IACR News item: 11 April 2012
Xin Li
ePrint ReportPreviously, there are only two known constructions of non-malleable extractors \\cite{DLWZ11, CRS11}. 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. If the non-malleable extractor works for small min-entropy and has a short seed length with respect to the error, then 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 $\\alpha>0$.
We also generalize non-malleable extractors to the case where there are more than one adversarial seeds, and show a similar connection between the generalized non-malleable extractors and two-source extractors.
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}.
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