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Salim El Rouayheb

Publications and source records attributed to Salim El Rouayheb.

At least 55 records · Page 3Linked to original sources
arXiv2018

Staircase-PIR: Universally Robust Private Information Retrieval

We consider the problem of designing private information retrieval (PIR) schemes on data of $m$ files replicated on $n$ servers that can possibly collude. We focus on devising robust PIR schemes that can tolerate stragglers, i.e., slow or unresponsive servers. In many settings, the number of stragglers is not known a priori or may change with time. We define universally robust PIR as schemes that achieve PIR capacity asymptotically in $m$ and simultaneously for any number of stragglers up to a given threshold. We introduce Staircase-PIR schemes and prove that they are universally robust. Towards that end, we establish an equivalence between robust PIR and communication efficient secret sharing.

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arXiv2018

The Computational Wiretap Channel

We present the computational wiretap channel: Alice has some data x and wants to share some computation h(x) with Bob. To do this, she sends f(x), where f is some sufficient statistic for h. An eavesdropper, Eve, is interested in computing another function g(x). We show that, under some conditions on f and g, this channel can be approximated, from Eve's point of view, by the classic Wyner wiretap channel.

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arXiv2018

On the Capacity of Single-Server Multi-Message Private Information Retrieval with Side Information

We study Private Information Retrieval with Side Information (PIR-SI) in the single-server multi-message setting. In this setting, a user wants to download $D$ messages from a database of $K\geq D$ messages, stored on a single server, without revealing any information about the identities of the demanded messages to the server. The goal of the user is to achieve information-theoretic privacy by leveraging the side information about the database. The side information consists of a random subset of $M$ messages in the database which could have been obtained in advance from other users or from previous interactions with the server. The identities of the messages forming the side information are initially unknown to the server. Our goal is to characterize the capacity of this setting, i.e., the maximum achievable download rate. In our previous work, we have established the PIR-SI capacity for the special case in which the user wants a single message, i.e., $D=1$ and showed that the capacity can be achieved through the Partition and Code (PC) scheme. In this paper, we focus on the case when the user wants multiple messages, i.e., $D>1$. Our first result is that if the user wants more messages than what they have as side information, i.e., $D>M$, then the capacity is $\frac{D}{K-M}$, and it can be achieved using a scheme based on the Generalized Reed-Solomon (GRS) codes. In this case, the user must learn all the messages in the database in order to obtain the desired messages. Our second result shows that this may not be necessary when $D\leq M$, and the capacity in this case can be higher. We present a lower bound on the capacity based on an achievability scheme which we call Generalized Partition and Code (GPC).

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arXiv2018

Lifting Private Information Retrieval from Two to any Number of Messages

We study private information retrieval (PIR) on coded data with possibly colluding servers. Devising PIR schemes with optimal download rate in the case of collusion and coded data is still open in general. We provide a lifting operation that can transform what we call one-shot PIR schemes for two messages into schemes for any number of messages. We apply this lifting operation on existing PIR schemes and describe two immediate implications. First, we obtain novel PIR schemes with improved download rate in the case of MDS coded data and server collusion. Second, we provide a simplified description of existing optimal PIR schemes on replicated data as lifted secret sharing based PIR.

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arXiv2018

Guess & Check Codes for Deletions, Insertions, and Synchronization

We consider the problem of constructing codes that can correct $δ$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes, dating back to 1965, are zero-error single deletion $(δ=1)$ correcting codes, and have an asymptotically optimal redundancy. Finding similar codes for $δ\geq 2$ deletions remains an open problem. In this work, we relax the standard zero-error (i.e., worst-case) decoding requirement by assuming that the positions of the $δ$ deletions (or insertions) are independent of the codeword. Our contribution is a new family of explicit codes, that we call Guess & Check (GC) codes, that can correct with high probability up to a constant number of $δ$ deletions (or insertions). GC codes are systematic; and have deterministic polynomial time encoding and decoding algorithms. We also describe the application of GC codes to file synchronization.

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arXiv2018

Private Information Retrieval from MDS Coded Data in Distributed Storage Systems

The problem of providing privacy, in the private information retrieval (PIR) sense, to users requesting data from a distributed storage system (DSS), is considered. The DSS is coded by an $(n,k,d)$ Maximum Distance Separable (MDS) code to store the data reliably on unreliable storage nodes. Some of these nodes can be spies which report to a third party, such as an oppressive regime, which data is being requested by the user. An information theoretic PIR scheme ensures that a user can satisfy its request while revealing, to the spy nodes, no information on which data is being requested. A user can trivially achieve PIR by downloading all the data in the DSS. However, this is not a feasible solution due to its high communication cost. We construct PIR schemes with low download communication cost. When there is $b=1$ spy node in the DSS, we construct PIR schemes with download cost $\frac{1}{1-R}$ per unit of requested data ($R=k/n$ is the code rate), achieving the information theoretic limit for linear schemes. The proposed schemes are universal since they depend on the code rate, but not on the generator matrix of the code. Also, when $b\leq n-δk$, for some $δ\in \mathbb{N^+}$, we construct linear PIR schemes with $cPoP = \frac{b+δk}δ$.

