# 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(k 2/nd) bits that are 2 -nΩ(1) -close to uniform, provided d ≤ o(log n) and k ≥ n 2/3+γ (for arbitrarily small constants γ > 0). Using our result, we also improve a result of Viola [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+ n 1-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. © 2012 ACM 1942-3462/2012/03-ART3 $10.00.

## Publication Title

ACM Transactions on Computation Theory

## Recommended Citation

De, A., & Watson, T.
(2012). Extractors and lower bounds for locally samplable sources.* ACM Transactions on Computation Theory**, 4* (1)
https://doi.org/10.1145/2141938.2141941