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Card-Minimal Protocols for Three-Input Functions with Standard Playing Cards

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Progress in Cryptology - AFRICACRYPT 2022 (AFRICACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13503))

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Abstract

A protocol realizing a secure computation using a deck of physical cards is called a card-based cryptographic protocol. Since Niemi and Renvall first proposed a few protocols using a commercially available deck of playing cards in 1999, several protocols for the two-input AND and XOR functions have been proposed. By combining these existing protocols, one can construct a protocol for any Boolean function using a standard deck of playing cards. However, the minimal numbers of cards needed for Boolean functions having more than two inputs have not been revealed so much. Recently, Koyama et al. developed a card-minimal three-input AND protocol. In this study, by extending Koyama’s AND protocol, we construct a card-minimal protocol for the three-input majority function. Furthermore, carrying the idea behind these protocols further, we provide a generic card-minimal three-input protocol, which covers many important three-input Boolean functions.

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Notes

  1. 1.

    This paper (and the literature) assume the encoding (2), i.e., a two-card-per-bit encoding, when discussing the card-minimality of protocols; thus, an n-input (Boolean function) protocol always needs 2n cards for input commitments, and such a protocol using only 2n cards is card-minimal.

  2. 2.

    A NOT protocol can be simply constructed: swapping two cards comprising a commitment produces a commitment to the negation.

  3. 3.

    This step was proposed by Koch et al. [6], reducing the number of shuffles.

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Acknowledgements

We thank the anonymous referees, whose comments have helped us improve the presentation of the paper. We also thank Hiroto Koyama for his cooperation in preparing a Japanese draft version of Sect. 3 at an earlier stage of this work. This work was supported in part by JSPS KAKENHI Grant Numbers JP21K11881 and JP19H01104.

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Authors and Affiliations

  1. Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara, 630-0192, Japan

    Rikuo Haga & Yuichi Hayashi

  2. The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan

    Daiki Miyahara

  3. Tohoku University, Aramak-Aza-Aoba, Aoba, Sendai, 980-8576, Japan

    Takaaki Mizuki

  4. National Institute of Advanced Industrial Science and Technology (AIST), 2-3-26 Aomi, Koto-ku, Tokyo, 135-0064, Japan

    Yuichi Hayashi, Daiki Miyahara & Takaaki Mizuki

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  1. Rikuo Haga
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  4. Takaaki Mizuki
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Correspondence to Rikuo Haga .

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Editors and Affiliations

  1. Radboud University, Nijmegen, The Netherlands

    Lejla Batina

  2. Radboud University, Nijmegen, Gelderland, The Netherlands

    Joan Daemen

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Haga, R., Hayashi, Y., Miyahara, D., Mizuki, T. (2022). Card-Minimal Protocols for Three-Input Functions with Standard Playing Cards. In: Batina, L., Daemen, J. (eds) Progress in Cryptology - AFRICACRYPT 2022. AFRICACRYPT 2022. Lecture Notes in Computer Science, vol 13503. Springer, Cham. https://doi.org/10.1007/978-3-031-17433-9_19

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