Solvable black-box group problems are low for PP

Abstract

Let B = B m m>0 be a countable family of finite groups whose elements are uniquely encoded as strings of uniform length, and group operations are computable in time bounded by a polynomial in m. A black-box group over a group family B is a subgroup of some member of B and is presented by a generator set. In this paper we study the complexity of several algorithmic problems for abelian black-box groups and solvable black-box groups. We design a suitable oracle algorithm that computes an independent set of generators for a given abelian black-box group. Using this we show that the problems of Membership Testing, Group Intersection, Order Verification, and Group Isomorphism over abelian black-box groups are in SPP. We also show that Group Factorization, Coset Intersection, and Double Coset Membership problems over abelian black-box groups are in LWPP. As a consequence, all these problems are low for PP and CP. We define the notion of canonical generator sets for classes of groups and show that solvable black-box groups have canonical generator sets. We design a suitable randomized oracle algorithm that computes a canonical generator set for a given solvable black-box group. Using this algorithm we show that Membership Testing, Order Verification, and Group Isomorphism for solvable groups are in ZPP SPP and hence low for PP. We also show that Group Intersection, Group Factorization, Coset Intersection, and Double Coset Membership for solvable groups are low for PP.