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arXiv2018

Minimizing Latency for Secure Coded Computing Using Secret Sharing via Staircase Codes

We consider the setting of a Master server, M, who possesses confidential data (e.g., personal, genomic or medical data) and wants to run intensive computations on it, as part of a machine learning algorithm for example. The Master wants to distribute these computations to untrusted workers who have volunteered or are incentivized to help with this task. However, the data must be kept private and not revealed to the individual workers. Some of the workers may be stragglers, e.g., slow or busy, and will take a random time to finish the task assigned to them. We are interested in reducing the delays experienced by the Master. We focus on linear computations as an essential operation in many iterative algorithms such as principal component analysis, support vector machines and other gradient-descent based algorithms. A classical solution is to use a linear secret sharing scheme, such as Shamir's scheme, to divide the data into secret shares on which the workers can perform linear computations. However, classical codes can provide straggler mitigation assuming a worst-case scenario of a fixed number of stragglers. We propose a solution based on new secure codes, called Staircase codes, introduced previously by two of the authors. Staircase codes allow flexibility in the number of stragglers up to a given maximum, and universally achieve the information theoretic limit on the download cost by the Master, leading to latency reduction. Under the shifted exponential model, we find upper and lower bounds on the Master's mean waiting time. We derive the distribution of the Master's waiting time, and its mean, for systems with up to two stragglers. For systems with any number of stragglers, we derive an expression that can give the exact distribution, and the mean, of the waiting time of the Master. We show that Staircase codes always outperform classical secret sharing codes.

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arXiv2017

Private Information Retrieval with Side Information

We study the problem of Private Information Retrieval (PIR) in the presence of prior side information. The problem setup includes a database of $K$ independent messages possibly replicated on several servers, and a user that needs to retrieve one of these messages. In addition, the user has some prior side information in the form of a subset of $M$ messages, not containing the desired message and unknown to the servers. This problem is motivated by practical settings in which the user can obtain side information opportunistically from other users or has previously downloaded some messages using classical PIR schemes. The objective of the user is to retrieve the required message without revealing its identity while minimizing the amount of data downloaded from the servers. We focus on achieving information-theoretic privacy in two scenarios: (i) the user wants to protect jointly its demand and side information; (ii) the user wants to protect only the information about its demand, but not the side information. To highlight the role of side information, we focus first on the case of a single server (single database). In the first scenario, we prove that the minimum download cost is $K-M$ messages, and in the second scenario it is $\lceil \frac{K}{M+1}\rceil$ messages, which should be compared to $K$ messages, the minimum download cost in the case of no side information. Then, we extend some of our results to the case of the database replicated on multiple servers. Our proof techniques relate PIR with side information to the index coding problem. We leverage this connection to prove converse results, as well as to design achievability schemes.

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arXiv2017

Robust Private Information Retrieval on Coded Data

We consider the problem of designing PIR scheme on coded data when certain nodes are unresponsive. We provide the construction of $ν$-robust PIR schemes that can tolerate up to $ν$ unresponsive nodes. These schemes are adaptive and universally optimal in the sense of achieving (asymptotically) optimal download cost for any number of unresponsive nodes up to $ν$.

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arXiv2017

Private Information Retrieval Schemes for Coded Data with Arbitrary Collusion Patterns

In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size $t$ colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups.

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arXiv2017

Guess & Check Codes for Deletions and Synchronization

We consider the problem of constructing codes that can correct $δ$ deletions occurring in an arbitrary binary string of length $n$ bits. Varshamov-Tenengolts (VT) codes can correct all possible single deletions $(δ=1)$ with an asymptotically optimal redundancy. Finding similar codes for $δ\geq 2$ deletions is an open problem. We propose a new family of codes, that we call Guess & Check (GC) codes, that can correct, with high probability, a constant number of deletions $δ$ occurring at uniformly random positions within an arbitrary string. The GC codes are based on MDS codes and have an asymptotically optimal redundancy that is $Θ(δ\log n)$. We provide deterministic polynomial time encoding and decoding schemes for these codes. We also describe the applications of GC codes to file synchronization.

