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

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Provable and Practical Security (ProvSec 2021)

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Abstract

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.

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Notes

  1. 1.

    An encoding rule representing a positive integer in this manner was first considered by Shinagawa et al. in 2015 [31].

  2. 2.

    One might think that the resulting order could be easily known because there are only six possibilities. One possible implementation is to put piles of cards into a box or ball whose inside is invisible from outside and then throw it up to randomize the order of them (cf. [32]).

  3. 3.

    All other sequences in U start with illegal witness subsequences.

  4. 4.

    Very recently, Ruangwises and Itoh [25, 28] implied that a card-based ZKP protocol for the Hamiltonian cycle problem can be constructed in a similar way to our protocol for the graph isomorphism problem.

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Acknowledgements

We thank the anonymous referees, whose comments have helped us improve the presentation of the paper. We would like to thank Hideaki Sone for his cooperation in preparing a Japanese draft version at an earlier stage of this work. We would also like to thank Kazumasa Shinagawa for his idea improving a protocol for the 3-coloring problem. The second author is grateful to Haruka Mizuta for helpful discussions at the beginning of this work. This work was supported in part by JSPS KAKENHI Grant Numbers JP19J21153 and JP21K11881.

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Authors and Affiliations

  1. The University of Electro-Communications, Tokyo, Japan

    Daiki Miyahara, Hiromichi Haneda & Takaaki Mizuki

  2. National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

    Daiki Miyahara & Takaaki Mizuki

Authors
  1. Daiki Miyahara
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  2. Hiromichi Haneda
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  3. Takaaki Mizuki
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Correspondence to Daiki Miyahara or Takaaki Mizuki .

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Editors and Affiliations

  1. South China Agricultural University, Guangzhou, China

    Qiong Huang

  2. Shanghai Jiao Tong University, Shanghai, China

    Yu Yu

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Miyahara, D., Haneda, H., Mizuki, T. (2021). Card-Based Zero-Knowledge Proof Protocols for Graph Problems and Their Computational Model. In: Huang, Q., Yu, Y. (eds) Provable and Practical Security. ProvSec 2021. Lecture Notes in Computer Science(), vol 13059. Springer, Cham. https://doi.org/10.1007/978-3-030-90402-9_8

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