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A functional $k$ -batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^{s}-1$ servers. This conjecture is verified for $s \leqslant 5$ but previous work could only show that codes with $n=2^{s}-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left \lfloor{ \frac { 2^{s/2}}{\sqrt {24}} }\right \rfloor $ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac {5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes. [ABSTRACT FROM AUTHOR]

There has been much recent interest in private information retrieval (PIR) in models where a database is stored across several servers using coding techniques from distributed storage, rather than being a simply replicated. In particular, a recent breakthrough result of Fazelli, Vardy, and Yaakobi introduces the notion of a PIR code and a PIR array code, and uses this notion to produce efficient PIR protocols. In this paper, we are interested in designing PIR array codes. We consider the case when we have $m$ servers, with each server storing a fraction $(1/s)$ of the bits of the database; here $s$ is a fixed rational number with $s > 1$. A PIR array code with the $k$ -PIR property enables a $k$ -server PIR protocol (with $k\leq m$) to be emulated on $m$ servers, with the overall storage requirements of the protocol being reduced. The communication complexity of a PIR protocol reduces as $k$ grows, so the virtual server rate, defined to be $k/m$ , is an important parameter. We study the maximum virtual server rate of a PIR array code with the $k$ -PIR property. We present upper bounds on the achievable virtual server rate, some constructions, and ideas how to obtain the PIR array codes with the highest possible virtual server rate. In particular, we present constructions that asymptotically meet our upper bounds and the exact largest virtual server rate is obtained when $1 < s \leq 2$. A $k$ -PIR code (and similarly a $k$ -PIR array code) is also a locally repairable code with symbol availability $k-1$. Such a code ensures $k$ parallel reads for each information symbol. So the virtual server rate is very closely related to the symbol availability of the code when used as a locally repairable code. The results of this paper are discussed also in this context where subspace codes also have an important role. [ABSTRACT FROM AUTHOR]

In this paper, we study two families of codes with availability, namely, private information retrieval (PIR) codes and batch codes. While the former requires that every information symbol has $k$ mutually disjoint recovering sets, the latter imposes this property for each multiset request of $k$ information symbols. The main problem under this paradigm is to minimize the number of redundancy symbols. We denote this value by $r_{P}(n,k)$ and $r_{B}(n,k)$ , for PIR codes and batch codes, respectively, where $n$ is the number of information symbols. Previous results showed that for any constant $k$ , $r_{P}(n,k) = \Theta (\sqrt {n})$ and $r_{B}(n,k)= {\mathcal{ O}}(\sqrt {n}\log (n))$. In this paper, we study the asymptotic behavior of these codes for non-constant $k$ and specifically for $k=\Theta (n^\epsilon)$. We also study the largest value of $k$ such that the rate of the codes approaches 1 and show that for all $\epsilon < 1$ , $r_{P}(n,n^\epsilon) = o(n)$ and $r_{B}(n,n^\epsilon) = o(n)$. Furthermore, several more results are proved for the case of fixed $k$. [ABSTRACT FROM AUTHOR]

In the classical model for (information theoretically secure) Private Information Retrieval (PIR) due to Chor, Goldreich, Kushilevitz and Sudan, a user wishes to retrieve one bit of a database that is stored on a set of ${n}$ servers, in such a way that no individual server gains information about which bit the user is interested in. The aim is to design schemes that minimise the total communication between the user and the servers. More recently, there have been moves to consider more realistic models where the total storage of the set of servers, or the per server storage, should be minimised (possibly using techniques from distributed storage), and where the database is divided into ${R}$ -bit records with ${R}>1$ , and the user wishes to retrieve one record rather than one bit. When ${R}$ is large, downloads from the servers to the user dominate the communication complexity and so the aim is to minimise the total number of downloaded bits. Work of Shah, Rashmi and Ramchandran shows that at least ${R}+1$ bits must be downloaded from servers in the worst case, and provides PIR schemes meeting this bound. Sun and Jafar have considered the download cost of a scheme, defined as the ratio of the message length ${R}$ and the total number of bits downloaded. They determine the best asymptotic download cost of a PIR scheme (as ${R}\rightarrow \infty $) when a database of ${k}$ messages is stored by ${n}$ servers. This paper provides various bounds on the download complexity of a PIR scheme, generalising those of Shah et al. to the case when the number ${n}$ of servers is bounded, and providing links with classical techniques due to Chor et al. The paper also provides a range of constructions for PIR schemes that are either simpler or perform better than previously known schemes. These constructions include explicit schemes that achieve the best asymptotic download complexity of Sun and Jafar with significantly lower upload complexity, and general techniques for constructing a scheme with good worst case download complexity from a scheme with good download complexity on average. [ABSTRACT FROM AUTHOR]

Consider a continuous signal that cannot be observed directly. Instead, one has access to multiple corrupted versions of the signal. The available corrupted signals are correlated because they carry information about the common remote signal. The goal is to reconstruct the original signal from the data collected from its corrupted versions. Known as the indirect or remote reconstruction problem, it has been mainly studied in the literature from an information theoretic perspective. A variant of this problem for a class of Gaussian signals, known as the “Gaussian CEO problem,” has received particular attention; for example, it has been shown that the problem of recovering the remote signal is equivalent with the problem of recovering the set of corrupted signals (separation principle). The information theoretic formulation of the remote reconstruction problem assumes that the corrupted signals are uniformly sampled and the focus is on optimal compression of the samples. On the other hand, in this paper, we revisit this problem from a sampling perspective. More specifically, assuming restrictions on the sampling rate from each corrupted signal, we look at the problem of finding the best sampling locations for each signal to minimize the total reconstruction distortion of the remote signal. In finding the sampling locations, one can take advantage of the correlation among the corrupted signals. The statistical model of the original signal and its corrupted versions adopted in this paper are similar to the one considered for the Gaussian CEO problem; i.e., we restrict to a class of Gaussian signals. Our main contribution is a fundamental lower bound on the reconstruction distortion for any arbitrary nonuniform sampling strategy. This lower bound is valid for any sampling rate. Furthermore, it is tight and matches the optimal reconstruction distortion in low and high sampling rates. Moreover, it is shown that in the low sampling rate region, it is optimal to use a certain nonuniform sampling scheme on all the signals. On the other hand, in the high sampling rate region, it is optimal to uniformly sample all the signals. We also consider the problem of finding the optimal sampling locations to recover the set of corrupted signals, rather than the remote signal. Unlike the information theoretic formulation of the problem in which these two problems were equivalent, we show that they are not equivalent in our setting. Finally, another contribution of this paper is a new reverse majorization inequality that might be of independent interest. [ABSTRACT FROM PUBLISHER]

