Card-Based Zero-Knowledge Proof Protocols for Graph Problems and Their Computational Model

Zero-Knowledge Proof (ZKP) is a cryptographic technique that enables a prover to convince a verifier that a given statement is true without revealing any information other than its truth. It is known that ZKP can be realized by physical objects such as a deck of cards; recently, many such “card-based” ZKP protocols for pencil puzzles (such as Sudoku and Numberlink) have been devised. In this paper, we shift our attention to graph theory problems from pencil puzzles: We propose card-based ZKP protocols for the graph 3-coloring problem and the graph isomorphism problem. Similar to most of the existing card-based ZKP protocols, our two protocols have no soundness error. The proposed protocols can be implemented without any technical knowledge, and the principle of zero-knowledge proof is easy to understand. In particular, our protocol for the graph isomorphism problem requires only three shuffles regardless of the sizes of a pair of given graphs. In addition, to deal with our proposed protocols more rigorously, we present a formal framework for card-based ZKP protocols which are non-interactive and have no soundness error.