diff --git a/src/circuit.rs b/src/circuit.rs
index d06ed8b..7b20ad6 100644
--- a/src/circuit.rs
+++ b/src/circuit.rs
@@ -12,7 +12,9 @@ use super::{
   gadgets::{
     ecc::AllocatedPoint,
     r1cs::{AllocatedR1CSInstance, AllocatedRelaxedR1CSInstance},
-    utils::{alloc_num_equals, alloc_zero, conditionally_select, le_bits_to_num},
+    utils::{
+      alloc_num_equals, alloc_scalar_as_base, alloc_zero, conditionally_select, le_bits_to_num,
+    },
   },
   poseidon::{NovaPoseidonConstants, PoseidonROGadget},
   r1cs::{R1CSInstance, RelaxedR1CSInstance},
@@ -32,26 +34,29 @@ use ff::Field;
 pub struct NIFSVerifierCircuitParams {
   limb_width: usize,
   n_limbs: usize,
+  is_primary_circuit: bool, // A boolean indicating if this is the primary circuit
 }
 
 impl NIFSVerifierCircuitParams {
   #[allow(dead_code)]
-  pub fn new(limb_width: usize, n_limbs: usize) -> Self {
+  pub fn new(limb_width: usize, n_limbs: usize, is_primary_circuit: bool) -> Self {
     Self {
       limb_width,
       n_limbs,
+      is_primary_circuit,
     }
   }
 }
 
+#[derive(Debug)]
 pub struct NIFSVerifierCircuitInputs<G: Group> {
-  params: G::Base, // Hash(Shape of u2, Gens for u2). Needed for computing the challenge.
+  params: G::Scalar, // Hash(Shape of u2, Gens for u2). Needed for computing the challenge.
   i: G::Base,
   z0: G::Base,
-  zi: G::Base,
-  U: RelaxedR1CSInstance<G>,
-  u: R1CSInstance<G>,
-  T: Commitment<G>,
+  zi: Option<G::Base>,
+  U: Option<RelaxedR1CSInstance<G>>,
+  u: Option<R1CSInstance<G>>,
+  T: Option<Commitment<G>>,
 }
 
 impl<G> NIFSVerifierCircuitInputs<G>
@@ -61,13 +66,13 @@ where
   /// Create new inputs/witness for the verification circuit
   #[allow(dead_code, clippy::too_many_arguments)]
   pub fn new(
-    params: G::Base,
+    params: G::Scalar,
     i: G::Base,
     z0: G::Base,
-    zi: G::Base,
-    U: RelaxedR1CSInstance<G>,
-    u: R1CSInstance<G>,
-    T: Commitment<G>,
+    zi: Option<G::Base>,
+    U: Option<RelaxedR1CSInstance<G>>,
+    u: Option<R1CSInstance<G>>,
+    T: Option<Commitment<G>>,
   ) -> Self {
     Self {
       params,
@@ -131,46 +136,46 @@ where
     SynthesisError,
   > {
     // Allocate the params
-    let params = AllocatedNum::alloc(cs.namespace(|| "params"), || Ok(self.inputs.get()?.params))?;
+    let params = alloc_scalar_as_base::<G, _>(
+      cs.namespace(|| "params"),
+      self.inputs.get().map_or(None, |inputs| Some(inputs.params)),
+    )?;
 
     // Allocate i
     let i = AllocatedNum::alloc(cs.namespace(|| "i"), || Ok(self.inputs.get()?.i))?;
-
     // 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))?;
-
+    // Allocate zi. If inputs.zi is not provided (base case) allocate default value 0
+    let z_i = AllocatedNum::alloc(cs.namespace(|| "zi"), || {
+      Ok(self.inputs.get()?.zi.unwrap_or_else(G::Base::zero))
+    })?;
     // Allocate the running instance
     let U: AllocatedRelaxedR1CSInstance<G> = AllocatedRelaxedR1CSInstance::alloc(
       cs.namespace(|| "Allocate U"),
-      self
-        .inputs
-        .get()
-        .map_or(None, |inputs| Some(inputs.U.clone())),
+      self.inputs.get().map_or(None, |inputs| {
+        inputs.U.get().map_or(None, |U| Some(U.clone()))
+      }),
       self.params.limb_width,
       self.params.n_limbs,
     )?;
-
     // 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.u.clone())),
+      self.inputs.get().map_or(None, |inputs| {
+        inputs.u.get().map_or(None, |u| Some(u.clone()))
+      }),
     )?;
 
