Testing low-degree polynomials over GF (2)

N Alon; T Kaufman; M Krivelevich; S Litsyn; D Ron

Testing low-degree polynomials over GF (2)

N Alon, T Kaufman, M Krivelevich, S Litsyn, D Ron

Department of Computer Science

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

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(1ϵ+k4k)O(1ϵ+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(1ϵ+2k)Ω(1ϵ+2k) queries.

Original language	American English
Title 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
Editors	Sanjeev Arora, Klaus Jansen, José D. P. Rolim, Amit Sahai
Publisher	Springer Berlin Heidelberg
State	Published - 2003

Bibliographical note

Place of conference:USA

Cite this

Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., & Ron, D. (2003). Testing low-degree polynomials over GF (2). In S. Arora, K. Jansen, J. D. P. Rolim, & A. Sahai (Eds.), 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 Springer Berlin Heidelberg.

Alon, N ; Kaufman, T ; Krivelevich, M et al. / Testing low-degree polynomials over GF (2). 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. editor / Sanjeev Arora ; Klaus Jansen ; José D. P. Rolim ; Amit Sahai. Springer Berlin Heidelberg, 2003.

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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(1ϵ+k4k)O(1ϵ+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(1ϵ+2k)Ω(1ϵ+2k) queries.",

author = "N Alon and T Kaufman and M Krivelevich and S Litsyn and D Ron",

note = "Place of conference:USA",

year = "2003",

language = "American English",

editor = "Sanjeev Arora and Klaus Jansen and Rolim, {Jos{\'e} D. P.} and Amit Sahai",

booktitle = "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",

publisher = "Springer Berlin Heidelberg",

}

Alon, N, Kaufman, T, Krivelevich, M, Litsyn, S & Ron, D 2003, Testing low-degree polynomials over GF (2). in S Arora, K Jansen, JDP Rolim & A Sahai (eds), 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. Springer Berlin Heidelberg.

Testing low-degree polynomials over GF (2). / Alon, N; Kaufman, T; Krivelevich, M et al.
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. ed. / Sanjeev Arora; Klaus Jansen; José D. P. Rolim; Amit Sahai. Springer Berlin Heidelberg, 2003.

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

TY - GEN

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

AU - Alon, N

AU - Kaufman, T

AU - Krivelevich, M

AU - Litsyn, S

AU - Ron, D

N1 - Place of conference:USA

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(1ϵ+k4k)O(1ϵ+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(1ϵ+2k)Ω(1ϵ+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(1ϵ+k4k)O(1ϵ+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(1ϵ+2k)Ω(1ϵ+2k) queries.

UR - https://scholar.google.co.il/scholar?q=Testing+Low+Degree+Polynomials+Over+GF%282%29%2C+Tali+Kaufman&btnG=&hl=en&as_sdt=0%2C5

M3 - Conference contribution

BT - 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

A2 - Arora, Sanjeev

A2 - Jansen, Klaus

A2 - Rolim, José D. P.

A2 - Sahai, Amit

PB - Springer Berlin Heidelberg

ER -

Alon N, Kaufman T, Krivelevich M, Litsyn S, Ron D. Testing low-degree polynomials over GF (2). In Arora S, Jansen K, Rolim JDP, Sahai A, editors, 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. Springer Berlin Heidelberg. 2003

Testing low-degree polynomials over GF (2)

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