This paper concerns the problem of recovering an unknown but structured signal ${x}\in \mathbb {R} ^{n}$ from $m$ -quadratic measurements of the form ${y}_{r}= \left |{\langle {a}_{r}, {x}\rangle }\right |^{{2}}$ for ${r}={1},{2},\ldots, {m}$. We focus on the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal (${m} < < {n}$). We formulate the recovery problem as a nonconvex optimization problem where prior structural information about the signal is enforced through constrains on the optimization variables. We prove the projected gradient descent, when initialized in a neighborhood of the desired signal, converges to the unknown signal at a linear rate. These results hold for any closed constraint set (convex or non-convex) providing convergence guarantees to the global optimum even when the objective function and constraint set are nonconvex. Furthermore, these results hold with a number of measurements that are only a constant factor away from the minimal number of measurements required to uniquely identify the unknown signal. Our results provide the first provably tractable algorithm for this data-poor regime, breaking local sample complexity barriers that have emerged in this paper. In this paper, we utilize and further develop powerful tools for uniform convergence of empirical processes that may have broader implications for rigorous understanding of constrained nonconvex optimization heuristics. [ABSTRACT FROM AUTHOR]

Information retrieval in associative memories was studied in a recent paper by Yaakobi and Bruck (2012). Associations between memory entries give us the t -neighbourhood of an entry. In their model, an information unit is retrieved from the memory with the aid of input clues, which are chosen from a reference set. In this paper, we consider the situation where the information unit is found unambiguously using the associated t -neighbourhoods of the input clues. A varying number of input clues are allowed, but a limit mu on the maximum number of them is imposed. Of course, we would like mu to be as small as possible. We consider the problem over the binary Hamming space \mathbb F^n and focus on the minimum of mu , denoted by $\nu (n;t)$ . Using linear reference sets, we show that $\nu (n;2) \leq 5$ for any $n \geq 9$ . We also give infinite families of reference sets, which provide good bounds on $\nu (n;t)$ for $t=3$ . In addition, efficient methods are given to obtain bounds on $\nu (n;t)$ for any $t$ from known reference sets. We also discuss the applications of this model to the Levenshtein’s sequence reconstruction problem and the sensor network monitoring. [ABSTRACT FROM AUTHOR]

This paper presents and analyzes a novel concatenated coding scheme for enabling error resilience in two distributed storage settings: one being storage using existing regenerating codes and the second being storage using locally repairable codes. The concatenated coding scheme brings together a maximum rank distance code as an outer code and either a globally regenerating or a locally repairable code as an inner code. In addition, error resilience for combination of locally repairable codes with regenerating codes is considered. This concatenated coding system is designed to handle two different types of adversarial errors: the first type includes an adversary that can replace the content of an affected node only once; while the second type studies an adversary that is capable of polluting data an unbounded number of times. The paper establishes an upper bound on the resilience capacity for a locally repairable code. This paper also proves that the proposed concatenated coding approach attains the upper bound on the resilience capacity in the presence of the first type of adversary for both minimum storage regenerating codes and locally repairable codes. Further, this paper presents mechanisms that combine the presented concatenated coding scheme with subspace signatures to achieve error resilience for the second type of errors. [ABSTRACT FROM PUBLISHER]

A private proximity retrieval (PPR) scheme is a protocol which allows a user to retrieve the identities of all records in a database that are within some distance $r$ from the user’s record $x$. The user’s privacy at each server is given by the fraction of the record $x$ that is kept private. In this paper, this research is initiated and protocols that offer trade-offs between privacy, computational complexity, and storage are studied. In particular, we assume that each server stores a copy of the database and study the required minimum number of servers by our protocol which provides a given privacy level. Each server receives a query in the protocol and the set of queries forms a code. The main focus in the paper is dedicated to studying the family of codes generated by the set of queries. These codes will be shown to satisfy a specific covering property and will be called private proximity retrieval intersection covering codes. In particular, since the query every server receives is a codeword, the goal is to minimize the number of codewords in such a code which is the minimum number of servers required by the protocol. These codes are closely related to a family of codes known as covering designs. We introduce several lower bounds on the sizes of such codes as well as several constructions. This work focuses on the case when the records are binary vectors together with the Hamming distance. Other metrics such as the Johnson metric are also investigated. [ABSTRACT FROM AUTHOR]

We consider the problem of single-round private information retrieval (PIR) from $N$ replicated databases. We consider the case when $B$ databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of $M$ messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the $T$ -privacy constraint in this paper, where any $T$ databases can collude, and exchange the queries submitted by the user. We derive the information-theoretic capacity of this problem, which is the maximum number of correct symbols that can be retrieved privately (under the $T$ -privacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be $C=(N-2B)/N \cdot (1-T/(N-2B))/(1-(T/(N-2B))^{M})$ , if $2B+T < N$. This capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing $2B$ databases from the system, with a penalty factor of $(N-2B)/N$ , which signifies that even though the number of databases needed for PIR is effectively $N-2B$ , the user still needs to access the entire $N$ databases. The result shows that for the unsynchronized PIR problem, if the user does not have any knowledge about the fraction of the messages that are mis-synchronized, the single-round capacity is the same as the BPIR capacity. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the errors introduced by the Byzantine databases as opposed to erasures in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes. [ABSTRACT FROM AUTHOR]

