Diameters of Cayley graphs of Chevalley groups

We show that for integers k≥2k≥2 and n≥3n≥3, the diameter of the Cayley graph of SLn(Z/kZ) associated with a standard two-element generating set is at most a constant times n2lnkn2lnk. This answers a question of A. Lubotzky concerning SLn(Fp) and is unexpected because these Cayley graphs do not form an expander family. Our proof amounts to a quick algorithm for finding short words representing elements of SLn(Z/kZ). We generalize our results to other Chevalley groups over Z/kZZ/kZ.