Extended Addition Protocol and Efficient Voting Protocols Using Regular Polygon Cards

Card-based cryptography is a research field for realizing cryptographic protocols using a deck of physical cards. Shinagawa et al. proposed a regular n-sided polygon card, which can hold a value from 0 to n - 1, and constructed an addition protocol over Z/nZ and a voting protocol with v voters and c candidates when v < n. In this paper, we propose an addition protocol over Z/mnZ using regular n-sided polygon cards. Technically, we introduce a cyclic integer encoding and a rot-and-shift shuffle to extend the modulus from n to mn. In addition, we construct two voting protocols with v voters and c candidates using regular n-sided polygon cards. Our first voting protocol requires c([v+1/n] + v + 1) cards and v + 1 shuffles without restriction. Our second voting protocol reduces the number of cards to [v+1/n] n + v + 1 when v < n and c <= n.