Abstract. We present a general framework for private information retrieval (PIR) from arbitrary coded databases that allows one to adjust the rate of the scheme to the suspected number of colluding servers. If the storage code is a generalized Reed-Solomon code of length n and dimension k, we design PIR schemes that achieve a PIR rate of n−(k+t−1) n while protecting against any t colluding servers, for any 1 ≤ t ≤ n − k. This interpolates between the previously studied cases of t = 1 and k = 1 and achieves PIR capacity in both of these cases asymptotically as the number of files in the database grows.Key words. private information retrieval, distributed storage systems, generalized Reed-Solomon codes AMS subject classifications. 68P20, 68P30, 94B27, 14G50 DOI. 10.1137/16M11025621. Introduction. Private information retrieval (PIR) addresses the question of how to retrieve data items from a database without disclosing information about the identity of the data items retrieved, and was introduced by Chor et al. in [4,5]. The classic PIR model of [5] views the database as an m-bit binary string x = [x 1 · · · x m ] ∈ {0, 1} m and assumes that the user wants to retrieve a single bit x i without revealing any information about the index i. We consider a natural extension of this model, wherein the database is a string x = [x 1 · · · x m ] of files x i , which are themselves bit strings, and the user wants to download one of the files x i without revealing its index.The rate of a PIR scheme in this model is measured as the ratio of the gained information over the downloaded information, while upload costs of the requests are usually ignored. The trivial solution is to download the entire database. This, however, incurs a significant communication overhead whenever the database is large and is therefore not useful in practice. While the trivial solution is the only way to guarantee information-theoretic privacy in the case of a single server [5], this problem can be remedied by replicating the database onto n servers that do not communicate.The study of PIR recently received renewed attention when Shah et al. introduced a model of coded private information retrieval (cPIR) [10,11]. Here, all files are distributed over the
The problem of Private Information Retrieval (PIR) from coded storage systems with colluding, byzantine, and unresponsive servers is considered. An explicit scheme using an [n, k] Reed-Solomon storage code is designed, protecting against t-collusion and handling up to b byzantine and r unresponsive servers, when n > k + t + 2b + r − 1. This scheme achieves a PIR rate of n−r−(k+2b+t−1) n−r . In the case where the capacity is known, namely when k = 1, it is asymptotically capacity-achieving as the number of files grows. Lastly, the scheme is adapted to symmetric PIR.
In this paper, locally repairable codes with all-symbol locality are studied.
Methods to modify already existing codes are presented. Also, it is shown that
with high probability, a random matrix with a few extra columns guaranteeing
the locality property, is a generator matrix for a locally repairable code with
a good minimum distance. The proof of this also gives a constructive method to
find locally repairable codes. Constructions are given of three infinite
classes of optimal vector-linear locally repairable codes over an alphabet of
small size, not depending on the size of the code.Comment: 32 pages. Second code construction in Section V is corrected in this
version. Also, some typos are corrected. The results remain the same.
Submitted to IEEE Transactions on Information Theory. This is extended,
generalized, and clarified version of arXiv:1408.018
In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size t colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups.
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