Proving a Shuffle Using Representations of the Symmetric Group

Abstract

Shuffling protocol proposed in Crypto 2005 by Peng et al. is improved so that the number of communication rounds for the verification is reduced. We use an idea of linear representations of the symmetric group and a property of the incidence matrices of 1-subsets and 2-subsets of a finite set. The proposed protocol is valid for mix networks implemented with Paillier encryption schemes with which we can apply some known zero-knowledge proofs following the same line of approaches of Peng et al. [24]. The overall cost for the verification, if we fully implement our idea, is more expensive than that of the original protocol by Peng et al. We, however , can control the level of computation cost for the verification by using the idea of ***-designs properly.