1 理论背景
本文主要解析基于论文《Polynomial Commitments∗》在代码库中的实现细节。 代码的实现流程基于论文《Marlin:Preprocessing zkSNARKs with Universal and Updatable SRS》中的6.1节和B.3。 
zexe的bench-utils的print-trace features支持对各个阶段的代码运行效率进行profiling。调用的方法可为：
cargo test end_to_end_test --features print-trace -- --nocaputre
为了便于对比，修改了程序，要求bls12-377、bls12-381、mnt6、sw6每组曲线都对相同阶（如18）的多项式进行commit。
end_to_end_test::().expect("test failed for bls12-377");
end_to_end_test::().expect("test failed for bls12-381");
end_to_end_test::().expect("test failed for MNT6");
end_to_end_test::().expect("test failed for SW6");
会有类似的运行结果：【由此可知，总体来说bls12-377的运算性能最高，bls12-377>bls12-381>mnt6>sw6】【evaluating polynomial的时间不是这样的，可能跟evaluate point有关系。固定在相同的点evaluate 试下。】 
let (ck, vk) = SinglePC::setup(degree, rng)?;
https://github.com/scipr-lab/poly-commit/blob/master/src/single_pc/kzg10.rs中，setup代码实现流程如下：
/// Constructs public parameters when given as input the maximum degree `degree`
/// for the polynomial commitment scheme.
fn setup(
degree: usize,
rng: &mut R,
) -> Result {
if degree < 1 {
return Err(Error::UnsupportedDegree);
}
let setup_time = start_timer!(|| format!("Started KZG10::Setup with degree {}", degree));
let beta = E::Fr::rand(rng);
let g = E::G1Projective::rand(rng);
let gamma_g = E::G1Projective::rand(rng);
let h = E::G2Projective::rand(rng);
let mut powers_of_beta = vec![E::Fr::one()];
let mut cur = beta;
// TODO: can optimize by computing
// 1, beta, beta^2, beta^3, ..., beta^d and using those to compute arbitrary
// powers of beta faster via a bit decomposition of each scalar.
// But that seems like it would be slower? log(D) additions instead of
// one multiplication.
// Anyway it shouldn't be a bottleneck?
// If we use MPC to run the setup, we can consider this optimization.
for _ in 0..degree {
powers_of_beta.push(cur);
cur *= β
}
let window_size = FixedBaseMSM::get_mul_window_size(degree + 1);
let scalar_bits = E::Fr::size_in_bits();
let g_time = start_timer!(|| "Generating powers of G");
let g_table = FixedBaseMSM::get_window_table(scalar_bits, window_size, g);
let mut powers_of_g = FixedBaseMSM::multi_scalar_mul::(
scalar_bits,
window_size,
&g_table,
&powers_of_beta,
);
end_timer!(g_time);
let gamma_g_time = start_timer!(|| "Generating powers of gamma * G");
let gamma_g_table = FixedBaseMSM::get_window_table(scalar_bits, window_size, gamma_g);
let mut powers_of_gamma_g = FixedBaseMSM::multi_scalar_mul::(
scalar_bits,
window_size,
&gamma_g_table,
&powers_of_beta,
);
end_timer!(gamma_g_time);
E::G1Projective::batch_normalization(powers_of_g.as_mut_slice());
E::G1Projective::batch_normalization(powers_of_gamma_g.as_mut_slice());
let ck = CommitterKey {
powers_of_g: powers_of_g.into_iter().map(|e| e.into_affine()).collect(),
powers_of_gamma_g: powers_of_gamma_g
.into_iter()
.map(|e| e.into_affine())
.collect(),
degree,
};
let beta_h = h.mul(&beta).into_affine();
let h = h.into_affine();
let prepared_h = h.prepare();
let prepared_beta_h = beta_h.prepare();
let vk = VerifierKey {
g: g.into_affine(),
gamma_g: gamma_g.into_affine(),
h,
beta_h,
prepared_h,
prepared_beta_h,
degree,
};
end_timer!(setup_time);
Ok((ck, vk))
}
let p = Polynomial::rand(degree, rng);
pub use ff_fft::DensePolynomial as Polynomial;
https://github.com/scipr-lab/zexe/blob/master/ff-fft/src/polynomial/dense.rs中的
/// Outputs a polynomial of degree `d` where each coefficient is sampled uniformly at random
/// from the field `F`.
