Card-based Secure Sorting Protocols based on the Sorting Networks

Card-based cryptography enables us to realize secure computation using physical cards with simple manual operations. The efficiency of a card-based protocol is evaluated by the number of shuffles and cards. The Haga et al. (IWSEC2022) is the only work to address card-based secure sorting. They proposed two protocols: a Las Vegas protocol and a finite-runtime protocol. The finite-runtime means that the number of shuffles is deterministic. In this paper, we propose a general converter that transforms an arbitrary sorting network into a finite-runtime card-based sorting protocol. The converter uses Haga et al.'s finite-runtime protocol for $n=2$ as its building block. Furthermore, we show card-based sorting protocol obtained by applying the converter to the AKS sorts. As a result, we improve the number of additional cards from $O(n^{2}+m)$ to $O(n+m)$ compared to Haga et al. On the other hand, the number of shuffles increases from $O(nm)$ to $O(nm\log n)$.