Extractors for Affine Sources with Polylogarithmic Entropy

We give the first explicit construction of deterministic extractors for affine sources over F2, with entropy k ≥ log n for some large enough constant C, where n is the length of the source. Previously the best known results are by Bourgain [Bou07], Yehudayoff [Yeh11] and Li [Li11b], which require the affine source to have entropy at least Ω(n/ √ log log n). Our extractor outputs one bit with error n. Our construction is obtained by reducing an affine source to a non-oblivious bit-fixing source, and then applying a deterministic extractor for such sources in the recent breakthrough result of two-source extractors by Chattopadhyay and Zuckerman [CZ15]. To reduce an affine source to a non-oblivious bit-fixing source, we adapt the alternating extraction based approach in previous work on independent source extractors [Li13a] to the affine setting. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 121 (2015)