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Physical ZKP for Makaro Using a Standard Deck of Cards

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Theory and Applications of Models of Computation (TAMC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13571))

Abstract

Makaro is a logic puzzle with an objective to fill numbers into a rectangular grid to satisfy certain conditions. In 2018, Bultel et al. developed a physical zero-knowledge proof (ZKP) protocol for Makaro using a deck of cards, which allows a prover to physically convince a verifier that he/she knows a solution of the puzzle without revealing it. However, their protocol requires several identical copies of some cards, making it impractical as a deck of playing cards found in everyday life typically consists of all different cards. In this paper, we propose a new ZKP protocol for Makaro that can be implemented using a standard deck (a deck consisting of all different cards). Our protocol also uses asymptotically less cards than the protocol of Bultel et al. Most importantly, we develop a general method to encode a number with a sequence of all different cards. This allows us to securely compute several numerical functions using a standard deck, such as verifying that two given numbers are different and verifying that a number is the largest one among the given numbers.

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Notes

  1. 1.

    Although a “standard deck” of playing cards found in everyday life typically consists of 52 different cards, in theory we study a general setting where the deck is arbitrarily large, consisting of all different cards.

  2. 2.

    Assume that we have \(\ell \) cards with different numbers, e.g. cards with numbers \(1,2,...,\ell \). In the example in Fig. 2, we can, for instance, regard cards 1, 2, 3 on cells with numbers 1, 2, 3 in the top-left room as \(\alpha _1,\alpha _2,\alpha _3\), cards 4, 5 on cells with numbers 1, 2 in the top-center room as \(\beta _1,\beta _2\), cards 6, 7, 8, 9, 10 on cells with numbers 1, 2, 3, 4, 5 in the top-right room as \(\gamma _1,\gamma _2,\gamma _3,\gamma _4,\gamma _5\), and so on.

  3. 3.

    Some of the cells may be in the same room, but this does not affect the conversion as we apply the conversion protocol to each cell card one by one.

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Authors and Affiliations

  1. Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan

    Suthee Ruangwises & Toshiya Itoh

Authors
  1. Suthee Ruangwises
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  2. Toshiya Itoh
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Correspondence to Suthee Ruangwises .

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Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. University of New Brunswick, Fredericton, NB, Canada

    Donglei Du

  3. Tianjin University of Technology, Tianjin, China

    Chenchen Wu

  4. Beijing University of Technology, Beijing, China

    Dachuan Xu

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Ruangwises, S., Itoh, T. (2022). Physical ZKP for Makaro Using a Standard Deck of Cards. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Theory and Applications of Models of Computation. TAMC 2022. Lecture Notes in Computer Science, vol 13571. Springer, Cham. https://doi.org/10.1007/978-3-031-20350-3_5

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  • DOI: https://doi.org/10.1007/978-3-031-20350-3_5

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  • Print ISBN: 978-3-031-20349-7

  • Online ISBN: 978-3-031-20350-3

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