Gabizon等人2019年论文《PLONK: permutations over lagrange-bases for oecumenical noninteractive arguments of knowledge》。
本文主要针对的代码库为：
- https://github.com/dusk-network/plonk 【rust语言实现，在v0.1.0版本配有examples】
Plonk论文中，constraint 类型分为：【可参考Module dusk_plonk::notes ::prove_verify 文档】
- gate constraint：用于表示add,mul,and,xor等常规gate的运算关系。
- copy constraint：用于表示wires之间的关系，which have equality, from the entire circuit。copy constraint可通过permutation argument来验证。本质上，可通过Verifier给出的随机值来验证wires是否是不重复的。
主要见https://github.com/dusk-network/plonk/tree/master/src/commitment_scheme，采用的曲线为bls12-381。
BlsScalar：表示scalar fieldG1Affine，G1Projective：表示group point，实际commitment采用G1Affine来表示，而借助.into()函数可将G1Projective转换为G1Affine形式。
identity是指无穷远点 ( 0 , 1 , 1 ) (0,1,1) (0,1,1)。满足任意点与该identity相加均为该点。
#[derive(Copy, Clone, Debug, Eq, PartialEq)]
/// Holds a commitment to a polynomial in a form of a `G1Affine` Bls12_381 point.
pub struct Commitment(
/// The commitment is a group element.
pub G1Affine,
);
https://github.com/dusk-network/plonk代码库中，实际实现是参考了 Kate等人2010年论文[kzg10]《Polynomial Commitments∗》中 不具有hiding属性的 3.2节“
P
o
l
y
C
o
m
m
i
t
D
L
PolyCommit_{DL}
PolyCommitDL” 的属性，只是实际实现，采用的不是 symmetric pairing
e
:
G
×
G
→
G
T
e:\mathbb{G}\times\mathbb{G}\rightarrow\mathbb{G}_T
e:G×G→GT，而 unsymmetric pairing
e
:
G
1
×
G
2
→
G
T
e:\mathbb{G}_1\times \mathbb{G}_2\rightarrow \mathbb{G}_T
e:G1×G2→GT。
以degree 为 t t t的单个多项式 ϕ ( x ) ∈ Z p [ x ] \phi(x)\in\mathbb{Z}_p[x] ϕ(x)∈Zp[x]为例，构建polynomial commitment的数学背景为： ϕ ( x ) − ϕ ( z ) x − z \frac{\phi(x)-\phi(z)}{x-z} x−zϕ(x)−ϕ(z) 不存在余数 for z ∈ Z p z\in\mathbb{Z}_p z∈Zp。 g g g为generator of G 1 \mathbb{G}_1 G1， h h h为generator of G 2 \mathbb{G}_2 G2。
- S e t u p ( t ) → p p Setup(t)\rightarrow pp Setup(t)→pp：输出的public parameters p p pp pp中包含 Prover用于生成proof的commit_key 和 Verifier用于verify的opening_key。 public parameters 又称为 structured reference string (srs)。 选择 β ∈ R Z p \beta\in_R\mathbb{Z_p} β∈RZp为secret key S K SK SK，相应的： commit_key为： ( g , g β , ⋯ , g β t ) ∈ G 1 t + 1 (g,g^{\beta},\cdots,g^{\beta^t})\in\mathbb{G}_1^{t+1} (g,gβ,⋯,gβt)∈G1t+1 【Prover使用】 opening_key为： g ∈ G 1 , ( h , h β ) ∈ G 2 2 g\in\mathbb{G}_1,(h,h^{\beta})\in\mathbb{G}_2^2 g∈G1,(h,hβ)∈G22 【Verifier使用】
/// The Public Parameters can also be referred to as the Structured Reference String (SRS).
/// It is available to both the prover and verifier and allows the verifier to
/// efficiently verify and make claims about polynomials up to and including a configured degree.
#[derive(Debug, Clone)]
pub struct PublicParameters {
/// Key used to generate proofs for composed circuits.
pub commit_key: CommitKey,
/// Key used to verify proofs for composed circuits.
pub opening_key: OpeningKey,
}
/// Setup generates the public parameters using a random number generator.
/// This method will in most cases be used for testing and exploration.
/// In reality, a `Trusted party` or a `Multiparty Computation` will used to generate the SRS.
/// Returns an error if the configured degree is less than one.
pub fn setup(
max_degree: usize,
mut rng: &mut R,
) -> Result {
// Cannot commit to constants
if max_degree < 1 {
return Err(Error::DegreeIsZero);
}
// Generate the secret scalar beta
let beta = util::random_scalar(&mut rng);
// Compute powers of beta up to and including beta^max_degree
let powers_of_beta = util::powers_of(&beta, max_degree);
// Powers of G1 that will be used to commit to a specified polynomial
let g = util::random_g1_point(&mut rng);
let powers_of_g: Vec =
util::slow_multiscalar_mul_single_base(&powers_of_beta, g);
assert_eq!(powers_of_g.len(), max_degree + 1);
// Normalise all projective points
let mut normalised_g = vec![G1Affine::identity(); max_degree + 1];
G1Projective::batch_normalize(&powers_of_g, &mut normalised_g);
// Compute beta*G2 element and stored cached elements for verifying multiple proofs.
let h: G2Affine = util::random_g2_point(&mut rng).into();
let beta_h: G2Affine = (h * beta).into();
Ok(PublicParameters {
commit_key: CommitKey {
powers_of_g: normalised_g,
},
opening_key: OpeningKey::new(g.into(), h, beta_h),
})
}
- C o m m i t ( commit_key , ϕ ( x ) ) → C Commit(\text{commit\_key},\phi(x))\rightarrow \mathcal{C} Commit(commit_key,ϕ(x))→C：对多项式 ϕ ( x ) = ∑ j = 0 d e g ( ϕ ) ϕ j x j \phi(x)=\sum_{j=0}^{deg(\phi)}\phi_jx^j ϕ(x)=∑j=0deg(ϕ)ϕjxj进行commit，输出相应的commitment C = g ϕ ( β ) = ∏ j = 0 d e g ( ϕ ) ( g β j ) ϕ j \mathcal{C}=g^{\phi(\beta)}=\prod_{j=0}^{deg(\phi)}(g^{\beta^j})^{\phi_j} C=gϕ(β)=∏j=0deg(ϕ)(gβj)ϕj。
/// Commits to a polynomial returning the corresponding `Commitment`.
///
/// Returns an error if the polynomial's degree is more than the max degree of the commit key.
pub fn commit(&self, polynomial: &Polynomial) -> Result {
// Check whether we can safely commit to this polynomial
self.check_commit_degree_is_within_bounds(polynomial.degree())?;
// Compute commitment
let commitment = msm_variable_base(&self.powers_of_g, &polynomial.coeffs);
Ok(Commitment::from_projective(commitment))
}
- O p e n _ s i n g l e ( commit_key , ϕ ( x ) , v , z ) → π Open\_single(\text{commit\_key},\phi(x), v, z)\rightarrow \pi Open_single(commit_key,ϕ(x),v,z)→π：即证明 ϕ ( z ) = v \phi(z)=v ϕ(z)=v，输出为opening proof π \pi π。 实际的witness为 ψ ( x ) = ϕ ( x ) − ϕ ( z ) x − z \psi(x)=\frac{\phi(x)-\phi(z)}{x-z} ψ(x)=x−zϕ(x)−ϕ(z) 对应的commitment π = g ψ ( β ) \pi=g^{\psi(\beta)} π=gψ(β)。
/// Creates an opening proof that a polynomial `p` was correctly evaluated at p(z) and produced the value
/// `v`. ie v = p(z).
/// Returns an error if the polynomials degree is too large.
pub fn open_single(
&self,
polynomial: &Polynomial,
value: &BlsScalar,
point: &BlsScalar,
) -> Result {
let witness_poly = self.compute_single_witness(polynomial, point);
Ok(Proof {
commitment_to_witness: self.commit(&witness_poly)?,
evaluated_point: *value,
commitment_to_polynomial: self.commit(polynomial)?,
})
}
/// 这段注释有问题。。。。。。
/// For a given polynomial `p` and a point `z`, compute the witness
/// for p(z) using Ruffini's method for simplicity.
/// The Witness is the quotient of f(x) - f(z) / x-z.
/// However we note that the quotient polynomial is invariant under the value f(z)
/// ie. only the remainder changes. We can therefore compute the witness as f(x) / x - z
/// and only use the remainder term f(z) during verification.
pub fn compute_single_witness(&self, polynomial: &Polynomial, point: &BlsScalar) -> Polynomial {
// Computes `f(x) / x-z`, returning it as the witness poly
polynomial.ruffini(*point)
}
- V e r i f y E v a l _ s i n g l e ( opening_key , π , z , v , C ) → 0 or 1 VerifyEval\_single(\text{opening\_key},\pi,z,v,\mathcal{C})\rightarrow 0\text{ or } 1 VerifyEval_single(opening_key,π,z,v,C)→0 or 1：Verifier根据Prover提供的proof和之前对polynomial的commitment C \mathcal{C} C 来确认相应的evaluation是否正确。 即仅需验证： e ( C / g v , h ) ⋅ e ( π , h ( z − β ) ) = 1 e(\mathcal{C}/g^v, h)\cdot e(\pi,h^{(z-\beta)})=1 e(C/gv,h)⋅e(π,h(z−β))=1 是否成立即可。
/// Checks that a polynomial `p` was evaluated at a point `z` and returned the value specified `v`.
/// ie. v = p(z).
pub fn check(&self, point: BlsScalar, proof: Proof) -> bool {
let inner_a: G1Affine =
(proof.commitment_to_polynomial.0 - (self.g * proof.evaluated_point)).into();
let inner_b: G2Affine = (self.beta_h - (self.h * point)).into();
let prepared_inner_b = G2Prepared::from(-inner_b);
let pairing = dusk_bls12_381::multi_miller_loop(&[
(&inner_a, &self.prepared_h),
(&proof.commitment_to_witness.0, &prepared_inner_b),
])
.final_exponentiation();
pairing == dusk_bls12_381::Gt::identity()
}
[kzg10]单个多项式 polynomial commitment scheme示例为：
#[test]
fn test_basic_commit() {
let degree = 25;
let (proving_key, opening_key) = setup_test(degree);
let point = BlsScalar::from(10);
let poly = Polynomial::rand(degree, &mut rand::thread_rng());
let value = poly.evaluate(&point);
let proof = proving_key.open_single(&poly, &value, &point).unwrap();
let ok = opening_key.check(point, proof);
assert!(ok);
}
proof 个数为
O
(
1
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O(1)
O(1)，与points个数无关。 
具体测试用例为：
#[test]
fn test_aggregate_witness() {
let max_degree = 27;
let (proving_key, opening_key) = setup_test(max_degree);
let point = BlsScalar::from(10);
// Committer's View
let aggregated_proof = {
// Compute secret polynomials and their evaluations
let poly_a = Polynomial::rand(25, &mut rand::thread_rng());
let poly_a_eval = poly_a.evaluate(&point);
let poly_b = Polynomial::rand(26 + 1, &mut rand::thread_rng());
let poly_b_eval = poly_b.evaluate(&point);
let poly_c = Polynomial::rand(27, &mut rand::thread_rng());
let poly_c_eval = poly_c.evaluate(&point);
proving_key
.open_multiple(
&[poly_a, poly_b, poly_c],
vec![poly_a_eval, poly_b_eval, poly_c_eval],
&point,
&mut Transcript::new(b"agg_flatten"),
)
.unwrap()
};
// Verifier's View
let ok = {
let flattened_proof = aggregated_proof.flatten(&mut Transcript::new(b"agg_flatten"));
opening_key.check(point, flattened_proof)
};
assert!(ok);
}
/// Creates an opening proof that multiple polynomials were evaluated at the same point
/// and that each evaluation produced the correct evaluation point.
