Asymmetric topological matrix overall encryption technology of local area network

Abstract

In order to integrally encrypt a local area network and provide an asymmetric topological password capable of resisting attack of a quantum computer for a post-quantum cryptography family, a topological matrix of the invention is composed of two different mathematical fields of'topological structure 'and'mathematical constraint', a topological coding moment diagram and the topological coding matrix are input into a computer, and the topological coding moment diagram and the topological coding matrix are input into the computer; a full-coloring topological coding matrix, an edge coloring adjacent matrix, a full-coloring adjacent matrix, a parameter full-coloring topological coding matrix and a generic topological coding matrix are designed. Wherein the edge coloring adjacent matrix and the full coloring adjacent matrix provide unique identity signatures, the topological matrix can export massive character strings for encryption and decryption, and the character strings cannot be traced back to the original topological matrix, so that all practical algorithms using the technology of the invention have calculation security and provable security. The graph group and topological matrix group overall coloring technology generates a feasible and effective encryption algorithm, thereby implementing asymmetric overall encryption on the local area network.

Description

[0001] 1. Technical field: Information encryption

[0002] 2. Background technology: topological coding, graph theory, algebra 3. Summary of the invention:

[0003] It has been mathematically proven that classical public key cryptographic algorithms will be broken by quantum computers, and the attack of artificial intelligence with quantum computing will crack most of the passwords that currently ensure the security of our digital communications. In order to cope with the severe challenges brought by the advancement of quantum computer technology to information security, scholars are committed to building encryption technology that resists quantum computing in the future quantum computer era, also known as post-quantum cryptography, so that algorithms that replace classical public key cryptographic algorithms can effectively resist attacks from quantum computers.

[0004] As we all know, matrix application algorithms are widely used in the field of science and technology. The patented "asymmetric topological matrix encryption technology" of this invention has the following main technical features:

[0005] ●Based on various topological matrices, the topological coding diagram is input into the computer for digital processing; the electronic files are encrypted and decrypted using the topological character strings derived from various topological matrices.

[0006] ●Asymmetric topological matrix encryption belongs to the discipline of topological coding, which is a new technology and new field of information science. Topological coding is a branch of mathematics.

[0007] ●The mathematical theory of asymmetric topological matrix encryption technology comes from graph theory, number theory, set theory, algebra, abstract algebra, linear algebra, probability theory, computer science and other disciplines.

[0008] ●The fully colored adjacent matrix and topological coding matrix in the topological matrix can generate a topological coding graph.

[0009] ●Topological coding graphs and topological matrices are composed of two different mathematical fields: "topological structure" and "mathematical constraints". Each topological coding graph naturally becomes a "topological signature". Topological signatures have functions such as identity authentication, generating various forms of strings, and encrypting files. Topological signature verification is performed between colored graphs, adjacent matrices, and topological coding matrices or various strings generated by them. Various forms of topological strings are generated through various matrices of topological coding graphs.

[0010] ● Practical algorithms designed using asymmetric topological matrix encryption technology are computationally secure and provably secure. Not only do they not require proof every time, they can also provide key technologies for authentication, confidentiality, integrity, and non-repudiation.

[0011] Based on the asymmetric topological matrix encryption technology, this invention patent can provide an asymmetric topological cipher that approximates "one-time pad", and provide an asymmetric topological cipher that can resist quantum computer attacks for the post-quantum cipher family. IV. DESCRIPTION OF THE DRAWINGS:

[0012] Figure - 1. A topological coding graph explaining the adjacent matrix, edge-colored adjacent matrix, fully-colored adjacent matrix, and fully-colored topological coding matrix. Each topological coding graph L i admits an odd-graceful total coloring f i (i ∈ [1, 7]).

[0013] Figure - 2. (a) Graph Y 1 = K 4 P 3 admits 704 distinct graceful labelings; (b) Graph Y 2 = K 5 P 2 admits a unique graceful labeling; (c) Graph Y 3 = B(5, 2, 2) admits a unique graceful labeling. (Karen Petrie and Barbara Smith, 2003)

[0014] Figure - 3. The topological structures of the topological coding graphs L 1 , L 2 ,..., L 7 in Figure - 1, which are pairwise non-isomorphic.

[0015] Figure - 4. Four Chinese character coding graphs T 1 , T 2 , T 3 , T 4 and their union graph T = T 1 ∪T 2 ∪T 3 ∪T 4 .

[0016] Figure - 5. Ten labelings of the Chinese character graph of "five".

[0017] Figure - 6. Four labelings of the Chinese character graph of "five".

[0018] Figure - 7. (a) A simple directed topological coding graph with an edge-difference magic total labeling f (b) An uncolored directed graph with multiple arcs and self-loops

[0019] Figure - 8. An example of an infinite hybrid topological coding graph group.

[0020] Figure - 9. For each integer k ∈ [2, 17], there is a color-preserving graph homomorphism And the topological coding graph G k Form a Chinese character group.

[0021] Figure 10: For each integer k∈[18,25] and the topological coding graph G in Figure 9 1 , each topological coding graph G k is a tree with color-preserving graph isomorphism

[0022] Figure 11, (b) Star Tree K 1，m Consider a set of ordered graceful labelings with 12 leaves; (a) the caterpillar tree T and (c) the caterpillar tree H both have 12 leaves.

[0023] Figure 12. An example of a growing algorithm to explain the set-ordered graceful labeling of a caterpillar tree. A caterpillar tree T with set-ordered graceful full coloring 16 Isomorphic to the uncolored caterpillar tree T in Figure 11. V. Specific implementation methods:

[0024] The specific implementation method is divided into 6 parts: (1) basic concepts, terms, theorems; (2) advantages of asymmetric topological matrix encryption technology; (3) topological matrix; (4) matching topological matrix; (5) topological matrix group; (6) encryption algorithm based on topological matrix. We use definitions to express objects and concepts, theorems and corollaries to express theories, and algorithms to express practical technologies.

[0025] 1. Basic concepts, terms, and theorems

[0026] For simplicity and accuracy, we introduce the following terms and concepts: The number of elements in a set S is denoted by |S|. For non-negative integers a<b, the notation [a, b] represents the set of integers {a, a+1, ..., b}, and the notation [2a, 2b] represents the set of integers {a, a+1, ..., b}. e is the set of even integers {2a, 2(a+1), ..., 2b}, denoted [2a+1, 2b+1] o represents the set of odd integers {2a+1, 2a+3, ..., 2b+1}. The "graph" used here refers to the graph in graph theory, and the "string" represents a string of numbers or text.

[0027] A tree is a connected graph that does not contain cycles. A graph with vertex or edge coloring is called a topologically encoded graph, also called a topological signature. A topologically encoded graph G is said to admit a W-constrained coloring if its coloring satisfies a special constraint W.

[0028] The vertex set of the bipartite graph G is V(G) = X∪Y, and Let the two endpoints u and v of each edge uv∈E(G) of the bipartite graph G satisfy u∈X and v∈Y. Let the bipartite graph G admit a W-constrained coloring h: →M, where the subset The coloring set of the elements of is denoted as h(S) = {h(w): w∈S}. If the vertex shading set satisfies

[0029]

[0030] We call this W-constrained coloring h of the even graph G a set-ordered W-constrained coloring.

[0031] A T is the transposed matrix of matrix A. The matrix with m rows and n columns is denoted as M m×n =(A 1 , A 2 , ..., A m ) T =(B 1 , B 2 , ..., B n ), where the row vector A i =(a i，1 , a i，2 , ..., a i，n )(i∈[1,m]), column vector B j =(a 1，j , a 2，j , ..., a m，j ) T (j∈[1,n]). Coefficient matrix I m×n =(a i，j ) m×n Each element a i，j = 1. The zero matrix whose elements are all zero is denoted by O m×n The order "m×n" in the lower corner of the matrix is ​​sometimes omitted if there is no confusion.

[0032] Definition 1: The adjacency matrix of a graph G with p vertices is defined as A(G)=(a i，j ) p×p , when edge x i y j ∈E(G)(i≠j), then there is a i，j =1, otherwise a i，j = 0. Since the adjacency matrix is ​​a symmetric matrix, the adjacency matrix of graph G can be written as

[0033]

[0034] Among them, each row vector A i =(a i，1 , ai，2 , ..., a i，p )(i∈[1,p]).

[0035] Example 1: The adjacency matrix of the topological coding graph L in Figure 1 is A(L)=(A 1 , A 2 , ..., A 10 ) T , the 10 row vectors are

[0036] A 1 =(0,0,0,0,0,0,0,1,1,1),A 2 =(0,0,0,0,0,0,0,0,1,0),A 3 =(0,0,0,0,0,0,1,1,1,0),

[0037] A 4 =(0,0,0,0,0,1,1,1,0,0),A 5 =(0,0,0,0,0,1,1,0,0,0),A 6 =(0,0,0,1,1,0,0,0,0,0),

[0038] A 7 =(0,0,1,1,1,0,0,0,0,0),A 8 =(1,0,1,1,0,0,0,0,0,0),A 9 =(1,1,1,0,0,0,0,0,0,0),

[0039] A 10 =(1,0,0,0,0,0,0,0,0,0).

[0040] In Figure 1, except for the adjacent matrix A(L) and the adjacent matrix A(L 1 ) are similar, any other two topological coding graphs L j and L i (i, j∈[1,7) are not identical, and their corresponding adjacent matrices all satisfy A(L j ) is not similar to A(L i )(i≠j), that is, there is no invertible matrix Q such that A(L j )=QA(L i )Q -1 Established.

[0041] Theorem 1. (1) For a graph G, there are two adjacency matrices A(G) and B(G). Then the adjacency matrix A(G) is similar to the adjacency matrix B(G), that is, there exists a reversible matrix Q such that A(G) = QB(G)Q -1 Established.

[0042] (2) If the graph G is not structured in the same way as the graph H, then for any reversible matrix P, A(H)≠PA(G)P -1 .

[0043] According to Theorem 1, in the sense of similar adjacency matrices, a graph has a one-to-one correspondence with its own adjacency matrix. The mathematical concepts, coloring, and labeling techniques used here are all taken from the following literature:

[0044] [1] JABondy, USRMurty.Graph Theory.Springer London, 2008.D0I: 10.1007 / 978-1-84628-970-5.

[0045] [2]Joseph A. Gallian. A Dynamic Survey of Graph Labeling. The electronic journal of combinatorics, #DS6, Twenty-fourth edition, December 9, 2021. (576 pages, 3028 references, over 200 graph numbers)

[0046] [3] Bing Yao, Hongyu Wang. Recent Colorings And Labelings In Topological Coding. arXiv:2106.15254v1[cs.IT]29 Jun 2021.1-247. https: / / doi.org / 10.48550 / arXiv.2106.15254. (247 pages, 123 references, more than 150 figure numbers and colorings)

[0047] [4] Bing Yao, Meimei Zhao, Xiaohui Zhang, Yarong Mu, Yirong Sun, Mingjun Zhang, Sihua Yang, Fei Ma, Jing Su, Xiaomin Wang, Hongyu Wang, Hui Sun. Topological Coding and Topological Matrices Towards Network Overall Security. arXiv:1909.01587v2 [cs.IT] 15Sep2019.1-29. https: / / doi.org / 10.48550 / arXiv.1909.01587

[0048] (II) Advantages of asymmetric topology matrix encryption technology

[0049] To explain the characteristics of asymmetric topology matrix encryption technology, let's look at a simple example. According to Figure 1, A and B-k (k∈[1,7])) represent communication users in a local area network.

