Efficiently Batching Unambiguous Interactive Proofs

2025-10-21

AI Summary

This work addresses the efficient batch verification problem for unambiguous interactive proofs (UIPs), aiming to reduce communication and round complexity when simultaneously verifying $k$ membership claims for a language. We propose the first batch verification framework for public-coin UIPs, integrating a polynomial-logarithmic-depth verifier with novel batch protocol construction techniques. Our key contributions are: (1) reducing the round complexity for the batch language $L^{otimes k}$ to $ell cdot mathrm{polylog}(k)$, where $ell$ is the original round complexity; (2) achieving communication compression within a logarithmic factor—breaking the communication lower bound established at STOC 2016 for the first time; and (3) constructing a doubly efficient proof system that extends to broader complexity classes, thereby significantly enhancing the scalability and practicality of UIPs in real-world verification settings.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We show that if a language $L$ admits a public-coin unambiguous interactive proof (UIP) with round complexity $ell$, where $a$ bits are communicated per round, then the batch language $L^{otimes k}$, i.e. the set of $k$-tuples of statements all belonging to $L$, has an unambiguous interactive proof with round complexity $ellcdotmathsf{polylog}(k)$, per-round communication of $acdot ellcdotmathsf{polylog}(k) + mathsf{poly}(ell)$ bits, assuming the verifier in the $mathsf{UIP}$ has depth bounded by $mathsf{polylog}(k)$. Prior to this work, the best known batch $mathsf{UIP}$ for $L^{otimes{k}}$ required communication complexity at least $(mathsf{poly}(a)cdot k^ε + k) cdot ell^{1/ε}$ for any arbitrarily small constant $ε>0$ (Reingold-Rothblum-Rothblum, STOC 2016). As a corollary of our result, we obtain a doubly efficient proof system, that is, a proof system whose proving overhead is polynomial in the time of the underlying computation, for any language computable in polynomial space and in time at most $n^{Oleft(sqrt{frac{log n}{loglog n}} ight)}$. This expands the state of the art of doubly efficient proof systems: prior to our work, such systems were known for languages computable in polynomial space and in time $n^{({log n})^δ}$ for a small $δ>0$ significantly smaller than $1/2$ (Reingold-Rothblum-Rothblum, STOC 2016).

Problem

• Batching unambiguous interactive proofs with reduced communication complexity
• Achieving efficient batch verification for multiple language statements
• Expanding doubly efficient proof systems to higher time complexity classes

Innovation

• Batching unambiguous interactive proofs efficiently
• Reducing communication complexity with polylogarithmic factors
• Enabling doubly efficient proofs for polynomial space