Network systems often rely on distributed algorithms to achieve a global computation goal with iterative local information exchanges between neighbor nodes. To preserve data privacy, a node may add a random noise to its original data for information exchange at each iteration. Nevertheless, an eavesdropping node can estimate other’s original data based on the information it received. The estimation accuracy and data privacy can be measured in terms of $(\epsilon, \delta)$ -data-privacy, defined as the probability of $\epsilon $ -accurate estimate (the difference of an estimation and the original data is within $\epsilon $ ) is no larger than $\delta $ (the disclosure probability). How to optimize the estimation and analyze data privacy is a critical and open issue. In this paper, a theoretical framework is developed to investigate how to optimize the estimation of neighbor’s original data using the local information received, named optimal distributed estimation. Then, we study the disclosure probability under the optimal estimation for data privacy analysis. We further apply the developed framework to analyze the data privacy of the privacy-preserving average consensus algorithm and identify the optimal noises for the algorithm. [ABSTRACT FROM AUTHOR]

We consider the minimax estimation problem of a discrete distribution with support size $k$ under privacy constraints. A privatization scheme is applied to each raw sample independently, and we need to estimate the distribution of the raw samples from the privatized samples. A positive number $\epsilon $ measures the privacy level of a privatization scheme. For a given $\epsilon $ , we consider the problem of constructing optimal privatization schemes with $\epsilon $ -privacy level, i.e., schemes that minimize the expected estimation loss for the worst-case distribution. Two schemes known in the literature provide order optimal performance in the high privacy regime where $\epsilon $ is very close to 0, and in the low privacy regime where $e^{\epsilon }\approx k$ , respectively. In this paper, we propose a new family of schemes which substantially improve the performance of the existing schemes in the medium privacy regime when $1\ll e^{\epsilon } \ll k$. More concretely, we prove that when $3.8 < \epsilon <\ln (k/9) $ , our schemes reduce the expected estimation loss by 50% under $\ell _{2}^{2}$ metric and by 30% under $\ell _{1}$ metric over the existing schemes. We also prove a lower bound for the region $e^{\epsilon } \ll k$ , which implies that our schemes are order optimal in this regime. [ABSTRACT FROM AUTHOR]

Distributed storage systems (DSSs) in the presence of an external wiretapper are considered. A DSS is parameterized by $(n, k, d)$ , in which the data are stored across $n$ nodes (each with storage capacity $\alpha $ ), and must be recoverable by accessing the contents stored on any $k$ out of $n$ nodes. If a node fails, any $d \geq k$ out of $(n-1)$ nodes help in the repair (regeneration) of the failed node (by sending $d\beta $ units of repair data, where $\beta \leq \alpha $ ), so that the data can still be recovered from the DSS. For such a $(n, k, d)$ -DSS, security from the two types of wiretappers is investigated: 1) Type-I (node data) wiretapper, which can read the data stored on any $\ell

In a locally recoverable or repairable code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage nodes as possible. In this paper, we bound the minimum distance of a code in terms of its length, size, and locality. Unlike the previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence, the Simplex codes are the first example of an optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds. [ABSTRACT FROM PUBLISHER]

In a Private Information Retrieval (PIR) protocol, a user can download a file from a database without revealing the identity of the file to each individual server. A PIR protocol is called $t$ -private if the identity of the file remains concealed even if $t$ of the servers collude. Graph based replication is a simple technique, which is prevalent in both theory and practice, for achieving robustness in storage systems. In this technique each file is replicated on two or more storage servers, giving rise to a (hyper-)graph structure. In this paper we study private information retrieval protocols in graph based replication systems. The main interest of this work is understanding the collusion structures which emerge in the underlying graph. Our main contribution is a 2-replication scheme which guarantees perfect privacy from acyclic sets in the graph, and guarantees partial-privacy in the presence of cycles. Furthermore, by providing an upper bound, it is shown that the PIR rate of this scheme is at most a factor of two from its optimal value for regular graphs. Lastly, we extend our results to larger replication factors and to graph-based coding, a generalization of graph based replication that induces smaller storage overhead and larger PIR rate in many cases. [ABSTRACT FROM AUTHOR]

Fractional repetition (FR) codes form a special class of minimum bandwidth regenerating codes by providing uncoded repairs via a table-based repair model. For a given file size, it is desirable to design FR codes that minimize the reconstruction degree, which is defined as the number of storage nodes required for data retrieval. In this paper, we first consider a lower bound on the reconstruction degree of FR codes obtained by Silberstein and Etzion. We present several families of FR codes that attain this lower bound, which are derived from combinatorial designs and regular graphs. We further provide a new lower bound on the reconstruction degree of FR codes, which is tighter than the existing one. Moreover, we show that for an FR code with reconstruction degree achieving the lower bounds, the corresponding dual code attains upper bounds on the supported file size. [ABSTRACT FROM AUTHOR]

A user wants to retrieve a file from a database without revealing the identity of the file retrieved to the operator of the database (server), which is known as the problem of private information retrieval (PIR). If it is further required that the user obtains no information about the other files in the database, the concept of symmetric PIR (SPIR) is introduced to guarantee privacy for both parties. For SPIR, the server(s) need to access some randomness independent of the database, to protect the content of undesired files from the user. The information-theoretic capacity of SPIR is defined as the maximum number of information bits of the desired file retrieved per downloaded bit. In this paper, the problem of SPIR is studied for a distributed storage system with $N$ servers (nodes), where all data (including the files and the randomness) are stored in a distributed way. Specifically, the files are stored by an $(N,K_{C})$ -MDS storage code. The randomness is distributedly stored such that any $K_{C}$ servers store independent randomness information. We consider two scenarios regarding to the ability of the storage nodes to cooperate. In the first scenario considered, the storage nodes do not communicate or collude. It is shown that the SPIR capacity for MDS-coded storage (hence called MDS-SPIR) is $1-\frac {K_{C}}{N}$ , when the amount of the total randomness of distributed nodes (unavailable at the user) is at least $\frac {K_{C}}{N - K_{\vphantom {R_{l}}C}}$ times the file size. Otherwise, the MDS-SPIR capacity equals zero. The second scenario considered is the $T$ -colluding SPIR problem (hence called TSPIR). Specifically, any $T$ out of $N$ servers may collude, that is, they may communicate their interactions with the user to guess the identity of the requested file. In the special case with $K_{C}=1$ , i.e., the database is replicated at each node, the capacity of TSPIR is shown to be $1-\frac {T}{N}$ , with the ratio of the total randomness size relative to the file size be at least $\frac {T}{\vphantom {R_{l}}N - T}$. For TSPIR with MDS-coded storage (called MDS-TSPIR for short), when restricted to schemes with additive randomness where the servers add the randomness to the answers regardless of the queries received, the capacity is proved to equal $1-\frac {K_{C} + T - 1}{N}$ , with total randomness at least $\frac {K_{C} + T - 1}{N - K_{\vphantom {R_{l}}C} - T + 1}$ times the file size. The MDS-TSPIR capacity for general schemes remains an open problem. [ABSTRACT FROM AUTHOR]

