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Non-interactive Zero Knowledge Proofs in the Random Oracle Model

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Book cover Codes, Cryptology and Information Security (C2SI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11445))

Abstract

The Fiat-Shamir (FS) transform is a well known and widely used technique to convert any constant-round public-coin honest-verifier zero-knowledge (HVZK) proof or argument system \(\mathsf {HVZK}=(\mathcal {P},\mathcal {V})\) in a non-interactive zero-knowledge (NIZK) argument system

\(\mathsf {NIZK}=(\mathsf {NIZK}.\mathsf {Prove}, \mathsf {NIZK}.\mathsf{Verify})\). The FS transform is secure in the random oracle (RO) model and is extremely efficient: it adds an evaluation of the RO for every message played by \(\mathcal {V}\).

While a major effort has been done to attack the soundness of the transform when the RO is instantiated with a “secure” hash function, here we focus on a different limitation of the FS transform that exists even when there is a secure instantiation of the random oracle: the soundness of \(\mathsf {NIZK}\) holds against polynomial-time adversarial provers only. Therefore even when \(\mathsf {HVZK}\) is a proof system, \(\mathsf {NIZK}\) is only an argument system.

In this paper we propose a new transform from 3-round public-coin HVZK proof systems for several practical relations to NIZK proof systems in the RO model. Our transform outperforms the FS transform protecting the honest verifier from unbounded adversarial provers with no restriction on the number of RO queries. The protocols our transform can be applied to are the ones for proving membership to the range of a one-way group homomorphism as defined by [Maurer - Design, Codes and Cryptography 2015] except that we additionally require the function to be endowed with a trapdoor and other natural properties. For instance, we obtain new efficient instantiations of NIZK proofs for relations related to quadratic residuosity and the RSA function.

As a byproduct, with our transform we obtain essentially for free the first efficient non-interactive zap (i.e., 1-round non-interactive witness indistinguishable proof system) for several practical languages in the non-programmable RO model and in an ideal-PUF model.

Our approach to NIZK proofs can be seen as an abstraction of the celebrated work of [Feige, Lapidot and Shamir - FOCS 1990].

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Notes

  1. 1.

    In literature this difference is often overlooked. Despite this subtle difference, for simplicity we will call proof the string generated by the prover, irrespective of whether the prover be part of a proof or an argument system. We will however be precise on using the words “proof system” and“argument system”.

  2. 2.

    When the challenge c is computed as H(a), the FS transform offers weaker security guarantees (see [BPW12, CPS+16]). In this work, we will consider the strong FS transform.

  3. 3.

    This follows from the fact that \(\lim _{\lambda \rightarrow \infty } 2^{k(\lambda )}=\infty \) and thus \(\lim _{\lambda \rightarrow \infty } (1-\frac{1}{2^{k(\lambda )}})^{-2^{k(\lambda )}}=e.\).

  4. 4.

    Our transform cannot be applied to Chaum and Pedersen’s protocol. However there are examples of natural 3-round public-coin HVZK protocols that have a big ratio between space of commitments and space of challenges and can be made non-interactive through our transform (e.g., quadratic residuosity).

  5. 5.

    This holds for NIZKAs resulting from the strong FS transform, not for the weak FS one [BPW12].

  6. 6.

    Note that also the FS transform leads to statistically sound proof systems against computationally unbounded provers constrained to a polynomial number of RO queries. In this paper, we deem a non-interactive system in the RO a proof system only if it enjoys statistical soundness against unbounded adversaries without any limitation on the number of RO queries.

  7. 7.

    Specifically, it does not hold for all negligible functions but does hold for functions like \(2^{-c\cdot \lambda }\) for some constant \(c>0\).

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Authors and Affiliations

  1. University of Luxembourg, Luxembourg City, Luxembourg

    Vincenzo Iovino

  2. DIEM, University of Salerno, Fisciano, Italy

    Ivan Visconti

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Correspondence to Vincenzo Iovino .

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  1. Université Paris 8, Saint-Denis, France

    Claude Carlet

  2. Institut MINES-TELECOM, Paris, France

    Sylvain Guilley

  3. Université de Caen, Caen, France

    Abderrahmane Nitaj

  4. Mohammed V University, Rabat, Morocco

    El Mamoun Souidi

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Iovino, V., Visconti, I. (2019). Non-interactive Zero Knowledge Proofs in the Random Oracle Model. In: Carlet, C., Guilley, S., Nitaj, A., Souidi, E. (eds) Codes, Cryptology and Information Security. C2SI 2019. Lecture Notes in Computer Science(), vol 11445. Springer, Cham. https://doi.org/10.1007/978-3-030-16458-4_9

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  • DOI: https://doi.org/10.1007/978-3-030-16458-4_9

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  • Print ISBN: 978-3-030-16457-7

  • Online ISBN: 978-3-030-16458-4

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