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arXiv2017

Minimizing Latency for Secure Distributed Computing

We consider the setting of a master server who possesses confidential data (genomic, medical data, etc.) and wants to run intensive computations on it, as part of a machine learning algorithm for example. The master wants to distribute these computations to untrusted workers who have volunteered or are incentivized to help with this task. However, the data must be kept private (in an information theoretic sense) and not revealed to the individual workers. The workers may be busy, or even unresponsive, and will take a random time to finish the task assigned to them. We are interested in reducing the aggregate delay experienced by the master. We focus on linear computations as an essential operation in many iterative algorithms. A known solution is to use a linear secret sharing scheme to divide the data into secret shares on which the workers can compute. We propose to use instead new secure codes, called Staircase codes, introduced previously by two of the authors. We study the delay induced by Staircase codes which is always less than that of secret sharing. The reason is that secret sharing schemes need to wait for the responses of a fixed fraction of the workers, whereas Staircase codes offer more flexibility in this respect. For instance, for codes with rate $R=1/2$ Staircase codes can lead to up to $40\%$ reduction in delay compared to secret sharing.

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arXiv2016

Staircase Codes for Secret Sharing with Optimal Communication and Read Overheads

We study the communication efficient secret sharing (CESS) problem introduced by Huang, Langberg, Kliewer and Bruck. A classical threshold secret sharing scheme randomly encodes a secret into $n$ shares given to $n$ parties, such that any set of at least $t$, $t n$. We describe two explicit constructions of Staircase codes that achieve minimum communication and read overheads respectively for a fixed $d$, and universally for all possible values of $d, t\leq d\leq n$.

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arXiv2015

Index Coding and Network Coding via Rank Minimization

Index codes reduce the number of bits broadcast by a wireless transmitter to a number of receivers with different demands and with side information. It is known that the problem of finding optimal linear index codes is NP-hard. We investigate the performance of different heuristics based on rank minimization and matrix completion methods, such as alternating projections and alternating minimization, for constructing linear index codes over the reals. As a summary of our results, the alternating projections method gives the best results in terms of minimizing the number of broadcast bits and convergence rate and leads to up to 13% savings in average communication cost compared to graph coloring algorithms studied in the literature. Moreover, we describe how the proposed methods can be used to construct linear network codes for non-multicast networks. Our computer code is available online.

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arXiv2015

Securing Data against Limited-Knowledge Adversaries in Distributed Storage Systems

We study the problem of constructing secure regenerating codes that protect data integrity in distributed storage systems (DSS) in which some nodes may be compromised by a malicious adversary. The adversary can corrupt the data stored on and transmitted by the nodes under its control. The "damage" incurred by the actions of the adversary depends on how much information it knows about the data in the whole DSS. We focus on the limited-knowledge model in which the adversary knows only the data on the nodes under its control. The only secure capacity-achieving codes known in the literature for this model are for the bandwidth-limited regime and repair degree $d=n-1$, i.e., when a node fails in a DSS with $n$ nodes all the remaining $n-1$ nodes are contacted for repair. We extend these results to the more general case of $d\leq n-1$ in the bandwidth-limited regime. Our capacity-achieving scheme is based on the use of product-matrix codes with special hashing functions and allow the identification of the compromised nodes and their elimination from the DSS while preserving the data integrity.

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arXiv2015

Efficient Algorithms for the Data Exchange Problem

In this paper we study the data exchange problem where a set of users is interested in gaining access to a common file, but where each has only partial knowledge about it as side-information. Assuming that the file is broken into packets, the side-information considered is in the form of linear combinations of the file packets. Given that the collective information of all the users is sufficient to allow recovery of the entire file, the goal is for each user to gain access to the file while minimizing some communication cost. We assume that users can communicate over a noiseless broadcast channel, and that the communication cost is a sum of each user's cost function over the number of bits it transmits. For instance, the communication cost could simply be the total number of bits that needs to be transmitted. In the most general case studied in this paper, each user can have any arbitrary convex cost function. We provide deterministic, polynomial-time algorithms (in the number of users and packets) which find an optimal communication scheme that minimizes the communication cost. To further lower the complexity, we also propose a simple randomized algorithm inspired by our deterministic algorithm which is based on a random linear network coding scheme.

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arXiv2014

Synchronizing Edits in Distributed Storage Networks

We consider the problem of synchronizing data in distributed storage networks under an edit model that includes deletions and insertions. We present two modifications of MDS, regenerating and locally repairable codes that allow updates in the parity-check values to be performed with one round of communication at low bit rates and using small storage overhead. Our main contributions are novel protocols for synchronizing both hot and semi-static data and protocols for data deduplication applications, based on intermediary permutation, Vandermonde and Cauchy matrix coding.

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