@@ -243,7 +243,7 @@ let (pk, vk): (ProverKey<G1>, VerifierKey<Bn254>) = KZGSetup::<Bn254>::setup(rng, n);
         let v: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(rng)).take(n).collect();
         let cm = KZGProver::<G1>::commit(&pk, &v, &Fr::zero()).unwrap();
         let (eval, proof) =
-            KZGProver::<G1>::prove(&pk, transcript_p, &cm, &v, &Fr::zero()).unwrap();
+            KZGProver::<G1>::prove(&pk, transcript_p, &cm, &v, &Fr::zero(), None).unwrap();
         let template = KZG10Verifier::from(
             &vk,
             &pk.powers_of_g[..5],

@@ -0,0 +1,681 @@
+/// IPA implements the modified Inner Product Argument described in
+/// [Halo](https://eprint.iacr.org/2019/1021.pdf). The variable names used follow the paper
+/// notation in order to make it more readable.
+///
+/// The implementation does the following optimizations in order to reduce the amount of
+/// constraints in the circuit:
+/// i. <s, b> computation is done in log time following a modification of the equation 3 in section
+/// 3.2 from the paper.
+/// ii. s computation is done in 2^{k+1}-2 instead of k*2^k.
+use ark_ec::{AffineRepr, CurveGroup};
+use ark_ff::{Field, PrimeField};
+use ark_r1cs_std::{
+    alloc::{AllocVar, AllocationMode},
+    boolean::Boolean,
+    fields::{nonnative::NonNativeFieldVar, FieldVar},
+    groups::GroupOpsBounds,
+    prelude::CurveVar,
+    ToBitsGadget,
+};
+use ark_relations::r1cs::{Namespace, SynthesisError};
+use ark_std::{
+    cfg_iter,
+    rand::{Rng, RngCore},
+    UniformRand, Zero,
+};
+use core::{borrow::Borrow, marker::PhantomData};
+use rayon::iter::{IndexedParallelIterator, IntoParallelRefIterator, ParallelIterator};
+
+use super::{pedersen::Params as PedersenParams, CommitmentProver};
+use crate::transcript::Transcript;
+use crate::utils::{
+    powers_of,
+    vec::{vec_add, vec_scalar_mul},
+};
+use crate::Error;
+
+/// IPA implements the Inner Product Argument protocol. The `H` parameter indicates if to use the
+/// commitment in hiding mode or not.
+#[derive(Debug, Clone, Eq, PartialEq)]
+pub struct IPA<C: CurveGroup, const H: bool = false> {
+    _c: PhantomData<C>,
+}
+
+#[derive(Debug, Clone, Eq, PartialEq)]
+pub struct Proof<C: CurveGroup> {
+    a: C::ScalarField,
+    l: Vec<C::ScalarField>,
+    r: Vec<C::ScalarField>,
+    L: Vec<C>,
+    R: Vec<C>,
+}
+
+impl<C: CurveGroup, const H: bool> IPA<C, H> {
+    pub fn new_params<R: Rng>(rng: &mut R, max: usize) -> PedersenParams<C> {
+        let generators: Vec<C::Affine> = std::iter::repeat_with(|| C::Affine::rand(rng))
+            .take(max.next_power_of_two())
+            .collect();
+        PedersenParams::<C> {
+            h: C::rand(rng),
+            generators,
+        }
+    }
+}
+
+// Implement the CommitmentProver trait for IPA
+impl<C: CurveGroup, const H: bool> CommitmentProver<C, H> for IPA<C, H> {
+    type Params = PedersenParams<C>;
+    type Proof = Proof<C>;
+    fn commit(
+        params: &PedersenParams<C>,
+        a: &[C::ScalarField],
+        r: &C::ScalarField, // blinding factor
+    ) -> Result<C, Error> {
+        if params.generators.len() < a.len() {
+            return Err(Error::PedersenParamsLen(params.generators.len(), a.len()));
+        }
+        // h⋅r + <g, a>
+        // use msm_unchecked because we already ensured at the if that lengths match
+        if !H {
+            return Ok(C::msm_unchecked(&params.generators[..a.len()], a));
+        }
+        Ok(params.h.mul(r) + C::msm_unchecked(&params.generators[..a.len()], a))
+    }
+
+    fn prove(
+        params: &Self::Params,
+        transcript: &mut impl Transcript<C>,
+        P: &C, // commitment
+        a: &[C::ScalarField],
+        x: &C::ScalarField,
+        rng: Option<&mut dyn RngCore>,
+    ) -> Result<Self::Proof, Error> {
+        if !a.len().is_power_of_two() {
+            return Err(Error::NotPowerOfTwo("a".to_string(), a.len()));
+        }
+        let d = a.len();
+        let k = (f64::from(d as u32).log2()) as usize;
+
+        if params.generators.len() < a.len() {
+            return Err(Error::PedersenParamsLen(params.generators.len(), a.len()));
+        }
+        // blinding factors
+        let l: Vec<C::ScalarField>;
+        let r: Vec<C::ScalarField>;
+        if H {
+            let rng = rng.ok_or(Error::MissingRandomness)?;
+            l = std::iter::repeat_with(|| C::ScalarField::rand(rng))
+                .take(k)
+                .collect();
+            r = std::iter::repeat_with(|| C::ScalarField::rand(rng))
+                .take(k)
+                .collect();
+        } else {
+            l = vec![];
+            r = vec![];
+        }
+
+        transcript.absorb_point(P)?;
+        let s = transcript.get_challenge();
+        let U = C::generator().mul(s);
+
+        let mut a = a.to_owned();
+        let mut b = powers_of(*x, d);
+        let mut G = params.generators.clone();
+
+        let mut L: Vec<C> = vec![C::zero(); k];
+        let mut R: Vec<C> = vec![C::zero(); k];
+
+        // u challenges
+        let mut u: Vec<C::ScalarField> = vec![C::ScalarField::zero(); k];
+        for j in (0..k).rev() {
+            let m = a.len() / 2;
+
+            if H {
+                L[j] = C::msm_unchecked(&G[m..], &a[..m])
+                    + params.h.mul(l[j])
+                    + U.mul(inner_prod(&a[..m], &b[m..])?);
+                R[j] = C::msm_unchecked(&G[..m], &a[m..])
+                    + params.h.mul(r[j])
+                    + U.mul(inner_prod(&a[m..], &b[..m])?);
+            } else {
+                L[j] = C::msm_unchecked(&G[m..], &a[..m]) + U.mul(inner_prod(&a[..m], &b[m..])?);
+                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])?;
+            u[j] = transcript.get_challenge();
+
+            let uj = u[j];
+            let uj_inv = u[j]
+                .inverse()
+                .ok_or(Error::Other("error on computing inverse".to_string()))?;
+
+            // a_hi * uj^-1 + a_lo * uj
+            a = vec_add(
+                &vec_scalar_mul(&a[..m], &uj),
+                &vec_scalar_mul(&a[m..], &uj_inv),
+            )?;
+            // b_lo * uj^-1 + b_hi * uj
+            b = vec_add(
+                &vec_scalar_mul(&b[..m], &uj_inv),
+                &vec_scalar_mul(&b[m..], &uj),
+            )?;
+            // G_lo * uj^-1 + G_hi * uj
+            G = cfg_iter!(G[..m])
+                .map(|e| e.into_group().mul(uj_inv))
+                .zip(cfg_iter!(G[m..]).map(|e| e.into_group().mul(uj)))
+                .map(|(a, b)| (a + b).into_affine())
+                .collect::<Vec<C::Affine>>();
+        }
+
+        if a.len() != 1 {
+            return Err(Error::NotExpectedLength(a.len(), 1));
+        }
+        if b.len() != 1 {
+            return Err(Error::NotExpectedLength(b.len(), 1));
+        }
+        if G.len() != 1 {
+            return Err(Error::NotExpectedLength(G.len(), 1));
+        }
+
+        Ok(Self::Proof {
+            a: a[0],
+            l: l.clone(),
+            r: r.clone(),
+            L,
+            R,
+        })
+    }
+}
+
+impl<C: CurveGroup, const H: bool> IPA<C, H> {
+    #[allow(clippy::too_many_arguments)]
+    pub fn verify(
+        params: &PedersenParams<C>,
+        transcript: &mut impl Transcript<C>,
+        x: C::ScalarField, // evaluation point
+        v: C::ScalarField, // value at evaluation point
+        P: C,              // commitment
+        p: Proof<C>,
+        r: C::ScalarField, // blinding factor
+        k: usize,          // k = log2(d), where d is the degree of the committed polynomial
+    ) -> Result<bool, Error> {
+        if !H && (!p.l.is_empty() || !p.r.is_empty()) {
+            return Err(Error::CommitmentVerificationFail);
+        }
+        if H && (p.l.len() != k || p.r.len() != k) {
+            return Err(Error::CommitmentVerificationFail);
+        }