pub fn rand(d: usize, rng: &mut R) -> Self {
let mut random_coeffs = Vec::new();
for _ in 0..(d + 1) {
random_coeffs.push(F::rand(rng));
}
Self::from_coefficients_vec(random_coeffs)
}
/// Constructs a new polynomial from a list of coefficients.
pub fn from_coefficients_vec(mut coeffs: Vec) -> Self {
// While there are zeros at the end of the coefficient vector, pop them off.
while coeffs.last().map_or(false, |c| c.is_zero()) {
coeffs.pop();
}
// Check that either the coefficients vec is empty or that the last coeff is non-zero.
assert!(coeffs.last().map_or(true, |coeff| !coeff.is_zero()));
Self { coeffs }
}
/// Outputs a commitment to `polynomial`.
fn commit(
ck: &Self::CommitterKey,
polynomial: &Polynomial,
hiding_bound: Option,
rng: Option,
) -> Result {
Error::check_degree(polynomial.degree(), ck.max_degree())?;
let commit_time = start_timer!(|| format!(
"Committing to polynomial of degree {} with hiding_bound: {:?}",
polynomial.degree(),
hiding_bound,
));
let mut skip_zeros = 0;
while polynomial.coeffs[skip_zeros].is_zero() && skip_zeros < polynomial.coeffs.len() {
skip_zeros += 1;
}
let from_mont_repr_time =
start_timer!(|| "Converting plaintext polynomial from Montgomery repr");
let plain_coeffs = polynomial.coeffs[skip_zeros..]
.par_iter()
.map(|s| s.into_repr())
.collect::();
end_timer!(from_mont_repr_time);
let msm_time = start_timer!(|| "MSM to compute commitment to plaintext poly");
let mut commitment =
VariableBaseMSM::multi_scalar_mul(&ck.powers_of_g[skip_zeros..], &plain_coeffs);
end_timer!(msm_time);
let mut randomness = Randomness::empty();
if let Some(hiding_degree) = hiding_bound {
let mut rng = rng.ok_or(Error::MissingRng)?;
let sample_random_poly_time = start_timer!(|| format!(
"Sampling a random polynomial of degree {}",
hiding_degree
));
randomness = Randomness::rand(hiding_degree, &mut rng);
if randomness.blinding_polynomial.degree() > ck.max_degree() {
eprintln!("The hiding bound is too large for the commitment key.");
Err(Error::PolynomialDegreeTooLarge {
poly_degree: randomness.blinding_polynomial.degree(),
max_degree: ck.max_degree(),
})?;
}
end_timer!(sample_random_poly_time);
}
let from_mont_repr_time =
start_timer!(|| "Converting random polynomial from Montgomery repr");
let random_ints = randomness
.blinding_polynomial
.coeffs
.par_iter()
.map(|s: &E::Fr| s.into_repr())
.collect::();
end_timer!(from_mont_repr_time);
let msm_time = start_timer!(|| "MSM to compute commitment to random poly");
let random_commitment =
VariableBaseMSM::multi_scalar_mul(&ck.powers_of_gamma_g, random_ints.as_slice())
.into_affine();
end_timer!(msm_time);
commitment.add_assign_mixed(&random_commitment);
end_timer!(commit_time);
Ok((Commitment(commitment.into()), randomness))
}
impl PCRandomness for Randomness {
fn empty() -> Self {
Self {
blinding_polynomial: Polynomial::zero(),
}
}
fn rand(d: usize, rng: &mut R) -> Self {
let mut randomness = Randomness::empty();
randomness.blinding_polynomial = Polynomial::rand(d + 1, rng);
randomness
}
}
let point = F::rand(rng);
let value = p.evaluate(point);
/// Evaluates `self` at the given `point` in the field.
pub fn evaluate(&self, point: F) -> F {
if self.is_zero() {
return F::zero();
}
let mut powers_of_point = vec![F::one()];
let mut cur = point;
for _ in 0..self.degree() {
powers_of_point.push(cur);
cur *= &point;
}
assert_eq!(powers_of_point.len(), self.coeffs.len());
let zero = F::zero();
powers_of_point
.into_par_iter()
.zip(&self.coeffs)
.map(|(power, coeff)| power * coeff)
.reduce(|| zero, |a, b| a + &b)
}
let proof = SinglePC::open(&ck, &p, point, &rand)?;
/// On input a polynomial `p` and a point `point`, outputs a proof for the same.
