Extracting randomness using few independent sources

In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ > 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0,1}ⁿ each having min-entropy δn. These extractors output n bits that are 2^-n close to uniform. This construction uses several results from additive number theory, and in particular a recent result of Bourgain et al. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with δ-entropy ω(log n), and outputs every possible m-bit string with positive probability. The main tool we use is a variant of the "stepping-up lemma" of Erdos and Hajnal used in establishing a lower bound on the Ramsey number for hypergraphs.