     // Allocate T
     let T = AllocatedPoint::alloc(
       cs.namespace(|| "allocate T"),
-      self
-        .inputs
-        .get()
-        .map_or(None, |inputs| Some(inputs.T.comm.to_coordinates())),
+      self.inputs.get().map_or(None, |inputs| {
+        inputs
+          .T
+          .get()
+          .map_or(None, |T| Some(T.comm.to_coordinates()))
+      }),
     )?;
-
     Ok((params, i, z_0, z_i, U, u, T))
   }
 
@@ -178,12 +183,24 @@ where
   fn synthesize_base_case<CS: ConstraintSystem<<G as Group>::Base>>(
     &self,
     mut cs: CS,
+    u: AllocatedR1CSInstance<G>,
   ) -> Result<AllocatedRelaxedR1CSInstance<G>, SynthesisError> {
-    let U_default: AllocatedRelaxedR1CSInstance<G> = AllocatedRelaxedR1CSInstance::default(
-      cs.namespace(|| "Allocate U_default"),
-      self.params.limb_width,
-      self.params.n_limbs,
-    )?;
+    let U_default: AllocatedRelaxedR1CSInstance<G> = if self.params.is_primary_circuit {
+      // The primary circuit just returns the default R1CS instance
+      AllocatedRelaxedR1CSInstance::default(
+        cs.namespace(|| "Allocate U_default"),
+        self.params.limb_width,
+        self.params.n_limbs,
+      )?
+    } else {
+      // The secondary circuit returns the incoming R1CS instance
+      AllocatedRelaxedR1CSInstance::from_r1cs_instance(
+        cs.namespace(|| "Allocate U_default"),
+        u,
+        self.params.limb_width,
+        self.params.n_limbs,
+      )?
+    };
     Ok(U_default)
   }
 
@@ -250,7 +267,7 @@ where
     let is_base_case = alloc_num_equals(cs.namespace(|| "Check if base case"), &i.clone(), &zero)?; //TODO: maybe optimize this?
 
     // 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 Unew_base = self.synthesize_base_case(cs.namespace(|| "base case"), u.clone())?;
 
     // Synthesize the circuit for the non-base case and get the new running
     // instance along with a boolean indicating if all checks have passed
@@ -336,7 +353,6 @@ mod tests {
   use crate::constants::{BN_LIMB_WIDTH, BN_N_LIMBS};
   use crate::{
     bellperson::r1cs::{NovaShape, NovaWitness},
-    commitments::CommitTrait,
     traits::HashFuncConstantsTrait,
   };
   use ff::PrimeField;
@@ -361,22 +377,24 @@ mod tests {
 
   #[test]
   fn test_verification_circuit() {
-    // We experiment with 8 limbs of 32 bits each
-    let params = NIFSVerifierCircuitParams::new(BN_LIMB_WIDTH, BN_N_LIMBS);
-    // The first circuit that verifies G2
+    // In the following we use 1 to refer to the primary, and 2 to refer to the secondary circuit
+    let params1 = NIFSVerifierCircuitParams::new(BN_LIMB_WIDTH, BN_N_LIMBS, true);
+    let params2 = NIFSVerifierCircuitParams::new(BN_LIMB_WIDTH, BN_N_LIMBS, false);
     let poseidon_constants1: NovaPoseidonConstants<<G2 as Group>::Base> =
       NovaPoseidonConstants::new();
+    let poseidon_constants2: NovaPoseidonConstants<<G1 as Group>::Base> =
+      NovaPoseidonConstants::new();
+
+    // Initialize the shape and gens for the primary
     let circuit1: NIFSVerifierCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
       NIFSVerifierCircuit::new(
-        params.clone(),
+        params1.clone(),
         None,
         TestCircuit {
           _p: Default::default(),
         },
         poseidon_constants1.clone(),
       );
-
-    // First create the shape
     let mut cs: ShapeCS<G1> = ShapeCS::new();
     let _ = circuit1.synthesize(&mut cs);
     let (shape1, gens1) = (cs.r1cs_shape(), cs.r1cs_gens());
@@ -385,19 +403,16 @@ mod tests {
       cs.num_constraints()
     );
 