+        if p.L.len() != k || p.R.len() != k {
+            return Err(Error::CommitmentVerificationFail);
+        }
+
+        // absorbs & get challenges
+        transcript.absorb_point(&P)?;
+        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])?;
+            u[i] = transcript.get_challenge();
+        }
+
+        let P = P + U.mul(v);
+
+        let mut q_0 = P;
+        let mut r = r;
+
+        // compute u[i]^-1 once
+        let mut u_invs = vec![C::ScalarField::zero(); u.len()];
+        for (j, u_j) in u.iter().enumerate() {
+            u_invs[j] = u_j
+                .inverse()
+                .ok_or(Error::Other("error on computing inverse".to_string()))?;
+        }
+
+        // compute b & G from s
+        let s = build_s(&u, &u_invs, k)?;
+        // b = <s, b_vec> = <s, [1, x, x^2, ..., x^d-1]>
+        let b = s_b_inner(&u, &x)?;
+        let d: usize = 2_u64.pow(k as u32) as usize;
+        if params.generators.len() < d {
+            return Err(Error::PedersenParamsLen(params.generators.len(), d));
+        }
+        let G = C::msm_unchecked(&params.generators, &s);
+
+        for (j, u_j) in u.iter().enumerate() {
+            let uj2 = u_j.square();
+            let uj_inv2 = u_invs[j].square();
+
+            q_0 = q_0 + p.L[j].mul(uj2) + p.R[j].mul(uj_inv2);
+            if H {
+                r = r + p.l[j] * uj2 + p.r[j] * uj_inv2;
+            }
+        }
+
+        let q_1 = if H {
+            G.mul(p.a) + params.h.mul(r) + U.mul(p.a * b)
+        } else {
+            G.mul(p.a) + U.mul(p.a * b)
+        };
+
+        Ok(q_0 == q_1)
+    }
+}
+
+/// Computes s such that
+/// s = (
+///   u₁⁻¹ u₂⁻¹ … uₖ⁻¹,
+///   u₁   u₂⁻¹ … uₖ⁻¹,
+///   u₁⁻¹ u₂   … uₖ⁻¹,
+///   u₁   u₂   … uₖ⁻¹,
+///   ⋮    ⋮      ⋮
+///   u₁   u₂   … uₖ
+/// )
+/// Uses Halo2 approach computing $g(X) = \prod\limits_{i=0}^{k-1} (1 + u_{k - 1 - i} X^{2^i})$,
+/// taking 2^{k+1}-2.
+/// src: https://github.com/zcash/halo2/blob/81729eca91ba4755e247f49c3a72a4232864ec9e/halo2_proofs/src/poly/commitment/verifier.rs#L156
+fn build_s<F: PrimeField>(u: &[F], u_invs: &[F], k: usize) -> Result<Vec<F>, Error> {
+    let d: usize = 2_u64.pow(k as u32) as usize;
+    let mut s: Vec<F> = vec![F::one(); d];
+    for (len, (u_j, u_j_inv)) in u
+        .iter()
+        .zip(u_invs)
+        .enumerate()
+        .map(|(i, u_j)| (1 << i, u_j))
+    {
+        let (left, right) = s.split_at_mut(len);
+        let right = &mut right[0..len];
+        right.copy_from_slice(left);
+        for s in left {
+            *s *= u_j_inv;
+        }
+        for s in right {
+            *s *= u_j;
+        }
+    }
+    Ok(s)
+}
+
+/// Computes (in-circuit) s such that
+/// s = (
+///   u₁⁻¹ u₂⁻¹ … uₖ⁻¹,
+///   u₁   u₂⁻¹ … uₖ⁻¹,
+///   u₁⁻¹ u₂   … uₖ⁻¹,
+///   u₁   u₂   … uₖ⁻¹,
+///   ⋮    ⋮      ⋮
+///   u₁   u₂   … uₖ
+/// )
+/// Uses Halo2 approach computing $g(X) = \prod\limits_{i=0}^{k-1} (1 + u_{k - 1 - i} X^{2^i})$,
+/// taking 2^{k+1}-2.
+/// src: https://github.com/zcash/halo2/blob/81729eca91ba4755e247f49c3a72a4232864ec9e/halo2_proofs/src/poly/commitment/verifier.rs#L156
+fn build_s_gadget<F: PrimeField, CF: PrimeField>(
+    u: &[NonNativeFieldVar<F, CF>],
+    u_invs: &[NonNativeFieldVar<F, CF>],
+    k: usize,
+) -> Result<Vec<NonNativeFieldVar<F, CF>>, SynthesisError> {
+    let d: usize = 2_u64.pow(k as u32) as usize;
+    let mut s: Vec<NonNativeFieldVar<F, CF>> = vec![NonNativeFieldVar::one(); d];
+    for (len, (u_j, u_j_inv)) in u
+        .iter()
+        .zip(u_invs)
+        .enumerate()
+        .map(|(i, u_j)| (1 << i, u_j))
+    {
+        let (left, right) = s.split_at_mut(len);
+        let right = &mut right[0..len];
+        right.clone_from_slice(left);
+        for s in left {
+            *s *= u_j_inv;
+        }
+        for s in right {
+            *s *= u_j;
+        }
+    }
+    Ok(s)
+}
+
+fn inner_prod<F: PrimeField>(a: &[F], b: &[F]) -> Result<F, Error> {
+    if a.len() != b.len() {
+        return Err(Error::NotSameLength(
+            "a".to_string(),
+            a.len(),
+            "b".to_string(),
+            b.len(),
+        ));
+    }
+    let c = cfg_iter!(a)
+        .zip(cfg_iter!(b))
+        .map(|(a_i, b_i)| *a_i * b_i)
+        .sum();
+    Ok(c)
+}
+
+// g(x, u_1, u_2, ..., u_k) = <s, b>, naively takes linear, but can compute in log time through
+// g(x, u_1, u_2, ..., u_k) = \Prod u_i x^{2^i} + u_i^-1
+fn s_b_inner<F: PrimeField>(u: &[F], x: &F) -> Result<F, Error> {
+    let mut c: F = F::one();
+    let mut x_2_i = *x; // x_2_i is x^{2^i}, starting from x^{2^0}=x
+    for u_i in u.iter() {
+        c *= (*u_i * x_2_i)
+            + u_i
+                .inverse()
+                .ok_or(Error::Other("error on computing inverse".to_string()))?;
+        x_2_i *= x_2_i;
+    }
+    Ok(c)
+}
+
+// g(x, u_1, u_2, ..., u_k) = <s, b>, naively takes linear, but can compute in log time through
+// g(x, u_1, u_2, ..., u_k) = \Prod u_i x^{2^i} + u_i^-1
+fn s_b_inner_gadget<F: PrimeField, CF: PrimeField>(
+    u: &[NonNativeFieldVar<F, CF>],
+    x: &NonNativeFieldVar<F, CF>,
+) -> Result<NonNativeFieldVar<F, CF>, SynthesisError> {
+    let mut c: NonNativeFieldVar<F, CF> = NonNativeFieldVar::<F, CF>::one();
+    let mut x_2_i = x.clone(); // x_2_i is x^{2^i}, starting from x^{2^0}=x
+    for u_i in u.iter() {
+        c *= u_i.clone() * x_2_i.clone() + u_i.inverse()?;
+        x_2_i *= x_2_i.clone();
+    }
+    Ok(c)
+}
+
+pub type CF<C> = <<C as CurveGroup>::BaseField as Field>::BasePrimeField;
+
+pub struct ProofVar<C: CurveGroup, GC: CurveVar<C, CF<C>>> {
+    a: NonNativeFieldVar<C::ScalarField, CF<C>>,
+    l: Vec<NonNativeFieldVar<C::ScalarField, CF<C>>>,
+    r: Vec<NonNativeFieldVar<C::ScalarField, CF<C>>>,
+    L: Vec<GC>,
+    R: Vec<GC>,
+}
+impl<C, GC> AllocVar<Proof<C>, CF<C>> for ProofVar<C, GC>
+where
+    C: CurveGroup,
+    GC: CurveVar<C, CF<C>>,
+    <C as ark_ec::CurveGroup>::BaseField: PrimeField,
+{
+    fn new_variable<T: Borrow<Proof<C>>>(
+        cs: impl Into<Namespace<CF<C>>>,
+        f: impl FnOnce() -> Result<T, SynthesisError>,
+        mode: AllocationMode,
+    ) -> Result<Self, SynthesisError> {
+        f().and_then(|val| {
+            let cs = cs.into();
+
+            let a = NonNativeFieldVar::<C::ScalarField, CF<C>>::new_variable(
+                cs.clone(),
+                || Ok(val.borrow().a),
+                mode,
+            )?;
+            let l: Vec<NonNativeFieldVar<C::ScalarField, CF<C>>> =
+                Vec::new_variable(cs.clone(), || Ok(val.borrow().l.clone()), mode)?;
+            let r: Vec<NonNativeFieldVar<C::ScalarField, CF<C>>> =
+                Vec::new_variable(cs.clone(), || Ok(val.borrow().r.clone()), mode)?;
+            let L: Vec<GC> = Vec::new_variable(cs.clone(), || Ok(val.borrow().L.clone()), mode)?;
+            let R: Vec<GC> = Vec::new_variable(cs.clone(), || Ok(val.borrow().R.clone()), mode)?;
+
+            Ok(Self { a, l, r, L, R })
+        })
+    }
+}
+
+/// IPAGadget implements the circuit that verifies an IPA Proof. The `H` parameter indicates if to
+/// use the commitment in hiding mode or not, reducing a bit the number of constraints needed in
+/// the later case.
+pub struct IPAGadget<C, GC, const H: bool = false>
+where
+    C: CurveGroup,
+    GC: CurveVar<C, CF<C>>,
+{
+    _cf: PhantomData<CF<C>>,
+    _c: PhantomData<C>,
+    _gc: PhantomData<GC>,
+}
+
+impl<C, GC, const H: bool> IPAGadget<C, GC, H>
+where
+    C: CurveGroup,
+    GC: CurveVar<C, CF<C>>,
+
+    <C as ark_ec::CurveGroup>::BaseField: ark_ff::PrimeField,
+    for<'a> &'a GC: GroupOpsBounds<'a, C, GC>,
+{
+    /// Verify the IPA opening proof, K=log2(d), where d is the degree of the committed polynomial,
+    /// and H indicates if the commitment is in hiding mode and thus uses blinding factors, if not,
+    /// there are some constraints saved.
