Unconditional pseudorandom generators for low degree polynomials

Abstract: We give an explicit construction of a pseudorandom generator against lowdegree polynomials over finite fields. Pseudorandom generators against linear polynomials, known as small-bias generators, were first introduced by Naor and Naor (STOC 1990). We show that the sum of 2d independent small-bias generators with error ε2O(d) is a pseudorandom generator against degree-d polynomials with error ε. This gives a generator with seed length 2O(d) log(n/ε) against degree-d polynomails. Our construction follows the breakthrough result of Bogdanov and Viola (FOCS 2007). Their work shows that the sum of d small-bias generators is a pseudo-random generator against degree-d polynomials, assuming a conjecture in additive combinatorics, known as the inverse conjecture for the Gowers norm. However, this conjecture was proven only for d = 2,3. The main advantage of this work is that it does not rely on any unproven conjectures. Subsequently, the inverse conjecture for the Gowers norm was...

• Type: Journal
• Year: 2007
• Where: ECCC
• Authors: Shachar Lovett
• Keywords: Conjecture, ECCC 2007, Polynomials, Pseudorandom Generator