Non-Malleable Extractors for Entropy Rate <1/2

Dodis and Wichs [DW09] introduced the notion of a non-malleable extractor to study the problem of privacy amplification with an active adversary. A non-malleable extractor is a much stronger version of a strong extractor. Given a weakly-random string x and a uniformly random seed y as the inputs, the non-malleable extractor nmExt has the property that nmExt(x, y) appears uniform even given y as well as nmExt(x,A(y)), for an arbitrary function A with A(y) 6= y. Dodis and Wichs showed that such an object can be used to give optimal privacy amplification protocols with an active adversary. Previously, there are only two known explicit constructions of non-malleable extractors [DLWZ11, CRS11]. Both constructions only work for (n, k)-sources with k > n/2 and thus they also only achieve optimal privacy amplification protocols for k > n/2. In this paper, we give the first construction of non-malleable extractors for k < n/2. Specifically, we give two unconditional constructions for min-entropy k = (1/2 − δ)n for some constant δ > 0, and a conditional construction that can potentially achieve k = αn for any constant α > 0. Using our non-malleable extractor, we also obtain the first optimal privacy amplification protocol for min-entropy k = (1/2 − δ)n, with an active adversary. Our constructions mainly use the bilinear property of the inner product function, and involve appropriate encodings of the seed y as well as ideas from additive combinatorics. ∗Partially supported by NSF Grants CCF-0634811, CCF-0916160, THECB ARP Grant 003658-0113-2007, and a Simons postdoctoral fellowship. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 166 (2011)