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Register a new userTopological Coding and Topological Matrices Toward Network Overall Security
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Added by
Bing Yao
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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A mathematical topology with matrix is a natural representation of a coding relational structure that is found in many fields of the world. Matrices are very important in computation of real applications, s ce matrices are easy saved in computer and run quickly, as well as matrices are convenient to deal with communities of current networks, such as Laplacian matrices, adjacent matrices in graph theory. Motivated from convenient, useful and powerful matrices used in computation and investigation of todays networks, we have introduced Topcode-matrices, which are matrices of order $3times q$ and differ from popular matrices applied in linear algebra and computer science. Topcode-matrices can use numbers, letters, Chinese characters, sets, graphs, algebraic groups emph{etc.} as their elements. One important thing is that Topcode-matrices of numbers can derive easily number strings, since number strings are text-based passwords used in information security. Topcode-matrices can be used to describe topological graphic passwords (Topsnut-gpws) used in information security and graph connected properties for solving some problems coming in the investigation of Graph Networks and Graph Neural Networks proposed by GoogleBrain and DeepMind. Our topics, in this article, are: Topsnut-matrices, Topcode-matrices, Hanzi-matrices, adjacency ve-value matrices and pan-Topcode-matrices, and some connections between these Topcode-matrices will be proven. We will discuss algebraic groups obtained from the above matrices, graph groups, graph networking groups and number string groups for encrypting different communities of dynamic networks. The operations and results on our matrices help us to set up our overall security mechanism to protect networks.
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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).
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.
In this paper we introduce Neural Network Coding(NNC), a data-driven approach to joint source and network coding. In NNC, the encoders at each source and intermediate node, as well as the decoder at each destination node, are neural networks which are all trained jointly for the task of communicating correlated sources through a network of noisy point-to-point links. The NNC scheme is application-specific and makes use of a training set of data, instead of making assumptions on the source statistics. In addition, it can adapt to any arbitrary network topology and power constraint. We show empirically that, for the task of transmitting MNIST images over a network, the NNC scheme shows improvement over baseline schemes, especially in the low-SNR regime.
Topological Coding consists of two different kinds of mathematics: topological structure and mathematical relation. The colorings and labelings of graph theory are main techniques in topological coding applied in asymmetric encryption system. Topsnut-gpws (also, colored graphs) have the following advantages: (1) Run fast in communication networks because they are saved in computer by popular matrices rather than pictures. (2) Produce easily text-based (number-based) strings for encrypt files. (3) Diversity of asymmetric ciphers, one public-key corresponds to more private-keys, or more public-keys correspond more private-keys. (4) Irreversibility, Topsnut-gpws can generate quickly text-based (number-based) strings with bytes as long as desired, but these strings can not reconstruct the original Topsnut-gpws. (5) Computational security, since there are many non-polynomial (NP-complete, NP-hard) algorithms in creating Topsnut-gpws. (6) Provable security, since there are many mathematical conjectures (open problems) in graph labelings and graph colorings. We are committed to create more kinds of new Topsnut-gpws to approximate practical applications and antiquantum computation, and try to use algebraic method and Topsnut-gpws to establish graphic group, graphic lattice, graph homomorphism etc.
Lattice-based Cryptography is considered to have the characteristics of classical computers and quantum attack resistance. We will design various graphic lattices and matrix lattices based on knowledge of graph theory and topological coding, since many problems of graph theory can be expressed or illustrated by (colored) star-graphic lattices. A new pair of the leaf-splitting operation and the leaf-coinciding operation will be introduced, and we combine graph colorings and graph labellings to design particular proper total colorings as tools to build up various graphic lattices, graph homomorphism lattice, graphic group lattices and Topcode-matrix lattices. Graphic group lattices and (directed) Topcode-matrix lattices enable us to build up connections between traditional lattices and graphic lattices. We present mathematical problems encountered in researching graphic lattices, some problems are: Tree topological authentication, Decompose graphs into Hanzi-graphs, Number String Decomposition Problem, $(p,s)$-gracefully total numbers.
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