+    #[allow(clippy::too_many_arguments)]
+    pub fn verify<const K: usize>(
+        g: &Vec<GC>,                                  // params.generators
+        h: &GC,                                       // params.h
+        x: &NonNativeFieldVar<C::ScalarField, CF<C>>, // evaluation point
+        v: &NonNativeFieldVar<C::ScalarField, CF<C>>, // value at evaluation point
+        P: &GC,                                       // commitment
+        p: &ProofVar<C, GC>,
+        r: &NonNativeFieldVar<C::ScalarField, CF<C>>, // blinding factor
+        u: &[NonNativeFieldVar<C::ScalarField, CF<C>>; K], // challenges
+        U: &GC,                                       // challenge
+    ) -> Result<Boolean<CF<C>>, SynthesisError> {
+        if p.L.len() != K || p.R.len() != K {
+            return Err(SynthesisError::Unsatisfiable);
+        }
+
+        let P_ = P + U.scalar_mul_le(v.to_bits_le()?.iter())?;
+        let mut q_0 = P_;
+        let mut r = r.clone();
+
+        // compute u[i]^-1 once
+        let mut u_invs = vec![NonNativeFieldVar::<C::ScalarField, CF<C>>::zero(); u.len()];
+        for (j, u_j) in u.iter().enumerate() {
+            u_invs[j] = u_j.inverse()?;
+        }
+
+        // compute b & G from s
+        // let d: usize = 2_u64.pow(K as u32) as usize;
+        let s = build_s_gadget(u, &u_invs, K)?;
+        // b = <s, b_vec> = <s, [1, x, x^2, ..., x^K-1]>
+        let b = s_b_inner_gadget(u, x)?;
+        // ensure that generators.len() === s.len():
+        if g.len() < K {
+            return Err(SynthesisError::Unsatisfiable);
+        }
+
+        // msm: G=<G, s>
+        let mut G = GC::zero();
+        for (i, s_i) in s.iter().enumerate() {
+            G += g[i].scalar_mul_le(s_i.to_bits_le()?.iter())?;
+        }
+
+        for (j, u_j) in u.iter().enumerate() {
+            let uj2 = u_j.square()?;
+            let uj_inv2 = u_invs[j].square()?; // cheaper square than inversing the uj2
+
+            q_0 = q_0
+                + p.L[j].scalar_mul_le(uj2.to_bits_le()?.iter())?
+                + p.R[j].scalar_mul_le(uj_inv2.to_bits_le()?.iter())?;
+            if H {
+                r = r + &p.l[j] * &uj2 + &p.r[j] * &uj_inv2;
+            }
+        }
+
+        let q_1 = if H {
+            G.scalar_mul_le(p.a.to_bits_le()?.iter())?
+                + h.scalar_mul_le(r.to_bits_le()?.iter())?
+                + U.scalar_mul_le((p.a.clone() * b).to_bits_le()?.iter())?
+        } else {
+            G.scalar_mul_le(p.a.to_bits_le()?.iter())?
+                + U.scalar_mul_le((p.a.clone() * b).to_bits_le()?.iter())?
+        };
+        // q_0 == q_1
+        q_0.is_eq(&q_1)
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    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_relations::r1cs::ConstraintSystem;
+    use ark_std::UniformRand;
+    use std::ops::Mul;
+
+    use super::*;
+    use crate::transcript::poseidon::{poseidon_test_config, PoseidonTranscript};
+
+    #[test]
+    fn test_ipa() {
+        test_ipa_opt::<false>();
+        test_ipa_opt::<true>();
+    }
+    fn test_ipa_opt<const hiding: bool>() {
+        let mut rng = ark_std::test_rng();
+
+        const k: usize = 4;
+        const d: usize = 2_u64.pow(k as u32) as usize;
+
+        // setup params
+        let params = IPA::<Projective, hiding>::new_params(&mut rng, d);
+
+        let poseidon_config = poseidon_test_config::<Fr>();
+        // init Prover's transcript
+        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        // init Verifier's transcript
+        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+
+        let a: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(&mut rng))
+            .take(d)
+            .collect();
+        let r_blind: Fr = Fr::rand(&mut rng);
+        let cm = IPA::<Projective, hiding>::commit(&params, &a, &r_blind).unwrap();
+
+        // evaluation point
+        let x = Fr::rand(&mut rng);
+
+        let proof = IPA::<Projective, hiding>::prove(
+            &params,
+            &mut transcript_p,
+            &cm,
+            &a,
+            &x,
+            Some(&mut rng),
+        )
+        .unwrap();
+
+        let b = powers_of(x, d);
+        let v = inner_prod(&a, &b).unwrap();
+        assert!(IPA::<Projective, hiding>::verify(
+            &params,
+            &mut transcript_v,
+            x,
+            v,
+            cm,
+            proof,
+            r_blind,
+            k,
+        )
+        .unwrap());
+    }
+
+    #[test]
+    fn test_ipa_gadget() {
+        test_ipa_gadget_opt::<false>();
+        test_ipa_gadget_opt::<true>();
+    }
+    fn test_ipa_gadget_opt<const hiding: bool>() {
+        let mut rng = ark_std::test_rng();
+
+        const k: usize = 4;
+        const d: usize = 2_u64.pow(k as u32) as usize;
+
+        // setup params
+        let params = IPA::<Projective, hiding>::new_params(&mut rng, d);
+
+        let poseidon_config = poseidon_test_config::<Fr>();
+        // init Prover's transcript
+        let mut transcript_p = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        // init Verifier's transcript
+        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+
+        let mut a: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(&mut rng))
+            .take(d / 2)
+            .collect();
+        a.extend(vec![Fr::zero(); d / 2]);
+        let r_blind: Fr = Fr::rand(&mut rng);
+        let cm = IPA::<Projective, hiding>::commit(&params, &a, &r_blind).unwrap();
+
+        // evaluation point
+        let x = Fr::rand(&mut rng);
+
+        let proof = IPA::<Projective, hiding>::prove(
+            &params,
+            &mut transcript_p,
+            &cm,
+            &a,
+            &x,
+            Some(&mut rng),
+        )
+        .unwrap();
+
+        let b = powers_of(x, d);
+        let v = inner_prod(&a, &b).unwrap();
+        assert!(IPA::<Projective, hiding>::verify(
+            &params,
+            &mut transcript_v,
+            x,
+            v,
+            cm,
+            proof.clone(),
+            r_blind,
+            k,
+        )
+        .unwrap());
+
+        // circuit
+        let cs = ConstraintSystem::<Fq>::new_ref();
+
+        let mut transcript_v = PoseidonTranscript::<Projective>::new(&poseidon_config);
+        transcript_v.absorb_point(&cm).unwrap();
+        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.L[i]).unwrap();
+            transcript_v.absorb_point(&proof.R[i]).unwrap();
+            u[i] = transcript_v.get_challenge();
+        }
+
+        // prepare inputs
+        let gVar = Vec::<GVar>::new_constant(cs.clone(), params.generators).unwrap();
+        let hVar = GVar::new_constant(cs.clone(), params.h).unwrap();
+        let xVar = NonNativeFieldVar::<Fr, Fq>::new_witness(cs.clone(), || Ok(x)).unwrap();
+        let vVar = NonNativeFieldVar::<Fr, Fq>::new_witness(cs.clone(), || Ok(v)).unwrap();
+        let cmVar = GVar::new_witness(cs.clone(), || Ok(cm)).unwrap();
+        let proofVar = ProofVar::<Projective, GVar>::new_witness(cs.clone(), || Ok(proof)).unwrap();
+        let r_blindVar =
+            NonNativeFieldVar::<Fr, Fq>::new_witness(cs.clone(), || Ok(r_blind)).unwrap();
+        let uVar_vec = Vec::<NonNativeFieldVar<Fr, Fq>>::new_witness(cs.clone(), || Ok(u)).unwrap();
+        let uVar: [NonNativeFieldVar<Fr, Fq>; k] = uVar_vec.try_into().unwrap();
+        let UVar = GVar::new_witness(cs.clone(), || Ok(U)).unwrap();
+
+        let v = IPAGadget::<Projective, GVar, hiding>::verify::<k>(
+            // &mut transcriptVar,
+            &gVar,
+            &hVar,
+            &xVar,
+            &vVar,
+            &cmVar,
+            &proofVar,
+            &r_blindVar,
+            &uVar,
+            &UVar,
+        )
+        .unwrap();
+        v.enforce_equal(&Boolean::TRUE).unwrap();
+        assert!(cs.is_satisfied().unwrap());
+    }
+}