/// Returns an error if any of the polynomial's degrees are too large.
pub fn open_multiple(
&self,
polynomials: &[Polynomial],
evaluations: Vec,
point: &BlsScalar,
transcript: &mut Transcript,
) -> Result {
// Commit to polynomials
let mut polynomial_commitments = Vec::with_capacity(polynomials.len());
for poly in polynomials.iter() {
polynomial_commitments.push(self.commit(poly)?)
}
// Compute the aggregate witness for polynomials
let witness_poly = self.compute_aggregate_witness(polynomials, point, transcript);
// Commit to witness polynomial
let witness_commitment = self.commit(&witness_poly)?;
let aggregate_proof = AggregateProof {
commitment_to_witness: witness_commitment,
evaluated_points: evaluations,
commitments_to_polynomials: polynomial_commitments,
};
Ok(aggregate_proof)
}
/// Computes a single witness for multiple polynomials at the same point, by taking
/// a random linear combination of the individual witnesses.
/// We apply the same optimisation mentioned in when computing each witness; removing f(z).
pub(crate) fn compute_aggregate_witness(
&self,
polynomials: &[Polynomial],
point: &BlsScalar,
transcript: &mut Transcript,
) -> Polynomial {
let challenge = transcript.challenge_scalar(b"aggregate_witness");
let powers = util::powers_of(&challenge, polynomials.len() - 1);
assert_eq!(powers.len(), polynomials.len());
let numerator: Polynomial = polynomials
.iter()
.zip(powers.iter())
.map(|(poly, challenge)| poly * challenge)
.sum();
numerator.ruffini(*point)
}
Verifier 进行 flatten的主要作用为：
- 获取相同的随机值 γ \gamma γ，计算 ( 1 , γ , ⋯ , γ t − 1 ) (1,\gamma,\cdots,\gamma^{t-1}) (1,γ,⋯,γt−1)。
flattened_poly_commitments对应为 F F F，flattened_poly_evaluations对应为 v v v。
/// Flattens an `AggregateProof` into a `Proof`.
/// The transcript must have the same view as the transcript that was used to aggregate the witness in the proving stage.
pub fn flatten(&self, transcript: &mut Transcript) -> Proof {
let challenge = transcript.challenge_scalar(b"aggregate_witness");
let powers = powers_of(&challenge, self.commitments_to_polynomials.len() - 1);
// Flattened polynomial commitments using challenge
let flattened_poly_commitments: G1Projective = self
.commitments_to_polynomials
.iter()
.zip(powers.iter())
.map(|(poly, challenge)| poly.0 * challenge)
.sum();
// Flattened evaluation points
let flattened_poly_evaluations: BlsScalar = self
.evaluated_points
.iter()
.zip(powers.iter())
.map(|(eval, challenge)| eval * challenge)
.fold(BlsScalar::zero(), |acc, current_val| acc + current_val);
Proof {
commitment_to_witness: self.commitment_to_witness,
evaluated_point: flattened_poly_evaluations,
commitment_to_polynomial: Commitment::from_projective(flattened_poly_commitments),
}
}
当proof 个数与points个数一致时，即为每个point的evaluation生成一个proof，对应示例为：
#[test]
fn test_batch_verification() {
let degree = 25;
let (proving_key, vk) = setup_test(degree);
let point_a = BlsScalar::from(10);
let point_b = BlsScalar::from(11);
// Compute secret polynomial a
let poly_a = Polynomial::rand(degree, &mut rand::thread_rng());
let value_a = poly_a.evaluate(&point_a);
let proof_a = proving_key
.open_single(&poly_a, &value_a, &point_a)
.unwrap();
assert!(vk.check(point_a, proof_a));
// Compute secret polynomial b
let poly_b = Polynomial::rand(degree, &mut rand::thread_rng());
let value_b = poly_b.evaluate(&point_b);
let proof_b = proving_key
.open_single(&poly_b, &value_b, &point_b)
.unwrap();
assert!(vk.check(point_b, proof_b));
assert!(vk
.batch_check(
&[point_a, point_b],
&[proof_a, proof_b],
&mut Transcript::new(b""),
)
.is_ok());
}
相应batch算法实现为：
- V e r i f y E v a l _ b a t c h ( opening_key , π 1 , ⋯ , π n , z 1 , ⋯ , z n , v 1 . ⋯ , v n , C 1 , ⋯ , C n ) → 0 or 1 VerifyEval\_batch(\text{opening\_key},\pi_1,\cdots,\pi_n,z_1,\cdots,z_n,v_1.\cdots,v_n,\mathcal{C}_1,\cdots,\mathcal{C}_n)\rightarrow 0\text{ or } 1 VerifyEval_batch(opening_key,π1,⋯,πn,z1,⋯,zn,v1.⋯,vn,C1,⋯,Cn)→0 or 1：Verifier根据Prover提供的proofs π 1 , ⋯ , π n \pi_1,\cdots,\pi_n π1,⋯,πn和之前对polynomials f 1 , ⋯ , f n f_1,\cdots,f_n f1,⋯,fn的commitments C 1 , ⋯ , C n \mathcal{C}_1,\cdots,\mathcal{C}_n C1,⋯,Cn 来确认相应的evaluations v 1 = f 1 ( z 1 ) , ⋯ , v n = f n ( z n ) v_1=f_1(z_1),\cdots,v_n=f_n(z_n) v1=f1(z1),⋯,vn=fn(zn)是否正确。 Verifier选择随机值 γ \gamma γ，并计算 ( 1 , γ , ⋯ , γ n − 1 ) (1,\gamma,\cdots,\gamma^{n-1}) (1,γ,⋯,γn−1)。 计算： C s u m = ∑ i = 1 n γ i − 1 ( C i + π i ⋅ z i ) − ∑ i = 1 n γ i − 1 v i = ∑ i = 1 n γ i − 1 β ( f i ( β ) − f i ( z i ) ) β − z i \mathcal{C}_{sum}=\sum_{i=1}^{n}\gamma^{i-1}(\mathcal{C}_{i}+\pi_{i}\cdot z_i)-\sum_{i=1}^n\gamma^{i-1}v_i=\sum_{i=1}^n\gamma^{i-1}\frac{\beta(f_i(\beta)-f_i(z_i))}{\beta-z_i} Csum=∑i=1nγi−1(Ci+πi⋅zi)−∑i=1nγi−1vi=∑i=1nγi−1β−ziβ(fi(β)−fi(zi)) π s u m = − ∑ i = 1 n γ i − 1 π i = − ∑ i = 1 n γ i − 1 f i ( β ) − f i ( z i ) β − z i \pi_{sum}=-\sum_{i=1}^{n}\gamma^{i-1}\pi_i=-\sum_{i=1}^{n}\gamma^{i-1}\frac{f_i(\beta)-f_i(z_i)}{\beta-z_i} πsum=−∑i=1nγi−1πi=−∑i=1nγi−1β−zifi(β)−fi(zi) 即仅需验证： e ( C s u m , h ) ⋅ e ( π s u m , h β ) = 1 e(\mathcal{C}_{sum}, h)\cdot e(\pi_{sum},h^{\beta})=1 e(Csum,h)⋅e(πsum,hβ)=1 是否成立即可。
/// Checks whether a batch of polynomials evaluated at different points, returned their specified value.
pub fn batch_check(
&self,
points: &[BlsScalar],
proofs: &[Proof],
transcript: &mut Transcript,
) -> Result {
let mut total_c = G1Projective::identity();
let mut total_w = G1Projective::identity();
let challenge = transcript.challenge_scalar(b"batch"); // XXX: Verifier can add their own randomness at this point
let powers = util::powers_of(&challenge, proofs.len() - 1);
// Instead of multiplying g and gamma_g in each turn, we simply accumulate
// their coefficients and perform a final multiplication at the end.
let mut g_multiplier = BlsScalar::zero();
for ((proof, challenge), point) in proofs.iter().zip(powers).zip(points) {
let mut c = G1Projective::from(proof.commitment_to_polynomial.0);
let w = proof.commitment_to_witness.0;
c += w * point;
g_multiplier += challenge * proof.evaluated_point;
total_c += c * challenge;
total_w += w * challenge;
}
total_c -= self.g * g_multiplier;
let affine_total_w = G1Affine::from(-total_w);
let affine_total_c = G1Affine::from(total_c);
let pairing = dusk_bls12_381::multi_miller_loop(&[
(&affine_total_w, &self.prepared_beta_h),
(&affine_total_c, &self.prepared_h),
])
.final_exponentiation();
if pairing != dusk_bls12_381::Gt::identity() {
return Err(Error::PairingCheckFailure);
};
Ok(())
}
实际，Plonk中针对的场景为，evaluation points中实际仅有2组不同，分别表示为 z , z ′ z,z' z,z′，具体示例为：【即proof 个数与不同的points个数一致】
#[test]
fn test_batch_with_aggregation() {
let max_degree = 28;
let (proving_key, opening_key) = setup_test(max_degree);
let point_a = BlsScalar::from(10);
let point_b = BlsScalar::from(11);
// Committer's View
let (aggregated_proof, single_proof) = {
// Compute secret polynomial and their evaluations
let poly_a = Polynomial::rand(25, &mut rand::thread_rng());
let poly_a_eval = poly_a.evaluate(&point_a);
let poly_b = Polynomial::rand(26, &mut rand::thread_rng());
let poly_b_eval = poly_b.evaluate(&point_a);
let poly_c = Polynomial::rand(27, &mut rand::thread_rng());
let poly_c_eval = poly_c.evaluate(&point_a);
let poly_d = Polynomial::rand(28, &mut rand::thread_rng());
let poly_d_eval = poly_d.evaluate(&point_b);
let aggregated_proof = proving_key
.open_multiple(
&[poly_a, poly_b, poly_c],
vec![poly_a_eval, poly_b_eval, poly_c_eval],
&point_a,
&mut Transcript::new(b"agg_batch"),
)
.unwrap();
let single_proof = proving_key
.open_single(&poly_d, &poly_d_eval, &point_b)
.unwrap();
(aggregated_proof, single_proof)
};
// Verifier's View
let ok = {
let mut transcript = Transcript::new(b"agg_batch");
let flattened_proof = aggregated_proof.flatten(&mut transcript);
opening_key.batch_check(
&[point_a, point_b],
&[flattened_proof, single_proof],
&mut transcript,
)
};
assert!(ok.is_ok());
}
(I)FFT 仅适于 multiplicative subgroup of size that is a power-of-2。 具体代码见https://github.com/3for/plonk/blob/master/src/fft/ 目录：
/// Performs O(nlogn) multiplication of polynomials if F is smooth.
impl Mul for &'b Polynomial {
type Output = Polynomial;
#[inline]
fn mul(self, other: &'a Polynomial) -> Polynomial {
if self.is_zero() || other.is_zero() {
Polynomial::zero()
} else {
let domain = EvaluationDomain::new(self.coeffs.len() + other.coeffs.len())
.expect("field is not smooth enough to construct domain");
let mut self_evals = Evaluations::from_vec_and_domain(domain.fft(&self.coeffs), domain);
let other_evals = Evaluations::from_vec_and_domain(domain.fft(&other.coeffs), domain);
self_evals *= &other_evals;
self_evals.interpolate()
}
}
}
/// Defines a domain over which finite field (I)FFTs can be performed. Works
/// only for fields that have a large multiplicative subgroup of size that is
/// a power-of-2.
