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halo2/src/plonk.rs

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//! This module provides an implementation of a variant of (Turbo)[PLONK][plonk]
//! that is designed specifically for the polynomial commitment scheme described
//! in the [Halo][halo] paper.
//!
//! [halo]: https://eprint.iacr.org/2019/1021
//! [plonk]: https://eprint.iacr.org/2019/953
use blake2b_simd::Params as Blake2bParams;
use crate::arithmetic::{CurveAffine, FieldExt};
use crate::poly::{
commitment::Params, Coeff, EvaluationDomain, ExtendedLagrangeCoeff, LagrangeCoeff,
PinnedEvaluationDomain, Polynomial,
};
use crate::transcript::{ChallengeScalar, Transcript};
mod circuit;
mod keygen;
mod lookup;
pub(crate) mod permutation;
mod vanishing;
mod prover;
mod verifier;
pub use circuit::*;
pub use keygen::*;
pub use prover::*;
pub use verifier::*;
use std::io;
/// This is a verifying key which allows for the verification of proofs for a
/// particular circuit.
#[derive(Debug)]
pub struct VerifyingKey<C: CurveAffine> {
domain: EvaluationDomain<C::Scalar>,
fixed_commitments: Vec<C>,
permutations: Vec<permutation::VerifyingKey<C>>,
cs: ConstraintSystem<C::Scalar>,
}
impl<C: CurveAffine> VerifyingKey<C> {
/// Writes a verifying key to a buffer.
pub fn write<W: io::Write>(&self, writer: &mut W) -> io::Result<()> {
for commitment in &self.fixed_commitments {
writer.write_all(&commitment.to_bytes())?;
}
for permutation in &self.permutations {
permutation.write(writer)?;
}
Ok(())
}
/// Reads a verification key from a buffer.
pub fn read<R: io::Read, ConcreteCircuit: Circuit<C::Scalar>>(
reader: &mut R,
params: &Params<C>,
) -> io::Result<Self> {
let (domain, cs, _) = keygen::create_domain::<C, ConcreteCircuit>(params);
let fixed_commitments: Vec<_> = (0..cs.num_fixed_columns)
.map(|_| C::read(reader))
.collect::<Result<_, _>>()?;
let permutations: Vec<_> = cs
.permutations
.iter()
.map(|argument| permutation::VerifyingKey::read(reader, argument))
.collect::<Result<_, _>>()?;
Ok(VerifyingKey {
domain,
fixed_commitments,
permutations,
cs,
})
}
/// Hashes a verification key into a transcript.
pub fn hash_into<T: Transcript<C>>(&self, transcript: &mut T) -> io::Result<()> {
let mut hasher = Blake2bParams::new()
.hash_length(64)
.personal(b"Halo2-Verify-Key")
.to_state();
let s = format!("{:?}", self.pinned());
hasher.update(&(s.len() as u64).to_le_bytes());
hasher.update(s.as_bytes());
// Hash in final Blake2bState
transcript.common_scalar(C::Scalar::from_bytes_wide(hasher.finalize().as_array()))?;
Ok(())
}
/// Obtains a pinned representation of this verification key that contains
/// the minimal information necessary to reconstruct the verification key.
pub fn pinned(&self) -> PinnedVerificationKey<'_, C> {
PinnedVerificationKey {
base_modulus: C::Base::MODULUS,
scalar_modulus: C::Scalar::MODULUS,
domain: self.domain.pinned(),
fixed_commitments: &self.fixed_commitments,
permutations: &self.permutations,
cs: self.cs.pinned(),
}
}
}
/// Minimal representation of a verification key that can be used to identify
/// its active contents.
