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