Adjoint representations of black box groups ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q)$

Given a black box group $\mathsf{Y}$ encrypting ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{F})$ 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}$)$,itssubgroup$$\mathsf{Y}$^\circ$ofindex$2$isomorphicto$$\mathsf{Y}$$andaprobabilisticpolynomialin$\log E$ time isomorphism ${\mathsf{Y}}^{\circ }⟶\mathsf{Y}$;
2. a black box field $\mathsf{K}$, and
3. polynomial time, in $\log E$, isomorphisms $${\mathrm{S}\mathrm{O}}_3(\mathsf{K})⟶\mathsf{X}⟶{\mathrm{S}\mathrm{O}}_3(\mathsf{K})\mathrm{.}$$ If, in addition, we know $p$ and the standard explicitly given finite field $\mathbb{F}$ isomorphic to $\mathbb{F}$ then we construct, in time polynomial in $\log E$, isomorphism $${\mathrm{S}\mathrm{O}}_3(\mathbb{F})⟶{\mathrm{S}\mathrm{O}}_3(\mathsf{K})\mathrm{.}$$ 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 ${\mathrm{S}\mathrm{L}}_2$-oracle in the black box group theory. We implemented our algorithms in GAP and tested them for groups such as ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{F})$ for $\mathrm{\mid }\mathbb{F}\mathrm{\mid }=115756986668303657898962467957$ (a prime number).