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 ${ \rm PGL}_2(\mathbb{ F})$, ${ \rm PSL}_2(\mathbb{ F})$.
Categories: math.GR