PSEUDORANDOM BITS FOR POLYNOMIALS
We present a new approach to constructing pseudorandom generators that fool low-degree 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 finite 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 epsilon, with seed length s = O(log(|F|) n) + f(epsilon, F), where f depends only on epsilon and F (not on n), 3. assuming the "inverse conjecture for the Gowers norm," for every d a generator that fools degree-d polynomials to within error epsilon, with seed length s = O(d . log(|F|) n) + f(d, epsilon, F), where f depends only on d, epsilon, 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 componentwise 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 [J. Naor and M. Naor, SIAM J. Comput., 22 (1993), pp. 838-856]. In fact, over small fields such as F-2 = {0, 1}, our results constitute the first progress on these problems since the long-standing generator by Luby, Velickovic, and Wigderson [Deterministic approximate counting of depth-2 circuits, in Proceedings of the 2nd Israeli Symposium on Theoretical Computer Science (ISTCS), 1993, pp. 18-24], whose seed length is much bigger: s = exp (Omega (root log n)), even for the case of degree-2 polynomials over F-2.
polynomial pseudorandom generator gowers norm degree bias