Bulletin of the
Korean Mathematical Society BKMS

We use an idea of linear representations of the symmetric group to reduce the number of communication rounds in the verification protocol, proposed in Crypto 2005 by Peng et al., of a shuffling. We assume Paillier encryption scheme with which we can apply some known zero-knowledge proofs following the same line of approaches of Peng et al. Incidence matrices of $1$-subsets and $2$-subsets of a finite set is intensively used for the implementation, and the idea of $\lambda$-designs is employed for the improvement of the computational complexity.

Keywords: proof of shuffle, mix-net, representation of symmetric group, homomorphic encryption, $\lambda$-designs

MSC numbers: Primary 94A60; Secondary 11T71