-    // The second circuit that verifies G1
-    let poseidon_constants2: NovaPoseidonConstants<<G1 as Group>::Base> =
-      NovaPoseidonConstants::new();
+    // Initialize the shape and gens for the secondary
     let circuit2: NIFSVerifierCircuit<G1, TestCircuit<<G1 as Group>::Base>> =
       NIFSVerifierCircuit::new(
-        params.clone(),
+        params2.clone(),
         None,
         TestCircuit {
           _p: Default::default(),
         },
-        poseidon_constants2,
+        poseidon_constants2.clone(),
       );
-    // First create the shape
     let mut cs: ShapeCS<G2> = ShapeCS::new();
     let _ = circuit2.synthesize(&mut cs);
     let (shape2, gens2) = (cs.r1cs_shape(), cs.r1cs_gens());
@@ -406,35 +421,56 @@ mod tests {
       cs.num_constraints()
     );
 
-    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(
-      <<G2 as Group>::Base as Field>::zero(), // TODO: provide real inputs
-      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
+    // Execute the base case for the primary
+    let zero1 = <<G2 as Group>::Base as Field>::zero();
+    let mut cs1: SatisfyingAssignment<G1> = SatisfyingAssignment::new();
+    let inputs1: NIFSVerifierCircuitInputs<G2> = NIFSVerifierCircuitInputs::new(
+      shape2.get_digest(),
+      zero1,
+      zero1, // TODO: Provide real input for z0
+      None,
+      None,
+      None,
+      None,
     );
-
-    let circuit: NIFSVerifierCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
+    let circuit1: NIFSVerifierCircuit<G2, TestCircuit<<G2 as Group>::Base>> =
       NIFSVerifierCircuit::new(
-        params,
-        Some(inputs),
+        params1,
+        Some(inputs1),
         TestCircuit {
           _p: Default::default(),
         },
         poseidon_constants1,
       );
-    let _ = circuit.synthesize(&mut cs);
-    let (inst, witness) = cs.r1cs_instance_and_witness(&shape1, &gens1).unwrap();
-
+    let _ = circuit1.synthesize(&mut cs1);
+    let (inst1, witness1) = cs1.r1cs_instance_and_witness(&shape1, &gens1).unwrap();
     // Make sure that this is satisfiable
-    assert!(shape1.is_sat(&gens1, &inst, &witness).is_ok());
+    assert!(shape1.is_sat(&gens1, &inst1, &witness1).is_ok());
+
+    // Execute the base case for the secondary
+    let zero2 = <<G1 as Group>::Base as Field>::zero();
+    let mut cs2: SatisfyingAssignment<G2> = SatisfyingAssignment::new();
+    let inputs2: NIFSVerifierCircuitInputs<G1> = NIFSVerifierCircuitInputs::new(
+      shape1.get_digest(),
+      zero2,
+      zero2,
+      None,
+      None,
+      Some(inst1),
+      None,
+    );
+    let circuit: NIFSVerifierCircuit<G1, TestCircuit<<G1 as Group>::Base>> =
+      NIFSVerifierCircuit::new(
+        params2,
+        Some(inputs2),
+        TestCircuit {
+          _p: Default::default(),
+        },
+        poseidon_constants2,
+      );
+    let _ = circuit.synthesize(&mut cs2);
+    let (inst2, witness2) = cs2.r1cs_instance_and_witness(&shape2, &gens2).unwrap();
+    // Make sure that it is satisfiable
+    assert!(shape2.is_sat(&gens2, &inst2, &witness2).is_ok());
   }
 }
diff --git a/src/gadgets/ecc.rs b/src/gadgets/ecc.rs
index 9f2a488..3e10353 100644
--- a/src/gadgets/ecc.rs
+++ b/src/gadgets/ecc.rs
@@ -29,14 +29,19 @@ where
   Fp: PrimeField,
 {
   /// Allocates a new point on the curve using coordinates provided by `coords`.
+  /// If coords = None, it allocates the default infinity point
   pub fn alloc<CS>(mut cs: CS, coords: Option<(Fp, Fp, bool)>) -> Result<Self, SynthesisError>
   where
     CS: ConstraintSystem<Fp>,
   {
-    let x = AllocatedNum::alloc(cs.namespace(|| "x"), || Ok(coords.unwrap().0))?;
-    let y = AllocatedNum::alloc(cs.namespace(|| "y"), || Ok(coords.unwrap().1))?;
+    let x = AllocatedNum::alloc(cs.namespace(|| "x"), || {
+      Ok(coords.map_or(Fp::zero(), |c| c.0))
+    })?;
+    let y = AllocatedNum::alloc(cs.namespace(|| "y"), || {
+      Ok(coords.map_or(Fp::zero(), |c| c.0))
+    })?;
     let is_infinity = AllocatedNum::alloc(cs.namespace(|| "is_infinity"), || {
-      Ok(if coords.unwrap().2 {
+      Ok(if coords.map_or(true, |c| c.2) {
         Fp::one()
       } else {
         Fp::zero()
diff --git a/src/gadgets/r1cs.rs b/src/gadgets/r1cs.rs
index 7cab720..990121a 100644
--- a/src/gadgets/r1cs.rs
+++ b/src/gadgets/r1cs.rs
@@ -3,7 +3,7 @@ use crate::{
   gadgets::{
     ecc::AllocatedPoint,
     utils::{
-      alloc_bignat_constant, alloc_scalar_as_base, conditionally_select,
+      alloc_bignat_constant, alloc_one, alloc_scalar_as_base, conditionally_select,
       conditionally_select_bignat, le_bits_to_num,
     },
   },
@@ -114,16 +114,25 @@ where
       inst.get().map_or(None, |inst| Some(inst.u)),
     )?;
 