[0050] Example 2: In asymmetric topological matrix encryption technology, A sends the public key topological signature L in Figure 1 to B-k k (k∈[1,7]), save the private key topological signature L and the private key adjacent matrix A(L) by yourself 10×10 Each public key topological signature L k (k∈[1,7]) corresponds to its own public key adjacent matrix A(L 1 ) 10×10 , A(L 2 ) 12×12 , A(L 3 ) 13×13 , A(L 4 ) 12×12 , A(L 5 ) 14×14 , A(L 6 ) 16×16 and A(L 7 ) 14×14 (See definition 1).

[0051] A's private key topological signature L recognizes a set of ordered odd graceful labels f and has its own adjacent matrix A(L) 10×10 , edge-colored adjacency matrix E adj *L, f) 10×10 (see Definition 2) and the full coloring topological encoding matrix T code (L, f) 3×12 (See Definition 4). The full coloring topological encoding matrix Tcode (L, f) 3×12 Can export (36)! (>2 107 ) different digital strings s i =c i，1 c i，2 ... 1 c i，36 =s i，pri ∪s i，pub (i∈[1,(36)!]), where s i，pri =c i，1 c i，2 ...c i，18 is the private key digital string, s i，pub =c i，19 c i，20 ... 1 c i，36 Is a public key digital string, or a private key digital string s i，pri =c i，1 c i，3 ... 1 c i，35 and the public key string s i，pub =c i，2 c i，4 ...c i，36 . A will string the numbers s i，pri Send the public key string s to B-k as your own private key string i，pub . The full coloring topological encoding matrix T code (L, f) 3×12 Is the private key digital string s i，pri and the public key string s i，pub Topology authentication A uth [s i，pub ,s i，pri ]. Topological authentication verifies that: the private key topological signature L recognizes a set of ordered odd graceful labels f, so that no two vertices of the private key topological signature L have the same coloring, and each edge x i y j ∈E(L) satisfies the coloring f(x i y j )=f(y j )-f(x i )>0, and the edge coloring set of L is f(E(L))=[1,12].

[0052] There are (100)! algorithms that can generate the adjacent matrix A(L) of L from the topological signature of A's private key. 10×10 Export (100)! (≈2 524 ) public key digital strings. For each positive integer j∈[1,(100)! ], each algorithm θ j From the adjacency matrix A(L)10×10 Export a public key digital string a j =r j，1 r j，2 ...r j，100 , algorithm θ j Color the adjacency matrix E from the edges adj (L, f) 10×10 Export private key digital string b j =t j，1 t j，2 ...t j，100 , and the public key digital string a j and private key string b j The order of arrangement is the same, and their topological authentication A uth [a j , b j ] is a shading transformation based on the adjacency matrix E adj (L, f) 10×10 =f[A(L) 10×10 ].

[0053] B-k (k∈[1,7]) uses the public key digital string s sent by A i，pub and public key string a j To encrypt a plaintext F, and send the double-encrypted ciphertext to A. After receiving the ciphertext, A uses the private key digital string s i，pri and private key string b j To decrypt the ciphertext, when the adjacent matrix A(L k ) 12×12 Topology authentication A uth [a j , b j ] After (confirming the identity of B-k, according to Theorem 1, this topological signature is unique), the full coloring topological encoding matrix T code (L, f) 3×12 Topology authentication A uth [s i，pub ,s i，pri ] is passed (also called topological digital authentication), and A decrypts the ciphertext.

[0054] From the above examples, we can summarize the advantages of asymmetric topology matrix encryption technology as follows:

[0055] (1) Asymmetric topological matrix encryption is a dual encryption system: one is a topological coding map (topological signature) with identity authentication function, and the other is topological string encryption.

[0056] (2) The topological coding diagram of asymmetric topological matrix encryption relies on the matrix to be stored in the computer, and the calculation is fast and accurate. The topological matrix can also be used to derive digital strings and character strings for making key pairs, which is beneficial to the implementation of algorithms for practical applications. The larger the order of various topological matrices, the longer the byte length of the derived character string, and the higher the calculation security.

[0057] (3) In the field of graph theory, there are a large number of graphs with complex topological structures. There is no polynomial algorithm that can draw all mutually isomorphic graphs with a specified number of vertices, because it has been proven that subgraph isomorphism is an NP-complete problem. The number of graphs with 23 vertices G 23 and the number of 24-vertex graphs G 24 They are as follows:

[0058] G 23 =559946939699792080597976380819462179812276348458981632~2 179

[0059] G 24 =195704906302078447922174862416726256004122075267063365754368~2 197

[0060] Note that the factorial number (12 2 )! =(144)!≈2 829 . Online report: The estimated number of stars in the universe is 3.10 22 ~2 74 , the number of atoms in the solar system is about 10 57 ~2 189 atoms.

[0061] (4) There are thousands of W-constraint colorings and W-constraint labelings in graph theory, and new W-constraint colorings and W-constraint labelings are being created every day. For a graph and a special W-constraint coloring, there is no report that there is a polynomial algorithm that can give all the coloring schemes of the graph with respect to this W-constraint coloring. For a positive integer q, Sheppard proved (1976): There are q! graceful graphs with q edges, among which The graceful graphs correspond to different graceful notations of the same graceful graph. Figure 2 gives examples of the complexity of graceful graph coloring and graceful graph notation.

[0062] (5) Since a graph allows multiple colorings and labelings, it results in a graph corresponding to multiple topological coding matrices (see Definition 4), and each topological coding matrix derives a huge number of strings.

[0063] (6) Unlike the adjacency matrix, a topological coding matrix corresponds to multiple topological coding graphs with different topological structures, see the examples in Figures 1 and 3.

[0064] (7) Problems in graph theory such as point clusters, feedback point sets, vertex cover problems, subgraph isomorphism, Hamiltonian path problems, and graph planarity are all NP-complete problems. The topological structure of the topological matrix is ​​closely related to these graph structures and is the cornerstone of the provable security of the technology of the present invention.

[0065] (8) The discipline of graph theory contains long-standing open problems such as the graceful tree conjecture, edge magic total coloring conjecture, odd graceful tree conjecture, total coloring conjecture, vertex coloring conjecture, and uniform total coloring conjecture. These conjectures are all related to NP-type problems and are the guarantee of the computational security of the technology of the present invention.

[0066] (9) The topological coding matrix in the topological coding graph can quickly generate digital strings and character strings. However, the character strings are not related to the topological structure, that is, the inverse process of tracing back from the character strings and reconstructing the original topological coding graph does not exist.

[0067] (10) To decipher the topological coding graph, it is necessary to cross at least two different mathematical fields: “topological structure” and “mathematical constraints”. Some studies have reported that even AI attacks equipped with quantum computers cannot cross two different mathematical fields to cause effective damage.

[0068] (11) Since Chinese characters are natural topological graphs, spoken and written Chinese characters can be quickly converted into topological coding graphs and topological coding graph matrices (see Figure 1, Figure 3, Figure 4, Figure 5, Figure 6, and Figure 9), which lays a foundation for a wide range of applications for the patent of this invention.

[0069] (12) Asymmetric topological matrix encryption can achieve data compressibility, data anti-modification, data irreversibility, ultra-long bit string keys, and has the potential to resist quantum computing, or become a member of the post-quantum cryptography family (see the notes on Definition 6).

[0070] In summary, the present invention is not only practical and convenient, but also computationally secure and provably secure, so that a practical algorithm designed using the technology of the present invention does not need to prove its computationally secure and provably secure every time.

[0071] (III) Topological Matrix

[0072] Matrices associated with topological structures are collectively called topological matrices. The adjacency matrix in Definition 1 is one of the topological matrices.

[0073] (3.1) Edge-colored adjacent matrix, fully colored adjacent matrix

[0074] Definition 2: Let the dual graph G = (X, Y) have p vertices and q edges, and its vertex set is V(G) = X∪Y, satisfying X={u i :i∈[1,s]},Y={v j :j∈[1,t]}, the number of vertices of the bipartite graph G is p=s+t(s=|X|,t=|Y|). The bipartite graph G=(X,Y) admits a set-ordered W-constrained total coloring h, satisfying

[0075] (i) Set ordered constraints: max{h(u i ):u i ∈X}=max h(X)<min h(Y)=min{h(v j ):v j ∈Y};

[0076] (ii) W-constraint: Each edge u i v j ∈E(H) satisfies the W-constraint, that is, h(u i v j )=W[h(u i ), h(v j )];

[0077] (iii) Vertex coloring matrix: X - The vertex coloring matrix is ​​M(X, h) = (h(u 1 ), h(u 2 ), ..., h(u s )) 1×s , the Y-vertex shading matrix is ​​M(Y,h)=(h(v 1 ), h(v 2 ), ..., h(v t )) 1×t ; Overall vertex shading matrix

[0078] M(V(G),h)=M(X,h)∪M(Y,h)=(h(u 1 ), h(u 2 ), ..., h(u s ), h(v 1 ), h(v 2 ), ..., h(v t )) 1×p

[0079] (iv) Edge coloring matrix: The edge coloring matrix E of the bipartite graph G M (G, h) = (h(a i，j )) s×t It is defined as: if there is an edge u i vj ∈E(G)(i≠j, i∈[1,s], j∈[1,t]), then define h(a i，j )=h(u i v j ); otherwise h(a i , j )=0.

[0080] We obtain the following adjacency matrix:

[0081] (1) The edge-colored adjacency matrix E of the bipartite graph G = (X, Y) adj (G, h) is defined as

[0082]

[0083] (2) The fully colored adjacency matrix of the bipartite graph G = (X, Y) is defined as

[0084]

[0085] (3) The subtotal coloring adjacency matrix of the bipartite graph G = (X, Y) is defined as

[0086]

[0087] Note that the sub-total coloring adjacent matrix It is small in size, not only has a one-to-one correspondence with the even graph G, but also fully reflects the topological structure and W-constrained total coloring h of the even graph G.

