) = match self.cmE.into_affine().xy() {
+            Some((&x, &y)) => (x, y),
+            None => (C::BaseField::zero(), C::BaseField::zero()),
+        };
+        let (cmW_x, cmW_y) = match self.cmW.into_affine().xy() {
+            Some((&x, &y)) => (x, y),
+            None => (C::BaseField::zero(), C::BaseField::zero()),
+        };
+        cmE_x.to_sponge_field_elements(dest);
+        cmE_y.to_sponge_field_elements(dest);
+        cmW_x.to_sponge_field_elements(dest);
+        cmW_y.to_sponge_field_elements(dest);
+    }
+}
+
 impl<C: CurveGroup> CommittedInstance<C>
 where
     <C as Group>::ScalarField: Absorb,
@@ -66,67 +113,21 @@ where
     /// nova/circuits.rs::CommittedInstanceVar.hash.
     /// Returns `H(i, z_0, z_i, U_i)`, where `i` can be `i` but also `i+1`, and `U_i` is the
     /// `CommittedInstance`.
-    pub fn hash(
+    pub fn hash<T: Transcript<C::ScalarField>>(
         &self,
-        poseidon_config: &PoseidonConfig<C::ScalarField>,
+        sponge: &T,
         pp_hash: C::ScalarField, // public params hash
         i: C::ScalarField,
         z_0: Vec<C::ScalarField>,
         z_i: Vec<C::ScalarField>,
-    ) -> Result<C::ScalarField, Error> {
-        let (cmE_x, cmE_y) = nonnative_affine_to_field_elements::<C>(self.cmE)?;
-        let (cmW_x, cmW_y) = nonnative_affine_to_field_elements::<C>(self.cmW)?;
-
-        CRH::<C::ScalarField>::evaluate(
-            poseidon_config,
-            vec![
-                vec![pp_hash, i],
-                z_0,
-                z_i,
-                vec![self.u],
-                self.x.clone(),
-                cmE_x,
-                cmE_y,
-                cmW_x,
-                cmW_y,
-            ]
-            .concat(),
-        )
-        .map_err(|e| Error::Other(e.to_string()))
-    }
-}
-
-impl<C: CurveGroup> ToConstraintField<C::BaseField> for CommittedInstance<C>
-where
-    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField + Absorb,
-{
-    fn to_field_elements(&self) -> Option<Vec<C::BaseField>> {
-        let u = nonnative_field_to_field_elements(&self.u);
-        let x = self
-            .x
-            .iter()
-            .flat_map(nonnative_field_to_field_elements)
-            .collect::<Vec<_>>();
-        let (cmE_x, cmE_y, cmE_is_inf) = match self.cmE.into_affine().xy() {
-            Some((&x, &y)) => (x, y, C::BaseField::zero()),
-            None => (
-                C::BaseField::zero(),
-                C::BaseField::zero(),
-                C::BaseField::one(),
-            ),
-        };
-        let (cmW_x, cmW_y, cmW_is_inf) = match self.cmW.into_affine().xy() {
-            Some((&x, &y)) => (x, y, C::BaseField::zero()),
-            None => (
-                C::BaseField::zero(),
-                C::BaseField::zero(),
-                C::BaseField::one(),
-            ),
-        };
-        // Concatenate `cmE_is_inf` and `cmW_is_inf` to save constraints for CRHGadget::evaluate in the corresponding circuit
-        let is_inf = cmE_is_inf.double() + cmW_is_inf;
-
-        Some([u, x, vec![cmE_x, cmE_y, cmW_x, cmW_y, is_inf]].concat())
+    ) -> C::ScalarField {
+        let mut sponge = sponge.clone();
+        sponge.absorb(&pp_hash);
+        sponge.absorb(&i);
+        sponge.absorb(&z_0);
+        sponge.absorb(&z_i);
+        sponge.absorb(&self);
+        sponge.squeeze_field_elements(1)[0]
     }
 }
 
@@ -137,16 +138,15 @@ where
     /// hash_cyclefold implements the committed instance hash compatible with the gadget implemented in
     /// nova/cyclefold.rs::CycleFoldCommittedInstanceVar.hash.
     /// Returns `H(U_i)`, where `U_i` is the `CommittedInstance` for CycleFold.
-    pub fn hash_cyclefold(
+    pub fn hash_cyclefold<T: Transcript<C::BaseField>>(
         &self,
-        poseidon_config: &PoseidonConfig<C::BaseField>,
+        sponge: &T,
         pp_hash: C::BaseField, // public params hash
-    ) -> Result<C::BaseField, Error> {
-        CRH::<C::BaseField>::evaluate(
-            poseidon_config,
-            [vec![pp_hash], self.to_field_elements().unwrap()].concat(),
-        )
-        .map_err(|e| Error::Other(e.to_string()))
+    ) -> C::BaseField {
+        let mut sponge = sponge.clone();
+        sponge.absorb(&pp_hash);
+        sponge.absorb_nonnative(self);
+        sponge.squeeze_field_elements(1)[0]
     }
 }
 
@@ -453,6 +453,11 @@ where
         _rng: impl RngCore,
         external_inputs: Vec<C1::ScalarField>,
     ) -> Result<(), Error> {
+        // `sponge` is for digest computation.
+        let sponge = PoseidonSponge::<C1::ScalarField>::new(&self.poseidon_config);
+        // `transcript` is for challenge generation.
+        let mut transcript = sponge.clone();
+
         let augmented_F_circuit: AugmentedFCircuit<C1, C2, GC2, FC>;
 
         if self.z_i.len() != self.F.state_len() {
@@ -488,12 +493,12 @@ where
 
         // r_bits is the r used to the RLC of the F' instances
         let r_bits = ChallengeGadget::<C1>::get_challenge_native(
-            &self.poseidon_config,
+            &mut transcript,
             self.pp_hash,
             self.U_i.clone(),
             self.u_i.clone(),
             cmT,
-        )?;
+        );
         let r_Fr = C1::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits))
             .ok_or(Error::OutOfBounds)?;
         let r_Fq = C1::BaseField::from_bigint(BigInteger::from_bits_le(&r_bits))
@@ -507,19 +512,17 @@ where
         // folded instance output (public input, x)
         // u_{i+1}.x[0] = H(i+1, z_0, z_{i+1}, U_{i+1})
         let u_i1_x = U_i1.hash(
-            &self.poseidon_config,
+            &sponge,
             self.pp_hash,
             self.i + C1::ScalarField::one(),
             self.z_0.clone(),
             z_i1.clone(),
-        )?;
+        );
         // u_{i+1}.x[1] = H(cf_U_{i+1})
         let cf_u_i1_x: C1::ScalarField;
 
         if self.i == C1::ScalarField::zero() {
-            cf_u_i1_x = self
-                .cf_U_i
-                .hash_cyclefold(&self.poseidon_config, self.pp_hash)?;
+            cf_u_i1_x = self.cf_U_i.hash_cyclefold(&sponge, self.pp_hash);
             // base case
             augmented_F_circuit = AugmentedFCircuit::<C1, C2, GC2, FC> {
                 _gc2: PhantomData,
@@ -584,16 +587,22 @@ where
 
             // fold self.cf_U_i + cfW_U -> folded running with cfW
             let (_cfW_w_i, cfW_u_i, cfW_W_i1, cfW_U_i1, cfW_cmT, _) = self.fold_cyclefold_circuit(
+                &mut transcript,
                 self.cf_W_i.clone(), // CycleFold running instance witness
                 self.cf_U_i.clone(), // CycleFold running instance
                 cfW_u_i_x,
                 cfW_circuit,
             )?;
             // fold [the output from folding self.cf_U_i + cfW_U] + cfE_U = folded_running_with_cfW + cfE
-            let (_cfE_w_i, cfE_u_i, cf_W_i1, cf_U_i1, cf_cmT, _) =
-                self.fold_cyclefold_circuit(cfW_W_i1, cfW_U_i1.clone(), cfE_u_i_x, cfE_circuit)?;
+            let (_cfE_w_i, cfE_u_i, cf_W_i1, cf_U_i1, cf_cmT, _) = self.fold_cyclefold_circuit(
+                &mut transcript,
+                cfW_W_i1,
+                cfW_U_i1.clone(),
+                cfE_u_i_x,
+                cfE_circuit,
+            )?;
 
-            cf_u_i1_x = cf_U_i1.hash_cyclefold(&self.poseidon_config, self.pp_hash)?;
+            cf_u_i1_x = cf_U_i1.hash_cyclefold(&sponge, self.pp_hash);
 
             augmented_F_circuit = AugmentedFCircuit::<C1, C2, GC2, FC> {
                 _gc2: PhantomData,
@@ -697,6 +706,8 @@ where
         incoming_instance: Self::IncomingInstance,
         cyclefold_instance: Self::CFInstance,
     ) -> Result<(), Error> {
+        let sponge = PoseidonSponge::<C1::ScalarField>::new(&vp.poseidon_config);
+
         if num_steps == C1::ScalarField::zero() {
             if z_0 != z_i {
                 return Err(Error::IVCVerificationFail);
@@ -716,12 +727,12 @@ where
 
