diff --git a/src/circuit.rs b/src/circuit.rs
index f4a1724..002a7c2 100644
--- a/src/circuit.rs
+++ b/src/circuit.rs
@@ -1,36 +1,31 @@
-//! There are two Verification Circuits. Each of them is over a Pasta curve but
-//! only one of them executes the next step of the computation by applying the inner function F.
-//! There are also two running relaxed r1cs instances.
-//!
-//! When we build a circuit we denote u1 the running relaxed r1cs instance of
-//! the circuit and u2 the running relaxed r1cs instance of the other circuit.
-//! The circuit takes as input two hashes h1 and h2.
-//! If the circuit applies the inner function F, then
-//! h1 = H(params = H(shape, gens), u2, i, z0, zi) and h2 = H(u1, i)
-//! otherwise
-//! h1 = H(u2, i) and h2 = H(params = H(shape, gens), u1, i, z0, zi)
+//! There are two Verification Circuits. The primary and the secondary.
+//! Each of them is over a Pasta curve but
+//! only the primary executes the next step of the computation.
+//! TODO: The base case is different for the primary and the secondary.
+//! We have two running instances. Each circuit takes as input 2 hashes: one for each
+//! of the running instances. Each of these hashes is
+//! H(params = H(shape, gens), i, z0, zi, U). Each circuit folds the last invocation of
+//! the other into the running instance
 
 use super::{
   commitments::Commitment,
   gadgets::{
     ecc::AllocatedPoint,
-    utils::{
-      alloc_bignat_constant, alloc_num_equals, alloc_one, alloc_zero, conditionally_select,
-      conditionally_select_bignat, le_bits_to_num,
-    },
+    r1cs::{AllocatedR1CSInstance, AllocatedRelaxedR1CSInstance},
+    utils::{alloc_num_equals, alloc_zero, conditionally_select, le_bits_to_num},
   },
   poseidon::{NovaPoseidonConstants, PoseidonROGadget},
-  r1cs::RelaxedR1CSInstance,
+  r1cs::{R1CSInstance, RelaxedR1CSInstance},
   traits::{Group, StepCircuit},
 };
 use bellperson::{
-  gadgets::{boolean::Boolean, num::AllocatedNum, Assignment},
+  gadgets::{
+    boolean::{AllocatedBit, Boolean},
+    num::AllocatedNum,
+    Assignment,
+  },
   Circuit, ConstraintSystem, SynthesisError,
 };
-use bellperson_nonnative::{
-  mp::bignat::BigNat,
-  util::{convert::f_to_nat, num::Num},
-};
 use ff::{Field, PrimeField, PrimeFieldBits};
 
 #[derive(Debug, Clone)]
@@ -53,15 +48,13 @@ pub struct NIFSVerifierCircuitInputs<G>
 where
   G: Group,
 {
-  h1: G::Base,
-  h2: G::Base,
-  u2: RelaxedR1CSInstance<G>,
+  params: G::Base, // Hash(Shape of u2, Gens for u2). Needed for computing the challenge.
   i: G::Base,
   z0: G::Base,
   zi: G::Base,
-  params: G::Base, // Hash(Shape of u2, Gens for u2). Needed for computing the challenge.
+  U: RelaxedR1CSInstance<G>,
+  u: R1CSInstance<G>,
   T: Commitment<G>,
-  w: Commitment<G>, // The commitment to the witness of the fresh r1cs instance
 }
 
 impl<G> NIFSVerifierCircuitInputs<G>
@@ -71,26 +64,22 @@ where
   /// Create new inputs/witness for the verification circuit
   #[allow(dead_code, clippy::too_many_arguments)]
   pub fn new(
-    h1: G::Base,
-    u2: RelaxedR1CSInstance<G>,
+    params: G::Base,
     i: G::Base,
     z0: G::Base,
     zi: G::Base,
-    h2: G::Base,
-    params: G::Base,
+    U: RelaxedR1CSInstance<G>,
+    u: R1CSInstance<G>,
     T: Commitment<G>,
-    w: Commitment<G>,
   ) -> Self {
     Self {
-      h1,
-      u2,
+      params,
       i,
       z0,
       zi,
-      h2,
-      params,
+      U,
+      u,
       T,
-      w,
     }
   }
 }
@@ -111,7 +100,8 @@ where
 impl<G, SC> NIFSVerifierCircuit<G, SC>
 where
   G: Group,
-  <G as Group>::Base: ff::PrimeField,
+  <G as Group>::Base: PrimeField + PrimeFieldBits,
+  <G as Group>::Scalar: PrimeField + PrimeFieldBits,
   SC: StepCircuit<G::Base>,
 {
   /// Create a new verification circuit for the input relaxed r1cs instances
@@ -132,133 +122,56 @@ where
       poseidon_constants,
     }
   }
-}
-
-impl<G, SC> Circuit<<G as Group>::Base> for NIFSVerifierCircuit<G, SC>
-where
-  G: Group,
-  <G as Group>::Base: PrimeField + PrimeFieldBits,
-  <G as Group>::Scalar: PrimeFieldBits,
-  SC: StepCircuit<G::Base>,
-{
-  fn synthesize<CS: ConstraintSystem<<G as Group>::Base>>(
-    self,
-    cs: &mut CS,
-  ) -> Result<(), SynthesisError> {
-    /***********************************************************************/
-    // Allocate h1
-    /***********************************************************************/
-
-    let h1 = AllocatedNum::alloc(cs.namespace(|| "allocate h1"), || Ok(self.inputs.get()?.h1))?;
-    let h1_bn = BigNat::from_num(
-      cs.namespace(|| "allocate h1_bn"),
-      Num::from(h1.clone()),
-      self.params.limb_width,
-      self.params.n_limbs,
-    )?;
 
-    /***********************************************************************/
-    // This circuit does not modify h2 but it outputs it.
-    // Allocate it and output it.
-    /***********************************************************************/
+  /// Allocate all witnesses and return
+  fn alloc_witness<CS: ConstraintSystem<<G as Group>::Base>>(
+    &self,
+    mut cs: CS,
+  ) -> Result<
+    (
+      AllocatedNum<G::Base>,
+      AllocatedNum<G::Base>,
+      AllocatedNum<G::Base>,
+      AllocatedNum<G::Base>,
+      AllocatedRelaxedR1CSInstance<G>,
+      AllocatedR1CSInstance<G>,
+      AllocatedPoint<G::Base>,
+    ),
+    SynthesisError,
+  > {
+    // Allocate the params
+    let params = AllocatedNum::alloc(cs.namespace(|| "params"), || Ok(self.inputs.get()?.params))?;
 
-    // Allocate h2 as a big number with 8 limbs
-    let h2 = AllocatedNum::alloc(cs.namespace(|| "allocate h2"), || Ok(self.inputs.get()?.h2))?;
-    let h2_bn = BigNat::from_num(
-      cs.namespace(|| "allocate h2_bn"),
-      Num::from(h2.clone()),
-      self.params.limb_width,
-      self.params.n_limbs,
-    )?;
+    // Allocate i
+    let i = AllocatedNum::alloc(cs.namespace(|| "i"), || Ok(self.inputs.get()?.i))?;
 
-    let _ = h2.inputize(cs.namespace(|| "Output 1"))?;
+    // Allocate z0
+    let z_0 = AllocatedNum::alloc(cs.namespace(|| "z0"), || Ok(self.inputs.get()?.z0))?;
 
-    /***********************************************************************/
-    // Allocate u2 by allocating W_r, E_r, u_r, X_r
-    /***********************************************************************/
+    // Allocate zi
+    let z_i = AllocatedNum::alloc(cs.namespace(|| "zi"), || Ok(self.inputs.get()?.zi))?;
 
