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Add InnerCircuit abstraction

Abstract signature gadget into InnerCircuit. So that PODs just need to
implement the InnerCircuit and then plug and recurse.
main
arnaucube 6 months ago
parent
commit
53ce944df5
4 changed files with 280 additions and 230 deletions
  1. +73
    -0
      src/example_innercircuit.rs
  2. +1
    -1
      src/lib.rs
  3. +0
    -105
      src/sig_gadget.rs
  4. +206
    -124
      src/tree_recursion.rs

+ 73
- 0
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(())
}
}

+ 1
- 1
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;

+ 0
- 105
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(())
}
}

+ 206
- 124
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):
FFFF
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(),

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