@@ -17,7 +17,7 @@ univariate::{DenseOrSparsePolynomial, DensePolynomial},
     DenseUVPolynomial, EvaluationDomain, Evaluations, GeneralEvaluationDomain, Polynomial,
 };
 use ark_poly_commit::kzg10::{VerifierKey, KZG10};
-use ark_std::rand::Rng;
+use ark_std::rand::{Rng, RngCore};
 use ark_std::{borrow::Cow, fmt::Debug};
 use ark_std::{One, Zero};
 use core::marker::PhantomData;
@@ -66,11 +66,11 @@ }
 
 /// KZGProver implements the CommitmentProver trait for the KZG commitment scheme.
 #[derive(Debug, Clone, Default, Eq, PartialEq)]
-pub struct KZGProver<'a, C: CurveGroup> {
+pub struct KZGProver<'a, C: CurveGroup, const H: bool = false> {
     _a: PhantomData<&'a ()>,
     _c: PhantomData<C>,
 }
-impl<'a, C> CommitmentProver<C> for KZGProver<'a, C>
+impl<'a, C, const H: bool> CommitmentProver<C, H> for KZGProver<'a, C, H>
 where
     C: CurveGroup,
 {
@@ -87,8 +87,8 @@ params: &Self::Params,
         v: &[C::ScalarField],
         _blind: &C::ScalarField,
     ) -> Result<C, Error> {
-        if !_blind.is_zero() {
-            return Err(Error::NotSupportedYet("blinding factors".to_string()));
+        if !_blind.is_zero() || H {
+            return Err(Error::NotSupportedYet("hiding".to_string()));
         }
 
