Trading group theory for randomness

In a previous paper [BS] we proved, using the elements of the theory of nilpotent groups, that some of the fundamental computational problems in matriz groups belong to NP. These problems were also shown to belong to coNP, assuming an unproven hypothesis concerning finite simple groups.

The aim of this paper is to replace most of the (proven and unproven) group theory of [BS] by elementary combinatorial arguments. The result we prove is that relative to a random oracle B, the mentioned matrix group problems belong to (NP∩coNP)^B.

The problems we consider are membership in and order of a matrix group given by a list of generators. These problems can be viewed as multidimensional versions of a close relative of the discrete logarithm problem. Hence NP∩coNP might be the lowest natural complexity class they may fit in.

We remark that the results remain valid for black box groups where group operations are performed by an oracle.

The tools we introduce seem interesting in their own right. We define a new hierarchy of complexity classes AM(k) “just above NP”, introducing Arthur vs. Merlin games, the bounded-away version of Papdimitriou's Games against Nature. We prove that in spite of their analogy with the polynomial time hierarchy, the finite levels of this hierarchy collapse to AM=AM(2). Using a combinatorial lemma on finite groups [BE], we construct a game by which the nondeterministic player (Merlin) is able to convince the random player (Arthur) about the relation [G]=N provided Arthur trusts conclusions based on statistical evidence (such as a Slowly-Strassen type “proof” of primality).

One can prove that AM consists precisely of those languages which belong to NP^B for almost every oracle B.

Our hierarchy has an interesting, still unclarified relation to another hierarchy, obtained by removing the central ingredient from the User vs. Expert games of Goldwasser, Micali and Rackoff.