Classical and Quantum Algorithms for Exponential Congruences

Authors:
Wim Dam;Igor E. Shparlinski
Affiliations:
Department of Computer Science, Department of Physics, University of California, Santa Barbara, USA CA 93106-5110;Department of Computing, Macquarie University, Australia NSW 2109
Venue:
Theory of Quantum Computation, Communication, and Cryptography
Year:
2008

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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.