Classical and quantum algorithms for exponential congruences

Abstract

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 . A quantum algorithm with time complexity q 3/8 (logq) O(1) is presented. While still superpolynomial in logq, this quantum algorithm is significantly faster than the best known classical algorithm, which has time complexity q 9/8 (logq) O(1). Thus it gives an example of a natural problem where quantum algorithms provide about a cubic speed-up over classical ones. © 2008 Springer Berlin Heidelberg.