Add InnerCircuit abstraction

Abstract signature gadget into InnerCircuit. So that PODs just need to
implement the InnerCircuit and then plug and recurse.

src/example_innercircuit.rs

@@ -0,0 +1,73 @@
+/// This file contains a simple example implementing the InnerCircuit trait, by a circuit that
+/// checks a signature over the given msg.
+use anyhow::Result;
+use plonky2::iop::target::BoolTarget;
+use plonky2::iop::witness::PartialWitness;
+use plonky2::plonk::circuit_builder::CircuitBuilder;
+
+use sch::schnorr::*;
+use sch::schnorr_prover::*;
+
+use super::tree_recursion::{selector_gate, InnerCircuit};
+use super::{C, D, F};
+
+pub struct ExampleGadgetInput {
+    pub pk: SchnorrPublicKey,
+    pub sig: SchnorrSignature,
+}
+
+pub struct ExampleGadgetTargets {
+    pub pk_targ: SchnorrPublicKeyTarget,
+    pub sig_targ: SchnorrSignatureTarget,
+}
+
+/// The logic of this gadget verifies the given signature if `selector==0`.
+///
+/// It implements the InnerCircuit trait, so it contains the methods to `add_targets` (ie. create
+/// the targets, the logic of the circuit), and `set_targets` (ie. set the specific values to be
+/// used for the previously created targets).
+pub struct ExampleGadget {}
+
+impl InnerCircuit for ExampleGadget {
+    type Input = ExampleGadgetInput;
+    type Targets = ExampleGadgetTargets;
+
+    fn add_targets(
+        mut builder: &mut CircuitBuilder<F, D>,
+        selector_booltarg: &BoolTarget,
+        msg_targ: &MessageTarget,
+    ) -> Result<Self::Targets> {
+        // signature verification:
+        let sb: SchnorrBuilder = SchnorrBuilder {};
+        let pk_targ = SchnorrPublicKeyTarget::new_virtual(&mut builder);
+        let sig_targ = SchnorrSignatureTarget::new_virtual(&mut builder);
+        let sig_verif_targ = sb.verify_sig::<C>(&mut builder, &sig_targ, &msg_targ, &pk_targ);
+
+        // if selector==0: verify the signature; else: don't check it. ie:
+        //   if selector=0: check that sig_verif==1
+        //   if selector=1: check that one==1
+        let one = builder.one();
+        let expected = selector_gate(
+            builder,
+            sig_verif_targ.target,
+            one,
+            selector_booltarg.target,
+        );
+        let one_2 = builder.one();
+        builder.connect(expected, one_2);
+
+        Ok(Self::Targets { pk_targ, sig_targ })
+    }
+
+    fn set_targets(
+        pw: &mut PartialWitness<F>,
+        targets: &Self::Targets,
+        pod: &Self::Input,
+    ) -> Result<()> {
+        // set signature related values:
+        targets.pk_targ.set_witness(pw, &pod.pk).unwrap();
+        targets.sig_targ.set_witness(pw, &pod.sig).unwrap();
+
+        Ok(())
+    }
+}

src/lib.rs

@@ -3,7 +3,7 @@
 #![allow(non_upper_case_globals)]
 #![allow(non_camel_case_types)]
 
-pub mod sig_gadget;
+pub mod example_innercircuit;
 pub mod tree_recursion;
 
 use plonky2::field::goldilocks_field::GoldilocksField;

