Paper 2025/1536

Inner-Product Commitments Over Integers With Applications to Succinct Arguments

Abstract

Proving statements over integers is crucial in modern cryptographic protocols because certain computations, such as range proofs and Diophantine satisfiability, are more efficiently expressed over integers. Currently, the prevailing approach to achieve this is to convert the integer relations into statements tractable for proof systems over a finite field $\mathbb{Z}_p$. However, finding these corresponding tractable statements over $\mathbb{Z}_p$ is not always straightforward, and in practical schemes, the conversion often introduces computational overheads. Therefore, there is a growing interest in proving the statements directly over integers. Due to the significant applicability of inner-product arguments (IPA) in constructing succinct proof systems, in this work, we extend them to work natively in the integer setting. We introduce and construct inner-product commitment schemes over integers that allow a prover to open two committed integer vectors to a claimed inner product. The commitment size is constant and the verification proof size is logarithmic in the vector length. The construction significantly improves the slackness parameter of witness extraction, surpassing the existing state-of-the-art approach. Our construction is based on the folding techniques for Pedersen commitments defined originally over $\mathbb{Z}_p$. We develop general-purpose techniques to make it work properly over $\mathbb{Z}$, which may be of independent interest. Building upon our IPAs, we first present a novel batchable argument of knowledge of nonnegativity of exponents that can be used to further reduce the proof size of Dew-PCS (Arun et al., PKC 2023). Second, we present a construction for range proofs that allows for extremely efficient batch verification of a large number of range proofs over much larger intervals. We also provide a succinct zero-knowledge argument of knowledge with a logarithmic-size proof for more general arithmetic circuit satisfiability over integers.