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2019 | OriginalPaper | Chapter

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

Authors : Shuichi Katsumata, Ryo Nishimaki, Shota Yamada, Takashi Yamakawa

Published in: Advances in Cryptology – EUROCRYPT 2019

Publisher: Springer International Publishing

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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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previous chapter Reusable Designated-Verifier NIZKs for all NP from CDH

next chapter Building an Efficient Lattice Gadget Toolkit: Subgaussian Sampling and More

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.

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.

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 \(p^{-1}\cdot \mathsf {negl}(\kappa )\).

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

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.

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

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Title: Designated Verifier/Prover and Preprocessing NIZKs from Diffie-Hellman Assumptions
Authors: Shuichi Katsumata
Ryo Nishimaki
Shota Yamada
Takashi Yamakawa
Publisher: Springer International Publishing
Book: Advances in Cryptology – EUROCRYPT 2019
Print ISBN: 978-3-030-17655-6

Electronic ISBN: 978-3-030-17656-3

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-17656-3_22

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