#[derive(Copy, Clone, Eq, PartialEq)]
pub struct EvaluationDomain {
/// The size of the domain.
pub size: u64,
/// `log_2(self.size)`.
pub log_size_of_group: u32,
/// Size of the domain as a field element.
pub size_as_field_element: BlsScalar,
/// Inverse of the size in the field.
pub size_inv: BlsScalar,
/// A generator of the subgroup.
pub group_gen: BlsScalar,
/// Inverse of the generator of the subgroup.
pub group_gen_inv: BlsScalar,
/// Multiplicative generator of the finite field.
pub generator_inv: BlsScalar,
}
coset FFT的主要作用是将n域的系数扩展至4n域内，从而加速 点值表示方式下 的求商运算 得 quotient polynomial：
t
(
X
)
=
z
(
X
)
−
z
H
(
X
)
z
H
(
X
)
=
f
(
X
)
X
n
−
1
t(X)=\frac{z(X)-z_H(X)}{z_H(X)}=\frac{f(X)}{X^n-1}
t(X)=zH(X)z(X)−zH(X)=Xn−1f(X)。缺点是增加了Prover需要维护的信息量，由n扩充至4n。 
假设order为 p p p的有限域内，相应的generator为 g g g，有 g p − 1 ≡ 1 m o d p g^{p-1}\equiv 1\mod p gp−1≡1modp。 以domain_n n = 8 n=8 n=8为例，假设有 w 8 ≡ 1 m o d p w^8\equiv 1\mod p w8≡1modp。 则相应的domain_4n 中有， v 32 ≡ 1 m o d p v^{32}\equiv 1\mod p v32≡1modp。
let q_m_eval_4n =
Evaluations::from_vec_and_domain(domain_4n.coset_fft(&selectors.q_m), domain_4n);
其中domain_n的q_m多项式：
- 以系数表示为 ( c 0 , c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 ) (c_0,c_1,c_2,c_3,c_4,c_5,c_6,c_7) (c0,c1,c2,c3,c4,c5,c6,c7)；
- 以点值表示为 ( ( 1 , q m 0 ) , ( w , q m 1 ) , ( w 2 , q m 2 ) , ⋯ , ( w 7 , q m 7 ) ) ((1,qm_0), (w,qm_1),(w^2,qm_2),\cdots,(w^7,qm_7)) ((1,qm0),(w,qm1),(w2,qm2),⋯,(w7,qm7))
coset_fft()函数是将系数扩充至domain_4n，多项式q_m_4n：
- 以系数表示为 ( c 0 , c 1 g , c 2 g 2 , c 3 g 3 , c 4 g 4 , c 5 g 5 , c 6 g 6 , c 7 g 7 , 0 , ⋯ , 0 ) (c_0,c_1g,c_2g^2,c_3g^3,c_4g^4,c_5g^5,c_6g^6,c_7g^7,0,\cdots,0) (c0,c1g,c2g2,c3g3,c4g4,c5g5,c6g6,c7g7,0,⋯,0)【总长度为8*4=32】
- 以点值表示为
(
(
1
,
q
m
4
n
0
)
,
(
v
,
q
m
4
n
1
)
,
⋯
,
(
v
31
,
q
m
4
n
31
)
)
((1,qm4n_0),(v,qm4n_1),\cdots,(v^{31},qm4n_{31}))
((1,qm4n0),(v,qm4n1),⋯,(v31,qm4n31))。【代码中
q_m_eval_4n即为相应的点值表示。】
// Compute 4n evaluations for X^n -1
v_h_coset_4n: domain_4n.compute_vanishing_poly_over_coset(domain.size() as u64),
compute_vanishing_poly_over_coset()中的 evaluations v_h 为：
[
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
g
8
−
1
g
8
v
8
−
1
g
8
v
16
−
1
g
8
v
24
−
1
]
\begin{bmatrix} g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1\\ g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1\\ g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1\\ g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1\\ g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1\\ g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1\\ g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1\\ g^8-1 & g^8v^8-1 & g^8v^{16}-1 & g^8v^{24}-1 \end{bmatrix}
⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡g8−1g8−1g8−1g8−1g8−1g8−1g8−1g8−1g8v8−1g8v8−1g8v8−1g8v8−1g8v8−1g8v8−1g8v8−1g8v8−1g8v16−1g8v16−1g8v16−1g8v16−1g8v16−1g8v16−1g8v16−1g8v16−1g8v24−1g8v24−1g8v24−1g8v24−1g8v24−1g8v24−1g8v24−1g8v24−1⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤
相应的插值点为： [ 1 v v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 12 v 13 v 14 v 15 v 16 v 17 v 18 v 19 v 20 v 21 v 22 v 23 v 24 v 25 v 26 v 27 v 28 v 29 v 30 v 31 ] \begin{bmatrix} 1 & v & v^2 & v^3\\ v^4 & v^5 & v^6 & v^7\\ v^8 & v^9 & v^{10} & v^{11}\\ v^{12} & v^{13} & v^{14} & v^{15}\\ v^{16} & v^{17} & v^{18} & v^{19}\\ v^{20} & v^{21} & v^{22} & v^{23}\\ v^{24} & v^{25} & v^{26} & v^{27}\\ v^{28} & v^{29} & v^{30} & v^{31} \end{bmatrix} ⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡1v4v8v12v16v20v24v28vv5v9v13v17v21v25v29v2v6v10v14v18v22v26v30v3v7v11v15v19v23v27v31⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤
4. permutation argument具体代码实现见：https://github.com/3for/plonk/blob/master/src/permutation/permutation.rs。
粒度在Variable（以此为键值），每个变量可为多个门的左侧输入、右侧输入、输出或者是Fourth值，这些信息维护在相应的Vec中。
/// Permutation provides the necessary state information and functions
/// to create the permutation polynomial. In the literature, Z(X) is the "accumulator",
/// this is what this codebase calls the permutation polynomial.
#[derive(Debug)]
pub struct Permutation {
// Maps a variable to the wires that it is associated to
pub(crate) variable_map: HashMap,
}
/// The value is a reference to the actual value that was added to the constraint system
#[derive(Debug, Eq, PartialEq, Clone, Copy, Hash)]
pub struct Variable(pub(crate) usize);
impl Into for Variable {
fn into(self) -> (BlsScalar, Variable) {
(BlsScalar::one(), self)
}
}
/// Stores the data for a specific wire in an arithmetic circuit
/// This data is the gate index and the type of wire
/// Left(1) signifies that this wire belongs to the first gate and is the left wire
#[derive(Copy, Clone, PartialEq, Eq, Debug)]
pub enum WireData {
/// Left Wire of n'th gate
Left(usize),
/// Right Wire of n'th gate
Right(usize),
/// Output Wire of n'th gate
Output(usize),
/// Fourth Wire of n'th gate
Fourth(usize),
}
假设总共有 n n n个gate，令 ω \omega ω为 n n n-th root of unity，即在scalar 域内，满足 ω n = 1 \omega^n=1 ωn=1。 令 H = { 1 , ω , ⋯ , ω n − 1 } H=\{1,\omega,\cdots,\omega^{n-1}\} H={1,ω,⋯,ωn−1}，取 k 1 , k 2 , k 3 ∈ F k_1,k_2,k_3\in\mathbb{F} k1,k2,k3∈F，满足 H , k 1 ⋅ H , k 2 ⋅ H , k 3 ⋅ H H,k_1\cdot H,k_2\cdot H, k_3\cdot H H,k1⋅H,k2⋅H,k3⋅H 为distinct cosets of H H H in F ∗ \mathbb{F}^* F∗。
/// Constants used in the permutation argument to ensure that the wire subsets are disjoint.