[0088] Example 3: The edge-colored adjacent matrix E corresponding to the topological coding graph L in Figure 1 adj (L, f) is

[0089]

[0090] Among them, the edge coloring adjacency matrix E adj The 10 row vectors of (L, f) are

[0091] B 1 =(0,0,0,0,0,0,0,10,11,12),B 2 =(0,0,0,0,0,0,0,0,9,0),B 3 =(0,0,0,0,0,0,5,7,8,0),B 4 =(0,0,0,0,0,2,4,6,0,0),B 5 =(0,0,0,0,0,1,3,0,0,0),B 6 =(0,0,0,2,1,0,0,0,0,0),B 7=(0,0,5,4,3,0,0,0,0,0),B 8 =(10,0,7,6,0,0,0,0,0,0),B 9 =(11,9,8,0,0,0,0,0,0,0),B 10 =(12,0,0,0,0,0,0,0,0,0)

[0092] Since the edge-colored adjacency matrix E adj (L, f) is the matrix obtained by coloring the adjacent matrix A(L) with W-constrained total coloring f, so it can be written as the topological matrix coloring transformation E adj (L, f) = f[A(L)] Edge-colored adjacent matrix E adj (L, f) can be derived (10 2 )! (≈2 524 ) different digital strings, which are different from the digital strings derived from the adjacent matrix A(L). According to the topological coding graph L in Figure 1, we get the matrix:

[0093]

[0094] According to Definition 2, the edge-colored adjacent matrix E of the topological coding graph L in Figure 1 adj (L, f) can be abbreviated as

[0095]

[0096] Example 4: Due to the topological coding graph L in Figure 1 1 The fully colored adjacent matrix T adj (L 1 , f 1 )for

[0097]

[0098] Among them, the fully colored adjacent matrix T adj (L 1 , f 1 ) are the 11 row vectors of

[0099] C 1 = (0, 0, 4, 6, 8, 10, 11, 15, 19, 21, 23), C 2 = (0, 0, 0, 0, 0, 0, 0, 0, 19, 21, 23), C 3 =(4,0,0,0,0,0,0,0,0,17,0),C 4 =(6,0,0,0,0,0,0,9,13,15,0),C 5=(8,0,0,0,0,0,3,7,11,0,0),C 6 =(10,0,0,0,0,0,1,5,0,0,0),C 7 =(11,0,0,0,3,1,0,0,0,0,0),C 8 =(15,0,0,9,7,5,0,0,0,0,0),C 9 = (19, 19, 0, 13, 11, 0, 0, 0, 0, 0, 0), C 10 = (21, 21, 17, 15, 0, 0, 0, 0, 0, 0, 0), C 11 =(23,23,0,0,0,0,0,0,0,0,0).

[0100] The topological coding graph L in Figure 1 1 The corresponding sub-total coloring adjacent matrix is

[0101]

[0102] According to Definition 2, the topologically encoded bigraph L 1 The fully colored adjacent matrix T adj (L 1 , f 1 ) can be written as

[0103]

[0104] Among them, the matrix M(V(L 1 ), f 1 )=(0,4,6,8,10,11,15,19,21,23) is the topological coding graph L 1 The total vertex coloring matrix, the total coloring adjacent matrix T adj (L 1 , f 1 ) can be exported (11 2 )! Different digital strings. Note that the topological coding diagram L in Figure 1 1 , L 2 , ...L 7 They are not mutually isomorphic, so their corresponding edge-colored adjacent matrices and fully-colored adjacent matrices are not similar to each other.

[0105] Suppose that the topological coding graph G that allows full coloring h has two edge-coloring adjacent matrices E adj (G, h) and E′ adj (G, h), according to Theorem 1, there exists a reversible matrix Q such that E′ adj (G, h) = QE adj (G,h))Q -1If it holds, then the edge-colored adjacent matrix E′ is called adj (G, h) is similar to the edge-colored adjacency matrix E adj (G, h)), and vice versa. Under the constraint of similarity matrices, a topologically coded graph that allows full coloring always has a one-to-one correspondence with its own edge-colored adjacent matrix. Definition 2 can be generalized to general topologically coded graphs.

[0106] Definition 3: Let the topological coding graph H with p vertices admit a W-constrained total coloring f, so that the total vertex coloring matrix M(V(H), f) 1×p =(f(u 1 ), f(u 2 ), ..., f(u p )), then the fully colored adjacency matrix of the topological coding graph H is defined as

[0107]

[0108] Edge-colored adjacency matrix E adj (H, f) = (f(b i，j )) p×p Defined as: opposite side u i u j ∈E(H)(i≠j),f(b i，j )=f(u i u j ); otherwise f(b i，j )=0.

[0109] In Definition 3, since the edge-colored adjacent matrix E adj (H, f) is the matrix obtained by coloring the elements of the adjacent matrix A(H) using the W-constrained full coloring f, so it can be written as the topological matrix coloring transformation E adj (H, f) p×p =f[A(H)] p×p ,as well as

[0110]

[0111] Theorem 2: According to Cayley's theorem in graph theory, the complete graph K n The fully colored adjacent matrix T adj (K n , f) can be derived n n-2 different calibrated spanning trees T s The fully colored adjacency matrix

[0112]

[0113] Among them, each f s (s∈[1,n n-2 ]) is the calibration support tree Ts Full coloring of.

[0114] (3.2) Fully colored topological encoding matrix

[0115] Definition 4: Suppose a topological coding graph G with q edges admits a W-constrained total coloring f: V(G)∪E(G)→[a, b], and the edge set of the topological coding graph G is E(G)={e i =x i y j : i∈[1,q]}. Matrix

[0116]

[0117] It is called the full coloring topological coding matrix of the topological coding graph G, where the vertex coloring vector X f =(f(x 1 ), f(x 2 ), ..., f(x q )) and the vertex shading vector Y f =(f(y 1 ), f(y 2 ), ..., f(y q )), and the edge coloring vector E f =(f(e 1 ), f(e 2 ), ..., f(e q )); each edge e i ∈E(G) satisfies the W-constraint f(e i )=W[f(x i ), f(y i )](i∈[1,q]).

[0118] (3.3) Parameterized fully colored topological encoding matrix

[0119] Define the unit total coloring topological encoding matrix as So that v-vector X 0 =(0, 0, ..., 0) 1×q and v-vector Y 0 =(1, 1, ..., 1) 1×q , and the e-vector E 0 =(1, 1, ..., 1) 1×q .

[0120] Definition 5. Let a q-edge bipartite graph G = (X, Y) admit a set-ordered W-constrained total coloring h: V(G) ∪ E(G) → [a, b], such that the two vertex coloring sets satisfy the set-ordered constraint max h(X) < min h(Y) (see formula (1)). Then the parametric total coloring topological encoding matrix of the graph G is defined as

[0121]

[0122] Here k and d are both non-negative integers, T code (G, h) 3×q is the total coloring topological coding matrix of the topologically coded bigraph G.

[0123] Example 5: The even graph T in Figure 4 1 Admit a set ordered odd graceful label h 1 , edge coloring set h 1 (E(T 1 )) = [1, 9] o , even graph T 1 The parameterized full coloring topological encoding matrix is

[0124]

[0125] Among them, the parameter full coloring topological coding matrix in formula (16) determines the topological coding bigraph T 1 A parameter full coloring F of .

[0126] (3.4) General topological encoding matrix

[0127] Definition 6. [4] A topological encoding matrix is ​​defined as

[0128]

[0129] Where X = (x 1 , x 2 , ..., x q ) and Y=(y 1 ,y 2 , ..., y q ) is called the v-vector, E=(e 1 , e 2 , ..., e q ) is called the e-vector, and q is the topological encoding matrix T code The scale of e k 、x k and k (k∈[1,q]) are all non-negative integers, x k and k They are called e k If there is a function W such that e k =W(x k ,y k )(k∈[1,q]), then the topological coding matrix T code Satisfy W-constraints. Each group of x k , ek ,y k Construct the topological encoding matrix T code The meta-coding matrix (x k , e k ,y k ) T , then there is a meta-coding matrix and operation

[0130] Notes on Definition 6:

[0131] (1) If the topological encoding matrix T in Definition 6 code is the topological coding matrix of a topological coding graph, then this matrix T code is a graph topological encoding matrix.

[0132] (2) If max{x 1 , x 2 , ..., x q}<min{y 1 ,y 2 , ..., y q}, then the topological coding matrix T code is a set-ordered W-constrained topological encoding matrix.

[0133] (3) The notation (XY)* is the v-vector X=(x 1 , x 2 , ..., x q ) and v-vector Y=(y 1 ,y 2 , ..., y q ) is a set of mutually different components, the symbol E* is the e-vector E = (e 1 , e 2 , ..., e q ) is a set of distinct components.

[0134] (4) Meta-coding matrix (x k , e k ,y k ) T and the meta-coding matrix (y k , e k , x k ) T equal.

[0135] (5) For every number w∈(XY)*, there is a matrix If from the topological encoding matrix T code Remove the matrix D egree (w), so that the remaining matrix T code \D egree If there is no number w in (w), then Degree (w) 3×d is the degree encoding matrix, d is the degree of w, denoted as d = d egree (w). Extreme value min{d egree (w): w∈(XY)*} and max{d egree (w): w∈(XY)*} are called topological encoding matrices T code The minimum and maximum degrees of code ) and Δ(T code ).

[0136] (6) Definitions 4 and 5 are special cases of Definition 6.

[0137] (7) General topological coding matrix class. Define the topological coding matrix T in 6. code The element x i ,y i and e i Replace them with:

[0138] ① Topological coding diagram of graph collection and The resulting matrix is ​​called the graph coloring topology encoding matrix;

[0139] ②String of string collection and The resulting matrix is ​​called the string-colored topological encoding matrix;

[0140] ③Matrix of matrix sets and The resulting matrix is ​​called a matrix-colored topological encoding matrix (similar to a tensor matrix);

[0141] ④Vector of vector set and The resulting matrix is ​​called the vector coloring topology encoding matrix;

[0142] ⑤The elements of a set are sets and The resulting matrix is ​​called a set-coloring topological encoding matrix, or a hypergraph-coloring topological encoding matrix;

[0143] ⑥ General things that are a collection of things and The resulting matrix is ​​called the object shading matrix.

[0144] Theorem 3: Let every number u i ∈(XY)*={u 1 ,u 2 , ..., u p} appears in the topological encoding matrix T code The number of times in is di . Then the topological coding matrix T code The graphability problem can be reduced to the degree sequence graphability problem in graph theory: the sequence d = (d 1 , d 2 , ..., d p )(d i ≥d i+1 , i∈[1,p-1]) is the degree sequence of a graph G with p vertices if and only if is an even number and satisfies

[0145] (3.5) Structural and colored topological coding matrices

[0146] Using the meta-coding matrix in Definition 6, we can characterize some structural topological coding matrices.

[0147] (1) Perfect match. If there is a meta-encoding matrix So that any two element encoding matrices (x i , e i ,y i ) T and (y j , e j , x j ) T There is no common endpoint, and 2m = |(XY)*|, then T code There is a perfect matching sub-topology encoding matrix T*, called T code is the perfect matching topology encoding matrix.

[0148] (2) Complete topological encoding matrix. If any number x i ∈(XY)* corresponds to a number y j ∈(XY)*\{x i}, so that x i and j It is a certain e ij ∈E*, that is, the topological encoding matrix T code Contains the meta-coding matrix (x i , e ij ,y j ) T , and p=|(XY)*| and 2q=p(p-1), then T code is the complete graph topology encoding matrix.

[0149] (3) Link sub-topology coding matrix and circle topology coding matrix. If there is a meta-coding matrix Make y i =x i+1 (i∈[1,m-1]), then T codeThere is a link sub-topology encoding matrix T*, x 1 is called the starting point of the link, y m is called the end point of the link; if there is x in the union T* of the element coding matrices 1 =y m , then the topological coding matrix T code Contains the circle topology encoding matrix T*; if any pair of different w i , w j ∈(XY)* are T code The starting and ending points of a link sub-topology encoding matrix, then T code is the connected topological encoding matrix; if T code There is a circle topology encoding matrix T* that contains every number in the set (XY)*, then it is called T code is the Hamiltonian topological encoding matrix.

[0150] (4) Support sub-topological coding matrix. If the topological coding matrix T code Contains a connected sub-topology encoding matrix So that each number in (XY)* is a certain element encoding matrix (x i , e i ,y i ) T e in i The endpoint of T* is called T code If the supporting sub-topology encoding matrix T* does not contain the circle topology encoding matrix, T* is called T code The linear support subtopology encoding matrix of .

