International Association
for Cryptologic Research

A monotone policy batch $\mathsf{NP}$ language $\mathcal{L}_{\mathcal{R}, P}$ is parameterized by a monotone policy $P \colon \{0,1\}^k \to \{0,1\}$ and an $\mathsf{NP}$ relation $\mathcal{R}$. A statement $(x_1, \ldots, x_k)$ is a YES instance if there exists $w_1, \ldots, w_k$ where $P(\mathcal{R}(x_1, w_1), \ldots, \mathcal{R}(x_k, w_k)) = 1$. For example, we might say that an instance $(x_1, \ldots, x_k)$ is a YES instance if a majority of the statements are true. A monotone policy batch argument (BARG) for $\mathsf{NP}$ allows a prover to prove that $(x_1, \ldots, x_k) \in \mathcal{L}_{\mathcal{R}, P}$ with a proof of size $\mathsf{poly}(\lambda, |\mathcal{R}|, \log k)$, where $\lambda$ is the security parameter, $|\mathcal{R}|$ is the size of the Boolean circuit that computes $\mathcal{R}$, and $k$ is the number of instances. Recently, Brakerski, Brodsky, Kalai, Lombardi, and Paneth (CRYPTO 2023) gave the first monotone policy BARG for $\mathsf{NP}$ from the learning with errors (LWE) assumption.

In this work, we describe a generic approach for constructing monotone policy BARGs from any BARG for $\mathsf{NP}$ together with an additively homomorphic encryption scheme. This yields the first constructions of monotone policy BARGs from the $k$-Lin assumption in prime-order pairing groups as well as the (subexponential) DDH assumption in pairing-free groups. Central to our construction is a notion of a zero-fixing hash function, which is a relaxed version of a predicate-extractable hash function from the work of Brakerski et al. Our relaxation enables a direct realization of zero-fixing hash functions from standard BARGs for $\mathsf{NP}$ and additively homomorphic encryption, whereas the previous notion relied on leveled homomorphic encryption, and by extension, the LWE assumption.