-    // W_r = (x, y, infinity)
-    let W_r = AllocatedPoint::alloc(
-      cs.namespace(|| "allocate W_r"),
+    // Allocate the running instance
+    let U: AllocatedRelaxedR1CSInstance<G> = AllocatedRelaxedR1CSInstance::alloc(
+      cs.namespace(|| "Allocate U"),
       self
         .inputs
         .get()
-        .map_or(None, |inputs| Some(inputs.u2.comm_W.comm.to_coordinates())),
+        .map_or(None, |inputs| Some(inputs.U.clone())),
+      self.params.limb_width,
+      self.params.n_limbs,
     )?;
 
-    // E_r = (x, y, infinity)
-    let E_r = AllocatedPoint::alloc(
-      cs.namespace(|| "allocate E_r"),
+    // Allocate the instance to be folded in
+    let u = AllocatedR1CSInstance::alloc(
+      cs.namespace(|| "allocate instance u to fold"),
       self
         .inputs
         .get()
-        .map_or(None, |inputs| Some(inputs.u2.comm_E.comm.to_coordinates())),
+        .map_or(None, |inputs| Some(inputs.u.clone())),
     )?;
 
-    // u_r << |G::Base| despite the fact that u_r is a scalar.
-    // So we parse all of its bytes as a G::Base element
-    let u_r = AllocatedNum::alloc(cs.namespace(|| "u_r"), || {
-      let u_bits = self.inputs.get()?.u2.u.to_le_bits();
-      let mut mult = G::Base::one();
-      let mut u = G::Base::zero();
-      for bit in u_bits {
-        if bit {
-          u += mult;
-        }
-        mult = mult + mult;
-      }
-      Ok(u)
-    })?;
-
-    // The running X is two items! the running h1 and the running h2
-    let Xr0 = BigNat::alloc_from_nat(
-      cs.namespace(|| "allocate X_r[0]"),
-      || Ok(f_to_nat(&self.inputs.get()?.u2.X[0])),
-      self.params.limb_width,
-      self.params.n_limbs,
-    )?;
-
-    // Analyze Xr0 as limbs to use later
-    let Xr0_bn = Xr0
-      .as_limbs::<CS>()
-      .iter()
-      .enumerate()
-      .map(|(i, limb)| {
-        limb
-          .as_sapling_allocated_num(cs.namespace(|| format!("convert limb {} of X_r[0] to num", i)))
-      })
-      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
-
-    let Xr1 = BigNat::alloc_from_nat(
-      cs.namespace(|| "allocate X_r[1]"),
-      || Ok(f_to_nat(&self.inputs.get()?.u2.X[1])),
-      self.params.limb_width,
-      self.params.n_limbs,
-    )?;
-
-    // Analyze Xr1 as limbs to use later
-    let Xr1_bn = Xr1
-      .as_limbs::<CS>()
-      .iter()
-      .enumerate()
-      .map(|(i, limb)| {
-        limb
-          .as_sapling_allocated_num(cs.namespace(|| format!("convert limb {} of X_r[1] to num", i)))
-      })
-      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
-
-    /***********************************************************************/
-    // Allocate i
-    /***********************************************************************/
-
-    let i = AllocatedNum::alloc(cs.namespace(|| "i"), || Ok(self.inputs.get()?.i))?;
-
-    /***********************************************************************/
     // Allocate T
-    /***********************************************************************/
-
-    // T = (x, y, infinity)
-
     let T = AllocatedPoint::alloc(
       cs.namespace(|| "allocate T"),
       self
@@ -267,200 +180,125 @@ where
         .map_or(None, |inputs| Some(inputs.T.comm.to_coordinates())),
     )?;
 
-    /***********************************************************************/
-    // Allocate params
-    /***********************************************************************/
-
-    let params = AllocatedNum::alloc(cs.namespace(|| "params"), || Ok(self.inputs.get()?.params))?;
-
-    /***********************************************************************/
-    // Allocate W
-    /***********************************************************************/
-
-    // W = (x, y, infinity)
-    let W = AllocatedPoint::alloc(
-      cs.namespace(|| "allocate W"),
-      self
-        .inputs
-        .get()
-        .map_or(None, |inputs| Some(inputs.w.comm.to_coordinates())),
-    )?;
-
-    /***********************************************************************/
-    // U2' = default if i == 0, otherwise NIFS.V(pp, u_new, U, T)
-    /***********************************************************************/
-
-    // Allocate 0 and 1
-    let zero = alloc_zero(cs.namespace(|| "zero"))?;
-    let one = alloc_one(cs.namespace(|| "one"))?;
-
-    // Compute default values of U2':
-    // W_default and E_default are a commitment to zero
-    let W_default = AllocatedPoint::new(zero.clone(), zero.clone(), one);
-    let E_default = W_default.clone();
-
-    // u_default = 0
-    let u_default = zero.clone();
-
-    // X_default = 0
-    let X0_default = BigNat::alloc_from_nat(
-      cs.namespace(|| "allocate x_default[0]"),
-      || Ok(f_to_nat(&G::Scalar::zero())),
-      self.params.limb_width,
-      self.params.n_limbs,
-    )?;
+    Ok((params, i, z_0, z_i, U, u, T))
+  }
 
-    let X1_default = BigNat::alloc_from_nat(
-      cs.namespace(|| "allocate x_default[1]"),
-      || Ok(f_to_nat(&G::Scalar::zero())),
+  /// Synthesizes base case and returns the new relaxed R1CSInstance
+  fn synthesize_base_case<CS: ConstraintSystem<<G as Group>::Base>>(
+    &self,
+    mut cs: CS,
+  ) -> Result<AllocatedRelaxedR1CSInstance<G>, SynthesisError> {
+    let U_default: AllocatedRelaxedR1CSInstance<G> = AllocatedRelaxedR1CSInstance::default(
+      cs.namespace(|| "Allocate U_default"),
       self.params.limb_width,
       self.params.n_limbs,
     )?;
+    Ok(U_default)
+  }
 
-    // Compute r:
-
+  /// Synthesizes non base case and returns the new relaxed R1CSInstance
+  /// And a boolean indicating if all checks pass
+  #[allow(clippy::too_many_arguments)]
+  fn synthesize_non_base_case<CS: ConstraintSystem<<G as Group>::Base>>(
+    &self,
+    mut cs: CS,
+    params: AllocatedNum<G::Base>,
+    i: AllocatedNum<G::Base>,
+    z_0: AllocatedNum<G::Base>,
+    z_i: AllocatedNum<G::Base>,
+    U: AllocatedRelaxedR1CSInstance<G>,
+    u: AllocatedR1CSInstance<G>,
+    T: AllocatedPoint<G::Base>,
+  ) -> Result<(AllocatedRelaxedR1CSInstance<G>, AllocatedBit), SynthesisError> {
+    // Check that u.x[0] = Hash(params, U,i,z0,zi)
     let mut ro: PoseidonROGadget<G::Base> = PoseidonROGadget::new(self.poseidon_constants.clone());
+    ro.absorb(params);
+    ro.absorb(i);
+    ro.absorb(z_0);
+    ro.absorb(z_i);
+    let _ = U.absorb_in_ro(cs.namespace(|| "absorb U"), &mut ro)?;
 