[0151] Definition 7: Coloring type fully colored topological coding matrix. Suppose the 3×q-order topological coding matrix T in Definition 6 is code , and |(XY)^*|=p, we have the following coloring constraints:

[0152] Constraint -1.(XY)*=[0,p-1], p≤q+1.

[0153] Constraint-2. (XY)*=[0, p-1], |(XY)*|=p=q+1.

[0154] Constraints - 3.

[0155] Constraints - 4.

[0156] Constraints - 5.

[0157] Constraint-6. (XY)*∪E*=[1, p+q].

[0158] Constraint-7.E*=[1,2q-1] o .

[0159] Constraint-8. (XY)*=[1, p], E*=[p+1, p+q].

[0160] Constraint-9. Set ordered constraints max{x i :i∈[1,q]}<min{y j :j∈[1,q]}, then T code is the set-ordered topological encoding matrix.

[0161] Constraint-10.e i =|x i -y i |, i∈[1,q], E*=[1,q].

[0162] Constraint-11.e i =|x i -y i |,i∈[1,q],E*=[1,2q-1] o .

[0163] Constraint-12.e i =x i +y i (mod 2q), i∈[1, q], E*=[1, 2q-1] o .

[0164] Constraint-13.e i =x i +y i (mod 2q), i∈[1, q], E*=[0, q-1].

[0165] Constraint-14. There exists a constant k such that x i +e i +y i =k,i∈[1,q].

[0166] Constraint-15. There exists a constant k such that e i +|x i -y i |=k,i∈[1,q].

[0167] Constraint-16. There exists a constant k such that ||x i -y i |-e i |=k,i∈[1,q].

[0168] Constraint-17. There exists a constant k such that |x i +yi -e i |=k,i∈[1,q].

[0169] According to the above constraints, we get the following W-constrained topological encoding matrix T code :

[0170] Coloring matrix-1. When constraints-2 and constraints-10 hold, W-constraint = linear grace.

[0171] Coloring matrix-2. When constraints-2, constraints-9 and constraints-10 hold, W-constraint = set ordered linear graceful.

[0172] Coloring matrix-3. When constraints-3 and constraints-10 hold, W-constraint = graceful.

[0173] Coloring matrix-4. When constraints-3, constraints-9 and constraints-10 hold, W-constraints = set ordered gracefully.

[0174] Coloring matrix-5. When constraints-4 and constraints-11 hold, W-constraint = odd-graceful.

[0175] Coloring matrix-6. When constraints-4, constraints-9 and constraints-11 hold, W-constraint = set ordered odd graceful.

[0176] Coloring matrix-7. When constraints-4, constraints-7 and constraints-14 are met, W-constraint = odd-edge magic full.

[0177] Coloring matrix-8. When constraints-4, constraints-7, constraints-9 and constraints-14 are established, W-constraint = set ordered odd edge magic full.

[0178] Coloring Matrix-9. When constraint-1 and constraint-13 hold, W-constraint = elegant.

[0179] Coloring Matrix-10. When constraint-1, constraint-9 and constraint-13 hold, W-constraint = set ordered elegant.

[0180] Coloring Matrix-11. When Constraint-4 and Constraint-12 hold, W-Constraint = Odd Elegant.

[0181] Coloring matrix-12. When constraints-4, constraints-9 and constraints-12 hold, W-constraint = set ordered odd elegant.

[0182] Coloring Matrix-13. When Constraint-3 and Constraint-13 hold, W-Constraint = Harmony.

[0183] Coloring Matrix-14. When constraint-3, constraint-9 and constraint-13 hold, W-constraint = set ordered and harmonious.

[0184] Coloring matrix-15. When constraint-4, constraint-7 and {|ab|:a, b∈(XY)*}=[1,|(XY)*|] hold, W-constraint=perfect odd-elegant.

[0185] Coloring matrix-16. When constraint-6 and constraint-14 are established, W-constraint = edge magic full.

[0186] Coloring Matrix-17. When Constraint-8 and Constraint-14 are established, W-Constraint = Super Edge Magic All.

[0187] Coloring matrix-18. When constraints-6, constraints-9 and constraints-14 are established, W-constraint = set ordered edge magic full.

[0188] Coloring matrix-19. When constraint-6 and constraint-15 are met, W-constraint = edge difference.

[0189] Coloring matrix-20. When constraints-6, constraints-9 and constraints-15 are established, W-constraint = set ordered edge difference.

[0190] Coloring matrix-21. When constraint-6 and constraint-16 are established, W-constraint = graceful difference.

[0191] Coloring matrix-22. When constraints-6, constraints-9 and constraints-16 hold, W-constraint = set ordered graceful difference.

[0192] Coloring Matrix-23. When constraint-6 and constraint-17 hold, W-constraint = happiness difference.

[0193] Coloring matrix-24. When constraint-6, constraint-9 and constraint-17 hold, W-constraint = set ordered happiness difference.

[0194] Note to Definition 7: If the topological encoding matrix T of the topological encoding graph G code (G) is a W-constrained topological encoding matrix in Definition 7, then the topological encoding graph G is said to admit a W-constrained label h, because every edge uv∈E(G) satisfies h(u)≠h(v).

[0195] Example 6: According to Definition 7, the topological coding graph W in Figure 5 k Corresponding to a W-constrained topological encoding matrix T code (W k , f k )(k∈[1,5]), then: k=1, W-constraint = set ordered graceful; k=2, W-constraint = set ordered happiness difference; k=3, W-constraint = set ordered edge magic; k=4, W-constraint = set ordered edge difference; k=5, W-constraint = set ordered graceful difference. Further, there is a matrix coloring transformation T code (W i+1, f i+1 )=θ i (T code (W i , f i ))(i∈[1,4]), and the matrix coloring transformation θ i The expression is as follows:

[0196] θ 1 :f 2 (y j ) = f 1 (y j ), f 2 (x i y j ) = f 1 (x i y j ), f 2 (x i )=4-f 1 (x i );

[0197] θ 2 :f 3 (y j ) = f 2 (y j ), f 3 (x i y j )=11-f 2 (x i y j ), f 3 (x i ) = f 2 (x i );

[0198] θ 4 :f 4 (y j ) = f 3 (y j ), f 4 (x i y j ) = f 3 (x i y j ), f 4 (x i )=4-f 3 (x i );

[0199] θ 5 :f 5 (y j )=15-f 4 (y j ), f5 (x i y j ) = f 4 (x i y j ), f 5 (x i ) = f 4 (x i ).

[0200] Topological coding diagram D in Figure 5 s Corresponding to a W-constrained topological encoding matrix T code (D s ,h s )(s∈[1,5]), then: s=1, W-constraint=set ordered odd graceful; s=2, W-constraint=set ordered odd edge happiness difference; s=3, W-constraint=set ordered odd edge magic full; s=4, W-constraint=set ordered odd edge edge difference; s=5, W-constraint=set ordered odd edge graceful difference. Furthermore, there exists a matrix coloring transformation T code (D j+1 ,h j+1 )=θ j (T code (D j ,h j ))(j∈[1,4]), where the matrix coloring transformation θ j The expression is as follows:

[0201] θ 1 :h 2 (y j )=h 1 (y j ), h 2 (x i y j )=h 1 (x i y j ), h 2 (x i )=8-h 1 (x i ).

[0202] θ 2 :h 3 (y j )=h 2 (y j ), h 3 (x i y j )=20-h 2 (x i y j ), h 3 (x i)=h 2 (x i ).

[0203] θ 3 :h 4 (y j )=h 3 (y j ), h 4 (x i y j )=h 3 (x i y j ), h 4 (x i )=8-h 3 (x i ).

[0204] θ 4 :h 5 (y j )=28-h 4 (y j ), h 5 (x i y j )=h 4 (x i y j ), h 5 (x i )=h 4 (x i ).

[0205] In addition, there are h 1 (x i )=2f 1 (x i ), h 1 (y j )=2f 1 (y j )-1,h 1 (x i y j )=2f 1 (x i y j )-1=2[f 1 (y j )-f 1 (x i )]=h 1 (y j )-h 1 (x i ).

[0206] Each topological coding graph Q in Figure 6 i Corresponding topological encoding matrix T code(Q i , α i )(i∈[1,6]), then

[0207] Topologically encoded graph Q 1 Admit an elegant label 1 , satisfying γ 1 (uv) = γ 1 (u)+γ 1 (v)(uv∈E(Q 1 )).

[0208] Topologically encoded graph Q 2 Admit a harmonious label γ 2 , satisfying γ 2 (uv) = γ 2 (u)+γ 2 (v)(mod 10)(uv∈E(Q 2 )).

[0209] Topologically encoded graph Q 3 Admit a strange elegant label γ 3 , satisfying γ 3 (uv) = γ 3 (u)+γ 3 (v)(uv∈E(Q 3 )).

[0210] Topologically encoded graph Q 4 Admit an odd-harmonic label γ 4 , satisfying γ 4 (uv) = γ 4 (u)+γ 4 (v)(mod 20)(uv∈E(Q 4 )).

[0211] Theorem 4: If the 3×q-order topological coding matrix T in Definition 6 code is the set-ordered linear graceful topological encoding matrix in Definition 7, then there is a transformation β k , so that the matrix β k (T code ) is each coloring type matrix -k (k∈[3,24]) in Definition 7.

[0212] (3.6) Directed topological encoding matrix

[0213] Definition 8. [4] A directed topological encoding matrix is ​​defined as

[0214]

[0215] Wherein, the head vector X=(x 1 , x2 , ..., x q ) and tail vector Y = (y 1 ,y 2 , ..., y q ), arc vector So that each arc e i The head is x i , and the tail is y i (i∈[1,q]), number q is the directed topological encoding matrix The scale of is a directed meta-topological encoding matrix.

[0216] Note that the directed element topological encoding matrix Cannot be written Figure 7 shows a directed topological coding diagram This graph corresponds to a directed topological encoding matrix In the directed graph theory of graph theory, the out-degree of a directed arc is marked as "+" and the in-degree is marked as "-". For example, the directed topological coding graph The vertex x 5 There is out-degree d + (x 5 )=5 and in-degree d - (x 5 )=0; vertex x 8 Has in-degree d - (x 8 )=3 and out-degree d + (x 8 )=0, etc. The adjacency matrix of an undirected graph is symmetric, but in general, the adjacency matrix of a directed graph is not symmetric.

[0217] Example 7: Uncolored directed graph The directed adjacent matrix of is

[0218]

[0219] Observe Figure 7(b) and the directed adjacent matrix Vertex v 1 The out-degree d + (v 1 )=3 and in-degree d - (v 1 )=1; vertex v 2 The out-degree d + (v 2 )=1 and in-degree d - (v 2 )=3; vertex v 3 The out-degree d + (v 3 )=1 and in-degree d -(v 3 )=3; vertex v 4 The out-degree d + (v 4 )=1 and in-degree d - (v 4 )=1.

[0220] Directed Topological Coding Graph The directed adjacency matrix of The row vector is

[0221] A 1 =(0,0,0,0,0,0,0,0,0,0,0,0,0),A 1 =A 2 =A 6 =A 8 =A 9 =A 10 =A 11 =A 12 ,

[0222] A 4 =(0,0,0,0,0,0,0,1,0,0,0,0,0),A 5 =(1,1,0,0,0,1,0,0,1,1,0,0,0),

[0223] A 7 =(0,0,0,0,0,1,0,1,0,0,1,1,0),A 13 =(0,0,0,0,0,0,0,1,0,0,0,0,0,0).