A functional PIR array code is a coding scheme which encodes some $s$ information bits into a $t\times m$ array such that every linear combination of the $s$ information bits has $k$ mutually disjoint recovering sets. Every recovering set consists of some of the array’s columns while it is allowed to read at most $\ell $ encoded bits from every column in order to receive the requested linear combination of the information bits. Functional batch array codes impose a stronger property where every multiset request of $k$ linear combinations has $k$ mutually disjoint recovering sets. Locality functional array codes demand that the size of every recovering set is restrained to be at most $r$. Given the values of $s, k, t, \ell,r$ , the goal of this paper is to study the optimal value of the number of columns $m$ such that these codes exist. Several lower bounds are presented as well as explicit constructions for several of these parameters. [ABSTRACT FROM AUTHOR]

In order to scale economically, data centers are increasingly evolving their data storage methods from simple data replication to more powerful erasure codes, which provide the same level of reliability as replication, but at a significantly lower storage cost. In particular, it is well known that maximum-distance-separable (MDS) codes, such as Reed–Solomon codes, can achieve a target reliability with the maximum storage efficiency. While the use of codes for providing improved reliability in archival storage systems, where data is less frequently accessed (or so-called “cold data”), is well understood, the role of codes in storing more frequently accessed and active “hot data”, where latency is the key metric, is less clear. In this paper, we study data storage systems based on MDS codes through the lens of queueing theory, and term the queueing system arising under codes as an “MDS queue.” We provide lower and upper bounds on the average job latency for both centralized and decentralized versions of MDS queues. We also provide extensive simulations to corroborate our analysis as well as obtain additional insights. [ABSTRACT FROM PUBLISHER]

Partial syndrome matrices and weak-locator polynomials have received considerable attention in recent years due to their applications in decoding cyclic codes. The technical contribution of this paper is threefold: 1) cyclic codes can be decoded by the newly proposed partial syndrome matrices, which generalize the previously known results on the determination of error positions; 2) a new type of polynomial associated with error locations, called radical-locator polynomial, is defined, which includes the weak-locator polynomial as a special case; and 3) a novel class of matrices with many zero entries, called row-echelon partial syndrome matrices, is presented, based on the Newton identities, for efficiently decoding cyclic codes. It is also shown that the radical-locator polynomials can be obtained from the determinants of the above-mentioned (row-echelon) partial syndrome matrices. [ABSTRACT FROM AUTHOR]

Private information retrieval (PIR) protocols allow a user to retrieve entries of a database without revealing the index of the desired item. Information-theoretical privacy can be achieved by the use of several servers and specific retrieval algorithms. Most known PIR protocols focus on decreasing the number of bits exchanged between the client and the server(s) during the retrieval process. On another side, Fazeli et al. introduced so-called PIR codes in order to reduce the storage overhead on the servers. However, few works address the issue of the computation complexity of the servers. In this paper, we show that a specific encoding of the database yields PIR protocols with reasonable communication complexity, low storage overhead, and optimal computational complexity for the servers. This encoding is based on incidence matrices of transversal designs, from which a natural and efficient recovering algorithm is derived. We also present several instances for our construction, which make use of finite geometries and orthogonal arrays. We finally give a generalization of our main construction in order to resist collusions of servers. [ABSTRACT FROM AUTHOR]

This paper investigates reducing sub-packetization of capacity-achieving schemes for uncoded Storage Constrained Private Information Retrieval (SC-PIR) systems. In the SC-PIR system, a user aims to download one out of K files from N servers while revealing nothing about the identity of the requested file to any individual server, in which the K files are stored at the N servers in an uncoded form and each server can store up to μK equivalent files, where μ is the normalized storage capacity of each server. We first prove that there exists a capacity-achieving SC-PIR scheme for a given storage design if and only if all the packets are stored exactly at M ≜ μN servers for μ such that M = μN ∈ {2,3, . . . ,N}. Then, the optimal sub-packetization for capacity-achieving linear SC-PIR schemes is characterized as the solution to an optimization problem, which is typically hard to solve since it involves non-continuous indicator functions. Moreover, a new notion of array called Storage Design Array (SDA) is introduced for the SC-PIR system. With any given SDA, an associated capacity-achieving SC-PIR scheme is constructed. Next, the SC-PIR schemes that have equal-size packets are investigated. Furthermore, the optimal equal-size sub-packetization among all capacity-achieving linear SC-PIR schemes characterized by Woolsey et al. is proved to be N(M−1)/gcd (N,M), which is achieved by a construction of SDA. Finally, by allowing unequal size of packets, a greedy SDA construction is proposed, where the sub-packetization of the associated SC-PIR scheme is upper bounded by N(M−1)/gcd (N,M). Among all capacity-achieving linear SC-PIR schemes, the sub-packetization is optimal when min{M,N−M}|N or M = N, and within a multiplicative gap min{M,N−M}/gcd (N,M) of the optimal one in general. In particular, for the special case N = d ⋅ M ± 1 where the positive integer d ≥ 2, we propose another SDA construction to obtain lower sub-packetization. [ABSTRACT FROM AUTHOR]

