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Explicit Rate-1 Non-malleable Codes for Local Tampering

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Book cover Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11692))

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Abstract

This paper constructs high-rate non-malleable codes in the information-theoretic plain model against tampering functions with bounded locality. We consider \(\delta \)-local tampering functions; namely, each output bit of the tampering function is a function of (at most) \(\delta \) input bits. This work presents the first explicit and efficient rate-1 non-malleable code for \(\delta \)-local tampering functions, where \(\delta =\xi \lg n\) and \(\xi <1\) is any positive constant. As a corollary, we construct the first explicit rate-1 non-malleable code against NC\(^0\) tampering functions.

Before our work, no explicit construction for a constant-rate non-malleable code was known even for the simplest 1-local tampering functions. Ball et al. (EUROCRYPT–2016), and Chattopadhyay and Li (STOC–2017) provided the first explicit non-malleable codes against \(\delta \)-local tampering functions. However, these constructions are rate-0 even when the tampering functions have 1-locality. In the CRS model, Faust et al. (EUROCRYPT–2014) constructed efficient rate-1 non-malleable codes for \(\delta = O(\log n)\) local tampering functions.

Our main result is a general compiler that bootstraps a rate-0 non-malleable code against leaky input and output local tampering functions to construct a rate-1 non-malleable code against \(\xi \lg n\)-local tampering functions, for any positive constant \(\xi < 1\). Our explicit construction instantiates this compiler using an appropriate encoding by Ball et al. (EUROCRYPT–2016).

H. K. Maji—The research effort is supported in part by an NSF CRII Award CNS–1566499, an NSF SMALL Award CNS–1618822, and an REU CNS–1724673.

H. K. Maji and M. Wang—The research effort is supported in part by a Purdue Research Foundation grant.

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Notes

  1. 1.

    Tampering functions can access the CRS; however, they cannot tamper the CRS.

  2. 2.

    This construction is also an efficient rate-1 construction in the CRS model.

  3. 3.

    Error-correcting codes can be converted into error-correcting secret sharing schemes using standard share-packing techniques [9, 21, 34].

  4. 4.

    An ECSS of independence t has the property that any t shares are uniformly and independently random.

  5. 5.

    An ECSS with distance d ensures that, for two different secrets, at least d secret shares are different.

  6. 6.

    If the tampering function flips the input bit then the probability of disagreement is 1; otherwise, the probability of disagreement is 1/2.

  7. 7.

    Similar to [6], hash function families with sufficiently high independence also suffice in this context.

  8. 8.

    Note that at this point, the original seed \(s^L\) and \(s^R\) and their input neighbors \(c_Q\) from main codeword c is already fixed.

  9. 9.

    If \(\widetilde{c^L}\) or \(\widetilde{c^R}\) are not contained in \(\widetilde{\alpha ^L}\) or \(\widetilde{\alpha ^R}\), \(f_0\) will simply set g to be a \(\bot \) function.

  10. 10.

    Note that those places in \(\alpha ^L,\alpha ^R\) that are not used to store \(c^L\) and \(c^R\) are also fixed (to be 0 by the compiler).

  11. 11.

    Note that, by the definition of V, all the output bits from \([n]\backslash V\) are fixed to some values with no input neighbors. Hence, it suffices to have the neighbor of V to finish the hybrid completely.

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Authors and Affiliations

  1. Microsoft Research, Bangalore, India

    Divya Gupta

  2. Department of Computer Science, Purdue University, West Lafayette, USA

    Hemanta K. Maji & Mingyuan Wang

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  1. Divya Gupta
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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

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Gupta, D., Maji, H.K., Wang, M. (2019). Explicit Rate-1 Non-malleable Codes for Local Tampering. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11692. Springer, Cham. https://doi.org/10.1007/978-3-030-26948-7_16

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