Pseudorandom bits for polynomials

Abstract

We present a new approach to constructing pseudorandom generators that fool low- degree polynomials over finite fields, based on the Gowers norm. sUsing this approach, we obtain the following main constructions of explicitly computable generators G : Fs → Fn that fool polynomials over a finite field F: 1. a generator that fools degree-2 (i.e., quadratic) polyno mials to within error 1/n, with seed length s = 0(log n), 2. a generator that fools degree-3 (i.e., cubic) polynomials to within error ∈, with seed length s = O(log|F| n) + f (∈,F), where f depends only on ∈ 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 ∈, with seed length s = 0(d · log|F| n) + f (d,∈,F), where f depends only on d, ∈,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 componentw ise 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 F2 = {0, 1}, our results constitute the first progress on these problems since the long-standing generator by Luby, Veličković, 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 (ω (√log n)), even for the case of degree-2 polynomials over 7Fdbl;2. © 2010 Society for Industrial and Applied Mathematics.