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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References
Bacon, D., Childs, A.M., van Dam, W.: From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 469–478 (2005)
Berndt, B., Evans, R., Williams, K.S.: Gauss and Jacobi Sums. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 21. John Wiley & Sons, Chichester (1998)
Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik 46, 493–505 (1998)
Crandall, R., Pomerance, C.: Prime numbers: A computational perspective. Springer, Berlin (2005)
Dobrowolski, E., Williams, K.S.: An upper bound for the sum \(\sum\sp {a+H}\sb {n=a+1}f(n)\) for a certain class of functions f. Proceedings of the American Mathematical Society 114, 29–35 (1992)
Grover, L.: A fast quantum-mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC 1996), pp. 212–219 (1996)
Kohel, D.R., Shparlinski, I.E.: Exponential sums and group generators for elliptic curves over finite fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 395–404. Springer, Heidelberg (2000)
Lenstra, A., de Weger, B.: On the possibility of constructing meaningful hash collisions for public keys. In: Boyd, C., González Nieto, J.M. (eds.) ACISP 2005. LNCS, vol. 3574, pp. 267–279. Springer, Heidelberg (2005)
Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Cambridge University Press, Cambridge (1997)
Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing 26, 1484–1509 (1997)
Storer, T.: Cyclotomy and Difference Sets. Lectures in Advanced Mathematics. Markham Publishing Company (1967)
Yu, H.B.: Estimates of character sums with exponential function. Acta Arithmetica 97, 211–218 (2001)
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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
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