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From here - Batch Opening of KZG PCS


One can prove multiple evaluations $(\phi(e_i) = y_i)_{i\in I}$,for arbitrary points $e_i$ using a constant-sized KZG batch proof, $\pi_I = g^{q_I(\tau)}$, where

\begin{align} \label{eq:batch-proof-rel} q_I(X) &=\frac{\phi(X)-R_I(X)}{A_I(X)}\\ A_I(X) &=\prod_{i\in I} (X - e_i)\\ R_I(e_i) &= y_i,\forall i\in I\\ \end{align}

$R_I(X)$ can be interpolated via Lagrange interpolation in $O(\vert I\vert\log^2{\vert I\vert})$ time as:

\begin{align} R_I(X)=\sum_{i\in I} y_i \prod_{j\in I,j\ne i}\frac{X - e_j}{e_i - e_j} \end{align}


My question here is as to why Lagrange interpolation is needed for finding $R_I(X)$?

All the $e_i$'s are known, so $A_I(X)$ is a known polynomial. If you use long division to divide $\phi(X)$ by $A_I(X)$, you will get $q_I(X)$ with $R_I(X)$ as the reminder. So why Lagrange Interpolation is needed here?

Down below, the page also says that $A_I(X)$ is also interpolated. Again why?

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It's computational efficiency rather than mathematical necessity.

Using interpolation means that the computation can be done with complexity independent of the degree of $\phi(X)$. For $\phi(X)$ of large degree relative to the size of $|I|$, interpolation will be much quicker than polynomial division.

The reference to interpolating $A_I(X)$ simply refers to constructing $A_I(X)$ using the product tree described (here interpolation just means the construction of a polynomial satisfying constraints rather than specifically Lagrange interpolation).

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  • $\begingroup$ Hey, one more question on the same thing. The original Kate Paper says $R(x)$ is handed by the Prover to the verifier. However other documents like the one I linked to in my question say that the verifier also figures out $R(x)$ by Lagrange's Interpolation. I understand how the verifier can do this, but I wondering about the inconsistency. Is this an optimisation - i.e. to reduce the size of communication between prover & verifier. Also, it's a trade-off since the verifier would need to spend time doing the interpolation. $\endgroup$
    – user93353
    Jan 23, 2023 at 12:03
  • $\begingroup$ As well as the trade off, with interpolation, the verifier has the choice to only verify a subset or to amalgamate proofs. I could see how this might lead to more efficiency. $\endgroup$
    – Daniel S
    Jan 24, 2023 at 10:12
  • $\begingroup$ Thank you ..... $\endgroup$
    – user93353
    Jan 24, 2023 at 12:10

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