Adjoint representations of black box groups [formula omitted

Abstract

Given a black box group Y encrypting PSL2(F) over an unknown field F of unknown odd characteristic p and a global exponent E for Y (that is, an integer E such that yE=1 for all y∈Y), we present a Las Vegas algorithm which constructs a unipotent element in 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. All our algorithms are randomized.Furthermore, we construct, in time polynomial in log⁡E,•a black box group X encrypting PGL2(F)≅SO3(F) over the same field as Y, its subgroup Y∘ of index 2 isomorphic to Y and a polynomial in log⁡E time isomorphism Y∘⟶Y;•a black box field K, and•polynomial time, in log⁡E, isomorphismsSO3(K)⟶X⟶SO3(K).If, in addition, we know p and the standard explicitly given finite field Fq isomorphic to the field on which Y is defined then we construct, in time polynomial in log⁡E, isomorphismSO3(Fq)⟶SO3(K).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 SL2-oracle in the black box group theory.We implemented our algorithms in GAP and tested them for groups such as PSL2(F) for|F|=115756986668303657898962467957 (a prime number).