         // check that u_i's output points to the running instance
         // u_i.X[0] == H(i, z_0, z_i, U_i)
-        let expected_u_i_x = U_i.hash(&vp.poseidon_config, pp_hash, num_steps, z_0, z_i.clone())?;
+        let expected_u_i_x = U_i.hash(&sponge, pp_hash, num_steps, z_0, z_i.clone());
         if expected_u_i_x != u_i.x[0] {
             return Err(Error::IVCVerificationFail);
         }
         // u_i.X[1] == H(cf_U_i)
-        let expected_cf_u_i_x = cf_U_i.hash_cyclefold(&vp.poseidon_config, pp_hash)?;
+        let expected_cf_u_i_x = cf_U_i.hash_cyclefold(&sponge, pp_hash);
         if expected_cf_u_i_x != u_i.x[1] {
             return Err(Error::IVCVerificationFail);
         }
@@ -790,8 +801,9 @@ where
 {
     // folds the given cyclefold circuit and its instances
     #[allow(clippy::type_complexity)]
-    fn fold_cyclefold_circuit(
+    fn fold_cyclefold_circuit<T: Transcript<C1::ScalarField>>(
         &self,
+        transcript: &mut T,
         cf_W_i: Witness<C2>,           // witness of the running instance
         cf_U_i: CommittedInstance<C2>, // running instance
         cf_u_i_x: Vec<C2::ScalarField>,
@@ -808,7 +820,7 @@ where
         Error,
     > {
         fold_cyclefold_circuit::<C1, GC1, C2, GC2, FC, CS1, CS2>(
-            &self.poseidon_config,
+            transcript,
             self.cf_r1cs.clone(),
             self.cf_cs_pp.clone(),
             self.pp_hash,
diff --git a/folding-schemes/src/folding/nova/nifs.rs b/folding-schemes/src/folding/nova/nifs.rs
index 9e5a1be..725247e 100644
--- a/folding-schemes/src/folding/nova/nifs.rs
+++ b/folding-schemes/src/folding/nova/nifs.rs
@@ -183,7 +183,7 @@ where
     }
 
     pub fn prove_commitments(
-        tr: &mut impl Transcript<C>,
+        tr: &mut impl Transcript<C::ScalarField>,
         cs_prover_params: &CS::ProverParams,
         w: &Witness<C>,
         ci: &CommittedInstance<C>,
@@ -200,7 +200,10 @@ where
 #[cfg(test)]
 pub mod tests {
     use super::*;
-    use ark_crypto_primitives::sponge::poseidon::PoseidonConfig;
+    use ark_crypto_primitives::sponge::{
+        poseidon::{PoseidonConfig, PoseidonSponge},
+        CryptographicSponge,
+    };
     use ark_ff::{BigInteger, PrimeField};
     use ark_pallas::{Fr, Projective};
     use ark_std::{ops::Mul, UniformRand};
@@ -209,7 +212,7 @@ pub mod tests {
     use crate::commitment::pedersen::{Params as PedersenParams, Pedersen};
     use crate::folding::nova::circuits::ChallengeGadget;
     use crate::folding::nova::traits::NovaR1CS;
-    use crate::transcript::poseidon::{poseidon_canonical_config, PoseidonTranscript};
+    use crate::transcript::poseidon::poseidon_canonical_config;
 
     #[allow(clippy::type_complexity)]
     pub(crate) fn prepare_simple_fold_inputs<C>() -> (
@@ -258,16 +261,16 @@ pub mod tests {
                 .unwrap();
 
         let poseidon_config = poseidon_canonical_config::<C::ScalarField>();
+        let mut transcript = PoseidonSponge::<C::ScalarField>::new(&poseidon_config);
 
         let pp_hash = C::ScalarField::from(42u32); // only for test
         let r_bits = ChallengeGadget::<C>::get_challenge_native(
-            &poseidon_config,
+            &mut transcript,
             pp_hash,
             ci1.clone(),
             ci2.clone(),
             cmT,
-        )
-        .unwrap();
+        );
         let r_Fr = C::ScalarField::from_bigint(BigInteger::from_bits_le(&r_bits)).unwrap();
 
         let (w3, ci3) =
@@ -364,9 +367,9 @@ pub mod tests {
             .unwrap();
 
         // init Prover's transcript
-        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p = PoseidonSponge::<Fr>::new(&poseidon_config);
         // init Verifier's transcript
-        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         // prove the ci3.cmE, ci3.cmW, cmT commitments
         let cm_proofs = NIFS::<Projective, Pedersen<Projective>>::prove_commitments(
diff --git a/folding-schemes/src/folding/protogalaxy/folding.rs b/folding-schemes/src/folding/protogalaxy/folding.rs
index f12d07b..e9b78da 100644
--- a/folding-schemes/src/folding/protogalaxy/folding.rs
+++ b/folding-schemes/src/folding/protogalaxy/folding.rs
@@ -6,12 +6,10 @@ use ark_poly::{
     univariate::{DensePolynomial, SparsePolynomial},
     DenseUVPolynomial, EvaluationDomain, Evaluations, GeneralEvaluationDomain, Polynomial,
 };
-use ark_std::log2;
-use ark_std::{cfg_into_iter, Zero};
+use ark_std::{cfg_into_iter, log2, Zero};
 use rayon::iter::{IntoParallelIterator, ParallelIterator};
 use std::marker::PhantomData;
 
-use super::traits::ProtoGalaxyTranscript;
 use super::utils::{all_powers, betas_star, exponential_powers};
 use super::ProtoGalaxyError;
 use super::{CommittedInstance, Witness};
@@ -36,7 +34,7 @@ where
     #![allow(clippy::type_complexity)]
     /// implements the non-interactive Prover from the folding scheme described in section 4
     pub fn prove(
-        transcript: &mut (impl Transcript<C> + ProtoGalaxyTranscript<C>),
+        transcript: &mut impl Transcript<C::ScalarField>,
         r1cs: &R1CS<C::ScalarField>,
         // running instance
         instance: &CommittedInstance<C>,
@@ -81,10 +79,8 @@ where
         }
 
         // absorb the committed instances
-        transcript.absorb_committed_instance(instance)?;
-        for ci in vec_instances.iter() {
-            transcript.absorb_committed_instance(ci)?;
-        }
+        transcript.absorb(instance);
+        transcript.absorb(&vec_instances);
 
         let delta = transcript.get_challenge();
         let deltas = exponential_powers(delta, t);
@@ -95,7 +91,7 @@ where
         let F_X: SparsePolynomial<C::ScalarField> =
             calc_f_from_btree(&f_w, &instance.betas, &deltas).expect("Error calculating F[x]");
         let F_X_dense = DensePolynomial::from(F_X.clone());
-        transcript.absorb_vec(&F_X_dense.coeffs);
+        transcript.absorb(&F_X_dense.coeffs);
 
         let alpha = transcript.get_challenge();
 
@@ -187,7 +183,7 @@ where
             return Err(Error::ProtoGalaxy(ProtoGalaxyError::RemainderNotZero));
         }
 
-        transcript.absorb_vec(&K_X.coeffs);
+        transcript.absorb(&K_X.coeffs);
 
         let gamma = transcript.get_challenge();
 
@@ -223,7 +219,7 @@ where
 
     /// implements the non-interactive Verifier from the folding scheme described in section 4
     pub fn verify(
-        transcript: &mut (impl Transcript<C> + ProtoGalaxyTranscript<C>),
+        transcript: &mut impl Transcript<C::ScalarField>,
         r1cs: &R1CS<C::ScalarField>,
         // running instance
         instance: &CommittedInstance<C>,
@@ -237,15 +233,13 @@ where
         let n = r1cs.A.n_cols;
 
         // absorb the committed instances
-        transcript.absorb_committed_instance(instance)?;
-        for ci in vec_instances.iter() {
-            transcript.absorb_committed_instance(ci)?;
-        }
+        transcript.absorb(instance);
+        transcript.absorb(&vec_instances);
 
         let delta = transcript.get_challenge();
         let deltas = exponential_powers(delta, t);
 
-        transcript.absorb_vec(&F_coeffs);
+        transcript.absorb(&F_coeffs);
 
         let alpha = transcript.get_challenge();
         let alphas = all_powers(alpha, n);
@@ -266,7 +260,7 @@ where
         let K_X: DensePolynomial<C::ScalarField> =
             DensePolynomial::<C::ScalarField>::from_coefficients_vec(K_coeffs);
 
-        transcript.absorb_vec(&K_X.coeffs);
+        transcript.absorb(&K_X.coeffs);
 
         let gamma = transcript.get_challenge();
 
@@ -380,12 +374,14 @@ fn eval_f<F: PrimeField>(r1cs: &R1CS<F>, w: &[F]) -> Result<Vec<F>, Error> {
 #[cfg(test)]
 mod tests {
     use super::*;
+    use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
+    use ark_crypto_primitives::sponge::CryptographicSponge;
     use ark_pallas::{Fr, Projective};
     use ark_std::UniformRand;
 
     use crate::arith::r1cs::tests::{get_test_r1cs, get_test_z};
     use crate::commitment::{pedersen::Pedersen, CommitmentScheme};
-    use crate::transcript::poseidon::{poseidon_canonical_config, PoseidonTranscript};
+    use crate::transcript::poseidon::poseidon_canonical_config;
 
     pub(crate) fn check_instance<C: CurveGroup>(
         r1cs: &R1CS<C::ScalarField>,
@@ -495,7 +491,7 @@ mod tests {
             .unwrap();
             let instance_i = CommittedInstance::<Projective> {
                 phi: phi_i,
-                betas: betas.clone(),
+                betas: vec![],
                 e: Fr::zero(),
             };
             witnesses.push(witness_i);
@@ -513,8 +509,8 @@ mod tests {
 
