Securely Computing the $n$-Variable Equality Function with $2n$ Cards

Research in the area of secure multi-party computation using a deck of playing cards, often called card-based cryptography, started from the introduction of the five-card trick protocol to compute the logical AND function by den Boar in 1989. Since then, many card-based protocols to compute various functions have been developed. In this paper, we propose two new protocols that securely compute the $n$-variable equality function (determining whether all inputs are equal) $E: \{0,1\}^n \rightarrow \{0,1\}$ using $2n$ cards. The first protocol can be generalized to compute any doubly symmetric function $f: \{0,1\}^n \rightarrow \mathbb{Z}$ using $2n$ cards, and any symmetric function $f: \{0,1\}^n \rightarrow \mathbb{Z}$ using $2n+2$ cards. The second protocol can be generalized to compute the $k$-candidate $n$-variable equality function $E: (\mathbb{Z}/k\mathbb{Z})^n \rightarrow \{0,1\}$ using $2 \lceil \lg k \rceil n$ cards.