Testing low-degree polynomials over GF(2)

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Abstract

We describe an efficient randomized algorithm to test if a given binary function f : {0, 1}n → {0, 1} is a low-degree polynomial (that is, a sum of low-degree monomials). For a given integer k ≥ 1 and a given real ε > 0, the algorithm queries f at O(1/ε + k4k) points. If f is a polynomial of degree at most k, the algorithm always accepts, and if the value of f has to be modified on at least an ε fraction of all inputs in order to transform it to such a polynomial, then the algorithm rejects with probability at least 2/3. Our result is essentially tight: Any algorithm for testing degree-k polynomials over GF(2) must perform Ω(1/e + 2k) queries. © Springer-Verlag Berlin Heidelberg 2003.

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Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., & Ron, D. (2003). Testing low-degree polynomials over GF(2). Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2764, 188–199. https://doi.org/10.1007/978-3-540-45198-3_17

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