Pseudorandom Bits for Polynomials

• Authors: Andrej Bogdanov;Emanuele Viola
• Venue: FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2007

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 : \mathbb{F}^s\to \mathbb{F}^n that fool polynomials over a prime field \mathbb{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 \in, with seed length {\text{s=O(log}}\left| \mathbb{F} \right|n) + f( \in ,\mathbb{F}) where f depends only on\in and \mathbb{F} (not on n), 3. assuming the "Gowers inverse conjecture," for every d a generator that fools degree-d polynomials to within error \in, with seed length {\text{s=O(d\cdotlog}}\left| \mathbb{F} \right|n) + f{\text{(d,}} \in ,\mathbb{F}) where f depends only on {\text{d,}} \in , and \mathbb{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 \mathbb{F}_2 = {0, 1}, our results constitute the first progress on these problems since the long-standing generator by Luby, Veli隆ckovíc andWigderson (ISTCS 1993), whose seed length is much bigger: s = exp (\Omega (\sqrt {\log n} )), even for the case of degree-2 polynomials over \mathbb{F}_2.