Bounded Round Interactive Proofs in Finite Groups

Abstract

This paper considers “black box groups,” i.e., finite groups whose elements are uniquely encoded by strings of uniform length, with group operations being performed by a group oracleB. Let G, H be such groups, each given by a list of generators. It is known that the problem of membership in G belongs to ${\text{NP}}^B $ [L. Babai and E. Szemeredi, Proceedings of the 25th IEEE Symposium on the Foundation of Computer Science, 1984, pp. 229–240]. The following problems are shown to belong to the complexity class ${\text{AM}}^B $; i.e., they possess bounded-round randomized interactive proofs (Arthur–Merlin protocols): nonmembership in G, the verification of the order of G, isomorphism of G and H, and checking the list of composition factors of G. A group oracle B is constructed, under which none of these problems belongs to ${\text{NP}}^B $, even for abelian groups.All the results extend to “black box factor groups,” i.e., groups defined as factor groups $G/N$, where G is a black box group, $N \triangleleft ...