Information-theoretic formulations of the private information retrieval (PIR) problem have been investigated under a variety of scenarios. Symmetric private information retrieval (SPIR) is a variant where a user is able to privately retrieve one out of K messages from N non-colluding replicated databases without learning anything about the remaining K−1 messages. However, the goal of perfect privacy can be too taxing for certain applications. In this paper, we investigate if the information-theoretic capacity of SPIR (equivalently, the inverse of the minimum download cost) can be increased by relaxing both user and DB privacy definitions. Such relaxation is relevant in applications where privacy can be traded for communication efficiency. We introduce and investigate the Asymmetric Leaky PIR (AL-PIR) model with different privacy leakage budgets in each direction. For user privacy leakage, we bound the probability ratios between all possible realizations of DB queries by a function of a non-negative constant ε. For DB privacy, we bound the mutual information between the undesired messages, the queries, and the answers, by a function of a non-negative constant &#948. We propose a general AL-PIR scheme that achieves an upper bound on the optimal download cost for arbitrary ε and δ. We show that the optimal download cost of AL-PIR is upper-bounded as D*(ε,δ) ≤ 1+ 1/N−1 − δeε/NK−1−1. Second, we obtain an information-theoretic lower bound on the download cost as D*(ε,δ) ≥ 1 + 1/Neε−1 − δ/(Neε)K−1 −1. The gap analysis between the two bounds shows that our AL-PIR scheme is optimal when ε = 0, i.e., under perfect user privacy and it is optimal within a maximum multiplicative gap of N−e−ε/N−1 for any ε > 0 and δ > 0. [ABSTRACT FROM AUTHOR]

The Levenshtein’s channel model for substitution errors is relevant in information retrieval where information is received through many noisy channels. In each of the channels there can occur at most t errors and the decoder tries to recover the information with the aid of the channel outputs. Recently, Yaakobi and Bruck considered the problem where the decoder provides a list instead of a unique output. If the underlying code C ⊆ F2n has error-correcting capability e, we write t = e + ℓ, (ℓ ≥ 1). In this paper, we provide new (constant) bounds on the size of the list. In particular, we give using the Sauer-Shelah lemma the upper bound ℓ + 1 on the list size for large enough n provided that we have a sufficient number of channels. We also show that the bound ℓ + 1 is the best possible. Most of our other new results rely on constant weight codes. [ABSTRACT FROM AUTHOR]

The problem of reconstructing a string from its error-prone copies, the trace reconstruction problem, was introduced by Vladimir Levenshtein two decades ago. While there has been considerable theoretical work on trace reconstruction, practical solutions have only recently started to emerge in the context of two rapidly developing research areas: immunogenomics and DNA data storage. In immunogenomics, traces correspond to mutated copies of genes, with mutations generated naturally by the adaptive immune system. In DNA data storage, traces correspond to noisy copies of DNA molecules that encode digital data, with errors being artifacts of the data retrieval process. In this paper, we introduce several new trace generation models and open questions relevant to trace reconstruction for immunogenomics and DNA data storage, survey theoretical results on trace reconstruction, and highlight their connections to computational biology. Throughout, we discuss the applicability and shortcomings of known solutions and suggest future research directions. [ABSTRACT FROM AUTHOR]

Motivated by applications in DNA-based storage, we introduce the new problem of code design in the Damerau metric. The Damerau metric is a generalization of the Levenshtein distance which, in addition to deletions, insertions, and substitution errors also accounts for adjacent transposition edits. We first provide constructions for codes that may correct either a single deletion or a single adjacent transposition and then proceed to extend these results to codes that can simultaneously correct a single deletion and multiple adjacent transpositions. We conclude with constructions for joint block deletion and adjacent block transposition error-correcting codes.11

Parts of the results were presented at the International Symposium on Information Theory in Barcelona, 2016.

In the context of the phase retrieval problem, it is known that certain natural classes of measurements, such as Fourier measurements and random Bernoulli measurements, do not lead to the unique reconstruction of all possible signals, even in combination with certain practically feasible random masks. To avoid this difficulty, the analysis is often restricted to measurement ensembles (or masks) that satisfy a small-ball probability condition, in order to ensure that the reconstruction is unique. This paper shows a complementary result: for random Bernoulli measurements, there is still a large class of signals that can be reconstructed uniquely, namely, those signals that are non-peaky. In fact, this result is much more general: it holds for random measurements sampled from any subgaussian distribution \mathcal D , without any small-ball conditions. This is demonstrated in two ways: 1) a proof of stability and uniqueness and 2) a uniform recovery guarantee for the PhaseLift algorithm. In all of these cases, the number of measurements $m$ approaches the information-theoretic lower bound. Finally, for random Bernoulli measurements with erasures, it is shown that PhaseLift achieves uniform recovery of all signals (including peaky ones). [ABSTRACT FROM PUBLISHER]

The ability to store data in the DNA of a living organism has applications in a variety of areas including synthetic biology and watermarking of patented genetically modified organisms. Data stored in this medium are subject to errors arising from various mutations, such as point mutations, indels, and tandem duplication, which need to be corrected to maintain data integrity. In this paper, we provide error-correcting codes for errors caused by tandem duplications, which create a copy of a block of the sequence and insert it in a tandem manner, i.e., next to the original. In particular, we present two families of codes for correcting errors due to tandem duplications of a fixed length: the first family can correct any number of errors, while the second corrects a bounded number of errors. We also study codes for correcting tandem duplications of length up to a given constant $k$ , where we are primarily focused on the cases of $k=2,3$ . Finally, we provide a full classification of the sets of lengths allowed in tandem duplication that result in a unique root for all sequences. [ABSTRACT FROM AUTHOR]

The support recovery problem consists of determining a sparse subset of variables that is relevant in generating a set of observations. In this paper, we study the support recovery problem in the phase retrieval model consisting of noisy phaseless measurements, which arises in a diverse range of settings such as optical detection, X-ray crystallography, electron microscopy, and coherent diffractive imaging. Our focus is on information-theoretic fundamental limits under an approximate recovery criterion, considering both discrete and Gaussian models for the sparse non-zero entries, along with Gaussian measurement matrices. In both cases, our bounds provide sharp thresholds with near-matching constant factors in several scaling regimes on the sparsity and signal-to-noise ratio. As a key step towards obtaining these results, we develop new concentration bounds for the conditional information content of log-concave random variables, which may be of independent interest. [ABSTRACT FROM AUTHOR]

