Recent Advances in Complexity Theory

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, thismatches the parameters of the current best known construction [LRV03]; for lossless condensers,the previous best constructions achieved optimality to within a constant factor in one parameteronly at the expense of a polynomial loss in the other.Our constructions are based on the Parvaresh-Vardy codes [PV05], and our proof technique isinspired by the list-decoding algorithm for those codes. The main object we construct is a condenserthat loses only the entropy of its seed plus one bit, while condensing to entropy rate 1 − α for anydesired constant α > 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-efficienthash functions (to obtain a lossless condenser) or expander walks (to obtain an extractor).Our techniques also show for the first time that a natural construction based on univariatepolynomials (i.e., Reed-Solomon codes) yields a condenser that retains a 1−α fraction of the sourcemin-entropy, for any desired constant α > 0, while condensing to constant entropy rate and using aseed length that is optimal to within constant factors.