         // init Prover & Verifier's transcript
         let poseidon_config = poseidon_canonical_config::<Fr>();
-        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
-        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        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,
@@ -553,8 +549,8 @@ mod tests {
 
         // init Prover & Verifier's transcript
         let poseidon_config = poseidon_canonical_config::<Fr>();
-        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
-        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        let mut transcript_p = PoseidonSponge::<Fr>::new(&poseidon_config);
+        let mut transcript_v = PoseidonSponge::<Fr>::new(&poseidon_config);
 
         let (mut running_witness, mut running_instance, _, _) = prepare_inputs(0);
 
diff --git a/folding-schemes/src/folding/protogalaxy/mod.rs b/folding-schemes/src/folding/protogalaxy/mod.rs
index f2f244f..1d8ab6f 100644
--- a/folding-schemes/src/folding/protogalaxy/mod.rs
+++ b/folding-schemes/src/folding/protogalaxy/mod.rs
@@ -1,8 +1,11 @@
 /// Implements the scheme described in [ProtoGalaxy](https://eprint.iacr.org/2023/1106.pdf)
 use ark_ec::CurveGroup;
 use ark_ff::PrimeField;
+use ark_r1cs_std::fields::fp::FpVar;
 use thiserror::Error;
 
+use super::circuits::nonnative::affine::NonNativeAffineVar;
+
 pub mod folding;
 pub mod traits;
 pub(crate) mod utils;
@@ -14,6 +17,13 @@ pub struct CommittedInstance<C: CurveGroup> {
     e: C::ScalarField,
 }
 
+#[derive(Clone, Debug)]
+pub struct CommittedInstanceVar<C: CurveGroup> {
+    phi: NonNativeAffineVar<C>,
+    betas: Vec<FpVar<C::ScalarField>>,
+    e: FpVar<C::ScalarField>,
+}
+
 #[derive(Clone, Debug)]
 pub struct Witness<F: PrimeField> {
     w: Vec<F>,
diff --git a/folding-schemes/src/folding/protogalaxy/traits.rs b/folding-schemes/src/folding/protogalaxy/traits.rs
index ff943e1..db96638 100644
--- a/folding-schemes/src/folding/protogalaxy/traits.rs
+++ b/folding-schemes/src/folding/protogalaxy/traits.rs
@@ -1,23 +1,42 @@
-use ark_crypto_primitives::sponge::Absorb;
-use ark_ec::{CurveGroup, Group};
+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_relations::r1cs::SynthesisError;
 
-use super::CommittedInstance;
-use crate::transcript::{poseidon::PoseidonTranscript, Transcript};
-use crate::Error;
+use super::{CommittedInstance, CommittedInstanceVar};
+use crate::transcript::AbsorbNonNative;
 
-/// ProtoGalaxyTranscript extends [`Transcript`] with the method to absorb ProtoGalaxy's
-/// CommittedInstance.
-pub trait ProtoGalaxyTranscript<C: CurveGroup>: Transcript<C> {
-    fn absorb_committed_instance(&mut self, ci: &CommittedInstance<C>) -> Result<(), Error> {
-        self.absorb_point(&ci.phi)?;
-        self.absorb_vec(&ci.betas);
-        self.absorb(&ci.e);
-        Ok(())
+// Implements the trait for absorbing ProtoGalaxy's CommittedInstance.
+impl<C: CurveGroup> Absorb for CommittedInstance<C>
+where
+    C::ScalarField: Absorb,
+{
+    fn to_sponge_bytes(&self, _dest: &mut Vec<u8>) {
+        unimplemented!()
+    }
+
+    fn to_sponge_field_elements<F: PrimeField>(&self, dest: &mut Vec<F>) {
+        self.phi
+            .to_native_sponge_field_elements_as_vec()
+            .to_sponge_field_elements(dest);
+        self.betas.to_sponge_field_elements(dest);
+        self.e.to_sponge_field_elements(dest);
     }
 }
 
-// Implements ProtoGalaxyTranscript for PoseidonTranscript
-impl<C: CurveGroup> ProtoGalaxyTranscript<C> for PoseidonTranscript<C> where
-    <C as Group>::ScalarField: Absorb
-{
+// Implements the trait for absorbing ProtoGalaxy's CommittedInstanceVar in-circuit.
+impl<C: CurveGroup> AbsorbGadget<C::ScalarField> for CommittedInstanceVar<C> {
+    fn to_sponge_bytes(&self) -> Result<Vec<UInt8<C::ScalarField>>, SynthesisError> {
+        unimplemented!()
+    }
+
+    fn to_sponge_field_elements(&self) -> Result<Vec<FpVar<C::ScalarField>>, SynthesisError> {
+        Ok([
+            self.phi.to_constraint_field()?,
+            self.betas.to_sponge_field_elements()?,
+            self.e.to_sponge_field_elements()?,
+        ]
+        .concat())
+    }
 }
diff --git a/folding-schemes/src/frontend/circom/mod.rs b/folding-schemes/src/frontend/circom/mod.rs
index 9259a94..303a5de 100644
--- a/folding-schemes/src/frontend/circom/mod.rs
+++ b/folding-schemes/src/frontend/circom/mod.rs
@@ -207,8 +207,7 @@ impl<F: PrimeField> CircomFCircuit<F> {
 pub mod tests {
     use super::*;
     use ark_bn254::Fr;
-    use ark_r1cs_std::alloc::AllocVar;
-    use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
+    use ark_relations::r1cs::ConstraintSystem;
 
     // Tests the step_native function of CircomFCircuit.
     #[test]
diff --git a/folding-schemes/src/frontend/circom/utils.rs b/folding-schemes/src/frontend/circom/utils.rs
index 4bdab8b..9ee7dfb 100644
--- a/folding-schemes/src/frontend/circom/utils.rs
+++ b/folding-schemes/src/frontend/circom/utils.rs
@@ -110,7 +110,6 @@ mod tests {
     use ark_circom::circom::{CircomBuilder, CircomConfig};
     use ark_circom::CircomCircuit;
     use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
-    use std::path::PathBuf;
 
     //To generate .r1cs and .wasm files, run the below command in the terminal.
     //bash ./folding-schemes/src/frontend/circom/test_folder/compile.sh
diff --git a/folding-schemes/src/frontend/mod.rs b/folding-schemes/src/frontend/mod.rs
index 59f18eb..15fcc74 100644
--- a/folding-schemes/src/frontend/mod.rs
+++ b/folding-schemes/src/frontend/mod.rs
@@ -52,9 +52,7 @@ pub mod tests {
     use super::*;
     use ark_bn254::Fr;
     use ark_r1cs_std::{alloc::AllocVar, eq::EqGadget};
-    use ark_relations::r1cs::{
-        ConstraintSynthesizer, ConstraintSystem, ConstraintSystemRef, SynthesisError,
-    };
+    use ark_relations::r1cs::{ConstraintSynthesizer, ConstraintSystem};
     use core::marker::PhantomData;
 
     /// CubicFCircuit is a struct that implements the FCircuit trait, for the R1CS example circuit
diff --git a/folding-schemes/src/transcript/mod.rs b/folding-schemes/src/transcript/mod.rs
index 0e22a11..2c728d4 100644
--- a/folding-schemes/src/transcript/mod.rs
+++ b/folding-schemes/src/transcript/mod.rs
@@ -1,31 +1,103 @@
-use crate::Error;
+use ark_crypto_primitives::sponge::{constraints::CryptographicSpongeVar, CryptographicSponge};
 use ark_ec::CurveGroup;
 use ark_ff::PrimeField;
-use ark_r1cs_std::{boolean::Boolean, fields::fp::FpVar};
-use ark_relations::r1cs::{ConstraintSystemRef, SynthesisError};
-use ark_std::fmt::Debug;
+use ark_r1cs_std::{
+    boolean::Boolean, fields::fp::FpVar, groups::CurveVar, ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::SynthesisError;
 
 pub mod poseidon;
 
-pub trait Transcript<C: CurveGroup> {
-    type TranscriptConfig: Debug;
+/// An interface for objects that can be absorbed by a `Transcript`.
+///
+/// Matches `Absorb` in `ark-crypto-primitives`.
+pub trait AbsorbNonNative<F: PrimeField> {
+    /// Converts the object into field elements that can be absorbed by a `Transcript`.
+    /// Append the list to `dest`
+    fn to_native_sponge_field_elements(&self, dest: &mut Vec<F>);
 