-    ro.absorb(h1.clone());
-    ro.absorb(h2);
-    ro.absorb(W.x.clone());
-    ro.absorb(W.y.clone());
-    ro.absorb(W.is_infinity.clone());
-    ro.absorb(T.x.clone());
-    ro.absorb(T.y.clone());
-    ro.absorb(T.is_infinity.clone());
-    // absorb each of the limbs of X_r[0]
-    for limb in Xr0_bn.clone().into_iter() {
-      ro.absorb(limb);
-    }
-
-    // absorb each of the limbs of X_r[1]
-    for limb in Xr1_bn.clone().into_iter() {
-      ro.absorb(limb);
-    }
-
-    let r_bits = ro.get_challenge(cs.namespace(|| "r bits"))?;
-    let r = le_bits_to_num(cs.namespace(|| "r"), r_bits.clone())?;
-
-    // W_fold = W_r + r * W
-    let rW = W.scalar_mul(cs.namespace(|| "r * W"), r_bits.clone())?;
-    let W_fold = W_r.add(cs.namespace(|| "W_r + r * W"), &rW)?;
-
-    // E_fold = E_r + r * T
-    let rT = T.scalar_mul(cs.namespace(|| "r * T"), r_bits)?;
-    let E_fold = E_r.add(cs.namespace(|| "E_r + r * T"), &rT)?;
-
-    // u_fold = u_r + r
-    let u_fold = AllocatedNum::alloc(cs.namespace(|| "u_fold"), || {
-      Ok(*u_r.get_value().get()? + r.get_value().get()?)
-    })?;
-    cs.enforce(
-      || "Check u_fold",
-      |lc| lc,
-      |lc| lc,
-      |lc| lc + u_fold.get_variable() - u_r.get_variable() - r.get_variable(),
-    );
-
-    // Fold the IO:
-    // Analyze r into limbs
-    let r_bn = BigNat::from_num(
-      cs.namespace(|| "allocate r_bn"),
-      Num::from(r.clone()),
-      self.params.limb_width,
-      self.params.n_limbs,
+    let hash_bits = ro.get_hash(cs.namespace(|| "Input hash"))?;
+    let hash = le_bits_to_num(cs.namespace(|| "bits to hash"), hash_bits)?;
+    let check_pass = alloc_num_equals(
+      cs.namespace(|| "check consistency of u.X[0] with H(params, U, i, z0, zi)"),
+      u.X0.clone(),
+      hash,
     )?;
 
-    // Allocate the order of the non-native field as a constant
-    let m_bn = alloc_bignat_constant(
-      cs.namespace(|| "alloc m"),
-      &G::get_order(),
+    // Run NIFS Verifier
+    let U_fold = U.fold_with_r1cs(
+      cs.namespace(|| "compute fold of U and u"),
+      u,
+      T,
+      self.poseidon_constants.clone(),
       self.params.limb_width,
       self.params.n_limbs,
     )?;
 
-    // First the fold h1 with X_r[0];
-    let (_, r_0) = h1_bn.mult_mod(cs.namespace(|| "r*h1"), &r_bn, &m_bn)?;
-    // add X_r[0]
-    let r_new_0 = Xr0.add::<CS>(&r_0)?;
-    // Now reduce
-    let Xr0_fold = r_new_0.red_mod(cs.namespace(|| "reduce folded X_r[0]"), &m_bn)?;
-
-    // First the fold h2 with X_r[1];
-    let (_, r_1) = h2_bn.mult_mod(cs.namespace(|| "r*h2"), &r_bn, &m_bn)?;
-    // add X_r[1]
-    let r_new_1 = Xr1.add::<CS>(&r_1)?;
-    // Now reduce
-    let Xr1_fold = r_new_1.red_mod(cs.namespace(|| "reduce folded X_r[1]"), &m_bn)?;
-
-    // Now select the default values if i == 0 otherwise the fold values
-    let is_base_case = Boolean::from(alloc_num_equals(
-      cs.namespace(|| "Check if base case"),
-      i.clone(),
-      zero,
-    )?);
+    Ok((U_fold, check_pass))
+  }
+}
 
-    let W_new = AllocatedPoint::conditionally_select(
-      cs.namespace(|| "W_new"),
-      &W_default,
-      &W_fold,
-      &is_base_case,
-    )?;
+impl<G, SC> Circuit<<G as Group>::Base> for NIFSVerifierCircuit<G, SC>
+where
+  G: Group,
+  <G as Group>::Base: PrimeField + PrimeFieldBits,
+  <G as Group>::Scalar: PrimeFieldBits,
+  SC: StepCircuit<G::Base>,
+{
+  fn synthesize<CS: ConstraintSystem<<G as Group>::Base>>(
+    self,
+    cs: &mut CS,
+  ) -> Result<(), SynthesisError> {
+    // Allocate all witnesses
+    let (params, i, z_0, z_i, U, u, T) =
+      self.alloc_witness(cs.namespace(|| "allocate the circuit witness"))?;
 
-    let E_new = AllocatedPoint::conditionally_select(
-      cs.namespace(|| "E_new"),
-      &E_default,
-      &E_fold,
-      &is_base_case,
-    )?;
+    // Compute variable indicating if this is the base case
+    let zero = alloc_zero(cs.namespace(|| "zero"))?;
+    let is_base_case = alloc_num_equals(cs.namespace(|| "Check if base case"), i.clone(), zero)?; //TODO: maybe optimize this?
 
-    let u_new = conditionally_select(cs.namespace(|| "u_new"), &u_default, &u_fold, &is_base_case)?;
+    // Synthesize the circuit for the base case and get the new running instance
+    let Unew_base = self.synthesize_base_case(cs.namespace(|| "base case"))?;
 
-    let Xr0_new = conditionally_select_bignat(
-      cs.namespace(|| "X_r_new[0]"),
-      &X0_default,
-      &Xr0_fold,
-      &is_base_case,
+    // Synthesize the circuit for the non-base case and get the new running
+    // instance along with a boolean indicating if all checks have passed
+    let (Unew_non_base, check_non_base_pass) = self.synthesize_non_base_case(
+      cs.namespace(|| "synthesize non base case"),
+      params.clone(),
+      i.clone(),
+      z_0.clone(),
+      z_i.clone(),
+      U,
+      u.clone(),
+      T,
     )?;
 
-    // Analyze Xr0_new as limbs to use later
-    let Xr0_new_bn = Xr0_new
-      .as_limbs::<CS>()
-      .iter()
-      .enumerate()
-      .map(|(i, limb)| {
-        limb.as_sapling_allocated_num(
-          cs.namespace(|| format!("convert limb {} of X_r_new[0] to num", i)),
-        )
-      })
-      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
-
-    let Xr1_new = conditionally_select_bignat(
-      cs.namespace(|| "X_r_new[1]"),
-      &X1_default,
-      &Xr1_fold,
+    // Either check_non_base_pass=true or we are in the base case
+    let should_be_false = AllocatedBit::nor(
+      cs.namespace(|| "check_non_base_pass nor base_case"),
+      &check_non_base_pass,
       &is_base_case,
     )?;
+    cs.enforce(
+      || "check_non_base_pass nor base_case = false",
+      |lc| lc + should_be_false.get_variable(),
+      |lc| lc + CS::one(),
+      |lc| lc,
+    );
 
-    // Analyze Xr1_new as limbs to use later
-    let Xr1_new_bn = Xr1_new
-      .as_limbs::<CS>()
-      .iter()
-      .enumerate()
-      .map(|(i, limb)| {
-        limb.as_sapling_allocated_num(
-          cs.namespace(|| format!("convert limb {} of X_r_new[1] to num", i)),
-        )
-      })
-      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
-
-    /***********************************************************************/
-    // Compute i + 1
-    /***********************************************************************/
+    // Compute the U_new
+    let Unew = Unew_base.conditionally_select(
+      cs.namespace(|| "compute U_new"),
+      Unew_non_base,
+      &Boolean::from(is_base_case.clone()),
+    )?;
 
