arXiv subjects

Xuan Guang

Publications and source records attributed to Xuan Guang.

At least 19 recordsLinked to original sources

arXiv2025

Uniquely-Decodable Coding for Zero-Error Network Function Computation

We consider uniquely-decodable coding for zero-error network function computation, where in a directed acyclic graph, the single sink node is required to compute with zero error a target function multiple times, whose arguments are the information sources generated at a set of source nodes. We are interested in the computing capacity from the information theoretic point of view, which is defined as the infimum of the maximum expected number of bits transmitted on all the edges for computing the target function once on average. We first prove some new results on clique entropy, in particular, the substitution lemma of clique entropy for probabilistic graphs with a certain condition. With them, we prove a lower bound on the computing capacity associated with clique entropies of the induced characteristic graphs, where the obtained lower bound is applicable to arbitrary network topologies, arbitrary information sources, and arbitrary target functions. By refining the probability distribution of information sources, we further strictly improve the obtained lower bound. In addition, we compare uniquely-decodable network function-computing coding and fixed-length network function-computing coding, and show that the former indeed outperforms the latter in terms of the computing capacity. Therein, we provide a novel graph-theoretic explanation of the key parameter in the best known bound on the computing capacity for fixed-length network function-computing codes, which would be helpful to improve the existing results.

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arXiv2025

Distributed Source Coding for Compressing Vector-Linear Functions

Inspired by mobile satellite communication systems and the important and prevalent applications of computational tasks, we consider a distributed source coding model for compressing vector-linear functions, which consists of multiple sources, multiple encoders and a decoder linked to all the encoders. Each encoder has access to a certain subset of the sources and the decoder is required to compute with zero error a vector-linear function of the source information, which corresponds to a matrix $T$. The connectivity state between the sources and the encoders and the vector-linear function are all arbitrary. In the paper, we are interested in the function-compression capacity to measure the efficiency of using the system. We first present a general lower bound on the function-compression capacity applicable to arbitrary connectivity states and vector-linear functions. Next, we confine to the nontrivial models with only three sources and no more than three encoders, and prove that all the $3\times2$ column-full-rank matrices $T$ can be divided into two types $T_1$ and $T_2$, for which the function-compression capacities are identical if the matrices $T$ have the same type. We explicitly characterize the function-compression capacities for two most nontrivial models associated with $T_2$ by a novel approach of both upper bounding and lower bounding the size of image sets of encoding functions. This shows that the lower bound thus obtained is not always tight. Rather, by completely characterizing their capacities, the lower bound is tight for all the models associated with $T_1$ and all the models associated with $T_2$ except for the two most nontrivial models. We finally apply the obtained results to network function computation and answer the open problem whether the best known upper bound proved by Guang et. al. (2019) on computing capacity is in general asymptotically tight.

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arXiv2025

Secure Network Function Computation for Linear Functions, Part II: Target-Function Security

In this Part II of a two-part paper, we put forward secure network function computation, where in a directed acyclic network, a sink node is required to compute a target function of which the inputs are generated as source messages at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, is not allowed to obtain any information about a security function of the source messages. In Part I of the two-part paper, we have investigated securely computing linear functions with the wiretapper who can eavesdrop any edge subset up to a certain size r, referred to as the security level, where the security function is the identity function. The notion of this security is called source security. In the current paper, we consider another interesting model which is the same as the above one except that the security function is identical to the target function, i.e., we need to protect the information on the target function from being leaked to the wiretapper. The notion of this security is called target-function security. We first prove a non-trivial upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. In particular, when the security level r is equal to 0, the upper bound reduces to the computing capacity without security consideration. Further, from an algebraic point of view, we prove two equivalent conditions for target-function security and source security for the existence of the corresponding linear function-computing secure network codes. With them, for any linear function over a given finite field, we develop a code construction of linear secure network codes for target-function security and thus obtain a lower bound on the secure computing capacity; and also generalize the code construction developed in Part I for source security.

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arXiv2023

Zero-Error Distributed Function Compression for Binary Arithmetic Sum

In this paper, we put forward the model of zero-error distributed function compression system of two binary memoryless sources X and Y, where there are two encoders En1 and En2 and one decoder De, connected by two channels (En1, De) and (En2, De) with the capacity constraints C1 and C2, respectively. The encoder En1 can observe X or (X,Y) and the encoder En2 can observe Y or (X,Y) according to the two switches s1 and s2 open or closed (corresponding to taking values 0 or 1). The decoder De is required to compress the binary arithmetic sum f(X,Y)=X+Y with zero error by using the system multiple times. We use (s1s2;C1,C2;f) to denote the model in which it is assumed that C1 \geq C2 by symmetry. The compression capacity for the model is defined as the maximum average number of times that the function f can be compressed with zero error for one use of the system, which measures the efficiency of using the system. We fully characterize the compression capacities for all the four cases of the model (s1s2;C1,C2;f) for s1s2= 00,01,10,11. Here, the characterization of the compression capacity for the case (01;C1,C2;f) with C1>C2 is highly nontrivial, where a novel graph coloring approach is developed. Furthermore, we apply the compression capacity for (01;C1,C2;f) to an open problem in network function computation that whether the best known upper bound of Guang et al. on computing capacity is in general tight.

