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.

Graphic Lattices and Matrix Lattices Of Topological Coding.

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 today's networks, we have introduced Topcode-matrices, which are matrices of order $3\times q$ 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.

pdf/topological-coding-and-topological-matrices-toward-network-4gxdab9ve5.pdf

Topological Coding and Topological Matrices Toward Network Overall Security

It is always an open, demanding and difficult task for generating available model to simulate dynamical functions and reveal inner principles from complex systems and networks. In this article, due to lots of real-life and artificial networks are built from series of simple and small groups (components), we discuss some interesting and helpful network-operation to generate more realistic network models. In view of community structure (modular topology), we present a class of sparse network models N ( t , m ) . At the moment, we capture the fact the N ( t , 4 ) has not only scale-free feature, which means that the probability that a randomly selected vertex with degree k decays as a power-law, following P ( k ) ∼ k − γ , where γ is the degree exponent, but also small-world property, which indicates that the typical distance between two uniform randomly chosen vertices grows proportionally to logarithm of the order of N ( t , 4 ) , namely, relatively shorter diameter and lower average path length, simultaneously displays higher clustering coefficient. Next, as a new topological parameter correlating to reliability, synchronization capability and diffusion properties of networks, the number of spanning trees over a network is studied in more detail, an exact analytical solution for the number of spanning trees of the N ( t , 4 ) is obtained. Based on the network-operation, part hub-vertex linking with each other will be helpful for structuring various network models and investigating the rules related with real-life networks.

The relations between network-operation and topological-property in a scale-free and small-world network with community structure

Since lattice-based cryptosystem is resisting classical computers and quantum computers, we, by means of graph operations and algebraic technique, will construct graphic lattices, graphic group lattices, Topcode-matrix lattices, matching-type graphic lattices, graphic lattice sequences in this article. Graphic lattices introduced here are various sets of Topsnut-gpws (topological structures plus number theory) in topological coding. We propose a way to realize connecting graphic lattices and traditional lattices. Clearly, these lattices are high-quality network sets, since dynamic graphs are just dynamic networks, meanwhile, these graphic lattices also lead to many new problems in mathematics.

Graphic Lattices for Constructing High-Quality Networks

Graphical passwords (GPWs) are in many areas of the current world.
Topological graphic passwords (Topsnut-gpws) are a new type of cryptography,
and they differ from the existing GPWs. A Topsnut-gpw consists of two parts:
one is a topological structure (graph), and one is a set of discrete elements
(a graph labelling, or coloring), the topological structure connects these
discrete elements together to form an interesting "story". Our idea is to
transform Chinese characters into computer and electronic equipments with touch
screen by speaking, writing and keyboard for forming Hanzi-graphs and
Hanzi-gpws. We will use Hanzigpws to produce text-based passwords (TB-paws). We
will introduce flawed graph labellings on disconnected Hanzi-graphs.

pdf/using-chinese-characters-to-generate-text-based-passwords-2uh09rd0lp.pdf

Using Chinese Characters To Generate Text-Based Passwords For Information Security

Topological graphic passwords (Topsnut-gpws) are one of graph-type passwords, but differ from the existing graphical passwords, since Topsnut-gpws are saved in computer by algebraic matrices We focus on the transformation between text-based passwords (TB-paws) and Topsnut-gpws in this article Several methods for generating TB-paws from Topsnut-gpws are introduced; these methods are based on topological structures and graph coloring/labellings, such that authentications must have two steps: one is topological structure authentication, and another is text-based authentication Four basic topological structure authentications are introduced and many text-based authentications follow Topsnut-gpws Our methods are based on algebraic, number theory and graph theory, many of them can be transformed into polynomial algorithms A new type of matrices for describing Topsnut-gpws is created here, and such matrices can produce TB-paws in complex forms and longer bytes Estimating the space of TB-paws made by Topsnut-gpws is very important for application We propose to encrypt dynamic networks and try to face: (1) thousands of nodes and links of dynamic networks; (2) large numbers of Topsnut-gpws generated by machines rather than human's hands As a try, we apply spanning trees of dynamic networks and graphic groups (Topsnut-groups) to approximate the solutions of these two problems We present some unknown problems in the end of the article for further research

pdf/text-based-passwords-generated-from-topological-graphic-fjhwysmvum.pdf

Text-based Passwords Generated From Topological Graphic Passwords

The dynamical phenomena of complex networks are not easy to model and to characterize by current methods of mathematics. Newman, Barab´ asi and Watts pointed out the direction of re- searching complex networks by graph theory, which was successfully applied in current investigation of complex networks, and Newman's network-based methods have been applied to a variety of field- s. Some well-known and new methods of mathematics are provided in this article, associated with construction and problems on scale-free models from real networks.

/pdf/methods-and-problems-attempt-in-scale-free-models-from-53e4ua3syc.pdf

Methods And Problems Attempt in Scale-Free Models From Complex Networks

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Papers

Topological Coding and Topological Matrices Toward Network Overall Security

Text-based Passwords Generated From Topological Graphic Passwords

Methods And Problems Attempt in Scale-Free Models From Complex Networks

Using Chinese Characters To Generate Text-Based Passwords For Information Security