The problem of $X$ -secure $T$ -private information retrieval from MDS coded storage is studied in this paper, where the user wishes to privately retrieve one out of $K$ independent messages that are distributed over $N$ servers according to an MDS code. It is guaranteed that any group of up to $X$ colluding servers learn nothing about the messages and that any group of up to $T$ colluding servers learn nothing about the identity of desired message. A lower bound of achievable rates is proved by presenting a novel scheme based on cross-subspace alignment and a successive decoding with interference cancellation strategy. For large number of messages $(K\rightarrow \infty)$ the achieved rate, which we conjecture to be optimal, improves upon the best known rates previously reported in the literature by Raviv and Karpuk, and generalizes an achievable rate for MDS-TPIR previously found by Freij-Hollanti et al. that is also conjectured to be asymptotically optimal. The setting is then expanded to allow unresponsive and Byzantine servers. Finally, the scheme is applied to find a new lower convex hull of (download, upload) pairs of secure and private distributed matrix multiplication that generalizes, and in certain asymptotic settings strictly improves upon the best known previous results. [ABSTRACT FROM AUTHOR]

Recently, information-theoretic private information retrieval (PIR) from coded storage systems has gained a lot of attention, and a general star product PIR scheme was proposed. In this paper, the star product scheme is adopted, with appropriate modifications, to the case of private (e.g., video) streaming. It is assumed that the files to be streamed are stored on $n$ servers in a coded form, and the streaming is carried out via a convolutional code. The star product scheme is defined for this special case, and various properties are analyzed for two channel models related to straggling and Byzantine servers, both in the baseline case as well as with colluding servers. The achieved PIR rates for the given models are derived and, for the cases where the capacity is known, the first model is shown to be asymptotically optimal, when the number of stripes in a file is large. The second scheme introduced in this work is shown to be the equivalent of block convolutional codes in the PIR setting. For the Byzantine server model, it is shown to outperform the trivial scheme of downloading stripes of the desired file separately without memory. [ABSTRACT FROM AUTHOR]

In distributed storage systems, regenerating codes can achieve the optimal tradeoff between storage capacity and repair bandwidth. However, a critical drawback of existing regenerating codes, in general, is the high coding and repair complexity, since the coding and repair processes involve expensive multiplication operations in finite field. In this paper, we present a design framework of regenerating codes, which employ binary addition and bitwise cyclic shift as the elemental operations, named BASIC regenerating codes. The proposed BASIC regenerating codes can be regarded as a concatenated code with the outer code being a binary parity-check code, and the inner code being a regenerating code utilizing the binary parity-check code as the alphabet. We show that the proposed functional-repair BASIC regenerating codes can achieve the fundamental tradeoff curve between the storage and repair bandwidth asymptotically of functional-repair regenerating codes with less computational complexity. Furthermore, we demonstrate that the existing exact-repair product-matrix construction of regenerating codes can be modified to exact-repair BASIC product-matrix regenerating codes with much less encoding, repair, and decoding complexity from the theoretical analysis, and with less encoding time, repair time, and decoding time from the implementation results. [ABSTRACT FROM AUTHOR]

Recently, symbol-pair codes are proposed to protect against pair errors in symbol-pair read channels. One main task in symbol-pair coding theory is to design codes with large minimum pair distance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense they attain maximal minimum pair distance. In this paper, based on constacyclic codes, we construct some new MDS symbol-pair codes with minimum pair-distance five and six. Compared with classical $q$ -ary MDS codes, the constructed MDS symbol-pair codes have length up to q^{2}+q+1 . [ABSTRACT FROM PUBLISHER]

Suppose $M$ records are replicated in $N$ servers (each storing all $M$ records), a user wants to privately retrieve one record by accessing the servers such that the identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a $T$ -private information retrieval (PIR) scheme. In practice, capacity-achieving and small sub-packetization are both desired for PIR schemes, because the former implies the highest download rate and the latter means simple realization. Meanwhile, sub-packetization is the key technique for achieving capacity. In this paper, we characterize the optimal sub-packetization for linear capacity-achieving $T$ -PIR schemes. First, a lower bound on the sub-packetization $L$ for linear capacity-achieving $T$ -PIR schemes is proved, i.e., $L\geq dn^{M-1}$ , where $d={\mathrm{ gcd}}(N,T)$ and $n=N/d$. Then, for general values of $M$ and $N>T\geq 1$ , a linear capacity-achieving $T$ -PIR scheme with sub-packetization $dn^{M-1}$ is designed. Comparing with the first capacity-achieving $T$ -PIR scheme given by Sun and Jafar in 2016, our scheme reduces the sub-packetization from $N^{M}$ to the optimal and further reduces the field size by a factor of $Nd^{M-2}$. [ABSTRACT FROM AUTHOR]

Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. Optimizing NB-LDPC codes to overcome their error floor is one of the main code design challenges facing storage engineers upon deploying such codes in practice. Furthermore, the increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems, e.g., multi-level Flash systems, exacerbate the error floor problem by widening the spectrum of problematic objects that contribute to the error floor of an NB-LDPC code. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective combinatorial NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. The WCM framework was used to optimize codes for asymmetric Flash channels, MR channels that have intrinsic memory, in addition to canonical symmetric additive white Gaussian noise channels. In this paper, we provide an in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs) as the detrimental objects of interest. In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal in the sense that it operates on the minimum number of matrices, which are the WCMs, to remove a GAST. Then, we enumerate WCMs and demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. In addition, we propose a new set of problematic objects, namely oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with additional soft information, and spatially coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels. [ABSTRACT FROM AUTHOR]

We study a generalization of the setting of regenerating codes, motivated by applications to storage systems consisting of clusters of storage nodes. There are $n$ clusters in total, with $m$ nodes per cluster. A data file is coded and stored across the $mn$ nodes, with each node storing $\alpha $ symbols. For availability of data, we require that the file be retrievable by downloading the entire content from any subset of $k$ clusters. Nodes represent entities that can fail. We distinguish between intra-cluster and inter-cluster bandwidth (BW) costs during node repair. Node-repair in a cluster is accomplished by downloading $\beta $ symbols each from any set of $d$ other clusters, dubbed remote helper clusters, and also up to $\alpha $ symbols each from any set of $\ell $ surviving nodes, dubbed local helper nodes, in the host cluster. We first identify the optimal trade-off between storage-overhead and inter-cluster repair-bandwidth under functional repair, and also present optimal exact-repair code constructions for a class of parameters. The new trade-off is strictly better than what is achievable via space-sharing existing coding solutions, whenever $\ell > 0$. We then obtain sharp lower bounds on the necessary intra-cluster repair BW to achieve optimal trade-off. Under functional repair, random linear network codes (RLNCs) simultaneously optimize usage of both inter- and intra-cluster repair BW; simulation results based on RLNCs suggest optimality of the bounds on intra-cluster repair-bandwidth. Our bounds reveal the interesting fact that, while it is beneficial to increase the number of local helper nodes $\ell $ in order to improve the storage-vs-inter-cluster-repair-BW trade-off, increasing $\ell $ not only increases intra-cluster BW in the host-cluster, but also increases the intra-cluster BW in the remote helper clusters. We also analyze resilience of the clustered storage system against passive eavesdropping by providing file-size bounds and optimal code constructions. [ABSTRACT FROM AUTHOR]

