Adjoint representations of black box groups ${\rm PSL}_2(\mathbb{F}_q)$

TL;DR

This paper presents a polynomial-time Las Vegas algorithm for constructing unipotent elements and determining the characteristic of the underlying field in black box groups encrypting PSL_2 over an unknown field, answering a question from 1999.

Contribution

It introduces a novel algorithm that constructs unipotent elements and identifies the field characteristic in black box groups without additional oracles, advancing black box group theory.

Findings

Constructed unipotent elements in polynomial time

Determined the field characteristic efficiently

Built isomorphisms between black box groups and classical groups

Abstract

Given a black box group $\mathsf{Y}$ encrypting $\rm{PSL}_2(\mathbb{F})$ over an unknown field $\mathbb{F}$ of unknown odd characteristic $p$ and a global exponent $E$ for $\mathsf{Y}$ (that is, an integer $E$ such that $\mathsf{y}^E=1$ for all $\mathsf{y} \in \mathsf{Y}$), we present a Las Vegas algorithm which constructs a unipotent element in $\mathsf{Y}$. The running time of our algorithm is polynomial in $\log E$. This answers the question posed by Babai and Beals in 1999. We also find the characteristic of the underlying field in time polynomial in $\log E$ and linear in $p$. Furthermore, we construct, in probabilistic time polynomial in $\log E$, 1. a black box group $\mathsf{X}$ encrypting $\rm{PGL}_2(\mathbb{F}) \cong\rm{SO}_3(\mathbb{F})$, its subgroup $\mathsf{Y}^\circ$ of index $2$ isomorphic to $\mathsf{Y}$ and a probabilistic polynomial in $\log E$ time isomorphism $\mathsf{Y}^\circ \longrightarrow \mathsf{Y}$; 2. a black box field $\mathsf{K}$, and 3. polynomial time, in $\log E$, isomorphisms \[ \rm{SO}_3(\mathsf{K}) \longrightarrow \mathsf{X} \longrightarrow \rm{SO}_3(\mathsf{K}). \] If, in addition, we know $p$ and the standard explicitly given finite field $\mathbb{F}$ isomorphic to $\mathbb{F}$ then we construct, in time polynomial in $\log E$, isomorphism \[ \rm{SO}_3(\mathbb{F})\longrightarrow \rm{SO}_3(\mathsf{K}). \] Unlike many papers on black box groups, our algorithms make no use of additional oracles other than the black box group operations. Moreover, our result acts as an $\rm{SL}_2$-oracle in the black box group theory. We implemented our algorithms in GAP and tested them for groups such as $\rm{PSL}_2(\mathbb{F})$ for $|\mathbb{F}|=115756986668303657898962467957$ (a prime number).