fn open(
ck: &Self::CommitterKey,
p: &Polynomial,
point: E::Fr,
randomness: &Self::Randomness,
) -> Result {
Error::check_degree(p.degree(), ck.max_degree())?;
let open_time = start_timer!(|| format!("Opening polynomial of degree {}", p.degree()));
let eval_time = start_timer!(|| "Evaluating polynomial");
let value = p.evaluate(point);
end_timer!(eval_time);
let witness_time = start_timer!(|| "Computing witness polynomial");
let witness_polynomial = &(p - &Polynomial::from_coefficients_vec(vec![value]))
/ &Polynomial::from_coefficients_vec(vec![-point, E::Fr::one()]);
end_timer!(witness_time);
let convert_time = start_timer!(|| "Converting witness polynomial from Montgomery repr");
let witness_coeffs = witness_polynomial
.coeffs
.into_par_iter()
.map(|s| s.into_repr())
.collect::();
end_timer!(convert_time);
let witness_comm_time = start_timer!(|| "Computing commitment to witness polynomial");
let mut w = VariableBaseMSM::multi_scalar_mul(&ck.powers_of_g, &witness_coeffs);
end_timer!(witness_comm_time);
let mut random_v = E::Fr::zero();
if randomness.is_hiding() {
let random_p = &randomness.blinding_polynomial;
let rand_eval_time = start_timer!(|| "Evaluating random polynomial");
let random_value = random_p.evaluate(point);
end_timer!(rand_eval_time);
let witness_time = start_timer!(|| "Computing random witness polynomial");
let random_witness_polynomial = &(random_p
- &Polynomial::from_coefficients_vec(vec![random_value]))
/ &Polynomial::from_coefficients_vec(vec![-point, E::Fr::one()]);
end_timer!(witness_time);
let random_witness_coeffs = random_witness_polynomial
.coeffs
.into_par_iter()
.map(|s| s.into_repr())
.collect::();
let witness_comm_time =
start_timer!(|| "Computing commitment to random witness polynomial");
w += &VariableBaseMSM::multi_scalar_mul(&ck.powers_of_gamma_g, &random_witness_coeffs);
end_timer!(witness_comm_time);
random_v = random_value;
}
end_timer!(open_time);
Ok(Proof {
w: w.into_affine(),
random_v,
})
}
// 多项式除法的实现如下：
impl Div for &'b DensePolynomial {
type Output = DensePolynomial;
#[inline]
fn div(self, divisor: &'a DensePolynomial) -> DensePolynomial {
let a: DenseOrSparsePolynomial = self.into();
let b: DenseOrSparsePolynomial = divisor.into();
a.divide_with_q_and_r(&b).expect("division failed").0
}
}
/// Divide self by another (sparse or dense) polynomial, and returns the quotient and remainder.
pub fn divide_with_q_and_r(&self, divisor: &Self) -> Option {
if self.is_zero() {
Some((DensePolynomial::zero(), DensePolynomial::zero()))
} else if divisor.is_zero() {
panic!("Dividing by zero polynomial")
} else if self.degree() < divisor.degree() {
Some((DensePolynomial::zero(), self.clone().into()))
} else {
// Now we know that self.degree() >= divisor.degree();
let mut quotient = vec![F::zero(); self.degree() - divisor.degree() + 1];
let mut remainder: DensePolynomial = self.clone().into();
// Can unwrap here because we know self is not zero.
let divisor_leading_inv = divisor.leading_coefficient().unwrap().inverse().unwrap();
while !remainder.is_zero() && remainder.degree() >= divisor.degree() {
let cur_q_coeff = *remainder.coeffs.last().unwrap() * &divisor_leading_inv;
let cur_q_degree = remainder.degree() - divisor.degree();
quotient[cur_q_degree] = cur_q_coeff;
for (i, div_coeff) in divisor.iter_with_index() {
remainder[cur_q_degree + i] -= &(cur_q_coeff * &div_coeff);
}
while let Some(true) = remainder.coeffs.last().map(|c| c.is_zero()) {
remainder.coeffs.pop();
}
}
Some((DensePolynomial::from_coefficients_vec(quotient), remainder))
}
}
assert!(
SinglePC::check(&vk, &comm, point, value, &proof)?,
"proof was incorrect for max_degree = {}, polynomial_degree = {}, hiding_bound = {:?}",
degree,
p.degree(),
hiding_bound,
);
/// Verifies that `value` is the evaluation at `x` of the polynomial
/// committed inside `comm`.
