Testing low-degree polynomials over GF(2)

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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
Title of host publicationApproximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques
Subtitle of host publication6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2003 and 7th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2003, Princeton, NJ, USA, August 24-26, 2003. Proceedings
EditorsSanjeev Asora, Klaus Jansen, Jose D.P. Rolim, Amit Sahai
Place of PublicationBerlin Heidelberg
PublisherSpringer Verlag
Number of pages12
ISBN (Electronic)978-3-540-45198-3
ISBN (Print)3540407707, 9783540407706
StatePublished - 2003

Publication series

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


  • linear code
  • Israel
  • Dual distance


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