Non-Malleable Extractors, Two-Source Extractors and Privacy Amplification

Dodis and Wichs 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. Previously, there are only two known constructions of non-malleable extractors. Both constructions only work for (n, k)-sources with k>n/2. Interestingly, both constructions are also two-source extractors. In this paper, we present a strong connection between non-malleable extractors and two-source extractors. The first part of the connection shows that non-malleable extractors can be used to construct two-source extractors. With appropriate parameters the resulted two-source extractor beats the best known construction of two-source extractors. This partially explains why previous constructions of non-malleable extractors only work for sources with entropy rate >1/2, and why explicit non-malleable extractors for small min-entropy may be hard to get. The second part of the connection shows that certain two-source extractors can be used to construct non-malleable extractors. Using this connection, we obtain the first construction of non-malleable extractors for k < n/2. Specifically, we give an unconditional construction for min-entropy k=(1/2-\delta)n for some constant \delta>0, and a conditional (semi-explicit) construction that can potentially achieve k=\alpha n for any constant \alpha>0. Finally, despite the lack of explicit non-malleable extractors for arbitrarily linear entropy, we give the first 2-round privacy amplification protocol with asymptotically optimal entropy loss and communication complexity for (n, k) sources with k=\alpha n for any constant \alpha>0. This dramatically improves previous results and answers an open problem in \cite{DLWZ11}.