## A polynomial-time theory of black-box groups I (1998)

Citations: | 40 - 6 self |

### BibTeX

@MISC{Babai98apolynomial-time,

author = {László Babai and Robert Beals},

title = {A polynomial-time theory of black-box groups I},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomial-time solutions due to number theoretic obstacles such as factoring integers and discrete logarithm. While these and other “abelian obstacles ” persist, we demonstrate that the “nonabelian normal structure ” of matrix groups over finite fields can be mapped out in great detail by polynomial-time randomized (Monte Carlo) algorithms. The methods are based on statistical results on finite simple groups. We indicate the elements of a project under way towards a more complete “recognition” of such groups in polynomial time. In particular, under a now plausible hypothesis, we are able to determine the names of all nonabelian composition factors of a matrix group over a finite field. Our context is actually far more general than matrix groups: most of the algorithms work for “black-box groups ” under minimal assumptions. In a black-box group, the group elements are encoded by strings of uniform length, and the group operations are performed by a “black box.”