Private Information Retrieval from Locally Repairable Databases with Colluding Servers
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In this work, information-theoretical private information retrieval (PIR) is considered from a coded database with colluding servers. The considered 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 (and in N if r is fixed). When N - k - rt = 0 , the rate R = 1/N is optimal (asymptotically in the number of files) and coincides with that of previous PIR schemes that work for an arbitrary MDS storage code. When 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 LRC or linearized Reed-Solomon storage codes) and is always strictly higher than that of known PIR schemes that work for arbitrary MDS storage codes. Finally, the obtained PIR scheme can also be adapted to the case where communication between the user and each server is through a linearly-coded network, achieving the same rate as previous known constructions but with polynomial field sizes in the number of servers.
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