-    fn new(config: &Self::TranscriptConfig) -> Self;
-    fn absorb(&mut self, v: &C::ScalarField);
-    fn absorb_vec(&mut self, v: &[C::ScalarField]);
-    fn absorb_point(&mut self, v: &C) -> Result<(), Error>;
-    fn get_challenge(&mut self) -> C::ScalarField;
+    /// Converts the object into field elements that can be absorbed by a `Transcript`.
+    /// Return the list as `Vec`
+    fn to_native_sponge_field_elements_as_vec(&self) -> Vec<F> {
+        let mut result = Vec::new();
+        self.to_native_sponge_field_elements(&mut result);
+        result
+    }
+}
+
+/// An interface for objects that can be absorbed by a `TranscriptVar` whose constraint field
+/// is `F`.
+///
+/// Matches `AbsorbGadget` in `ark-crypto-primitives`.
+pub trait AbsorbNonNativeGadget<F: PrimeField> {
+    /// Converts the object into field elements that can be absorbed by a `TranscriptVar`.
+    fn to_native_sponge_field_elements(&self) -> Result<Vec<FpVar<F>>, SynthesisError>;
+}
+
+pub trait Transcript<F: PrimeField>: CryptographicSponge {
+    /// `absorb_point` is for absorbing points whose `BaseField` is the field of
+    /// the sponge, i.e., the type `C` of these points should satisfy
+    /// `C::BaseField = F`.
+    ///
+    /// If the sponge field `F` is `C::ScalarField`, call `absorb_nonnative`
+    /// instead.
+    fn absorb_point<C: CurveGroup<BaseField = F>>(&mut self, v: &C);
+    /// `absorb_nonnative` is for structs that contain non-native (field or
+    /// group) elements, including:
+    ///
+    /// - A field element of type `T: PrimeField` that will be absorbed into a
+    ///   sponge that operates in another field `F != T`.
+    /// - A group element of type `C: CurveGroup` that will be absorbed into a
+    ///   sponge that operates in another field `F != C::BaseField`, e.g.,
+    ///   `F = C::ScalarField`.
+    /// - A `CommittedInstance` on the secondary curve (used for CycleFold) that
+    ///   will be absorbed into a sponge that operates in the (scalar field of
+    ///   the) primary curve.
+    ///
+    ///   Note that although a `CommittedInstance` for `AugmentedFCircuit` on
+    ///   the primary curve also contains non-native elements, we still regard
+    ///   it as native, because the sponge is on the same curve.
+    fn absorb_nonnative<V: AbsorbNonNative<F>>(&mut self, v: &V);
+
+    fn get_challenge(&mut self) -> F;
     /// get_challenge_nbits returns a field element of size nbits
     fn get_challenge_nbits(&mut self, nbits: usize) -> Vec<bool>;
-    fn get_challenges(&mut self, n: usize) -> Vec<C::ScalarField>;
+    fn get_challenges(&mut self, n: usize) -> Vec<F>;
 }
 
-pub trait TranscriptVar<F: PrimeField> {
-    type TranscriptVarConfig: Debug;
+pub trait TranscriptVar<F: PrimeField, S: CryptographicSponge>:
+    CryptographicSpongeVar<F, S>
+{
+    /// `absorb_point` is for absorbing points whose `BaseField` is the field of
+    /// the sponge, i.e., the type `C` of these points should satisfy
+    /// `C::BaseField = F`.
+    ///
+    /// If the sponge field `F` is `C::ScalarField`, call `absorb_nonnative`
+    /// instead.
+    fn absorb_point<C: CurveGroup<BaseField = F>, GC: CurveVar<C, F> + ToConstraintFieldGadget<F>>(
+        &mut self,
+        v: &GC,
+    ) -> Result<(), SynthesisError>;
+    /// `absorb_nonnative` is for structs that contain non-native (field or
+    /// group) elements, including:
+    ///
+    /// - A field element of type `T: PrimeField` that will be absorbed into a
+    ///   sponge that operates in another field `F != T`.
+    /// - A group element of type `C: CurveGroup` that will be absorbed into a
+    ///   sponge that operates in another field `F != C::BaseField`, e.g.,
+    ///   `F = C::ScalarField`.
+    /// - A `CommittedInstance` on the secondary curve (used for CycleFold) that
+    ///   will be absorbed into a sponge that operates in the (scalar field of
+    ///   the) primary curve.
+    ///
+    ///   Note that although a `CommittedInstance` for `AugmentedFCircuit` on
+    ///   the primary curve also contains non-native elements, we still regard
+    ///   it as native, because the sponge is on the same curve.
+    fn absorb_nonnative<V: AbsorbNonNativeGadget<F>>(
+        &mut self,
+        v: &V,
+    ) -> Result<(), SynthesisError>;
 
-    fn new(cs: ConstraintSystemRef<F>, poseidon_config: &Self::TranscriptVarConfig) -> Self;
-    fn absorb(&mut self, v: FpVar<F>) -> Result<(), SynthesisError>;
-    fn absorb_vec(&mut self, v: &[FpVar<F>]) -> Result<(), SynthesisError>;
     fn get_challenge(&mut self) -> Result<FpVar<F>, SynthesisError>;
     /// returns the bit representation of the challenge, we use its output in-circuit for the
     /// `GC.scalar_mul_le` method.
diff --git a/folding-schemes/src/transcript/poseidon.rs b/folding-schemes/src/transcript/poseidon.rs
index 55e7f4f..0496a17 100644
--- a/folding-schemes/src/transcript/poseidon.rs
+++ b/folding-schemes/src/transcript/poseidon.rs
@@ -3,114 +3,86 @@ use ark_crypto_primitives::sponge::{
     poseidon::{constraints::PoseidonSpongeVar, PoseidonConfig, PoseidonSponge},
     Absorb, CryptographicSponge,
 };
-use ark_ec::{AffineRepr, CurveGroup, Group};
-use ark_ff::{BigInteger, Field, PrimeField};
-use ark_r1cs_std::{boolean::Boolean, fields::fp::FpVar};
-use ark_relations::r1cs::{ConstraintSystemRef, SynthesisError};
-use ark_std::Zero;
-
-use crate::transcript::Transcript;
-use crate::Error;
-
-use super::TranscriptVar;
-
-/// PoseidonTranscript implements the Transcript trait using the Poseidon hash
-pub struct PoseidonTranscript<C: CurveGroup>
-where
-    <C as Group>::ScalarField: Absorb,
-{
-    sponge: PoseidonSponge<C::ScalarField>,
-}
+use ark_ec::{AffineRepr, CurveGroup};
+use ark_ff::{BigInteger, PrimeField};
+use ark_r1cs_std::{
+    boolean::Boolean, fields::fp::FpVar, groups::CurveVar, ToConstraintFieldGadget,
+};
+use ark_relations::r1cs::SynthesisError;
 
-impl<C: CurveGroup> Transcript<C> for PoseidonTranscript<C>
-where
-    <C as Group>::ScalarField: Absorb,
-{
-    type TranscriptConfig = PoseidonConfig<C::ScalarField>;
+use super::{AbsorbNonNative, AbsorbNonNativeGadget, Transcript, TranscriptVar};
 
-    fn new(poseidon_config: &Self::TranscriptConfig) -> Self {
-        let sponge = PoseidonSponge::<C::ScalarField>::new(poseidon_config);
-        Self { sponge }
-    }
-    fn absorb(&mut self, v: &C::ScalarField) {
-        self.sponge.absorb(&v);
+impl<F: PrimeField + Absorb> Transcript<F> for PoseidonSponge<F> {
+    // Compatible with the in-circuit `TranscriptVar::absorb_point`
+    fn absorb_point<C: CurveGroup<BaseField = F>>(&mut self, p: &C) {
+        let (x, y) = match p.into_affine().xy() {
+            Some((&x, &y)) => (x, y),
+            None => (C::BaseField::zero(), C::BaseField::zero()),
+        };
+        self.absorb(&x);
+        self.absorb(&y);
     }
-    fn absorb_vec(&mut self, v: &[C::ScalarField]) {
-        self.sponge.absorb(&v);
+    fn absorb_nonnative<V: AbsorbNonNative<F>>(&mut self, v: &V) {
+        self.absorb(&v.to_native_sponge_field_elements_as_vec());
     }
-    fn absorb_point(&mut self, p: &C) -> Result<(), Error> {
-        self.sponge.absorb(&prepare_point(p)?);
-        Ok(())
-    }
-    fn get_challenge(&mut self) -> C::ScalarField {
-        let c = self.sponge.squeeze_field_elements(1);
-        self.sponge.absorb(&c[0]);
+    fn get_challenge(&mut self) -> F {
+        let c = self.squeeze_field_elements(1);
+        self.absorb(&c[0]);
         c[0]
     }
     fn get_challenge_nbits(&mut self, nbits: usize) -> Vec<bool> {
-        self.sponge.squeeze_bits(nbits)
+        let bits = self.squeeze_bits(nbits);
+        self.absorb(&F::from(F::BigInt::from_bits_le(&bits)));
+        bits
     }
-    fn get_challenges(&mut self, n: usize) -> Vec<C::ScalarField> {
-        let c = self.sponge.squeeze_field_elements(n);
-        self.sponge.absorb(&c);
+    fn get_challenges(&mut self, n: usize) -> Vec<F> {
+        let c = self.squeeze_field_elements(n);
+        self.absorb(&c);
         c
     }
 }
 
