Automorphism Shuffles for Graphs and Hypergraphs and Its Applications

TL;DR

This paper introduces improved graph and hypergraph shuffles for card-based cryptography, reducing card requirements and expanding applications, including implementing cyclic group shuffles and hypergraph automorphisms.

Contribution

It presents a new protocol for graph shuffles with fewer cards, demonstrates that cyclic group shuffles are graph shuffles, and introduces hypergraph shuffles with implementation methods.

Findings

New graph shuffle protocol with 2n+m cards

Cyclic group shuffles are a subset of graph shuffles

Hypergraph shuffles can be implemented with pile-scramble shuffles

Abstract

In card-based cryptography, a deck of physical cards is used to achieve secure computation. A shuffle, which randomly permutes a card-sequence along with some probability distribution, ensures the security of a card-based protocol. The authors proposed a new class of shuffles called graph shuffles, which randomly permutes a card-sequence by an automorphism of a directed graph (New Generation Computing 2022). For a directed graph $G$ with $n$ vertices and $m$ edges, such a shuffle could be implemented with pile-scramble shuffles with $2(n+m)$ cards. In this paper, we study graph shuffles and give an implementation, an application, and a slight generalization of them. First, we propose a new protocol for graph shuffles with $2n+m$ cards. Second, as a new application of graph shuffles, we show that any cyclic group shuffle, which is a shuffle over a cyclic group, is a graph shuffle associated with some graph. Third, we define a hypergraph shuffle, which is a shuffle by an automorphism of a hypergraph, and show that any hypergraph shuffle can also be implemented with pile-scramble shuffles.