#[derive(Debug)]
pub struct PinnedVerificationKey<'a, C: CurveAffine> {
base_modulus: &'static str,
scalar_modulus: &'static str,
domain: PinnedEvaluationDomain<'a, C::Scalar>,
cs: PinnedConstraintSystem<'a, C::Scalar>,
fixed_commitments: &'a Vec<C>,
permutations: &'a Vec<permutation::VerifyingKey<C>>,
}
/// This is a proving key which allows for the creation of proofs for a
/// particular circuit.
#[derive(Debug)]
pub struct ProvingKey<C: CurveAffine> {
vk: VerifyingKey<C>,
// TODO: get rid of this?
l0: Polynomial<C::Scalar, ExtendedLagrangeCoeff>,
fixed_values: Vec<Polynomial<C::Scalar, LagrangeCoeff>>,
fixed_polys: Vec<Polynomial<C::Scalar, Coeff>>,
fixed_cosets: Vec<Polynomial<C::Scalar, ExtendedLagrangeCoeff>>,
permutations: Vec<permutation::ProvingKey<C>>,
}
/// This is an error that could occur during proving or circuit synthesis.
// TODO: these errors need to be cleaned up
#[derive(Debug)]
pub enum Error {
/// This is an error that can occur during synthesis of the circuit, for
/// example, when the witness is not present.
SynthesisError,
/// The structured reference string or the parameters are not compatible
/// with the circuit being synthesized.
IncompatibleParams,
/// The constraint system is not satisfied.
ConstraintSystemFailure,
/// Out of bounds index passed to a backend
BoundsFailure,
/// Opening error
OpeningError,
/// Transcript error
TranscriptError,
}
impl<C: CurveAffine> ProvingKey<C> {
/// Get the underlying [`VerifyingKey`].
pub fn get_vk(&self) -> &VerifyingKey<C> {
&self.vk
}
}
impl<C: CurveAffine> VerifyingKey<C> {
/// Get the underlying [`EvaluationDomain`].
pub fn get_domain(&self) -> &EvaluationDomain<C::Scalar> {
&self.domain
}
}
#[derive(Clone, Copy, Debug)]
struct Theta;
type ChallengeTheta<F> = ChallengeScalar<F, Theta>;
#[derive(Clone, Copy, Debug)]
struct Beta;
type ChallengeBeta<F> = ChallengeScalar<F, Beta>;
#[derive(Clone, Copy, Debug)]
struct Gamma;
type ChallengeGamma<F> = ChallengeScalar<F, Gamma>;
#[derive(Clone, Copy, Debug)]
struct Y;
type ChallengeY<F> = ChallengeScalar<F, Y>;
#[derive(Clone, Copy, Debug)]
struct X;
type ChallengeX<F> = ChallengeScalar<F, X>;
#[test]
fn test_proving() {
use crate::arithmetic::{Curve, FieldExt};
use crate::dev::MockProver;
use crate::pasta::{EqAffine, Fp};
use crate::poly::{
commitment::{Blind, Params},
Rotation,
};
use crate::transcript::{Blake2bRead, Blake2bWrite};
use circuit::{Advice, Column, Fixed};
use std::marker::PhantomData;
const K: u32 = 5;
/// This represents an advice column at a certain row in the ConstraintSystem
#[derive(Copy, Clone, Debug)]
pub struct Variable(Column<Advice>, usize);
// Initialize the polynomial commitment parameters
let params: Params<EqAffine> = Params::new(K);
#[derive(Copy, Clone)]
struct PLONKConfig {
a: Column<Advice>,
b: Column<Advice>,
c: Column<Advice>,
d: Column<Advice>,
e: Column<Advice>,
