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.

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Multiple-input double-output (MIDO) codes are important in the near-future wireless communications, where the portable end-user device is physically small and will typically contain at most two receive antennas. Especially tempting is the 4 × 2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of (4 × 4) matrices. Unfortunately, such codes are in general very complex to decode, hence setting forth a call for constructions with reduced complexity.Recently, some reduced complexity constructions have been proposed, but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes. In this paper, it will be shown that a family of division algebra based MIDO codes will always result in at least 37.5% worst-case complexity reduction, while maintaining full diversity and, for the first time, the non-vanishing determinant (NVD) property. The reduction follows from the fact that, similarly to the Alamouti code, the codes will be subsets of matrix rings of the Hamiltonian quaternions, hence allowing simplified decoding. At the moment, such reductions are among the best known for rate-two MIDO codes [4], [5]. Several explicit constructions are presented and shown to have excellent performance through computer simulations. Index Terms-Coding gain, cyclic division algebra, digital video broadcasting next generation handheld (DVB-NGH), fast maximum-likelihood (ML) sphere decoding, Hamiltonian quaternions, Hasse invariants, lattices, lowcomplexity space-time block codes (STBCs), multiple-input single/double/multiple-output (MISO/MIDO/MIMO), nonvanishing determinant (NVD), orders.