Abstract
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:
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For any \(s \ge 2\), we give an explicit construction of a s-source non-malleable extractor for min-entropy \(\varOmega (n)\) and error \(2^{-n^{\varOmega (1)}}\) 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).
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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 \(t \le n\) 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.
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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 \(t \le n\) in the overlapping joint tampering model. This is the first construction of a threshold NMSS in the overlapping joint tampering model.
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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 \(\varOmega (n)\) 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 \(t = \varOmega (p)\) faulty processors and failed to extract uniform random bits for a fraction of the honest processors.
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Notes
- 1.
A multi-tamperable non-malleable extractor introduced in [CGL16] considers several sets of split-state tampering functions and requires the output of the extractor to be random even conditioned on all the tampered outputs generated by each split-state tampering function. An equivalent way to view the multi tamperable (or, t tamperable) non-malleable extractor is to allow the split-state tampering functions to have t sets of outputs and we require the real output to be close to random even conditioned on joint distribution of the t tampered outputs.
- 2.
This is where we need the stronger property that for every source j there exists at least one other source that is not tampered together with this source.
- 3.
We note that even for the case of disjoint tampering, the work of Goyal and Kumar [GK18a] assumes that the partitioned subsets must be of unequal length.
- 4.
Similar to the construction of multi-source non-malleable extractor in Sect. 6.2, we need this condition since in proof, we need the fact that there exists \(\mathsf {L}^*\) such that for every \(s\in \{0,1\}^{3m}\) there exists an \(R_{s}\) such that \(\mathsf {2SLNMExt}(L^*, R_{s}) = s\).
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Acknowledgements
We thank the anonymous reviewers of Eurocrypt 2021 for useful comments on our manuscript. The first author was supported in part by NSF grant 1916939, a gift from Ripple, a JP Morgan Faculty Fellowship, and a Cylab seed funding award. Work partially done while the second author was at UC Berkeley and supported in part from AFOSR Award FA9550-19-1-0200, AFOSR YIP Award, NSF CNS Award 1936826, DARPA and SPAWAR under contract N66001-15-C-4065, a Hellman Award and research grants by the Okawa Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). The work was partially done while the second and third authors were visiting CMU. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies.
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Goyal, V., Srinivasan, A., Zhu, C. (2021). Multi-source Non-malleable Extractors and Applications. In: Canteaut, A., Standaert, FX. (eds) Advances in Cryptology – EUROCRYPT 2021. EUROCRYPT 2021. Lecture Notes in Computer Science(), vol 12697. Springer, Cham. https://doi.org/10.1007/978-3-030-77886-6_16
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