+    // Compute i + 1
     let i_new = AllocatedNum::alloc(cs.namespace(|| "i + 1"), || {
       Ok(*i.get_value().get()? + G::Base::one())
     })?;
-
     cs.enforce(
       || "check i + 1",
       |lc| lc,
@@ -468,113 +306,32 @@ where
       |lc| lc + i_new.get_variable() - CS::one() - i.get_variable(),
     );
 
-    /***********************************************************************/
-    // Allocate z0
-    /***********************************************************************/
-
-    let z_0 = AllocatedNum::alloc(cs.namespace(|| "z0"), || Ok(self.inputs.get()?.z0))?;
-
-    /***********************************************************************/
-    // Allocate zi
-    /***********************************************************************/
-
-    let z_i = AllocatedNum::alloc(cs.namespace(|| "zi"), || Ok(self.inputs.get()?.zi))?;
-
-    /***********************************************************************/
-    //Check that if i == 0, z0 = zi, that is (i == 0) AND (z0 != zi) = false
-    /***********************************************************************/
-
-    let z0_is_zi = Boolean::from(alloc_num_equals(
-      cs.namespace(|| "z0 = zi"),
-      z_0.clone(),
-      z_i.clone(),
-    )?);
-
-    cs.enforce(
-      || "i == 0 and z0 != zi = false",
-      |_| is_base_case.lc(CS::one(), G::Base::one()),
-      |_| z0_is_zi.not().lc(CS::one(), G::Base::one()),
-      |lc| lc,
-    );
-
-    /***********************************************************************/
-    // Check that h1 = Hash(params, u2,i,z0,zi)
-    /***********************************************************************/
-
-    let mut h1_hash: PoseidonROGadget<G::Base> =
-      PoseidonROGadget::new(self.poseidon_constants.clone());
-
-    h1_hash.absorb(params.clone());
-    h1_hash.absorb(W_r.x);
-    h1_hash.absorb(W_r.y);
-    h1_hash.absorb(W_r.is_infinity);
-    h1_hash.absorb(E_r.x);
-    h1_hash.absorb(E_r.y);
-    h1_hash.absorb(E_r.is_infinity);
-    h1_hash.absorb(u_r.clone());
-
-    // absorb each of the limbs of X_r[0]
-    for limb in Xr0_bn.into_iter() {
-      h1_hash.absorb(limb);
-    }
-
-    // absorb each of the limbs of X_r[1]
-    for limb in Xr1_bn.into_iter() {
-      h1_hash.absorb(limb);
-    }
-
-    h1_hash.absorb(i.clone());
-    h1_hash.absorb(z_0.clone());
-    h1_hash.absorb(z_i.clone());
-
-    let hash_bits = h1_hash.get_challenge(cs.namespace(|| "Input hash"))?; // TODO: use get_hash method
-    let hash = le_bits_to_num(cs.namespace(|| "bits to hash"), hash_bits)?;
-
-    cs.enforce(
-      || "check h1",
-      |lc| lc,
-      |lc| lc,
-      |lc| lc + h1.get_variable() - hash.get_variable(),
-    );
-
-    /***********************************************************************/
     // Compute z_{i+1}
-    /***********************************************************************/
-
+    let z_input = conditionally_select(
+      cs.namespace(|| "select input to F"),
+      &z_0,
+      &z_i,
+      &Boolean::from(is_base_case),
+    )?;
     let z_next = self
       .step_circuit
-      .synthesize(&mut cs.namespace(|| "F"), z_i)?;
-
-    /***********************************************************************/
-    // Compute the new hash H(params, u2_new, i+1, z0, z_{i+1})
-    /***********************************************************************/
-
-    h1_hash.flush_state();
-    h1_hash.absorb(params);
-    h1_hash.absorb(W_new.x.clone());
-    h1_hash.absorb(W_new.y.clone());
-    h1_hash.absorb(W_new.is_infinity);
-    h1_hash.absorb(E_new.x.clone());
-    h1_hash.absorb(E_new.y.clone());
-    h1_hash.absorb(E_new.is_infinity);
-    h1_hash.absorb(u_new);
-
-    // absorb each of the limbs of X_r_new[0]
-    for limb in Xr0_new_bn.into_iter() {
-      h1_hash.absorb(limb);
-    }
-
-    // absorb each of the limbs of X_r_new[1]
-    for limb in Xr1_new_bn.into_iter() {
-      h1_hash.absorb(limb);
-    }
-
-    h1_hash.absorb(i_new.clone());
-    h1_hash.absorb(z_0);
-    h1_hash.absorb(z_next);
-    let h1_new_bits = h1_hash.get_challenge(cs.namespace(|| "h1_new bits"))?; // TODO: use get_hash method
-    let h1_new = le_bits_to_num(cs.namespace(|| "h1_new"), h1_new_bits)?;
-    let _ = h1_new.inputize(cs.namespace(|| "output h1_new"))?;
+      .synthesize(&mut cs.namespace(|| "F"), z_input)?;
+
+    // Compute the new hash H(params, Unew, i+1, z0, z_{i+1})
+    let mut ro: PoseidonROGadget<G::Base> = PoseidonROGadget::new(self.poseidon_constants);
+    ro.absorb(params);
+    ro.absorb(i_new.clone());
+    ro.absorb(z_0);
+    ro.absorb(z_next);
+    let _ = Unew.absorb_in_ro(cs.namespace(|| "absorb U_new"), &mut ro)?;
+    let hash_bits = ro.get_hash(cs.namespace(|| "output hash bits"))?;
+    let hash = le_bits_to_num(cs.namespace(|| "convert hash to num"), hash_bits)?;
+
+    // Outputs the computed hash and u.X[1] that corresponds to the hash of the other circuit
+    let _ = hash.inputize(cs.namespace(|| "output new hash of this circuit"))?;
+    let _ = u
+      .X1
+      .inputize(cs.namespace(|| "Output unmodified hash of the other circuit"))?;
 
     Ok(())
   }
@@ -628,6 +385,7 @@ mod tests {
         },
         poseidon_constants1.clone(),
       );
+
     // First create the shape
     let mut cs: ShapeCS<G1> = ShapeCS::new();
     let _ = circuit1.synthesize(&mut cs);
@@ -658,27 +416,20 @@ mod tests {
       cs.num_constraints()
     );
 
-    // TODO: We need to hardwire default hash or give it as input
-    let default_hash = <<G2 as Group>::Base as ff::PrimeField>::from_str_vartime(
-      "332553638888022689042501686561503049809",
-    )
-    .unwrap();
-
+    let zero = <<G2 as Group>::Base as Field>::zero();
+    let zero_fq = <<G2 as Group>::Scalar as Field>::zero();
     let T = vec![<G2 as Group>::Scalar::zero()].commit(&gens2.gens_E);
     let w = vec![<G2 as Group>::Scalar::zero()].commit(&gens2.gens_E);
-
     // Now get an assignment
     let mut cs: SatisfyingAssignment<G1> = SatisfyingAssignment::new();
     let inputs: NIFSVerifierCircuitInputs<G2> = NIFSVerifierCircuitInputs::new(
-      default_hash,
-      RelaxedR1CSInstance::default(&gens2, &shape2),
       <<G2 as Group>::Base as Field>::zero(), // TODO: provide real inputs
-      <<G2 as Group>::Base as Field>::zero(), // TODO: provide real inputs
-      <<G2 as Group>::Base as Field>::zero(), // TODO: provide real inputs
-      <<G2 as Group>::Base as Field>::zero(), // TODO: provide real inputs
-      <<G2 as Group>::Base as Field>::zero(), // TODO: provide real inputs
-      T,                                      // TODO: provide real inputs
-      w,
+      zero,                                   // TODO: provide real inputs
+      zero,                                   // TODO: provide real inputs
+      zero,                                   // TODO: provide real inputs
+      RelaxedR1CSInstance::default(&gens2, &shape2),
+      R1CSInstance::new(&shape2, &w, &[zero_fq, zero_fq]).unwrap(),
+      T, // TODO: provide real inputs
     );
 
