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

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 is an element of{0, 1, 2}).