Recently, the problem of privacy amplification with an active adversary has received a lot of attention. Given a shared n-bit weak random source X with min-entropy k and a security parameter s, the main goal is to construct an explicit 2-round privacy amplification protocol that achieves entropy loss O(s). Dodis and Wichs [1] showed that optimal protocols can be achieved by constructing explicit non-malleable extractors. However, the best known explicit non-malleable extractor only achieves k = 0.49n [2] and evidence in [2] suggests that constructing explicit non-malleable extractors for smaller min-entropy may be hard. In an alternative approach, Li [3] introduced the notion of a non-malleable condenser and showed that explicit non-malleable condensers also give optimal privacy amplification protocols. In this paper, we give the first construction of non-malleable condensers for arbitrary min-entropy. Using our construction, we obtain a 2-round privacy amplification protocol with optimal entropy loss for security parameter up to s = Ω(√ k). This is the first protocol that simultaneously achieves optimal round complexity and optimal entropy loss for arbitrary min-entropy k. We also generalize this result to obtain a protocol that runs in O(s/√ k) rounds with optimal entropy loss, for security parameter up to s = Ω(k). This significantly improves the protocol in [4]. Finally, we give a better non-malleable condenser for linear min-entropy, and in this case obtain a 2-round protocol with optimal entropy loss for security parameter up to s = Ω(k), which improves the entropy loss and communication complexity of the protocol in [2].
CITATION STYLE
Li, X. (2015). Non-malleable condensers for arbitrary min-entropy, and almost optimal protocols for privacy amplification. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9014, pp. 502–531). Springer Verlag. https://doi.org/10.1007/978-3-662-46494-6_21
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