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.
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van Dam, W., Shparlinski, I.E. (2008). Classical and Quantum Algorithms for Exponential Congruences. In: Kawano, Y., Mosca, M. (eds) Theory of Quantum Computation, Communication, and Cryptography. TQC 2008. Lecture Notes in Computer Science, vol 5106. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89304-2_1
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DOI: https://doi.org/10.1007/978-3-540-89304-2_1
Publisher Name: Springer, Berlin, Heidelberg
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