-// Returns the point coordinates in Fr, so it can be absorbed by the transcript. It does not work
-// over bytes in order to have a logic that can be reproduced in-circuit.
-fn prepare_point<C: CurveGroup>(p: &C) -> Result<Vec<C::ScalarField>, Error> {
-    let affine = p.into_affine();
-    let zero_point = (&C::BaseField::zero(), &C::BaseField::zero());
-    let xy = affine.xy().unwrap_or(zero_point);
-
-    let x_bi =
-        xy.0.to_base_prime_field_elements()
-            .next()
-            .expect("a")
-            .into_bigint();
-    let y_bi =
-        xy.1.to_base_prime_field_elements()
-            .next()
-            .expect("a")
-            .into_bigint();
-    Ok(vec![
-        C::ScalarField::from_le_bytes_mod_order(x_bi.to_bytes_le().as_ref()),
-        C::ScalarField::from_le_bytes_mod_order(y_bi.to_bytes_le().as_ref()),
-    ])
-}
-
-/// PoseidonTranscriptVar implements the gadget compatible with PoseidonTranscript
-pub struct PoseidonTranscriptVar<F: PrimeField> {
-    sponge: PoseidonSpongeVar<F>,
-}
-impl<F: PrimeField> TranscriptVar<F> for PoseidonTranscriptVar<F> {
-    type TranscriptVarConfig = PoseidonConfig<F>;
-
-    fn new(cs: ConstraintSystemRef<F>, poseidon_config: &Self::TranscriptVarConfig) -> Self {
-        let sponge = PoseidonSpongeVar::<F>::new(cs, poseidon_config);
-        Self { sponge }
+impl<F: PrimeField> TranscriptVar<F, PoseidonSponge<F>> for PoseidonSpongeVar<F> {
+    fn absorb_point<
+        C: CurveGroup<BaseField = F>,
+        GC: CurveVar<C, F> + ToConstraintFieldGadget<F>,
+    >(
+        &mut self,
+        v: &GC,
+    ) -> Result<(), SynthesisError> {
+        let mut vec = v.to_constraint_field()?;
+        // The last element in the vector tells whether the point is infinity,
+        // but we can in fact avoid absorbing it without loss of soundness.
+        // This is because the `to_constraint_field` method internally invokes
+        // [`ProjectiveVar::to_afine`](https://github.com/arkworks-rs/r1cs-std/blob/4020fbc22625621baa8125ede87abaeac3c1ca26/src/groups/curves/short_weierstrass/mod.rs#L160-L195),
+        // which guarantees that an infinity point is represented as `(0, 0)`,
+        // but the y-coordinate of a non-infinity point is never 0 (for why, see
+        // https://crypto.stackexchange.com/a/108242 ).
+        vec.pop();
+        self.absorb(&vec)
     }
-    fn absorb(&mut self, v: FpVar<F>) -> Result<(), SynthesisError> {
-        self.sponge.absorb(&v)
-    }
-    fn absorb_vec(&mut self, v: &[FpVar<F>]) -> Result<(), SynthesisError> {
-        self.sponge.absorb(&v)
+    fn absorb_nonnative<V: AbsorbNonNativeGadget<F>>(
+        &mut self,
+        v: &V,
+    ) -> Result<(), SynthesisError> {
+        self.absorb(&v.to_native_sponge_field_elements()?)
     }
     fn get_challenge(&mut self) -> Result<FpVar<F>, SynthesisError> {
-        let c = self.sponge.squeeze_field_elements(1)?;
-        self.sponge.absorb(&c[0])?;
+        let c = self.squeeze_field_elements(1)?;
+        self.absorb(&c[0])?;
         Ok(c[0].clone())
     }
 
     /// returns the bit representation of the challenge, we use its output in-circuit for the
     /// `GC.scalar_mul_le` method.
     fn get_challenge_nbits(&mut self, nbits: usize) -> Result<Vec<Boolean<F>>, SynthesisError> {
-        self.sponge.squeeze_bits(nbits)
+        let bits = self.squeeze_bits(nbits)?;
+        self.absorb(&Boolean::le_bits_to_fp_var(&bits)?)?;
+        Ok(bits)
     }
     fn get_challenges(&mut self, n: usize) -> Result<Vec<FpVar<F>>, SynthesisError> {
-        let c = self.sponge.squeeze_field_elements(n)?;
-        self.sponge.absorb(&c)?;
+        let c = self.squeeze_field_elements(n)?;
+        self.absorb(&c)?;
         Ok(c)
     }
 }
@@ -147,12 +119,17 @@ pub fn poseidon_canonical_config<F: PrimeField>() -> PoseidonConfig<F> {
 
 #[cfg(test)]
 pub mod tests {
+    use crate::folding::circuits::nonnative::affine::NonNativeAffineVar;
+
     use super::*;
-    use ark_bn254::{constraints::GVar, Fq, Fr, G1Projective as G1};
-    use ark_grumpkin::Projective;
-    use ark_r1cs_std::{alloc::AllocVar, groups::CurveVar, R1CSVar};
+    use ark_bn254::{constraints::GVar, g1::Config, Fq, Fr, G1Projective as G1};
+    use ark_ec::Group;
+    use ark_ff::UniformRand;
+    use ark_r1cs_std::{
+        alloc::AllocVar, groups::curves::short_weierstrass::ProjectiveVar, R1CSVar,
+    };
     use ark_relations::r1cs::ConstraintSystem;
-    use std::ops::Mul;
+    use ark_std::test_rng;
 
     // Test with value taken from https://github.com/iden3/circomlibjs/blob/43cc582b100fc3459cf78d903a6f538e5d7f38ee/test/poseidon.js#L32
     #[test]
@@ -178,19 +155,69 @@ pub mod tests {
         );
     }
 
+    #[test]
+    fn test_transcript_and_transcriptvar_absorb_native_point() {
+        // use 'native' transcript
+        let config = poseidon_canonical_config::<Fq>();
+        let mut tr = PoseidonSponge::<Fq>::new(&config);
+        let rng = &mut test_rng();
+
+        let p = G1::rand(rng);
+        tr.absorb_point(&p);
+        let c = tr.get_challenge();
+
+        // use 'gadget' transcript
+        let cs = ConstraintSystem::<Fq>::new_ref();
+        let mut tr_var = PoseidonSpongeVar::<Fq>::new(cs.clone(), &config);
+        let p_var = ProjectiveVar::<Config, FpVar<Fq>>::new_witness(
+            ConstraintSystem::<Fq>::new_ref(),
+            || Ok(p),
+        )
+        .unwrap();
+        tr_var.absorb_point(&p_var).unwrap();
+        let c_var = tr_var.get_challenge().unwrap();
+
+        // assert that native & gadget transcripts return the same challenge
+        assert_eq!(c, c_var.value().unwrap());
+    }
+
+    #[test]
+    fn test_transcript_and_transcriptvar_absorb_nonnative_point() {
+        // use 'native' transcript
+        let config = poseidon_canonical_config::<Fr>();
+        let mut tr = PoseidonSponge::<Fr>::new(&config);
+        let rng = &mut test_rng();
+
+        let p = G1::rand(rng);
+        tr.absorb_nonnative(&p);
+        let c = tr.get_challenge();
+
+        // use 'gadget' transcript
+        let cs = ConstraintSystem::<Fr>::new_ref();
+        let mut tr_var = PoseidonSpongeVar::<Fr>::new(cs.clone(), &config);
+        let p_var =
+            NonNativeAffineVar::<G1>::new_witness(ConstraintSystem::<Fr>::new_ref(), || Ok(p))
+                .unwrap();
+        tr_var.absorb_nonnative(&p_var).unwrap();
+        let c_var = tr_var.get_challenge().unwrap();
+
+        // assert that native & gadget transcripts return the same challenge
+        assert_eq!(c, c_var.value().unwrap());
+    }
+
     #[test]
     fn test_transcript_and_transcriptvar_get_challenge() {
         // use 'native' transcript
         let config = poseidon_canonical_config::<Fr>();
-        let mut tr = PoseidonTranscript::<G1>::new(&config);
+        let mut tr = PoseidonSponge::<Fr>::new(&config);
         tr.absorb(&Fr::from(42_u32));
         let c = tr.get_challenge();
 
         // use 'gadget' transcript
         let cs = ConstraintSystem::<Fr>::new_ref();
-        let mut tr_var = PoseidonTranscriptVar::<Fr>::new(cs.clone(), &config);
+        let mut tr_var = PoseidonSpongeVar::<Fr>::new(cs.clone(), &config);
         let v = FpVar::<Fr>::new_witness(cs.clone(), || Ok(Fr::from(42_u32))).unwrap();
-        tr_var.absorb(v).unwrap();
+        tr_var.absorb(&v).unwrap();
         let c_var = tr_var.get_challenge().unwrap();
 
         // assert that native & gadget transcripts return the same challenge
@@ -203,7 +230,7 @@ pub mod tests {
 
         // use 'native' transcript
         let config = poseidon_canonical_config::<Fq>();
-        let mut tr = PoseidonTranscript::<Projective>::new(&config);
+        let mut tr = PoseidonSponge::<Fq>::new(&config);
         tr.absorb(&Fq::from(42_u32));
 
         // get challenge from native transcript
@@ -211,9 +238,9 @@ pub mod tests {
 
         // use 'gadget' transcript
         let cs = ConstraintSystem::<Fq>::new_ref();
-        let mut tr_var = PoseidonTranscriptVar::<Fq>::new(cs.clone(), &config);
+        let mut tr_var = PoseidonSpongeVar::<Fq>::new(cs.clone(), &config);
         let v = FpVar::<Fq>::new_witness(cs.clone(), || Ok(Fq::from(42_u32))).unwrap();
-        tr_var.absorb(v).unwrap();
+        tr_var.absorb(&v).unwrap();
 
         // get challenge from circuit transcript
         let c_var = tr_var.get_challenge_nbits(nbits).unwrap();
@@ -226,7 +253,7 @@ pub mod tests {
 
         // native c*P
         let c_Fr = Fr::from_bigint(BigInteger::from_bits_le(&c_bits)).unwrap();
-        let cP_native = P.mul(c_Fr);
+        let cP_native = P * c_Fr;
 
