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Card-Based Zero-Knowledge Proof Protocol for Pancake Sorting

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Innovative Security Solutions for Information Technology and Communications (SecITC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13809))

Abstract

Assume that, given a sequence of n integers from 1 to n arranged in random order, we want to sort them, provided that the only acceptable operation is a prefix reversal, which means to take any number of integers (sub-sequence) from the left of the sequence, reverse the order of the sub-sequence, and return them to the original sequence. This problem is called “pancake sorting,” and sorting an arbitrary sequence with the minimum number of operations restricted in this way is known to be NP-hard. In this paper, we consider applying the concept of zero-knowledge proofs to the pancake sorting problem. That is, we design a physical zero-knowledge proof protocol in which a user (the prover) who knows how to sort a given sequence with \(\ell \) operations can convince another user (the verifier) that the prover knows this information without divulging it.

Y. Komano—Presently, the author is with Chiba Institute of Technology, Japan.

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Notes

  1. 1.

    In this way, we sometimes use the terms “sequence” and “permutation” interchangeably.

  2. 2.

    https://oeis.org/A058986.

  3. 3.

    Here, \(E_n(y_1)\) is the second row, \(E_n(y_2)\) is the third row, and so on.

  4. 4.

    Instead of a pile-scramble shuffle, one may use a “pile-shifting shuffle” [47, 72].

  5. 5.

    This generalization was pointed out by Koji Nuida.

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Acknowledgements

We thank the anonymous referees, whose comments have helped us improve the presentation of the paper. This work was supported by Grant-in-Aid for Scientific Research (JP18H05289, JP21K11881). We thank Koji Nuida for advising us to generalize the problem, as described in the third paragraph of Sect. 6.

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Authors and Affiliations

  1. Toshiba Corporation, 1 Komukai-Toshiba-Cho, Saiwai-ku, Kawasaki, Japan

    Yuichi Komano

  2. Tohoku University, 6–3 Aramaki-Aza-Aoba, Aoba-ku, Sendai, Japan

    Takaaki Mizuki

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  1. Yuichi Komano
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  2. Takaaki Mizuki
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Correspondence to Yuichi Komano .

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Editors and Affiliations

  1. University of Catania, Catania, Italy

    Giampaolo Bella

  2. Bucharest University of Economic Studies, Bucharest, Romania

    Mihai Doinea

  3. Edith Cowan University, Joondalup, WA, Australia

    Helge Janicke

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© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

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Komano, Y., Mizuki, T. (2023). Card-Based Zero-Knowledge Proof Protocol for Pancake Sorting. In: Bella, G., Doinea, M., Janicke, H. (eds) Innovative Security Solutions for Information Technology and Communications. SecITC 2022. Lecture Notes in Computer Science, vol 13809. Springer, Cham. https://doi.org/10.1007/978-3-031-32636-3_13

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  • DOI: https://doi.org/10.1007/978-3-031-32636-3_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-32635-6

  • Online ISBN: 978-3-031-32636-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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