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

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 ${\rm PGL}_2(\mathbb{F})$, ${\rm PSL}_2(\mathbb{F})$.

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