Abstract
We consider the problem of determining a rational function f over a finite field \(\mathbb{F}_p\) of p elements given a noisy black box \({\mathcal B}\), which for each \(t \in \mathbb{F}_p\) returns several most significant bits of the residue of f(t) modulo the prime p.
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Boneh, D., Halevi, S., Howgrave-Graham, N.: The modular inversion hidden number problem. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 36–51. Springer, Heidelberg (2001)
Chalk, J.H.H.: Polynomial congruences over incomplete residue systems modulo k. Proc. Kon. Ned. Acad. Wetensch. 92, 49–62 (1989)
van Dam, W.: Quantum algorithms for weighing matrices and quadratic residues. Algorithmica 34, 413–428 (2002)
van Dam, W., Hallgren, S., Ip, L.: Quantum algorithms for hidden coset problems. In: Proc. 14th ACM-SIAM Symp. on Discr. Algorithms, pp. 489–498. SIAM, Philadelphia (2003)
Grigoriev, D.: Testing shift-equivalence of polynomials by deterministic, probabilistic and quantum machines. Theor. Comp. Sci. 180, 217–228 (1997)
Hales, L., Hallgren, S.: An improved quantum Fourier transform algorithm and applications. In: Proc 41th IEEE Symp. on Found. of Comp. Sci., pp. 515–525. IEEE, Los Alamitos (2000)
Hallgren, S.: Polynomial-time quantum algorithms for Pell’s equation and the principal ideal problem. In: Proc. 34th ACM Symp. on Theory of Comp., pp. 653–658. ACM, New York (2002)
Kitaev, Y., Shen, A.H., Vyalyi, M.N.: Classical and quantum computation. Graduate Studies in Mathematics, vol. 47. Amer. Math. Soc., Providence (2002)
Li, W.-C.W.: Number theory with qpplications. World Scientific, Singapore (1996)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Moreno, C.J., Moreno, O.: Exponential sums and Goppa codes, 1. Proc. Amer. Math. Soc. 111, 523–531 (1991)
Mosca, M., Ekert, A.: The hidden subgroup problem and eigenvalue estimation on a quantum computer. In: Williams, C.P. (ed.) QCQC 1998. LNCS, vol. 1509, pp. 174–188. Springer, Heidelberg (1999)
Niederreiter, H., Schnorr, C.P.: Local randomness in polynomial random number and random function generators. SIAM J. Comp. 13, 684–694 (1993)
Nielsen, M., Chuang, I.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2002)
Russell, C., Shparlinski, I.E.: Classical and quantum algorithms for function reconstruction via character evaluation. J. Compl. 20, 404–422 (2004)
Shor, P.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comp. 26, 1484–1509 (1997)
Shor, P.: Quantum information theory: Results and open problems. Geometric and Functional Analysis 2, 816–838 (2000)
Shparlinski, E.: Sparse polynomial approximation in finite fields. In: Proc. 33rd ACM Symp. on Theory of Comput., Crete, Greece, July 6-8, pp. 209–215 (2001)
Shparlinski, E., Winterhof, A.: Noisy interpolation of sparse polynomials in finite fields. Appl. Algebra in Engin., Commun. and Computing (to appear)
Simon, D.R.: On the power of quantum computation. SIAM J. Comp. 26, 1474–1483 (1997)
Weil, A.: Basic number theory. Springer, New York (1974)
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Hallgren, S., Russell, A., Shparlinski, I.E. (2005). Quantum Noisy Rational Function Reconstruction. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_43
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DOI: https://doi.org/10.1007/11533719_43
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28061-3
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