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arXiv2022

Secure Network Function Computation for Linear Functions -- Part I: Source Security

In this paper, we put forward secure network function computation over a directed acyclic network. In such a network, a sink node is required to compute with zero error a target function of which the inputs are generated as source messages at multiple source nodes, while a wiretapper, who can access any one but not more than one wiretap set in a given collection of wiretap sets, is not allowed to obtain any information about a security function of the source messages. The secure computing capacity for the above model is defined as the maximum average number of times that the target function can be securely computed with zero error at the sink node with the given collection of wiretap sets and security function for one use of the network. The characterization of this capacity is in general overwhelmingly difficult. In the current paper, we consider securely computing linear functions with a wiretapper who can eavesdrop any subset of edges up to a certain size r, referred to as the security level, with the security function being the identity function. We first prove an upper bound on the secure computing capacity, which is applicable to arbitrary network topologies and arbitrary security levels. When the security level r is equal to 0, our upper bound reduces to the computing capacity without security consideration. We discover the surprising fact that for some models, there is no penalty on the secure computing capacity compared with the computing capacity without security consideration. We further obtain an equivalent expression of the upper bound by using a graph-theoretic approach, and accordingly we develop an efficient approach for computing this bound. Furthermore, we present a construction of linear function-computing secure network codes and obtain a lower bound on the secure computing capacity.

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arXiv2021

Linear Network Error Correction Coding: A Revisit

We consider linear network error correction (LNEC) coding when errors may occur on edges of a communication network of which the topology is known. In this paper, we first revisit and explore the framework of LNEC coding, and then unify two well-known LNEC coding approaches. Furthermore, by developing a graph-theoretic approach to the framework of LNEC coding, we obtain a significantly enhanced characterization of the error correction capability of LNEC codes in terms of the minimum distances at the sink nodes. In LNEC coding, the minimum required field size for the existence of LNEC codes, in particular LNEC maximum distance separable (MDS) codes which are a type of most important optimal codes, is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of the coding scheme in terms of computational complexity and storage requirement. By applying the graph-theoretic approach, we obtain an improved upper bound on the minimum required field size. The improvement over the existing results is in general significant. The improved upper bound, which is graph-theoretic, depends only on the network topology and requirement of the error correction capability but not on a specific code construction. However, this bound is not given in an explicit form. We thus develop an efficient algorithm that can compute the bound in linear time. In developing the upper bound and the efficient algorithm for computing this bound, various graph-theoretic concepts are introduced. These concepts appear to be of fundamental interest in graph theory and they may have further applications in graph theory and beyond.

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arXiv2020

Vector Gaussian Successive Refinement With Degraded Side Information

We investigate the problem of the successive refinement for Wyner-Ziv coding with degraded side information and obtain a complete characterization of the rate region for the quadratic vector Gaussian case. The achievability part is based on the evaluation of the Tian-Diggavi inner bound that involves Gaussian auxiliary random vectors. For the converse part, a matching outer bound is obtained with the aid of a new extremal inequality. Herein, the proof of this extremal inequality depends on the integration of the monotone path argument and the doubling trick as well as information-estimation relations.

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arXiv2018

Local-Encoding-Preserving Secure Network Coding---Part I: Fixed Security Level

Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. In the current paper (i.e., Part I of the two-part paper), we focus on the problem for a fixed security level and a flexible rate. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. We present an efficient approach for constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same security level and the rate reduced by one. By applying this approach repeatedly, we can obtain a family of local-encoding-preserving SLNCs with a fixed security level and multiple rates. We also develop a polynomial-time algorithm for efficient implementation of this approach. Furthermore, it is proved that the proposed approach incurs no penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. The result in this paper will be used as a building block for efficiently constructing a family of local-encoding-preserving SLNCs for all possible pairs of rate and security level, which will be discussed in the companion paper (i.e., Part II of the two-part paper).