src/sig_gadget.rs

@@ -1,105 +0,0 @@
-use anyhow::Result;
-use plonky2::iop::target::{BoolTarget, Target};
-use plonky2::iop::witness::{PartialWitness, WitnessWrite};
-use plonky2::plonk::circuit_builder::CircuitBuilder;
-
-use sch::schnorr::*;
-use sch::schnorr_prover::*;
-
-use super::{C, D, F};
-
-/// if s==0: returns x
-/// if s==1: returns y
-/// Warning: this method assumes all input values are ensured to be \in {0,1}
-fn selector_gate(builder: &mut CircuitBuilder<F, D>, x: Target, y: Target, s: Target) -> Target {
-    // z = x + s(y-x)
-    let y_x = builder.sub(y, x);
-    // z = x+s(y-x) <==> mul_add(s, yx, x)=s*(y-x)+x
-    builder.mul_add(s, y_x, x)
-}
-
-/// ensures b \in {0,1}
-fn binary_check(builder: &mut CircuitBuilder<F, D>, b: Target) {
-    let zero = builder.zero();
-    let one = builder.one();
-    // b * (b-1) == 0
-    let b_1 = builder.sub(b, one);
-    let r = builder.mul(b, b_1);
-    builder.connect(r, zero);
-}
-
-pub struct PODInput {
-    pub pk: SchnorrPublicKey,
-    pub sig: SchnorrSignature,
-}
-
-/// The logic of this gadget verifies the given signature if `selector==0`.
-/// We reuse this gadget for all the the signature verifications in the node of the recursion tree.
-///
-/// Contains the methods to `add_targets` (ie. create the targets, the logic of the circuit), and
-/// `set_targets` (ie. set the specific values to be used for the previously created targets).
-pub struct PODGadgetTargets {
-    pub selector_targ: Target,
-    pub selector_booltarg: BoolTarget,
-
-    pub pk_targ: SchnorrPublicKeyTarget,
-    pub sig_targ: SchnorrSignatureTarget,
-}
-
-impl PODGadgetTargets {
-    pub fn add_targets(
-        mut builder: &mut CircuitBuilder<F, D>,
-        msg_targ: &MessageTarget,
-    ) -> Result<Self> {
-        let selector_targ = builder.add_virtual_target();
-        // ensure that selector_booltarg is \in {0,1}
-        binary_check(builder, selector_targ);
-        let selector_booltarg = BoolTarget::new_unsafe(selector_targ);
-
-        // signature verification:
-        let sb: SchnorrBuilder = SchnorrBuilder {};
-        let pk_targ = SchnorrPublicKeyTarget::new_virtual(&mut builder);
-        let sig_targ = SchnorrSignatureTarget::new_virtual(&mut builder);
-        let sig_verif_targ = sb.verify_sig::<C>(&mut builder, &sig_targ, &msg_targ, &pk_targ);
-
-        // - if selector=0
-        //     verify_sig==1 && proof_enabled=0
-        // - if selector=1
-        //     verify_sig==NaN && proof_enabled=1   (don't check the sig)
-        //
-        // if selector=0: check that sig_verif==1
-        // if selector=1: check that one==1
-        let one = builder.one();
-        let expected = selector_gate(
-            builder,
-            sig_verif_targ.target,
-            one,
-            selector_booltarg.target,
-        );
-        let one_2 = builder.one();
-        builder.connect(expected, one_2);
-
-        Ok(Self {
-            selector_targ,
-            selector_booltarg,
-            pk_targ,
-            sig_targ,
-        })
-    }
-    pub fn set_targets(
-        &mut self,
-        pw: &mut PartialWitness<F>,
-        // if `selector` set to 0 will verify the given signature, if set to 1 won't (and the
-        // recursion layer will verify the respective plonky2 proof)
-        selector: F,
-        pod: &PODInput,
-    ) -> Result<()> {
-        pw.set_target(self.selector_targ, selector)?;
-
-        // set signature related values:
-        self.pk_targ.set_witness(pw, &pod.pk).unwrap();
-        self.sig_targ.set_witness(pw, &pod.sig).unwrap();
-
-        Ok(())
-    }
-}

