arXiv subjects

Alex Sprintson

Publications and source records attributed to Alex Sprintson.

At least 37 records · Page 2Linked to original sources

arXiv2021

Single-Server Private Linear Transformation: The Individual Privacy Case

This paper considers the single-server Private Linear Transformation (PLT) problem with individual privacy guarantees. In this problem, there is a user that wishes to obtain $L$ independent linear combinations of a $D$-subset of messages belonging to a dataset of $K$ messages stored on a single server. The goal is to minimize the download cost while keeping the identity of each message required for the computation individually private. The individual privacy requirement ensures that the identity of each individual message required for the computation is kept private. This is in contrast to the stricter notion of joint privacy that protects the entire set of identities of all messages used for the computation, including the correlations between these identities. The notion of individual privacy captures a broad set of practical applications. For example, such notion is relevant when the dataset contains information about individuals, each of them requires privacy guarantees for their data access patterns. We focus on the setting in which the required linear transformation is associated with a maximum distance separable (MDS) matrix. In particular, we require that the matrix of coefficients pertaining to the required linear combinations is the generator matrix of an MDS code. We establish lower and upper bounds on the capacity of PLT with individual privacy, where the capacity is defined as the supremum of all achievable download rates. We show that our bounds are tight under certain conditions.

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arXiv2021

Download Time Analysis for Distributed Storage Codes with Locality and Availability

The paper presents techniques for analyzing the expected download time in distributed storage systems that employ systematic availability codes. These codes provide access to hot data through the systematic server containing the object and multiple recovery groups. When a request for an object is received, it can be replicated (forked) to the systematic server and all recovery groups. We first consider the low-traffic regime and present the close-form expression for the download time. By comparison across systems with availability, maximum distance separable (MDS), and replication codes, we demonstrate that availability codes can reduce download time in some settings but are not always optimal. In the high-traffic regime, the system consists of multiple inter-dependent Fork-Join queues, making exact analysis intractable. Accordingly, we present upper and lower bounds on the download time, and an M/G/1 queue approximation for several cases of interest. Via extensive numerical simulations, we evaluate our bounds and demonstrate that the M/G/1 queue approximation has a high degree of accuracy.

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arXiv2021

Private Linear Transformation: The Individual Privacy Case

This paper considers the single-server Private Linear Transformation (PLT) problem when individual privacy is required. In this problem, there is a user that wishes to obtain $L$ linear combinations of a $D$-subset of messages belonging to a dataset of $K$ messages stored on a single server. The goal is to minimize the download cost while keeping the identity of every message required for the computation individually private. The individual privacy requirement implies that, from the perspective of the server, every message is equally likely to belong to the $D$-subset of messages that constitute the support set of the required linear combinations. We focus on the setting in which the matrix of coefficients pertaining to the required linear combinations is the generator matrix of a Maximum Distance Separable code. We establish lower and upper bounds on the capacity of PLT with individual privacy, where the capacity is defined as the supremum of all achievable download rates. We show that our bounds are tight under certain divisibility conditions. In addition, we present lower bounds on the capacity of the settings in which the user has a prior side information about a subset of messages.

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arXiv2021

Minimizing the alphabet size in codes with restricted error sets

This paper focuses on error-correcting codes that can handle a predefined set of specific error patterns. The need for such codes arises in many settings of practical interest, including wireless communication and flash memory systems. In many such settings, a smaller field size is achievable than that offered by MDS and other standard codes. We establish a connection between the minimum alphabet size for this generalized setting and the combinatorial properties of a hypergraph that represents the prespecified collection of error patterns. We also show a connection between error and erasure correcting codes in this specialized setting. This allows us to establish bounds on the minimum alphabet size and show an advantage of non-linear codes over linear codes in a generalized setting. We also consider a variation of the problem which allows a small probability of decoding error and relate it to an approximate version of hypergraph coloring.

