Testing low-degree polynomials over GF(2)

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

Research output: Chapter in Book/Report/Conference proceedingChapter

59 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 + 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.

Original languageEnglish (US)
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
EditorsSanjeev Asora, Amit Sahai, Klaus Jansen, Jose D.P. Rolim
PublisherSpringer Verlag
Pages188-199
Number of pages12
ISBN (Print)3540407707, 9783540407706
DOIs
StatePublished - 2003
Externally publishedYes

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2764
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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  • 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