src/tree_recursion.rs

@@ -1,97 +1,150 @@
-/// N-arity tree of recursion with conditionals.
-///
-///                   p_root
-///                   ▲
-///                   │
-///               ┌────────┐
-///               │   F    │
-///               └────────┘
-///                ▲ ▲  ▲ ▲
-///              ┌─┘ │  │ └─┐
-///         ┌────┘ ┌─┘  └┐  └───┐
-///         │      │ ... │      │
-///     ┌────────┐┌┴┐┌─┐┌┴┐ ┌────────┐
-///     │   F    ││.││.││.│ │   F    │
-///     └────────┘└─┘└─┘└─┘ └────────┘
-///     ▲ ▲  ▲  ▲            ▲ ▲  ▲  ▲
-///   ┌─┘ │  └┐ └─┐        ┌─┘┌┘  └┐ └┐
-///   │   │   │   │        │  │    │  │
-///  p_1 p_2 ... p_n     p'_1 p'_2... p'_n
-///
-///
-/// where each p_i is either
-///    - signature verification
-///    - recursive plonky2 proof (proof that verifies previous proof)
-///            (generated by `RecursiveCircuit::prove_step` method)
-/// in other words, each p_i is checking:
-///   `(signature proof OR recursive proof)`
-///
-/// Each node of the recursion tree, ie. each F, verifies the N incoming p_i's, that is
-///   `(signature proof OR recursive proof) AND ... AND (signature proof OR recursive proof)`
-/// and produces a new proof.
-///
-/// For example, if N is set to N=2, then we work with a binary recursion tree:
-///           p_root
-///            ▲
-///            │
-///          ┌─┴─┐
-///          │ F │
-///          └───┘
-///           ▲ ▲
-///         ┌─┘ └─┐
-///     ┌───┘     └───┐
-///     │p_5          │p_6
-///   ┌─┴─┐         ┌─┴─┐
-///   │ F │         │ F │
-///   └───┘         └───┘
-///    ▲ ▲           ▲ ▲
-///  ┌─┘ └─┐       ┌─┘ └─┐
-///  │     │       │     │
-/// p_1   p_2     p_3   p_4
-///
-/// So that each node (F box) is verifying 2 p_i's, ie:
-///   `(signature proof OR recursive proof) AND (signature proof OR recursive proof)`
-///
-///
-/// With N=3, each node will be verifying 3 p_i's.
-///   `(signature proof OR recursive proof) AND (signature proof OR recursive proof) AND (signature proof OR recursive proof)`
-///
-///
-///
-/// Also, notice that if we set N=1, it is directly a linear chain of recursive proofs ('tree' of
-/// arity 1):
-///        ┌─┐     ┌─┐     ┌─┐     ┌─┐
-///  ─────►│F├────►│F├────►│F├────►│F├────►
-///   p_1  └─┘ p_2 └─┘ p_3 └─┘ p_4 └─┘ p_5
-///
-/// where each p_i is proving: `(signature proof OR recursive proof)`.
-///
-///
-/// To run the tests that checks this logic:
-/// cargo test --release test_tree_recursion -- --nocapture
+/*
+ N-arity tree of recursion with conditionals.
+
+                   p_root
+                   ▲
+                   │
+               ┌────────┐
+               │   F    │
+               └────────┘
+                ▲ ▲  ▲ ▲
+              ┌─┘ │  │ └─┐
+         ┌────┘ ┌─┘  └┐  └───┐
+         │      │ ... │      │
+     ┌────────┐┌┴┐┌─┐┌┴┐ ┌────────┐
+     │   F    ││.││.││.│ │   F    │
+     └────────┘└─┘└─┘└─┘ └────────┘
+     ▲ ▲  ▲  ▲            ▲ ▲  ▲  ▲
+   ┌─┘ │  └┐ └─┐        ┌─┘┌┘  └┐ └┐
+   │   │   │   │        │  │    │  │
+  p_1 p_2 ... p_n     p'_1 p'_2... p'_n
+
+
+ where each p_i is either
+    - InnerCircuit verification
+    - recursive plonky2 proof (proof that verifies previous proof)
+            (generated by `RecursiveCircuit::prove_step` method)
+ in other words, each p_i is checking:
+   `(InnerCircuit OR recursive proof verify)`
+
+ Each node of the recursion tree, ie. each F, verifies the N incoming p_i's, that is
+   `(InnerCircuit OR recursive proof verify) AND ... AND (InnerCircuit OR recursive proof verify)`
+ and produces a new proof.
+
+ For example, if N is set to N=2, then we work with a binary recursion tree:
+           p_root
+            ▲
+            │
+          ┌─┴─┐
+          │ F │
+          └───┘
+           ▲ ▲
+         ┌─┘ └─┐
+     ┌───┘     └───┐
+     │p_5          │p_6
+   ┌─┴─┐         ┌─┴─┐
+   │ F │         │ F │
+   └───┘         └───┘
+    ▲ ▲           ▲ ▲
+  ┌─┘ └─┐       ┌─┘ └─┐
+  │     │       │     │
+ p_1   p_2     p_3   p_4
+
+ p_i: `(InnerCircuit OR recursive-proof-verification)`
+
+
+ So that each node (F box) is verifying 2 p_i's, ie:
+   `(InnerCircuit OR recursive-proof-verification) AND (InnerCircuit OR recursive-proof-verification)`
+
+
+ With N=3, each node will be verifying 3 p_i's.
+   `(InnerCircuit OR recursive-proof-verification) AND (InnerCircuit OR recursive-proof-verification) AND (InnerCircuit OR recursive-proof-verification)`
+
+
+
+ Also, notice that if we set N=1, it is directly a linear chain of recursive proofs ('tree' of
+ arity 1):
+        ┌─┐     ┌─┐     ┌─┐     ┌─┐
+  ─────►│F├────►│F├────►│F├────►│F├────►
+   p_1  └─┘ p_2 └─┘ p_3 └─┘ p_4 └─┘ p_5
+
+ where each p_i is proving: `(InnerCircuit OR recursive-proof-verification)`.
+
+
+ To run the tests that checks this logic:
+ cargo test --release test_tree_recursion -- --nocapture
+*/
 use anyhow::{anyhow, Result};
 use plonky2::field::types::Field;
 use plonky2::gates::noop::NoopGate;
+use plonky2::iop::target::{BoolTarget, Target};
 use plonky2::iop::witness::{PartialWitness, WitnessWrite};
 use plonky2::plonk::circuit_builder::CircuitBuilder;
 use plonky2::plonk::circuit_data::{
     CircuitConfig, CircuitData, VerifierCircuitData, VerifierCircuitTarget,
 };
 use plonky2::plonk::proof::{ProofWithPublicInputs, ProofWithPublicInputsTarget};
+use std::marker::PhantomData;
 use std::time::Instant;
 