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arXiv2018

Local-Encoding-Preserving Secure Network Coding---Part II: Flexible Rate and Security Level

In the two-part paper, we consider the problem of secure network coding when the information rate and the security level can change over time. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving (LEP) if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. In this paper (Part II), we first consider the design of a family of LEP SLNCs for a fixed rate and a flexible security level. We present a novel and efficient approach for constructing upon an SLNC that exists an LEP SLNC with the same rate and the security level increased by one. Next, we consider the design of a family of LEP SLNCs for a fixed dimension (equal to the sum of rate and security level) and a flexible pair of rate and security level. We propose another novel approach for designing an SLNC such that the same SLNC can be applied for all the rate and security-level pairs with the fixed dimension. Also, two polynomial-time algorithms are developed for efficient implementations of our two approaches, respectively. Furthermore, we prove that both approaches do not incur any penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. Finally, we consider the ultimate problem of designing a family of LEP SLNCs that can be applied to all possible pairs of rate and security level. By combining the construction of a family of LEP SLNCs for a fixed security level and a flexible rate (obtained in Part I) with the constructions of the two families of LEP SLNCs in the current paper in suitable ways, we can obtain a family of LEP SLNCs that can be applied for all possible pairs of rate and security level. Three possible such constructions are presented.

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arXiv2017

Improved Upper Bound on the Network Function Computing Capacity

The problem of network function computation over a directed acyclic network is investigated in this paper. In such a network, a sink node desires to compute with zero error a {\em target function}, of which the inputs are generated at multiple source nodes. The edges in the network are assumed to be error-free and have limited capacity. The nodes in the network are assumed to have unbounded computing capability and be able to perform network coding. The {\em computing rate} of a network code that can compute the target function over the network is the average number of times that the target function is computed with zero error for one use of the network. In this paper, we obtain an improved upper bound on the computing capacity, which is applicable to arbitrary target functions and arbitrary network topologies. This improved upper bound not only is an enhancement of the previous upper bounds but also is the first tight upper bound on the computing capacity for computing an arithmetic sum over a certain non-tree network, which has been widely studied in the literature. We also introduce a multi-dimensional array approach that facilitates evaluation of the improved upper bound. Furthermore, we apply this bound to the problem of computing a vector-linear function over a network. With this bound, we are able to not only enhance a previous result on computing a vector-linear function over a network but also simplify the proof significantly. Finally, we prove that for computing the binary maximum function over the reverse butterfly network, our improved upper bound is not achievable. This result establishes that in general our improved upper bound is non achievable, but whether it is asymptotically achievable or not remains open.

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arXiv2017

Practical Inner Codes for Batched Sparse Codes in Wireless Multihop Networks

Batched sparse (BATS) code is a promising technology for reliable data transmission in multi-hop wireless networks. As a BATS code consists of an outer code and an inner code that typically is a random linear network code, one main research topic for BATS codes is to design an inner code with good performance in transmission efficiency and complexity. In this paper, this issue is addressed with a focus on the problem of minimizing the total number of packets transmitted by the source and intermediate nodes. Subsequently, the problem is formulated as a mixed integer nonlinear programming (MINLP) problem that is NP-hard in general. By exploiting the properties of inner codes and the incomplete beta function, we construct a nonlinear programming (NLP) problem that gives a valid upper bound on the best performance that can be achieved by any feasible solutions. Moreover, both centralized and decentralized real-time optimization strategies are developed. In particular, the decentralized approach is performed independently by each node to find a feasible solution in linear time with the use of look-up tables. Numerical results show that the gap in performance between our proposed approaches and the upper bound is very small, which demonstrates that all feasible solutions developed in the paper are near-optimal with a guaranteed performance bound.

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arXiv2017

Comments on Cut-Set Bounds on Network Function Computation

A function computation problem in directed acyclic networks has been considered in the literature, where a sink node wants to compute a target function with the inputs generated at multiple source nodes. The network links are error-free but capacity-limited, and the intermediate network nodes perform network coding. The target function is required to be computed with zero error. The computing rate of a network code is measured by the average number of times that the target function can be computed for one use of the network, i.e., each link in the network is used at most once. In the papers [1], [2], two cut-set bounds were proposed on the computing rate. However, we show in this paper that these bounds are not valid for general network function computation problems. We analyze the arguments that lead to the invalidity of these bounds and fix the issue with a new cut-set bound, where a new equivalence relation associated with the inputs of the target function is used. Our bound is qualified for general target functions and network topologies. We also show that our bound is tight for some special cases where the computing capacity is known. Moreover, some results in [11], [12] were proved using the invalid upper bound in [1] and hence their correctness needs further justification. We also justify their validity in the paper.

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arXiv2016

Alphabet Size Reduction for Secure Network Coding: A Graph Theoretic Approach

We consider a communication network where there exist wiretappers who can access a subset of channels, called a wiretap set, which is chosen from a given collection of wiretap sets. The collection of wiretap sets can be arbitrary. Secure network coding is applied to prevent the source information from being leaked to the wiretappers. In secure network coding, the required alphabet size is an open problem not only of theoretical interest but also of practical importance, because it is closely related to the implementation of such coding schemes in terms of computational complexity and storage requirement. In this paper, we develop a systematic graph-theoretic approach for improving Cai and Yeung's lower bound on the required alphabet size for the existence of secure network codes. The new lower bound thus obtained, which depends only on the network topology and the collection of wiretap sets, can be significantly smaller than Cai and Yeung's lower bound. A polynomial-time algorithm is devised for efficient computation of the new lower bound.

