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Designated-Verifier Pseudorandom Generators, and Their Applications

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Advances in Cryptology – EUROCRYPT 2019 (EUROCRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11477))

Abstract

We provide a generic construction of non-interactive zero-knowledge (NIZK) schemes. Our construction is a refinement of Dwork and Naor’s (FOCS 2000) implementation of the hidden bits model using verifiable pseudorandom generators (VPRGs). Our refinement simplifies their construction and relaxes the necessary assumptions considerably.

As a result of this conceptual improvement, we obtain interesting new instantiations:

  • A designated-verifier NIZK (with unbounded soundness) based on the computational Diffie-Hellman (CDH) problem. If a pairing is available, this NIZK becomes publicly verifiable. This constitutes the first fully secure CDH-based designated-verifier NIZKs (and more generally, the first fully secure designated-verifier NIZK from a non-generic assumption which does not already imply publicly-verifiable NIZKs), and it answers an open problem recently raised by Kim and Wu (CRYPTO 2018).

  • A NIZK based on the learning with errors (LWE) assumption, and assuming a non-interactive witness-indistinguishable (NIWI) proof system for bounded distance decoding (BDD). This simplifies and improves upon a recent NIZK from LWE that assumes a NIZK for BDD (Rothblum et al., PKC 2019).

G. Couteau—Supported by ERC grant “PREP-CRYPTO” (724307).

D. Hofheinz—Supported by ERC grant “PREP-CRYPTO” (724307) and DFG project GZ HO 4304/4-2.

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Notes

  1. 1.

    In the HBM, there exist unconditionally secure \(\textsf {NIZK}\) proofs [17].

  2. 2.

    A formal argument requires a little care in choosing parameters, and in randomizing \(\mathsf {hb} \) with an additional component in the \(\textsf {NIZK}\) common reference string.

  3. 3.

    One might wonder why we do not follow another route to obtain VPRGs from NIWI proofs for \(\mathsf {BDD}\). Specifically, [3, 23] construct even verifiable random functions from a NIWI for a (complex) LWE-related language. However, these constructions inherently use disjunctions, and it seems unlikely that the corresponding NIWIs can be reduced to the \(\mathsf {BDD}\) language.

  4. 4.

    The actual construction of [37] relies on a new notion of public-key encryption with prover-assisted oblivious ciphertext sampling, but the high level idea is comparable to the \(\mathsf {VPRG}\)-based approach. A side contribution of our construction is that it is conceptually much simpler and straightforward than the construction of [37].

  5. 5.

    Since n is the order of \(\mathbb {G}\) and \(\mathbb {G}\) is a group in which CDH is assumed to hold, 2 / n is negligible in the security parameter.

  6. 6.

    Strictly speaking, this may not be true, depending on \(G_m\): the LWE parameters may depend on the size of the \(G_{m,i}\), which in turn may depend on \(m\). However, here we implicitly assume that \(m\le 2^\lambda \) and a suitable \(G_m\) (e.g., one in which \(G_{m,i}(s)\) is of the form \(F_s(i)\) for a PRF \(F:\{0,1\} ^\lambda \rightarrow \{0,1\} \)).

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Authors and Affiliations

  1. KIT, Karlsruhe, Germany

    Geoffroy Couteau & Dennis Hofheinz

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  1. Geoffroy Couteau
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Correspondence to Geoffroy Couteau .

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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Couteau, G., Hofheinz, D. (2019). Designated-Verifier Pseudorandom Generators, and Their Applications. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11477. Springer, Cham. https://doi.org/10.1007/978-3-030-17656-3_20

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