Physical ZKP for Connected Spanning Subgraph: Applications to Bridges Puzzle and Other Problems

TL;DR

This paper introduces a physical zero-knowledge proof protocol using a deck of cards to verify that a subgraph is connected and spanning without revealing it, with applications to NP-complete problems and puzzles.

Contribution

It presents an unconventional physical ZKP protocol with practical applications to complex graph problems and puzzles, expanding ZKP methods beyond digital implementations.

Findings

Protocol effectively verifies connected spanning subgraphs physically.

Applications include NP-complete problems and logic puzzles.

Demonstrates practical feasibility of physical ZKPs.

Abstract

An undirected graph $G$ is known to both the prover $P$ and the verifier $V$, but only $P$ knows a subgraph $H$ of $G$. Without revealing any information about $H$, $P$ wants to convince $V$ that $H$ is a connected spanning subgraph of $G$, i.e. $H$ is connected and contains all vertices of $G$. In this paper, we propose an unconventional zero-knowledge proof protocol using a physical deck of cards, which enables $P$ to physically show that $H$ satisfies the condition without revealing it. We also show applications of this protocol to verify solutions of three well-known NP-complete problems: the Hamiltonian cycle problem, the maximum leaf spanning tree problem, and a popular logic puzzle called Bridges.