AI 中文总结
基于数论猜想，提出一种稀疏S-解码多项式构造，实现指数级更少服务器的PIR协议，显著降低通信复杂度。
AI 中文摘要
我们证明，在一个合理的数论猜想下，对于任何常数s，存在一个s-服务器私有信息检索（PIR）协议，该协议在n位数据库上需要通信量为exp(O((log n)^{1/s} (log log n)^{1-1/s})))。先前的构造实现相同通信量需要2^{O(s)}个服务器。我们的数论猜想由现有的猜想所暗示，即广义重位猜想和Schinzel的假设H（其中任何一个单独的猜想就足够）。我们的结果建立在“匹配向量族+S-解码多项式”框架上，该框架由Efremenko（STOC 2009）率先提出，并最近由Ghasemi、Kopparty和Sudan（STOC 2025）进一步完善。主要成分是一个框架，用于构造具有仅k+1个非零系数的S-解码多项式，模特殊乘积的k个素数，解决了Ghasemi和Kopparty（ITCS 2026）提出的问题。通过Ghasemi和Kopparty展示的下界，这是可实现的最小稀疏度。我们还通过经验验证我们的构造，并使我们的结果对所有s ≤ 15成为无条件的。我们还应用我们的技术到s随n增长的领域，证明在我们数论猜想的更强变体下，s-服务器匹配向量PIR的通信复杂度可以从之前的最先进水平超多项式地减少，对于任何s ≤ exp(o(sqrt(log log n/log log log n)))。主要结果对于s=O(1)及其证明是在作者 prompted by GPT-5.5 Pro对话中发现的。
英文摘要
We show that under a plausible number-theoretic conjecture, for any constant $s$ there exists an $s$-server private information retrieval (PIR) protocol that on an $n$-bit database requires communication $\exp(O((\log n)^{1/s} (\log \log n)^{1-1/s}))$. Previous constructions attaining the same communication required $2^{O(s)}$ servers. Our number-theoretic conjecture is implied by existing conjectures, namely the generalized repunit conjecture and Schinzel's hypothesis H (either one of these conjectures would suffice alone). Our result builds on the ``matching vector family + $S$-decoding polynomials'' framework pioneered by Efremenko (STOC 2009) and recently refined by Ghasemi, Kopparty, and Sudan (STOC 2025). The main ingredient is a framework for constructing $S$-decoding polynomials with only $k+1$ nonzero coefficients modulo special products of $k$ primes, resolving an open problem posed by Ghasemi and Kopparty (ITCS 2026). By the lower bound shown by Ghasemi and Kopparty, this is the minimum achievable sparsity. We also empirically validate our construction and make our result unconditional for all $s \leq 15$. We also apply our techniques to regimes where $s$ grows with $n$, showing under a stronger variant of our number-theoretic conjecture that the communication complexity of $s$-server matching-vector PIR can be superpolynomially reduced from the previous state of the art for any $s \leq \exp(o(\sqrt{\log \log n/\log \log \log n}))$. The main result for $s = O(1)$ and its proof were discovered in a GPT-5.5 Pro conversation prompted by the authors.
