Natural Representations of Black Box Groups SL2(Fq)

In this paper, we make one step further in the recognition of black box groups of Lie type: given a black box group encrypting a special linear group of dimension 2 over a finite field of an unknown odd characteristic, we construct a black box field and a polynomial time isomorphism from the special linear group of dimension 2 over this new field to the black box, which can be made polynomial time-reversible for small characteristics at the expense of constructing a look-up table for the prime field. Our result opens a way to constructing structural proxies for black box groups of Lie type.