Efficiently Batching Unambiguous Interactive Proofs
Bonnie Berger, Rohan Goyal, Matthew M. Hong, Yael Tauman Kalai
TL;DR
The paper presents a framework for batch unambiguous interactive proofs (UIP) that drastically improves the efficiency of proving multiple statements from a single UIP. It introduces execution-wise distance and leverages the GKR protocol, distance-generation via checksums, and RR-style instance reduction to batch verify $L^{\otimes k}$ with round complexity $\ell \cdot \mathsf{polylog}(k)$ and per-round communication near $a \ell \mathsf{polylog}(k)$. The approach yields a doubly efficient proof system for languages decidable in polynomial space and subexponential time, significantly extending prior results that covered shallower time bounds. The construction integrates Distance Generation and Instance Reduction into Batch-UIP, with explicit UIPs for the intermediate language $\mathcal{L}'_{\bm x}$ and tight complexity analyses for both prover and verifier. This work advances practical, scalable proof systems for space-bounded computation and strengthens the connections between UIP, RR, and GKR techniques in a coherent batching framework.
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 $\ell\cdot\mathsf{polylog}(k)$, per-round communication of $a\cdot \ell\cdot\mathsf{polylog}(k) + \mathsf{poly}(\ell)$ 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 $n^{O\left(\sqrt{\frac{\log n}{\log\log n}}\right)}$. 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).