sa: Column<Fixed>,
sb: Column<Fixed>,
sc: Column<Fixed>,
sm: Column<Fixed>,
sp: Column<Fixed>,
sl: Column<Fixed>,
sl2: Column<Fixed>,
perm: usize,
perm2: usize,
}
trait StandardCS<FF: FieldExt> {
fn raw_multiply<F>(&mut self, f: F) -> Result<(Variable, Variable, Variable), Error>
where
F: FnOnce() -> Result<(FF, FF, FF), Error>;
fn raw_add<F>(&mut self, f: F) -> Result<(Variable, Variable, Variable), Error>
where
F: FnOnce() -> Result<(FF, FF, FF), Error>;
fn copy(&mut self, a: Variable, b: Variable) -> Result<(), Error>;
fn public_input<F>(&mut self, f: F) -> Result<Variable, Error>
where
F: FnOnce() -> Result<FF, Error>;
fn lookup_table(&mut self, values: &[Vec<FF>]) -> Result<(), Error>;
}
#[derive(Clone)]
struct MyCircuit<F: FieldExt> {
a: Option<F>,
lookup_tables: Vec<Vec<F>>,
}
struct StandardPLONK<'a, F: FieldExt, CS: Assignment<F> + 'a> {
cs: &'a mut CS,
config: PLONKConfig,
current_gate: usize,
_marker: PhantomData<F>,
}
impl<'a, FF: FieldExt, CS: Assignment<FF>> StandardPLONK<'a, FF, CS> {
fn new(cs: &'a mut CS, config: PLONKConfig) -> Self {
StandardPLONK {
cs,
config,
current_gate: 0,
_marker: PhantomData,
}
}
}
impl<'a, FF: FieldExt, CS: Assignment<FF>> StandardCS<FF> for StandardPLONK<'a, FF, CS> {
fn raw_multiply<F>(&mut self, f: F) -> Result<(Variable, Variable, Variable), Error>
where
F: FnOnce() -> Result<(FF, FF, FF), Error>,
{
let index = self.current_gate;
self.current_gate += 1;
let mut value = None;
self.cs.assign_advice(
|| "lhs",
self.config.a,
index,
|| {
value = Some(f()?);
Ok(value.ok_or(Error::SynthesisError)?.0)
},
)?;
self.cs.assign_advice(
|| "lhs^4",
self.config.d,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.0.square().square()),
)?;
self.cs.assign_advice(
|| "rhs",
self.config.b,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.1),
)?;
self.cs.assign_advice(
|| "rhs^4",
self.config.e,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.1.square().square()),
)?;
self.cs.assign_advice(
|| "out",
self.config.c,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.2),
)?;
self.cs
.assign_fixed(|| "a", self.config.sa, index, || Ok(FF::zero()))?;
self.cs
.assign_fixed(|| "b", self.config.sb, index, || Ok(FF::zero()))?;
self.cs
.assign_fixed(|| "c", self.config.sc, index, || Ok(FF::one()))?;
self.cs
.assign_fixed(|| "a * b", self.config.sm, index, || Ok(FF::one()))?;
Ok((
Variable(self.config.a, index),
Variable(self.config.b, index),
Variable(self.config.c, index),
))
}
fn raw_add<F>(&mut self, f: F) -> Result<(Variable, Variable, Variable), Error>
where
F: FnOnce() -> Result<(FF, FF, FF), Error>,
{
let index = self.current_gate;
self.current_gate += 1;
let mut value = None;
self.cs.assign_advice(
|| "lhs",
self.config.a,
index,
|| {
value = Some(f()?);
Ok(value.ok_or(Error::SynthesisError)?.0)
},
)?;
self.cs.assign_advice(
|| "lhs^4",
