Private Information Retrieval from Locally Repairable Databases with Colluding Servers

Information-theoretical private information retrieval (PIR) is considered from a coded database with colluding servers. The storage code is a locally repairable code (LRC) with maximal recoverability (MR), and in particular, with optimal global minimum distance, for arbitrary code parameters: Number of local groups g, locality r, local distance δ, dimension k ≤ gr and length n = g(r + δ - 1). Servers are identified bijectively with local groups, and only locally non-redundant information is considered and downloaded from each server, that is, only r nodes (out of r + δ - 1) are considered per server. When the remaining MDS code, after removing all locally redundant nodes, is a linearized Reed-Solomon code, a PIR scheme is provided achieving the (download) rate R = (N - k - rt + 1)/N, where N = gr = n - g(δ - 1) is the length of the restricted MDS code, for any t colluding servers such that k + rt ≤ N. The field size is roughly g^r, polynomial in the number of servers g. Assume an arbitrarily large number of stored files. If N - k - rt = 0, the rate R = 1/N is the highest known and coincides with that of previous PIR schemes that work for any MDS storage code. If N - k - rt > 0, the achieved rate R > 1/N coincides with the best known rate of PIR schemes for MDS storage codes (but which do not work for LRCs or linearized Reed-Solomon storage codes) and is always strictly higher than that of known PIR schemes that work for arbitrary MDS storage codes.