Card-Minimal Protocols for Symmetric Boolean Functions of More than Seven Inputs

Secure computations enable us to obtain the output value of a predetermined function while keeping its input values secret. Card-based cryptography realizes secure computations using a deck of physical cards. Because each input bit is typically encoded with two cards, an obvious lower bound on the number of required cards is 2 n when securely computing an n -input Boolean function. Although card-based protocols often require helping cards (aside from 2 n cards needed for input), there exist several protocols that require no helping card, namely, helping-card-free protocols. For example, there are helping-card-free protocols for several fundamental functions, such as the AND, XOR, and three-input majority functions. However, in general, it remains an open problem whether all Boolean functions have their helping-card-free protocols. In this study, we focus our attention on symmetric functions: Whereas the best known result is that any n -input symmetric function can be securely computed using two helping cards, we present a helping-card-free protocol for an arbitrary n -input symmetric function such that n > 7 . Because much attention has been drawn to constructing card-based protocols using the minimum number of cards, our protocol, which is card-minimal, would be of interest to the research area of card-based cryptography.