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arXiv2021

Private Linear Transformation: The Joint Privacy Case

We introduce the problem of Private Linear Transformation (PLT). This problem includes a single (or multiple) remote server(s) storing (identical copies of) $K$ messages and a user who wants to compute $L$ linear combinations of a $D$-subset of these messages by downloading the minimum amount of information from the server(s) while protecting the privacy of the entire set of $D$ messages. This problem generalizes the Private Information Retrieval and Private Linear Computation problems. In this work, we focus on the single-server case. For the setting in which the coefficient matrix of the required $L$ linear combinations generates a Maximum Distance Separable (MDS) code, we characterize the capacity -- defined as the supremum of all achievable download rates, for all parameters $K, D, L$. In addition, we present lower and/or upper bounds on the capacity for the settings with non-MDS coefficient matrices and the settings with a prior side information.

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arXiv2020

Joint Index Coding and Incentive Design for Selfish Clients

The index coding problem includes a server, a group of clients, and a set of data chunks. While each client wants a subset of the data chunks and already has another subset as its side information, the server transmits some uncoded data chunks or coded data chunks to the clients over a noiseless broadcast channel. The objective of the problem is to satisfy the demands of all clients with the minimum number of transmissions. In this paper, we investigate the index coding setting from a game-theoretical perspective. We consider selfish clients, where each selfish client has private side information and a private valuation of each data chunk it wants. In this context, our objectives are following: 1) to motivate each selfish client to reveal the correct side information and true valuation of each data chunk it wants; 2) to maximize the social welfare, i.e., the total valuation of the data chunks recovered by the clients minus the total cost incurred by the transmissions from the server. Our main contribution is to jointly develop coding schemes and incentive schemes for achieving the first objective perfectly and achieving the second objective optimally or approximately with guaranteed approximation ratios (potentially within some restricted sets of coding matrices).

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arXiv2020

Efficient Storage Schemes for Desired Service Rate Regions

A major concern in cloud/edge storage systems is serving a large number of users simultaneously. The service rate region is introduced recently as an important performance metric for coded distributed systems, which is defined as the set of all data access requests that can be simultaneously handled by the system. This paper studies the problem of designing a coded distributed storage system storing k files where a desired service rate region R of the system is given and the goal is 1) to determine the minimum number of storage nodes n(R) (or a lower bound on n(R)) for serving all demand vectors inside the set R and 2) to design the most storage-efficient redundancy scheme with the service rate region covering R. Towards this goal, we propose three general lower bounds for n(R). Also, for k=2, we characterize n(R), i.e., we show that the proposed lower bounds are tight via designing a novel storage-efficient redundancy scheme with n(R) storage nodes and the service rate region covering R.

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arXiv2020

Minimizing the alphabet size of erasure codes with restricted decoding sets

A Maximum Distance Separable code over an alphabet $F$ is defined via an encoding function $C:F^k \rightarrow F^n$ that allows to retrieve a message $m \in F^k$ from the codeword $C(m)$ even after erasing any $n-k$ of its symbols. The minimum possible alphabet size of general (non-linear) MDS codes for given parameters $n$ and $k$ is unknown and forms one of the central open problems in coding theory. The paper initiates the study of the alphabet size of codes in a generalized setting where the coding scheme is required to handle a pre-specified subset of all possible erasure patterns, naturally represented by an $n$-vertex $k$-uniform hypergraph. We relate the minimum possible alphabet size of such codes to the strong chromatic number of the hypergraph and analyze the tightness of the obtained bounds for both the linear and non-linear settings. We further consider variations of the problem which allow a small probability of decoding error.

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arXiv2020

A Combinatorial View of the Service Rates of Codes Problem, its Equivalence to Fractional Matching and its Connection with Batch Codes

We propose a novel technique for constructing a graph representation of a code through which we establish a significant connection between the service rate problem and the well-known fractional matching problem. Using this connection, we show that the service capacity of a coded storage system equals the fractional matching number in the graph representation of the code, and thus is lower bounded and upper bounded by the matching number and the vertex cover number, respectively. This is of great interest because if the graph representation of a code is bipartite, then the derived upper and lower bounds are equal, and we obtain the capacity. Leveraging this result, we characterize the service capacity of the binary simplex code whose graph representation, as we show, is bipartite. Moreover, we show that the service rate problem can be viewed as a generalization of the multiset primitive batch codes problem.

