Constructive membership testing in black-box classical groups

TL;DR

This paper develops algorithms for the constructive membership problem in black-box classical groups, enabling representation of group elements as words without relying on specific group representations.

Contribution

It introduces black-box algorithms for constructive membership testing in quasisimple classical groups, applicable without detailed knowledge of group representations.

Findings

Algorithms successfully determine membership in black-box classical groups.

Elements can be expressed as straight-line programs (SLPs).

The methods are representation-agnostic and rely on oracles.

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; 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.