[0224] In practical applications, we can use the directed topological coding matrix in (20) Export the following string

[0225] s 1 =251131322423313144232351516623227161782222918191022221120201221

[0226] s 2 =229182511313224233131442322112023515227161782219102220122116623

[0227]

[0228] Obviously, the string s 1 and 2 The simple directed topological coding diagram in Figure 7 The topological structures of are unrelated, but their encryption authentication is done through the directed topological encoding matrix in (20) The directed topological coding graph Admit a set of ordered edge difference magic full label f: For each arc Always Note that the directed topological encoding matrix It is possible to export (36)! unique strings.

[0229] The present invention patent proposes the problem of digital strings in directed topologically coded graphs: To crack a directed topologically coded graph The generated digital string s = c 1 c 2 ...c n , where each number c i ∈[0,9]. First, we need to rewrite this string of numbers into 3q small strings of numbers a 1 , a 2 , ..., a 3q , so that each small number string Each number c in the original number string s i Only in a unique string of small numbers a j Appears in c i Cannot appear in other small number strings a k (k≠j). Next, we need to find a non-negative integer k 0 >0 and d 0 > 0, so that every small number string a j can be expressed as an integer indeterminate equation a j =k 0 α j +d 0 β j Then, we need to find a directed topologically encoded bigraph with p vertices and q arcs. And this directed topological coding graph admits a parameter total coloring F. Finally, a parameter directed topological coding matrix is ​​generated

[0230]

[0231] Where f is a directed topologically encoded bigraph Admit a set of ordered W-constrained total colorings such that the string of numbers a 1 a 2 ...a 3q =s=c 1 c 2 ...c n The directed topological encoding matrix is ​​exactly Export, that is, obtain topology certification.

[0232] Solving the digital string problem of the directed topological coding graph above will encounter the following difficult problems:

[0233] (1) Digital string decomposition problem. Because there are multiple topological coding matrices of different topological coding graphs that can derive the same digital string or character string. Even for a digital string, there is no general way to rewrite the digital string into 3q small digital strings and associate them with a certain directed topological coding matrix. Note that there are (3q)! 1 a 2 ...a 3q When q is large, this work is non-polynomial.

[0234] (2) Solve indeterminate equations. Find 3q indeterminate equations a j =k o α j +d 0 β j Integer solutions for (j∈[1,3q]) are difficult.

[0235] (3) NP-complete problem. Finding a directed topologically coded bigraph We will encounter the NP-complete problem of subgraph isomorphism.

[0236] (4) Number of directed graphs. The edge orientation of an undirected graph G with p vertices and q edges can generate 2 q There is a directed graph G with p vertices and q arcs. k (k∈[1,2 q ]), so that their topological structures are the same, that is, This shows that the number of directed graphs is much larger than the number of undirected graphs.

[0237] (5) Find W-constrained coloring. The parameterized directed topological encoding matrix (21) is associated with the solution of the indeterminate equation (k 0 , d 0 ) and directed topologically encoded bigraphs However, it is difficult to find a directed topologically encoded bigraph from thousands of W-constrained colorings. Admit an ordered W-constrained coloring f. For example, such that is the coloring type W-constraint topological encoding matrix in Definition 7. This is a computational task that cannot be completed in polynomial time, and is even an NP-class problem.

[0238] In summary, the digital string s = c 1 c 2 ...c n Unable to find the original directed topologically encoded bigraph and its parameterized fully colored topological coding matrix, thus obtaining the computational security and provable security of directed topological coding graph and directed topological coding matrix encryption technology.

[0239] (IV) Matching topology matrix

[0240] The matching topological matrix can form a private key topological matrix and a public key topological matrix, which not only have functions similar to topological signatures and digital signatures, but also can generate key graph set pairs. The matching pairs of topological coding matrices can form key pairs of private key graph sets and public key graph sets.

[0241] Definition 9. Matrix T code =(X, E, Y) T is the 3×q-order topological coding matrix in Definition 6. Let the parameter set X 0 ={0,d,2d,...,(q-1)d} and S k，d ={k, k+d, k+2d, ..., k+(q-1)d}, where the integers k, d ≥ 1 and q ≥ 2. We have the following topological matrix matching pairs:

[0242] Matching-1. If two topological encoding matrices T code =(X, E, Y) T and satisfy Then it is called is the topological encoding matrix T code The (k, d)-complement topological encoding matrix, T code and is a twin (k, d)-topological encoding matrix matching pair.

[0243] Matching-2. Let the topological encoding matrix in Definition 6 be corresponds to a topologically encoded graph G with p vertices and q edges, and satisfies the set and e t，i -λ=x t，i +y t，i -λ(mod qd)(t=1,2;i∈[1,q]), where λ is a constant. 1，i +e 2，i =2k+(q-1)d(i∈[1,q]), then and is a (k, d)-harmonious mirror topology encoding matrix matching pair.

[0244] Matching-3. Let the topological encoding matrix in Definition 6 be corresponds to a topological coding graph G with p vertices and q edges, and satisfies e k，i =|xk，i -y k，i |(k=1,2;i∈[1,q]). If there exists a constant α such that e 1，i +e 2，i =α(i∈[1,q]), then and is a mirror topological encoding matrix matching pair, called topological encoding matrix is the topological encoding matrix Mirror image.

[0245] Match-4. Set S adj (G, f) is the edge-colored adjacency matrix E adj (G, f) p×p All exported numeric strings , let S adj (G, h) is the edge-colored adjacency matrix E adj (G, h) p×p All exported numeric strings If there exists a coloring transformation θ such that h = θ(f), then the two string sets S adj (G, f) and S adj (G, h) forms a string set matching pair based on the coloring transformation θ, with B ad (j) = θ(A ad (j))(j∈[1,(p 2 )! ]).

[0246] Matching-5. Topological encoding matrix T code =(X, E, Y) T There are 3 inverted row vectors: X -1 =(x q , x q-1 , ..., x 1 ), E -1 =(e q , e q-1 , ..., e 1 ) and Y -1 =(y q ,y q-1 , ..., y 1 ), generating an inverted topological encoding matrix Therefore, T code and Form topological encoding matrix matching pairs.

[0247] Matching-6. Adjacency matrix A(L) of a topologically encoded graph L with p vertices and q edges p×p and one of its edge-colored adjacency matrices E adj(L, f) p×p Form adjacent matrix matching pairs.

[0248] Matching-7. For every integer n∈[1,M], the parametric full coloring topological coding matrix P(G, f|k n , d n ) = k n I 0 +d n ·T code (G, f), where T code (G, f) is the full coloring topological encoding matrix in Definition 4, the matrix Satisfy row vector and the row vector Suppose there is an integer point d on the curve y = f(x) n =f(k n )(n∈[1,M]), parameterized full coloring topological encoding matrix P(G,f|k n , d n ) and the curve y = f(x) form a matching pair of parameter-full-coloring topological encoding matrix and curve. A parameter-full-coloring topological encoding matrix can form a "one-to-many" matrix-curve matching pair with multiple curves, and vice versa.

[0249] (V) Topological Matrix Group

[0250] Graph group and topology matrix group technologies are the basic technologies for overall encryption of local area networks.

[0251] (5.1) Topological coding matrix group

[0252] Definition 10: Let a topologically coded graph G with q edges admit a W-constrained total coloring f: V(G)∪E(G)→[a,b], and M=max{f(w):w∈V(G)∪E(G)}. Then there is a topologically coded graph set F f (G) = {G 1 , G 2 , ..., G M}, each topological coding graph G k Admit a W-constrained full coloring h k , defined as h k (w)=f(w)+k(mod M)(w∈V(G)∪E(G)), and Finite module M Abelian additive operation Defined as h i (w)+h j (w)-h k (w) = h λ(w)(w∈V(G)∪E(G)), where G k is an arbitrarily selected zero element, λ=i+jk(mod M), so that G λ ∈F f (G). It is not difficult to verify the following facts:

[0253] (i) Zero-ary. Each topological encoding graph G k ∈F f (G) can be used as zero, so that

[0254] (ii) Inverse element. For i+j=2k(mod M), we have

[0255] (iii) Closure law. For λ = i + jk (mod M), we have

[0256] (iv) Associative law.

[0257] (v) Commutative law.

[0258] Then the topological coding graph set F f (G) is a finite-module Abelian additive operation topological coding graph group with zero elements everywhere, denoted by

[0259] Note to Definition 10: According to Definition 10 and finite modular Abelian additive operations Definition, we define the topological encoding matrix set T code (G,F code )={T code (G i ,h i ): finite modular Abelian additive operation "T code (G i ,h i )[+]T code (G j ,h j )[-]T code (G k ,h k )=T code (G λ ,h λ )"for:

[0260] (i)T code (G k ,h k )∈T code (G,F code ) is an arbitrarily chosen zero;

[0261] (ii)h i (w)+h j (w)-h k (w) = h λ (w)(w∈V(G i )∪E(G i )=V(G)∪E(G)), so that λ=i+jk(mod M), and G λ ∈F f (G). Thus, the topological coding matrix set T code (G,F code ) under the finite-module Abelian additive operation, a topological coding matrix group with zero elements everywhere is formed.

[0262] Definition 11: Let a topological coding graph G with q edges admit a W-constrained total coloring f: V(G)∪E(G)→[a,b], and M=max{f(w):w∈V(G)∪E(G)}. According to the topological coding graph group in Definition 10 We have the following matrix group:

[0263] Matrix Group-1. Based on the set of fully colored adjacent matrices T under finite modulus Abelian additive operations adj (G,F adj )={T adj (G i ,h i ): i∈[1,M]} forms a group of adjacent matrices with zero elements everywhere

[0264] Matrix group-2. Since the topological coding graph G corresponds to the topological coding matrix T code (G, f), topologically encoded graph group Each topological coding graph G in i Corresponding topological encoding matrix T code (G i ,h i ), in the finite modulus Abelian additive operation "T code (G i ,h i )[+]T code (G j ,h j )[-]T code (G k ,h k )=T code (G λ ,h λ )”, we get the topological coding matrix set T code (G,F code )={T code (Gi ,h i ): a group of topological coding matrices with zero elements everywhere on i∈[1,M]}, denoted by

[0265] Matrix group-3. Since each topological encoding matrix Corresponding to a topological coding graph set Then the set of topologically coded graph sets is A set of topologically coded graphs with zero elements formed under finite-module Abelian additive operations

[0266] Matrix Group-4. Parameterized Full Coloring Topological Encoding Matrix Set P code (G, L|k, d) = {P code (G 1 , L 1 |k,d),P code (G 2 , L 2 |k,d),...,P code (G M , L M |k, d)}, each parameter full coloring topological encoding matrix is In the finite-module Abelian addition operation "P code (G i ,h i )[+]P code (G j ,h j )[-]P code (G k ,h k )=P code (G λ ,h μ )” forms a group of topological coding matrices with zero parameters everywhere, denoted as

[0267] Definition 12: According to the topological coding graph group of Definition 10 and the total coloring adjacent matrix group, topological coding matrix group, topological coding graph set group and parameter total coloring topological coding matrix group of Definition 11, define group coloring: Graph H admits a W-group total coloring F X : V(H)∪E(H)→W-group, so that every edge uv satisfies F X (u)≠F X (v) and satisfies the finite modulus Abelian operation “F X (x)[+]F X (y)[-]F X (w) = F X (z)∈W-group”, where F X(w) is an arbitrarily selected zero, F X =F adj , F gra , F mat , F param , F set . We have:

[0268] Group Coloring - 1. Graph H admits a total coloring adjacency matrix group total coloring F adj : So that the topological encoding matrix T code (H,F adj ) is a fully colored adjacent matrix, i.e., the topological encoding matrix T code (H,F adj ) is the topological encoding matrix of the fully colored adjacency matrix, similar to a tensor matrix.