fn check(
vk: &Self::VerifierKey,
comm: &Self::Commitment,
point: E::Fr,
value: E::Fr,
proof: &Self::Proof,
) -> Result {
let check_time = start_timer!(|| "Checking evaluation");
let inner = comm.0.into_projective()
- &vk.g.into_projective().mul(&value)
- &vk.gamma_g.into_projective().mul(&proof.random_v);
let lhs = E::pairing(inner, vk.h);
let inner = vk.beta_h.into_projective() - &vk.h.into_projective().mul(&point);
let rhs = E::pairing(proof.w, inner);
end_timer!(check_time, || format!("Result: {}", lhs == rhs));
Ok(lhs == rhs)
}
fn batch_check(
vk: &Self::VerifierKey,
commitments: &[Self::Commitment],
points: &[E::Fr],
values: &[E::Fr],
proofs: &[Self::Proof],
rng: &mut R,
) -> Result {
let check_time =
start_timer!(|| format!("Checking {} evaluation proofs", commitments.len()));
let g = vk.g.into_projective();
let gamma_g = vk.gamma_g.into_projective();
let mut total_c = ::zero();
let mut total_w = ::zero();
let combination_time = start_timer!(|| "Combining commitments and proofs");
let mut randomizer = E::Fr::one();
// 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 = E::Fr::zero();
let mut gamma_g_multiplier = E::Fr::zero();
for (((c, z), v), proof) in commitments.iter().zip(points).zip(values).zip(proofs) {
let mut c = c.0.into_projective();
let w = proof.w.into_projective();
c += &w.mul(z);
g_multiplier += &(randomizer * &v);
gamma_g_multiplier += &(randomizer * &proof.random_v);
total_c += &c.mul(&randomizer);
total_w += &w.mul(&randomizer);
// We don't need to sample randomizers from the full field,
// only from 128-bit strings.
randomizer = u128::rand(rng).into();
}
total_c -= &g.mul(&g_multiplier);
total_c -= &gamma_g.mul(&gamma_g_multiplier);
end_timer!(combination_time);
let to_affine_time = start_timer!(|| "Converting results to affine for pairing");
let mut to_affine = [-total_w, total_c];
E::G1Projective::batch_normalization(&mut to_affine);
let [total_w, total_c] = to_affine;
let total_w = total_w.into_affine();
let total_c = total_c.into_affine();
end_timer!(to_affine_time);
let pairing_time = start_timer!(|| "Performing product of pairings");
let result = E::product_of_pairings(&[
(&total_w.prepare(), &vk.prepared_beta_h),
(&total_c.prepare(), &vk.prepared_h),
]) == E::Fqk::one();
end_timer!(pairing_time);
end_timer!(check_time, || format!("Result: {}", result));
Ok(result)
}
另关注下，Halo中polynomial commitment的实现细节。
借用博客polynomial commitment及实现方式对比3.2节内容：
-
论文《Constant-Size Commitments to Polynomials and Their Applications》： provided protocols to commit to polynomials and then evaluate them at a given point in a verifiable way. Their protocols only require a constant number of commitments but security relies on pairing assumptions。 基于的数学原理为：（论文《Sonic: Zero-Knowledge SNARKs from Linear-Size Universal and Updatable Structured Reference Strings》中的polynomial commitment也是基于此原理。）
-
论文《Efficient Zero-Knowledge Arguments for Arithmetic Circuits in the Discrete Log Setting》： Our polynomial commitment protocol has square root communication complexity but relies solely on the discrete logarithm assumption. 基于的原理是：（将系数拆分为矩阵，每行作为一个vector，将polynomial commitment拆分由vector commitment组成。）
参考资料： [1] 论文《Polynomial Commitments∗》 [2] 代码库：https://github.com/scipr-lab/poly-commit [3] 博客polynomial commitment及实现方式对比 [4] 博客椭圆曲线形式下的Pedersen commitment——vector commitmnt和polynomial commitment
