Extractors for Turing-Machine Sources

We obtain the first deterministic randomness extractors for n-bit sources with minentropy ≥ n1−α generated (or sampled) by single-tape Turing machines running in time n2−16α, for all sufficiently small α > 0. We also show that such machines cannot sample a uniform n-bit input to the Inner Product function together with the output. The proofs combine a variant of the crossing-sequence technique by Hennie [SWCT 1965] with extractors for block sources, especially those by Chor and Goldreich [SICOMP 1988] and by Kamp, Rao, Vadhan, and Zuckerman [JCSS 2011]. ∗Supported by NSF grant CCF-0845003. Email: [email protected]