         // native c*P using mul_bits_be (notice the .rev to convert the LE to BE)
         let cP_native_bits = P.mul_bits_be(c_bits.into_iter().rev());
diff --git a/folding-schemes/src/utils/espresso/sum_check/mod.rs b/folding-schemes/src/utils/espresso/sum_check/mod.rs
index 86842c4..638719d 100644
--- a/folding-schemes/src/utils/espresso/sum_check/mod.rs
+++ b/folding-schemes/src/utils/espresso/sum_check/mod.rs
@@ -13,7 +13,7 @@ use crate::{
     transcript::Transcript,
     utils::virtual_polynomial::{VPAuxInfo, VirtualPolynomial},
 };
-use ark_ec::CurveGroup;
+use ark_crypto_primitives::sponge::Absorb;
 use ark_ff::PrimeField;
 use ark_poly::univariate::DensePolynomial;
 use ark_poly::{DenseMultilinearExtension, DenseUVPolynomial, Polynomial};
@@ -22,7 +22,6 @@ use std::{fmt::Debug, marker::PhantomData, sync::Arc};
 
 use crate::utils::sum_check::structs::IOPProverMessage;
 use crate::utils::sum_check::structs::IOPVerifierState;
-use ark_ff::Field;
 use espresso_subroutines::poly_iop::prelude::PolyIOPErrors;
 use structs::{IOPProof, IOPProverState};
 
@@ -31,7 +30,7 @@ pub mod structs;
 pub mod verifier;
 
 /// A generic sum-check trait over a curve group
-pub trait SumCheck<C: CurveGroup> {
+pub trait SumCheck<F: PrimeField> {
     type VirtualPolynomial;
     type VPAuxInfo;
     type MultilinearExtension;
@@ -40,27 +39,27 @@ pub trait SumCheck<C: CurveGroup> {
     type SumCheckSubClaim: Clone + Debug + Default + PartialEq;
 
     /// Extract sum from the proof
-    fn extract_sum(proof: &Self::SumCheckProof) -> C::ScalarField;
+    fn extract_sum(proof: &Self::SumCheckProof) -> F;
 
     /// Generate proof of the sum of polynomial over {0,1}^`num_vars`
     ///
     /// The polynomial is represented in the form of a VirtualPolynomial.
     fn prove(
         poly: &Self::VirtualPolynomial,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<F>,
     ) -> Result<Self::SumCheckProof, PolyIOPErrors>;
 
     /// Verify the claimed sum using the proof
     fn verify(
-        sum: C::ScalarField,
+        sum: F,
         proof: &Self::SumCheckProof,
         aux_info: &Self::VPAuxInfo,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<F>,
     ) -> Result<Self::SumCheckSubClaim, PolyIOPErrors>;
 }
 
 /// Trait for sum check protocol prover side APIs.
-pub trait SumCheckProver<C: CurveGroup>
+pub trait SumCheckProver<F: PrimeField>
 where
     Self: Sized,
 {
@@ -77,12 +76,12 @@ where
     /// Main algorithm used is from section 3.2 of [XZZPS19](https://eprint.iacr.org/2019/317.pdf#subsection.3.2).
     fn prove_round_and_update_state(
         &mut self,
-        challenge: &Option<C::ScalarField>,
+        challenge: &Option<F>,
     ) -> Result<Self::ProverMessage, PolyIOPErrors>;
 }
 
 /// Trait for sum check protocol verifier side APIs.
-pub trait SumCheckVerifier<C: CurveGroup> {
+pub trait SumCheckVerifier<F: PrimeField> {
     type VPAuxInfo;
     type ProverMessage;
     type Challenge;
@@ -100,7 +99,7 @@ pub trait SumCheckVerifier<C: CurveGroup> {
     fn verify_round_and_update_state(
         &mut self,
         prover_msg: &Self::ProverMessage,
-        transcript: &mut impl Transcript<C>,
+        transcript: &mut impl Transcript<F>,
     ) -> Result<Self::Challenge, PolyIOPErrors>;
 
     /// This function verifies the deferred checks in the interactive version of
@@ -113,7 +112,7 @@ pub trait SumCheckVerifier<C: CurveGroup> {
     /// Larger field size guarantees smaller soundness error.
     fn check_and_generate_subclaim(
         &self,
-        asserted_sum: &C::ScalarField,
+        asserted_sum: &F,
     ) -> Result<Self::SumCheckSubClaim, PolyIOPErrors>;
 }
 
@@ -129,42 +128,42 @@ pub struct SumCheckSubClaim<F: PrimeField> {
 }
 
 #[derive(Clone, Debug, Default, Copy, PartialEq, Eq)]
-pub struct IOPSumCheck<C: CurveGroup, T: Transcript<C>> {
+pub struct IOPSumCheck<F: PrimeField, T: Transcript<F>> {
     #[doc(hidden)]
-    phantom: PhantomData<C>,
+    phantom: PhantomData<F>,
     #[doc(hidden)]
     phantom2: PhantomData<T>,
 }
 
-impl<C: CurveGroup, T: Transcript<C>> SumCheck<C> for IOPSumCheck<C, T> {
-    type SumCheckProof = IOPProof<C::ScalarField>;
-    type VirtualPolynomial = VirtualPolynomial<C::ScalarField>;
-    type VPAuxInfo = VPAuxInfo<C::ScalarField>;
-    type MultilinearExtension = Arc<DenseMultilinearExtension<C::ScalarField>>;
-    type SumCheckSubClaim = SumCheckSubClaim<C::ScalarField>;
+impl<F: PrimeField + Absorb, T: Transcript<F>> SumCheck<F> for IOPSumCheck<F, T> {
+    type SumCheckProof = IOPProof<F>;
+    type VirtualPolynomial = VirtualPolynomial<F>;
+    type VPAuxInfo = VPAuxInfo<F>;
+    type MultilinearExtension = Arc<DenseMultilinearExtension<F>>;
+    type SumCheckSubClaim = SumCheckSubClaim<F>;
 
-    fn extract_sum(proof: &Self::SumCheckProof) -> C::ScalarField {
+    fn extract_sum(proof: &Self::SumCheckProof) -> F {
         let start = start_timer!(|| "extract sum");
         let poly = DensePolynomial::from_coefficients_vec(proof.proofs[0].coeffs.clone());
-        let res = poly.evaluate(&C::ScalarField::ONE) + poly.evaluate(&C::ScalarField::ZERO);
+        let res = poly.evaluate(&F::ONE) + poly.evaluate(&F::ZERO);
         end_timer!(start);
         res
     }
 
     fn prove(
-        poly: &VirtualPolynomial<C::ScalarField>,
-        transcript: &mut impl Transcript<C>,
-    ) -> Result<IOPProof<C::ScalarField>, PolyIOPErrors> {
-        transcript.absorb(&C::ScalarField::from(poly.aux_info.num_variables as u64));
-        transcript.absorb(&C::ScalarField::from(poly.aux_info.max_degree as u64));
-        let mut prover_state: IOPProverState<C> = IOPProverState::prover_init(poly)?;
-        let mut challenge: Option<C::ScalarField> = None;
-        let mut prover_msgs: Vec<IOPProverMessage<C::ScalarField>> =
+        poly: &VirtualPolynomial<F>,
+        transcript: &mut impl Transcript<F>,
+    ) -> Result<IOPProof<F>, PolyIOPErrors> {
+        transcript.absorb(&F::from(poly.aux_info.num_variables as u64));
+        transcript.absorb(&F::from(poly.aux_info.max_degree as u64));
+        let mut prover_state: IOPProverState<F> = IOPProverState::prover_init(poly)?;
+        let mut challenge: Option<F> = None;
+        let mut prover_msgs: Vec<IOPProverMessage<F>> =
             Vec::with_capacity(poly.aux_info.num_variables);
         for _ in 0..poly.aux_info.num_variables {
-            let prover_msg: IOPProverMessage<C::ScalarField> =
+            let prover_msg: IOPProverMessage<F> =
                 IOPProverState::prove_round_and_update_state(&mut prover_state, &challenge)?;
-            transcript.absorb_vec(&prover_msg.coeffs);
+            transcript.absorb(&prover_msg.coeffs);
             prover_msgs.push(prover_msg);
             challenge = Some(transcript.get_challenge());
         }
@@ -178,17 +177,17 @@ impl<C: CurveGroup, T: Transcript<C>> SumCheck<C> for IOPSumCheck<C, T> {
     }
 
     fn verify(
-        claimed_sum: C::ScalarField,
-        proof: &IOPProof<C::ScalarField>,
-        aux_info: &VPAuxInfo<C::ScalarField>,
-        transcript: &mut impl Transcript<C>,
-    ) -> Result<SumCheckSubClaim<C::ScalarField>, PolyIOPErrors> {
-        transcript.absorb(&C::ScalarField::from(aux_info.num_variables as u64));
-        transcript.absorb(&C::ScalarField::from(aux_info.max_degree as u64));
+        claimed_sum: F,
+        proof: &IOPProof<F>,
+        aux_info: &VPAuxInfo<F>,
+        transcript: &mut impl Transcript<F>,
+    ) -> Result<SumCheckSubClaim<F>, PolyIOPErrors> {
+        transcript.absorb(&F::from(aux_info.num_variables as u64));
+        transcript.absorb(&F::from(aux_info.max_degree as u64));
         let mut verifier_state = IOPVerifierState::verifier_init(aux_info);
         for i in 0..aux_info.num_variables {
             let prover_msg = proof.proofs.get(i).expect("proof is incomplete");
-            transcript.absorb_vec(&prover_msg.coeffs);
+            transcript.absorb(&prover_msg.coeffs);
             IOPVerifierState::verify_round_and_update_state(
                 &mut verifier_state,
                 prover_msg,
@@ -204,16 +203,15 @@ impl<C: CurveGroup, T: Transcript<C>> SumCheck<C> for IOPSumCheck<C, T> {
 pub mod tests {
     use std::sync::Arc;
 
