Black Box White Arrow
Alexandre Borovik, \c{S}\"ukr\"u Yal\c{c}inkaya

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 . 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…
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Taxonomy
TopicsFinite Group Theory Research · Coding theory and cryptography · Algebraic Geometry and Number Theory
