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Structure-Preserving Signatures on Equivalence Classes from Standard Assumptions

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Advances in Cryptology – ASIACRYPT 2019 (ASIACRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11923))

Abstract

Structure-preserving signatures on equivalence classes (SPS-EQ) introduced at ASIACRYPT 2014 are a variant of SPS where a message is considered as a projective equivalence class, and a new representative of the same class can be obtained by multiplying a vector by a scalar. Given a message and corresponding signature, anyone can produce an updated and randomized signature on an arbitrary representative from the same equivalence class. SPS-EQ have proven to be a very versatile building block for many cryptographic applications.

In this paper, we present the first EUF-CMA secure SPS-EQ scheme under standard assumptions. So far only constructions in the generic group model are known. One recent candidate under standard assumptions are the weakly secure equivalence class signatures by Fuchsbauer and Gay (PKC’18), a variant of SPS-EQ satisfying only a weaker unforgeability and adaption notion. Fuchsbauer and Gay show that this weaker unforgeability notion is sufficient for many known applications of SPS-EQ. Unfortunately, the weaker adaption notion is only proper for a semi-honest (passive) model and as we show in this paper, makes their scheme unusable in the current models for almost all of their advertised applications of SPS-EQ from the literature.

We then present a new EUF-CMA secure SPS-EQ scheme with a tight security reduction under the SXDH assumption providing the notion of perfect adaption (under malicious keys). To achieve the strongest notion of perfect adaption under malicious keys, we require a common reference string (CRS), which seems inherent for constructions under standard assumptions. However, for most known applications of SPS-EQ we do not require a trusted CRS (as the CRS can be generated by the signer during key generation). Technically, our construction is inspired by a recent work of Gay et al. (EUROCRYPT’18), who construct a tightly secure message authentication code and translate it to an SPS scheme adapting techniques due to Bellare and Goldwasser (CRYPTO’89).

M. Khalili—Work partly done while visiting Universitat Pompeu Fabra, Barcelona, Spain.

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Notes

  1. 1.

    For the group signatures in [12] it will only work with our construction when relying on a CRS, or by using the construction secure in the GGM in [38].

  2. 2.

    https://z.cash/.

  3. 3.

    See e.g., https://z.cash/blog/the-design-of-the-ceremony/ or https://www.zfnd.org/blog/conclusion-of-powers-of-tau/.

  4. 4.

    We note that we can only instantiate our construction for \(k=1\), i.e., under the SXDH assumption, and leave the construction of SPS-EQ under the more general Matrix Decision Diffie-Hellman assumption as an interesting open problem.

  5. 5.

    Thus, we will formally have a NIZK argument, but in the text we will usually not make a distinction between NIZK proofs and arguments.

  6. 6.

    Even if all involved proof systems provide zero-knowledge definitions in the style of composable zero-knowledge [50], i.e., even if the adversary knows the trapdoor and still simulated and honestly computed proofs cannot be distinguished, we still have the problem of maliciously generated proofs and thus we cannot avoid a CRS.

  7. 7.

    One syntactical difference is that for EQS they do not input the message \([\mathbf{m}]_i\) in their \(\mathsf {ChgRep}\) algorithm, but this does not matter for our discussion.

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Acknowledgments

We are grateful to the anonymous reviewers from ASIACRYPT 2019 and Romain Gay for their careful reading of the paper, their valuable feedback and suggestions to improve the presentation. We also thanks Carla Ràfols and Alonso González for their comments on earlier versions of this work. This work was supported by the EU’s Horizon 2020 ECSEL Joint Undertaking project SECREDAS under grant agreement n\(\circ \)783119 and by the Austrian Science Fund (FWF) and netidee SCIENCE project PROFET (grant agreement P31621-N38).

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Authors and Affiliations

  1. Isfahan University of Technology, Isfahan, Iran

    Mojtaba Khalili & Mohammad Dakhilalian

  2. AIT Austrian Institute of Technology, Vienna, Austria

    Daniel Slamanig

Authors
  1. Mojtaba Khalili
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  2. Daniel Slamanig
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  3. Mohammad Dakhilalian
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Correspondence to Mojtaba Khalili .

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Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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© 2019 International Association for Cryptologic Research

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Khalili, M., Slamanig, D., Dakhilalian, M. (2019). Structure-Preserving Signatures on Equivalence Classes from Standard Assumptions. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11923. Springer, Cham. https://doi.org/10.1007/978-3-030-34618-8_3

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  • DOI: https://doi.org/10.1007/978-3-030-34618-8_3

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  • Online ISBN: 978-3-030-34618-8

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