Efficiently Batching Unambiguous Interactive Proofs

• URL: http://arxiv.org/abs/2510.19075v1
• Date: Tue, 21 Oct 2025 21:04:10 GMT
• Title: Efficiently Batching Unambiguous Interactive Proofs
• Authors: Bonnie Berger, Rohan Goyal, Matthew M. Hong, Yael Tauman Kalai,
• Abstract summary: We show that if a language $L$ admits a publiccoin interactive proof (UIP) with round $ell$, where $a$ bits are communicated per round, then the batch language $Lotimes k$, i.e. the set of $k$-tuples of statements all belonging to $Lotimes k$, has unambiguous interactive proof with complexity $ellcdotmathsfpolylog(k)$, per-round communication of $acdot ellcdotmathsfpolylog(k
• Score: 8.993111413196559
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $(\mathsf{poly}(a)\cdot k^{\epsilon} + k) \cdot \ell^{1/\epsilon}$ for any arbitrarily small constant $\epsilon>0$ (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 $n^{({\log n})^\delta}$ for a small $\delta>0$ significantly smaller than $1/2$ (Reingold-Rothblum-Rothblum, STOC 2016).

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