Extractors and condensers from univariate polynomials

13 years 4 months ago

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We give new constructions of randomness extractors and lossless condensers that are optimal to within constant factors in both the seed length and the output length. For extractors, this matches the parameters of the current best known construction [LRVW03], with an improvement in case the error parameter is small (e.g. 1/poly(n)). For lossless condensers, the previous best constructions achieved optimality to within a constant factor in one parameter only at the expense of a polynomial loss in the other. Our constructions are based on the Parvaresh-Vardy codes [PV05], and our proof technique is inspired by the list-decoding algorithm for those codes. The main object we construct is a condenser that loses only the entropy of its seed plus one bit, while condensing to entropy rate 1 - for any desired constant > 0. This construction is simple to describe, and has a short and completely self-contained analysis. Our other results only require, in addition, standard uses of randomness...

Venkatesan Guruswami, Christopher Umans, Salil P.

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