Efficient mental card shuffling via optimised arbitrary-sized benes permutation network

Abstract

The presumption of player distrust and untrustworthiness in mental card gaming results in the formulation of complex and compute-intensive protocols, particularly for shuffling. We present a robust, verifiable and efficient card shuffling protocol based on an optimisation of Chang-Melham arbitrary-sized (AS) Benes permutation network (PN), which can flexibly accommodates variable pack sizes, achieving optimal shuffling performance. We also outline the use of these PNs in a distributed (among η players) construction, which combines the best attributes of Abe and Jakobsson-Juels mix-net formalisms. Card shuffling can therefore be executed on a structurally simple mix-net – with only t + 1 PNs required for operational robustness against collusion by t cheating players, and efficient zero knowledge proofs (ZKP) to verify correct shuffling by each player. Shuffling efficiency is also enhanced by our limited application of verifiable secret sharing (VSS) on the ElGamal keys. The resultant protocol achieves an asymptotic complexity of O(tN lg N) for N inputs; which is comparable or superior to previous schemes.