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TR15-183 | 16th November 2015 23:01

#### Non-Malleable Extractors - New Tools and Improved Constructions

TR15-183
Authors: Gil Cohen
Publication: 18th November 2015 00:08
Keywords:

Abstract:

A non-malleable extractor is a seeded extractor with a very strong guarantee - the output of a non-malleable extractor obtained using a typical seed is close to uniform even conditioned on the output obtained using any other seed. The first contribution of this paper consists of two new and improved constructions of non-malleable extractors:

* We construct a non-malleable extractor with seed-length $O(\log{n} \cdot \log\log{n})$ that works for entropy $\Omega(\log{n})$. This improves upon a recent exciting construction by Chattopadhyay, Goyal, and Li (ECCC'15) that has seed length $O(\log^{2}{n})$ and requires entropy $\Omega(\log^{2}{n})$.

* Secondly, we construct a non-malleable extractor with optimal seed length $O(\log{n})$ for entropy $n/\mathrm{polylog}n$. Prior to this construction, non-malleable extractors with a logarithmic seed length, due to Li (FOCS'12), required entropy $0.49n$. Even non-malleable condensers with seed length $O(\log{n})$, by Li (STOC'12), could only support linear entropy.

We further devise several tools for enhancing a given non-malleable extractor in a black-box manner. One such tool is an algorithm that reduces the entropy requirement of a non-malleable extractor at the expense of a slightly longer seed. A second algorithm increases the output length of a non-malleable extractor from constant to linear in the entropy of the source. We also devise an algorithm that transforms a non-malleable extractor to the so-called $t$-non-malleable extractor for any desired $t$. Besides being useful building blocks for our constructions, we consider these modular tools to be of independent interest.

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