  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 \cite{DW09} 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 \cite{Li12b} and
evidence in \cite{Li12b} suggests that constructing explicit non-malleable
extractors for smaller min-entropy may be hard. In an alternative approach, Li
\cite{Li12} 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=\Omega(\sqrt{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/\sqrt{k})
rounds with optimal entropy loss, for security parameter up to s=\Omega(k).
This significantly improves the protocol in \cite{ckor}. 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=\Omega(k), which improves the entropy loss and communication complexity of
the protocol in \cite{Li12b}.
