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IACR International Workshop on Public Key Cryptography

PKC 2019: Public-Key Cryptography – PKC 2019 pp 519–548Cite as

Upper and Lower Bounds for Continuous Non-Malleable Codes

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11442))

Abstract

Recently, Faust et al. (TCC’14) introduced the notion of continuous non-malleable codes (CNMC), which provides stronger security guarantees than standard non-malleable codes, by allowing an adversary to tamper with the codeword in a continuous way instead of one-time tampering. They also showed that CNMC with information theoretic security cannot be constructed in the 2-split-state tampering model, and presented a construction in the common reference string (CRS) model from collision-resistant hash functions and non-interactive zero-knowledge proofs.

In this work, we ask if it is possible to construct CNMC from weaker assumptions. We answer this question by presenting lower as well as upper bounds. We show that it is impossible to construct 2-split-state CNMC, with no CRS, for one-bit messages from any falsifiable assumption, thus establishing the lower bound. We additionally provide an upper bound by constructing 2-split-state CNMC for one-bit messages, assuming only the existence of a family of injective one way functions. We note that in a recent work, Ostrovsky et al. (CRYPTO’18) considered the construction of a relaxed notion of 2-split-state CNMC from minimal assumptions.

We also present a construction of 4-split-state CNMC for multi-bit messages in CRS model from the same assumptions. Additionally, we present definitions of the following new primitives: (1) One-to-one commitments, and (2) Continuous Non-Malleable Randomness Encoders, which may be of independent interest.

D. Dachman-Soled—This work is supported in part by NSF grants #CNS-1840893, #CNS-1453045 (CAREER), by a research partnership award from Cisco and by financial assistance award 70NANB15H328 from the U.S. Department of Commerce, National Institute of Standards and Technology.

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Notes

  1. 1.

    A similar relaxed definition was previously given for a variant of CNMC, known as R- CNMC [38], but in this setting it was shown that it is actually impossible to achieve the stronger notion.

  2. 2.

    Note that our extraction technique is inefficient. This is ok, since the goal of the extraction technique is simply to show that the view of the adversary can be simulated given a small amount of leakage on each of the two split-states. Then, information-theoretic properties of the encoding are used to show that the view of the adversary must be independent of the random encoded value.

  3. 3.

    Recall that \(\mathcal {S}'_L \subseteq \mathcal {S}_L\), and \(\mathcal {S}_L\) contains all the values of \(\hat{L'}\) which occur with probability at least \(\epsilon \). Therefore \(| \mathcal {S}_L | \le 1/ \epsilon \) (and thus \(| \mathcal {S}'_L | \le 1/ \epsilon \)), since otherwise the sum of the probabilities would exceed 1. A similar argument is true for \(\mathcal {S}'_R\).

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Acknowledgments

We thank the anonymous PKC 2019 reviewers for pointing out an error and fix to our lower bound proof. We also thank them for extensive comments that helped to significantly improve our presentation.

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  1. University of Maryland, College Park, USA

    Dana Dachman-Soled & Mukul Kulkarni

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  1. Dana Dachman-Soled
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    Dongdai Lin

  2. Security Research Laboratories, NEC Corporation, Kawasaki, Japan

    Kazue Sako

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Dachman-Soled, D., Kulkarni, M. (2019). Upper and Lower Bounds for Continuous Non-Malleable Codes. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11442. Springer, Cham. https://doi.org/10.1007/978-3-030-17253-4_18

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