Testing low-degree polynomials over GF(2)

Noga Alon; Tali Kaufman; Michael Krivelevich; Simon Litsyn; Dana Ron

doi:10.1007/978-3-540-45198-3_17

Testing low-degree polynomials over GF(2)

Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, Dana Ron

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

71 Scopus citations

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 + k4^k) 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 + 2^k) queries.

Original language	English
Title of host publication	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Editors	Sanjeev Asora, Amit Sahai, Klaus Jansen, Jose D.P. Rolim
Publisher	Springer Verlag
Pages	188-199
Number of pages	12
ISBN (Print)	3540407707, 9783540407706
DOIs	https://doi.org/10.1007/978-3-540-45198-3_17
State	Published - 2003
Externally published	Yes

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	2764
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

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10.1007/978-3-540-45198-3_17

Cite this

Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., & Ron, D. (2003). Testing low-degree polynomials over GF(2). In S. Asora, A. Sahai, K. Jansen, & J. D. P. Rolim (Eds.), Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (pp. 188-199). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2764). Springer Verlag. https://doi.org/10.1007/978-3-540-45198-3_17

Alon, Noga ; Kaufman, Tali ; Krivelevich, Michael et al. / Testing low-degree polynomials over GF(2). Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). editor / Sanjeev Asora ; Amit Sahai ; Klaus Jansen ; Jose D.P. Rolim. Springer Verlag, 2003. pp. 188-199 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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Alon, N, Kaufman, T, Krivelevich, M, Litsyn, S & Ron, D 2003, Testing low-degree polynomials over GF(2). in S Asora, A Sahai, K Jansen & JDP Rolim (eds), Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2764, Springer Verlag, pp. 188-199. https://doi.org/10.1007/978-3-540-45198-3_17

Testing low-degree polynomials over GF(2). / Alon, Noga; Kaufman, Tali; Krivelevich, Michael et al.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). ed. / Sanjeev Asora; Amit Sahai; Klaus Jansen; Jose D.P. Rolim. Springer Verlag, 2003. p. 188-199 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2764).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

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AB - 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.

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Alon N, Kaufman T, Krivelevich M, Litsyn S, Ron D. Testing low-degree polynomials over GF(2). In Asora S, Sahai A, Jansen K, Rolim JDP, editors, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Springer Verlag. 2003. p. 188-199. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-540-45198-3_17

Testing low-degree polynomials over GF(2)

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