 use sch::schnorr_prover::*;
 
-use super::{
-    sig_gadget::{PODGadgetTargets, PODInput},
-    PlonkyProof, C, D, F,
-};
+use super::{PlonkyProof, C, D, F};
+
+/// if s==0: returns x
+/// if s==1: returns y
+/// Warning: this method assumes all input values are ensured to be \in {0,1}
+pub fn selector_gate(
+    builder: &mut CircuitBuilder<F, D>,
+    x: Target,
+    y: Target,
+    s: Target,
+) -> Target {
+    // z = x + s(y-x)
+    let y_x = builder.sub(y, x);
+    // z = x+s(y-x) <==> mul_add(s, yx, x)=s*(y-x)+x
+    builder.mul_add(s, y_x, x)
+}
+
+/// ensures b \in {0,1}
+pub fn binary_check(builder: &mut CircuitBuilder<F, D>, b: Target) {
+    let zero = builder.zero();
+    let one = builder.one();
+    // b * (b-1) == 0
+    let b_1 = builder.sub(b, one);
+    let r = builder.mul(b, b_1);
+    builder.connect(r, zero);
+}
+
+/// InnerCircuit is the trait that is used to define the logic of the circuit that is used at each
+/// node of the recursive tree.
+pub trait InnerCircuit {
+    type Input;
+    type Targets;
+
+    fn add_targets(
+        builder: &mut CircuitBuilder<F, D>,
+        selector_booltarg: &BoolTarget,
+        msg_targ: &MessageTarget,
+    ) -> Result<Self::Targets>;
 
-/// Contains the methods to `add_targets` (ie. create the targets, the logic of the circuit), and
-/// `set_targets` (ie. set the specific values to be used for the previously created targets).
-pub struct RecursiveCircuit<const N: usize> {
+    fn set_targets(
+        pw: &mut PartialWitness<F>,
+        targets: &Self::Targets,
+        input: &Self::Input,
+    ) -> Result<()>;
+}
+
+/// RecursiveCircuit defines the circuit used on each node of the recursion tree, which is doing
+/// `(InnerCircuit OR recursive-proof-verification)` N times, and generating a new proof that can
+/// be verified by the same circuit itself.
+///
+/// It contains the methods to `add_targets` (ie. create the targets, the logic of the circuit),
+/// and `set_targets` (ie. set the specific values to be used for the previously created targets).
+pub struct RecursiveCircuit<I: InnerCircuit, const N: usize> {
     msg_targ: MessageTarget,
-    sigs_targ: Vec<PODGadgetTargets>,
+    selectors_targ: Vec<Target>,
+    inner_circuit_targ: Vec<I::Targets>,
     proofs_targ: Vec<ProofWithPublicInputsTarget<D>>,
     // the next two are common for all the given proofs. It is the data for this circuit itself
     // (cyclic circuit).
@@ -99,7 +152,7 @@ pub struct RecursiveCircuit {
     verifier_data: VerifierCircuitData<F, C, D>,
 }
 
