arXiv subjects

Son Hoang Dau

Publications and source records attributed to Son Hoang Dau.

At least 37 records · Page 2Linked to original sources

arXiv2023

$k$-server Byzantine-Resistant PIR Scheme with Optimal Download Rate and Optimal File Size

We consider the problem of designing a Private Information Retrieval (PIR) scheme on $m$ files replicated on $k$ servers that can collude or, even worse, can return incorrect answers. Our goal is to correctly retrieve a specific message while keeping its identity private from the database servers. We consider the asymptotic information-theoretic capacity of this problem defined as the maximum ratio of the number of correctly retrieved symbols to the downloaded one for a large enough number of stored files. We propose an achievable scheme with a small file size and prove that such a file size is minimal for the fixed number of retrieved symbols, solving the problem pointed out by Banawan and Ulukus.

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arXiv2022

Efficient Hamiltonian Reduction for Quantum Annealing on SatCom Beam Placement Problem

Beam Placement (BP) is a well-known problem in Low-Earth Orbit (LEO) satellite communication (SatCom) systems, which can be modelled as an NP-hard clique cover problem. Recently, quantum computing has emerged as a novel technology which revolutionizes how to solve challenging optimization problems by formulating Quadratic Unconstrained Binary Optimization (QUBO), then preparing Hamiltonians as inputs for quantum computers. In this paper, we study how to use quantum computing to solve BP problems. However, due to limited hardware resources, existing quantum computers are unable to tackle large optimization spaces. Therefore, we propose an efficient Hamiltonian Reduction method that allows quantum processors to solve large BP instances encountered in LEO systems. We conduct our simulations on real quantum computers (D-Wave Advantage) using a real dataset of vessel locations in the US. Numerical results show that our algorithm outperforms commercialized solutions of D-Wave by allowing existing quantum annealers to solve 17.5 times larger BP instances while maintaining high solution quality. Although quantum computing cannot theoretically overcome the hardness of BP problems, this work contributes early efforts to applying quantum computing in satellite optimization problems, especially applications formulated as clique cover/graph coloring problems.

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arXiv2022

Practical Considerations in Repairing Reed-Solomon Codes

The issue of repairing Reed-Solomon codes currently employed in industry has been sporadically discussed in the literature. In this work we carry out a systematic study of these codes and investigate important aspects of repairing them under the trace repair framework, including which evaluation points to select and how to implement a trace repair scheme efficiently. In particular, we employ different heuristic algorithms to search for low-bandwidth repair schemes for codes of short lengths with typical redundancies and establish three tables of current best repair schemes for $[n, k]$ Reed-Solomon codes over GF(256) with $4 \leq n \leq 16$ and $r = n - k \in \{2,3,4\}$. The tables cover most known codes currently used in the distributed storage industry.

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arXiv2016

Latent Network Features and Overlapping Community Discovery via Boolean Intersection Representations

We propose a new latent Boolean feature model for complex networks that captures different types of node interactions and network communities. The model is based on a new concept in graph theory, termed the Boolean intersection representation of a graph, which generalizes the notion of an intersection representation. We mostly focus on one form of Boolean intersection, termed cointersection, and describe how to use this representation to deduce node feature sets and their communities. We derive several general bounds on the minimum number of features used in cointersection representations and discuss graph families for which exact cointersection characterizations are possible. Our results also include algorithms for finding optimal and approximate cointersection representations of a graph.

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arXiv2015

Erasure codes with symbol locality and group decodability for distributed storage

We introduce a new family of erasure codes, called group decodable code (GDC), for distributed storage system. Given a set of design parameters {α; β; k; t}, where k is the number of information symbols, each codeword of an (α; β; k; t)-group decodable code is a t-tuple of strings, called buckets, such that each bucket is a string of βsymbols that is a codeword of a [β; α] MDS code (which is encoded from αinformation symbols). Such codes have the following two properties: (P1) Locally Repairable: Each code symbol has locality (α; β-α+ 1). (P2) Group decodable: From each bucket we can decode αinformation symbols. We establish an upper bound of the minimum distance of (α; β; k; t)-group decodable code for any given set of {α; β; k; t}; We also prove that the bound is achievable when the coding field F has size |F| > n-1 \choose k-1.

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arXiv2015

Local Codes with Addition Based Repair

We consider the complexities of repair algorithms for locally repairable codes and propose a class of codes that repair single node failures using addition operations only, or codes with addition based repair. We construct two families of codes with addition based repair. The first family attains distance one less than the Singleton-like upper bound, while the second family attains the Singleton-like upper bound.

