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 > 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/) number of distributions over {0, 1}n , each having min-entropy n. These extractors output n bits, which are 2-n close to uniform. This construction uses several results from additive number theory, and in particular a recent one by Bourgain, Katz and Tao [BKT03] and of Konyagin [Kon03]. 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 min-entropy (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" used in establishing lower bound on the Ramsey number for hypergraphs (Erdos and Hajnal, [GRS80])....

• Type: Conference
• Year: 2004
• Where: FOCS
• Authors: Boaz Barak, Russell Impagliazzo, Avi Wigderson
• Keywords: Additive Number Theory, Constant Number, Constant Output Length, FOCS 2004, Theoretical Computer Science