         let polynomial = poly_from_vec(v.to_vec())?;
@@ -113,9 +113,10 @@ transcript: &mut impl Transcript<C>,
         cm: &C,
         v: &[C::ScalarField],
         _blind: &C::ScalarField,
+        _rng: Option<&mut dyn RngCore>,
     ) -> Result<Self::Proof, Error> {
-        if !_blind.is_zero() {
-            return Err(Error::NotSupportedYet("blinding factors".to_string()));
+        if !_blind.is_zero() || H {
+            return Err(Error::NotSupportedYet("hiding".to_string()));
         }
 
         let polynomial = poly_from_vec(v.to_vec())?;
@@ -213,7 +214,7 @@ let v: Vec<Fr> = std::iter::repeat_with(|| Fr::rand(rng)).take(n).collect();
         let cm = KZGProver::<G1>::commit(&pk, &v, &Fr::zero()).unwrap();
 
         let (eval, proof) =
-            KZGProver::<G1>::prove(&pk, transcript_p, &cm, &v, &Fr::zero()).unwrap();
+            KZGProver::<G1>::prove(&pk, transcript_p, &cm, &v, &Fr::zero(), None).unwrap();
 
         // verify the proof:
         // get evaluation challenge

@@ -1,14 +1,17 @@
 use ark_ec::CurveGroup;
 use ark_std::fmt::Debug;
+use ark_std::rand::RngCore;
 
 use crate::transcript::Transcript;
 use crate::Error;
 
+pub mod ipa;
 pub mod kzg;
 pub mod pedersen;
 
-/// CommitmentProver defines the vector commitment scheme prover trait.
-pub trait CommitmentProver<C: CurveGroup>: Clone + Debug {
+/// CommitmentProver defines the vector commitment scheme prover trait. Where `H` indicates if to
+/// use the commitment in hiding mode or not.
+pub trait CommitmentProver<C: CurveGroup, const H: bool = false>: Clone + Debug {
     type Params: Clone + Debug;
     type Proof: Clone + Debug;
 
@@ -23,6 +26,7 @@ transcript: &mut impl Transcript<C>,
         cm: &C,
         v: &[C::ScalarField],
         blind: &C::ScalarField,
+        rng: Option<&mut dyn RngCore>,
     ) -> Result<Self::Proof, Error>;
 }
 
@@ -65,7 +69,15 @@ let cm_3 = cm_1 + cm_2.mul(r);
         let v_3: Vec<C::ScalarField> = v_1.iter().zip(v_2).map(|(a, b)| *a + (r * b)).collect();
 
         let transcript = &mut PoseidonTranscript::<C>::new(poseidon_config);
-        let proof = CP::prove(params, transcript, &cm_3, &v_3, &C::ScalarField::zero()).unwrap();
+        let proof = CP::prove(
+            params,
+            transcript,
+            &cm_3,
+            &v_3,
+            &C::ScalarField::zero(),
+            None,
+        )
+        .unwrap();
 
         Ok((cm_3, proof))
     }

@@ -1,8 +1,12 @@
 use ark_ec::CurveGroup;
 use ark_ff::Field;
-use ark_r1cs_std::{boolean::Boolean, groups::GroupOpsBounds, prelude::CurveVar};
+use ark_r1cs_std::{groups::GroupOpsBounds, prelude::CurveVar};
 use ark_relations::r1cs::SynthesisError;
-use ark_std::{rand::Rng, UniformRand};
+use ark_std::Zero;
+use ark_std::{
+    rand::{Rng, RngCore},
+    UniformRand,
+};
 use core::marker::PhantomData;
 
 use super::CommitmentProver;
@@ -24,25 +28,24 @@ pub generators: Vec<C::Affine>,
 }
 
 #[derive(Debug, Clone, Eq, PartialEq)]
-pub struct Pedersen<C: CurveGroup> {
+pub struct Pedersen<C: CurveGroup, const H: bool = false> {
     _c: PhantomData<C>,
 }
 