pub(crate) const K1: BlsScalar = BlsScalar::from_raw([7, 0, 0, 0]);
pub(crate) const K2: BlsScalar = BlsScalar::from_raw([13, 0, 0, 0]);
pub(crate) const K3: BlsScalar = BlsScalar::from_raw([17, 0, 0, 0]);
以
n
=
4
n=4
n=4为例，具体见test_permutation_compute_sigmas_only_left_wires，采用lagrange插值，插值点为
(
1
,
ω
,
⋯
,
ω
3
)
(1,\omega,\cdots,\omega^3)
(1,ω,⋯,ω3)，使得：
- 对于Left wire有： L 0 = L ( 1 ) = 1 , L 1 = L ( ω ) = ω , L 2 = L ( ω 2 ) = ω 2 , L 3 = L ( ω 3 ) = ω 3 L_0=L(1)=1,L_1=L(\omega)=\omega,L_2=L(\omega^2)=\omega^2,L_3=L(\omega^3)=\omega^3 L0=L(1)=1,L1=L(ω)=ω,L2=L(ω2)=ω2,L3=L(ω3)=ω3。
- 对于Right wire有： R 0 = R ( 1 ) = K 1 , R 1 = R ( ω ) = K 1 ω , R 2 = R ( ω 2 ) = K 1 ω 2 , R 3 = R ( ω 3 ) = K 1 ω 3 R_0=R(1)=K_1,R_1=R(\omega)=K_1\omega,R_2=R(\omega^2)=K_1\omega^2,R_3=R(\omega^3)=K_1\omega^3 R0=R(1)=K1,R1=R(ω)=K1ω,R2=R(ω2)=K1ω2,R3=R(ω3)=K1ω3。
- 对于Output wire有： O 0 = O ( 1 ) = K 2 , O 1 = O ( ω ) = K 2 ω , O 2 = O ( ω 2 ) = K 2 ω 2 , O 3 = O ( ω 3 ) = K 2 ω 3 O_0=O(1)=K_2,O_1=O(\omega)=K_2\omega,O_2=O(\omega^2)=K_2\omega^2,O_3=O(\omega^3)=K_2\omega^3 O0=O(1)=K2,O1=O(ω)=K2ω,O2=O(ω2)=K2ω2,O3=O(ω3)=K2ω3。
- 对于Fourth wire有： F 0 = F ( 1 ) = K 3 , F 1 = F ( ω ) = K 3 ω , F 2 = F ( ω 2 ) = K 3 ω 2 , F 3 = F ( ω 3 ) = K 3 ω 3 F_0=F(1)=K_3,F_1=F(\omega)=K_3\omega,F_2=F(\omega^2)=K_3\omega^2,F_3=F(\omega^3)=K_3\omega^3 F0=F(1)=K3,F1=F(ω)=K3ω,F2=F(ω2)=K3ω2,F3=F(ω3)=K3ω3。
permutation是根据实际各个wire与Variable之间的逻辑关系，进行了调整后再插值：
let num_wire_mappings = 4;
// ！此处即为实际各个wire与Variable之间的逻辑关系！
// Add four wire mappings
perm.add_variables_to_map(var_zero, var_zero, var_five, var_nine, 0);
perm.add_variables_to_map(var_zero, var_two, var_six, var_nine, 1);
perm.add_variables_to_map(var_zero, var_three, var_seven, var_nine, 2);
perm.add_variables_to_map(var_zero, var_four, var_eight, var_nine, 3);
/*
var_zero = {L0, R0, L1, L2, L3}
var_two = {R1}
var_three = {R2}
var_four = {R3}
var_five = {O0}
var_six = {O1}
var_seven = {O2}
var_eight = {O3}
var_nine = {F0, F1, F2, F3}
Left_sigma = {R0, L2, L3, L0}
Right_sigma = {L1, R1, R2, R3}
Out_sigma = {O0, O1, O2, O3}
Fourth_sigma = {F1, F2, F3, F0}
*/
let sigmas = perm.compute_sigma_permutations(num_wire_mappings);
let left_sigma = &sigmas[0];
let right_sigma = &sigmas[1];
let out_sigma = &sigmas[2];
let fourth_sigma = &sigmas[3];
// Check the left sigma polynomial
assert_eq!(left_sigma[0], WireData::Right(0));
assert_eq!(left_sigma[1], WireData::Left(2));
assert_eq!(left_sigma[2], WireData::Left(3));
assert_eq!(left_sigma[3], WireData::Left(0));
// Check the right sigma polynomial
assert_eq!(right_sigma[0], WireData::Left(1));
assert_eq!(right_sigma[1], WireData::Right(1));
assert_eq!(right_sigma[2], WireData::Right(2));
assert_eq!(right_sigma[3], WireData::Right(3));
// Check the output sigma polynomial
assert_eq!(out_sigma[0], WireData::Output(0));
assert_eq!(out_sigma[1], WireData::Output(1));
assert_eq!(out_sigma[2], WireData::Output(2));
assert_eq!(out_sigma[3], WireData::Output(3));
// Check the output sigma polynomial
assert_eq!(fourth_sigma[0], WireData::Fourth(1));
assert_eq!(fourth_sigma[1], WireData::Fourth(2));
assert_eq!(fourth_sigma[2], WireData::Fourth(3));
assert_eq!(fourth_sigma[3], WireData::Fourth(0));
let domain = EvaluationDomain::new(num_wire_mappings).unwrap();
let w: Fr = domain.group_gen;
let w_squared = w.pow(&[2, 0, 0, 0]);
let w_cubed = w.pow(&[3, 0, 0, 0]);
// Check the left sigmas have been encoded properly
// Left_sigma = {R0, L2, L3, L0}
// Should turn into {1 * K1, w^2, w^3, 1}
let encoded_left_sigma = perm.compute_permutation_lagrange(left_sigma, &domain);
assert_eq!(encoded_left_sigma[0], Fr::one() * &K1);
assert_eq!(encoded_left_sigma[1], w_squared);
assert_eq!(encoded_left_sigma[2], w_cubed);
assert_eq!(encoded_left_sigma[3], Fr::one());
// Check the right sigmas have been encoded properly
// Right_sigma = {L1, R1, R2, R3}
// Should turn into {w, w * K1, w^2 * K1, w^3 * K1}
let encoded_right_sigma = perm.compute_permutation_lagrange(right_sigma, &domain);
assert_eq!(encoded_right_sigma[0], w);
assert_eq!(encoded_right_sigma[1], w * &K1);
assert_eq!(encoded_right_sigma[2], w_squared * &K1);
assert_eq!(encoded_right_sigma[3], w_cubed * &K1);
// Check the output sigmas have been encoded properly
// Out_sigma = {O0, O1, O2, O3}
// Should turn into {1 * K2, w * K2, w^2 * K2, w^3 * K2}
let encoded_output_sigma = perm.compute_permutation_lagrange(out_sigma, &domain);
assert_eq!(encoded_output_sigma[0], Fr::one() * &K2);
assert_eq!(encoded_output_sigma[1], w * &K2);
assert_eq!(encoded_output_sigma[2], w_squared * &K2);
assert_eq!(encoded_output_sigma[3], w_cubed * &K2);
// Check the fourth sigmas have been encoded properly
// Out_sigma = {F1, F2, F3, F0}
// Should turn into {w * K3, w^2 * K3, w^3 * K3, 1 * K3}
let encoded_fourth_sigma = perm.compute_permutation_lagrange(fourth_sigma, &domain);
assert_eq!(encoded_fourth_sigma[0], w * &K3);
assert_eq!(encoded_fourth_sigma[1], w_squared * &K3);
assert_eq!(encoded_fourth_sigma[2], w_cubed * &K3);
assert_eq!(encoded_fourth_sigma[3], K3);
fn compute_permutation_lagrange(
&self,
sigma_mapping: &[WireData],
domain: &EvaluationDomain,
) -> Vec {
let roots: Vec = domain.elements().collect();
let lagrange_poly: Vec = sigma_mapping
.iter()
.map(|x| match x {
WireData::Left(index) => {
let root = &roots[*index];
*root
}
WireData::Right(index) => {
let root = &roots[*index];
K1 * root
}
WireData::Output(index) => {
let root = &roots[*index];
K2 * root
}
WireData::Fourth(index) => {
let root = &roots[*index];
K3 * root
}
})
.collect();
lagrange_poly
}
test_correct_permutation_poly中：【仍然以
n
=
4
n=4
n=4为例，
ω
4
=
1
\omega^4=1
ω4=1】
- numerator_components为： ( n 0 , n 1 , n 2 , n 3 ) (n_0,n_1,n_2,n_3) (n0,n1,n2,n3)
- denominator_components为： ( d 0 , d 1 , d 2 , d 3 ) (d_0,d_1,d_2,d_3) (d0,d1,d2,d3)
- 根据permutation，有： n 0 n 1 n 2 n 3 d 0 d 1 d 2 d 3 = 1 \frac{n_0n_1n_2n_3}{d_0d_1d_2d_3}=1 d0d1d2d3n0n1n2n3=1。
- z_vec为： ( 1 , n 0 d 0 , n 0 n 1 d 0 d 1 , n 0 n 1 n 2 d 0 d 1 d 2 ) (1,\frac{n_0}{d_0},\frac{n_0n_1}{d_0d_1},\frac{n_0n_1n_2}{d_0d_1d_2}) (1,d0n0,d0d1n0n1,d0d1d2n0n1n2)
- z_poly多项式的点值表示为：
(
(
1
,
1
)
,
(
ω
,
n
0
d
0
)
,
(
ω
2
,
n
0
n
1
d
0
d
1
)
,
(
ω
3
,
n
0
n
1
n
2
d
0
d
1
d
2
)
)
=
(
(
1
,
Z
(
1
)
)
,
(
ω
,
Z
(
ω
)
)
,
(
ω
2
,
Z
(
ω
2
)
)
,
(
ω
3
,
Z
(
ω
3
)
)
)
((1,1),(\omega,\frac{n_0}{d_0}),(\omega^2,\frac{n_0n_1}{d_0d_1}),(\omega^3,\frac{n_0n_1n_2}{d_0d_1d_2}))=((1,Z(1)),(\omega,Z(\omega)),(\omega^2,Z(\omega^2)),(\omega^3,Z(\omega^3)))
((1,1),(ω,d0n0),(ω2,d0d1n0n1),(ω3,d0d1d2n0n1n2))=((1,Z(1)),(ω,Z(ω)),(ω2,Z(ω2)),(ω3,Z(ω3)))。【
compute_permutation_poly()函数实际计算的即为z_poly多项式。】 - 从而有
Z
(
ω
i
)
Z
(
ω
i
+
1
)
=
d
i
n
i
\frac{Z(\omega^i)}{Z(\omega^{i+1})}=\frac{d_i}{n_i}
Z(ωi+1)Z(ωi)=nidi，for
1
≤
i
<
n
1\leq i BlsScalar::zero(),
_ => unreachable!(),
})
+ qrange
* (delta(c - four * d)
+ delta(b - four * c)
+ delta(a - four * b)
+ delta(d_next - four * a));
assert_eq!(k, BlsScalar::zero(), "Check failed at gate {}", i,);
其中
delta函数用于保证f值仅为0 或 1 或 2 或 3，使得最终delta函数的输出为0值：【原因在于logic_gate中是同时对两个bit进行and或xor操作。与ebfull/halo中的xor方案不同，ebfull中是对单个bit进行and或xor操作。】// Computes f(f-1)(f-2)(f-3) let delta = |f: BlsScalar| -> BlsScalar { let f_1 = f - BlsScalar::one(); let f_2 = f - BlsScalar::from(2); let f_3 = f - BlsScalar::from(3); f * f_1 * f_2 * f_3 };具体的结构体设计为：
/// A composer is a circuit builder /// and will dictate how a circuit is built /// We will have a default Composer called `StandardComposer` #[derive(Debug)] pub struct StandardComposer { // n represents the number of arithmetic gates in the circuit pub(crate) n: usize, // Selector vectors // // Multiplier selector pub(crate) q_m: Vec, // Left wire selector pub(crate) q_l: Vec, // Right wire selector pub(crate) q_r: Vec, // Output wire selector pub(crate) q_o: Vec, // Fourth wire selector pub(crate) q_4: Vec, // Constant wire selector pub(crate) q_c: Vec, // Arithmetic wire selector pub(crate) q_arith: Vec, // Range selector pub(crate) q_range: Vec, // Logic selector pub(crate) q_logic: Vec, // Fixed base group addition selector pub(crate) q_fixed_group_add: Vec, // Variable base group addition selector pub(crate) q_variable_group_add: Vec, /// Public inputs vector pub public_inputs: Vec, // Witness vectors pub(crate) w_l: Vec, pub(crate) w_r: Vec, pub(crate) w_o: Vec, pub(crate) w_4: Vec, /// A zero variable that is a part of the circuit description. /// We reserve a variable to be zero in the system /// This is so that when a gate only uses three wires, we set the fourth wire to be /// the variable that references zero pub(crate) zero_var: Variable, // These are the actual variable values // N.B. They should not be exposed to the end user once added into the composer pub(crate) variables: HashMap, pub(crate) perm: Permutation, }对于常规的3-wire门，表示为：【其中 p i pi pi 表示 public inputs。】
/// Adds a width-3 poly gate. /// This gate gives total freedom to the end user to implement the corresponding /// circuits in the most optimized way possible because the under has access to the /// whole set of variables, as well as selector coefficients that take part in the /// computation of the gate equation. /// /// The final constraint added will force the following: /// `(a * b) * q_m + a * q_l + b * q_r + q_c + PI + q_o * c = 0`. pub fn poly_gate( &mut self, a: Variable, b: Variable, c: Variable, q_m: BlsScalar, q_l: BlsScalar, q_r: BlsScalar, q_o: BlsScalar, q_c: BlsScalar, pi: BlsScalar, ) -> (Variable, Variable, Variable) { self.w_l.push(a); self.w_r.push(b); self.w_o.push(c); self.w_4.push(self.zero_var); self.q_l.push(q_l); self.q_r.push(q_r); // Add selector vectors self.q_m.push(q_m); self.q_o.push(q_o); self.q_c.push(q_c); self.q_4.push(BlsScalar::zero()); self.q_arith.push(BlsScalar::one()); self.q_range.push(BlsScalar::zero()); self.q_logic.push(BlsScalar::zero()); self.q_fixed_group_add.push(BlsScalar::zero()); self.q_variable_group_add.push(BlsScalar::zero()); self.public_inputs.push(pi); self.perm .add_variables_to_map(a, b, c, self.zero_var, self.n); self.n += 1; (a, b, c) }支持的constraint类型有：
