We introduce and construct a pseudorandom object which we call a local correlation breaker (LCB). Informally speaking, an LCB is a function that gets as input a sequence of φ (arbitrarily correlated) random variables and an independent weak-source. The output of the LCB is a sequence of φ random variables with the following property. If the ith input random variable is uniform, then the ith output variable is uniform even given a bounded number of any other output variables. That is, an LCB uses the weak-source to "break" local correlations between random variables. Using our construction of LCBs, we obtain the following results: (1) We construct a threesource extractor where one of the sources is only assumed to have a double-logarithmic entropy. More precisely, for any integer n and constant σ > 0, we construct a three-source extractor for entropies σn, O(log n), and O(log log n). This result improves the three-source extractor of Raz [Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, 2005, pp. 11-20] and is incomparable with the recent three-source extractor by Li [Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), 2015, pp. 863-882]. As the third source is required to have tantalizingly low entropy, we hope that further ideas can be used to eliminate the need for this source altogether. (2) We construct a merger with weak-seeds that merges r random variables using an independent (n, k)-weak-source with k = Õ (r) E log log n. A previous construction by Barak et al. [Ann. of Math., 176 (2012), pp. 1483.1544] assumes k ≥ Ω(r2) + polylog(n).
All Science Journal Classification (ASJC) codes
- Computer Science(all)
- Multi-source extractors
- Nonmalleable extractors