# The Power of Distributed Verifiers in Interactive Proofs

### Abstract:

We explore the power of interactive proofs with a distributed verifier. In this setting, the verifier consists of $n$ nodes and a graph $G$ that defines their communication pattern. The prover is a single entity that communicates with all nodes by short messages. The goal is to verify that the graph $G$ belongs to some language in a small number of rounds, and with small communication bound, \ie the proof size.

This interactive model was introduced by Kol, Oshman and Saxena (PODC 2018) as a generalization of non-interactive distributed proofs. They demonstrated the power of interaction in this setting by constructing protocols for problems as Graph Symmetry and Graph Non-Isomorphism -- both of which require proofs of $\Omega(n^2)$-bits without interaction.

In this work, we provide a new general framework for distributed interactive proofs that allows one to translate standard interactive protocols (i.e., with a centralized verifier) to ones where the verifier is distributed with a proof size that depends on the computational complexity of the verification algorithm run by the centralized verifier. We show the following:

• Every (centralized) computation performed in time $O(n)$ on a RAM can be translated into three-round distributed interactive protocol with $O(\log n)$ proof size. This implies that many graph problems for sparse graphs have succinct proofs (\eg testing planarity).
• Every (centralized) computation implemented by either a small space or by uniform NC circuit can be translated into a distributed protocol with $O(1)$ rounds and $O(\log n)$ bits proof size for the low space case and $\polylog(n)$ many rounds and proof size for NC.
• We show that for Graph Non-Isomorphism, one of the striking demonstrations of the power of interaction, there is a 4-round protocol with $O(\log n)$ proof size, improving upon the $O(n \log n)$ proof size of Kol et al.
• For many problems, we show how to reduce proof size below the seemingly natural barrier of $\log n$. By employing our RAM compiler, we get a 5-round protocol with proof size $O(\log \log n)$ for a family of problems including Fixed Automorphism, Clique and Leader Election (for the latter two problems we actually get $O(1)$ proof size). \item Finally, we discuss how to make these proofs non-interactive {\em arguments} via random oracles.
• Our compilers capture many natural problems and demonstrate the difficulty in showing lower bounds in these regimes.

Paper: PDF. Slides: ppt

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