self.config.d,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.0.square().square()),
)?;
self.cs.assign_advice(
|| "rhs",
self.config.b,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.1),
)?;
self.cs.assign_advice(
|| "rhs^4",
self.config.e,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.1.square().square()),
)?;
self.cs.assign_advice(
|| "out",
self.config.c,
index,
|| Ok(value.ok_or(Error::SynthesisError)?.2),
)?;
self.cs
.assign_fixed(|| "a", self.config.sa, index, || Ok(FF::one()))?;
self.cs
.assign_fixed(|| "b", self.config.sb, index, || Ok(FF::one()))?;
self.cs
.assign_fixed(|| "c", self.config.sc, index, || Ok(FF::one()))?;
self.cs
.assign_fixed(|| "a * b", self.config.sm, index, || Ok(FF::zero()))?;
Ok((
Variable(self.config.a, index),
Variable(self.config.b, index),
Variable(self.config.c, index),
))
}
fn copy(&mut self, left: Variable, right: Variable) -> Result<(), Error> {
let left_column = match left.0 {
x if x == self.config.a => 0,
x if x == self.config.b => 1,
x if x == self.config.c => 2,
_ => unreachable!(),
};
let right_column = match right.0 {
x if x == self.config.a => 0,
x if x == self.config.b => 1,
x if x == self.config.c => 2,
_ => unreachable!(),
};
self.cs
.copy(self.config.perm, left_column, left.1, right_column, right.1)?;
self.cs.copy(
self.config.perm2,
left_column,
left.1,
right_column,
right.1,
)
}
fn public_input<F>(&mut self, f: F) -> Result<Variable, Error>
where
F: FnOnce() -> Result<FF, Error>,
{
let index = self.current_gate;
self.current_gate += 1;
self.cs
.assign_advice(|| "value", self.config.a, index, || f())?;
self.cs
.assign_fixed(|| "public", self.config.sp, index, || Ok(FF::one()))?;
Ok(Variable(self.config.a, index))
}
fn lookup_table(&mut self, values: &[Vec<FF>]) -> Result<(), Error> {
for (&value_0, &value_1) in values[0].iter().zip(values[1].iter()) {
let index = self.current_gate;
self.current_gate += 1;
self.cs
.assign_fixed(|| "table col 1", self.config.sl, index, || Ok(value_0))?;
self.cs
.assign_fixed(|| "table col 2", self.config.sl2, index, || Ok(value_1))?;
}
Ok(())
}
}
impl<F: FieldExt> Circuit<F> for MyCircuit<F> {
type Config = PLONKConfig;
fn configure(meta: &mut ConstraintSystem<F>) -> PLONKConfig {
let e = meta.advice_column();
let a = meta.advice_column();
let b = meta.advice_column();
let sf = meta.fixed_column();
let c = meta.advice_column();
let d = meta.advice_column();
let p = meta.instance_column();
let perm = meta.permutation(&[a.into(), b.into(), c.into()]);
let perm2 = meta.permutation(&[a.into(), b.into(), c.into()]);
let sm = meta.fixed_column();
let sa = meta.fixed_column();
let sb = meta.fixed_column();
let sc = meta.fixed_column();
let sp = meta.fixed_column();
let sl = meta.fixed_column();
let sl2 = meta.fixed_column();
/*
* A B ... sl sl2
* [
* instance 0 ... 0 0
* a a ... 0 0
* a a^2 ... 0 0
* a a ... 0 0
* a a^2 ... 0 0
* ... ... ... ... ...
