 
Summary: Testing LowDegree Polynomials over GF(2)
Noga Alon
Tali Kaufman
Michael Krivelevich
Simon Litsyn §
Dana Ron¶
July 9, 2003
Abstract
We describe an efficient randomized algorithm to test if a given binary function f : {0, 1}n
{0, 1} is a lowdegree polynomial (that is, a sum of lowdegree 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
degreek polynomials over GF(2) must perform (1
+ 2k
) queries.