-impl<const N: usize> RecursiveCircuit<N> {
+impl<I: InnerCircuit, const N: usize> RecursiveCircuit<I, N> {
     pub fn prepare_public_inputs(
         verifier_data: VerifierCircuitData<F, C, D>,
         msg: Vec<F>,
@@ -130,11 +183,23 @@ impl RecursiveCircuit {
         // set msg as public input
         builder.register_public_inputs(&msg_targ.msg);
 
-        // build the signature verification logic
-        let mut sigs_targ: Vec<PODGadgetTargets> = vec![];
+        // build the InnerCircuit logic. Also set the selectors, used both by the InnerCircuit and
+        // by the recursive proofs verifications.
+        let mut selectors_targ: Vec<Target> = vec![];
+        let mut selectors_bool_targ: Vec<BoolTarget> = vec![];
+        let mut inner_circuit_targ: Vec<I::Targets> = vec![];
         for _ in 0..N {
-            let sig_targets = PODGadgetTargets::add_targets(builder, &msg_targ)?;
-            sigs_targ.push(sig_targets);
+            // selectors:
+            let selector_F_targ = builder.add_virtual_target();
+            // ensure that selector_booltarg is \in {0,1}
+            binary_check(builder, selector_F_targ);
+            let selector_bool_targ = BoolTarget::new_unsafe(selector_F_targ);
+            selectors_targ.push(selector_F_targ);
+            selectors_bool_targ.push(selector_bool_targ);
+
+            // inner circuits:
+            let inner_circuit_targets = I::add_targets(builder, &selector_bool_targ, &msg_targ)?;
+            inner_circuit_targ.push(inner_circuit_targets);
         }
 
         // proof verification:
@@ -146,7 +211,7 @@ impl RecursiveCircuit {
         for i in 0..N {
             let proof_targ = builder.add_virtual_proof_with_pis(&common_data);
             builder.conditionally_verify_cyclic_proof_or_dummy::<C>(
-                sigs_targ[i].selector_booltarg,
+                selectors_bool_targ[i],
                 &proof_targ,
                 &common_data,
             )?;
@@ -155,7 +220,8 @@ impl RecursiveCircuit {
 
         Ok(Self {
             msg_targ,
-            sigs_targ,
+            selectors_targ,
+            inner_circuit_targ,
             proofs_targ,
             verifier_data_targ,
             verifier_data,
@@ -166,18 +232,19 @@ impl RecursiveCircuit {
         &mut self,
         pw: &mut PartialWitness<F>,
         msg: &Vec<F>,
-        // if selectors[i]==0: verify pods[i] signature. if selectors[i]==1: verify
-        // recursive_proof[i]
+        // if selectors[i]==0: verify InnerCircuit. if selectors[i]==1: verify recursive_proof[i]
         selectors: Vec<F>,
-        pods_input: Vec<PODInput>,
+        inner_circuit_input: Vec<I::Input>,
         recursive_proofs: &Vec<PlonkyProof>,
     ) -> Result<()> {
-        // set the msg value (used by all N sig gadgets)
+        // set the msg value (used by all N InnerCircuit gadgets)
         self.msg_targ.set_witness(pw, &msg).unwrap();
 
-        // set the signature related values
+        // set the InnerCircuit related values
         for i in 0..N {
-            self.sigs_targ[i].set_targets(pw, selectors[i], &pods_input[i])?;
+            pw.set_target(self.selectors_targ[i], selectors[i])?;
+
+            I::set_targets(pw, &self.inner_circuit_targ[i], &inner_circuit_input[i])?;
         }
 
         // set proof related values:
@@ -185,8 +252,10 @@ impl RecursiveCircuit {
         // recursive proofs verification
         pw.set_verifier_data_target(&self.verifier_data_targ, &self.verifier_data.verifier_only)?;
 
