# Extractors and lower bounds for locally samplable sources

## Abstract

We consider the problem of extracting randomness from sources that are efficiently samplable, in the sense that each output bit of the sampler only depends on some small number d of the random input bits. As our main result, we construct a deterministic extractor that, given any d-local source with min-entropy k on n bits, extracts Ω(k2/nd) bits that are 2 -nΩ(1)-close to uniform, provided d ≤ o(log n) and k ≥ n2/3+γ (for arbitrarily small constants γ > 0). Using our result, we also improve a result of Viola (FOCS 2010), who proved a 1/2 - O(1/log n) statistical distance lower bound for o(log n)-local samplers trying to sample input-output pairs of an explicit boolean function, assuming the samplers use at most n + n1-δ random bits for some constant δ > 0. Using a different function, we simultaneously improve the lower bound to 1/2 - 2-nΩ(1) and eliminate the restriction on the number of random bits. © 2011 Springer-Verlag.

## Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

## Recommended Citation

De, A., & Watson, T.
(2011). Extractors and lower bounds for locally samplable sources.* Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)**, 6845 LNCS*, 483-494.
https://doi.org/10.1007/978-3-642-22935-0_41