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Abstract

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 shown to be false for d ≥ 4 by Green and Tao (2008) and independently by the author, Roy Meshulam, and Alex Samorodnitsky (STOC 2008). A revised version of the conjecture was proved by Bergelson, Tao, and Ziegler (2009). Additionally, Viola (CCC 2008) showed the original construction of Bogdanov and Viola to hold unconditionally

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oai:CiteSeerX.psu:10.1.1.390.4324Last time updated on 10/22/2014

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