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Continuously Non-malleable Codes Against Bounded-Depth Tampering

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Advances in Cryptology – ASIACRYPT 2022 (ASIACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13794))

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Abstract

Non-malleable codes (Dziembowski, Pietrzak and Wichs, ICS 2010 & JACM 2018) allow protecting arbitrary cryptographic primitives against related-key attacks (RKAs). Even when using codes that are guaranteed to be non-malleable against a single tampering attempt, one obtains RKA security against poly-many tampering attacks at the price of assuming perfect memory erasures. In contrast, continuously non-malleable codes (Faust, Mukherjee, Nielsen and Venturi, TCC 2014) do not suffer from this limitation, as the non-malleability guarantee holds against poly-many tampering attempts. Unfortunately, there are only a handful of constructions of continuously non-malleable codes, while standard non-malleable codes are known for a large variety of tampering families including, e.g., NC0 and decision-tree tampering, AC0, and recently even bounded polynomial-depth tampering. We change this state of affairs by providing the first constructions of continuously non-malleable codes in the following natural settings:

  • Against decision-tree tampering, where, in each tampering attempt, every bit of the tampered codeword can be set arbitrarily after adaptively reading up to d locations within the input codeword. Our scheme is in the plain model, can be instantiated assuming the existence of one-way functions, and tolerates tampering by decision trees of depth \(d = O(n^{1/8})\), where n is the length of the codeword. Notably, this class includes NC0.

  • Against bounded polynomial-depth tampering, where in each tampering attempt the adversary can select any tampering function that can be computed by a circuit of bounded polynomial depth (and unbounded polynomial size). Our scheme is in the common reference string model, and can be instantiated assuming the existence of time-lock puzzles and simulation-extractable (succinct) non-interactive zero-knowledge proofs.

G. Brian—Supported by grant SPECTRA from Sapienza University of Rome. This work was partly done while G. Brian was visiting the University of Warsaw, Poland, supported by the Copernicus Award (agreement no. COP/01/2020) from the Foundation for Polish Science and by the Premia na Horyzoncie grant (agreement no. 512681/PnH2/2021) from the Polish Ministry of Education and Science.

S. Faust and E. Micheli—This work has been funded by the German Research Foundation (DFG) CRC 1119 CROSSING (project S7), by the German Federal Ministry of Education and Research and the Hessen State Ministry for Higher Education, Research and the Arts within their joint support of the National Research Center for Applied Cybersecurity ATHENE.

D. Venturi—Supported by grant SPECTRA from Sapienza University of Rome.

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Notes

  1. 1.

    Their attack is simple: The j-th tampering function tries to set the j-th bit of the secret key to 0: If the device returns an invalid output, the next function \(f_{j+1}\) additionally sets the j-th bit of the key to 1 and otherwise it sets it to 0.

  2. 2.

    To the best of our knowledge, this observation is new. Previous work in the setting of non-compartmentalized tampering implicitly circumvented the above attack by requiring each tampering function to have high min-entropy and few fixed points, or by assuming that the number of tampering queries is a-priori bounded [49].

  3. 3.

    We can take, e.g., the non-malleable code of [5] for a concrete instantiation.

  4. 4.

    As a bonus, we actually prove continuous super non-malleability.

  5. 5.

    The same observation holds true for the setting of AC0 tampering, but not for decision-tree tampering.

  6. 6.

    Note that we cannot extract the proof outside the leakage function, as the corresponding statement is the tampered modified codeword \(\tilde{\gamma }\) inside the leakage oracle.

  7. 7.

    In the literature, the latter flavor of non-malleability is sometimes known as strong non-malleability whereas the former flavor is also known as weak non-malleability. However, we find this terminology rather confusing due to the fact that a code can be at the same time weakly non-malleable and super non-malleable (as defined below).

  8. 8.

    The oracle additionally takes as input all the values that are required to evaluate the above predicates. We omit them for clarity.

  9. 9.

    Note that, e.g., \(\mathcal {F}_\textsf{split}\) is \(\mathcal {G} _\textsf{split}\)-leakage friendly for any \(q \in poly (\lambda )\).

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Authors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Gianluca Brian & Daniele Venturi

  2. Technische Universität Darmstadt, Darmstadt, Germany

    Sebastian Faust, Elena Micheli & Daniele Venturi

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  1. Gianluca Brian
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  2. Sebastian Faust
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  4. Daniele Venturi
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  1. Indian Institute of Technology Madras, Chennai, India

    Shweta Agrawal

  2. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

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Brian, G., Faust, S., Micheli, E., Venturi, D. (2022). Continuously Non-malleable Codes Against Bounded-Depth Tampering. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13794. Springer, Cham. https://doi.org/10.1007/978-3-031-22972-5_14

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