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Testing Low-Degree Polynomials over GF(2) Tali Kaufman

Summary: Testing Low-Degree Polynomials over GF(2)
Noga Alon
Tali Kaufman
Michael Krivelevich
Simon Litsyn §
Dana Ron¶
July 9, 2003
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 > 0, the algorithm queries f at O(1
+ 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 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 (1
+ 2k
) queries.


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics