## CryptoDB

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

Authors: Shuichi Katsumata Ryo Nishimaki Shota Yamada Takashi Yamakawa DOI: 10.1007/978-3-030-17656-3_22 (login may be required) Search ePrint Search Google 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 $|C| \cdot \mathsf {poly}(\kappa )$|C|·poly(κ), where |C| is the size of the circuit that computes the $\mathbf {NP}$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 $\mathbf {NP}$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 $\mathbf {NP}$NP with short proof size from a DH-type assumption over pairing groups. Here, the proof size has an additive-overhead $|C|+\mathsf {poly}(\kappa )$|C|+poly(κ) rather then an multiplicative-overhead $|C| \cdot \mathsf {poly}(\kappa )$|C|·poly(κ). 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 $\mathbf {NP}$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 $|C|+\mathsf {poly}(\kappa )$|C|+poly(κ). 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.
##### BibTeX
@article{eurocrypt-2019-29374,
title={Designated Verifier/Prover and Preprocessing NIZKs from Diffie-Hellman Assumptions},
booktitle={Advances in Cryptology – EUROCRYPT 2019},
series={Advances in Cryptology – EUROCRYPT 2019},
publisher={Springer},
volume={11477},
pages={622-651},
doi={10.1007/978-3-030-17656-3_22},
author={Shuichi Katsumata and Ryo Nishimaki and Shota Yamada and Takashi Yamakawa},
year=2019
}