* ... ... ... instance 0
* ... ... ... a a
* ... ... ... a a^2
* ... ... ... 0 0
* ]
*/
meta.lookup(&[a.into()], &[sl.into()]);
meta.lookup(&[a.into(), b.into()], &[sl.into(), sl2.into()]);
meta.create_gate("Combined add-mult", |meta| {
let d = meta.query_advice(d, Rotation::next());
let a = meta.query_advice(a, Rotation::cur());
let sf = meta.query_fixed(sf, Rotation::cur());
let e = meta.query_advice(e, Rotation::prev());
let b = meta.query_advice(b, Rotation::cur());
let c = meta.query_advice(c, Rotation::cur());
let sa = meta.query_fixed(sa, Rotation::cur());
let sb = meta.query_fixed(sb, Rotation::cur());
let sc = meta.query_fixed(sc, Rotation::cur());
let sm = meta.query_fixed(sm, Rotation::cur());
a.clone() * sa + b.clone() * sb + a * b * sm + (c * sc * (-F::one())) + sf * (d * e)
});
meta.create_gate("Public input", |meta| {
let a = meta.query_advice(a, Rotation::cur());
let p = meta.query_instance(p, Rotation::cur());
let sp = meta.query_fixed(sp, Rotation::cur());
sp * (a + p * (-F::one()))
});
PLONKConfig {
a,
b,
c,
d,
e,
sa,
sb,
sc,
sm,
sp,
sl,
sl2,
perm,
perm2,
}
}
fn synthesize(
&self,
cs: &mut impl Assignment<F>,
config: PLONKConfig,
) -> Result<(), Error> {
let mut cs = StandardPLONK::new(cs, config);
let _ = cs.public_input(|| Ok(F::one() + F::one()))?;
for _ in 0..10 {
let mut a_squared = None;
let (a0, _, c0) = cs.raw_multiply(|| {
a_squared = self.a.map(|a| a.square());
Ok((
self.a.ok_or(Error::SynthesisError)?,
self.a.ok_or(Error::SynthesisError)?,
a_squared.ok_or(Error::SynthesisError)?,
))
})?;
let (a1, b1, _) = cs.raw_add(|| {
let fin = a_squared.and_then(|a2| self.a.map(|a| a + a2));
Ok((
self.a.ok_or(Error::SynthesisError)?,
a_squared.ok_or(Error::SynthesisError)?,
fin.ok_or(Error::SynthesisError)?,
))
})?;
cs.copy(a0, a1)?;
cs.copy(b1, c0)?;
}
cs.lookup_table(&self.lookup_tables)?;
Ok(())
}
}
let a = Fp::from_u64(2834758237) * Fp::ZETA;
let a_squared = a * &a;
let instance = Fp::one() + Fp::one();
let lookup_table = vec![instance, a, a, Fp::zero()];
let lookup_table_2 = vec![Fp::zero(), a, a_squared, Fp::zero()];
let empty_circuit: MyCircuit<Fp> = MyCircuit {
a: None,
lookup_tables: vec![lookup_table.clone(), lookup_table_2.clone()],
};
let circuit: MyCircuit<Fp> = MyCircuit {
a: Some(a),
lookup_tables: vec![lookup_table, lookup_table_2],
};
// Initialize the proving key
let vk = keygen_vk(&params, &empty_circuit).expect("keygen_vk should not fail");
let pk = keygen_pk(&params, vk, &empty_circuit).expect("keygen_pk should not fail");
let mut pubinputs = pk.get_vk().get_domain().empty_lagrange();
pubinputs[0] = instance;
let pubinput = params
.commit_lagrange(&pubinputs, Blind::default())
.to_affine();
// Check this circuit is satisfied.
let prover = match MockProver::run(K, &circuit, vec![pubinputs.to_vec()]) {
Ok(prover) => prover,
Err(e) => panic!("{:?}", e),
};
assert_eq!(prover.verify(), Ok(()));
for _ in 0..100 {
let mut transcript = Blake2bWrite::init(vec![]);
// Create a proof
create_proof(
&params,
&pk,
