Polynomial-time theory of matrix groups

Abstract

We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles.Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log.One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity).As a by-product, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N , does the group generated by A have a semisimple quotient of order ≥ N ?We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.

On the Number ofp-Regular Elements in Finite Simple Groups

“…This result allows p-regular elements to be discovered in S efficiently by random sampling, a key ingredient to several recent algorithms for matrix groups [2,4,3]. We emphasize that p may or may not be equal to the characteristic of the field of definition of S when S is of Lie type.…”

Algorithms for Group Isomorphism via Group Extensions and Cohomology

“…The Babai-Beals filtration was defined and used in the context of algorithms for matrix groups [BB99,BBS09]-where the groups are given by a generating set of matrices, and the goal is algorithms which run in time polynomial in the input size, which can be polylogarithmic in |G|. In the context of GpI, it has also been used successfully in the polynomial-time algorithm for semisimple groups [BCGQ11,BCQ12].…”

Algorithms for matrix groups

“…In summary, the black-box approach aims to construct this characteristic series C for G ≤ GL(d, q) using black-box algorithms. In 2009, as a culmination of 25 years of work, Babai, Beals & Seress [10] proved that, subject to the existence of a discrete log oracle and the ability to factorise integers of the form q i − 1 for 1 ≤ i ≤ d, there exist black-box polynomial-time Las Vegas algorithms to construct C for a large class of matrix groups. Building on results of [9], [56], [81] and [93], they solve the major tasks identified in Section 3 (and others) for groups in this class.…”

“…Building on results of [9], [56], [81] and [93], they solve the major tasks identified in Section 3 (and others) for groups in this class. We refer the reader to [7] and [10] for details.…”

“…This work and the "reduction algorithms" of [25] and [33] suggest that, subject to the availability of discrete log and integer factorisation oracles, it may be possible using matrix group algorithms to construct in polynomial time the composition factors of G. We contrast this with the results obtained in the black-box context [10].…”