Randomness Extraction in AC0 and with Small Locality

We study two variants of seeded randomness extractors. The first one, as studied by Goldreich et al. [8], is seeded extractors that can be computed by AC circuits. The second one, as introduced by Bogdanov and Guo [3], is (strong) extractor families that consist of sparse transformations, i.e., functions that have a small number of overall input-output dependencies (called sparse extractor families). In this paper we focus on the stronger condition where any function in the family can be computed by local functions. The parameters here are the length of the source n, the min-entropy k = k(n), the seed length d = d(n), the output length m = m(n), the error ǫ = ǫ(n), and the locality of functions l = l(n). In the AC extractor case, our main results substantially improve the positive results in [8], where for k ≥ n/poly(logn) a seed length of O(m) is required to extract m bits with error 1/poly(n). We give constructions of strong seeded extractors for k = δn ≥ n/poly(logn), with seed length d = O(log n), output length m = k, and error any 1/poly(n). We can then boost the output length to Ω(δk) with seed length d = O(log n), or to (1−γ)k for any constant 0 < γ < 1 with d = O(1δ logn). In the special case where δ is a constant and ǫ = 1/poly(n), our parameters are essentially optimal. In addition, we can reduce the error to 2 at the price of increasing the seed length to d = poly(log n). In the case of sparse extractor families, Bogdanov and Guo [3] gave constructions for any minentropy k with locality at least O(n/k log(m/ǫ) log(n/m)), but the family size is quite large, i.e., 2. Equivalently, this means the seed length is at least nm. In this paper we significantly reduce the seed length. For k ≥ n/poly(log n) and error 1/poly(n), our AC extractor with output k also has small locality l = poly(logn), and the seed length is only O(log n). We then show that for k ≥ n/poly(log n) and ǫ ≥ 2−kΩ(1) , we can use our error reduction techniques to get a strong seeded extractor with seed length d = O(log n + log (1/ǫ) logn ), output length m = k Ω(1) and locality log(1/ǫ)poly(logn). Finally, for min-entropy k = Ω(log n) and error ǫ ≥ 2−kΩ(1) , we give a strong seeded extractor with seed length d = O(k), m = (1 − γ)k and locality nk log (1/ǫ)(logn)poly(log k). As an intermediate tool for this extractor, we construct a condenser that condenses an (n, k)-source into a (10k,Ω(k))-source with seed length d = O(k), error 2 and locality Θ(nk logn). [email protected]. Department of Computer Science, Johns Hopkins University. [email protected]. Department of Computer Science, Johns Hopkins University