- 1）constant constraint
- 2）equal constraint
- 3）dummy constraint
- 4）multiply constraint
- 5）add constraint
- 6）boolean constraint
- 7）fixed group add constraint
- 8）xor logic constraint
- 9）and logic constraint
- 10）range constraint
constant constraint：即某变量值等于某常数，如
a变量满足a - constant + pi = 0。/// Adds a gate which is designed to constrain a `Variable` to have /// a specific constant value which is sent as a `BlsScalar`. pub fn constrain_to_constant(&mut self, a: Variable, constant: BlsScalar, pi: BlsScalar) { self.poly_gate( a, a, a, BlsScalar::zero(), BlsScalar::one(), BlsScalar::zero(), BlsScalar::zero(), -constant, pi, ); }constant constraint可用于约束circuit中的某个point为某个特定的public point：【验证public info】
5.2 equal constraint/// Represents a JubJub point in the circuit #[derive(Debug, Clone, Copy)] pub struct Point { x: Variable, y: Variable, } /// Asserts that a point in the circuit is equal to a known public point pub fn assert_equal_public_point( &mut self, point: Point, public_point: dusk_jubjub::JubJubAffine, ) { self.constrain_to_constant(point.x, BlsScalar::zero(), -public_point.get_x()); self.constrain_to_constant(point.y, BlsScalar::zero(), -public_point.get_y()); }equal constraint：即某两个变量相等，
a = b。/// Asserts that two variables are the same // XXX: Instead of wasting a gate, we can use the permutation polynomial to do this pub fn assert_equal(&mut self, a: Variable, b: Variable) { self.poly_gate( a, b, self.zero_var, BlsScalar::zero(), BlsScalar::one(), -BlsScalar::one(), BlsScalar::zero(), BlsScalar::zero(), BlsScalar::zero(), ); }equal constraint可用于约束circuit中的某两个point是相等的：【验证private info】
5.3 dummy constraint/// Asserts that a point in the circuit is equal to another point in the circuit pub fn assert_equal_point(&mut self, point_a: Point, point_b: Point) { self.assert_equal(point_a.x, point_b.x); self.assert_equal(point_b.y, point_b.y); }对于有 n n n个gate的circuit，其constraint数量为 n + 3 n+3 n+3，其中那3个为：
- 1）默认设置circuit的第一个变量值为0。
// Reserve the first variable to be zero composer.zero_var = composer.add_witness_to_circuit_description(BlsScalar::zero());- 2）为witness polynomials的blinding属性引入了2个dummy constraint—— 一个constraint用于保证selector polynomials不全为 zero polynomial；另一个constraint用于保证permutation polynomial不为identity polynomial。 目前是通过dummy constraint的方式来为witness polynomials添加blinding factor：
/// This function is used to add a blinding factor to the witness polynomials /// XXX: Split this into two separate functions and document /// XXX: We could add another section to add random witness variables, with selector polynomials all zero pub fn add_dummy_constraints(&mut self) { // Add a dummy constraint so that we do not have zero polynomials self.q_m.push(BlsScalar::from(1)); self.q_l.push(BlsScalar::from(2)); self.q_r.push(BlsScalar::from(3)); self.q_o.push(BlsScalar::from(4)); self.q_c.push(BlsScalar::from(4)); self.q_4.push(BlsScalar::one()); self.q_arith.push(BlsScalar::one()); self.q_range.push(BlsScalar::zero()); self.q_logic.push(BlsScalar::zero()); self.q_fixed_group_add.push(BlsScalar::zero()); self.q_variable_group_add.push(BlsScalar::zero()); self.public_inputs.push(BlsScalar::zero()); let var_six = self.add_input(BlsScalar::from(6)); let var_one = self.add_input(BlsScalar::from(1)); let var_seven = self.add_input(BlsScalar::from(7)); let var_min_twenty = self.add_input(-BlsScalar::from(20)); self.w_l.push(var_six); self.w_r.push(var_seven); self.w_o.push(var_min_twenty); self.w_4.push(var_one); self.perm .add_variables_to_map(var_six, var_seven, var_min_twenty, var_one, self.n); self.n += 1; //Add another dummy constraint so that we do not get the identity permutation self.q_m.push(BlsScalar::from(1)); self.q_l.push(BlsScalar::from(1)); self.q_r.push(BlsScalar::from(1)); self.q_o.push(BlsScalar::from(1)); self.q_c.push(BlsScalar::from(127)); self.q_4.push(BlsScalar::zero()); self.q_arith.push(BlsScalar::one()); self.q_range.push(BlsScalar::zero()); self.q_logic.push(BlsScalar::zero()); self.q_fixed_group_add.push(BlsScalar::zero()); self.q_variable_group_add.push(BlsScalar::zero()); self.public_inputs.push(BlsScalar::zero()); self.w_l.push(var_min_twenty); self.w_r.push(var_six); self.w_o.push(var_seven); self.w_4.push(self.zero_var); self.perm .add_variables_to_map(var_min_twenty, var_six, var_seven, self.zero_var, self.n); self.n += 1; }即对于空circuit，默认3个constraint，相应的测试用例为：
#[test] /// Tests that a circuit initially has 3 gates fn test_initial_circuit_size() { let composer: StandardComposer = StandardComposer::new(); // Circuit size is n+3 because // - We have an extra gate which forces the first witness to be zero. This is used when the advice wire is not being used. // - We have two gates which ensure that the permutation polynomial is not the identity and // - Another gate which ensures that the selector polynomials are not all zeroes assert_eq!(3, composer.circuit_size()); composer.check_circuit_satisfied(); //打印调试信息。feature中增加"print-trace" }相应的circuit gate satisfied 打印信息为：
5.4 multiply constraint-------------------------------------------- #Gate Index = 0 #Selector Polynomials: - qm -> 0000000000000000000000000000000000000000000000000000000000000000 - ql -> 0100000000000000000000000000000000000000000000000000000000000000 - qr -> 0000000000000000000000000000000000000000000000000000000000000000 - q4 -> 0000000000000000000000000000000000000000000000000000000000000000 - qo -> 0000000000000000000000000000000000000000000000000000000000000000 - qc -> 0000000000000000000000000000000000000000000000000000000000000000 - q_arith -> 0100000000000000000000000000000000000000000000000000000000000000 - q_range -> 0000000000000000000000000000000000000000000000000000000000000000 - q_logic -> 0000000000000000000000000000000000000000000000000000000000000000 - q_fixed_group_add -> 0000000000000000000000000000000000000000000000000000000000000000 - q_variable_group_add -> 0000000000000000000000000000000000000000000000000000000000000000 # Witness polynomials: - w_l -> 0000000000000000000000000000000000000000000000000000000000000000 - w_r -> 0000000000000000000000000000000000000000000000000000000000000000 - w_o -> 0000000000000000000000000000000000000000000000000000000000000000 - w_4 -> 0000000000000000000000000000000000000000000000000000000000000000 -------------------------------------------- #Gate Index = 1 #Selector Polynomials: - qm -> 0100000000000000000000000000000000000000000000000000000000000000 - ql -> 0200000000000000000000000000000000000000000000000000000000000000 - qr -> 0300000000000000000000000000000000000000000000000000000000000000 - q4 -> 0100000000000000000000000000000000000000000000000000000000000000 - qo -> 0400000000000000000000000000000000000000000000000000000000000000 - qc -> 0400000000000000000000000000000000000000000000000000000000000000 - q_arith -> 0100000000000000000000000000000000000000000000000000000000000000 - q_range -> 0000000000000000000000000000000000000000000000000000000000000000 - q_logic -> 0000000000000000000000000000000000000000000000000000000000000000 - q_fixed_group_add -> 0000000000000000000000000000000000000000000000000000000000000000 - q_variable_group_add -> 0000000000000000000000000000000000000000000000000000000000000000 # Witness polynomials: - w_l -> 0600000000000000000000000000000000000000000000000000000000000000 - w_r -> 0700000000000000000000000000000000000000000000000000000000000000 - w_o -> edfffffffefffffffe5bfeff02a4bd5305d8a10908d83933487d9d2953a7ed73 - w_4 -> 0100000000000000000000000000000000000000000000000000000000000000 -------------------------------------------- #Gate Index = 2 #Selector Polynomials: - qm -> 0100000000000000000000000000000000000000000000000000000000000000 - ql -> 0100000000000000000000000000000000000000000000000000000000000000 - qr -> 0100000000000000000000000000000000000000000000000000000000000000 - q4 -> 0000000000000000000000000000000000000000000000000000000000000000 - qo -> 0100000000000000000000000000000000000000000000000000000000000000 - qc -> 7f00000000000000000000000000000000000000000000000000000000000000 - q_arith -> 0100000000000000000000000000000000000000000000000000000000000000 - q_range -> 0000000000000000000000000000000000000000000000000000000000000000 - q_logic -> 0000000000000000000000000000000000000000000000000000000000000000 - q_fixed_group_add -> 0000000000000000000000000000000000000000000000000000000000000000 - q_variable_group_add -> 0000000000000000000000000000000000000000000000000000000000000000 # Witness polynomials: - w_l -> edfffffffefffffffe5bfeff02a4bd5305d8a10908d83933487d9d2953a7ed73 - w_r -> 0600000000000000000000000000000000000000000000000000000000000000 - w_o -> 0700000000000000000000000000000000000000000000000000000000000000 - w_4 -> 0000000000000000000000000000000000000000000000000000000000000000支持left wire、right wire、output wire和fourth wire的加法门。 对于mul gate， q l , q r q_l,q_r ql,qr恒为0。
