A nonadaptive NC checker for permutation group intersection

In this paper we design a nonadaptive NC checker for permutation group intersection, sharpening a result from M. Blum and S. Kannan (1995). This is a consequence of two results. First we show that a nontrivial permutation in the intersection of two given permutation groups (described by lists of generators) can be computed by an NC algorithm with one round of parallel queries to the group intersection problem. Next we design a two-round interactive proof system for the complement of the group intersection problem, for which the honest prover can be simulated by an NC algorithm with one round of parallel queries to group intersection. As a consequence we also have nonadaptive NC checkers for some related group-theoretic problems. On the technical side, we define a generalization of wreath products of permutation groups. This product plays a crucial role in the design of the nonadaptive checkers.