Distributed storage systems for large-scale applications typically use replication for reliability. Recently, erasure codes were used to reduce the large storage overhead, while increasing data reliability. A main limitation of off-the-shelf erasure codes is their high-repair cost during single node failure events. A major open problem in this area has been the design of codes that: 1) are repair efficient and 2) achieve arbitrarily high data rates. In this paper, we explore the repair metric of locality, which corresponds to the number of disk accesses required during a single node repair. Under this metric, we characterize an information theoretic tradeoff that binds together the locality, code distance, and storage capacity of each node. We show the existence of optimal locally repairable codes (LRCs) that achieve this tradeoff. The achievability proof uses a locality aware flow-graph gadget, which leads to a randomized code construction. Finally, we present an optimal and explicit LRC that achieves arbitrarily high data rates. Our locality optimal construction is based on simple combinations of Reed–Solomon blocks. [ABSTRACT FROM AUTHOR]

In this paper, we extend the concept of maximum-distance separable (MDS) array codes to a larger class of codes, where the array columns contain a variable number of data and parity symbols and the codewords cannot be arranged, in general, in a regular array structure with equal column length. These new codes, named irregular MDS array codes, find applications in problems of distributed data storage with multiple sources of information generating data at unequal rates. We solve the problem of finding optimal parity symbol allocations that achieve minimum redundancy for a given level of protection against block erasures. We provide a classification of irregular MDS array codes according to the parameters of their parity symbol allocation and we show how regular MDS array codes are a special case of this wider class. We derive necessary and sufficient conditions for such irregular array codes to be MDS and extend the concept of the lowest density generator matrix. Finally, we show how a simple constructive method allows to design irregular lowest density MDS array codes with alphabet size independent of the size of the array columns. [ABSTRACT FROM AUTHOR]

Network coding can significantly improve the transmission rate of communication networks with packet loss compared with routing. However, using network coding usually incurs high computational and storage costs in the network devices and terminals. For example, some network coding schemes require the computational and/or storage capacities of an intermediate network node to increase linearly with the number of packets for transmission, making such schemes difficult to be implemented in a router-like device that has only constant computational and storage capacities. In this paper, we introduce batched sparse code (BATS code), which enables a digital fountain approach to resolve the above issue. BATS code is a coding scheme that consists of an outer code and inner code. The outer code is a matrix generation of a fountain code. It works with the inner code that comprises random linear coding at the intermediate network nodes. BATS codes preserve such desirable properties of fountain codes as ratelessness and low encoding/decoding complexity. The computational and storage capacities of the intermediate network nodes required for applying BATS codes are independent of the number of packets for transmission. Almost capacity-achieving BATS code schemes are devised for unicast networks and certain multicast networks. For general multicast networks, under different optimization criteria, guaranteed decoding rates for the destination nodes can be obtained. [ABSTRACT FROM AUTHOR]

Consider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, whereas other (heavy) parity symbols may depend on all data symbols. Such codes have been studied recently in the context of erasure coding for data storage, where the local parities facilitate fast recovery of any single symbol when it is erased, whereas the heavy parities provide tolerance to a large number of simultaneous erasures. A code as above is maximally recoverable, if it corrects all erasure patterns, which are information theoretically correctable given the prescribed dependence relations between data symbols and parity symbols. In this paper, we present explicit families of maximally recoverable codes with locality. We also initiate the general study of the tradeoff between maximal recoverability and alphabet size. [ABSTRACT FROM AUTHOR]

We consider the recovery of a (real- or complex-valued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a “non-lifting” relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is basis pursuit, which implies that the phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice. [ABSTRACT FROM AUTHOR]

Quantum private information retrieval (QPIR) is a protocol in which a user retrieves one of multiple files from n non-communicating servers by downloading quantum systems without revealing which file is retrieved. As variants of QPIR with stronger security requirements, symmetric QPIR is a protocol in which no other files than the target file are leaked to the user, and t-private QPIR is a protocol in which the identity of the target file is kept secret even if at most t servers may collude to reveal the identity. The QPIR capacity is the maximum ratio of the file size to the size of downloaded quantum systems, and we prove that the symmetric t-private QPIR capacity is min{1, 2(n−t)/n} for any 1 ≤ t < n. We construct a capacity-achieving QPIR protocol by the stabilizer formalism and prove the optimality of our protocol. The proposed capacity is greater than the classical counterpart. [ABSTRACT FROM AUTHOR]

Symmetric fix-free codes are prefix condition codes in which each codeword is required to be a palindrome. Their study is motivated by the topic of joint source-channel coding and by some information retrieval problems. Although they have been considered by a few communities they are not well understood. In earlier work, we used a collection of instances of Boolean satisfiability problems as a tool in the generation of all optimal binary symmetric fix-free codes with \(n\) codewords and observed that the number of different optimal codelength sequences grows slowly compared with the corresponding number for prefix condition codes. We demonstrate that all optimal symmetric fix-free codes can alternatively be obtained by sequences of codes generated by simple manipulations starting from one particular code. We also discuss simplifications in the process of searching for this set of codes as well as a conjecture, which if correct, together with the other results leads to a relatively fast algorithm which has been implemented in MATLAB to construct all optimal binary symmetric fix-free codes. [ABSTRACT FROM PUBLISHER]

In the captioned paper , and are based on incorrect numerical calculation. The corrected plots in look as follows: [ABSTRACT FROM AUTHOR]

We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs), where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed–Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al.. Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol. For all three protocols, we present an algorithm to optimize their PIR rates. [ABSTRACT FROM AUTHOR]

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. Finally, the scheme is adapted to symmetric PIR. [ABSTRACT FROM AUTHOR]

