Automorphism Shuffles for Graphs and Hypergraphs and Its Applications
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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.
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