Black Box White Arrow

TL;DR

This paper introduces a categorical framework for black box groups in computational group theory, enabling new classes of problems and automorphisms, with applications to Lie type groups and automorphisms over finite fields.

Contribution

It develops a systematic categorical approach to black box groups, allowing for richer structures and automorphisms, and applies this to construct Frobenius maps and inverse-transpose automorphisms.

Findings

Constructed Frobenius maps on black box groups of Lie type

Built inverse-transpose automorphisms for groups encrypting SL_n(F_q)

Explained constructions of involutions in black box groups

Abstract

The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Section 6) and inverse-transpose automorphisms on black box groups encrypting ${\rm (P)SL}_n(\mathbb{F}_q)$. One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is explanatory power of our methods; as an example, we explain Kantor's and Kassabov's construction of an involution in black box groups encrypting ${\rm SL}_2(2^n)$. Due to the nature of our work we also have to discuss a few methodological issues of the black box group theory. The paper is further development of our text "Fifty shades of black" [arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black box groups.