SNARGs for Monotone Policy Batch NP

We construct a succinct non-interactive argument ($\mathsf{SNARG}$) for the class of monotone policy batch $\mathsf{NP}$ languages under the Learning with Errors ($\mathsf{LWE}$) assumption. This class is a subclass of $\mathsf{NP}$ that is associated with a monotone function $f:\{0,1\}^k\rightarrow\{0,1\}$ and an $\mathsf{NP}$ language $\mathcal L$, and contains instances $(x_1,\ldots,x_k)$ such that $f(b_1,\ldots,b_k)=1$ where $b_j=1$ if and only if $x_j\in \mathcal L$. Our $\mathsf{SNARG}$s are arguments of knowledge in the non-adaptive setting, and satisfy a new notion of somewhere extractability against adaptive adversaries.

This is the first $\mathsf{SNARG}$ under standard hardness assumptions for a sub-class of $\mathsf{NP}$ that is not known to have a (computational) non-signaling $\mathsf{PCP}$ with small locality. Indeed, our approach necessarily departs from the known framework of constructing $\mathsf{SNARG}$s dating back to [Kalai-Raz-Rothblum, STOC '13]

Our construction combines existing quasi-arguments for $\mathsf{NP}$ (based on batch arguments for $\mathsf{NP}$) with a novel ingredient which we call a predicate-extractable hash ($\mathsf{PEH}$) family. This notion generalizes the notion of a somewhere extractable hash. Whereas a somewhere extractable hash allows to extract a single input coordinate, our $\mathsf{PEH}$ extracts a global property of the input. We view this primitive to be of independent interest, and believe that it will find other applications.