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 $|C| \cdot \poly(\secpar)$, where $|C|$ is the size of the circuit that computes the $\NP$ 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:
\begin{itemize}
\item DV-NIZKs for $\NP$ 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). \item DP-NIZKs for $\NP$ with short proof size from a DH-type assumption over pairing groups. Here, the proof size has an additive-overhead $|C|+\poly(\secpar)$ rather then an multiplicative-overhead $|C| \cdot \poly(\secpar)$. 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.
\item PP-NIZK for $\NP$ 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 $|C|+\poly(\secpar)$. This too serves as a solution to the open problem posed by Kim and Wu (CRYPTO'18). \end{itemize}
Along the way, we construct two new homomorphic authentication (HomAuth) schemes which may be of independent interest.

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