An obvious way to build a computational extractor is via the ``extract-then-prg" method: apply a statistical extractor and use its output to seed a PRG. This approach carries with it the entropy cost inherent to implementing statistical extractors, namely, the source entropy needs to be substantially higher than the PRG's seed length. It also requires a PRG and thus relies on one-way functions.
We study the necessity of one-way functions in the construction of computational extractors and determine matching lower and upper bounds on the ``black-box efficiency" of generic constructions of computational extractors that use a one-way permutation as an oracle. Under this efficiency measure we prove a direct correspondence between the complexity of computational extractors and that of pseudorandom generators, showing the optimality of the extract-then-prg approach for generic constructions of computational extractors and confirming the intuition that to build a computational extractor via a PRG one needs to make up for the entropy gap intrinsic to statistical extractors.
On the other hand, we show that with stronger cryptographic primitives one can have more entropy- and computationally-efficient constructions. In particular, we show a construction of a very practical computational extractor from any weak PRF without resorting to statistical extractors.Category / Keywords: foundations / randomness extractors, pseudo-randomness Publication Info: A conference version of this paper is published in TCC'2012. Date: received 28 Dec 2011 Contact author: hugo at ee technion ac il Available formats: PDF | BibTeX Citation Version: 20111231:153727 (All versions of this report) Discussion forum: Show discussion | Start new discussion