Abstract
In this paper we give improved constructions of several central objects in the literature of randomness extraction and tamper-resilient cryptography. Our main results are: (1) An explicit seeded non-malleable extractor with error ε and seed length d=O(log n)+O(log(1/ε)log log (1/ε)), that supports min-entropy k=Ω(d) and outputs Ω(k) bits. Combined with the protocol in \cite{DW09}, this gives a two round privacy amplification protocol with optimal entropy loss in the presence of an active adversary, for all security parameters up to Ω(k/log k). (2) An explicit non-malleable two-source extractor for min-entropy k ≥ (1-γ)n, some constant γ>0, that outputs Ω(k) bits with error 2^-Ω(n/log n). Combined with the connection in \cite{CG14b} this gives a non-malleable code in the two-split-state model with relative rate Ω(1/log n). This exponentially improves previous constructions, all of which only achieve rate n^-Ω(1).\footnote{The work of Aggarwal et. al \cite{ADKO15} had a construction which "achieves" constant rate, but recently the author found an error in their proof.} (3)A two-source extractor for min-entropy O(log n log log n), which also implies a K-Ramsey graph on N vertices with K=(log N)^O(log log log N). We also obtain a seeded non-malleable 9-source extractor with optimal seed length, which in turn gives a 10-source extractor for min-entropy O(log n). Previously the best known extractor for such min-entropy requires O(log log n) sources \cite{CohL16}. Independent of our work, Cohen \cite{Cohen16} obtained similar results to (1) and the two-source extractor, except the dependence on ε is log(1/ε)(log log (1/ε))^O(1) and the two-source extractor requires min-entropy log n (log log n)^O(1).