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 afx + bgy = 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.
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