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arXiv2015

Weakly Secure MDS Codes for Simple Multiple Access Networks

We consider a simple multiple access network (SMAN), where $k$ sources of unit rates transmit their data to a common sink via $n$ relays. Each relay is connected to the sink and to certain sources. A coding scheme (for the relays) is weakly secure if a passive adversary who eavesdrops on less than $k$ relay-sink links cannot reconstruct the data from each source. We show that there exists a weakly secure maximum distance separable (MDS) coding scheme for the relays if and only if every subset of $\ell$ relays must be collectively connected to at least $\ell+1$ sources, for all $0 < \ell < k$. Moreover, we prove that this condition can be verified in polynomial time in $n$ and $k$. Finally, given a SMAN satisfying the aforementioned condition, we provide another polynomial time algorithm to trim the network until it has a sparsest set of source-relay links that still supports a weakly secure MDS coding scheme.

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arXiv2015

Locally Encodable and Decodable Codes for Distributed Storage Systems

We consider the locality of encoding and decoding operations in distributed storage systems (DSS), and propose a new class of codes, called locally encodable and decodable codes (LEDC), that provides a higher degree of operational locality compared to currently known codes. For a given locality structure, we derive an upper bound on the global distance and demonstrate the existence of an optimal LEDC for sufficiently large field size. In addition, we also construct two families of optimal LEDC for fields with size linear in code length.

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arXiv2014

Secure Erasure Codes With Partial Decodability

The MDS property (aka the $k$-out-of-$n$ property) requires that if a file is split into several symbols and subsequently encoded into $n$ coded symbols, each being stored in one storage node of a distributed storage system (DSS), then an user can recover the file by accessing any $k$ nodes. We study the so-called $p$-decodable $μ$-secure erasure coding scheme $(1 \leq p \leq k - μ, 0 \leq μ< k, p | (k-μ))$, which satisfies the MDS property and the following additional properties: (P1) strongly secure up to a threshold: an adversary which eavesdrops at most $μ$ storage nodes gains no information (in Shannon's sense) about the stored file, (P2) partially decodable: a legitimate user can recover a subset of $p$ file symbols by accessing some $μ+ p$ storage nodes. The scheme is perfectly $p$-decodable $μ$-secure if it satisfies the following additional property: (P3) weakly secure up to a threshold: an adversary which eavesdrops more than $μ$ but less than $μ+p$ storage nodes cannot reconstruct any part of the file. Most of the related work in the literature only focused on the case $p = k - μ$. In other words, no partial decodability is provided: an user cannot retrieve any part of the file by accessing less than $k$ nodes. We provide an explicit construction of $p$-decodable $μ$-secure coding schemes over small fields for all $μ$ and $p$. That construction also produces perfectly $p$-decodable $μ$-secure schemes over small fields when $p = 1$ (for every $μ$), and when $μ= 0, 1$ (for every $p$). We establish that perfect schemes exist over \emph{sufficiently large} fields for almost all $μ$ and $p$.

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arXiv2014

On Block Security of Regenerating Codes at the MBR Point for Distributed Storage Systems

A passive adversary can eavesdrop stored content or downloaded content of some storage nodes, in order to learn illegally about the file stored across a distributed storage system (DSS). Previous work in the literature focuses on code constructions that trade storage capacity for perfect security. In other words, by decreasing the amount of original data that it can store, the system can guarantee that the adversary, which eavesdrops up to a certain number of storage nodes, obtains no information (in Shannon's sense) about the original data. In this work we introduce the concept of block security for DSS and investigate minimum bandwidth regenerating (MBR) codes that are block secure against adversaries of varied eavesdropping strengths. Such MBR codes guarantee that no information about any group of original data units up to a certain size is revealed, without sacrificing the storage capacity of the system. The size of such secure groups varies according to the number of nodes that the adversary can eavesdrop. We show that code constructions based on Cauchy matrices provide block security. The opposite conclusion is drawn for codes based on Vandermonde matrices.

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arXiv2014

On the Existence of MDS Codes Over Small Fields With Constrained Generator Matrices

We study the existence over small fields of Maximum Distance Separable (MDS) codes with generator matrices having specified supports (i.e. having specified locations of zero entries). This problem unifies and simplifies the problems posed in recent works of Yan and Sprintson (NetCod'13) on weakly secure cooperative data exchange, of Halbawi et al. (arxiv'13) on distributed Reed-Solomon codes for simple multiple access networks, and of Dau et al. (ISIT'13) on MDS codes with balanced and sparse generator matrices. We conjecture that there exist such $[n,k]_q$ MDS codes as long as $q \geq n + k - 1$, if the specified supports of the generator matrices satisfy the so-called MDS condition, which can be verified in polynomial time. We propose a combinatorial approach to tackle the conjecture, and prove that the conjecture holds for a special case when the sets of zero coordinates of rows of the generator matrix share with each other (pairwise) at most one common element. Based on our numerical result, the conjecture is also verified for all $k \leq 7$. Our approach is based on a novel generalization of the well-known Hall's marriage theorem, which allows (overlapping) multiple representatives instead of a single representative for each subset.

