We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups $ g $, locality $ r $, local distance $ \delta $ and dimension $ k $. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed--Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only $ r $ nodes (out of $ r+\delta-1 $) are downloaded per server. The PIR scheme achieves the (download) rate $ R = (N - k - rt + 1)/N $, where $ N = gr $ is the length of the MDS code obtained after removing the local parities, and for any $ t $ colluding servers such that $ k + rt \leq N $. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed--Solomon codes, but Reed--Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

翻译：我们考虑在共谋服务器场景下，从编码数据库中实现信息论安全的私有信息检索（PIR）。首次以本地可修复存储码（LRC）为研究对象，考虑任意数量的本地组$g$、局部性$r$、本地距离$\delta$及维度$k$。主要贡献是提出了一种针对基于线性化Reed-Solomon码的最大可恢复（MR）LRC的PIR方案，该方案在众多参数范围内实现了MR-LRC的最小域尺寸。方案中，节点对应码字符号，服务器对应节点本地组。每个服务器仅下载本地非冗余信息，即每台服务器仅下载$r$个节点（共$r+\delta-1$个节点）。该PIR方案实现的（下载）速率为$R = (N - k - rt + 1)/N$，其中$N = gr$为删除本地校验后的MDS码长度，且适用于满足$k + rt \leq N$的任意$t$个共谋服务器。当存储文件数量无界时，所得速率严格优于已知适用于任意MDS码的PIR方案。此外，该PIR方案可适配用户与服务器通过线性网络编码通信的场景，在多域尺寸从指数级降至多项式级的前提下，达到与先前方案相同的速率。本方案速率与面向Reed-Solomon码的PIR方案相当，但Reed-Solomon码不兼容MR-LRC特性或线性网络编码，因此本PIR方案在实际应用中限制更少。