[0269] Group Coloring-2. Graph H admits a topologically encoded graph group total coloring So that the topological encoding matrix T code (H,F gra ) is a topologically encoded graph G i , that is, the topological encoding matrix T code (H,F gra ) is the graph topology encoding matrix.

[0270] Group Coloring-3. Graph H admits a topological encoding matrix group total coloring F mat : So that the topological encoding matrix T code (H,F mat ) is a topological coding matrix, that is, the topological coding matrix T code (H,F mat ) is the tensor topology encoding matrix.

[0271] Group Coloring - 4. Graph H admits a parameterized total coloring topological encoding matrix group total coloring F param : So that the topological encoding matrix T code (H,F param ) is a parameterized fully colored topological encoding matrix.

[0272] Group Coloring-5. Graph H admits a topologically encoded graph set group total coloring F set : So that the topological encoding matrix T code (H,F set ) is a collection of topologically encoded graphs.

[0273] Theorem 5. Let the topological coding graph H admit a topological coding graph group full coloring F: Then the topological coding graph H admits a topological coding matrix group total coloring F: Admit a parameter total coloring topological encoding matrix group total coloring F param : Admit a topologically encoded graph set group total coloring h: And admit a parameter total coloring topological encoding matrix group total coloring F param : (See Definition 10, Definition 11 and Definition 12).

[0274] Corollary 6: If a graph H admits a topologically encoded graph group coloring, then there exists a set of graphs such that every graph in this set produces a topologically encoded graph group coloring admitted by graph H.

[0275] Theorem 7. According to the topological coding graph group of Definition 10, and the integer M>q, the tree T with q edges admits a topological coding graph group total coloring θ: For any specified zero There is always a set of subscripts for topologically encoded graphs

[0276] θ index (E(T))={k=i+ja(mod M):θ(u k v k )=G k ,u k v k ∈E(T)}=[1,q]

[0277] in and G k ∈F f (G) = {G 1 , G 2 , ..., G M}. For a general sequence of positive integers Then there is

[0278] (5.2) Topological coding matrix group based on hybrid topological coding graph group

[0279] Definition 13. Let a (p, q)-graph G admit a W-constrained normal total coloring f: V(G)uE(G)→[1, R], such that the vertex coloring set f(V(G)) = {f(x): x∈V(G)} and the edge coloring set f(E(G)) = {f(uv): uv∈E(G)} satisfy the W-constraint. Graph set R f (G) = {G s，k : s∈[1, p], k∈[1, q]} is called a set of mixed topological coding graphs. Each topological coding graph Gs，k isomorphic to G, and admits a W-constrained normal total coloring h s，k , so that h s，k (x) = f (x) + s (mod p) (x ∈ V (G s，k )) and h s，k (uv)=f(uv)+k(mod q)(uv∈E(G s，k )).

[0280] Optional Zero G a，b ∈R f (G), is the set of mixed topological coding graphs R in Definition 13 f (G) Define the finite modulus Abelian additive operation "G s，k [+]G i，j [-]G a，b =G λ，μ "as follows:

[0281] (i)h s，k (x)+h i，j (x)-h a，b (x) = h λ，μ (x), s+ia=λ(mod p), x∈V(G);

[0282] (ii)h s，k (e)+h i，j (e))-h a，b (e) = h λ，μ (e), k+jb=μ(mod q), e∈E(G)

[0283] Makes it possible to admit full coloring h λ，μ Hybrid topological coding graph G λ，μ Belongs to the graph set R f (G).

[0284] In the finite-module Abelian addition operation "G s，k [+]G i，j [-]G a，b =G λ，μ ", hybrid topological coding atlas R f Each topological coding graph in (G) can be used as the zero element of the finite module Abelian additive operation and satisfies: uniqueness, closure, each topological coding graph has its own inverse element, associative law, and commutative law. Therefore, the mixed topological coding graph set R f (G) is a group of mixed topological coding graphs with zero digits everywhere, denoted by

[0285] Definition 14. According to Definition 13, let the (p, q)-graph G admit a W-constrained normal total coloring f: V(G)uE(G)→[1, B], based on the mixed topological coding graph group and the finite modulus Abelian addition operation "G s，k [+]G i，j [-]G a，b =G λ，μ ", suppose the topological coding graph G admits a topological coding graph group full coloring F: Define the topological coding matrix of the topological coding graph G to be fully colored as R at (w) = T code (G s，k ,h s，k ), where F(w) = G s，k (w∈V(G)uE(G)). Since is a topologically coded graph group under finite-module Abelian additive operations, so the topological coding matrix set R at (G) = {R at (w) = T code (G s，k ,h s，k ): G s，k ∈M f (G)} also forms a zero-element mixed topological coding matrix group, denoted as

[0286] Definition 15. According to Definition 13, let the (p, q)-graph G admit a W-constrained normal total coloring f: V(G)∪E(G)→[1,B], in the finite modulo Abelian additive operation “G s，k [+]G i，j [-]G a，b =G λ，μ ", then there is a topological coding graph group and the topological coding matrix group Since each topological encoding matrix T code (G s，k ,h s，k ) is of order 3×q, and can derive (3q)! strings, then there are (3q)! algorithms, and each algorithm C θ (θ∈[1,(3q)!]) from the topological encoding matrix T code (G s，k ,h s，k ) to export a string s(C θ , T code (G s，k ,h s，k )). For every integer θ∈[1,(3q)! ], the string set S tring (θ)={s(Cθ , T code (G s，k ,h s，k )):G s，k ∈M f (G)} forms a group of zero-ary strings everywhere, denoted by

[0287] Theorem 8. Let the (p, q)-graph G admit a W-constrained normal total coloring f, based on the mixed topological encoding graph set R f (G) = {G s，k : s∈[1, p], k∈[1, q]} (see Definition 13) and the finite-module Abelian additive operation “G s，k [+]G i，j [-]G a，b =G λ，μ ", then there is a zero-ary mixed topological coding graph group (See definition]4), the group of mixed topological coding matrices with zero elements everywhere and (3q)! groups of zero-ary mixed strings (See definition 15).

[0288] Definition 16. Let the (p, q)-graph G admit a W-constrained normal total coloring h. According to Definition 13, we get an infinite set of mixed topological coding graphs Each of the mixed topological coding graphs G 0，0 =G, and each mixed topology encoding graph G s，k Admit a W-constrained normal full coloring h s，k (x) = h(x) + s(x∈V(G s，k )),h s，k (uv)=h(uv)+k(uv∈E(G s，k )), so that for any specified zero element Modulo-free Abelian addition Established, called set It is a group of infinite mixed topological coding graphs with zero elements everywhere.

[0289] From Definition 13, Definition 14, Definition 15 and Definition 16, we get:

[0290] (i) Group of mixed topological coding graphs with zero numerators everywhere. Group of mixed topological coding graphs is an infinite mixed topological coding graph group with zero elements everywhere A proper subset of .

[0291] (ii) Infinite topological coding matrix group with zero elements everywhere. Mixed topological coding matrix group is an infinite topological coding matrix group with zero elements everywhere A proper subset of .

[0292] (iii) The group of infinite strings with zero elements everywhere.

[0293]

[0294] (iv) Every integer point (s, k) on the xOy-plane corresponds to a topological coding graph H of the infinite mixed topological coding graph group with zero elements everywhere. s，k , and the topological coding graph H s，k The topological encoding matrix T code (G s，k ,h s，k ), and (3q)! more strings.

[0295] Figure 8 shows an example of an infinite mixed topological coding graph group.

[0296] Definition 17. An infinite 3-dimensional topological coding matrix group with zero elements everywhere. Let the vertex set of the dual graph T be V(T) = (X, Y), which admits a set-ordered W-constrained total coloring F and satisfies the set-ordered constraint max{F(x): x∈X} = max F(X) < minF(Y) = min{F(y): y∈Y}. Let the set Each topological coding graph T in i，j，k satisfy And admit that the full coloring f i，j，k (x) = F (x) + i (x∈X), f i，j，k (y) = F (y) + j (y∈Y), f i，j，k (xy) = F(xy) + k(xy∈E(T)), so that f i，j，k (x)<f i，j，k (y), non-modular Abelian additive operation "T i，j，k [+]T r，s，t [-]T a，b，c =T λ，μ，γ ” is defined as

[0297] (i)f i，j，k (x)+f r，s，t (x)-f a，b，c (x) = f λ，μ，γ (x), x∈X; (ii)f i，j，k (y)+f r，s，t (y)-f a，b，c (y) = f λ，μ，γ (y), y∈Y;

[0298] (iii)f i，j，k (xy)+f r，s，t (xy)-f a，b，c(xy) = f λ，μ，γ (xy), xy∈E(T)

[0299] Where T a，b，c is an arbitrarily specified zero, and there is We call the set It is an infinite 3-dimensional mixed topological coding graph group with zero elements everywhere.

[0300] Similarly, we can define the group of infinite 3-dimensional topological encoding matrices with zero elements everywhere and the group of infinite 3-dimensional strings with zero elements everywhere.

[0301] (VI) Encryption algorithm based on topological matrix

[0302] (6.1) Topological matrix homomorphism

[0303] Homomorphic mapping between topological matrices is one of the cloud topological encryption technologies. The patent of this invention designs the following color-preserving graph homomorphism, color-preserving adjacent matrix homomorphism, color-preserving fully colored adjacent matrix homomorphism, and color-preserving fully colored topological coding matrix homomorphism.

[0304] Definition 18: Let the topological coding graph G admit a set-ordered W-constrained total coloring f, and the topological coding graph H admit a set-ordered W-constrained total coloring h, if there exists a mapping F: V(H) → V(G) such that

[0305] (i) Every edge uv∈E(H) if and only if there is an edge F(u)F(v)∈E(G);

[0306] (ii) Every vertex x∈V(H) satisfies f(F(x))=h(x), and every edge uv∈E(H) satisfies f(F(u)F(v))=h(uv).

[0307] We call graph H a color-preserving graph isomorphic to graph G, denoted by And there is a color-preserving adjacent matrix homomorphism Color-preserving total coloring adjacency matrix homomorphism Color-preserving total coloring topological encoding matrix homomorphism

[0308] Definition 19: Let the topological coding graph set G be rap (T code ) corresponds to the 3×q-order topological coding matrix T in Definition 6. code , if there is a topological coding graph G∈G rap (T code ), so that each topological coding graph H∈G rap (T code ) all satisfy the color-preserving graph homomorphism Then the topological coding graph G is called the graphable topological coding matrix T code The gathering point.

[0309] Note to Definition 19: The 3×q-order topological encoding matrix T in Definition 6 code Corresponding to a topological coding graph set G rap (T code ), no network reports on calculating the topological coding graph set G rap (T code ) in the topological coding graph set G. rap (T code ) is an NP-complete problem. Finding a graphable topological coding matrix T code The clustering points involve color-preserving graph homomorphism, color-preserving adjacent matrix homomorphism, color-preserving full-coloring adjacent matrix homomorphism, and color-preserving full-coloring topological coding matrix homomorphism, which means that using the topological coding graph set G rap (T code )’s full coloring encryption technology has computational security and provable security.