&[circuit.clone(), circuit.clone()],
&[&[pubinputs.clone()], &[pubinputs.clone()]],
&mut transcript,
)
.expect("proof generation should not fail");
let proof: Vec<u8> = transcript.finalize();
let pubinput_slice = &[pubinput];
let pubinput_slice_copy = &[pubinput];
let msm = params.empty_msm();
let mut transcript = Blake2bRead::init(&proof[..]);
let guard = verify_proof(
&params,
pk.get_vk(),
msm,
&[pubinput_slice, pubinput_slice_copy],
&mut transcript,
)
.unwrap();
{
let msm = guard.clone().use_challenges();
assert!(msm.eval());
}
{
let g = guard.compute_g();
let (msm, _) = guard.clone().use_g(g);
assert!(msm.eval());
}
let msm = guard.clone().use_challenges();
assert!(msm.clone().eval());
let mut transcript = Blake2bRead::init(&proof[..]);
let mut vk_buffer = vec![];
pk.get_vk().write(&mut vk_buffer).unwrap();
let vk = VerifyingKey::<EqAffine>::read::<_, MyCircuit<Fp>>(&mut &vk_buffer[..], &params)
.unwrap();
let guard = verify_proof(
&params,
&vk,
msm,
&[pubinput_slice, pubinput_slice_copy],
&mut transcript,
)
.unwrap();
{
let msm = guard.clone().use_challenges();
assert!(msm.eval());
}
{
let g = guard.compute_g();
let (msm, _) = guard.clone().use_g(g);
assert!(msm.eval());
}
}
// Check that the verification key has not changed unexpectedly
{
assert_eq!(
format!("{:#?}", pk.vk.pinned()),
r#####"PinnedVerificationKey {
base_modulus: "0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001",
scalar_modulus: "0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001",
domain: PinnedEvaluationDomain {
k: 5,
extended_k: 7,
omega: 0x0cc3380dc616f2e1daf29ad1560833ed3baea3393eceb7bc8fa36376929b78cc,
},
cs: PinnedConstraintSystem {
num_fixed_columns: 8,
num_advice_columns: 5,
num_instance_columns: 1,
gates: [
Sum(
Sum(
Sum(
Sum(
Product(
Advice(
0,
),
Fixed(
3,
),
),
Product(
Advice(
1,
),
Fixed(
4,
),
),
),
Product(
Product(
Advice(
0,
),
Advice(
1,
),
),
Fixed(
6,
),
),
),
Scaled(
Product(
Advice(
2,
),
Fixed(
5,
),
),
0x40000000000000000000000000000000224698fc094cf91b992d30ed00000000,
),
),
Product(
Fixed(
2,
),
Product(
Advice(
3,
),
Advice(
4,
),
),
),
),
Product(
Fixed(
7,
),
Sum(
Advice(
0,
),
Scaled(
Instance(
0,
),
0x40000000000000000000000000000000224698fc094cf91b992d30ed00000000,
),
),
),
],
advice_queries: [
(
Column {
index: 1,
column_type: Advice,
},
Rotation(
0,
),
),
(
Column {
index: 2,
column_type: Advice,
},
Rotation(
0,
),
),
(
Column {
index: 3,
column_type: Advice,
},
Rotation(
0,
),
),
(
Column {
index: 4,
column_type: Advice,
},
Rotation(
1,
),
),
(
Column {
index: 0,
column_type: Advice,
},
Rotation(
-1,
),
),
],
instance_queries: [
(
Column {
index: 0,
column_type: Instance,
},
Rotation(
0,
),
),
],
fixed_queries: [
(
Column {
index: 6,
column_type: Fixed,
},
Rotation(
0,
),
),
(
Column {
index: 7,
column_type: Fixed,
},
Rotation(
0,
),
),
(
Column {
index: 0,
column_type: Fixed,
},
Rotation(
0,
),
),
(
Column {
index: 2,
column_type: Fixed,
},
Rotation(
0,
),
),
(
Column {
index: 3,
column_type: Fixed,
},
Rotation(
0,
),
),
(
Column {