+    use ark_crypto_primitives::sponge::poseidon::PoseidonSponge;
+    use ark_crypto_primitives::sponge::CryptographicSponge;
     use ark_ff::Field;
     use ark_pallas::Fr;
-    use ark_pallas::Projective;
     use ark_poly::DenseMultilinearExtension;
     use ark_poly::MultilinearExtension;
     use ark_std::test_rng;
 
     use crate::transcript::poseidon::poseidon_canonical_config;
-    use crate::transcript::poseidon::PoseidonTranscript;
-    use crate::transcript::Transcript;
     use crate::utils::sum_check::SumCheck;
     use crate::utils::virtual_polynomial::VirtualPolynomial;
 
@@ -227,20 +225,19 @@ pub mod tests {
         let poseidon_config = poseidon_canonical_config::<Fr>();
 
         // sum-check prove
-        let mut poseidon_transcript_prove: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
-        let sum_check = IOPSumCheck::<Projective, PoseidonTranscript<Projective>>::prove(
+        let mut poseidon_transcript_prove: PoseidonSponge<Fr> =
+            PoseidonSponge::<Fr>::new(&poseidon_config);
+        let sum_check = IOPSumCheck::<Fr, PoseidonSponge<Fr>>::prove(
             &virtual_poly,
             &mut poseidon_transcript_prove,
         )
         .unwrap();
 
         // sum-check verify
-        let claimed_sum =
-            IOPSumCheck::<Projective, PoseidonTranscript<Projective>>::extract_sum(&sum_check);
-        let mut poseidon_transcript_verify: PoseidonTranscript<Projective> =
-            PoseidonTranscript::<Projective>::new(&poseidon_config);
-        let res_verify = IOPSumCheck::<Projective, PoseidonTranscript<Projective>>::verify(
+        let claimed_sum = IOPSumCheck::<Fr, PoseidonSponge<Fr>>::extract_sum(&sum_check);
+        let mut poseidon_transcript_verify: PoseidonSponge<Fr> =
+            PoseidonSponge::<Fr>::new(&poseidon_config);
+        let res_verify = IOPSumCheck::<Fr, PoseidonSponge<Fr>>::verify(
             claimed_sum,
             &sum_check,
             &virtual_poly.aux_info,
diff --git a/folding-schemes/src/utils/espresso/sum_check/prover.rs b/folding-schemes/src/utils/espresso/sum_check/prover.rs
index 46d2e0c..24440d2 100644
--- a/folding-schemes/src/utils/espresso/sum_check/prover.rs
+++ b/folding-schemes/src/utils/espresso/sum_check/prover.rs
@@ -14,11 +14,9 @@ use crate::utils::{
     lagrange_poly::compute_lagrange_interpolated_poly, multilinear_polynomial::fix_variables,
     virtual_polynomial::VirtualPolynomial,
 };
-use ark_ec::CurveGroup;
-use ark_ff::Field;
 use ark_ff::{batch_inversion, PrimeField};
 use ark_poly::DenseMultilinearExtension;
-use ark_std::{cfg_into_iter, end_timer, start_timer, vec::Vec};
+use ark_std::{cfg_into_iter, end_timer, start_timer};
 use rayon::prelude::{IntoParallelIterator, IntoParallelRefIterator};
 use std::sync::Arc;
 
@@ -28,9 +26,9 @@ use espresso_subroutines::poly_iop::prelude::PolyIOPErrors;
 // #[cfg(feature = "parallel")]
 use rayon::iter::{IntoParallelRefMutIterator, ParallelIterator};
 
-impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
-    type VirtualPolynomial = VirtualPolynomial<C::ScalarField>;
-    type ProverMessage = IOPProverMessage<C::ScalarField>;
+impl<F: PrimeField> SumCheckProver<F> for IOPProverState<F> {
+    type VirtualPolynomial = VirtualPolynomial<F>;
+    type ProverMessage = IOPProverMessage<F>;
 
     /// Initialize the prover state to argue for the sum of the input polynomial
     /// over {0,1}^`num_vars`.
@@ -49,9 +47,7 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
             poly: polynomial.clone(),
             extrapolation_aux: (1..polynomial.aux_info.max_degree)
                 .map(|degree| {
-                    let points = (0..1 + degree as u64)
-                        .map(C::ScalarField::from)
-                        .collect::<Vec<_>>();
+                    let points = (0..1 + degree as u64).map(F::from).collect::<Vec<_>>();
                     let weights = barycentric_weights(&points);
                     (points, weights)
                 })
@@ -65,7 +61,7 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
     /// Main algorithm used is from section 3.2 of [XZZPS19](https://eprint.iacr.org/2019/317.pdf#subsection.3.2).
     fn prove_round_and_update_state(
         &mut self,
-        challenge: &Option<C::ScalarField>,
+        challenge: &Option<F>,
     ) -> Result<Self::ProverMessage, PolyIOPErrors> {
         // let start =
         //     start_timer!(|| format!("sum check prove {}-th round and update state",
@@ -90,7 +86,7 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
         //    g(r_1, ..., r_{m-1}, x_m ... x_n)
         //
         // eval g over r_m, and mutate g to g(r_1, ... r_m,, x_{m+1}... x_n)
-        let mut flattened_ml_extensions: Vec<DenseMultilinearExtension<C::ScalarField>> = self
+        let mut flattened_ml_extensions: Vec<DenseMultilinearExtension<F>> = self
             .poly
             .flattened_ml_extensions
             .par_iter()
@@ -124,7 +120,7 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
         self.round += 1;
 
         let products_list = self.poly.products.clone();
-        let mut products_sum = vec![C::ScalarField::ZERO; self.poly.aux_info.max_degree + 1];
+        let mut products_sum = vec![F::ZERO; self.poly.aux_info.max_degree + 1];
 
         // Step 2: generate sum for the partial evaluated polynomial:
         // f(r_1, ... r_m,, x_{m+1}... x_n)
@@ -134,8 +130,8 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
                 .fold(
                     || {
                         (
-                            vec![(C::ScalarField::ZERO, C::ScalarField::ZERO); products.len()],
-                            vec![C::ScalarField::ZERO; products.len() + 1],
+                            vec![(F::ZERO, F::ZERO); products.len()],
+                            vec![F::ZERO; products.len() + 1],
                         )
                     },
                     |(mut buf, mut acc), b| {
@@ -146,17 +142,17 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
                                 *eval = table[b << 1];
                                 *step = table[(b << 1) + 1] - table[b << 1];
                             });
-                        acc[0] += buf.iter().map(|(eval, _)| eval).product::<C::ScalarField>();
+                        acc[0] += buf.iter().map(|(eval, _)| eval).product::<F>();
                         acc[1..].iter_mut().for_each(|acc| {
                             buf.iter_mut().for_each(|(eval, step)| *eval += step as &_);
-                            *acc += buf.iter().map(|(eval, _)| eval).product::<C::ScalarField>();
+                            *acc += buf.iter().map(|(eval, _)| eval).product::<F>();
                         });
                         (buf, acc)
                     },
                 )
                 .map(|(_, partial)| partial)
                 .reduce(
-                    || vec![C::ScalarField::ZERO; products.len() + 1],
+                    || vec![F::ZERO; products.len() + 1],
                     |mut sum, partial| {
                         sum.iter_mut()
                             .zip(partial.iter())
@@ -168,7 +164,7 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
             let extraploation = cfg_into_iter!(0..self.poly.aux_info.max_degree - products.len())
                 .map(|i| {
                     let (points, weights) = &self.extrapolation_aux[products.len() - 1];
-                    let at = C::ScalarField::from((products.len() + 1 + i) as u64);
+                    let at = F::from((products.len() + 1 + i) as u64);
                     extrapolate(points, weights, &sum, &at)
                 })
                 .collect::<Vec<_>>();
@@ -184,7 +180,7 @@ impl<C: CurveGroup> SumCheckProver<C> for IOPProverState<C> {
             .map(|x| Arc::new(x.clone()))
             .collect();
 
-        let prover_poly = compute_lagrange_interpolated_poly::<C::ScalarField>(&products_sum);
+        let prover_poly = compute_lagrange_interpolated_poly::<F>(&products_sum);
         Ok(IOPProverMessage {
             coeffs: prover_poly.coeffs,
         })
diff --git a/folding-schemes/src/utils/espresso/sum_check/structs.rs b/folding-schemes/src/utils/espresso/sum_check/structs.rs
index b34ed81..de487d9 100644
--- a/folding-schemes/src/utils/espresso/sum_check/structs.rs
+++ b/folding-schemes/src/utils/espresso/sum_check/structs.rs
@@ -10,7 +10,6 @@
 //! This module defines structs that are shared by all sub protocols.
 
 use crate::utils::virtual_polynomial::VirtualPolynomial;
-use ark_ec::CurveGroup;
 use ark_ff::PrimeField;
 use ark_serialize::CanonicalSerialize;
 
