Black box exceptional groups of Lie type

TL;DR

This paper presents a Las Vegas algorithm for constructing isomorphisms of black box groups isomorphic to certain exceptional Lie type groups, improving the efficiency of permutation group algorithms.

Contribution

It introduces a new Las Vegas algorithm for exceptional Lie type groups, enhancing existing permutation group algorithms with nearly linear time complexity.

Findings

Provides a constructive isomorphism algorithm for specific Lie type groups.

Upgrades Monte Carlo permutation algorithms to Las Vegas algorithms.

Applicable to groups excluding certain twisted types.

Abstract

If a black box group is known to be isomorphic to an exceptional simple group of Lie type of (twisted) rank $>1$, other than any $^2F_4(q)$, over a field of known size, a Las Vegas algorithm is given to produce a constructive isomorphism. In view of its timing, this algorithm yields an upgrade of all known nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to any group $^2F_4(q)$ or $^2G_2(q)$.