Natural representations of black box groups encrypting $S{L}_2({\mathbb{F}}_q)$

Abstract

Given a global exponent $E$ for a black box group $\mathsf{Y}$ encrypting ${\mathrm{SL}}_2(\mathbb{F})$, 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 ${\mathrm{PGL}}_2(\mathbb{F})$, ${\mathrm{PSL}}_2(\mathbb{F})$.

References in corpus (1)

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