@@ -33,28 +32,28 @@ pub struct IOPProverMessage<F: PrimeField> {
 
 /// Prover State of a PolyIOP.
 #[derive(Debug)]
-pub struct IOPProverState<C: CurveGroup> {
+pub struct IOPProverState<F: PrimeField> {
     /// sampled randomness given by the verifier
-    pub challenges: Vec<C::ScalarField>,
+    pub challenges: Vec<F>,
     /// the current round number
     pub(crate) round: usize,
     /// pointer to the virtual polynomial
-    pub(crate) poly: VirtualPolynomial<C::ScalarField>,
+    pub(crate) poly: VirtualPolynomial<F>,
     /// points with precomputed barycentric weights for extrapolating smaller
     /// degree uni-polys to `max_degree + 1` evaluations.
     #[allow(clippy::type_complexity)]
-    pub(crate) extrapolation_aux: Vec<(Vec<C::ScalarField>, Vec<C::ScalarField>)>,
+    pub(crate) extrapolation_aux: Vec<(Vec<F>, Vec<F>)>,
 }
 
 /// Verifier State of a PolyIOP, generic over a curve group
 #[derive(Debug)]
-pub struct IOPVerifierState<C: CurveGroup> {
+pub struct IOPVerifierState<F: PrimeField> {
     pub(crate) round: usize,
     pub(crate) num_vars: usize,
     pub(crate) finished: bool,
     /// a list storing the univariate polynomial in evaluation form sent by the
     /// prover at each round
-    pub(crate) polynomials_received: Vec<Vec<C::ScalarField>>,
+    pub(crate) polynomials_received: Vec<Vec<F>>,
     /// a list storing the randomness sampled by the verifier at each round
-    pub(crate) challenges: Vec<C::ScalarField>,
+    pub(crate) challenges: Vec<F>,
 }
diff --git a/folding-schemes/src/utils/espresso/sum_check/verifier.rs b/folding-schemes/src/utils/espresso/sum_check/verifier.rs
index 55fafb2..8ff83b3 100644
--- a/folding-schemes/src/utils/espresso/sum_check/verifier.rs
+++ b/folding-schemes/src/utils/espresso/sum_check/verifier.rs
@@ -14,7 +14,7 @@ use super::{
     SumCheckSubClaim, SumCheckVerifier,
 };
 use crate::{transcript::Transcript, utils::virtual_polynomial::VPAuxInfo};
-use ark_ec::CurveGroup;
+use ark_crypto_primitives::sponge::Absorb;
 use ark_ff::PrimeField;
 use ark_poly::Polynomial;
 use ark_poly::{univariate::DensePolynomial, DenseUVPolynomial};
@@ -24,11 +24,11 @@ use espresso_subroutines::poly_iop::prelude::PolyIOPErrors;
 #[cfg(feature = "parallel")]
 use rayon::iter::{IndexedParallelIterator, IntoParallelIterator, ParallelIterator};
 
-impl<C: CurveGroup> SumCheckVerifier<C> for IOPVerifierState<C> {
-    type VPAuxInfo = VPAuxInfo<C::ScalarField>;
-    type ProverMessage = IOPProverMessage<C::ScalarField>;
-    type Challenge = C::ScalarField;
-    type SumCheckSubClaim = SumCheckSubClaim<C::ScalarField>;
+impl<F: PrimeField + Absorb> SumCheckVerifier<F> for IOPVerifierState<F> {
+    type VPAuxInfo = VPAuxInfo<F>;
+    type ProverMessage = IOPProverMessage<F>;
+    type Challenge = F;
+    type SumCheckSubClaim = SumCheckSubClaim<F>;
 
     /// Initialize the verifier's state.
     fn verifier_init(index_info: &Self::VPAuxInfo) -> Self {
@@ -46,9 +46,9 @@ impl<C: CurveGroup> SumCheckVerifier<C> for IOPVerifierState<C> {
 
     fn verify_round_and_update_state(
         &mut self,
-        prover_msg: &<IOPVerifierState<C> as SumCheckVerifier<C>>::ProverMessage,
-        transcript: &mut impl Transcript<C>,
-    ) -> Result<<IOPVerifierState<C> as SumCheckVerifier<C>>::Challenge, PolyIOPErrors> {
+        prover_msg: &<IOPVerifierState<F> as SumCheckVerifier<F>>::ProverMessage,
+        transcript: &mut impl Transcript<F>,
+    ) -> Result<<IOPVerifierState<F> as SumCheckVerifier<F>>::Challenge, PolyIOPErrors> {
         let start =
             start_timer!(|| format!("sum check verify {}-th round and update state", self.round));
 
@@ -83,7 +83,7 @@ impl<C: CurveGroup> SumCheckVerifier<C> for IOPVerifierState<C> {
 
     fn check_and_generate_subclaim(
         &self,
-        asserted_sum: &C::ScalarField,
+        asserted_sum: &F,
     ) -> Result<Self::SumCheckSubClaim, PolyIOPErrors> {
         let start = start_timer!(|| "sum check check and generate subclaim");
         if !self.finished {
@@ -136,8 +136,8 @@ impl<C: CurveGroup> SumCheckVerifier<C> for IOPVerifierState<C> {
             .take(self.num_vars)
         {
             let poly = DensePolynomial::from_coefficients_slice(coeffs);
-            let eval_at_one: C::ScalarField = poly.iter().sum();
-            let eval_at_zero: C::ScalarField = poly.coeffs[0];
+            let eval_at_one: F = poly.iter().sum();
+            let eval_at_zero: F = poly.coeffs[0];
             let eval = eval_at_one + eval_at_zero;
 
             // the deferred check during the interactive phase:
diff --git a/folding-schemes/src/utils/espresso/virtual_polynomial.rs b/folding-schemes/src/utils/espresso/virtual_polynomial.rs
index 24f0b02..acc843b 100644
--- a/folding-schemes/src/utils/espresso/virtual_polynomial.rs
+++ b/folding-schemes/src/utils/espresso/virtual_polynomial.rs
@@ -18,8 +18,6 @@ use rayon::prelude::*;
 use std::{cmp::max, collections::HashMap, marker::PhantomData, ops::Add, sync::Arc};
 use thiserror::Error;
 
-use ark_std::string::String;
-
 //-- aritherrors
 /// A `enum` specifying the possible failure modes of the arithmetics.
 #[derive(Error, Debug)]
diff --git a/folding-schemes/src/utils/lagrange_poly.rs b/folding-schemes/src/utils/lagrange_poly.rs
index f3dfaee..22a38e1 100644
--- a/folding-schemes/src/utils/lagrange_poly.rs
+++ b/folding-schemes/src/utils/lagrange_poly.rs
@@ -52,7 +52,7 @@ mod tests {
     use crate::utils::lagrange_poly::compute_lagrange_interpolated_poly;
     use ark_pallas::Fr;
     use ark_poly::{univariate::DensePolynomial, DenseUVPolynomial, Polynomial};
-    use ark_std::{vec::Vec, UniformRand};
+    use ark_std::UniformRand;
     use espresso_subroutines::poly_iop::prelude::PolyIOPErrors;
 
     #[test]
diff --git a/solidity-verifiers/src/verifiers/kzg.rs b/solidity-verifiers/src/verifiers/kzg.rs
index dab3be7..9dc156d 100644
--- a/solidity-verifiers/src/verifiers/kzg.rs
+++ b/solidity-verifiers/src/verifiers/kzg.rs
@@ -78,7 +78,8 @@ mod tests {
         utils::HeaderInclusion,
         ProtocolVerifierKey,
     };
-    use ark_bn254::{Bn254, Fr, G1Projective as G1};
+    use ark_bn254::{Bn254, Fr};
+    use ark_crypto_primitives::sponge::{poseidon::PoseidonSponge, CryptographicSponge};
     use ark_ec::{AffineRepr, CurveGroup};
     use ark_ff::{BigInteger, PrimeField};
     use ark_std::rand::{RngCore, SeedableRng};
@@ -89,10 +90,7 @@ mod tests {
 
     use folding_schemes::{
         commitment::{kzg::KZG, CommitmentScheme},
-        transcript::{
-            poseidon::{poseidon_canonical_config, PoseidonTranscript},
-            Transcript,
-        },
+        transcript::{poseidon::poseidon_canonical_config, Transcript},
     };
 
     use super::KZG10Verifier;
@@ -133,8 +131,8 @@ mod tests {
     fn kzg_verifier_accepts_and_rejects_proofs() {
         let mut rng = ark_std::rand::rngs::StdRng::seed_from_u64(test_rng().next_u64());
         let poseidon_config = poseidon_canonical_config::<Fr>();
-        let transcript_p = &mut PoseidonTranscript::<G1>::new(&poseidon_config);
-        let transcript_v = &mut PoseidonTranscript::<G1>::new(&poseidon_config);
+        let transcript_p = &mut PoseidonSponge::<Fr>::new(&poseidon_config);
+        let transcript_v = &mut PoseidonSponge::<Fr>::new(&poseidon_config);
 
         let (_, kzg_pk, kzg_vk, _, _, _) = setup(DEFAULT_SETUP_LEN);
         let kzg_vk = KZG10VerifierKey::from((kzg_vk.clone(), kzg_pk.powers_of_g[0..3].to_vec()));
@@ -159,7 +157,7 @@ mod tests {
         let (x_proof, y_proof) = proof_affine.xy().unwrap();
         let y = proof.eval.into_bigint().to_bytes_be();
 
-        transcript_v.absorb_point(&cm).unwrap();
+        transcript_v.absorb_nonnative(&cm);
         let x = transcript_v.get_challenge();
 
         let x = x.into_bigint().to_bytes_be();