An Entropy Lower Bound for Non-Malleable Extractors

A (k, ε)-non-malleable extractor is a function nmExt : {0, 1}×{0, 1} → {0, 1} that takes two inputs, a weak source X ∼ {0, 1} of min-entropy k and an independent uniform seed s ∈ {0, 1}, and outputs a bit nmExt(X, s) that is ε-close to uniform, even given the seed s and the value nmExt(X, s′) for an adversarially chosen seed s′ 6= s. Dodis and Wichs (STOC 2009) showed the existence of (k, ε)-non-malleable extractors with seed length d = log(n− k− 1) + 2 log(1/ε) + 6 that support sources of entropy k > log(d) + 2 log(1/ε) + 8. We show that the foregoing bound is essentially tight, by proving that any (k, ε)-nonmalleable extractor must satisfy the entropy bound k > log(d) + 2 log(1/ε) − log log(1/ε) − C for an absolute constant C. In particular, this implies that non-malleable extractors require min-entropy at least Ω(log log(n)). This is in stark contrast to the existence of strong seeded extractors that support sources of entropy k = O(log(1/ε)). 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 (k, ε)-non-malleable extractor with seed length d induces a code C ⊆ {0, 1}2k with relative distance 0.5− 2ε and rate d−1 2k .