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Matrioska: A Compiler for Multi-key Homomorphic Signatures

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Security and Cryptography for Networks (SCN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11035))

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Abstract

Multi-Key Homomorphic Signatures (MK-HS) enable clients in a system to sign and upload messages to an untrusted server. At any later point in time, the server can perform a computation \(C\) on data provided by \(t\) different clients, and return the output \(\mathsf {y}\) and a short signature \(\sigma _{C, \mathsf {y}}\) vouching for the correctness of \(\mathsf {y}\) as the output of the function \(C\) on the signed data. Interestingly, MK-HS enable verifiers to check the validity of the signature using solely the public keys of the signers whose messages were used in the computation. Moreover, the signatures \(\sigma _{C, \mathsf {y}}\) are succinct, namely their size depends at most linearly in the number of clients, and only logarithmically in the total number of inputs of \(C\). Existing MK-HS are constructed based either on standard assumptions over lattices (Fiore et al. ASIACRYPT’16), or on non-falsifiable assumptions (SNARKs) (Lai et al., ePrint’16). In this paper, we investigate connections between single-key and multi-key homomorphic signatures. We propose a generic compiler, called Matrioska, which turns any (sufficiently expressive) single-key homomorphic signature scheme into a multi-key scheme. Matrioska establishes a formal connection between these two primitives and is the first alternative to the only known construction under standard falsifiable assumptions. Our result relies on a novel technique that exploits the homomorphic property of a single-key HS scheme to compress an arbitrary number of signatures from t different users into only t signatures.

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Notes

  1. 1.

    This is the case if one aims for a generic single-key to multi-key construction. In contrast, knowing for example the algebraic structure of signatures can be of help, as exploited in [17].

  2. 2.

    If \(\mathsf {HS}\) works without these a-priori bounds, it is enough to run \(\mathsf {HS}.{\mathsf {Setup}}(1^\lambda )\).

  3. 3.

    The readers can consider the circuit \(\mathsf{{HSV}}_i\) to be the representation of \(\mathsf {HS}.{\mathsf {Verify}}( \mathcal {E}_{{i-1}}, \cdot , \cdot , 1)\) where \(\mathcal {E}_{{i-1}}\) is a labelled program for a circuit of size at most \(O( ({{\mathsf {n}}_{\mathsf{{HSV}}}}_{i-1}+{\mathsf {q}}_{\mathsf{{HSV}}_{i-1}}) \lg ({\mathsf {w}}_{\mathsf{{HSV}}_{i-1}}) )\).

  4. 4.

    With abuse of notation one can think that \(E_{{i}}( {\mathsf {m}} _{ {\mathsf {t}}_{i}}, \ldots , {\mathsf {m}} _{ {\mathsf {t}}_{i}+{\mathsf {n}}_i}) = M_{i}( {\mathsf {m}} _{ {\mathsf {t}}_{i}}, \ldots , {\mathsf {m}} _{ {\mathsf {t}}_{i}+{\mathsf {n}}_i}) \triangleright \ \mathsf{{HSV}}_i = \mathsf{{HSV}}_i (M_{i}( {\mathsf {m}} _{ {\mathsf {t}}_{i}}, \ldots , {\mathsf {m}} _{ {\mathsf {t}}_{i}+{\mathsf {n}}_i}))\). Since \(M_{i}( {\mathsf {m}} _{ {\mathsf {t}}_{i}}, \ldots , {\mathsf {m}} _{ {\mathsf {t}}_{i}+{\mathsf {n}}_{i}})=S_{i-1}\) the claim follows by the definition of \(\mathsf{{HSV}}_i\).

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Acknowledgements

This work was partially supported by the COST Action IC1306 through a STSM grant to Elena Pagnin. Dario Fiore was partially supported by the Spanish Ministry of Economy under project references TIN2015-70713-R (DEDETIS), RTC-2016-4930-7 (DataMantium), and by the Madrid Regional Government under project N-Greens (ref. S2013/ICE-2731).

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Authors and Affiliations

  1. IMDEA Software Institute, Madrid, Spain

    Dario Fiore

  2. Chalmers University of Technology, Gothenburg, Sweden

    Elena Pagnin

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  1. Dario Fiore
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  2. Elena Pagnin
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Correspondence to Elena Pagnin .

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Editors and Affiliations

  1. University of Catania, Catania, Italy

    Dario Catalano

  2. University of Salerno, Fisciano, Italy

    Roberto De Prisco

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Fiore, D., Pagnin, E. (2018). Matrioska: A Compiler for Multi-key Homomorphic Signatures. In: Catalano, D., De Prisco, R. (eds) Security and Cryptography for Networks. SCN 2018. Lecture Notes in Computer Science(), vol 11035. Springer, Cham. https://doi.org/10.1007/978-3-319-98113-0_3

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  • DOI: https://doi.org/10.1007/978-3-319-98113-0_3

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