Private Information Retrieval from Locally Repairable Databases with Colluding Servers (1901.02938v3)

Abstract: 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.