[0310] On Finding a Topologically Coding Graph Set G rap (T code ) can be seen from the examples in Figures 9 and 10. It can be seen that each topological coding graph G k The color-preserving graph is isomorphic to the topologically encoded graph G1, that is, Figure 9 G 1 , G 2 , ..., G 17 This is the Chinese character group, Figure 10 of G 18 , G 19 , ..., G 25 It is a tree topology coding graph. Topology coding graph G 26 In graph theory, it is called a forest. Under the vertex coincidence operation, the topological coding graph G 26 We can construct a graph set G rap (T code ) in each topological coding graph.

[0311] (6.2) Modular algorithm based on topological matrix

[0312] For every integer k∈[1,m], let p k vertices and q k The topological encoding graph G of the edges k The adjacent matrix is The edge-colored adjacency matrix is The full coloring topological encoding matrix is Assume that A and B-k (k∈[1,m]) are communication users in a local area network, we have the following modular algorithm:

[0313] Modular algorithm-1 is based on the adjacency matrix and the topological coding matrix.

[0314] Step-1-1. For each integer k∈[1,m], A uses the adjacent matrix and an algorithm associated with it To derive the private key topology signature digital string s k，pri and the public key topological signature digital string s k，pub . Then, A saves the private key topological signature digital string s k，pri , send the public key topological signature digital string s to B-k k，pub .

[0315] Step-1-2. For each integer k∈[1,m], A uses the full coloring topological encoding matrix and an algorithm associated with it α i (i∈[1,(3q k )! ]) to derive the private key digital string n k，pri and public key string n k，pub Next, A saves the private key string n k，pri , send the public key digital string n to B-k k，pub .

[0316] Step-1-3. B-k uses the public key topological signature digital string s k，pub and public key string n k，pub To encrypt a plaintext F, get a ciphertext D with 2-fold protection, and then send it to A.

[0317] Step-1-4. After receiving the ciphertext D, A first uses the private key topological signature digital string s k，pri To decrypt text D, when the adjacent matrix A through topology authentication uth [s k，pub ,s k，pri ], A knows that the ciphertext is sent by B-k, which means that the identity of B-k is confirmed.

[0318] Step-1-5. For the ciphertext with only 1-fold protection, A uses the private key string n k，pri To decrypt, when the full coloring topological encoding matrix A through topology authentication uth [n k，pub , n k，pri ], A obtains the plaintext F.

[0319] Modular Algorithm-2 is based on edge-coloring adjacency matrix and topological encoding matrix.

[0320] Step-2-1. For each integer k∈[1,m], A uses edge coloring to color the adjacent matrix and an algorithm associated with it To derive the private key topology signature digital string s k，pri and the public key topological signature digital string s k，pub Next, A saves the private key topological signature digital string s k，pri , send the public key topological signature digital string s to B-k k，pub .

[0321] Step-2-2. For each integer k∈[1,m], A uses the full coloring topological encoding matrix and an algorithm associated with it α j (j∈[1,(3q k )! ]) to derive the private key digital string n k，pri and public key string n k，pub . Then, A saves the private key digital string n k，pri , send the public key digital string n to B-k k，pub .

[0322] Step-2-3. B-k uses the public key topological signature digital string s k，pub and public key string n k，pub To encrypt a plaintext F, get a ciphertext D with 2-fold protection, and then send it to A.

[0323] Step-2-4. After receiving the ciphertext D, A first uses the private key topological signature digital string s k，pri To decrypt document D, when the edge-colored adjacent matrix is A through topology authentication uth [s k，pub ,s k，pri ]After that, A knows that the sender of this ciphertext is B-k.

[0324] Step-2-5. For the ciphertext with only 1-fold protection, A uses the private key string n k，pri To decrypt, when the full coloring topological encoding matrix A through topology authentication uth [n k，pub , n k，pri ], A obtains the plaintext F.

[0325] Annotation for Modular Algorithm-1 and Modular Algorithm-2: Topological Coding Graph G k With the adjacency matrix and edge-colored adjacency matrix is one-to-one, ensuring the uniqueness of the identity topology signature. However, the topology encoding graph G k With the full coloring topological encoding matrix However, there is no one-to-one correspondence, because a fully colored topological coding matrix can correspond to multiple topological coding graphs. These properties bring NP-complete problems and complexity overhead to code breakers.

[0326] Module Algorithm-3 is based on the full coloring adjacency matrix and the topological coding matrix.

[0327] Since the steps of module algorithm-3 are similar to those of module algorithm-1 and module algorithm-2, they will not be described again.

[0328] (6.3) Matrix algebraic operations, matching algorithms

[0329] (6.3.1) Algebraic operation algorithm of topological matrix. The example in Figure 4 can help understand the following topological coding matrix union operation

[0330]

[0331] It is worth noting that the topological encoding matrix T code The string s* derived from (T, h) is more complicated. It finds each topological encoding matrix T from the string s*. code (T k ,h k )(k∈[1,n]) is impossible because:

[0332] (i) It is extremely difficult to split a graphable topological encoding matrix into two or more small graphable topological encoding matrices.

[0333] (ii) Each topological encoding matrix T code (T k ,h k )(k∈[1,n]) corresponds to a set of topological coding graphs, and finds a special topological coding graph T in this set of topological coding graphs. k You will encounter NP-complete problems;

[0334] (iii) Find each topological coding graph T from a large number of W-constraint colorings and W-constraint labels k Accepted coloring h k It is a non-polynomial job;

[0335] (iv) There are other algebraic operations on topological matrices: intersection operations, linear operations, tearing, coincidence operations, splitting operations, coloring transformations, homomorphic operations, etc. Therefore, it is determined that the topological encoding matrix T code (T, h) is the topology for which matrix algebra operations are difficult.

[0336] (6.3.2) Twin parameter full coloring topological encoding matrix matching pair algorithm. Topological encoding matrix matching pairs form topological signature key pairs. For example, the topological encoding matrix and is a twin (k, d)-topological encoding matrix matching pair, the topological encoding matrix T code The corresponding topological coding graph set is G rap (T code ), the twin (k, d)-topological encoding matrix The corresponding topological coding graph set is So that two topological coding graph sets G rap (T code ) (as a private key atlas) and Topological encoding in (as a public key atlas) Figure 1 One-to-one correspondence, thus forming a topological signature key pair. Since the parameters k and d in the twin (k, d)-topological encoding matrix matching pair can be infinite, it provides an inexhaustible topological signature key pair for "one-time one-key". We designed the topological encoding authentication matrix Among them A G =(T code , O 3×m ),

[0337] (6.3.3) Apply the (k, d)-harmonious mirror topology encoding matrix matching pair, and the remaining steps are similar to those in (6.3.2).

[0338] (6.3.4) Apply the topological coding matrix T code The inverted topological encoding matrix A topological encoding matrix matching pair is formed, and the remaining steps are similar to those in (6.3.2).

[0339] (6.3.5) Apply parameterized full coloring topological encoding matrix P(G, f|k n , d n ) and the curve y = f(x) form a matching pair of parameterized full coloring topological encoding matrices and curves, where d n =f(k n ), and the remaining steps are similar to those in (6.3.2).

[0340] (6.3.6) Apply the adjacency matrix A(L) of a topologically coded graph L with p vertices and q edges p×p and one of its edge-colored adjacency matrices E adj (L, f) p×p The adjacent matrix matching pairs are formed, and the remaining steps are similar to those in (6.3.2).

[0341] (6.3.7) Single private key to multiple public key algorithm. The topological encoding matrix T in Definition 6 code The gathering point is the private key topological coding graph H, the topological coding matrix T code Corresponding to a topological coding graph set G raph (T code )={H i :i∈[1,M(T code )]}, where each topological coding graph H i Is the public key, forming H i The color-preserving graph is homomorphic to H, that is, Get a "one-to-many" key group.

[0342] (6.4) Approximate one-time pad algorithm

[0343] A holds the private key adjacency matrix A(G) n×n and the private key fully colored topological encoding matrix T code (G) 3×q , B-k holds the public key adjacent matrix A(H) n×n and the private key fully colored topological encoding matrix T code (H) 3×q , get the adjacent authentication matrix A uth [A(G), A(H)] = (X G , X H ) T , where X G =(A(G) n×n , O n×n ), X H =(O n×n , A(H) n×n ). We design a fully colored topological encoding authentication matrix A uth [T code (G), T code (H)]=(T G , T H ) T , where T G =(T code (G) 3×q , O 3×q ), T H =(O 3×q , T code (H) 3×q ).

[0344] Approximate one-time pad modular algorithm-I. A and B-k each have a procedure to generate a topological signature key pair: (i) have (n 2 )! Algorithm α k (k∈[1,(n 2 )! ]) From A's private key adjacent matrix A(G)n×n Export (n 2 )! A private key digital string S A，k (k∈[1,(n 2 )! ]). (ii) yes(n 2 )! Algorithm β k (k∈[1,(n 2 )! ]) From the public key adjacent matrix A(H) of B-k n×n Export (n 2 )! Public key digital string S B，k (k∈[1,(n 2 )! ]). Each topology signature key pair [S A，k , S B，k ] can be obtained through the adjacent authentication matrix A uth [A(G), A(H)] certified.

[0345] In the kth communication, B-k uses the public key digital string S B，k Encrypted plaintext D k , get the ciphertext F k , and then send it to A.

[0346] After receiving the ciphertext F, A uses the private key string S A，k To decrypt the ciphertext F k , when the adjacent authentication matrix A uth [A(G), A(H)] through the topological signature key pair [S A，k , S B，k ]After authentication, the ciphertext F k is untied, and A sees the plaintext D k .

[0347] Approximate one-time pad modular algorithm-II. A and B-k each have a program to generate a digital key pair: (i) There are (3q)! algorithms α k (k∈[1,(3q)!]) from A's private key fully colored topological encoding matrix T code (G) 3×q Export (3q)! Private key digital string n A，k (k∈[1,(3q)!]). (ii) There are (3q)! algorithms β k (k∈[1,(3q)!]) from the public key total coloring topological encoding matrix T of B-k code (H) 3×q Derive (3q)! Public key digital string n B，k (k∈[1,(3q)!]). So that every digital key pair [n A，k , n B，k ] can be verified by topological encoding matrix A uth [T code(G), T code (H)] certification.

[0348] In the kth communication, B-k uses the public key digital string n B，k Encrypted plaintext D k , get the ciphertext F k , and then sent to A.

[0349] After receiving the ciphertext F, A uses the private key digital string n A，k To decrypt the ciphertext F k , when the topological coding authentication matrix A uth [T code (G), T code (H)] through the topological signature key pair [n A，k , n B，k ]After authentication, decrypt the ciphertext F k After that, A gets the plaintext D k .

[0350] In the above approximate one-time pad modular algorithm-I and approximate one-time pad modular algorithm-II, due to the factorial number (n 2 )! and (3q)! are quite large, and A and B-k cannot use them all in their lifetime communication (n 2 )! and (3q)! topological signature key pairs. Therefore, in the kth communication, for j∈[1, k-1], the previous private key digital string S A，j (or n A，j ) and the public key digital string S B，j (or n B，j ) are no longer used, making the communication topology signature key pair of A and B-k have a "one-time one-key" effect.

