Polynomial Interpolation and Identity Testing from High Powers Over Finite Fields

Ivanyos, Gábor and Karpinski, M. and Santha, M. and Saxena, N. and Shparlinski, I.E. (2018) Polynomial Interpolation and Identity Testing from High Powers Over Finite Fields. ALGORITHMICA, 80 (2). pp. 560-575. ISSN 0178-4617

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Abstract

We consider the problem of recovering (that is, interpolating) and identity testing of a “hidden” monic polynomial f, given an oracle access to (Formula presented.) for (Formula presented.), where (Formula presented.) is finite field of q elements (extension fields access is not permitted). The naive interpolation algorithm needs (Formula presented.) queries and thus requires (Formula presented.). We design algorithms that are asymptotically better in certain cases; requiring only (Formula presented.) queries to the oracle. In the randomized (and quantum) setting, we give a substantially better interpolation algorithm, that requires only (Formula presented.) queries. Such results have been known before only for the special case of a linear f, called the hidden shifted power problem. We use techniques from algebra, such as effective versions of Hilbert’s Nullstellensatz, and analytic number theory, such as results on the distribution of rational functions in subgroups and character sum estimates. © 2017 Springer Science+Business Media New York

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