We show that for integers k >= 2 and n >= 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 112 Ink. 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/kZ. (c) 2005 Elsevier Ltd. All rights reserved.