Testing Low-Degree Polynomials over GF(2(
نویسندگان
چکیده
We describe an efficient randomized algorithm to test if a given binary function f : {0, 1} → {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 > 0, the algorithm queries f at O( 1 + k4 ) 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 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 + 2 ) queries. ∗Institute for Advanced Study, Princeton, NJ 08540, USA and Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported in part by a USA Israeli BSF grant and by a grant from the Israel Science Foundation †School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected], This work is part of the author’s Ph.D. thesis prepared at Tel Aviv University under the supervision of Prof. Noga Alon, and Prof. Michael Krivelevich. ‡Department of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported in part by a USA Israeli BSF grant and by a grant from the Israel Science Foundation. §Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported in part by a USA Israeli BSF grant and by a grant from the Israel Science Foundation. ¶Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected]. Research supported by the Israel Science Foundation (grant number 32/00-1).
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