What is the KZG PCS

$[\tau {]}_1$ stand for

$\tau G_1$

• Public Parameters:
  $\text{srs}=([1{]}_1,[\tau {]}_1,[{\tau }^2{]}_1,[{\tau }^3{]}_1,\dots ,[{\tau }^{n-1}{]}_1,[1{]}_2,[\tau {]}_2)$, where the
  $\tau$ is a random, and will be deleted to make sure no one knows it after set up.
• Prover knows:
  $f(X)=a_0+a_{1}X+a_2{X}^2+\cdots +a_{n-1}{X}^{n-1}$
• for a Polynomial
  $f(X)$, its KZG10 Commitment is
  $C_{f(X)}=f(\tau )G=a_{0}G+a_1\tau G+a_2{\tau }^{2}G+\cdots a_{n-1}{\tau }^{n-1}G=a_0[1{]}_1+a_1[\tau {]}_1+a_2[{\tau }^2{]}_1+\cdots +a^{n-1}[{\tau }^{n-1}{]}_1$, so the Prover can calculate
  $C_{f(X)}$ according to the srs, and public the
  $C_{f(X)}$ at first.

1. The verifier provides the challenge value
   $\zeta$ to prover, where
   $\zeta$ is a random value.
2. the prover calculate
   $f(\zeta )=y$.
3. the prover calculate
   $q(X)$ according to
   $f(X)-y=q(X)\cdot (X-\zeta )X\in {\mathbb{F}}_n$
4. the prover calculate
   $C_{q(X)}=q(\tau )G$ according to the srs.
5. the prover send the values
   $y,C_{q(X)}$ to verifier.
6. the verifier check
   $e(C_{f(x)}-y\cdot [1{]}_1,[1{]}_2)=e(C_{q(X)},[\tau {]}_2-\zeta \cdot [1{]}_2)$, because
   $(f(\tau )-y)\cdot 1=q(\tau )\cdot (\tau -\zeta )$ and
   $e([a{]}_1,[b{]}_1)=e(G,H{)}^{ab}$, so if
   $a_1{b}_1=a_2{b}_2\Rightarrow e([a_1{]}_1,[b_1{]}_2)=e([a_2{]}_1,[b_2{]}_2)$

Reference

• https://learn.z2o-k7e.world/plonk-intro-cn/plonk-polycom.html
• https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html

$$f(x)=\sum _{i=1}^8(f_i\cdot \prod _{1\le j\le 8}^{j\ne i}\frac{x-j}{i-j})$$

$$f_1\cdot \frac{x-2}{1-2}\frac{x-3}{1-3}\frac{x-4}{1-4}\frac{x-5}{1-5}\frac{x-6}{1-6}\frac{x-7}{1-7}\frac{x-8}{1-8}$$

$$c=f(s)G_1$$

$${\pi }_i=\frac{f(s)-f_i}{s-i}\cdot G_1$$