+    // Allocate X0 and X1. If the input instance is None, then allocate default values 0.
     let X0 = BigNat::alloc_from_nat(
       cs.namespace(|| "allocate X[0]"),
-      || Ok(f_to_nat(&inst.get()?.X[0])),
+      || {
+        Ok(f_to_nat(
+          &inst.clone().map_or(G::Scalar::zero(), |inst| inst.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])),
+      || {
+        Ok(f_to_nat(
+          &inst.clone().map_or(G::Scalar::zero(), |inst| inst.X[1]),
+        ))
+      },
       limb_width,
       n_limbs,
     )?;
@@ -160,6 +169,41 @@ where
     Ok(AllocatedRelaxedR1CSInstance { W, E, u, X0, X1 })
   }
 
+  /// Allocates the R1CS Instance as a RelaxedR1CSInstance in the circuit.
+  /// E = 0, u = 1
+  pub fn from_r1cs_instance<CS: ConstraintSystem<<G as Group>::Base>>(
+    mut cs: CS,
+    inst: AllocatedR1CSInstance<G>,
+    limb_width: usize,
+    n_limbs: usize,
+  ) -> Result<Self, SynthesisError> {
+    let E = AllocatedPoint::default(cs.namespace(|| "allocate W"))?;
+
+    let u = alloc_one(cs.namespace(|| "one"))?;
+
+    let X0 = BigNat::from_num(
+      cs.namespace(|| "allocate X0 from relaxed r1cs"),
+      Num::from(inst.X0.clone()),
+      limb_width,
+      n_limbs,
+    )?;
+
+    let X1 = BigNat::from_num(
+      cs.namespace(|| "allocate X1 from relaxed r1cs"),
+      Num::from(inst.X1.clone()),
+      limb_width,
+      n_limbs,
+    )?;
+
+    Ok(AllocatedRelaxedR1CSInstance {
+      W: inst.W,
+      E,
+      u,
+      X0,
+      X1,
+    })
+  }
+
   /// Absorb the provided instance in the RO
   pub fn absorb_in_ro<CS: ConstraintSystem<<G as Group>::Base>>(
     &self,
diff --git a/src/gadgets/utils.rs b/src/gadgets/utils.rs
index 7b28147..2034544 100644
--- a/src/gadgets/utils.rs
+++ b/src/gadgets/utils.rs
@@ -86,7 +86,7 @@ where
   CS: ConstraintSystem<<G as Group>::Base>,
 {
   AllocatedNum::alloc(cs.namespace(|| "allocate scalar as base"), || {
-    let input_bits = input.get()?.clone().to_le_bits();
+    let input_bits = input.unwrap_or_else(G::Scalar::zero).clone().to_le_bits();
     let mut mult = G::Base::one();
     let mut val = G::Base::zero();
     for bit in input_bits {