-impl<C: CurveGroup> Pedersen<C> {
+impl<C: CurveGroup, const H: bool> Pedersen<C, H> {
     pub fn new_params<R: Rng>(rng: &mut R, max: usize) -> Params<C> {
         let generators: Vec<C::Affine> = std::iter::repeat_with(|| C::Affine::rand(rng))
             .take(max.next_power_of_two())
             .collect();
-        let params: Params<C> = Params::<C> {
+        Params::<C> {
             h: C::rand(rng),
             generators,
-        };
-        params
+        }
     }
 }
 
 // implement the CommitmentProver trait for Pedersen
-impl<C: CurveGroup> CommitmentProver<C> for Pedersen<C> {
+impl<C: CurveGroup, const H: bool> CommitmentProver<C, H> for Pedersen<C, H> {
     type Params = Params<C>;
     type Proof = Proof<C>;
     fn commit(
@@ -55,6 +58,9 @@ return Err(Error::PedersenParamsLen(params.generators.len(), v.len()));
         }
         // h⋅r + <g, v>
         // use msm_unchecked because we already ensured at the if that lengths match
+        if !H {
+            return Ok(C::msm_unchecked(&params.generators[..v.len()], v));
+        }
         Ok(params.h.mul(r) + C::msm_unchecked(&params.generators[..v.len()], v))
     }
 
@@ -64,6 +70,7 @@ transcript: &mut impl Transcript<C>,
         cm: &C,
         v: &[C::ScalarField],
         r: &C::ScalarField, // blinding factor
+        _rng: Option<&mut dyn RngCore>,
     ) -> Result<Self::Proof, Error> {
         if params.generators.len() < v.len() {
             return Err(Error::PedersenParamsLen(params.generators.len(), v.len()));
@@ -75,7 +82,10 @@ let d = transcript.get_challenges(v.len());
 
         // R = h⋅r_1 + <g, d>
         // use msm_unchecked because we already ensured at the if that lengths match
-        let R: C = params.h.mul(r1) + C::msm_unchecked(&params.generators[..d.len()], &d);
+        let mut R: C = C::msm_unchecked(&params.generators[..d.len()], &d);
+        if H {
+            R += params.h.mul(r1);
+        }
 
         transcript.absorb_point(&R)?;
         let e = transcript.get_challenge();
@@ -83,13 +93,16 @@
         // u = d + v⋅e
         let u = vec_add(&vec_scalar_mul(v, &e), &d)?;
         // r_u = e⋅r + r_1
-        let r_u = e * r + r1;
+        let mut r_u = C::ScalarField::zero();
+        if H {
+            r_u = e * r + r1;
+        }
 
         Ok(Self::Proof { R, u, r_u })
     }
 }
 
-impl<C: CurveGroup> Pedersen<C> {
+impl<C: CurveGroup, const H: bool> Pedersen<C, H> {
     pub fn verify(
         params: &Params<C>,
         transcript: &mut impl Transcript<C>,
@@ -112,8 +125,10 @@
         // check that: R + cm⋅e == h⋅r_u + <g, u>
         let lhs = proof.R + cm.mul(e);
         // use msm_unchecked because we already ensured at the if that lengths match
-        let rhs = params.h.mul(proof.r_u)
-            + C::msm_unchecked(&params.generators[..proof.u.len()], &proof.u);
+        let mut rhs = C::msm_unchecked(&params.generators[..proof.u.len()], &proof.u);
+        if H {
+            rhs += params.h.mul(proof.r_u);
+        }
         if lhs != rhs {
             return Err(Error::CommitmentVerificationFail);
         }
@@ -123,7 +138,7 @@ }
 
 pub type CF<C> = <<C as CurveGroup>::BaseField as Field>::BasePrimeField;
 
-pub struct PedersenGadget<C, GC>
+pub struct PedersenGadget<C, GC, const H: bool = false>
 where
     C: CurveGroup,
     GC: CurveVar<C, CF<C>>,
@@ -133,7 +148,8 @@ _c: PhantomData<C>,
     _gc: PhantomData<GC>,
 }
 
-impl<C, GC> PedersenGadget<C, GC>
+use ark_r1cs_std::{fields::nonnative::NonNativeFieldVar, ToBitsGadget};
+impl<C, GC, const H: bool> PedersenGadget<C, GC, H>
 where
     C: CurveGroup,
     GC: CurveVar<C, CF<C>>,
@@ -144,13 +160,15 @@ {
     pub fn commit(
         h: GC,
         g: Vec<GC>,
-        v: Vec<Vec<Boolean<CF<C>>>>,
-        r: Vec<Boolean<CF<C>>>,
+        v: Vec<NonNativeFieldVar<C::ScalarField, CF<C>>>,
+        r: NonNativeFieldVar<C::ScalarField, CF<C>>,
     ) -> Result<GC, SynthesisError> {
         let mut res = GC::zero();
-        res += h.scalar_mul_le(r.iter())?;
+        if H {
+            res += h.scalar_mul_le(r.to_bits_le()?.iter())?;
+        }
         for (i, v_i) in v.iter().enumerate() {
-            res += g[i].scalar_mul_le(v_i.iter())?;
+            res += g[i].scalar_mul_le(v_i.to_bits_le()?.iter())?;
         }
         Ok(res)
     }
@@ -158,9 +176,8 @@ }
 
 #[cfg(test)]
 mod tests {
-    use ark_ff::{BigInteger, PrimeField};
     use ark_pallas::{constraints::GVar, Fq, Fr, Projective};
-    use ark_r1cs_std::{alloc::AllocVar, bits::boolean::Boolean, eq::EqGadget};
+    use ark_r1cs_std::{alloc::AllocVar, eq::EqGadget};
     use ark_relations::r1cs::ConstraintSystem;
     use ark_std::UniformRand;
 
@@ -186,15 +203,20 @@ .take(n)
             .collect();
         let r: Fr = Fr::rand(&mut rng);
         let cm = Pedersen::<Projective>::commit(&params, &v, &r).unwrap();
-        let proof = Pedersen::<Projective>::prove(&params, &mut transcript_p, &cm, &v, &r).unwrap();
+        let proof =
+            Pedersen::<Projective>::prove(&params, &mut transcript_p, &cm, &v, &r, None).unwrap();
         Pedersen::<Projective>::verify(&params, &mut transcript_v, cm, proof).unwrap();
     }
 
     #[test]
     fn test_pedersen_circuit() {
+        test_pedersen_circuit_opt::<false>();
+        test_pedersen_circuit_opt::<true>();
+    }
+    fn test_pedersen_circuit_opt<const hiding: bool>() {
         let mut rng = ark_std::test_rng();
 
-        let n: usize = 10;
+        let n: usize = 16;
         // setup params
         let params = Pedersen::<Projective>::new_params(&mut rng, n);
 
@@ -207,17 +229,9 @@
         // circuit
         let cs = ConstraintSystem::<Fq>::new_ref();
 
