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 ($k \in \{0,1,2\}$).

翻译：基于物理牌组的安全多方计算（通常称为卡牌密码学）在过去十年间得到了广泛研究。目前已开发出多种计算布尔函数的卡牌协议。由于每个输入位通常由两张牌编码，计算一个$n$变量布尔函数至少需要$2n$张牌。我们关注使用恰好$2n$张牌的最优协议，并重点研究对称函数。本文通过将所有对称布尔函数划分为若干NPN等价类，形式化描述了开发$2n$张牌协议以计算$n$变量对称布尔函数的问题。我们总结了现有能计算这些类中若干代表性函数的协议，并解决了$n=4,5,6,7$情况下的一些开放问题。特别地，我们开发了一个计算函数$k$Mod3的协议，该函数用于判断所有输入之和是否与$k$模3同余（$k \in \{0,1,2\}$）。