The Landscape of Computing Symmetric $n$-Variable Functions with $2n$ Cards

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 $k$Mod3, which determines whether the sum of all inputs is congruent to $k$ modulo 3 ($k \in {0,1,2}$).