arXiv subjects

Wentu Song

Publications and source records attributed to Wentu Song.

30 records · Page 2Linked to original sources

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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arXiv2015

On the Solvability of 3s/nt Sum-Network---A Region Decomposition and Weak Decentralized Code Method

We study the network coding problem of sum-networks with 3 sources and n terminals (3s/nt sum-network), for an arbitrary positive integer n, and derive a sufficient and necessary condition for the solvability of a family of so-called terminal-separable sum-network. Both the condition of terminal-separable and the solvability of a terminal-separable sum-network can be decided in polynomial time. Consequently, we give another necessary and sufficient condition, which yields a faster (O(|E|) time) algorithm than that of Shenvi and Dey ([18], (O(|E|^3) time), to determine the solvability of the 3s/3t sum-network. To obtain the results, we further develop the region decomposition method in [22], [23] and generalize the decentralized coding method in [21]. Our methods provide new efficient tools for multiple source multiple sink network coding problems.

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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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arXiv2014

Network Coding for $3$s$/n$t Sum-Networks

A sum-network is a directed acyclic network where each source independently generates one symbol from a given field $\mathbb F$ and each terminal wants to receive the sum $($over $\mathbb F)$ of the source symbols. For sum-networks with two sources or two terminals, the solvability is characterized by the connection condition of each source-terminal pair [3]. A necessary and sufficient condition for the solvability of the $3$-source $3$-terminal $(3$s$/3$t$)$ sum-networks was given by Shenvi and Dey [6]. However, the general case of arbitrary sources/sinks is still open. In this paper, we investigate the sum-network with three sources and $n$ sinks using a region decomposition method. A sufficient and necessary condition is established for a class of $3$s$/n$t sum-networks. As a direct application of this result, a necessary and sufficient condition of solvability is obtained for the special case of $3$s$/3$t sum-networks.

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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

Encoding Complexity of Network Coding with Two Simple Multicast Sessions

The encoding complexity of network coding for single multicast networks has been intensively studied from several aspects: e.g., the time complexity, the required number of encoding links, and the required field size for a linear code solution. However, these issues as well as the solvability are less understood for networks with multiple multicast sessions. Recently, Wang and Shroff showed that the solvability of networks with two unit-rate multicast sessions (2-URMS) can be decided in polynomial time. In this paper, we prove that for the 2-URMS networks: $1)$ the solvability can be determined with time $O(|E|)$; $2)$ a solution can be constructed with time $O(|E|)$; $3)$ an optimal solution can be obtained in polynomial time; $4)$ the number of encoding links required to achieve a solution is upper-bounded by $\max\{3,2N-2\}$; and $5)$ the field size required to achieve a linear solution is upper-bounded by $\max\{2,\lfloor\sqrt{2N-7/4}+1/2\rfloor\}$, where $|E|$ is the number of links and $N$ is the number of sinks of the underlying network. Both bounds are shown to be tight.

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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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arXiv2012

Error Correction for Cooperative Data Exchange

This paper considers the problem of error correction for a cooperative data exchange (CDE) system, where some clients are compromised or failed and send false messages. Assuming each client possesses a subset of the total messages, we analyze the error correction capability when every client is allowed to broadcast only one linearly-coded message. Our error correction capability bound determines the maximum number of clients that can be compromised or failed without jeopardizing the final decoding solution at each client. We show that deterministic, feasible linear codes exist that can achieve the derived bound. We also evaluate random linear codes, where the coding coefficients are drawn randomly, and then develop the probability for a client to withstand a certain number of compromised or failed peers and successfully deduce the complete message for any network size and any initial message distributions.

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arXiv2012

Exchanging Third-Party Information with Minimum Transmission Cost

In this paper, we consider the problem of minimizing the total transmission cost for exchanging channel state information. We proposed a network coded cooperative data exchange scheme, such that the total transmission cost is minimized while each client can decode all the channel information held by all other clients. In this paper, we first derive a necessary and sufficient condition for a feasible transmission. Based on the derived condition, there exists a feasible code design to guarantee that each client can decode the complete information. We further formulate the problem of minimizing the total transmission cost as an integer linear programming. Finally, we discuss the probability that each client can decode the complete information with distributed random linear network coding.

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