     let circuit: NIFSVerifierCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
diff --git a/src/gadgets/mod.rs b/src/gadgets/mod.rs
index c845d83..b8f53e0 100644
--- a/src/gadgets/mod.rs
+++ b/src/gadgets/mod.rs
@@ -1,2 +1,3 @@
 pub mod ecc;
+pub mod r1cs;
 pub mod utils;
diff --git a/src/gadgets/r1cs.rs b/src/gadgets/r1cs.rs
new file mode 100644
index 0000000..4d92dd0
--- /dev/null
+++ b/src/gadgets/r1cs.rs
@@ -0,0 +1,363 @@
+use crate::{
+  gadgets::{
+    ecc::AllocatedPoint,
+    utils::{
+      alloc_bignat_constant, alloc_one, alloc_scalar_as_base, alloc_zero, conditionally_select,
+      conditionally_select_bignat, le_bits_to_num,
+    },
+  },
+  poseidon::{NovaPoseidonConstants, PoseidonROGadget},
+  r1cs::{R1CSInstance, RelaxedR1CSInstance},
+  traits::Group,
+};
+use bellperson::{
+  gadgets::{boolean::Boolean, num::AllocatedNum, Assignment},
+  ConstraintSystem, SynthesisError,
+};
+use bellperson_nonnative::{
+  mp::bignat::BigNat,
+  util::{convert::f_to_nat, num::Num},
+};
+use ff::{Field, PrimeField, PrimeFieldBits};
+
+/// An Allocated R1CS Instance
+#[derive(Clone)]
+pub struct AllocatedR1CSInstance<G>
+where
+  G: Group,
+{
+  pub(crate) W: AllocatedPoint<G::Base>,
+  pub(crate) X0: AllocatedNum<G::Base>,
+  pub(crate) X1: AllocatedNum<G::Base>,
+}
+
+impl<G> AllocatedR1CSInstance<G>
+where
+  G: Group,
+  <G as Group>::Base: PrimeField + PrimeFieldBits,
+  <G as Group>::Scalar: PrimeFieldBits,
+{
+  /// Takes the r1cs instance and creates a new allocated r1cs instance
+  pub fn alloc<CS: ConstraintSystem<<G as Group>::Base>>(
+    mut cs: CS,
+    u: Option<R1CSInstance<G>>,
+  ) -> Result<Self, SynthesisError> {
+    // Check that the incoming instance has exactly 2 io
+    let W = AllocatedPoint::alloc(
+      cs.namespace(|| "allocate W"),
+      u.get()
+        .map_or(None, |u| Some(u.comm_W.comm.to_coordinates())),
+    )?;
+
+    let X0 = alloc_scalar_as_base::<G, _>(
+      cs.namespace(|| "allocate X[0]"),
+      u.get().map_or(None, |u| Some(u.X[0])),
+    )?;
+    let X1 = alloc_scalar_as_base::<G, _>(
+      cs.namespace(|| "allocate X[1]"),
+      u.get().map_or(None, |u| Some(u.X[1])),
+    )?;
+
+    Ok(AllocatedR1CSInstance { W, X0, X1 })
+  }
+
+  pub fn absorb_in_ro(&self, ro: &mut PoseidonROGadget<G::Base>) {
+    ro.absorb(self.W.x.clone());
+    ro.absorb(self.W.y.clone());
+    ro.absorb(self.W.is_infinity.clone());
+    ro.absorb(self.X0.clone());
+    ro.absorb(self.X1.clone());
+  }
+}
+
+/// An Allocated Relaxed R1CS Instance
+pub struct AllocatedRelaxedR1CSInstance<G>
+where
+  G: Group,
+  <G as Group>::Base: PrimeField + PrimeFieldBits,
+  <G as Group>::Scalar: PrimeFieldBits,
+{
+  pub(crate) W: AllocatedPoint<G::Base>,
+  pub(crate) E: AllocatedPoint<G::Base>,
+  pub(crate) u: AllocatedNum<G::Base>,
+  pub(crate) X0: BigNat<G::Base>,
+  pub(crate) X1: BigNat<G::Base>,
+}
+
+impl<G> AllocatedRelaxedR1CSInstance<G>
+where
+  G: Group,
+  <G as Group>::Base: PrimeField + PrimeFieldBits,
+  <G as Group>::Scalar: PrimeFieldBits,
+{
+  /// Allocates the given RelaxedR1CSInstance as a witness of the circuit
+  pub fn alloc<CS: ConstraintSystem<<G as Group>::Base>>(
+    mut cs: CS,
+    inst: Option<RelaxedR1CSInstance<G>>,
+    limb_width: usize,
+    n_limbs: usize,
+  ) -> Result<Self, SynthesisError> {
+    let W = AllocatedPoint::alloc(
+      cs.namespace(|| "allocate W"),
+      inst
+        .get()
+        .map_or(None, |inst| Some(inst.comm_W.comm.to_coordinates())),
+    )?;
+
+    let E = AllocatedPoint::alloc(
+      cs.namespace(|| "allocate E"),
+      inst
+        .get()
+        .map_or(None, |inst| Some(inst.comm_E.comm.to_coordinates())),
+    )?;
+
+    // u << |G::Base| despite the fact that u is a scalar.
+    // So we parse all of its bytes as a G::Base element
+    let u = alloc_scalar_as_base::<G, _>(
+      cs.namespace(|| "allocate u"),
+      inst.get().map_or(None, |inst| Some(inst.u)),
+    )?;
+
+    let X0 = BigNat::alloc_from_nat(
+      cs.namespace(|| "allocate X[0]"),
+      || Ok(f_to_nat(&inst.get()?.X[0])),
+      limb_width,
+      n_limbs,
+    )?;
+
+    let X1 = BigNat::alloc_from_nat(
+      cs.namespace(|| "allocate X[1]"),
+      || Ok(f_to_nat(&inst.get()?.X[1])),
+      limb_width,
+      n_limbs,
+    )?;
+
+    Ok(AllocatedRelaxedR1CSInstance { W, E, u, X0, X1 })
+  }
+
+  /// Allocates the hardcoded default RelaxedR1CSInstance in the circuit.
+  /// W = E = 0, u = 1, X0 = X1 = 0
+  pub fn default<CS: ConstraintSystem<<G as Group>::Base>>(
+    mut cs: CS,
+    limb_width: usize,
+    n_limbs: usize,
+  ) -> Result<Self, SynthesisError> {
+    let zero = alloc_zero(cs.namespace(|| "zero"))?;
+    let one = alloc_one(cs.namespace(|| "one"))?;
+
+    let W_default = AllocatedPoint::new(zero.clone(), zero.clone(), one);
+    let E_default = W_default.clone();
+
+    let u_default = zero;
+
+    let X0_default = BigNat::alloc_from_nat(
+      cs.namespace(|| "allocate x_default[0]"),
+      || Ok(f_to_nat(&G::Scalar::zero())),
+      limb_width,
+      n_limbs,
+    )?;
+
+    let X1_default = BigNat::alloc_from_nat(
+      cs.namespace(|| "allocate x_default[1]"),
+      || Ok(f_to_nat(&G::Scalar::zero())),
+      limb_width,
+      n_limbs,
+    )?;
+    Ok(AllocatedRelaxedR1CSInstance {
+      W: W_default,
+      E: E_default,
+      u: u_default,
+      X0: X0_default,
+      X1: X1_default,
+    })
+  }
+
+  pub fn absorb_in_ro<CS: ConstraintSystem<<G as Group>::Base>>(
+    &self,
+    mut cs: CS,
+    ro: &mut PoseidonROGadget<G::Base>,
+  ) -> Result<(), SynthesisError> {
+    ro.absorb(self.W.x.clone());
+    ro.absorb(self.W.y.clone());
+    ro.absorb(self.W.is_infinity.clone());
