In this paper, we construct new t-server Private Information Retrieval (PIR) schemes with communication complexity subpolynomial in the previously best known, for all but finitely many t. Our results are based on combining derivatives (in the spirit of Woodruff-Yekhanin) with the Matching Vector based PIRs of Yekhanin and Efremenko. Previously such a combination was achieved in an ingenious way by Dvir and Gopi, using polynomials and derivatives over certain exotic rings, en route to their fundamental result giving the first 2-server PIR with subpolynomial communication. Our improved PIRs are based on two ingredients: - We develop a new and direct approach to combine derivatives with Matching Vector based PIRs. This approach is much simpler than that of Dvir-Gopi: it works over the same field as the original PIRs, and only uses elementary properties of polynomials and derivatives. - A key subproblem that arises in the above approach is a higher-order polynomial interpolation problem. We show how "sparse S-decoding polynomials", a powerful tool from the original constructions of Matching Vector PIRs, can be used to solve this higher-order polynomial interpolation problem using surprisingly few higer-order evaluations. Using the known sparse S-decoding polynomials, in combination with our ideas leads to our improved PIRs. Notably, we get a 3-server PIR scheme with communication $2^{O^{\sim}( (\log n)^{1/3}) }$, improving upon the previously best known communication of $2^{O^{\sim}( \sqrt{\log n})}$ due to Efremenko.

翻译：本文针对除有限个情况外的所有t值，构建了通信复杂度低于先前最佳已知方案亚多项式级别的t服务器私有信息检索新方案。我们的研究结果通过将导数思想与Yekhanin及Efremenko提出的基于匹配向量的PIR方案相结合而实现。此前Dvir与Gopi曾通过多项式及特殊环上的导数，以巧妙方式实现了此类组合，并由此取得了首个具有亚多项式通信复杂度的2服务器PIR基础性成果。我们改进的PIR方案基于两个核心要素：首先，我们提出了一种将导数与基于匹配向量的PIR方案直接结合的新方法。该方法较Dvir-Gopi方案更为简洁：其在原始PIR方案的同域上操作，仅利用多项式与导数的基本性质。其次，该方法衍生的关键子问题是高阶多项式插值问题。我们证明了如何运用"稀疏S解码多项式"——这一源自匹配向量PIR原始构造的强大工具——通过极少的高阶求值即可解决该插值问题。将已知的稀疏S解码多项式与我们的创新思路结合，最终形成了改进的PIR方案。特别值得注意的是，我们提出的3服务器PIR方案通信复杂度为$2^{O^{\sim}( (\log n)^{1/3}) }$，较Efremenko先前最佳已知的$2^{O^{\sim}( \sqrt{\log n})}$通信复杂度实现了显著提升。