index: 4,
column_type: Fixed,
},
Rotation(
0,
),
),
(
Column {
index: 1,
column_type: Fixed,
},
Rotation(
0,
),
),
(
Column {
index: 5,
column_type: Fixed,
},
Rotation(
0,
),
),
],
permutations: [
Argument {
columns: [
Column {
index: 1,
column_type: Advice,
},
Column {
index: 2,
column_type: Advice,
},
Column {
index: 3,
column_type: Advice,
},
],
},
Argument {
columns: [
Column {
index: 1,
column_type: Advice,
},
Column {
index: 2,
column_type: Advice,
},
Column {
index: 3,
column_type: Advice,
},
],
},
],
lookups: [
Argument {
input_columns: [
Column {
index: 1,
column_type: Advice,
},
],
table_columns: [
Column {
index: 6,
column_type: Fixed,
},
],
},
Argument {
input_columns: [
Column {
index: 1,
column_type: Advice,
},
Column {
index: 2,
column_type: Advice,
},
],
table_columns: [
Column {
index: 6,
column_type: Fixed,
},
Column {
index: 7,
column_type: Fixed,
},
],
},
],
},
fixed_commitments: [
(0x3710f15f98bf0a7421343fdf390b9519506c67431a5c78678fbcc4db815c8547, 0x0a3c77f30ab2a2741b21cd45326c87dc7f7050b1a5e8c181a534b9c84a09cb02),
(0x115235d6bd2467772db857d66e7f44837cd38bb6ac0c7b412c2997dd47cbbc3c, 0x0f339c20c40e10bed476699e2ddb88256092447582b250f329b0cbf6c3f66f17),
(0x2a9ba245dcce53752e1de999b45c975472bb33b58aed8bcebdfcd185627895f0, 0x1252bf596b7acd19625f68845749f6672b99e71238cfabe91ca545103168d0f0),
(0x2a9ba245dcce53752e1de999b45c975472bb33b58aed8bcebdfcd185627895f0, 0x1252bf596b7acd19625f68845749f6672b99e71238cfabe91ca545103168d0f0),
(0x241d6d9c2060ce821d4b05ff2f9566c3947541f3d14a9aabcdb96c19158e8bc7, 0x39582cc6bdb1a4a88e89c050ad6db0ade34f45ec5791b07de6e694e9627ca66a),
(0x1ee805e20232ba31eeae1fa345bd88ac7df81dc43ffb3967a218ea4defc9d17d, 0x2838c83c064d44e87e5c8d05a234ad24d2d4a502a370acb514b430f516c0f0bf),
(0x37ead9904c760201ec4734ef398f0bdb5fe5a5e6e9db19c85e6b5483bdeb0a0b, 0x1dc08c38ed713b14f7a21a891a83b52160a3ffb0dccfbd70db7c7eb235dd193e),
(0x2dc3d20553691216c988ecbb596c4bda329f27d50bd8a7c2fb0d84b423da3cb4, 0x025b40e800020458e15e3a57268562e6c08c6971d71262bd67c72437cfc60b4c),
],
permutations: [
VerifyingKey {
commitments: [
(0x289f468bca3471a3d240169ec65047d0c4bb5e1135e81822523c74b596139fed, 0x1a585c821c71fb49c883859f1389bcae45f17593ddb5f9fee1781b27129e1b06),
(0x096ef96a7725c636e7ca645dfe539694cf2a988da1ca50a468320f419d008054, 0x1ac0b48a254f8e2311081f81aa4a8ff39e37e40aa4c003325f1ac1219d231818),
(0x254c9f0088599aba37607cfd1700c653ef3ec21bfd60b98c3d725678540bc4df, 0x134d9818929589052f3cd80d50140b851db26231b2b784a6b2528a64805598dc),
],
},
VerifyingKey {
commitments: [
(0x289f468bca3471a3d240169ec65047d0c4bb5e1135e81822523c74b596139fed, 0x1a585c821c71fb49c883859f1389bcae45f17593ddb5f9fee1781b27129e1b06),
(0x096ef96a7725c636e7ca645dfe539694cf2a988da1ca50a468320f419d008054, 0x1ac0b48a254f8e2311081f81aa4a8ff39e37e40aa4c003325f1ac1219d231818),
(0x254c9f0088599aba37607cfd1700c653ef3ec21bfd60b98c3d725678540bc4df, 0x134d9818929589052f3cd80d50140b851db26231b2b784a6b2528a64805598dc),
],
},
],
}"#####
);
}
}
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