Paper 2025/316

$\mathsf{Zinc}$: Succinct Arguments with Small Arithmetization Overheads from IOPs of Proximity to the Integers

Abstract

We introduce $\mathsf{Zinc}$, a hash-based succinct argument for integer arithmetic. $\mathsf{Zinc}$'s goal is to provide a practically efficient scheme that bypasses the arithmetization overheads that many succinct arguments present. These overheads can be of orders of magnitude in many applications. By enabling proving statements over the integers, we are able to arithmetize many operations of interest with almost no overhead. This includes modular operations involving any moduli, not necessarily prime, and possibly involving multiple moduli in the same statement. In particular, $\mathsf{Zinc}$ allows to prove statements for the ring $\mathbb{Z}/n\mathbb{Z}$ for arbitrary $n\geq 1$. Importantly, and departing from prior work, our schemes are purely code and hash-based, and do not require hidden order groups. In its final form, $\mathsf{Zinc}$ operates similarly to other hash-based schemes using Brakedown as their PCS, and at the same time it benefits from the arithmetization perks brought by working over $\mathbb{Z}$ (and $\mathbb{Q}$) natively. At its core, $\mathsf{Zinc}$ is a succinct argument for proving relations over the rational numbers $\mathbb{Q}$, even though when applied to integer statements, an honest prover and verifier will only operate with small integers. $\mathsf{Zinc}$ consists of two main components: 1) $\mathsf{Zinc}$-$\mathsf{PIOP}$, a framework for proving algebraic statements over the rationals by reducing modulo a randomly chosen prime $q$, followed by running a suitable PIOP over $\mathbb{F}_q$ (this is similar to the approach taken in prior works, with the difference that we use localizations of $\mathbb{Q}$ to enable prime modular projection); and 2) $\mathsf{Zip}$, a Brakedown-type polynomial commitment scheme built from an IOP of proximity to the integers, a novel primitive that we introduce. The latter primitive guarantees that a prover is using a polynomial with coefficients close to being integral. With these two primitives in place, one can use a lookup argument over the rationals to ensure that the witness contains only integer elements.

Note: Fixed typos and added orcid information.