Natural Representations of Black Box Groups SL2(Fq)

Borovik, Alexandre; Yalçınkaya, Şükrü

doi:10.3390/axioms14120895

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Open AccessArticle

Natural Representations of Black Box Groups SL₂(F_q)

by

Alexandre Borovik

^1,*

and

Şükrü Yalçınkaya

²

¹

Department of Mathematics, University of Manchester, Manchester M13 9PL, UK

²

Department of Mathematics, Istinye University, Istanbul 34396, Turkey

^*

Author to whom correspondence should be addressed.

Axioms 2025, 14(12), 895; https://doi.org/10.3390/axioms14120895 (registering DOI)

Submission received: 29 October 2025 / Revised: 21 November 2025 / Accepted: 25 November 2025 / Published: 1 December 2025

(This article belongs to the Section Algebra and Number Theory)

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Abstract

In this paper, we make one step further in the recognition of black box groups of Lie type: given a black box group encrypting a special linear group of dimension 2 over a finite field of an unknown odd characteristic, we construct a black box field and a polynomial time isomorphism from the special linear group of dimension 2 over this new field to the black box, which can be made polynomial time-reversible for small characteristics at the expense of constructing a look-up table for the prime field. Our result opens a way to constructing structural proxies for black box groups of Lie type.

Keywords: black box groups; probabilistic methods; structural proxy

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MDPI and ACS Style

Borovik, A.; Yalçınkaya, Ş. Natural Representations of Black Box Groups SL₂(F_q). Axioms 2025, 14, 895. https://doi.org/10.3390/axioms14120895

AMA Style

Borovik A, Yalçınkaya Ş. Natural Representations of Black Box Groups SL₂(F_q). Axioms. 2025; 14(12):895. https://doi.org/10.3390/axioms14120895

Chicago/Turabian Style

Borovik, Alexandre, and Şükrü Yalçınkaya. 2025. "Natural Representations of Black Box Groups SL₂(F_q)" Axioms 14, no. 12: 895. https://doi.org/10.3390/axioms14120895

APA Style

Borovik, A., & Yalçınkaya, Ş. (2025). Natural Representations of Black Box Groups SL₂(F_q). Axioms, 14(12), 895. https://doi.org/10.3390/axioms14120895

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Natural Representations of Black Box Groups SL₂(F_q)

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