Briefing
This research addresses the computational overhead inherent in adapting inner-product arguments (IPAs) for integer-based statements, a common challenge when proof systems primarily operate over finite fields. It introduces inner-product commitment schemes operating natively over integers, alongside general techniques to extend IPAs to this domain. This foundational breakthrough significantly improves the slackness parameter of witness extraction and enables the construction of highly efficient, batchable, logarithmic-size zero-knowledge proofs for integer arithmetic circuits and range proofs, enhancing privacy and scalability for real-world blockchain applications.
Context
Prior to this research, established succinct proof systems and inner-product arguments predominantly functioned over finite fields. This necessitated computationally intensive conversions for statements involving integer arithmetic, which introduced inefficiencies and limited the direct applicability of these powerful cryptographic tools to real-world scenarios where integer precision is paramount. The prevailing theoretical limitation centered on the lack of native, efficient mechanisms for constructing zero-knowledge proofs directly over integers, constraining performance and scalability.
Analysis
The paper’s core mechanism centers on developing inner-product commitment schemes that operate natively over integers. This involves adapting and extending folding techniques from Pedersen commitments, originally designed for finite fields, to directly support integer vectors. The new primitive allows a prover to commit to two integer vectors and then cryptographically demonstrate their inner product. This is achieved with a constant-size commitment and a verification proof that grows only logarithmically with the vector length.
This approach fundamentally differs from previous methods by eliminating the need for costly field conversions, directly addressing integer arithmetic. The research also presents a novel batchable argument for the non-negativity of exponents and an efficient construction for batch-verifiable range proofs over significantly larger intervals, enabling succinct zero-knowledge arguments for general integer arithmetic circuit satisfiability.
Parameters
- Core Concept ∞ Inner-Product Arguments over Integers
- New Primitive ∞ Inner-Product Commitment Schemes
- Proof Size ∞ Logarithmic
- Commitment Size ∞ Constant
- Application ∞ Arithmetic Circuit Satisfiability
- Application ∞ Range Proofs
- Key Technique ∞ Folding Techniques (extended Pedersen commitments)
- Key Authors ∞ Shihui Fu et al.
Outlook
This research paves the way for significantly more efficient and direct construction of succinct proof systems tailored for applications inherently reliant on integer arithmetic. Potential real-world applications within the next 3-5 years include enhanced privacy-preserving financial transactions, verifiable supply chain logistics, and scalable blockchain solutions demanding precise integer operations. This theoretical advancement reduces computational overhead and improves proof efficiency, thereby unlocking new capabilities for privacy and scalability across decentralized systems. It also opens new avenues for optimizing underlying integer arithmetic operations within cryptographic protocols.