Constructive Membership Testing in Black-Box Classical Groups

Abstract

The research described in this note aims at solving the constructive membership problem for the class of quasisimple classical groups. Our algorithms are developed in the black-box group model (see [HCGT, Section 3.1.4]); that is, they do not require specific characteristics of the representations in which the input groups are given. The elements of a black-box group are represented, not necessarily uniquely, as bit strings of uniform length. We assume the existence of oracles to compute the product of two elements, the inverse of an element, and to test if two strings represent the same element. Solving the constructive membership problem for a black-box group G requires to write every element of G as a word in a given generating set. In practice we write the elements of G as straight-line programs (SLPs) which can be viewed as a compact way of writing words; see [HCGT, Section 3.1.3].