An Entropy Lower Bound for Non-Malleable Extractors

A <inline-formula> <tex-math notation="LaTeX">$(k, \varepsilon )$ </tex-math></inline-formula>-non-malleable extractor is a function <inline-formula> <tex-math notation="LaTeX">$ {\sf nmExt}: \{0,1\}^{n} \times \{0,1\}^{d} \to \{0,1\}$ </tex-math></inline-formula> that takes two inputs, a weak source <inline-formula> <tex-math notation="LaTeX">$X \sim \{0,1\}^{n}$ </tex-math></inline-formula> of min-entropy <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> and an independent uniform seed <inline-formula> <tex-math notation="LaTeX">$s \in \{0,1\}^{d}$ </tex-math></inline-formula>, and outputs a bit <inline-formula> <tex-math notation="LaTeX">$ {\sf nmExt}(X, s)$ </tex-math></inline-formula> that is <inline-formula> <tex-math notation="LaTeX">$ \varepsilon $ </tex-math></inline-formula>-close to uniform, even given the seed <inline-formula> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> and the value <inline-formula> <tex-math notation="LaTeX">$ {\sf nmExt}(X, s')$ </tex-math></inline-formula> for an adversarially chosen seed <inline-formula> <tex-math notation="LaTeX">$s' \neq s$ </tex-math></inline-formula>. Dodis and Wichs (STOC 2009) showed the existence of <inline-formula> <tex-math notation="LaTeX">$(k, \varepsilon )$ </tex-math></inline-formula>-non-malleable extractors with seed length <inline-formula> <tex-math notation="LaTeX">$d = \log (n-k-1) + 2\log (1/ \varepsilon ) + 6$ </tex-math></inline-formula> that support sources of min-entropy <inline-formula> <tex-math notation="LaTeX">$k > \log (d) + 2 \log (1/ \varepsilon ) + 8$ </tex-math></inline-formula>. We show that the foregoing bound is essentially tight, by proving that any <inline-formula> <tex-math notation="LaTeX">$(k, \varepsilon )$ </tex-math></inline-formula>-non-malleable extractor must satisfy the min-entropy bound <inline-formula> <tex-math notation="LaTeX">$k > \log (d) + 2 \log (1/ \varepsilon ) - \log \log (1/ \varepsilon ) - C$ </tex-math></inline-formula> for an absolute constant <inline-formula> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula>. In particular, this implies that non-malleable extractors require min-entropy at least <inline-formula> <tex-math notation="LaTeX">$\Omega (\log \log (n))$ </tex-math></inline-formula>. This is in stark contrast to the existence of strong seeded extractors that support sources of min-entropy <inline-formula> <tex-math notation="LaTeX">$k = O(\log (1/ \varepsilon ))$ </tex-math></inline-formula>. Our techniques strongly rely on coding theory. In particular, we reveal an inherent connection between non-malleable extractors and error correcting codes, by proving a new lemma which shows that any <inline-formula> <tex-math notation="LaTeX">$(k, \varepsilon )$ </tex-math></inline-formula>-non-malleable extractor with seed length <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> induces a code <inline-formula> <tex-math notation="LaTeX">$ {\mathcal C} \subseteq \{0,1\}^{2^{k}}$ </tex-math></inline-formula> with relative distance <inline-formula> <tex-math notation="LaTeX">$ \frac {1}{2}- 2 \varepsilon $ </tex-math></inline-formula> and rate <inline-formula> <tex-math notation="LaTeX">$\frac {d-1}{2^{k}}$ </tex-math></inline-formula>.

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