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(Nondeterministic) Hardness vs. Non-malleability

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

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Abstract

We present the first truly explicit constructions of non-malleable codes against tampering by bounded polynomial size circuits. These objects imply unproven circuit lower bounds and our construction is secure provided \(\textsf{E}\) requires exponential size nondeterministic circuits, an assumption from the derandomization literature.

Prior works on NMC for polysize circuits, either required an untamperable CRS [Cheraghchi, Guruswami ITCS’14; Faust, Mukherjee, Venturi, Wichs EUROCRYPT’14] or very strong cryptographic assumptions [Ball, Dachman-Soled, Kulkarni, Lin, Malkin EUROCRYPT’18; Dachman-Soled, Komargodski, Pass CRYPTO’21]. Both of works in the latter category only achieve non-malleability with respect to efficient distinguishers and, more importantly, utilize cryptographic objects for which no provably secure instantiations are known outside the random oracle model. In this sense, none of the prior yields fully explicit codes from non-heuristic assumptions. Our assumption is not known to imply the existence of one-way functions, which suggests that cryptography is unnecessary for non-malleability against this class.

Technically, security is shown by non-deterministically reducing polynomial size tampering to split-state tampering. The technique is general enough that it allows us to construct the first seedless non-malleable extractors [Cheraghchi, Guruswami TCC’14] for sources sampled by polynomial size circuits [Trevisan, Vadhan FOCS’00] (resp. recognized by polynomial size circuits [Shaltiel CC’11]) and tampered by polynomial size circuits. Our construction is secure assuming \(\textsf{E}\) requires exponential size \(\varSigma _4\)-circuits (resp. \(\varSigma _3\)-circuits), this assumption is the state-of-the-art for extracting randomness from such sources (without non-malleability).

Several additional results are included in the full version of this paper [Eprint 2022/070]. First, we observe that non-malleable codes and non-malleable secret sharing [Goyal, Kumar STOC’18] are essentially equivalent with respect to polynomial size tampering. In more detail, assuming \(\textsf{E}\) is hard for exponential size nondeterministic circuits, any efficient secret sharing scheme can be made non-malleable against polynomial size circuit tampering.

Second, we observe that the fact that our constructions only achieve inverse polynomial (statistical) security is inherent. Extending a result from [Applebaum, Artemenko, Shaltiel, Yang CC’16] we show it is impossible to do better using black-box reductions. However, we extend the notion of relative error from [Applebaum, Artemenko, Shaltiel, Yang CC’16] to non-malleable extractors and show that they can be constructed from similar assumptions.

Third, we observe that relative-error non-malleable extractors can be utilized to render a broad class of cryptographic primitives tamper and leakage resilient, while preserving negligible security guarantees.

M. Ball—Part of this work was done while the author was a student at Columbia University and a postdoc at University of Washington. This material is based upon work supported by the National Science Foundation under Grant #2030859 to the Computing Research Association for the CIFellows Project. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation nor the Computing Research Association.

D. Dachman-Soled—Supported in part by NSF grants #CNS-1933033, #CNS-1453045 (CAREER), and by financial assistance awards 70NANB15H328 and 70NANB19H126 from the U.S. Department of Commerce, National Institute of Standards and Technology.

J. Loss—Part of this work was done while the author was a postdoc at the University of Maryland and Carnegie Mellon University.

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Notes

  1. 1.

    If \((\textsf{E},\textsf{D})\) is \(\epsilon \)-non-malleable code for \(n^c\)-size tampering, then \(\textsf{D}\) is hard-on-average for \(n^c-O(n)\) size circuits with respect to the distribution \(\textsf{E}(\mathcal {U}_{\{0,1\}})\), encodings of a random bit. In particular if there exists a small circuit C such that \(\Pr [C(\textsf{E}(\mathcal {U}))=\textsf{D}(\textsf{E}(\mathcal {U}))=\mathcal {U}]\ge 1/2+\epsilon \) then consider the \(C'\) that on input c outputs a fixed encoding of 0,\(c_0\), if \(C(c)=1\) and a fixed encoding of 1, \(c_1\) otherwise. Then we have \(\Pr [\textsf{D}(C'(\textsf{E}(\mathcal {U})))=1-\mathcal {U}]\ge 1/2+\epsilon \), breaking \(\epsilon \)-non-malleability.

