Multi-Source Non-Malleable Extractors and Applications.

We introduce a natural generalization of two-source non-malleable extractors (Cheragachi and Guruswami, TCC 2014) called as multi-source non-malleable extractors. Multi-source non-malleable extractors are special independent source extractors which satisfy an additional non-malleability property. This property requires that the output of the extractor remains close to uniform even conditioned on its output generated by tampering several sources together. We formally define this primitive, give a construction that is secure against a wide class of tampering functions, and provide applications. More specifically, we obtain the following results: - For any s >= 2, we give an explicit construction of a s-source non-malleable extractor for min-entropy Omega(n) and error 2(-n Omega(1)) in the overlapping joint tampering model. This means that each tampered source could depend on any strict subset of all the sources and the sets corresponding to each tampered source could be overlapping in a way that we define. Prior to our work, there were no known explicit constructions that were secure even against disjoint tampering (where the sets are required to be disjoint without any overlap). - We adapt the techniques used in the above construction to give a t-out-of-n non-malleable secret sharing scheme (Goyal and Kumar, STOC 2018) for any t <= n in the disjoint tampering model. This is the first general construction of a threshold non-malleable secret sharing (NMSS) scheme in the disjoint tampering model. All prior constructions had a restriction that the size of the tampered subsets could not be equal. - We further adapt the techniques used in the above construction to give a t-out-of-n non-malleable secret sharing scheme (Goyal and Kumar, STOC 2018) for any t <= n in the overlapping joint tampering model. This is the first construction of a threshold NMSS in the overlapping joint tampering model. - We show that a stronger notion of s-source non-malleable extractor that is multi-tamperable against disjoint tampering functions gives a single round network extractor protocol (Kalai et al., FOCS 2008) with attractive features. Plugging in with a new construction of multi-tamperable, 2-source non-malleable extractors provided in our work, we get a network extractor protocol for min-entropy Omega(n) that tolerates an optimum number (t = p - 2) of faulty processors and extracts random bits for every honest processor. The prior network extractor protocols could only tolerate t = Omega(p) faulty processors and failed to extract uniform random bits for a fraction of the honest processors.