Secure multi-party computation using a physical deck of cards, often called card-based cryptography, has been extensively studied during the past decade. Many card-based protocols to securely 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, where the output only depends on the number of 1s in the inputs. 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 of the open problems by developing protocols to compute particular functions in the cases $n=4$, $5$, $6$, and $7$.

翻译：利用实体扑克牌进行安全多方计算（通常称为基于卡牌的密码学）在过去十年中得到了广泛研究。目前已开发出多种基于卡牌的安全计算布尔函数的协议。由于每个输入比特通常由两张卡牌编码，因此计算 $n$ 变量布尔函数至少需要 $2n$ 张卡牌。我们关注恰好使用 $2n$ 张卡牌的最优协议，特别聚焦于对称函数——其输出仅取决于输入中 1 的个数。本文通过将所有对称函数划分为若干 NPN 等价类，系统阐述了为计算 $n$ 变量对称布尔函数而设计 $2n$ 张卡牌协议的问题。随后我们总结了现有协议中可计算这些等价类中部分代表性函数的方法，并针对 $n=4$、$5$、$6$ 和 $7$ 的情况开发了计算特定函数的协议，从而解决了部分开放性问题。