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. 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.

This paper proposes graph linear notations and an extension of them with regular expressions. Graph linear notations are a set of strings to represent labeled general graphs. They are extended with regular expressions to represent sets of graphs by specifying chosen parts for selections and repetitions of certain induced subgraphs. Methods for the conversion between graph linear notations and labeled general graphs are shown. The NP-completeness of the membership problem for graph regular expressions is proved.