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Card-based protocols using unequal division shuffles

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Abstract

Card-based cryptographic protocols can perform secure computation of Boolean functions. In 2013, Cheung et al. presented a protocol that securely produces a hidden AND value using five cards; however, it fails with a probability of 1/2. The protocol uses an unconventional shuffle operation called an unequal division shuffle; after a sequence of five cards is divided into a two-card portion and a three-card portion, these two portions are randomly switched so that nobody knows which is which. In this paper, we first show that the protocol proposed by Cheung et al. securely produces not only a hidden AND value but also a hidden OR value (with a probability of 1/2). We then modify their protocol such that, even when it fails, we can still evaluate the AND value in the clear. Furthermore, we present two five-card copy protocols (which can duplicate a hidden value) using unequal division shuffle. Because the most efficient copy protocol currently known requires six cards, our new protocols improve upon the existing results. We also design a general copy protocol that produces multiple copies using an unequal division shuffle. Furthermore, we show feasible implementations of unequal division shuffles by the use of card cases.

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Notes

  1. As seen in Sect. 4, we repeat applying a shuffle until is found, and the probability that appears is 1 / 2.

  2. If players have difficulty to shuffle the two boxes publicly, they may shuffle the two boxes behind their backs (Ueda et al. 2016).

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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 by JSPS KAKENHI Grant Nos. 25289068, 26330001, and 17K00001.

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Authors and Affiliations

  1. Sone-Mizuki Laboratory, Graduate School of Information Sciences, Tohoku University, 6-3 Aramaki-Aza-Aoba, Aoba, Sendai, 980-8578, Japan

    Akihiro Nishimura & Takuya Nishida

  2. Graduate School of Information Sciences, Nara Institute of Science and Technology, 8916–5 Takayama, Ikoma, Nara, 630-0192, Japan

    Yu-ichi Hayashi

  3. Cyberscience Center, Tohoku University, 6-3 Aramaki-Aza-Aoba, Aoba, Sendai, 980-8578, Japan

    Takaaki Mizuki & Hideaki Sone

Authors
  1. Akihiro Nishimura
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  2. Takuya Nishida
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  3. Yu-ichi Hayashi
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  4. Takaaki Mizuki
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  5. Hideaki Sone
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Correspondence to Takaaki Mizuki.

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The authors declare that they have no conflict of interest.

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This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by C.M. Vide, A.H. Dediu.

An earlier version of this study was presented at 4th International Conference on the Theory and Practice of Natural Computing, TPNC 2015, Spain, December 15–16, 2015, and appeared in Proc. TPNC 2015, Lecture Notes in Computer Science, Springer International Publishing, vol. 9477, pp. 109–120, 2015 (Nishimura et al. 2015).

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Nishimura, A., Nishida, T., Hayashi, Yi. et al. Card-based protocols using unequal division shuffles. Soft Comput 22, 361–371 (2018). https://doi.org/10.1007/s00500-017-2858-2

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