-        let public_inputs =
-            RecursiveCircuit::<N>::prepare_public_inputs(self.verifier_data.clone(), msg.clone());
+        let public_inputs = RecursiveCircuit::<I, N>::prepare_public_inputs(
+            self.verifier_data.clone(),
+            msg.clone(),
+        );
         for i in 0..N {
             pw.set_proof_with_pis_target(
                 &self.proofs_targ[i],
@@ -201,10 +270,9 @@ impl RecursiveCircuit {
     }
 }
 
-#[derive(Debug, Clone)]
-pub struct Recursion<const N: usize> {}
-
-pub fn common_data_for_recursion<const N: usize>(msg_len: usize) -> Result<CircuitData<F, C, D>> {
+pub fn common_data_for_recursion<I: InnerCircuit, const N: usize>(
+    msg_len: usize,
+) -> Result<CircuitData<F, C, D>> {
     // 1st
     let config = CircuitConfig::standard_recursion_config();
     let builder = CircuitBuilder::<F, D>::new(config);
@@ -226,7 +294,6 @@ pub fn common_data_for_recursion(msg_len: usize) -> Result
     let config = CircuitConfig::standard_recursion_config();
     let mut builder = CircuitBuilder::<F, D>::new(config.clone());
     let msg_targ = MessageTarget::new_with_size(&mut builder, msg_len);
-    // sigs verify
     builder.register_public_inputs(&msg_targ.msg);
 
     builder.add_gate(
@@ -238,8 +305,13 @@ pub fn common_data_for_recursion(msg_len: usize) -> Result
         vec![],
     );
 
-    let _ = PODGadgetTargets::add_targets(&mut builder, &msg_targ).unwrap();
-    let _ = PODGadgetTargets::add_targets(&mut builder, &msg_targ).unwrap();
+    // InnerCircuits targets
+    for _ in 0..N {
+        let selector_F_targ = builder.add_virtual_target();
+        binary_check(&mut builder, selector_F_targ);
+        let b = BoolTarget::new_unsafe(selector_F_targ);
+        let _ = I::add_targets(&mut builder, &b, &msg_targ).unwrap();
+    }
 
     // proofs verify
     let verifier_data = builder.add_verifier_data_public_inputs();
@@ -277,15 +349,20 @@ fn compute_num_gates() -> Result {
     Ok(n_gates)
 }
 
-impl<const N: usize> Recursion<N> {
+#[derive(Debug, Clone)]
+pub struct Recursion<I: InnerCircuit, const N: usize> {
+    _i: PhantomData<I>,
+}
+
+impl<I: InnerCircuit, const N: usize> Recursion<I, N> {
     /// returns the full-recursive CircuitData
     pub fn circuit_data(msg_len: usize) -> Result<CircuitData<F, C, D>> {
-        let mut data = common_data_for_recursion::<N>(msg_len)?;
+        let mut data = common_data_for_recursion::<I, N>(msg_len)?;
 
         // build the actual RecursiveCircuit circuit data
         let config = CircuitConfig::standard_recursion_config();
         let mut builder = CircuitBuilder::new(config);
-        let _ = RecursiveCircuit::<N>::add_targets(&mut builder, data.verifier_data(), msg_len)?;
+        let _ = RecursiveCircuit::<I, N>::add_targets(&mut builder, data.verifier_data(), msg_len)?;
         dbg!(builder.num_gates());
         data = builder.build::<C>();
 
@@ -294,22 +371,18 @@ impl Recursion {
 
     pub fn prove_step(
         verifier_data: VerifierCircuitData<F, C, D>,
-        msg: &Vec<F>, // will be an array of "pod roots (hashes)'
-        // if selectors[i]==0: verify pods[i] signature. if selectors[i]==1: verify
-        // recursive_proof[i]
+        msg: &Vec<F>,
+        // if selectors[i]==0: verify InnerCircuit. if selectors[i]==1: verify recursive_proof[i]
         selectors: Vec<F>,
-        pods_input: Vec<PODInput>,
+        inner_circuits_input: Vec<I::Input>,
         recursive_proofs: &Vec<PlonkyProof>,
     ) -> Result<PlonkyProof> {
         println!("prove_step:");
         for i in 0..N {
             if selectors[i].is_nonzero() {
-                println!("  (pods_input[{}].selector==1), verify {}-th proof", i, i);
+                println!("  (selectors[{}]==1), verify {}-th proof", i, i);
             } else {
-                println!(
-                    "  (pods_input[{}].selector==0), verify {}-th signature",
-                    i, i
-                );
+                println!("  (selectors[{}]==0), verify {}-th inner circuit", i, i);
             }
         }
 
