Abstract
Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Card-based protocols to compute various Boolean functions have been developed. As each input bit is typically encoded by two cards, computing an n-variable Boolean function requires at least 2n cards. We are interested in optimal protocols that use exactly 2n cards. In particular, we focus on symmetric functions. In this paper, we formulate the problem of developing 2n-card protocols to compute n-variable symmetric Boolean functions by classifying all such functions into several NPN-equivalence classes. We then summarize existing protocols that can compute some representative functions from these classes, and also solve some open problems in the cases \(n=4\), 5, 6, and 7. In particular, we develop a protocol to compute a function kMod3, which determines whether the sum of all inputs is congruent to k modulo 3 (\(k \in \{0,1,2\}\)).
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Notes
- 1.
The number of shuffles is defined to be the number of times we perform shuffling operations described in Sect. 4.1.
- 2.
A doubly symmetric function is a function \(S_X^n\) such that \(x \in X\) if and only if \(n-x \in X\) for every \(x \in \{0,1,...,n\}\).
- 3.
A Las Vegas protocol is a protocol that does not guarantee a finite number of shuffles, but has a finite expected number of shuffles.
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The author would like to thank Daiki Miyahara for a valuable discussion on this research.
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Ruangwises, S. (2023). The Landscape of Computing Symmetric n-Variable Functions with 2n Cards. In: Ábrahám, E., Dubslaff, C., Tarifa, S.L.T. (eds) Theoretical Aspects of Computing – ICTAC 2023. ICTAC 2023. Lecture Notes in Computer Science, vol 14446. Springer, Cham. https://doi.org/10.1007/978-3-031-47963-2_6
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