-        let v_bits: Vec<Vec<bool>> = v.iter().map(|val| val.into_bigint().to_bits_le()).collect();
-        let r_bits: Vec<bool> = r.into_bigint().to_bits_le();
-
         // prepare inputs
-        let vVar: Vec<Vec<Boolean<Fq>>> = v_bits
-            .iter()
-            .map(|val_bits| {
-                Vec::<Boolean<Fq>>::new_witness(cs.clone(), || Ok(val_bits.clone())).unwrap()
-            })
-            .collect();
-        let rVar = Vec::<Boolean<Fq>>::new_witness(cs.clone(), || Ok(r_bits)).unwrap();
+        let vVar = Vec::<NonNativeFieldVar<Fr, Fq>>::new_witness(cs.clone(), || Ok(v)).unwrap();
+        let rVar = NonNativeFieldVar::<Fr, Fq>::new_witness(cs.clone(), || Ok(r)).unwrap();
         let gVar = Vec::<GVar>::new_witness(cs.clone(), || Ok(params.generators)).unwrap();
         let hVar = GVar::new_witness(cs.clone(), || Ok(params.h)).unwrap();
         let expected_cmVar = GVar::new_witness(cs.clone(), || Ok(cm)).unwrap();

@@ -112,7 +112,7 @@ w: &Witness<C::ScalarField>,
     ) -> Result<(), Error> {
         // check that C is the commitment of w. Notice that this is not verifying a Pedersen
         // opening, but checking that the commitment comes from committing to the witness.
-        if self.C != Pedersen::commit(pedersen_params, &w.w, &w.r_w)? {
+        if self.C != Pedersen::<C>::commit(pedersen_params, &w.w, &w.r_w)? {
             return Err(Error::NotSatisfied);
         }
 

@@ -180,7 +180,7 @@ let gamma: Fr = Fr::rand(&mut rng);
         let r_x_prime: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
 
         // Initialize a multifolding object
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
         let (lcccs_instance, _) = ccs.to_lcccs(&mut rng, &pedersen_params, &z1).unwrap();
         let sigmas_thetas =
             compute_sigmas_and_thetas(&ccs, &[z1.clone()], &[z2.clone()], &r_x_prime);
@@ -224,7 +224,7 @@
         let r_x_prime: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
 
         // Initialize a multifolding object
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
         let (lcccs_instance, _) = ccs.to_lcccs(&mut rng, &pedersen_params, &z1).unwrap();
         let sigmas_thetas =
             compute_sigmas_and_thetas(&ccs, &[z1.clone()], &[z2.clone()], &r_x_prime);
@@ -267,7 +267,7 @@ let beta: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
         let r_x_prime: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
 
         // Initialize a multifolding object
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
         let (lcccs_instance, _) = ccs.to_lcccs(&mut rng, &pedersen_params, &z1).unwrap();
         let sigmas_thetas =
             compute_sigmas_and_thetas(&ccs, &[z1.clone()], &[z2.clone()], &r_x_prime);

@@ -46,7 +46,7 @@ z: &[C::ScalarField],
     ) -> Result<(LCCCS<C>, Witness<C::ScalarField>), Error> {
         let w: Vec<C::ScalarField> = z[(1 + self.l)..].to_vec();
         let r_w = C::ScalarField::rand(rng);
-        let C = Pedersen::commit(pedersen_params, &w, &r_w)?;
+        let C = Pedersen::<C>::commit(pedersen_params, &w, &r_w)?;
 
         let r_x: Vec<C::ScalarField> = (0..self.s).map(|_| C::ScalarField::rand(rng)).collect();
         let v = self.compute_v_j(z, &r_x);
@@ -96,8 +96,8 @@ ccs: &CCS<C>,
         w: &Witness<C::ScalarField>,
     ) -> Result<(), Error> {
         // check that C is the commitment of w. Notice that this is not verifying a Pedersen
-        // opening, but checking that the commitment comes from committing to the witness.
-        if self.C != Pedersen::commit(pedersen_params, &w.w, &w.r_w)? {
+        // opening, but checking that the Commmitment comes from committing to the witness.
+        if self.C != Pedersen::<C>::commit(pedersen_params, &w.w, &w.r_w)? {
             return Err(Error::NotSatisfied);
         }
 

@@ -430,7 +430,7 @@ let mut rng = test_rng();
 
         // Create a basic CCS circuit
         let ccs = get_test_ccs::<Projective>();
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
 
         // Generate a satisfying witness
         let z_1 = get_test_z(3);
@@ -489,7 +489,7 @@ let mut rng = test_rng();
 
         let ccs = get_test_ccs::<Projective>();
 
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
 
         // LCCCS witness
         let z_1 = get_test_z(2);
@@ -557,7 +557,7 @@ let mut rng = test_rng();
 
         // Create a basic CCS circuit
         let ccs = get_test_ccs::<Projective>();
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
 
         let mu = 10;
         let nu = 15;
@@ -639,7 +639,7 @@ let mut rng = test_rng();
 
         // Create a basic CCS circuit
         let ccs = get_test_ccs::<Projective>();
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
 
         let poseidon_config = poseidon_test_config::<Fr>();
         // Prover's transcript

@@ -290,7 +290,7 @@ let beta: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
         let r_x_prime: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
 
         // Initialize a multifolding object
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
         let (lcccs_instance, _) = ccs.to_lcccs(&mut rng, &pedersen_params, &z1).unwrap();
 
         let sigmas_thetas =
@@ -333,7 +333,7 @@ let gamma: Fr = Fr::rand(&mut rng);
         let beta: Vec<Fr> = (0..ccs.s).map(|_| Fr::rand(&mut rng)).collect();
 
         // Initialize a multifolding object
-        let pedersen_params = Pedersen::new_params(&mut rng, ccs.n - ccs.l - 1);
+        let pedersen_params = Pedersen::<Projective>::new_params(&mut rng, ccs.n - ccs.l - 1);
         let (lcccs_instance, _) = ccs.to_lcccs(&mut rng, &pedersen_params, &z1).unwrap();
 
         let mut sum_v_j_gamma = Fr::zero();

@@ -654,7 +654,6 @@ augmented_F_circuit
             .generate_constraints(cs.clone())
             .unwrap();
         cs.finalize();
-        println!("num_constraints={:?}", cs.num_constraints());
         let cs = cs.into_inner().unwrap();
         let r1cs = extract_r1cs::<Fr>(&cs);
         let (w, x) = extract_w_x::<Fr>(&cs);

@@ -459,7 +459,6 @@ )
         .unwrap();
         nifs_cf_check.enforce_equal(&Boolean::<Fq>::TRUE).unwrap();
         assert!(cs.is_satisfied().unwrap());
-        dbg!(cs.num_constraints());
     }
 