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arXiv2020

Private Computation with Individual and Joint Privacy

This paper considers the problem of single-server Private Computation (PC) in the presence of Side Information (SI). In this problem, there is a server that stores $K$ i.i.d. messages, and a user who has a subset of $M$ uncoded messages or a coded linear combination of them as side information, where the identities of these messages are unknown to the server. The user wants to privately compute (via downloading information from the server) a linear combination of a subset of $D$ other messages, where the identities of these messages must be kept private individually or jointly. For each setting, we define the capacity as the supremum of all achievable download rates. We characterize the capacity of both PC with coded and uncoded SI when individual privacy is required, for all $K, M, D$. Our results indicate that both settings have the same capacity. In addition, we establish a non-trivial lower bound on the capacity of PC with coded SI when joint privacy is required, for a range of parameters $K, M, D$. This lower bound is the same as the lower bound we previously established on the capacity of PC with uncoded SI when joint privacy is required.

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arXiv2019

The Role of Coded Side Information in Single-Server Private Information Retrieval

We study the role of coded side information in single-server Private Information Retrieval (PIR). An instance of the single-server PIR problem includes a server that stores a database of $K$ independently and uniformly distributed messages, and a user who wants to retrieve one of these messages from the server. We consider settings in which the user initially has access to a coded side information which includes a linear combination of a subset of $M$ messages in the database. We assume that the identities of the $M$ messages that form the support set of the coded side information as well as the coding coefficients are initially unknown to the server. We consider two different models, depending on whether the support set of the coded side information includes the requested message or not. We also consider the following two privacy requirements: (i) the identities of both the demand and the support set of the coded side information need to be protected, or (ii) only the identity of the demand needs to be protected. For each model and for each of the privacy requirements, we consider the problem of designing a protocol for generating the user's query and the server's answer that enables the user to decode the message they need while satisfying the privacy requirement. We characterize the (scalar-linear) capacity of each setting, defined as the ratio of the number of information bits in a message to the minimum number of information bits downloaded from the server over all (scalar-linear) protocols that satisfy the privacy condition. Our converse proofs rely on new information-theoretic arguments---tailored to the setting of single-server PIR and different from the commonly-used techniques in multi-server PIR settings. We also present novel capacity-achieving scalar-linear protocols for each of the settings being considered.

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arXiv2019

Non-adaptive Quantitative Group Testing Using Irregular Sparse Graph Codes

This paper considers the problem of Quantitative Group Testing (QGT) where there are some defective items among a large population of $N$ items. We consider the scenario in which each item is defective with probability $K/N$, independently from the other items. In the QGT problem, the goal is to identify all or a sufficiently large fraction of the defective items by testing groups of items, with the minimum possible number of tests. In particular, the outcome of each test is a non-negative integer which indicates the number of defective items in the tested group. In this work, we propose a non-adaptive QGT scheme for the underlying randomized model for defective items, which utilizes sparse graph codes over irregular bipartite graphs with optimized degree profiles on the left nodes of the graph as well as binary $t$-error-correcting BCH codes. We show that in the sub-linear regime, i.e., when the ratio $K/N$ vanishes as $N$ grows unbounded, the proposed scheme with ${m=c(t,d)K(t\log (\frac{\ell N}{c(t,d)K}+1)+1)}$ tests can identify all the defective items with probability approaching $1$, where $d$ and $\ell$ are the maximum and average left degree, respectively, and $c(t,d)$ depends only on $t$ and $d$ (and does not depend on $K$ and $N$). For any $t\leq 4$, the testing and recovery algorithms of the proposed scheme have the computational complexity of $\mathcal{O}(N\log \frac{N}{K})$ and $\mathcal{O}(K\log \frac{N}{K})$, respectively. The proposed scheme outperforms two recently proposed non-adaptive QGT schemes for the sub-linear regime, including our scheme based on regular bipartite graphs and the scheme of Gebhard et al., in terms of the number of tests required to identify all defective items with high probability.

