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Card-Minimal Protocols for Symmetric Boolean Functions of More than Seven Inputs

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Theoretical Aspects of Computing – ICTAC 2022 (ICTAC 2022)

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Abstract

Secure computations enable us to obtain the output value of a predetermined function while keeping its input values secret. Card-based cryptography realizes secure computations using a deck of physical cards. Because each input bit is typically encoded with two cards, an obvious lower bound on the number of required cards is 2n when securely computing an n-input Boolean function. Although card-based protocols often require helping cards (aside from 2n cards needed for input), there exist several protocols that require no helping card, namely, helping-card-free protocols. For example, there are helping-card-free protocols for several fundamental functions, such as the AND, XOR, and three-input majority functions. However, in general, it remains an open problem whether all Boolean functions have their helping-card-free protocols. In this study, we focus our attention on symmetric functions: Whereas the best known result is that any n-input symmetric function can be securely computed using two helping cards, we present a helping-card-free protocol for an arbitrary n-input symmetric function such that \(n > 7\). Because much attention has been drawn to constructing card-based protocols using the minimum number of cards, our protocol, which is card-minimal, would be of interest to the research area of card-based cryptography.

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Notes

  1. 1.

    There is another computation model where private actions are allowed, e.g. [2, 14,15,16,17, 29, 32].

  2. 2.

    This method is originated from the previous protocol [36] proposed by Shinagawa et al.

  3. 3.

    This protocol is inspired by the Mizuki–Kumamoto–Sone AND protocol [23]; the procedure is the same up to the middle.

  4. 4.

    The idea of adding two pos. encodings of the same color was suggested by Kazumasa Shinagawa.

  5. 5.

    Generally, there are two cards of the same color and one card of the other color.

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Acknowledgements

We thank the anonymous referees, whose comments have helped us improve the presentation of the paper. We thank Kazumasa Shinagawa for the idea of computing \(x_5+x_6\) for Case 2 in Sect. 4.1. This work was supported in part by JSPS KAKENHI Grant Number JP21K11881.

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Authors and Affiliations

  1. Tohoku University, Sendai, Japan

    Hayato Shikata, Kodai Toyoda & Takaaki Mizuki

  2. The University of Electro-Communications, Tokyo, Japan

    Daiki Miyahara

  3. National Institute of Advanced Industrial Science and Technology, Tokyo, Japan

    Daiki Miyahara & Takaaki Mizuki

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  1. Hayato Shikata
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Correspondence to Hayato Shikata .

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  1. TU München, München, Germany

    Helmut Seidl

  2. Southwest University, Chongqing, China

    Zhiming Liu

  3. NASA Ames Research Center, Moffett Field, CA, USA

    Corina S. Pasareanu

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Shikata, H., Toyoda, K., Miyahara, D., Mizuki, T. (2022). Card-Minimal Protocols for Symmetric Boolean Functions of More than Seven Inputs. In: Seidl, H., Liu, Z., Pasareanu, C.S. (eds) Theoretical Aspects of Computing – ICTAC 2022. ICTAC 2022. Lecture Notes in Computer Science, vol 13572. Springer, Cham. https://doi.org/10.1007/978-3-031-17715-6_25

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