We introduce the problem of private computation, comprised of $ {N}$ distributed and non-colluding servers, $ {K}$ independent datasets, and a user who wants to compute a function of the datasets privately, i.e., without revealing which function he wants to compute, to any individual server. This private computation problem is a strict generalization of the private information retrieval (PIR) problem, obtained by expanding the PIR message set (which consists of only independent messages) to also include functions of those messages. The capacity of private computation, $ {C}$ , is defined as the maximum number of bits of the desired function that can be retrieved per bit of total download from all servers. We characterize the capacity of private computation, for $ {N}$ servers and $ {K}$ independent datasets that are replicated at each server, when the functions to be computed are arbitrary linear combinations of the datasets. Surprisingly, the capacity, $ {C}=\left ({1+1/ {N}+\cdots +1/ {N}^{ {K}-1}}\right)^{-1}$ , matches the capacity of PIR with $ {N}$ servers and $ {K}$ messages. Thus, allowing arbitrary linear computations does not reduce the communication rate compared to pure dataset retrieval. The same insight is shown to hold even for arbitrary non-linear computations when the number of datasets $ {K}\rightarrow \infty $. [ABSTRACT FROM AUTHOR]

We consider the problem of private information retrieval (PIR) with colluding servers and eavesdroppers (abbreviated as ETPIR). The ETPIR problem is comprised of ${K}$ messages and ${N}$ servers where each server stores all ${K}$ messages, a user who wants to retrieve one of the ${K}$ messages without revealing the desired message index to any set of ${T}$ colluding servers, and an eavesdropper who can listen to the queries and answers of any ${E}$ servers but is prevented from learning any information about the messages. The information theoretic capacity of ETPIR is defined to be the maximum number of desired message symbols retrieved privately per information symbol downloaded. We show that the capacity of ETPIR is $ {C} = ({1}- ({{E}}/{{N}})) ({1} + ({ {T}- {E}}/{ {N}- {E}}) + \cdots + (({ {T}-\boldsymbol {E}}/{ {N}-{E}}))^{ {K}-{1}})^{-{1}}$ when $ {E} < {T}$ , and $ {C} = ({1} - ({ {E}}/{ {N}}))$ when $ {E} \geq {T}$. To achieve the capacity, the servers need to share a common random variable (independent of the messages), and its size must be at least $({{E}}/{ {N}}) \cdot ({{1}}/{ {C}})$ symbols per message symbol. Otherwise, with less amount of shared common randomness, ETPIR is not feasible and the capacity reduces to zero. An interesting observation is that the ETPIR capacity expression takes different forms in two regimes. When $ {E} < {T}$ , the capacity equals the inverse of a sum of a geometric series with ${K}$ terms and decreases with ${K}$ ; this form is typical for capacity expressions of PIR. When ${E} \geq {T}$ , the capacity does not depend on $ {K}$ , a typical form for capacity expressions of SPIR (symmetric PIR, which further requires data-privacy, i.e., the user learns no information about other undesired messages); the capacity does not depend on $ {T}$ either. In addition, the ETPIR capacity result includes multiple previous PIR and SPIR capacity results as special cases. [ABSTRACT FROM AUTHOR]

We consider the problem of private information retrieval (PIR) from $N$ non-colluding and replicated databases when the user is equipped with a cache that holds an uncoded fraction $r$ from each of the $K$ stored messages in the databases. We assume that the databases are unaware of the cache content. We investigate $D^{*}(r)$ the optimal download cost normalized with the message size as a function of $K$ , $N$ , and $r$. For a fixed $K$ and $N$ , we develop an inner bound (converse bound) for the $D^{*}(r)$ curve. The inner bound is a piece-wise linear function in $r$ that consists of $K$ line segments. For the achievability, we develop explicit schemes that exploit the cached bits as side information to achieve $K-1$ non-degenerate corner points. These corner points differ in the number of cached bits that are used to generate the one-side information equation. We obtain an outer bound (achievability) for any caching ratio by memory sharing between these corner points. Thus, the outer bound is also a piece-wise linear function in $r$ that consists of $K$ line segments. The inner and the outer bounds match in general for the cases of very low-caching ratio and very high-caching ratio. As a corollary, we fully characterize the optimal download cost caching ratio tradeoff for $K=3$. For general $K$ , $N$ , and $r$ , we show that the largest gap between the achievability and the converse bounds is 1/6. Our results show that the download cost can be reduced beyond memory sharing if the databases are unaware of the cached content. [ABSTRACT FROM AUTHOR]

We consider the problem of private information retrieval (PIR) and symmetric private information retrieval (SPIR) from replicated databases with colluding servers, in the presence of Byzantine adversaries and eavesdroppers. Specifically, there are $K$ messages replicatively stored at $N$ databases. A user wants to retrieve one message by communicating with the databases, without revealing the identity of the message retrieved. For $T$ -colluding databases, any $T$ out of $N$ databases may communicate their interactions with the user to guess the identity of the requested message. We consider the situation where the communication system can be vulnerable to attachers, namely, there is an adversary in the system that can tap in on or even try to corrupt the communication. The capacity is defined as the maximum number of information bits of the desired message retrieved per downloaded bit. For SPIR, it is further required that the user learns nothing about the other $K-1$ messages in the database. Three types of adversaries are considered: a Byzantine adversary who can overwrite the transmission of any $B$ servers to the user; a passive eavesdropper who can tap in on the incoming and outgoing transmissions of any $E$ servers; and a combination of both -- an adversary who can tap in on a set of any $E$ nodes, and overwrite the transmission of a set of any $B$ nodes. The problems of SPIR with colluding servers and the three types of adversaries are named T-BSPIR, T-ESPIR and T-BESPIR, respectively. We derive the capacities of the three secure SPIR problems. The results resemble those of secure network coding problems with adversaries and eavesdroppers. The capacity of $T$ -colluding PIR with Byzantine adversaries is characterized in. In this work, we consider $T$ -colluding PIR with an eavedropper (named T-EPIR). We derive the T-EPIR capacity when $E \geq T$ ; for the case where $E \leq T$ , we find an outer bound (converse bound) and an inner bound (achievability) on the optimal achievable rate. [ABSTRACT FROM AUTHOR]