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arXiv2013

Optimal Locally Repairable Linear Codes

Linear erasure codes with local repairability are desirable for distributed data storage systems. An [n, k, d] code having all-symbol (r, δ})-locality, denoted as (r, δ)a, is considered optimal if it also meets the minimum Hamming distance bound. The existing results on the existence and the construction of optimal (r, δ)a codes are limited to only the special case of δ = 2, and to only two small regions within this special case, namely, m = 0 or m >= (v+δ-1) > (δ-1), where m = n mod (r+δ-1) and v = k mod r. This paper investigates the existence conditions and presents deterministic constructive algorithms for optimal (r, δ)a codes with general r and δ. First, a structure theorem is derived for general optimal (r, δ)a codes which helps illuminate some of their structure properties. Next, the entire problem space with arbitrary n, k, r and δ is divided into eight different cases (regions) with regard to the specific relations of these parameters. For two cases, it is rigorously proved that no optimal (r, δ)a could exist. For four other cases the optimal (r, δ)a codes are shown to exist, deterministic constructions are proposed and the lower bound on the required field size for these algorithms to work is provided. Our new constructive algorithms not only cover more cases, but for the same cases where previous algorithms exist, the new constructions require a considerably smaller field, which translates to potentially lower computational complexity. Our findings substantially enriches the knowledge on (r, δ)a codes, leaving only two cases in which the existence of optimal codes are yet to be determined.

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arXiv2013

Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures

The min-rank of a graph was introduced by Haemers (1978) to bound the Shannon capacity of a graph. This parameter of a graph has recently gained much more attention from the research community after the work of Bar-Yossef et al. (2006). In their paper, it was shown that the min-rank of a graph G characterizes the optimal scalar linear solution of an instance of the Index Coding with Side Information (ICSI) problem described by the graph G. It was shown by Peeters (1996) that computing the min-rank of a general graph is an NP-hard problem. There are very few known families of graphs whose min-ranks can be found in polynomial time. In this work, we introduce a new family of graphs with efficiently computed min-ranks. Specifically, we establish a polynomial time dynamic programming algorithm to compute the min-ranks of graphs having simple tree structures. Intuitively, such graphs are obtained by gluing together, in a tree-like structure, any set of graphs for which the min-ranks can be determined in polynomial time. A polynomial time algorithm to recognize such graphs is also proposed.

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arXiv2013

Optimal Index Codes with Near-Extreme Rates

The min-rank of a digraph was shown by Bar-Yossef et al. (2006) to represent the length of an optimal scalar linear solution of the corresponding instance of the Index Coding with Side Information (ICSI) problem. In this work, the graphs and digraphs of near-extreme min-ranks are characterized. Those graphs and digraphs correspond to the ICSI instances having near-extreme transmission rates when using optimal scalar linear index codes. In particular, it is shown that the decision problem whether a digraph has min-rank two is NP-complete. By contrast, the same question for graphs can be answered in polynomial time. Additionally, a new upper bound on the min-rank of a digraph, the circuit-packing bound, is presented. This bound is often tighter than the previously known bounds. By employing this new bound, we present several families of digraphs whose min-ranks can be found in polynomial time.

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arXiv2013

Parity Declustering for Fault-Tolerant Storage Systems via $t$-designs

Parity declustering allows faster reconstruction of a disk array when some disk fails. Moreover, it guarantees uniform reconstruction workload on all surviving disks. It has been shown that parity declustering for one-failure tolerant array codes can be obtained via Balanced Incomplete Block Designs. We extend this technique for array codes that can tolerate an arbitrary number of disk failures via $t$-designs.

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arXiv2013

Delay Minimization in Varying-Bandwidth Direct Multicast with Side Information

We study the delay minimization in a direct multicast communication scheme where a base station wishes to transmit a set of original packets to a group of clients. Each of the clients already has in its cache a subset of the original packets, and requests for all the remaining packets. The base station communicates directly with the clients by broadcasting information to them. Assume that bandwidths vary between the station and different clients. We propose a method to minimize the total delay required for the base station to satisfy requests from all clients.

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arXiv2013

Balanced Sparsest Generator Matrices for MDS Codes

We show that given $n$ and $k$, for $q$ sufficiently large, there always exists an $[n, k]_q$ MDS code that has a generator matrix $G$ satisfying the following two conditions: (C1) Sparsest: each row of $G$ has Hamming weight $n - k + 1$; (C2) Balanced: Hamming weights of the columns of $G$ differ from each other by at most one.

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arXiv2011

Error Correction for Index Coding with Side Information

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $δ$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $α$-bound and the $κ$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another implicit bound on the length of an optimal linear ECIC. Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC. Finally, the notion of static ECIC, which is suitable for use with a family of instances of an index coding problem, is introduced.

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