@@ -319,13 +392,19 @@ impl Recursion {
         // assign the targets
         let start = Instant::now();
         let mut circuit =
-            RecursiveCircuit::<N>::add_targets(&mut builder, verifier_data.clone(), msg.len())?;
+            RecursiveCircuit::<I, N>::add_targets(&mut builder, verifier_data.clone(), msg.len())?;
         println!("RecursiveCircuit::add_targets(): {:?}", start.elapsed());
 
         // fill the targets
         let mut pw = PartialWitness::new();
         let start = Instant::now();
-        circuit.set_targets(&mut pw, msg, selectors, pods_input, recursive_proofs)?;
+        circuit.set_targets(
+            &mut pw,
+            msg,
+            selectors,
+            inner_circuits_input,
+            recursive_proofs,
+        )?;
         println!("circuit.set_targets(): {:?}", start.elapsed());
 
         let start = Instant::now();
@@ -367,6 +446,7 @@ mod tests {
     use std::time::Instant;
 
     use super::*;
+    use crate::example_innercircuit::{ExampleGadget, ExampleGadgetInput};
     use sch::schnorr::*;
 
     // this sets the plonky2 internal logs level
@@ -384,7 +464,7 @@ mod tests {
         // For testing: change the following `N` value to try different arities of the recursion tree:
         test_tree_recursion_opt::<2>()?; // N=2
 
-        // test_tree_recursion_opt::<3>()?; // N=3
+        test_tree_recursion_opt::<3>()?; // N=3
 
         Ok(())
     }
@@ -422,7 +502,7 @@ mod tests {
             .collect();
 
         // build the circuit_data & verifier_data for the recursive circuit
-        let circuit_data = Recursion::<N>::circuit_data(MSG_LEN)?;
+        let circuit_data = Recursion::<ExampleGadget, N>::circuit_data(MSG_LEN)?;
         let verifier_data = circuit_data.verifier_data();
 
         let dummy_proof_pis = cyclic_base_proof(
@@ -461,9 +541,9 @@ mod tests {
 
                 // prepare the inputs for the `Recursion::prove_step` call
                 let selectors = (0..N).into_iter().map(|_| proof_enabled.clone()).collect();
-                let pods_input: Vec<PODInput> = (0..N)
+                let innercircuits_input: Vec<ExampleGadgetInput> = (0..N)
                     .into_iter()
-                    .map(|k| PODInput {
+                    .map(|k| ExampleGadgetInput {
                         pk: pk_vec[j + k],
                         sig: sig_vec[j + k],
                     })
@@ -476,11 +556,11 @@ mod tests {
 
                 // do the recursive step
                 let start = Instant::now();
-                let new_proof = Recursion::<N>::prove_step(
+                let new_proof = Recursion::<ExampleGadget, N>::prove_step(
                     verifier_data.clone(),
                     &msg,
                     selectors,
-                    pods_input,
+                    innercircuits_input,
                     &proofs,
                 )?;
                 println!(
@@ -491,7 +571,7 @@ mod tests {
                 );
 
                 // verify the recursive proof
-                let public_inputs = RecursiveCircuit::<N>::prepare_public_inputs(
+                let public_inputs = RecursiveCircuit::<ExampleGadget, N>::prepare_public_inputs(
                     verifier_data.clone(),
                     msg.clone(),
                 );
@@ -509,8 +589,10 @@ mod tests {
         let last_proof = proofs_at_level_i[0].clone();
 
         // verify the last proof
-        let public_inputs =
-            RecursiveCircuit::<N>::prepare_public_inputs(verifier_data.clone(), msg.clone());
+        let public_inputs = RecursiveCircuit::<ExampleGadget, N>::prepare_public_inputs(
+            verifier_data.clone(),
+            msg.clone(),
+        );
         verifier_data.clone().verify(ProofWithPublicInputs {
             proof: last_proof.clone(),
             public_inputs: public_inputs.clone(),