Efficient Card-Based Protocols for Symmetric and Partially Doubly Symmetric Functions

Symmetric functions are an important class of functions that includes many interesting functions. Constructing efficient card-based protocols for symmetric functions is an important research topic in card-based cryptography. In 2020, Ruangwises and Itoh designed a protocol for symmetric functions f : {0, 1}n → R using 2n + 2 cards of a two-colored deck. Whether the number of cards in this protocol could be reduced is an open problem. In this paper, we design protocols for symmetric functions using 2n + 1 cards of a two-colored deck, which solve the above problem affirmatively. We also propose protocols for symmetric functions using 2n cards of a three-colored deck, while the existing 2n-card protocol requires a four-colored deck. By further restricting the class of functions to {0}-partially doubly symmetric functions, we can construct even more efficient protocols. In particular, we construct protocols for {0}-partially doubly symmetric functions using 2n cards of a two-colored deck, while the existing protocol requires 2n + 1 cards.