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International Workshop on Security

IWSEC 2023: Advances in Information and Computer Security pp 255–272Cite as

Check Alternating Patterns: A Physical Zero-Knowledge Proof for Moon-or-Sun

Part of the Lecture Notes in Computer Science book series (LNCS,volume 14128)

Abstract

A zero-knowledge proof (ZKP) allows a party to prove to another party that it knows some secret, such as the solution to a difficult puzzle, without revealing any information about it. We propose a physical zero-knowledge proof using only a deck of playing cards for solutions to a pencil puzzle called Moon-or-Sun. In this puzzle, one is given a grid of cells on which rooms, marked by thick black lines surrounding a connected set of cells, may contain a number of cells with a moon or a sun symbol. The goal is to find a loop passing through all rooms exactly once, and in each room either passes through all cells with a moon, or all cells with a sun symbol.

Finally, whenever the loop passes from one room to another, it must go through all cells with a moon if in the previous room it passed through all cells with a sun, and visa-versa. This last rule constitutes the main challenge for finding a physical zero-knowledge proof for this puzzle, as this must be verified without giving away through which borders the loop enters or leaves a given room. We design a card-based zero-knowledge proof of knowledge protocol for Moon-or-Sun solutions, together with an analysis of their properties. Our technique of verifying the alternation of a pattern along a non-disclosed path might be of independent interest for similar puzzles.

Keywords

  • Physical Zero-knowledge Proof
  • Pencil Puzzle
  • Card-based Cryptography
  • Moon-or-Sun
  • Nikoli Puzzle

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Notes

  1. 1.

    https://www.nikoli.co.jp/en/puzzles/moon_or_sun/.

  2. 2.

    Remember that the value of a commitment on a cell indicates the presence of line passing through the cell.

  3. 3.

    This means that rule 3 can be simultaneously verified for the target room.

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Acknowledgements

We thank the anonymous referees, whose comments have helped us improve the presentation of the paper. The fourth author was supported in part by Kayamori Foundation of Informational Science Advancement and JSPS KAKENHI Grant Number JP23H00479. The third and fifth authors were partially supported by the French ANR project ANR-18-CE39-0019 (MobiS5). Other programs also fund to write this paper, namely the French government research program “Investissements d’Avenir” through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) and the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01). Finally, the French ANR project DECRYPT (ANR-18-CE39-0007) and SEVERITAS (ANR-20-CE39-0009) also subsidize this work.

Author information

Authors and Affiliations

  1. University of Glasgow, Glasgow, UK

    Samuel Hand

  2. Paris Cité University, Paris, France

    Alexander Koch

  3. University Clermont Auvergne, LIMOS, CNRS UMR 6158, Aubière, France

    Pascal Lafourcade

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

    Daiki Miyahara

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

    Daiki Miyahara

  6. University of Limoges, XLIM, Limoges, France

    Léo Robert

Authors
  1. Samuel Hand
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  2. Alexander Koch
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  3. Pascal Lafourcade
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  4. Daiki Miyahara
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  5. Léo Robert
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Corresponding author

Correspondence to Daiki Miyahara .

Editor information

Editors and Affiliations

  1. Yokohama National University, Yokohama, Japan

    Junji Shikata

  2. Kobe University, Kobe, Japan

    Hiroki Kuzuno

A Full Description of XOR and Copy Protocols

A Full Description of XOR and Copy Protocols

XOR Protocol. Given commitments to \(a,b\in \{0,1\}\), the Mizuki–Sone XOR protocol [17] outputs a commitment to \(a\oplus b\):

figure an

This protocol proceeds as follows.

  1. 1.

    Rearrange the sequence: .

  2. 2.

    Apply a random bisection cut: .

  3. 3.

    Rearrange the sequence: .

  4. 4.

    Reveal the first and second cards in the sequence to obtain the output commitment as follows: .

Copy Protocol. Given a commitment to \(a\in \{0,1\}\) along with two commitments to 0, the Mizuki–Sone copy protocol [17] outputs two commitments to a:

figure as

This protocol proceeds as follows.

  1. 1.

    Rearrange the sequence as follows:

    figure at
  2. 2.

    Apply a random bisection cut to the sequence as follows:

    figure au
  3. 3.

    Rearrange the sequence as follows:

    figure av
  4. 4.

    Reveal the first and second cards in the sequence to obtain the output commitments as follows:

    figure aw

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Hand, S., Koch, A., Lafourcade, P., Miyahara, D., Robert, L. (2023). Check Alternating Patterns: A Physical Zero-Knowledge Proof for Moon-or-Sun. In: Shikata, J., Kuzuno, H. (eds) Advances in Information and Computer Security. IWSEC 2023. Lecture Notes in Computer Science, vol 14128. Springer, Cham. https://doi.org/10.1007/978-3-031-41326-1_14

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