Abstract

We present a new approach to constructing pseudorandom generators that fool lowdegree polynomials over finite fields, based on the Gowers norm. Using this approach, we obtain the following main constructions of explicitly computable generators G: F s → F n that fool polynomials over a prime field F: 1. a generator that fools degree-2 (i.e., quadratic) polynomials to within error 1/n, with seed length s = O(log n), 2. a generator that fools degree-3 (i.e., cubic) polynomials to within error ɛ, with seed length s = O(log |F | n) + f(ɛ, F) where f depends only on ɛ and F (not on n), 3. assuming the “Gowers inverse conjecture, ” for every d a generator that fools degree-d polynomials to within error ɛ, with seed length s = O(d · log |F | n) + f(d, ɛ, F) where f depends only on d, ɛ, and F (not on n). We stress that the results in (1) and (2) are unconditional, i.e. do not rely on any unproven assumption. Moreover, the results in (3) rely on a special case of the conjecture which may be easier to prove. Our generator for degree-d polynomials is the component-wise sum of d generators for degree-1 polynomials (on independent seeds). Prior to our work, generators with logarithmic seed length were only known for degree-1 (i.e., linear) polynomials (Naor and Naor; SIAM J. Comput., 1993). In fact, over small fields such as F2 = {0, 1}, our results constitute the first progress on these problems since the long-standing generator by Luby, Veličković and Wigderson (ISTCS 1993), whose seed length is much bigger: s = exp � Ω � √ log n � � , even for the case of degree-2 polynomials over F2.

Citations