# The Power of Distributed Verifiers in Interactive Proofs

## TL;DR

This paper introduces a framework for distributed interactive proofs that significantly reduces proof size and rounds for verifying complex computations and graph properties in distributed settings.

## Contribution

It develops a general method to convert standard interactive protocols into distributed ones with small proof size, improving efficiency for various computational and graph problems.

## Key findings

- Distributed protocols for $O(n)$ time computations with $O( ext{log} n)$ proof size.
- Protocols for small space and NC computations with $O(1)$ rounds and $O( ext{log} n)$ proof size.
- Improved protocol for Graph Non-Isomorphism with 4 rounds and $O( ext{log} n)$ proof size.

## 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, i.e., 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 to ones where the verifier is distributed with short proof size. We show the following: * Every (centralized) computation that can be performed in time $O(n)$ 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. * 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, 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 naturally seeming barrier of $\log n$. We get a 5-round protocols with proof size $O(\log \log n)$ for a family of problems.