/// Adds a width-3 add gate to the circuit linking the product of the /// provided inputs scaled by the selector coefficient `q_m` with the output /// provided scaled by `q_o`. /// /// Note that this gate requires to provide the actual result of the gate /// (output wire) since it will just add a `mul constraint` to the circuit. pub fn mul_gate( &mut self, a: Variable, b: Variable, c: Variable, q_m: BlsScalar, q_o: BlsScalar, q_c: BlsScalar, pi: BlsScalar, ) -> Variable { self.big_mul_gate(a, b, c, None, q_m, q_o, q_c, BlsScalar::zero(), pi) } /// Adds a width-4 `big_mul_gate` with the left, right and fourth inputs /// and it's scaling factors, computing & returning the output (result) /// `Variable` and adding the corresponding mul constraint. /// /// This type of gate is usually used when we need to have /// the largest amount of performance and the minimum circuit-size /// possible. Since it allows the end-user to setup all of the selector /// coefficients. /// /// Forces `q_m * (w_l * w_r) + w_4 * q_4 + q_c + PI = q_o * w_o`. /// /// `{w_l, w_r, w_o, w_4} = {a, b, c, d}` // XXX: Maybe make these tuples instead of individual field? pub fn big_mul_gate( &mut self, a: Variable, b: Variable, c: Variable, d: Option, q_m: BlsScalar, q_o: BlsScalar, q_c: BlsScalar, q_4: BlsScalar, pi: BlsScalar, ) -> Variable { // Check if advice wire has a value let d = match d { Some(var) => var, None => self.zero_var, }; self.w_l.push(a); self.w_r.push(b); self.w_o.push(c); self.w_4.push(d); // For a mul gate q_L and q_R is zero self.q_l.push(BlsScalar::zero()); self.q_r.push(BlsScalar::zero()); // Add selector vectors self.q_m.push(q_m); self.q_o.push(q_o); self.q_c.push(q_c); self.q_4.push(q_4); self.q_arith.push(BlsScalar::one()); self.q_range.push(BlsScalar::zero()); self.q_logic.push(BlsScalar::zero()); self.q_fixed_group_add.push(BlsScalar::zero()); self.q_variable_group_add.push(BlsScalar::zero()); self.public_inputs.push(pi); self.perm.add_variables_to_map(a, b, c, d, self.n); self.n += 1; c }以及
5.5 add constraint/// Adds a simple and basic addition to the circuit between to `Variable`s /// returning the resulting `Variable`. pub fn mul( &mut self, q_m: BlsScalar, a: Variable, b: Variable, q_c: BlsScalar, pi: BlsScalar, ) -> Variable { self.big_mul(q_m, a, b, None, q_c, pi) } /// Adds a width-4 `big_mul_gate` with the left, right and fourth inputs /// and it's scaling factors, computing & returning the output (result) /// `Variable` and adding the corresponding mul constraint. /// /// This type of gate is usually used when we don't need to have /// the largest amount of performance and the minimum circuit-size /// possible. Since it defaults some of the selector coeffs = 0 in order /// to reduce the verbosity and complexity. /// /// Forces `q_m * (w_l * w_r) + w_4 * q_4 + q_c + PI = w_o(computed by the gate)`. /// /// `{w_l, w_r, w_4} = {a, b, d}` // XXX: This API is not consistent. It should use tuples and not individual fields pub fn big_mul( &mut self, q_m: BlsScalar, a: Variable, b: Variable, q_4_d: Option, q_c: BlsScalar, pi: BlsScalar, ) -> Variable { let q_o = -BlsScalar::one(); // Check if advice wire is available let (q_4, d) = match q_4_d { Some((q_4, var)) => (q_4, var), None => (BlsScalar::zero(), self.zero_var), }; // Compute output wire let a_eval = self.variables[&a]; let b_eval = self.variables[&b]; let d_eval = self.variables[&d]; let c_eval = (q_m * a_eval * b_eval) + (q_4 * d_eval) + q_c + pi; let c = self.add_input(c_eval); self.big_mul_gate(a, b, c, Some(d), q_m, q_o, q_c, q_4, pi) }支持left wire、right wire、output wire和fourth wire的加法门。 对于add gate， q m q_m qm恒为0。
/// Adds a width-3 add gate to the circuit, linking the addition of the /// provided inputs, scaled by the selector coefficients with the output /// provided. pub fn add_gate( &mut self, a: Variable, b: Variable, c: Variable, q_l: BlsScalar, q_r: BlsScalar, q_o: BlsScalar, q_c: BlsScalar, pi: BlsScalar, ) -> Variable { self.big_add_gate(a, b, c, None, q_l, q_r, q_o, BlsScalar::zero(), q_c, pi) } /// Adds a width-4 add gate to the circuit and it's corresponding /// constraint. /// /// This type of gate is usually used when we need to have /// the largest amount of performance and the minimum circuit-size /// possible. Since it allows the end-user to set every selector coefficient /// as scaling value on the gate eq. pub fn big_add_gate( &mut self, a: Variable, b: Variable, c: Variable, d: Option, q_l: BlsScalar, q_r: BlsScalar, q_o: BlsScalar, q_4: BlsScalar, q_c: BlsScalar, pi: BlsScalar, ) -> Variable { // Check if advice wire has a value let d = match d { //若为None，则解析为零变量。 Some(var) => var, None => self.zero_var, }; self.w_l.push(a); self.w_r.push(b); self.w_o.push(c); self.w_4.push(d); // For an add gate, q_m is zero self.q_m.push(BlsScalar::zero()); // Add selector vectors self.q_l.push(q_l); self.q_r.push(q_r); self.q_o.push(q_o); self.q_c.push(q_c); self.q_4.push(q_4); self.q_arith.push(BlsScalar::one()); self.q_range.push(BlsScalar::zero()); self.q_logic.push(BlsScalar::zero()); self.q_fixed_group_add.push(BlsScalar::zero()); self.q_variable_group_add.push(BlsScalar::zero()); self.public_inputs.push(pi); self.perm.add_variables_to_map(a, b, c, d, self.n); self.n += 1; c }以及
5.6 boolean constraint 又称为 binary constraint/// Adds a `big_addition_gate` with the left and right inputs /// and it's scaling factors, computing & returning the output (result) /// `Variable`, and adding the corresponding addition constraint. /// /// This type of gate is usually used when we don't need to have /// the largest amount of performance as well as the minimum circuit-size /// possible. Since it defaults some of the selector coeffs = 0 in order /// to reduce the verbosity and complexity. /// /// Forces `q_l * w_l + q_r * w_r + q_c + PI = w_o(computed by the gate)`. pub fn add( &mut self, q_l_a: (BlsScalar, Variable), q_r_b: (BlsScalar, Variable), q_c: BlsScalar, pi: BlsScalar, ) -> Variable { self.big_add(q_l_a, q_r_b, None, q_c, pi) } /// Adds a `big_addition_gate` with the left, right and fourth inputs /// and it's scaling factors, computing & returning the output (result) /// `Variable` and adding the corresponding addition constraint. /// /// This type of gate is usually used when we don't need to have /// the largest amount of performance and the minimum circuit-size /// possible. Since it defaults some of the selector coeffs = 0 in order /// to reduce the verbosity and complexity. /// /// Forces `q_l * w_l + q_r * w_r + q_4 * w_4 + q_c + PI = w_o(computed by the gate)`. pub fn big_add( &mut self, q_l_a: (BlsScalar, Variable), q_r_b: (BlsScalar, Variable), q_4_d: Option, q_c: BlsScalar, pi: BlsScalar, ) -> Variable { // Check if advice wire is available let (q_4, d) = match q_4_d { Some((q_4, var)) => (q_4, var), None => (BlsScalar::zero(), self.zero_var), }; let (q_l, a) = q_l_a; let (q_r, b) = q_r_b; let q_o = -BlsScalar::one(); // Compute the output wire let a_eval = self.variables[&a]; let b_eval = self.variables[&b]; let d_eval = self.variables[&d]; let c_eval = (q_l * a_eval) + (q_r * b_eval) + (q_4 * d_eval) + q_c + pi; let c = self.add_input(c_eval); self.big_add_gate(a, b, c, Some(d), q_l, q_r, q_o, q_4, q_c, pi) }boolean constraint 又名 binary constraint，是指约束变量值要么为“0”，要么为“1”。 核心思想就是：若 a ( 1 − a ) = 0 a(1-a)=0 a(1−a)=0，即 a a a为0或1。
5.7 fixed group add constraint/// Adds a boolean constraint (also known as binary constraint) where /// the gate eq. will enforce that the `Variable` received is either `0` /// or `1` by adding a constraint in the circuit. /// /// Note that using this constraint with whatever `Variable` that is not /// representing a value equalling 0 or 1, will always force the equation to fail. pub fn boolean_gate(&mut self, a: Variable) -> Variable { self.w_l.push(a); self.w_r.push(a); self.w_o.push(a); self.w_4.push(self.zero_var); self.q_m.push(BlsScalar::one()); self.q_l.push(BlsScalar::zero()); self.q_r.push(BlsScalar::zero()); self.q_o.push(-BlsScalar::one()); self.q_c.push(BlsScalar::zero()); self.q_4.push(BlsScalar::zero()); self.q_arith.push(BlsScalar::one()); self.q_range.push(BlsScalar::zero()); self.q_logic.push(BlsScalar::zero()); self.q_fixed_group_add.push(BlsScalar::zero()); self.q_variable_group_add.push(BlsScalar::zero()); self.public_inputs.push(BlsScalar::zero()); self.perm .add_variables_to_map(a, a, a, self.zero_var, self.n); self.n += 1; a }#[derive(Debug, Clone, Copy)] /// Contains all of the components needed to verify that a bit scalar multiplication was computed correctly pub(crate) struct WnafRound { /// This is the accumulated x coordinate point that we wish to add (so far.. depends on where you are in the scalar mul) /// it is linked to the wnaf entry, so must not be revealed pub acc_x: Variable, /// This is the accumulated y coordinate pub acc_y: Variable, /// This is the wnaf accumulated entry /// For all intents and purposes, you can think of this as the secret bit pub accumulated_bit: Variable, /// This is the multiplication of x_\alpha * y_\alpha /// we need this as a distinct wire, so that the degree of the polynomial does not go over 4 pub xy_alpha: Variable, /// This is the possible x co-ordinate of the wnaf point we are going to add /// Actual x-co-ordinate = b_i * x_\beta pub x_beta: BlsScalar, /// This is the possible y co-ordinate of the wnaf point we are going to add /// Actual y coordinate = (b_i)^2 [y_\beta -1] + 1 pub y_beta: BlsScalar, /// This is the multiplication of x_\beta * y_\beta pub xy_beta: BlsScalar, }
