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Designated Verifier/Prover and Preprocessing NIZKs from Diffie-Hellman Assumptions

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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

In a non-interactive zero-knowledge (NIZK) proof, a prover can non-interactively convince a verifier of a statement without revealing any additional information. Thus far, numerous constructions of NIZKs have been provided in the common reference string (CRS) model (CRS-NIZK) from various assumptions, however, it still remains a long standing open problem to construct them from tools such as pairing-free groups or lattices. Recently, Kim and Wu (CRYPTO’18) made great progress regarding this problem and constructed the first lattice-based NIZK in a relaxed model called NIZKs in the preprocessing model (PP-NIZKs). In this model, there is a trusted statement-independent preprocessing phase where secret information are generated for the prover and verifier. Depending on whether those secret information can be made public, PP-NIZK captures CRS-NIZK, designated-verifier NIZK (DV-NIZK), and designated-prover NIZK (DP-NIZK) as special cases. It was left as an open problem by Kim and Wu whether we can construct such NIZKs from weak paring-free group assumptions such as DDH. As a further matter, all constructions of NIZKs from Diffie-Hellman (DH) type assumptions (regardless of whether it is over a paring-free or paring group) require the proof size to have a multiplicative-overhead \(|C| \cdot \mathsf {poly}(\kappa )\), where |C| is the size of the circuit that computes the \(\mathbf {NP}\) relation.

In this work, we make progress of constructing (DV, DP, PP)-NIZKs with varying flavors from DH-type assumptions. Our results are summarized as follows:

  • DV-NIZKs for \(\mathbf {NP}\) from the CDH assumption over pairing-free groups. This is the first construction of such NIZKs on pairing-free groups and resolves the open problem posed by Kim and Wu (CRYPTO’18).

  • DP-NIZKs for \(\mathbf {NP}\) with short proof size from a DH-type assumption over pairing groups. Here, the proof size has an additive-overhead \(|C|+\mathsf {poly}(\kappa )\) rather then an multiplicative-overhead \(|C| \cdot \mathsf {poly}(\kappa )\). This is the first construction of such NIZKs (including CRS-NIZKs) that does not rely on the LWE assumption, fully-homomorphic encryption, indistinguishability obfuscation, or non-falsifiable assumptions.

  • PP-NIZK for \(\mathbf {NP}\) with short proof size from the DDH assumption over pairing-free groups. This is the first PP-NIZK that achieves a short proof size from a weak and static DH-type assumption such as DDH. Similarly to the above DP-NIZK, the proof size is \(|C|+\mathsf {poly}(\kappa )\). This too serves as a solution to the open problem posed by Kim and Wu (CRYPTO’18).

Along the way, we construct two new homomorphic authentication (HomAuth) schemes which may be of independent interest.

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Notes

  1. 1.

    NIZK arguments are a relaxed notion of NIZK proofs where soundness only holds against computationally bounded adversaries. Throughout the introduction, we simply refer to them as NIZKs.

  2. 2.

    In fact, as we show in Table 1, all of these approaches lead to a much more succinct proof size of \(|w| + \mathsf {poly}(\kappa )\), where w is the witness.

  3. 3.

    Though a cheating prover can arbitrarily choose \(\tau \in \mathbb {Z}_p\), we can negligibly bound its success probability by the union bound if the success probability of a cheating prover of the underlying HBM-NIZK is bounded by \(p^{-1}\cdot \mathsf {negl}(\kappa )\).

  4. 4.

    Actually, these previous works prove the standard indistinguishability security notion rather than one-wayness. However, one-wayness is sufficient for our application.

  5. 5.

    Though the original construction by Catalano and Fiore [27] is based on PRF, we present an information theoretically secure variant of it in a simplified setting where the arity of an arithmetic circuit is bounded.

  6. 6.

    Though the scheme is not publicly verifiable, we call \(\varvec{\sigma }\) a “signature” for compatibility to HomSig.

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Acknowledgement

We would like to thank Geoffroy Couteau for helpful comments on related works and anonymous reviewers of Eurocrypt 2019 for their valuable comments. The first author was partially supported by JST CREST Grant Number JPMJCR1302 and JSPS KAKENHI Grant Number 17J05603. The third author was supported by JST CREST Grant No. JPMJCR1688 and JSPS KAKENHI Grant Number 16K16068.

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Authors and Affiliations

  1. AIST, Tokyo, Japan

    Shuichi Katsumata & Shota Yamada

  2. NTT Secure Platform Laboratories, Tokyo, Japan

    Ryo Nishimaki & Takashi Yamakawa

  3. The University of Tokyo, Tokyo, Japan

    Shuichi Katsumata

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  1. Shuichi Katsumata
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  2. Ryo Nishimaki
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  3. Shota Yamada
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  4. Takashi Yamakawa
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Correspondence to Shuichi Katsumata .

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  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Katsumata, S., Nishimaki, R., Yamada, S., Yamakawa, T. (2019). Designated Verifier/Prover and Preprocessing NIZKs from Diffie-Hellman Assumptions. 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_22

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