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

^*Corresponding author for this work

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

69 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	Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques
Subtitle of host publication	6th 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
Editors	Sanjeev Asora, Klaus Jansen, Jose D.P. Rolim, Amit Sahai
Place of Publication	Berlin Heidelberg
Publisher	Springer Verlag
Pages	188-199
Number of pages	12
ISBN (Electronic)	978-3-540-45198-3
ISBN (Print)	3540407707, 9783540407706
DOIs	https://doi.org/10.1007/978-3-540-45198-3_17
State	Published - 2003

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

Keywords

linear code
Israel
Dual distance

Access to Document

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, K. Jansen, J. D. P. Rolim, & A. Sahai (Eds.), Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques: 6th 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 (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). Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques: 6th 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. editor / Sanjeev Asora ; Klaus Jansen ; Jose D.P. Rolim ; Amit Sahai. Berlin Heidelberg : Springer Verlag, 2003. pp. 188-199 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{ed0e1860517f4d16ad985d0b3327cab4,

title = "Testing low-degree polynomials over GF(2)",

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.",

keywords = "linear code, Israel, Dual distance",

author = "Noga Alon and Tali Kaufman and Michael Krivelevich and Simon Litsyn and Dana Ron",

year = "2003",

doi = "10.1007/978-3-540-45198-3_17",

language = "אנגלית",

isbn = "3540407707",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "188--199",

editor = "Sanjeev Asora and Klaus Jansen and Rolim, {Jose D.P.} and Amit Sahai",

booktitle = "Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques",

}

Alon, N, Kaufman, T, Krivelevich, M , Litsyn, S & Ron, D 2003, Testing low-degree polynomials over GF(2). in S Asora, K Jansen, JDP Rolim & A Sahai (eds), Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques: 6th 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. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2764, Springer Verlag, Berlin Heidelberg, 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.
Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques: 6th 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. ed. / Sanjeev Asora; Klaus Jansen; Jose D.P. Rolim; Amit Sahai. Berlin Heidelberg: 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 › Conference contribution › peer-review

TY - GEN

T1 - Testing low-degree polynomials over GF(2)

AU - Alon, Noga

AU - Kaufman, Tali

AU - Krivelevich, Michael

AU - Litsyn, Simon

AU - Ron, Dana

PY - 2003

Y1 - 2003

N2 - 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.

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.

KW - linear code

KW - Israel

KW - Dual distance

UR - http://www.scopus.com/inward/record.url?scp=35248821354&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-45198-3_17

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

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:35248821354

SN - 3540407707

SN - 9783540407706

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 188

EP - 199

BT - Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques

A2 - Asora, Sanjeev

A2 - Jansen, Klaus

A2 - Rolim, Jose D.P.

A2 - Sahai, Amit

PB - Springer Verlag

CY - Berlin Heidelberg

ER -

Alon N, Kaufman T, Krivelevich M , Litsyn S , Ron D. Testing low-degree polynomials over GF(2). In Asora S, Jansen K, Rolim JDP, Sahai A, editors, Approximation, Randomization, and Combinatorial Optimization : Algorithms and Techniques: 6th 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. Berlin Heidelberg: 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)

Abstract

Publication series

Keywords

Access to Document

Other files and links

Fingerprint

Cite this