     #[test]
@@ -500,7 +499,6 @@ )
         .unwrap();
         nifs_check.enforce_equal(&Boolean::<Fq>::TRUE).unwrap();
         assert!(cs.is_satisfied().unwrap());
-        dbg!(cs.num_constraints());
     }
 
     #[test]

@@ -370,13 +370,6 @@ // imports here instead of at the top of the file, so we avoid having multiple
             // `#[cfg(not(test))]`
             use crate::commitment::pedersen::PedersenGadget;
             use crate::folding::nova::cyclefold::{CycleFoldCommittedInstanceVar, CF_IO_LEN};
-            use ark_r1cs_std::ToBitsGadget;
-
-            let cf_r1cs = R1CSVar::<
-                C1::BaseField,
-                CF1<C1>,
-                NonNativeFieldVar<C1::BaseField, CF1<C1>>,
-            >::new_witness(cs.clone(), || Ok(self.cf_r1cs.clone()))?;
 
             let cf_u_dummy_native = CommittedInstance::<C2>::dummy(CF_IO_LEN);
             let w_dummy_native = Witness::<C2>::new(
@@ -393,27 +386,23 @@
             // 5. check Pedersen commitments of cf_U_i.{cmE, cmW}
             let H = GC2::new_constant(cs.clone(), self.cf_pedersen_params.h)?;
             let G = Vec::<GC2>::new_constant(cs.clone(), self.cf_pedersen_params.generators)?;
-            let cf_W_i_E_bits: Vec<Vec<Boolean<CF1<C1>>>> = cf_W_i
-                .E
-                .iter()
-                .map(|E_i| E_i.to_bits_le().unwrap())
-                .collect();
-            let cf_W_i_W_bits: Vec<Vec<Boolean<CF1<C1>>>> = cf_W_i
-                .W
-                .iter()
-                .map(|W_i| W_i.to_bits_le().unwrap())
-                .collect();
 
             let computed_cmE = PedersenGadget::<C2, GC2>::commit(
                 H.clone(),
                 G.clone(),
-                cf_W_i_E_bits,
-                cf_W_i.rE.to_bits_le()?,
+                cf_W_i.E.clone(),
+                cf_W_i.rE,
             )?;
             cf_U_i.cmE.enforce_equal(&computed_cmE)?;
             let computed_cmW =
-                PedersenGadget::<C2, GC2>::commit(H, G, cf_W_i_W_bits, cf_W_i.rW.to_bits_le()?)?;
+                PedersenGadget::<C2, GC2>::commit(H, G, cf_W_i.W.clone(), cf_W_i.rW)?;
             cf_U_i.cmW.enforce_equal(&computed_cmW)?;
+
+            let cf_r1cs = R1CSVar::<
+                C1::BaseField,
+                CF1<C1>,
+                NonNativeFieldVar<C1::BaseField, CF1<C1>>,
+            >::new_witness(cs.clone(), || Ok(self.cf_r1cs.clone()))?;
 
             // 6. check RelaxedR1CS of cf_U_i
             let cf_z_U: Vec<NonNativeFieldVar<C2::ScalarField, CF1<C1>>> =
@@ -681,6 +670,5 @@
         // generate the constraints and check that are satisfied by the inputs
         decider_circuit.generate_constraints(cs.clone()).unwrap();
         assert!(cs.is_satisfied().unwrap());
-        dbg!(cs.num_constraints());
     }
 }

@@ -190,9 +190,9 @@ ci: &CommittedInstance<C>,
         T: Vec<C::ScalarField>,
         cmT: &C,
     ) -> Result<[CP::Proof; 3], Error> {
-        let cmE_proof = CP::prove(cm_prover_params, tr, &ci.cmE, &w.E, &w.rE)?;
-        let cmW_proof = CP::prove(cm_prover_params, tr, &ci.cmW, &w.W, &w.rW)?;
-        let cmT_proof = CP::prove(cm_prover_params, tr, cmT, &T, &C::ScalarField::one())?; // cm(T) is committed with rT=1
+        let cmE_proof = CP::prove(cm_prover_params, tr, &ci.cmE, &w.E, &w.rE, None)?;
+        let cmW_proof = CP::prove(cm_prover_params, tr, &ci.cmW, &w.W, &w.rW, None)?;
+        let cmT_proof = CP::prove(cm_prover_params, tr, cmT, &T, &C::ScalarField::one(), None)?; // cm(T) is committed with rT=1
         Ok([cmE_proof, cmW_proof, cmT_proof])
     }
 }

@@ -18,7 +18,8 @@ use super::{CommittedInstance, Witness};
 
 use crate::ccs::r1cs::R1CS;
 use crate::transcript::Transcript;
-use crate::utils::{bit::bit_decompose, vec::*};
+use crate::utils::vec::*;
+use crate::utils::virtual_polynomial::bit_decompose;
 use crate::Error;
 
 #[derive(Clone, Debug)]

@@ -40,10 +40,14 @@ #[error("Vectors should have the same length ({0}: {1}, {2}: {3})")]
     NotSameLength(String, usize, String, usize),
     #[error("Vector's length ({0}) is not the expected ({1})")]
     NotExpectedLength(usize, usize),
+    #[error("Vector ({0}) length ({1}) is not a power of two")]
+    NotPowerOfTwo(String, usize),
     #[error("Can not be empty")]
     Empty,
     #[error("Pedersen parameters length is not sufficient (generators.len={0} < vector.len={1} unsatisfied)")]
     PedersenParamsLen(usize, usize),
+    #[error("Randomness for blinding not found")]
+    MissingRandomness,
     #[error("Commitment verification failed")]
     CommitmentVerificationFail,
     #[error("IVC verification failed")]

@@ -1,9 +0,0 @@
-pub fn bit_decompose(input: u64, n: usize) -> Vec<bool> {
-    let mut res = Vec::with_capacity(n);
-    let mut i = input;
-    for _ in 0..n {
-        res.push(i & 1 == 1);
-        i >>= 1;
-    }
-    res
-}

@@ -1,4 +1,5 @@
-pub mod bit;
+use ark_ff::PrimeField;
+
 pub mod gadgets;
 pub mod hypercube;
 pub mod lagrange_poly;
@@ -10,3 +11,13 @@ pub mod espresso;
 pub use crate::utils::espresso::multilinear_polynomial;
 pub use crate::utils::espresso::sum_check;
 pub use crate::utils::espresso::virtual_polynomial;
+
+/// For a given x, returns [1, x^1, x^2, ..., x^n-1];
+pub fn powers_of<F: PrimeField>(x: F, n: usize) -> Vec<F> {
+    let mut c: Vec<F> = vec![F::zero(); n];
+    c[0] = F::one();
+    for i in 1..n {
+        c[i] = c[i - 1] * x;
+    }
+    c
+}