5.8 xor logic constraint/// Fixed group addition of a jubjub point pub(crate) fn fixed_group_add(&mut self, wnaf_round: WnafRound) { self.w_l.push(wnaf_round.acc_x); self.w_r.push(wnaf_round.acc_y); self.w_o.push(wnaf_round.xy_alpha); self.w_4.push(wnaf_round.accumulated_bit); self.q_l.push(wnaf_round.x_beta); self.q_r.push(wnaf_round.y_beta); self.q_c.push(wnaf_round.xy_beta); self.q_o.push(BlsScalar::zero()); self.q_fixed_group_add.push(BlsScalar::one()); self.q_variable_group_add.push(BlsScalar::zero()); self.q_m.push(BlsScalar::zero()); self.q_4.push(BlsScalar::zero()); self.q_arith.push(BlsScalar::zero()); self.q_range.push(BlsScalar::zero()); self.q_logic.push(BlsScalar::zero()); self.public_inputs.push(BlsScalar::zero()); self.perm.add_variables_to_map( wnaf_round.acc_x, wnaf_round.acc_y, wnaf_round.xy_alpha, wnaf_round.accumulated_bit, self.n, ); self.n += 1; }logic_gate中是同时对两个bit进行and或xor操作。具体见https://github.com/dusk-network/plonk/blob/master/src/proof_system/widget/logic/proverkey.rs中compute_quotient_i()函数：pub(crate) fn compute_quotient_i( &self, index: usize, logic_separation_challenge: &BlsScalar, w_l_i: &BlsScalar, w_l_i_next: &BlsScalar, w_r_i: &BlsScalar, w_r_i_next: &BlsScalar, w_o_i: &BlsScalar, w_4_i: &BlsScalar, w_4_i_next: &BlsScalar, ) -> BlsScalar { let four = BlsScalar::from(4); let q_logic_i = &self.q_logic.1[index]; let q_c_i = &self.q_c.1[index]; let kappa = logic_separation_challenge.square(); let kappa_sq = kappa.square(); let kappa_cu = kappa_sq * kappa; let kappa_qu = kappa_cu * kappa; let a = w_l_i_next - four * w_l_i; let c_0 = delta(a); let b = w_r_i_next - four * w_r_i; let c_1 = delta(b) * kappa; let d = w_4_i_next - four * w_4_i; let c_2 = delta(d) * kappa_sq; let w = w_o_i; let c_3 = (w - a * b) * kappa_cu; let c_4 = delta_xor_and(&a, &b, w, &d, &q_c_i) * kappa_qu; q_logic_i * (c_3 + c_0 + c_1 + c_2 + c_4) * logic_separation_challenge }其中的：
c_0,c_1,c_2,c_3constraint：是约束wire的相应取值仅允许为 [ 0 , 1 , 2 , 3 ] [0, 1, 2, 3] [0,1,2,3] 中之一。
// Computes f(f-1)(f-2)(f-3) fn delta(f: BlsScalar) -> BlsScalar { let f_1 = f - BlsScalar::one(); let f_2 = f - BlsScalar::from(2); let f_3 = f - BlsScalar::from(3); f * f_1 * f_2 * f_3 }c_4constraint：
// The identity we want to check is q_logic * A = 0 // A = B + E // B = q_c * [9c - 3(a+b)] // E = 3(a+b+c) - 2F // F = w[w(4w - 18(a+b) + 81) + 18(a^2 + b^2) - 81(a+b) + 83] #[allow(non_snake_case)] fn delta_xor_and( a: &BlsScalar, b: &BlsScalar, w: &BlsScalar, c: &BlsScalar, q_c: &BlsScalar, ) -> BlsScalar { let nine = BlsScalar::from(9); let two = BlsScalar::from(2); let three = BlsScalar::from(3); let four = BlsScalar::from(4); let eighteen = BlsScalar::from(18); let eighty_one = BlsScalar::from(81); let eighty_three = BlsScalar::from(83); let F = w * (w * (four * w - eighteen * (a + b) + eighty_one) + eighteen * (a.square() + b.square()) - eighty_one * (a + b) + eighty_three); let E = three * (a + b + c) - (two * F); let B = q_c * ((nine * c) - three * (a + b)); B + E }
5.9 and logic constraint/// Adds a logical XOR gate that performs the XOR between two values for the /// specified first `num_bits` returning a `Variable` holding the result. /// /// # Panics /// /// If the `num_bits` specified in the fn params is odd. pub fn xor_gate(&mut self, a: Variable, b: Variable, num_bits: usize) -> Variable { self.logic_gate(a, b, num_bits, true) }logic_gate中是同时对两个bit进行and或xor操作。 and logic constraint 具体与 xor logic constraint 类似，主要差别在于：w_o表示的逻辑分别为and或xor的结果值。
// The `out_quad` is the result of the bitwise ops `&` or `^` between // the left and right quads. The op is decided with a boolean flag set // as input of the function. let out_quad_fr = match is_xor_gate { true => BlsScalar::from((left_quad ^ right_quad) as u64), false => BlsScalar::from((left_quad & right_quad) as u64), };q_c和q_logic的取值不同：
match is_xor_gate { true => { self.q_c.push(-BlsScalar::one()); self.q_logic.push(-BlsScalar::one()); } false => { self.q_c.push(BlsScalar::one()); self.q_logic.push(BlsScalar::one()); } };
5.10 range constraint/// Adds a logical AND gate that performs the bitwise AND between two values /// for the specified first `num_bits` returning a `Variable` holding the result. /// /// # Panics /// /// If the `num_bits` specified in the fn params is odd. pub fn and_gate(&mut self, a: Variable, b: Variable, num_bits: usize) -> Variable { self.logic_gate(a, b, num_bits, false) }具体见
https://github.com/dusk-network/plonk/blob/master/src/constraint_system/range.rs中pub fn range_gate(&mut self, witness: Variable, num_bits: usize)函数，对应的constraints数量为：num_bits/8+1以32bit为例，相应的constraint数量为
6. proof system32/8+1=5。实际代码中，设计了六种不同类型的gate，分别为：
- arithmetic gate
- logic gate
- range gate
- ecc fixed base curve addition gate
- ecc variable base curve addition gate
- permutation check
- 借助
pad函数，通过附加零变量和零值的方式，使得circuit的constraint数量为power of two。
/// Prover composes a circuit and builds a proof #[allow(missing_debug_implementations)] pub struct Prover { /// ProverKey which is used to create proofs about a specific PLONK circuit pub prover_key: Option, pub(crate) cs: StandardComposer, /// Store the messages exchanged during the preprocessing stage /// This is copied each time, we make a proof pub preprocessed_transcript: Transcript, } /// PLONK circuit proving key #[derive(Debug, PartialEq, Eq, Clone)] pub struct ProverKey { /// Circuit size pub n: usize, /// ProverKey for arithmetic gate pub arithmetic: arithmetic::ProverKey, /// ProverKey for logic gate pub logic: logic::ProverKey, /// ProverKey for range gate pub range: range::ProverKey, /// ProverKey for fixed base curve addition gates pub fixed_base: ecc::scalar_mul::fixed_base::ProverKey, /// ProverKey for permutation checks pub permutation: permutation::ProverKey, /// ProverKey for variable base curve addition gates pub variable_base: ecc::curve_addition::ProverKey, // Pre-processes the 4n Evaluations for the vanishing polynomial, so they do not // need to be computed at the proving stage. // Note: With this, we can combine all parts of the quotient polynomial in their evaluation phase and // divide by the quotient polynomial without having to perform IFFT pub(crate) v_h_coset_4n: Evaluations, }实际实现时，根据circuit中gate类型分类不同，分别实现了不同的
ProverKey：- 1）对于 arithmetic gate，有：
#[derive(Debug, Eq, PartialEq, Clone)] pub struct ProverKey { pub q_m: (Polynomial, Evaluations), pub q_l: (Polynomial, Evaluations), pub q_r: (Polynomial, Evaluations), pub q_o: (Polynomial, Evaluations), pub q_c: (Polynomial, Evaluations), pub q_4: (Polynomial, Evaluations), pub q_arith: (Polynomial, Evaluations), }Round 3 第三轮 quotient polynomial 计算对应表示为：
(a(x)b(x)q_M(x) + a(x)q_L(x) + b(X)q_R(x) + c(X)q_O(X) + d(x)q_4(X) + Q_C(X)) * Q_Arith(X)Round 4 第四轮 linearisation polynomial 计算对应表示为：
(a_eval * b_eval * q_m_poly + a_eval * q_l + b_eval * q_r + c_eval * q_o + d_eval * q_4 + q_c) * q_arith_eval- 2）对于 logic gate，有： 【 logic gate详细设计思想参见 AztecProtocol中代码实现：
/* * Hoo boy, AND and XOR polynomials! * This transition constraint evaluates either an AND or an XOR relationship (but not an or in sight) between the *accumulating sums of three base-4 variables... * * Ok, so we want to evaluate a ^ b = c OR a & b = c . We can create a | b from a | b = (a ^ b) + (a & b) * * We also want the output memory cell to represent the actual result of the AND / XOR operation, * instead of a collection of bits / quads that need to be summed together. Who has time for that? * * We can also be super sneaky and evaluate both AND and XOR operations with a single selector polynomial. * * Let's call this selector 'S', it takes values in { -1, 0, 1} * * If S = -1, we're evaluating a XOR op * If S = 1, we're evaluating an AND op * If S = 0, we're evaluating nothing! This constraint is turned off * * We use 3 columns of program memory to represent accumulating sums of a, b, c. * * For example, we can represent a 32-bit 'A' via its quads * * 15 * === * \ i * A = / a . 4 * === i * i = 0 * * In program memory, we place an accumulating base-4 sum of A {A_0, ..., A_15}, where * * i * === * \ j * A = / a . 4 * i === (15 - j) * j = 0 * * * From this, we can extract a quad by validating that * * * A - 4 . A ϵ [0, 1, 2, 3] * i + 1 i * * Once we have validated the above, we can then extract an accumulator's implicit quad via: * * a = A - 4 . A ϵ [0, 1, 2, 3] * i i + 1 i * * * But of course it's not so simple! An AND/XOR polynomial identity with two input quads (plus selector) has a degree *of 8. To constrain the degree of our quotient polynomial T(X) we want our identity to have a degree of 5 * * We also have a spare column to work with, which we can use to store * * * w = a * b * i i * * For the polynomial identity, we use the following notation: * * (1) 'a' is the current round quad attributed to our operand a * (2) 'b' is the current round quad attributed to our operand b * (3) 'c' is the current round quad attributed to our output c * (4) 'w' = a * b * (5) 's' is the AND/XOR selector polynomial round value. * * The polynomial identity we're evaluating is... wait for it... * * 2 2 * s ⋅ (s ⋅ (9 ⋅ c - 3 ⋅ (a + b)) + 3 ⋅ (c + a + b) + w ⋅ (w ⋅ (4 ⋅ w - 18 ⋅ (a + b) + 81) + 18 ⋅ (a + b ) - 81 ⋅ (a + *b) + 83)) * * = * * 0 mod Z_H * * To simplify things, we *could* frankenstein integers out of the 4th roots of unity to make this simpler, * but then integer multiplication would be horrible. * So really, it's a question of picking ones poison, and blaming the Babylonians * for creating their number system out of the integers instead of a nice cyclic group. * * In addition to this nonsense, we also need to verify the following: * * (1) a is in the set { 0, 1, 2, 3 } * (2) b is in the set { 0, 1, 2, 3 } * (3) c is in the set { 0, 1, 2, 3 } * (4) w = a * c * * * We place our accumulating sums (A, B, C) in program memory in the following sequence: * * +-----+-----+-----+-----+ * | 1 | 2 | 3 | 4 | * +-----+-----+-----+-----+ * you are here --> | 0 | 0 | w1 | 0 | * | A1 | B1 | w2 | C1 | * | A2 | B2 | w3 | C2 | * | ... | ... | ... | ... | * | An | Bn | --- | Cn | --> exit * +-----+-----+-----+-----+ * * **/】
#[derive(Debug, Eq, PartialEq, Clone)] pub struct ProverKey { pub q_c: (Polynomial, Evaluations), pub q_logic: (Polynomial, Evaluations), }logic gate的逻辑为