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arXiv2016

Secure Network Coding over Small Fields

In the paradigm of network coding, the information-theoretic security problem is encountered in the presence of a wiretapper, who has capability of accessing an unknown channel-subset in communication networks. In order to combat this eavesdropping attack, secure network coding is introduced to prevent information from being leaked to adversaries. For any construction of secure linear network codes (SLNCs) over a wiretap network, the based field size is a very important index, because it largely determines the computational and space complexities, and further the efficiency of network transmission. Further, it is also very important for the process of secure network coding from theoretical research to practical applications. In the present paper, we proposed new lower bounds on the field size for constructing SLNCs, which shows that the field size can be reduced considerably without giving up any security and capacity. In addition, since the obtained lower bounds depend on network topology, how to efficiently determine them is another very important and attractive problem for explicit constructions of SLNCs. Motivated by a desire to develop methods with low complexity and empirical behavior to solve this problem, we propose efficient approaches and present a series of algorithms for implementation. Subsequently, the complexity of our algorithms is analyzed, which shows that they are polynomial-time.

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arXiv2015

On the Optimality of Secure Network Coding

In network communications, information transmission often encounters wiretapping attacks. Secure network coding is introduced to prevent information from being leaked to adversaries. The investigation of performance bounds on the numbers of source symbols and random symbols are two fundamental research problems. For an important case that each wiretap-set with cardinality not larger than $r$, Cai and Yeung proposed a coding scheme, which is optimal in the senses of maximizing the number of source symbols and at the same time minimizing the number of random symbols. In this letter, we further study achievable lower bound on the number of random key and show that it just depends on the security constraint, and particularly, is independent to the information amount for transmission. This implies that when the number of transmitted source message changes, we can't reduce the number of random key to keep the same security level. We further give an intuitive interpretation on our result. In addition, a similar construction of secure linear network codes is proposed, which achieves this lower bound on the number of random key no matter how much information is transmitted. At last, we also extend our result to imperfect security case.

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arXiv2015

Repairable Threshold Secret Sharing Schemes

In this paper, we propose a class of threshold secret sharing schemes with repairing function between shares without the help of the dealer, that we called repairable threshold secret sharing schemes. Specifically, if a share fails, such as broken or lost, it will be repaired just by some other shares. A construction of such repairable threshold secret sharing schemes is designed by applying linearized polynomials and regenerating codes in distributed storage systems. In addition, a new repairing rate is introduced to characterize the performance and efficiency of the repairing function. Then an achievable upper bound on the repairing rate is derived, which implies the optimality of the repair and describes the security between different shares. Under this optimality of the repair, we further discuss traditional information rate and also indicate its optimality, that can describe the efficiency of secret sharing schemes in the aspect of storage. Finally, by applying the minimum bandwidth regenerating (MBR) codes, our construction designs repairable threshold secret sharing schemes achieving both optimal repairing and information rates simultaneously.

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arXiv2014

Small Field Size for Secure Network Coding

In network coding, information transmission often encounters wiretapping attacks. Secure network coding is introduced to prevent information from being leaked to adversaries. For secure linear network codes (SLNCs), the required field size is a very important index, because it largely determines the computational and space complexities of a SLNC, and it is also very important for the process of secure network coding from theoretical research to practical application. In this letter, we further discuss the required field size of SLNCs, and obtain a new lower bound. This bound shows that the field size of SLNCs can be reduced further, and much smaller than the known results for almost all cases.

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arXiv2014

Distributed Storage over Unidirectional Ring Networks

In this paper, we study distributed storage problems over unidirectional ring networks, whose storage nodes form a directed ring and data is transmitted along the same direction. The original data is distributed to store on these nodes. Each user can connect one and only one storage node to download the total data. A lower bound on the reconstructing bandwidth to recover the original data for each user is proposed, and it is achievable for arbitrary parameters. If a distributed storage scheme can achieve this lower bound with equality for every user, we say it an optimal reconstructing distributed storage scheme (ORDSS). Furthermore, the repair problem for a failed storage node in ORDSSes is under consideration and a tight lower bound on the repair bandwidth is obtained. In particular, we indicate the fact that for any ORDSS, every storage node can be repaired with repair bandwidth achieving the lower bound with equality. In addition, we present two constructions for ORDSSes of arbitrary parameters, called MDS construction and ED construction, respectively. Particularly, ED construction, by using the concept of Euclidean division, is more efficient by our analysis in detail.

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