  We discuss classical and quantum algorithms for solvability testing and
finding integer solutions x,y of equations of the form af^x + bg^y = c over
finite fields GF(q). A quantum algorithm with time complexity q^(3/8) (log
q)^O(1) is presented. While still superpolynomial in log q, this quantum
algorithm is significantly faster than the best known classical algorithm,
which has time complexity q^(9/8) (log q)^O(1). Thus it gives an example of a
natural problem where quantum algorithms provide about a cubic speed-up over
classical ones.
