Extended Addition Protocol and Efficient Voting Protocols Using Regular Polygon Cards

Abstract

AbstractCard-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 $$n-1$$ n - 1 , and constructed an addition protocol over $$\mathbb {Z}/n\mathbb {Z}$$ Z / n Z and a voting protocol with v voters and c candidates when $$v<n$$ v < n . In this paper, we propose an addition protocol over $$\mathbb {Z}/mn\mathbb {Z}$$ Z / m n Z 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 $$c(\lceil \frac{v+1}{n} \rceil + v + 1)$$ c ( ⌈ v + 1 n ⌉ + v + 1 ) cards and $$v+1$$ v + 1 shuffles without restriction. Our second voting protocol reduces the number of cards to $$\lceil \frac{v+1}{n} \rceil n + v + 1$$ ⌈ v + 1 n ⌉ n + v + 1 when $$v < n$$ v < n and $$c\le n$$ c ≤ n .