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ZK-for-Z2K: MPC-in-the-Head Zero-Knowledge Proofs for \(\mathbb {Z}_{2^k}\)

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Cryptography and Coding (IMACC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14421))

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Abstract

In this work, we extend the MPC-in-the-Head framework, used in recent efficient zero-knowledge protocols, to work over the ring \(\mathbb {Z}_{2^k}\), which is the primary operating domain for modern CPUs. The proposed schemes are compatible with any threshold linear secret sharing scheme and draw inspiration from MPC protocols adapted for ring operations. Additionally, we explore various batching methodologies, leveraging Shamir’s secret sharing schemes and Galois ring extensions, and show the applicability of our approach in RAM program verification. Finally, we analyse different options for instantiating the resulting ZK scheme over rings and compare their communication costs.

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Notes

  1. 1.

    Many VOLE proofs can be split into an interactive, witness-independent preprocessing phase and a public-coin online phase, of which the latter can be made non-interactive. Note that this still requires the designated verifier to keep secret state.

  2. 2.

    Informally, an exceptional sequence of elements in a ring R is such that their pairwise difference is invertible. (See Sect. 2.2.).

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Acknowledgements

The work was partially supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001120C0085, by CyberSecurity Research Flanders with reference number VR20192203, by the FWO under an Odysseus project GOH9718N, and by the European Research Council (ERC) under the European Unions’s Horizon 2020 research and innovation programme under grant agreement No. 803096 (SPEC). Cyprien Delpech de Saint Guilhem is a Junior FWO Postdoctoral Fellow under project 1266123N. The work of the last author was conducted whilst they were a PhD student at KU Leuven.

Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of DARPA, the US Government, Cyber Security Research Flanders or the FWO. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.

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Authors and Affiliations

  1. Department of Computer Science, Aarhus University, Aarhus, Denmark

    Lennart Braun

  2. COSIC, KU Leuven, Leuven, Belgium

    Cyprien Delpech de Saint Guilhem, Robin Jadoul & Nigel P. Smart

  3. Department of Computing Sciences, Bocconi University, Milan, Italy

    Emmanuela Orsini

  4. Zama. Inc, Paris, France

    Nigel P. Smart & Titouan Tanguy

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  1. Lennart Braun
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  2. Cyprien Delpech de Saint Guilhem
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  3. Robin Jadoul
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Correspondence to Nigel P. Smart .

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  1. University of London, Egham, UK

    Elizabeth A. Quaglia

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Braun, L., de Saint Guilhem, C.D., Jadoul, R., Orsini, E., Smart, N.P., Tanguy, T. (2024). ZK-for-Z2K: MPC-in-the-Head Zero-Knowledge Proofs for \(\mathbb {Z}_{2^k}\). In: Quaglia, E.A. (eds) Cryptography and Coding. IMACC 2023. Lecture Notes in Computer Science, vol 14421. Springer, Cham. https://doi.org/10.1007/978-3-031-47818-5_8

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  • DOI: https://doi.org/10.1007/978-3-031-47818-5_8

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