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arXiv2019

On an Equivalence Between Single-Server PIR with Side Information and Locally Recoverable Codes

Private Information Retrieval (PIR) problem has recently attracted a significant interest in the information-theory community. In this problem, a user wants to privately download one or more messages belonging to a database with copies stored on a single or multiple remote servers. In the single server scenario, the user must have prior side information, i.e., a subset of messages unknown to the server, to be able to privately retrieve the required messages in an efficient way. In the last decade, there has also been a significant interest in Locally Recoverable Codes (LRC), a class of storage codes in which each symbol can be recovered from a limited number of other symbols. More recently, there is an interest in 'cooperative' locally recoverable codes, i.e., codes in which multiple symbols can be recovered from a small set of other code symbols. In this paper, we establish a relationship between coding schemes for the single-server PIR problem and LRCs. In particular, we show the following results: (i) PIR schemes designed for retrieving a single message are equivalent to classical LRCs; and (ii) PIR schemes for retrieving multiple messages are equivalent to cooperative LRCs. These equivalence results allow us to recover upper bounds on the download rate for PIR-SI schemes, and to obtain a novel rate upper bound on cooperative LRCs. We show results for both linear and non-linear codes.

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arXiv2019

Private Information Retrieval with Private Coded Side Information: The Multi-Server Case

In this paper, we consider the multi-server setting of Private Information Retrieval with Private Coded Side Information (PIR-PCSI) problem. In this problem, there is a database of $K$ messages whose copies are replicated across $N$ servers, and there is a user who knows a random linear combination of a random subset of $M$ messages in the database as side information. The user wishes to download one message from the servers, while protecting the identities of both the demand message and the messages forming the side information. We assume that the servers know the number of messages forming the user's side information in advance, whereas the indices of these messages and their coefficients in the side information are not known to any of the servers a priori. Our goal is to characterize (or derive a lower bound on) the capacity, i.e., the maximum achievable download rate, for the following two settings. In the first setting, the set of messages forming the linear combination available to the user as side information, does not include the user's demanded message. For this setting, we show that the capacity is equal to $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$. In the second setting, the demand message contributes to the linear combination available to the user as side information, i.e., the demand message is one of the messages that form the user's side information. For this setting, we show that the capacity is lower-bounded by $\left(1+{1}/{N}+\dots+{1}/{N^{K-M}}\right)^{-1}$. The proposed achievability schemes and proof techniques leverage ideas from both our recent methods proposed for the single-server PIR-PCSI problem as well as the techniques proposed by Sun and Jafar for multi-server private computation problem.

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arXiv2019

Multi-Server Private Information Retrieval with Coded Side Information

In this paper, we study the multi-server setting of the \emph{Private Information Retrieval with Coded Side Information (PIR-CSI)} problem. In this problem, there are $K$ messages replicated across $N$ servers, and there is a user who wishes to download one message from the servers without revealing any information to any server about the identity of the requested message. The user has a side information which is a linear combination of a subset of $M$ messages in the database. The parameter $M$ is known to all servers in advance, whereas the indices and the coefficients of the messages in the user's side information are unknown to any server \emph{a priori}. We focus on a class of PIR-CSI schemes, referred to as \emph{server-symmetric schemes}, in which the queries/answers to/from different servers are symmetric in structure. We define the \emph{rate} of a PIR-CSI scheme as its minimum download rate among all problem instances, and define the \emph{server-symmetric capacity} of the PIR-CSI problem as the supremum of rates over all server-symmetric PIR-CSI schemes. Our main results are as follows: (i) when the side information is not a function of the user's requested message, the capacity is given by ${(1+{1}/{N}+\dots+{1}/{N^{\left\lceil \frac{K}{M+1}\right\rceil -1}})^{-1}}$ for any ${1\leq M\leq K-1}$; and (ii) when the side information is a function of the user's requested message, the capacity is equal to $1$ for $M=2$ and $M=K$, and it is equal to ${N}/{(N+1)}$ for any ${3 \leq M \leq K-1}$. The converse proofs rely on new information-theoretic arguments, and the achievability schemes are inspired by our recently proposed scheme for single-server PIR-CSI as well as the Sun-Jafar scheme for multi-server PIR.