  2. 2.

    In addition to a variety of subexponentially secure variants of standard cryptographic assumptions, the work of [28, 29] also crucially requires a specific number-theoretic assumption (the non-uniform subexponential hardness of the repeated squaring assumption), while the work of [12] needs the same derandomization assumption in this work.

  3. 3.

    E.g. [21] suggests possibly instantiating keyless multi-collision resistant hash with an unstructured hash, such as SHA-2 (extended to arbitrarily large keys), with keys chosen according to digits of \(\pi \). Establishing the security of any such candidate is well beyond our current techniques, as we cannot even base the security of (extended) SHA-2 with randomly chosen keys to a natural computational problem.

  4. 4.

    Note that an \(\textsf{NP}\)-circuit is different than a nondeterministic circuit. The former is a nonuniform analogue of \(\textsf{P}^{\textsf{NP}}\) (which contains \(\textsf{coNP}\)) while the latter is an analogue of \(\textsf{NP}\).

  5. 5.

    Min-entropy measures the unpredictability of a random variable. In particular, X has min-entropy k if \(\forall x\) in the support of X, \(\Pr [X=x]\le 2^{-k}\).

  6. 6.

    Sources sampled by polynomial size quantum circuits seem a more appropriate model for physical sources of randomness. Nonetheless, (classical) samplable sources are an interesting and important subclass.

  7. 7.

    Note that with a random seed it is easy to extract from say \(X_1\) conditioned on \(X_2\).

  8. 8.

    In particular, the Decode function is hard with respect to the distribution formed by encoding a random bit. If this wasn’t the case, one could attack by computing the encoded value and outputting a fixed encoding of the opposite bit.

  9. 9.

    Note that ruling out reductions to 1-bit non-malleable codes also rules out reductions to k-bit non-malleable codes.

  10. 10.

    In fact, the precise leakage class we can handle is slightly more broad.

  11. 11.

    To see this, recall the characterization of non-malleability for a single bit (see previous footnote ). Note that for any tampering function f of size \(n^c\), one can define a function \(f'\) of size \(n^c+O(n)\) that has no fixed points and behaves identically to f on every x that is not a fixed point of f. Because, \(\Pr [D(f(\textsf{E}(b)=1-b]\le \Pr [D(f'(\textsf{E}(b))=1-b]\) we can deduce that \(\textsf{E},\textsf{D}\) is non-malleable with respect to circuits of size \(n^c-O(n)\), where \(\textsf{D}\) is \(\textrm{NMExt}\) and \(\textsf{E}\) simply performs rejection sampling to find a random (s, x) such that \(\textrm{NMExt}(s,x)=b\). Note that the resulting non-malleable code will not have perfect correctness because the rejection sampling procedure might fail.

  12. 12.

    We refer the reader to [51] for further discussion.

  13. 13.

    In actuality, this is too naive because these transformations only hold for worst-case notions of soundness and completeness. Thus in the body, we will instead show that there exists a constant round interactive proof for a promise problem (\(\varPi _Y,\varPi _N\)) such that \(\varPi _Y\) is dense in the pseudorandom distribution and \(\varPi _N\) is dense in the uniform distribution, and not vice-versa.

  14. 14.

    Cheraghchi and Guruswami [27] showed a similar lemma for the case of split-state tampering.

  15. 15.

    In the literature, leakage-resilient has been alternately used to refer to codes that handle leakage only to the distinguisher as well as code that handle leakage only between the tampering of each state.

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Author information

Authors and Affiliations

  1. New York University, New York, USA

    Marshall Ball

  2. University of Maryland, College Park, USA

    Dana Dachman-Soled

  3. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Julian Loss

Authors
  1. Marshall Ball
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  2. Dana Dachman-Soled
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  3. Julian Loss
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Correspondence to Dana Dachman-Soled .

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Editors and Affiliations

  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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© 2022 International Association for Cryptologic Research

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Ball, M., Dachman-Soled, D., Loss, J. (2022). (Nondeterministic) Hardness vs. Non-malleability. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13507. Springer, Cham. https://doi.org/10.1007/978-3-031-15802-5_6

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  • DOI: https://doi.org/10.1007/978-3-031-15802-5_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-15801-8

  • Online ISBN: 978-3-031-15802-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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