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Upper Bounds on the Number of Shuffles for Two-Helping-Card Multi-Input AND Protocols

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Cryptology and Network Security (CANS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14342))

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  • The original version of this chapter was revised: The change was updated and the correct chapter title is “Upper Bounds on the Number of Shuffles for Two-Helping-Card Multi-Input AND Protocols”. The correction to this chapter is available to https://doi.org/10.1007/978-981-99-7563-1_26

Abstract

Card-based cryptography uses a physical deck of cards to achieve secure computations. To evaluate the performance of card-based protocols, the numbers of helping cards and shuffles required to execute are often used as evaluation metrics. In this paper, we focus on n-input AND protocols that use at most two helping cards, and investigate how many shuffles suffice to construct such a two-helping-card AND protocol. Since the Mizuki–Sone two-input AND protocol uses two helping cards and it can be repeatedly applied \(n-1\) times to perform a secure n-input AND computation, an obvious upper bound on the number of required shuffles is \(n-1\). In this paper, to obtain better bounds (than \(n-1\)), we consider making use of the “batching” technique, which was developed by Shinagawa and Nuida in 2020 to reduce the number of shuffles. Specifically, we first formulate the class of two-helping-card n-input AND protocols obtained by applying the batching technique to the Mizuki–Sone AND protocol, and then show n-input AND protocols requiring the minimum number of shuffles (among the class) for the case of \(2\le n \le 500\).

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Change history

  • 08 January 2024

    A correction has been published.

Notes

  1. 1.

    These two cards are actually a commitment to \(\overline{x} \wedge y\).

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Acknowledgements

We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper. This work was supported in part by JSPS KAKENHI Grant Numbers JP21K11881 and JP23H00479.

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Authors and Affiliations

  1. National Institute of Technology, Ichinoseki College, Ichinoseki, Iwate, Japan

    Takuto Yoshida & Eikoh Chida

  2. Tohoku University, Sendai, Japan

    Kodai Tanaka, Keisuke Nakabayashi & Takaaki Mizuki

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  1. Takuto Yoshida
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  2. Kodai Tanaka
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Correspondence to Takuto Yoshida .

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Editors and Affiliations

  1. University of North Carolina, Greensboro, NC, USA

    Jing Deng

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

  3. Augusta University, Augusta, GA, USA

    Alexander A. Schwarzmann

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Yoshida, T., Tanaka, K., Nakabayashi, K., Chida, E., Mizuki, T. (2023). Upper Bounds on the Number of Shuffles for Two-Helping-Card Multi-Input AND Protocols. In: Deng, J., Kolesnikov, V., Schwarzmann, A.A. (eds) Cryptology and Network Security. CANS 2023. Lecture Notes in Computer Science, vol 14342. Springer, Singapore. https://doi.org/10.1007/978-981-99-7563-1_10

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  • DOI: https://doi.org/10.1007/978-981-99-7563-1_10

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