Natural representations of black box groups encrypting $SL_2(\mathbb{F}_q)$

TL;DR

This paper presents a probabilistic polynomial-time algorithm to construct explicit isomorphisms between black box groups encrypting SL_2 over an unknown finite field and their concrete matrix group counterparts, without additional oracles.

Contribution

The authors develop the first efficient algorithm for representing black box groups encrypting SL_2 over unknown finite fields as concrete matrix groups.

Findings

Constructs explicit isomorphisms in probabilistic polynomial time

Works for groups encrypting SL_2, PGL_2, and PSL_2 over unknown finite fields

No additional oracles required for the algorithm

Abstract

Given a global exponent $E$ for a black box group $\mathsf{Y}$ encrypting ${\rm SL}_2(\mathbb{F})$, where $\mathbb{F}$ is an unknown finite field of unknown odd characteristic, we construct, in probabilistic time polynomial in $\log E$, the isomorphisms \[ \mathsf{Y} \longleftrightarrow {\rm SL}_2(\mathsf{K}), \] where $\mathsf{K}$ is a black box field encrypting $\mathbb{F}$. Our algorithm makes no reference to any additional oracles. We also give similar algorithms for black box groups encrypting ${\rm PGL}_2(\mathbb{F})$, ${\rm PSL}_2(\mathbb{F})$.