q_logic=1表示为two bits and gate，q_logic=-1表示为two bits xor gate。- 对于 two bits and gate，有
q_c=1，c=a&b； - 对于 two bits xor gate，有
q_c=-1，c=a^b；
Round 3 第三轮 quotient polynomial 计算对应表示为：【
w=a*b】// The identity we want to check is q_logic * A = 0 // A = B + E // B = q_c * [9c - 3(a+b)] // E = 3(a+b+c) - 2F // F = w[w(4w - 18(a+b) + 81) + 18(a^2 + b^2) - 81(a+b) + 83]Round 4 第四轮 linearisation polynomial 计算对应表示为：【相应的evaluation *
q_logic_poly】- 3）对于 range gate，有： 【 range gate 详细设计思想参见 AztecProtocol中代码实现：
/* * The range constraint accumulates base 4 values into a sum. * We do this by evaluating a kind of 'raster scan', where we compare adjacent elements * and validate that their differences map to a base for value * * Let's say that we want to perform a 32-bit range constraint in 'x'. * We can represent x via 16 constituent base-4 'quads' {q_0, ..., q_15}: * * 15 * === * \ i * x = / q . 4 * === i * i = 0 * * In program memory, we place an accumulating base-4 sum of x {a_0, ..., a_15}, where * * i * === * \ j * a = / q . 4 * i === (15 - j) * j = 0 * * * From this, we can use our range transition constraint to validate that * * * a - 4 . a ϵ [0, 1, 2, 3] * i + 1 i * * * We place our accumulating sums in program memory in the following sequence: * * +-----+-----+-----+-----+ * | A | B | C | D | * +-----+-----+-----+-----+ * | a3 | a2 | a1 | 0 | * | a7 | a6 | a5 | a4 | * | a11 | a10 | a9 | a8 | * | a15 | a14 | a13 | a12 | * | --- | --- | --- | a16 | * +-----+-----+-----+-----+ * * Our range transition constraint on row 'i' * performs our base-4 range check on the follwing pairs: * * (D_{i}, C_{i}), (C_{i}, B_{i}), (B_{i}, A_{i}), (A_{i}, D_{i+1}) * * We need to start our raster scan at zero, so we simplify matters and just force the first value * to be zero. * * The output will be in the 4th column of an otherwise unused row. Assuming this row can * be used for a width-3 standard gate, the total number of gates for an n-bit range constraint * is (n / 8) gates * **/】
#[derive(Debug, Eq, PartialEq, Clone)] pub struct ProverKey { pub q_range: (Polynomial, Evaluations), }Round 3 第三轮 quotient polynomial 计算对应表示为：
Delta([c(X) - 4 * d(X)]) + Delta([b(X) - 4 * c(X)]) + Delta([a(X) - 4 * b(X)]) + Delta([d(Xg) - 4 * a(X)]) * Q_Range(X)Round 4 第四轮 linearisation polynomial 计算对应表示为：
Delta([c_eval - 4 * d_eval]) + Delta([b_eval - 4 * c_eval]) + Delta([a_eval - 4 * b_eval]) + Delta([d_next_eval - 4 * a_eval]) * Q_Range(X)- 4）对于 ecc fixed base curve addition gate，有：
#[derive(Debug, Eq, PartialEq, Clone)] pub struct ProverKey { pub q_l: (Polynomial, Evaluations), pub q_r: (Polynomial, Evaluations), pub q_c: (Polynomial, Evaluations), pub q_fixed_group_add: (Polynomial, Evaluations), }Round 3 第三轮 quotient polynomial 计算对应表示为：
Round 4 第四轮 linearisation polynomial 计算对应表示为：
- 5）对于 ecc variable base curve addition gate，有：
#[derive(Debug, Eq, PartialEq, Clone)] pub struct ProverKey { pub q_variable_group_add: (Polynomial, Evaluations), }Round 3 第三轮 quotient polynomial 计算对应表示为：
Round 4 第四轮 linearisation polynomial 计算对应表示为：
- 6）对于permutation check，主要用于验证circuit中wire之间的约束关系，有：
#[derive(Debug, Eq, PartialEq, Clone)] pub struct ProverKey { pub left_sigma: (Polynomial, Evaluations), pub right_sigma: (Polynomial, Evaluations), pub out_sigma: (Polynomial, Evaluations), pub fourth_sigma: (Polynomial, Evaluations), pub linear_evaluations: Evaluations, // Evaluations of f(x) = X [XXX: Remove this and benchmark if it makes a considerable difference -- These are just the domain elements] }Round 3 第三轮 quotient polynomial 计算对应表示为：
// 分子 // (a(x) + beta * X + gamma) (b(X) + beta * k1 * X + gamma) (c(X) + beta * k2 * X + gamma)(d(X) + beta * k3 * X + gamma)z(X) * alpha // 分母 // (a(x) + beta* Sigma1(X) + gamma) (b(X) + beta * Sigma2(X) + gamma) (c(X) + beta * Sigma3(X) + gamma)(d(X) + beta * Sigma4(X) + gamma) Z(X.omega) * alpha // 常量 // L_1(X)[Z(X) - 1]Round 4 第四轮 linearisation polynomial 计算对应表示为：
6.3 Verifier端// 分子 // (a_eval + beta * z_challenge + gamma)(b_eval + beta * K1 * z_challenge + gamma)(c_eval + beta * K2 * z_challenge + gamma) * alpha* z(X) // 分母 // -(a_eval + beta * sigma_1 + gamma)(b_eval + beta * sigma_2 + gamma) (c_eval + beta * sigma_3 + gamma) * beta *z_eval * alpha^2 * Sigma_4(X) // 常量 // Evaluate l_1(z)/// Verifier verifies a proof #[allow(missing_debug_implementations)] pub struct Verifier { /// VerificationKey which is used to verify a specific PLONK circuit pub verifier_key: Option, pub(crate) cs: StandardComposer, /// Store the messages exchanged during the preprocessing stage /// This is copied each time, we make a proof, so that we can use the same verifier to /// Verify multiple proofs from the same circuit. If this is not copied, then the verification procedure will modify /// the transcript, making it unusable for future proofs. pub preprocessed_transcript: Transcript, } /// PLONK circuit verification key #[derive(Debug, PartialEq, Eq, Copy, Clone)] pub struct VerifierKey { /// Circuit size pub n: usize, /// VerifierKey for arithmetic gates pub arithmetic: arithmetic::VerifierKey, /// VerifierKey for logic gates pub logic: logic::VerifierKey, /// VerifierKey for range gates pub range: range::VerifierKey, /// VerifierKey for fixed base curve addition gates pub fixed_base: ecc::scalar_mul::fixed_base::VerifierKey, /// VerifierKey for variable base curve addition gates pub variable_base: ecc::curve_addition::VerifierKey, /// VerifierKey for permutation checks pub permutation: permutation::VerifierKey, }实际实现时，根据circuit中gate类型分类不同，分别实现了不同的
VerifierKey：- 1）对于 arithmetic gate，有：
#[derive(Debug, PartialEq, Eq, Copy, Clone)] pub struct VerifierKey { pub q_m: Commitment, pub q_l: Commitment, pub q_r: Commitment, pub q_o: Commitment, pub q_c: Commitment, pub q_4: Commitment, pub q_arith: Commitment, }- 2）对于 logic gate，有：
#[derive(Debug, PartialEq, Eq, Copy, Clone)] pub struct VerifierKey { pub q_c: Commitment, pub q_logic: Commitment, }- 3）对于 range gate，有：
#[derive(Debug, PartialEq, Eq, Copy, Clone)] pub struct VerifierKey { pub q_range: Commitment, }- 4）对于 ecc fixed base curve addition gate，有：
#[derive(Debug, PartialEq, Eq, Copy, Clone)] pub struct VerifierKey { pub q_l: Commitment, pub q_r: Commitment, pub q_fixed_group_add: Commitment, }- 5）对于 ecc variable base curve addition gate，有：
#[derive(Debug, PartialEq, Eq, Copy, Clone)] pub struct VerifierKey { pub q_variable_group_add: Commitment, }- 6）对于permutation check，主要用于验证circuit中wire之间的约束关系，有：
6.4 proof#[derive(Debug, PartialEq, Eq, Copy, Clone)] pub struct VerifierKey { pub left_sigma: Commitment, pub right_sigma: Commitment, pub out_sigma: Commitment, pub fourth_sigma: Commitment, }proof的组成有：【根本目的是使得Verifier可
verify()通过。】- witness polynomials 的commitments 及 相应的evaluations；
- permutation polynomials 的commitments 及 相应的evaluations；
- quotient polynomials 的commitments 及 相应的evaluations；
- shifted polynomials 的commitments 及 相应的evaluations；
- opening polynomials 的commitments 及 相应的evaluations。
/// A Proof is a composition of `Commitments` to the witness, permutation, /// quotient, shifted and opening polynomials as well as the /// `ProofEvaluations`. /// /// It's main goal is to have a `verify()` method attached which contains the /// logic of the operations that the `Verifier` will need to do in order to /// formally verify the `Proof`. #[derive(Debug, Eq, PartialEq, Clone)] pub struct Proof { /// Commitment to the witness polynomial for the left wires. pub a_comm: Commitment, /// Commitment to the witness polynomial for the right wires. pub b_comm: Commitment, /// Commitment to the witness polynomial for the output wires. pub c_comm: Commitment, /// Commitment to the witness polynomial for the fourth wires. pub d_comm: Commitment, /// Commitment to the permutation polynomial. pub z_comm: Commitment, /// Commitment to the quotient polynomial. pub t_1_comm: Commitment, /// Commitment to the quotient polynomial. pub t_2_comm: Commitment, /// Commitment to the quotient polynomial. pub t_3_comm: Commitment, /// Commitment to the quotient polynomial. pub t_4_comm: Commitment, /// Commitment to the opening polynomial. pub w_z_comm: Commitment, /// Commitment to the shifted opening polynomial. pub w_zw_comm: Commitment, /// Subset of all of the evaluations added to the proof. pub evaluations: ProofEvaluations, } /// Proof Evaluations is a subset of all of the evaluations. These evaluations will be added to the proof #[derive(Debug, Eq, PartialEq, Clone)] pub struct ProofEvaluations { // Evaluation of the witness polynomial for the left wire at `z` pub a_eval: BlsScalar, // Evaluation of the witness polynomial for the right wire at `z` pub b_eval: BlsScalar, // Evaluation of the witness polynomial for the output wire at `z` pub c_eval: BlsScalar, // Evaluation of the witness polynomial for the fourth wire at `z` pub d_eval: BlsScalar, // pub a_next_eval: BlsScalar, // pub b_next_eval: BlsScalar, // Evaluation of the witness polynomial for the fourth wire at `z * root of unity` pub d_next_eval: BlsScalar, // Evaluation of the arithmetic selector polynomial at `z` pub q_arith_eval: BlsScalar, // pub q_c_eval: BlsScalar, // pub q_l_eval: BlsScalar, // pub q_r_eval: BlsScalar, // Evaluation of the left sigma polynomial at `z` pub left_sigma_eval: BlsScalar, // Evaluation of the right sigma polynomial at `z` pub right_sigma_eval: BlsScalar, // Evaluation of the out sigma polynomial at `z` pub out_sigma_eval: BlsScalar, // Evaluation of the linearisation sigma polynomial at `z` pub lin_poly_eval: BlsScalar, // (Shifted) Evaluation of the permutation polynomial at `z * root of unity` pub perm_eval: BlsScalar, }
