Securely computing the n-variable equality function with 2n cards

Abstract

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 Boer 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→{0,1} using 2n cards. The first protocol can be generalized to compute any doubly symmetric functionf:{0,1}n→Z using 2n cards, and any symmetric function f:{0,1}n→Z using 2n+2 cards. The second protocol can be generalized to compute the k-candidate n-variable equality function E:(Z/kZ)n→{0,1} using 2⌈lg⁡k⌉n cards.