+    ro.absorb(self.E.x.clone());
+    ro.absorb(self.E.y.clone());
+    ro.absorb(self.E.is_infinity.clone());
+    ro.absorb(self.u.clone());
+
+    // Analyze X0 as limbs
+    let X0_bn = self
+      .X0
+      .as_limbs::<CS>()
+      .iter()
+      .enumerate()
+      .map(|(i, limb)| {
+        limb
+          .as_sapling_allocated_num(cs.namespace(|| format!("convert limb {} of X_r[0] to num", i)))
+      })
+      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
+
+    // absorb each of the limbs of X[0]
+    for limb in X0_bn.into_iter() {
+      ro.absorb(limb);
+    }
+
+    // Analyze X1 as limbs
+    let X1_bn = self
+      .X1
+      .as_limbs::<CS>()
+      .iter()
+      .enumerate()
+      .map(|(i, limb)| {
+        limb
+          .as_sapling_allocated_num(cs.namespace(|| format!("convert limb {} of X_r[1] to num", i)))
+      })
+      .collect::<Result<Vec<AllocatedNum<G::Base>>, _>>()?;
+
+    // absorb each of the limbs of X[1]
+    for limb in X1_bn.into_iter() {
+      ro.absorb(limb);
+    }
+
+    Ok(())
+  }
+
+  /// Folds self with a relaxed r1cs instance and returns the result
+  pub fn fold_with_r1cs<CS: ConstraintSystem<<G as Group>::Base>>(
+    &self,
+    mut cs: CS,
+    u: AllocatedR1CSInstance<G>,
+    T: AllocatedPoint<G::Base>,
+    poseidon_constants: NovaPoseidonConstants<G::Base>,
+    limb_width: usize,
+    n_limbs: usize,
+  ) -> Result<AllocatedRelaxedR1CSInstance<G>, SynthesisError> {
+    // Compute r:
+    let mut ro: PoseidonROGadget<G::Base> = PoseidonROGadget::new(poseidon_constants);
+    u.absorb_in_ro(&mut ro);
+    ro.absorb(T.x.clone());
+    ro.absorb(T.y.clone());
+    ro.absorb(T.is_infinity.clone());
+    let r_bits = ro.get_challenge(cs.namespace(|| "r bits"))?;
+    let r = le_bits_to_num(cs.namespace(|| "r"), r_bits.clone())?;
+
+    // W_fold = self.W + r * u.W
+    let rW = u.W.scalar_mul(cs.namespace(|| "r * u.W"), r_bits.clone())?;
+    let W_fold = self.W.add(cs.namespace(|| "self.W + r * u.W"), &rW)?;
+
+    // E_fold = self.E + r * T
+    let rT = T.scalar_mul(cs.namespace(|| "r * T"), r_bits)?;
+    let E_fold = self.E.add(cs.namespace(|| "self.E + r * T"), &rT)?;
+
+    // u_fold = u_r + r
+    let u_fold = AllocatedNum::alloc(cs.namespace(|| "u_fold"), || {
+      Ok(*self.u.get_value().get()? + r.get_value().get()?)
+    })?;
+    cs.enforce(
+      || "Check u_fold",
+      |lc| lc,
+      |lc| lc,
+      |lc| lc + u_fold.get_variable() - self.u.get_variable() - r.get_variable(),
+    );
+
+    // Fold the IO:
+    // Analyze r into limbs
+    let r_bn = BigNat::from_num(
+      cs.namespace(|| "allocate r_bn"),
+      Num::from(r.clone()),
+      limb_width,
+      n_limbs,
+    )?;
+
+    // Allocate the order of the non-native field as a constant
+    let m_bn = alloc_bignat_constant(
+      cs.namespace(|| "alloc m"),
+      &G::get_order(),
+      limb_width,
+      n_limbs,
+    )?;
+
+    // Analyze X0 to bignat
+    let X0_bn = BigNat::from_num(
+      cs.namespace(|| "allocate X0_bn"),
+      Num::from(u.X0.clone()),
+      limb_width,
+      n_limbs,
+    )?;
+
+    // Fold self.X[0] + r * X[0]
+    let (_, r_0) = X0_bn.mult_mod(cs.namespace(|| "r*X[0]"), &r_bn, &m_bn)?;
+    // add X_r[0]
+    let r_new_0 = self.X0.add::<CS>(&r_0)?;
+    // Now reduce
+    let X0_fold = r_new_0.red_mod(cs.namespace(|| "reduce folded X[0]"), &m_bn)?;
+
+    // Analyze X1 to bignat
+    let X1_bn = BigNat::from_num(
+      cs.namespace(|| "allocate X1_bn"),
+      Num::from(u.X1.clone()),
+      limb_width,
+      n_limbs,
+    )?;
+
+    // Fold self.X[1] + r * X[1]
+    let (_, r_1) = X1_bn.mult_mod(cs.namespace(|| "r*X[1]"), &r_bn, &m_bn)?;
+    // add X_r[1]
+    let r_new_1 = self.X1.add::<CS>(&r_1)?;
+    // Now reduce
+    let X1_fold = r_new_1.red_mod(cs.namespace(|| "reduce folded X[1]"), &m_bn)?;
+
+    Ok(Self {
+      W: W_fold,
+      E: E_fold,
+      u: u_fold,
+      X0: X0_fold,
+      X1: X1_fold,
+    })
+  }
+
+  /// If the condition is true then returns this otherwise it returns the other
+  pub fn conditionally_select<CS: ConstraintSystem<<G as Group>::Base>>(
+    &self,
+    mut cs: CS,
+    other: AllocatedRelaxedR1CSInstance<G>,
+    condition: &Boolean,
+  ) -> Result<AllocatedRelaxedR1CSInstance<G>, SynthesisError> {
+    let W = AllocatedPoint::conditionally_select(
+      cs.namespace(|| "W = cond ? self.W : other.W"),
+      &self.W,
+      &other.W,
+      condition,
+    )?;
+
+    let E = AllocatedPoint::conditionally_select(
+      cs.namespace(|| "E = cond ? self.E : other.E"),
+      &self.E,
+      &other.E,
+      condition,
+    )?;
+
+    let u = conditionally_select(
+      cs.namespace(|| "u = cond ? self.u : other.u"),
+      &self.u,
+      &other.u,
+      condition,
+    )?;
+
+    let X0 = conditionally_select_bignat(
+      cs.namespace(|| "X[0] = cond ? self.X[0] : other.X[0]"),
+      &self.X0,
+      &other.X0,
+      condition,
+    )?;
+
+    let X1 = conditionally_select_bignat(
+      cs.namespace(|| "X[1] = cond ? self.X[1] : other.X[1]"),
+      &self.X1,
+      &other.X1,
+      condition,
+    )?;
+
+    Ok(AllocatedRelaxedR1CSInstance { W, E, u, X0, X1 })
+  }
+}
diff --git a/src/gadgets/utils.rs b/src/gadgets/utils.rs
index 54a742c..fea1ef0 100644
--- a/src/gadgets/utils.rs
+++ b/src/gadgets/utils.rs
@@ -1,3 +1,4 @@
+use crate::traits::Group;
 use bellperson::{
   gadgets::{
     boolean::{AllocatedBit, Boolean},
@@ -7,7 +8,7 @@ use bellperson::{
   ConstraintSystem, LinearCombination, SynthesisError,
 };
 use bellperson_nonnative::mp::bignat::{nat_to_limbs, BigNat};
-use ff::{PrimeField, PrimeFieldBits};
+use ff::{Field, PrimeField, PrimeFieldBits};
 use rug::Integer;
 
