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Zero-Knowledge Arguments for Subverted RSA Groups

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

This work investigates zero-knowledge protocols in subverted RSA groups where the prover can choose the modulus and where the verifier does not know the group order. We introduce a novel technique for extracting the witness from a general homomorphism over a group of unknown order that does not require parallel repetitions. We then present a NIZK range proof for general homomorphisms as Paillier encryptions in the designated verifier model that works under a subverted setup. The key ingredient of our proof is a constant sized NIZK proof of knowledge for a plaintext. Security is proven in the ROM assuming an IND-CPA additively homomorphic encryption scheme. The verifier’s public key can be maliciously generated and is reusable and linear in the number of proofs to be verified.

D. Kolonelos and M. Volkhov—Most of the work was done while the first and third authors were interns at Ethereum Foundation.

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Notes

  1. 1.

    By ‘natural’ we mean a protocol that works directly for the underlying language and does not involve NP-reductions.

  2. 2.

    For some relations (e.g. Paillier Encryptions) this can lead to fully reconstructing the witness.

  3. 3.

    https://github.com/alex-ozdemir/bellman-bignat.

  4. 4.

    https://research.protocol.ai/sites/snarks/.

  5. 5.

    Extracting k successful transcripts is no harder than extracting 2 [1].

  6. 6.

    Although the lifted ElGamal cryptosystem (alike ElGamal but the message is lifted in the exponent) is additively homomorphic, the decryption is not polynomial-time, unless one restricts the message space to polynomial size. This makes it unsuitable for most applications.

  7. 7.

    Its special PP is (q, 0), since \(Y^q = \psi (0)\); and the PP is non-trivial: \(q \ne 0 \mod p\).

  8. 8.

    From \(Y = G^m r^N\) we can derive \(Y^N = (G^m r^N)^N = G^0 (G^m r^N)^N\), so (N, (0, Y)) is a pseudo-preimage of degree N (and \(N \ne 0 \mod \phi (N^2)\)).

  9. 9.

    As long as all elements in \(\llbracket 2^{\lambda +1} \rrbracket \) have a multiplicative inverse in \(\mathcal {M}\).

  10. 10.

    We further assume that if \(\mathbb {Z}_N\) is the message space, then the largest factor of N is larger than \(2^{\lambda +1}\), which is the case for example in Paillier.

  11. 11.

    For ease of exposition we keep the description simple. We omit the technical details of special soundness extractors related to aborting senarios, that ensure termination in polynomial time(see lemma 5, [2]).

  12. 12.

    In case \(\phi (N_\textsf{cm})\) is unknown, sampling is statistically close.

  13. 13.

    The ± relaxation is artificially added in order to achieve a sound zero-knowledge proof of opening of c, which however does not affect the binding of the commitment scheme.

  14. 14.

    The implementation is available publicly on Github: https://github.com/volhovm/rsa-zkps-impl.

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Acknowledgements

The first author received funding from projects from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under project PICOCRYPT (grant agreement No. 101001283), from the Spanish Government under project PRODIGY (TED2021-132464B-I00), and from the Madrid Regional Government under project BLOQUES (S2018/TCS-4339). The last two projects are co-funded by European Union EIE, and NextGenerationEU/PRTR funds. The last author was partially funded by Input Output (iohk.io) through their funding of the Edinburgh Blockchain Technology Lab.

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Authors and Affiliations

  1. IMDEA Software Institute, Madrid, Spain

    Dimitris Kolonelos

  2. Universidad Politecnica de Madrid, Madrid, Spain

    Dimitris Kolonelos

  3. Ethereum Foundation, London, UK

    Mary Maller

  4. The University of Edinburgh, Edinburgh, UK

    Mikhail Volkhov

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Kolonelos, D., Maller, M., Volkhov, M. (2023). Zero-Knowledge Arguments for Subverted RSA Groups. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_18

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