Graph Automorphism Shuffles from Pile-Scramble Shuffles

Abstract

A pile-scramble shuffle is one of the most effective shuffles in card-based cryptography. Indeed, many card-based protocols are constructed from pile-scramble shuffles. This article aims to study the power of pile-scramble shuffles. In particular, for any directed graph G, we introduce a new protocol called "a graph shuffle protocol for G", and show that it is realized by using pile-scramble shuffles only. Our proposed protocol requires 2(n + m) cards, where n and m are the numbers of vertices and arrows of G, respectively. The number of pile-scramble shuffles is k + 1, where 1 ≤ k ≤ n is the number of distinct degrees of vertices in G. As an application, a random cut for n cards, which is also an important shuffle, can be realized by 2n cards and 2 pile-scramble shuffles.Secure computation; Card-based cryptography; Pile-scramble shuffles; Graph automorphisms Acknowledge. K. Miyamoto was partly supported by JSPS KAKENHI 20K14302. K. Shinagawa was partly supported by JSPS KAKENHI 21K17702. 1.2. Contribution. In this article, we show that a class of graph shuffles can be achieved by the use of PSSs only. Let G be a directed graph. A graph shuffle for G is a shuffle that arranges a sequence of cards according to an automorphism of G chosen uniformly at random. Our main contribution is to construct a card-based protocol that achieves a graph shuffle for any graph G. We call this a graph shuffle protocol for G. The number of cards in our protocol is 2(n + m), where n and m are the numbers of vertices and arrows of G, respectively. All shuffles in our protocol are PSSs.1 We regard a pile-shifting shuffle which is a pile version of a random cut as a random cut.