 /// Gets as input the little indian representation of a number and spits out the number
@@ -73,6 +74,30 @@ pub fn alloc_one<F: PrimeField, CS: ConstraintSystem<F>>(
   Ok(one)
 }
 
+/// Allocate a scalar as a base. Only to be used is the scalar fits in base!
+pub fn alloc_scalar_as_base<G, CS>(
+  mut cs: CS,
+  input: Option<G::Scalar>,
+) -> Result<AllocatedNum<G::Base>, SynthesisError>
+where
+  G: Group,
+  <G as Group>::Scalar: PrimeFieldBits,
+  CS: ConstraintSystem<<G as Group>::Base>,
+{
+  AllocatedNum::alloc(cs.namespace(|| "allocate scalar as base"), || {
+    let input_bits = input.get()?.clone().to_le_bits();
+    let mut mult = G::Base::one();
+    let mut val = G::Base::zero();
+    for bit in input_bits {
+      if bit {
+        val += mult;
+      }
+      mult = mult + mult;
+    }
+    Ok(val)
+  })
+}
+
 /// Allocate bignat a constant
 pub fn alloc_bignat_constant<F: PrimeField, CS: ConstraintSystem<F>>(
   mut cs: CS,
@@ -109,7 +134,6 @@ pub fn alloc_num_equals<F: PrimeField, CS: ConstraintSystem<F>>(
 ) -> Result<AllocatedBit, SynthesisError> {
   // Allocate and constrain `r`: result boolean bit.
   // It equals `true` if `a` equals `b`, `false` otherwise
-
   let r_value = match (a.get_value(), b.get_value()) {
     (Some(a), Some(b)) => Some(a == b),
     _ => None,
@@ -147,7 +171,6 @@ pub fn alloc_num_equals<F: PrimeField, CS: ConstraintSystem<F>>(
   // Allocate `t = delta * delta_inv`
   // If `delta` is non-zero (a != b), `t` will equal 1
   // If `delta` is zero (a == b), `t` cannot equal 1
-
   let t = AllocatedNum::alloc(cs.namespace(|| "t"), || {
     let mut tmp = *delta.get_value().get()?;
     tmp.mul_assign(&(*delta_inv.get_value().get()?));
@@ -215,7 +238,6 @@ pub fn conditionally_select<F: PrimeField, CS: ConstraintSystem<F>>(
 
   // a * condition + b*(1-condition) = c ->
   // a * condition - b*condition = c - b
-
   cs.enforce(
     || "conditional select constraint",
     |lc| lc + a.get_variable() - b.get_variable(),
@@ -278,7 +300,6 @@ pub fn conditionally_select2<F: PrimeField, CS: ConstraintSystem<F>>(
 
   // a * condition + b*(1-condition) = c ->
   // a * condition - b*condition = c - b
-
   cs.enforce(
     || "conditional select constraint",
     |lc| lc + a.get_variable() - b.get_variable(),
diff --git a/src/lib.rs b/src/lib.rs
index 2d0c2cc..8035469 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -57,7 +57,6 @@ impl<G: Group> StepSNARK<G> {
     NovaError,
   > {
     // append the protocol name to the transcript
-    //transcript.append_protocol_name(StepSNARK::protocol_name());
     transcript.append_message(b"protocol-name", StepSNARK::<G>::protocol_name());
 
     // compute a commitment to the cross-term
diff --git a/src/poseidon.rs b/src/poseidon.rs
index 9a8672c..d9d89c5 100644
--- a/src/poseidon.rs
+++ b/src/poseidon.rs
@@ -7,7 +7,7 @@ use bellperson::{
   ConstraintSystem, SynthesisError,
 };
 use ff::{PrimeField, PrimeFieldBits};
-use generic_array::typenum::{U24, U25, U27, U31};
+use generic_array::typenum::{U25, U27, U31, U8};
 use neptune::{
   circuit::poseidon_hash,
   poseidon::{Poseidon, PoseidonConstants},
@@ -22,7 +22,7 @@ pub struct NovaPoseidonConstants<F>
 where
   F: PrimeField,
 {
-  constants24: PoseidonConstants<F, U24>,
+  constants8: PoseidonConstants<F, U8>,
   constants25: PoseidonConstants<F, U25>,
   constants27: PoseidonConstants<F, U27>,
   constants31: PoseidonConstants<F, U31>,
@@ -35,12 +35,12 @@ where
 {
   /// Generate Poseidon constants for the arities that Nova uses
   pub fn new() -> Self {
-    let constants24 = PoseidonConstants::<F, U24>::new_with_strength(Strength::Strengthened);
+    let constants8 = PoseidonConstants::<F, U8>::new_with_strength(Strength::Strengthened);
     let constants25 = PoseidonConstants::<F, U25>::new_with_strength(Strength::Strengthened);
     let constants27 = PoseidonConstants::<F, U27>::new_with_strength(Strength::Strengthened);
     let constants31 = PoseidonConstants::<F, U31>::new_with_strength(Strength::Strengthened);
     Self {
-      constants24,
+      constants8,
       constants25,
       constants27,
       constants31,
@@ -71,12 +71,6 @@ where
     }
   }
 
-  #[allow(dead_code)]
-  /// Flush the state of the RO
-  pub fn flush_state(&mut self) {
-    self.state = Vec::new();
-  }
-
   /// Absorb a new number into the state of the oracle
   #[allow(dead_code)]
   pub fn absorb(&mut self, e: Scalar) {
@@ -85,8 +79,8 @@ where
 
   fn hash_inner(&mut self) -> Scalar {
     match self.state.len() {
-      24 => {
-        Poseidon::<Scalar, U24>::new_with_preimage(&self.state, &self.constants.constants24).hash()
+      8 => {
+        Poseidon::<Scalar, U8>::new_with_preimage(&self.state, &self.constants.constants8).hash()
       }
       25 => {
         Poseidon::<Scalar, U25>::new_with_preimage(&self.state, &self.constants.constants25).hash()
@@ -160,11 +154,6 @@ where
     }
   }
 
-  /// Flush the state of the RO
-  pub fn flush_state(&mut self) {
-    self.state = Vec::new();
-  }
-
   /// Absorb a new number into the state of the oracle
   #[allow(dead_code)]
   pub fn absorb(&mut self, e: AllocatedNum<Scalar>) {
@@ -176,10 +165,10 @@ where
     CS: ConstraintSystem<Scalar>,
   {
     let out = match self.state.len() {
-      24 => poseidon_hash(
+      8 => poseidon_hash(
         cs.namespace(|| "Posideon hash"),
         self.state.clone(),
-        &self.constants.constants24,
+        &self.constants.constants8,
       )?,
       25 => poseidon_hash(
         cs.namespace(|| "Poseidon hash"),
diff --git a/src/r1cs.rs b/src/r1cs.rs
index d3d9e7d..3ac1469 100644
--- a/src/r1cs.rs
+++ b/src/r1cs.rs
@@ -36,8 +36,8 @@ pub struct R1CSWitness<G: Group> {
 /// A type that holds an R1CS instance
 #[derive(Clone, Debug, PartialEq, Eq)]
 pub struct R1CSInstance<G: Group> {
-  comm_W: Commitment<G>,
-  X: Vec<G::Scalar>,
+  pub(crate) comm_W: Commitment<G>,
+  pub(crate) X: Vec<G::Scalar>,
 }
 
