Hitting Set Generators for Sparse Polynomials over Any Finite Fields

We consider the problem of constructing hitting set generators for sparse multivariate polynomials over any finite fields. Hitting set generators, just as pseudorandom generators, play a fundamental role in the study of derandomization. Pseudorandom generators with a logarithmic seed length are only known for polynomials of a constant degree cite{Lov09, Vio09}. On the other hand, hitting set generators with a logarithmic seed length are known for polynomials of larger degrees, but only over fields which are much larger than the degrees cite{KS01, Bog05}. Our main result is the construction of a hitting set generator with a seed length of O(log s), which works for s-term polynomials of any degrees over any finite fields of constant size. This gives the first optimal hitting set generator which allows the fields to be smaller than the degrees of polynomials. For larger fields, of non-constant size, we provide another hitting set generator with a seed length of O(log (sd)), which works for s-term polynomials of any degree d, as long as d is slightly smaller than the field size.