Repairable Block Failure Resilient Codes

In large scale distributed storage systems (DSS) deployed in cloud computing, correlated failures resulting in simultaneous failure (or, unavailability) of blocks of nodes are common. In such scenarios, the stored data or a content of a failed node can only be reconstructed from the available live nodes belonging to available blocks. To analyze the resilience of the system against such block failures, this work introduces the framework of Block Failure Resilient (BFR) codes, wherein the data (e.g., file in DSS) can be decoded by reading out from a same number of codeword symbols (nodes) from each available blocks of the underlying codeword. Further, repairable BFR codes are introduced, wherein any codeword symbol in a failed block can be repaired by contacting to remaining blocks in the system. Motivated from regenerating codes, file size bounds for repairable BFR codes are derived, trade-off between per node storage and repair bandwidth is analyzed, and BFR-MSR and BFR-MBR points are derived. Explicit codes achieving these two operating points for a wide set of parameters are constructed by utilizing combinatorial designs, wherein the codewords of the underlying outer codes are distributed to BFR codeword symbols according to projective planes

An MDS-PIR Capacity-Achieving Protocol for Distributed Storage Using Non-MDS Linear Codes

We propose a private information retrieval (PIR) protocol for distributed storage systems with noncolluding nodes where data is stored using an arbitrary linear code. An expression for the PIR rate, i.e., the ratio of the amount of retrieved data per unit of downloaded data, is derived, and a necessary and a sufficient condition for codes to achieve the maximum distance separable (MDS) PIR capacity are given. The necessary condition is based on the generalized Hamming weights of the storage code, while the sufficient condition is based on code automorphisms. We show that cyclic codes and Reed-Muller codes satisfy the sufficient condition and are thus MDS-PIR capacity-achieving.Comment: To be presented at 2018 IEEE International Symposium on Information Theory (ISIT). arXiv admin note: substantial text overlap with arXiv:1712.0389

Constructions of Batch Codes via Finite Geometry

A primitive $k$-batch code encodes a string $x$ of length $n$ into string $y$ of length $N$, such that each multiset of $k$ symbols from $x$ has $k$ mutually disjoint recovering sets from $y$. We develop new explicit and random coding constructions of linear primitive batch codes based on finite geometry. In some parameter regimes, our proposed codes have lower redundancy than previously known batch codes.Comment: 7 pages, 1 figure, 1 tabl

Derandomized Construction of Combinatorial Batch Codes

Combinatorial Batch Codes (CBCs), replication-based variant of Batch Codes introduced by Ishai et al. in STOC 2004, abstracts the following data distribution problem: $n$ data items are to be replicated among $m$ servers in such a way that any $k$ of the $n$ data items can be retrieved by reading at most one item from each server with the total amount of storage over $m$ servers restricted to $N$. Given parameters $m, c,$ and $k$, where $c$ and $k$ are constants, one of the challenging problems is to construct $c$-uniform CBCs (CBCs where each data item is replicated among exactly $c$ servers) which maximizes the value of $n$. In this work, we present explicit construction of $c$-uniform CBCs with $\Omega(m^{c-1+{1 \over k}})$ data items. The construction has the property that the servers are almost regular, i.e., number of data items stored in each server is in the range $[{nc \over m}-\sqrt{{n\over 2}\ln (4m)}, {nc \over m}+\sqrt{{n \over 2}\ln (4m)}]$. The construction is obtained through better analysis and derandomization of the randomized construction presented by Ishai et al. Analysis reveals almost regularity of the servers, an aspect that so far has not been addressed in the literature. The derandomization leads to explicit construction for a wide range of values of $c$ (for given $m$ and $k$) where no other explicit construction with similar parameters, i.e., with $n = \Omega(m^{c-1+{1 \over k}})$, is known. Finally, we discuss possibility of parallel derandomization of the construction

The capacity of symmetric Private information retrieval

Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of K messages from N non-communicating replicated databases (each holds all K messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other K - 1 messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is 1-1/N regardless of the number of messages K, if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least 1/(N - 1) bits per desired message bit, and zero otherwise

Combinatorial batch codes

In this paper, we study batch codes, which were introduced by Ishai, Kushilevitz, Ostrovsky and Sahai in [4]. A batch code specifies a method to distribute a database of [n] items among [m] devices (servers) in such a way that any [k] items can be retrieved by reading at most [t] items from each of the servers. It is of interest to devise batch codes that minimize the total storage, denoted by [N] , over all [m] servers. We restrict out attention to batch codes in which every server stores a subset of the items. This is purely a combinatorial problem, so we call this kind of batch code a ''combinatorial batch code''. We only study the special case [t=1] , where, for various parameter situations, we are able to present batch codes that are optimal with respect to the storage requirement, [N] . We also study uniform codes, where every item is stored in precisely [c] of the [m] servers (such a code is said to have rate [1/c] ). Interesting new results are presented in the cases [c = 2, k-2] and [k-1] . In addition, we obtain improved existence results for arbitrary fixed [c] using the probabilistic method