 /// A type that holds a witness for a given Relaxed R1CS instance
diff --git a/tests/bit.rs b/tests/bit.rs
index 7c29305..65bc1ba 100644
--- a/tests/bit.rs
+++ b/tests/bit.rs
@@ -9,7 +9,7 @@ use nova_snark::bellperson::{
 fn synthesize_alloc_bit<Fr: PrimeField, CS: ConstraintSystem<Fr>>(
   cs: &mut CS,
 ) -> Result<(), SynthesisError> {
-  //get two bits as input and check that they are indeed bits
+  // get two bits as input and check that they are indeed bits
   let a = AllocatedNum::alloc(cs.namespace(|| "a"), || Ok(Fr::one()))?;
   let _ = a.inputize(cs.namespace(|| "a is input"));
   cs.enforce(
@@ -33,18 +33,18 @@ fn synthesize_alloc_bit<Fr: PrimeField, CS: ConstraintSystem<Fr>>(
 fn test_alloc_bit() {
   type G = pasta_curves::pallas::Point;
 
-  //First create the shape
+  // First create the shape
   let mut cs: ShapeCS<G> = ShapeCS::new();
   let _ = synthesize_alloc_bit(&mut cs);
   let shape = cs.r1cs_shape();
   let gens = cs.r1cs_gens();
   println!("Mult mod constraint no: {}", cs.num_constraints());
 
-  //Now get the assignment
+  // Now get the assignment
   let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
   let _ = synthesize_alloc_bit(&mut cs);
   let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
 
-  //Make sure that this is satisfiable
+  // Make sure that this is satisfiable
   assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
 }
diff --git a/tests/nonnative.rs b/tests/nonnative.rs
index 007839b..d71c624 100644
--- a/tests/nonnative.rs
+++ b/tests/nonnative.rs
@@ -169,7 +169,7 @@ fn synthesize_add_mod<Fr: PrimeField, CS: ConstraintSystem<Fr>>(
 fn test_mult_mod() {
   type G = pasta_curves::pallas::Point;
 
-  //Set the inputs
+  // Set the inputs
   let a_val = Integer::from_str_radix(
     "11572336752428856981970994795408771577024165681374400871001196932361466228192",
     10,
@@ -196,19 +196,19 @@ fn test_mult_mod() {
   )
   .unwrap();
 
-  //First create the shape
+  // First create the shape
   let mut cs: ShapeCS<G> = ShapeCS::new();
   let _ = synthesize_mult_mod(&mut cs, &a_val, &b_val, &m_val, &q_val, &r_val, 32, 8);
   let shape = cs.r1cs_shape();
   let gens = cs.r1cs_gens();
   println!("Mult mod constraint no: {}", cs.num_constraints());
 
-  //Now get the assignment
+  // Now get the assignment
   let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
   let _ = synthesize_mult_mod(&mut cs, &a_val, &b_val, &m_val, &q_val, &r_val, 32, 8);
   let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
 
-  //Make sure that this is satisfiable
+  // Make sure that this is satisfiable
   assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
 }
 
@@ -216,7 +216,7 @@ fn test_mult_mod() {
 fn test_add() {
   type G = pasta_curves::pallas::Point;
 
-  //Set the inputs
+  // Set the inputs
   let a_val = Integer::from_str_radix(
     "11572336752428856981970994795408771577024165681374400871001196932361466228192",
     10,
@@ -229,19 +229,19 @@ fn test_add() {
   )
   .unwrap();
 
-  //First create the shape
+  // First create the shape
   let mut cs: ShapeCS<G> = ShapeCS::new();
   let _ = synthesize_add(&mut cs, &a_val, &b_val, &c_val, 32, 8);
   let shape = cs.r1cs_shape();
   let gens = cs.r1cs_gens();
   println!("Add mod constraint no: {}", cs.num_constraints());
 
-  //Now get the assignment
+  // Now get the assignment
   let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
   let _ = synthesize_add(&mut cs, &a_val, &b_val, &c_val, 32, 8);
   let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
 
-  //Make sure that this is satisfiable
+  // Make sure that this is satisfiable
   assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
 }
 
@@ -249,7 +249,7 @@ fn test_add() {
 fn test_add_mod() {
   type G = pasta_curves::pallas::Point;
 
-  //Set the inputs
+  // Set the inputs
   let a_val = Integer::from_str_radix(
     "11572336752428856981970994795408771577024165681374400871001196932361466228192",
     10,
@@ -267,19 +267,19 @@ fn test_add_mod() {
   )
   .unwrap();
 
-  //First create the shape
+  // First create the shape
   let mut cs: ShapeCS<G> = ShapeCS::new();
   let _ = synthesize_add_mod(&mut cs, &a_val, &b_val, &c_val, &m_val, 32, 8);
   let shape = cs.r1cs_shape();
   let gens = cs.r1cs_gens();
   println!("Add mod constraint no: {}", cs.num_constraints());
 
-  //Now get the assignment
+  // Now get the assignment
   let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
   let _ = synthesize_add_mod(&mut cs, &a_val, &b_val, &c_val, &m_val, 32, 8);
   let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
 
-  //Make sure that this is satisfiable
+  // Make sure that this is satisfiable
   assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
 }
 
@@ -287,21 +287,21 @@ fn test_add_mod() {
 fn test_equal() {
   type G = pasta_curves::pallas::Point;
 
-  //Set the inputs
+  // Set the inputs
   let a_val = Integer::from_str_radix("1157233675242885698197099479540877", 10).unwrap();
 
-  //First create the shape
+  // First create the shape
   let mut cs: ShapeCS<G> = ShapeCS::new();
   let _ = synthesize_is_equal(&mut cs, &a_val, 32, 8);
   let shape = cs.r1cs_shape();
   let gens = cs.r1cs_gens();
   println!("Equal constraint no: {}", cs.num_constraints());
 
-  //Now get the assignment
+  // Now get the assignment
   let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
   let _ = synthesize_is_equal(&mut cs, &a_val, 32, 8);
   let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
 
-  //Make sure that this is satisfiable
+  // Make sure that this is satisfiable
   assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
 }
diff --git a/tests/num.rs b/tests/num.rs
index a0f5b5a..55b58d1 100644
--- a/tests/num.rs
+++ b/tests/num.rs
@@ -42,18 +42,18 @@ fn synthesize_use_cs_one_after_inputize<Fr: PrimeField, CS: ConstraintSystem<Fr>
 fn test_use_cs_one() {
   type G = pasta_curves::pallas::Point;
 
-  //First create the shape
+  // First create the shape
   let mut cs: ShapeCS<G> = ShapeCS::new();
   let _ = synthesize_use_cs_one(&mut cs);
   let shape = cs.r1cs_shape();
   let gens = cs.r1cs_gens();
 
-  //Now get the assignment
+  // Now get the assignment
   let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
   let _ = synthesize_use_cs_one(&mut cs);
   let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
 
-  //Make sure that this is satisfiable
+  // Make sure that this is satisfiable
   assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
 }
 
@@ -61,17 +61,17 @@ fn test_use_cs_one() {
 fn test_use_cs_one_after_inputize() {
   type G = pasta_curves::pallas::Point;
 
-  //First create the shape
+  // First create the shape
   let mut cs: ShapeCS<G> = ShapeCS::new();
   let _ = synthesize_use_cs_one_after_inputize(&mut cs);
   let shape = cs.r1cs_shape();
   let gens = cs.r1cs_gens();
 
-  //Now get the assignment
+  // Now get the assignment
   let mut cs: SatisfyingAssignment<G> = SatisfyingAssignment::new();
   let _ = synthesize_use_cs_one_after_inputize(&mut cs);
   let (inst, witness) = cs.r1cs_instance_and_witness(&shape, &gens).unwrap();
 
-  //Make sure that this is satisfiable
+  // Make sure that this is satisfiable
   assert!(shape.is_sat(&gens, &inst, &witness).is_ok());
 }