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arXiv2019

Codes with Locality in the Rank and Subspace Metrics

We extend the notion of locality from the Hamming metric to the rank and subspace metrics. Our main contribution is to construct a class of array codes with locality constraints in the rank metric. Our motivation for constructing such codes stems from designing codes for efficient data recovery from correlated and/or mixed (i.e., complete and partial) failures in distributed storage systems. Specifically, the proposed local rank-metric codes can recover locally from 'crisscross errors and erasures', which affect a limited number of rows and/or columns of the storage system. We also derive a Singleton-like upper bound on the minimum rank distance of (linear) codes with 'rank-locality' constraints. Our proposed construction achieves this bound for a broad range of parameters. The construction builds upon Tamo and Barg's method for constructing locally repairable codes with optimal minimum Hamming distance. Finally, we construct a class of constant-dimension subspace codes (also known as Grassmannian codes) with locality constraints in the subspace metric. The key idea is to show that a Grassmannian code with locality can be easily constructed from a rank-metric code with locality by using the lifting method proposed by Silva et al. We present an application of such codes for distributed storage systems, wherein nodes are connected over a network that can introduce errors and erasures.

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arXiv2019

Sparse Graph Codes for Non-adaptive Quantitative Group Testing

This paper considers the problem of Quantitative Group Testing (QGT). Consider a set of $N$ items among which $K$ items are defective. The QGT problem is to identify (all or a sufficiently large fraction of) the defective items, where the result of a test reveals the number of defective items in the tested group. In this work, we propose a non-adaptive QGT algorithm using sparse graph codes over bi-regular bipartite graphs with left-degree $\ell$ and right degree $r$ and binary $t$-error-correcting BCH codes. The proposed scheme provides exact recovery with probabilistic guarantee, i.e. recovers all the defective items with high probability. In particular, we show that for the sub-linear regime where $\frac{K}{N}$ vanishes as $K,N\rightarrow\infty$, the proposed algorithm requires at most ${m=c(t)K\left(t\log_2\left(\frac{\ell N}{c(t)K}+1\right)+1\right)+1}$ tests to recover all the defective items with probability approaching one as ${K,N\rightarrow\infty}$, where $c(t)$ depends only on $t$. The results of our theoretical analysis reveal that the minimum number of required tests is achieved by $t=2$. The encoding and decoding of the proposed algorithm for any $t\leq 4$ have the computational complexity of $\mathcal{O}(K\log^2 \frac{N}{K})$ and $\mathcal{O}(K\log \frac{N}{K})$, respectively. Our simulation results also show that the proposed algorithm significantly outperforms a non-adaptive semi-quantitative group testing algorithm recently proposed by Abdalla \emph{et al.} in terms of the required number of tests for identifying all the defective items with high probability.

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arXiv2019

Single-Server Multi-Message Individually-Private Information Retrieval with Side Information

We consider a multi-user variant of the private information retrieval problem described as follows. Suppose there are $D$ users, each of which wants to privately retrieve a distinct message from a server with the help of a trusted agent. We assume that the agent has a random subset of $M$ messages that is not known to the server. The goal of the agent is to collectively retrieve the users' requests from the server. For protecting the privacy of users, we introduce the notion of individual-privacy -- the agent is required to protect the privacy only for each individual user (but may leak some correlations among user requests). We refer to this problem as Individually-Private Information Retrieval with Side Information (IPIR-SI). We first establish a lower bound on the capacity, which is defined as the maximum achievable download rate, of the IPIR-SI problem by presenting a novel achievability protocol. Next, we characterize the capacity of IPIR-SI problem for $M = 1$ and $D = 2$. In the process of characterizing the capacity for arbitrary $M$ and $D$ we present a novel combinatorial conjecture, that may be of independent interest.

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