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 $\delta >0$ we give an explicit construction for extracting randomness from a constant (depending polynomially on $1/\delta$) number of distributions over $\bits^n$, each having min-entropy $\delta n$. 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 requires min-entropy $\Omega(\log n)$, and outputs every possible $m$-bit string with positive probability. The main tool we use is a variant of the “stepping-up lemma” of Erdods and Hajnal used in establishing a lower bound on the Ramsey number for hypergraphs.