New Extractors for Interleaved Sources

We study how to extract randomness from a C-interleaved source, that is, a source comprised of C independent sources whose bits or symbols are interleaved. We describe a simple approach for constructing such extractors that yields: • For some δ > 0, c > 0, explicit extractors for 2-interleaved sources on {0, 1} when one source has min-entropy at least (1 − δ)n and the other has min-entropy at least c log n. The best previous construction, by Raz and Yehudayoff [RY11], worked only when both sources had entropy rate 1− δ. • For some c > 0 and any large enough prime p, explicit extractors for 2-interleaved sources on [p] when one source has min-entropy rate at least .51 and the other source has minentropy rate at least (c log n)/n. We use these to obtain the following applications: • We introduce the class of any-order-small-space sources, generalizing the class of smallspace sources studied by Kamp et al. [KRVZ11]. We construct extractors for such sources with min-entropy rate close to 1/2. Using the Raz-Yehudayoff construction would require entropy rate close to 1. • For any large enough prime p, we exhibit an explicit function f : [p] → {0, 1} such that the randomized best-partition communication complexity of f with error 1/2 − 2−Ω(n) is at least .24n log p. Previously this was known only for a tiny constant instead of .24, for p = 2 [RY11]. • We introduce non-malleable extractors in the interleaved model. For any large enough prime p, we give an explicit construction of a weak-seeded non-malleable extractor for sources over [p] with min-entropy rate .51. Nothing was known previously, even for almost full min-entropy. ∗Research supported in part by NSF Grant CCF-1218723. †Research supported in part by NSF Grant CCF-1218723. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 151 (2015)