[0351] (6.5) LAN overall encryption algorithm based on topology matrix group

[0352] According to the following Abelian additive operation:

[0353] ① Finite modular Abelian additive operation on graph groups i [+]G j [-]G k =G λ ”, “G s，k [+]G i，j [-]G a，b =G λ，μ ”;

[0354] ② Modular Abelian Additive Operations on Infinite Graph Groups s，k [+]G i，j [-]G a，b =G λ，μ ”,“T i，j，k[+]T r，s，t [-]T a，b，c =T λ，μ，γ ”;

[0355] ③ Finite modular Abelian additive operation of topological coding matrix group “T code (G i ,h i )[+]T code (G j ,h j )[-]T code (G k ,h k )=T code (G λ ,h λ )";

[0356] ④ Finite modular Abelian additive operations on parameterized fully colored topological coding matrix groups

[0357] P code (G i ,h i |k,d)[+]P code (G j ,h j |k,d)[-]P code (G k ,h k |k, d) = P code (G λ ,h μ |k,d)

[0358] Definitions 10 and 12 give the following matrix group:

[0359] (i) Topologically coded graph groups (ii) Fully Colored Adjacency Matrix Group

[0360] (iii) Topological coding matrix group (iv) Topologically coded graph collections

[0361] (v) Parameterized fully colored topological encoding matrix group

[0362] (6.5.1) Topological coding matrix group algorithm, overall zero element, user private topological signature

[0363] Let N(t)(t∈[a,b]) be a local area network, variable t is time, and each N k (t)(k∈[1,n]) is the community of N(t).

[0364] Step-5-1. Initialization (implementation of full coloring of the topological coding graph group). Encrypt the local area network N(t) as a whole, that is, perform topological coding on N(t) and fully color the graph group F g : Make each edge uv of the local area network N(t) satisfy F g (u)≠F g (v), and the finite-modular Abelian additive operation “F g (u)[+]F g (v)[-]F g (w 0 )=F g (uv)”, where F g (w 0 )=G k (w 0 ∈V(N(t))) is an arbitrarily selected global zero, F g (u) = G i 、F g (v) = G j and F g (uv)=G λ Satisfies λ=i+jk(mod M), and G λ ∈F f (G).

[0365] Step-5-2. Topological coding matrix group full coloring. Replace the topological coding graph group full coloring in initialization with the topological coding matrix group The topological encoding matrix group is completely colored F m : Make the coloring of each edge uv satisfy F m (u) = T code (G i ,h i ), F m (v) = T code (G j ,h j ) and F m (uv) = T code (G λ ,h λ ) satisfies the finite modulus Abelian addition operation of the topological encoding matrix "T code (G i ,h i )[+]T code (G j ,h j )[-]T code (G k ,h k)=T code (G λ ,h λ )”, where F m (w 0 )=T code (G k ,h k ) is zero yuan overall.

[0366] Step-5-3. User private topological signature. The coloring of vertex u is the topological encoding matrix T code (G i ,h i ) corresponds to a topologically encoded graph set G raph (T code (G i ,h i )); The coloring of vertex v is the topological encoding matrix T code (G j ,h j ) corresponds to a topologically encoded graph set G raph (T code (G j ,h j )), so that each topological coding graph H u ∈G raph (T code (G i ,h i )) is isomorphic to the unique topological coding graph H v ∈G raph (T code (G j ,h j )),Right now We choose a topological encoding graph H u ∈G raph (T code (G i ,h i )) is the topological signature of vertex u, and is selected to be isomorphic to the topological signature H u The topological coding graph H v ∈G raph (T code (G j ,h j )) as the topological signature of vertex v, so that there are no two vertices v adjacent to vertex u i and vertex v j The topological signature of isomorphism.

[0367] Step-5-4. File encryption. A uses the topological signature H of vertex u u The adjacent matrix A(H u ) and the topological encoding matrix T code (Hu ,h u ) derives the topological signature digital string, and B uses the topological signature H of vertex v v The adjacent matrix A(H v ) and the topological encoding matrix T code (H v ,h v ) to derive the encrypted digital string. Then, A and B use the algorithms in (6.1), (6.2) and (6.3) to communicate securely.

[0368] (6.5.2) Topological coding matrix group algorithm, community private zero-element, user private topological signature

[0369] In the overall topology encoding matrix encryption of the local area network N(t)(t∈[a,b]), the community N k The zeros of (t)(k∈[1,n]) are different from each other, and each community N k The zero inside (t)(t∈[a,b]) is different from the zero outside the community. The entire implementation process is the same as the steps in (6.5.1). When a community N k After the communication of (t) is destroyed, it will not affect the internal and inter-community communications of other communities in the local area network N(t).

[0370] (6.5.3) Topologically encoded matrix group algorithm, user private zero-ary and private topological signature.

[0371] The zero element between each edge uv in the local area network N(t)(t∈[a,b]) is different from the zero element between other edges xy. The other implementation steps are the same as those in (6.5.1) and (6.5.2). When a node in the local area network N(t) is disrupted in communication, it does not affect the communication between other nodes in the local area network N(t).

[0372] (6.6) Overall encryption algorithm for local area network based on fully colored adjacent matrix group

[0373] (6.6.1) Fully colored adjacent matrix group algorithm, overall zero element, user private topological signature.

[0374] Assume N(t)(t∈[a,b]) is a local area network, each N k (t)(k∈[1,n]) is a community of N(t).

[0375] Step-6-1. Initialization (implementation of full coloring of the topological coding graph group). Encrypt the entire local area network N(t) so that N(t) admits a color adjacent matrix group full coloring L adj : Make each edge uv of the local area network N(t) satisfy L adj (u)≠L adj (v), and the finite-modular Abelian additive operation of fully colored adjacency matrices "L adj (u)[+]L adj (v)[-]L adj (w 0 )=L adj (uv)”, where L adj (w 0 )=G k (w 0 ∈V(N(t))) is an arbitrarily selected global zero, L adj (u) = G i , L adj (v) = G j and L adj (uv)=G λ Satisfies λ=i+jk(mod M), and G λ ∈F f (G).

[0376] Step-6-2. Full coloring of adjacent matrix group. Replace the full coloring of the topological coding graph group in initialization with the full coloring of the adjacent matrix group. Total Coloring of Adjacent Matrix Group Total Coloring L adj : So that each edge uv is colored L adj (u) = T adj (G i ,h i )、L adj (v) = T adj (G j ,h j ) and L adj (uv) = T adj (G λ ,h λ ) satisfies the finite modulus Abelian additive operation of the fully colored adjacent matrix group "T adj (G i ,h i )[+]T adj (G j ,h j )[-]T adj (G k ,h k )=T adj (G λ ,h λ )”, where L adj (w 0 )=T adj (G k ,hk ) is zero yuan overall.

[0377] Step-6-3. User private topological signature. Fully colored adjacent matrix T of vertex u adj (G i ,h i ) corresponds to a topologically encoded graph set G raph (T adj (G i ,h i )); The full coloring adjacent matrix T of vertex v adj (G j ,h j ) corresponds to a topologically encoded graph set G raph (T adj (G j ,h j )), so that each topological coding graph J u ∈G raph (T adj (G i ,h i )) is isomorphic to the unique topological coding graph J v ∈G raph (T adj (G j ,h j )), that is, Select Topological Encoding Graph J u ∈G raph (T adj (G i ,h i )) as the topological signature of vertex u, select G raph (T adj (G j ,h j )) is isomorphic to the topological signature J u Topological coding graph J v As a topological signature of a vertex v, such that no two vertices v i and v j To increase the difficulty of deciphering, the topological signature J u and topological signature J v They are respectively taken as Chinese character topological graph groups, see Chinese character topological graph group G in Figure 9 j (j∈[2,17]).

[0378] Step-6-4. File encryption. A uses the topological signature J of vertex u u The adjacent matrix A(J u ) and the topological encoding matrix T code (J u ,h u) to derive the topological signature digital string, and B uses the topological signature J of vertex v v The adjacent matrix A(J v ) and the topological encoding matrix T code (J v ,h v ) to derive the encrypted digital string. Then, A and B communicate with each other using the algorithms in (6.1), (6.2) and (6.3).

[0379] (6.6.2) Fully colored adjacent matrix group algorithm, community private zero element, user private topological signature. The specific implementation steps are the same as those in (6.5.2).

[0380] (6.6.3) Fully colored adjacent matrix group algorithm, user private zero element and private topological signature. The specific implementation steps are the same as those in (6.5.3).

[0381] (6.7) Different matrix groups. In the overall encryption of the LAN, different communities use different topological matrix groups, and the communication between communities is achieved by coloring transformation of the topological matrix.

[0382] (6.8) Special technology for implementing matrix group overall encryption

[0383] (6.8.1) Extended finite modules

[0384] Extended finite module: The restriction of module M in the finite module Abelian additive operation in Definition 10 is enlarged to M≥max{f(w):w∈V(G)∪E(G)}. So that the topological coding graph set F f (G) = {G 1 , G 2 , ..., G M Each vertex of the local area network N(t)(t∈[a,b]) can be given a topological coding graph that is different from other vertices, that is, there is a graph group coloring h: The coloring of any two vertices u and v in the local area network N(t)(t∈[a,b]) always satisfies G i =h(u)≠h(v)=G j .

[0385] Infinite hybrid topological coding graph group The local area network N(t) can be graph group colored so that each vertex of the local area network N(t) has a topological coding graph that is different from other vertices.

[0386] (6.8.2) Increase the complexity of the algorithm: In step 5-3 of the topological coding matrix group algorithm (6.5.1), the topological signatures H of the communicating parties u and topological signature H vCan be taken as Chinese character topological graph groups, see 16 Chinese character topological graph groups G in Figure 9 2 , G 3 , ..., G 17 .

[0387] (6.8.3) Increase the number of bytes in the string: ① Use larger integers k and d, and parameter full coloring topological encoding matrix

[0388]

[0389] Not only can you get a long byte string, but also when T code (G, h) is the parameterized full coloring topological encoding matrix P for each coloring type matrix -k (k∈[3, 24]) in Definition 7 code (G, H|k, d) can preserve every coloring type matrix -k (k∈[3, 24]) in Definition 7, see Theorem 4.

[0390] ② Implement the coloring transformation of the reversible topological matrix The new topology matrix can be used to derive a long byte string S new , under the reversible transformation It is the string of the original topology matrix.

[0391] (6.8.4) string to generate a key pair (see Example 2).

[0392] (6.8.5) Pan-topological coding matrix technology (see Definition 6).

[0393] (6.8.6) Integer splitting problem, set-ordered graceful labeling algorithm for caterpillar trees

[0394] Integer splitting problem. A star tree K 1，m Infinitely many caterpillar trees with m leaves can be generated, see Figure 11. However, the problem of determining the number of such caterpillar trees with m leaves can be reduced to the integer splitting problem: suppose there are n(m, k) ways to split a positive integer m into k positive integers m 1 , m 2 , ..., m k The sum of m = m 1 +m 2 +…+m k (w i ≥1, 2≤k). However, the exact value of the computation number n(m, k) is similar to the Diophantine equation The solution of is associated with , so the calculation of the number n(m, k) is a difficult problem. In other words, it is difficult to solve the number of trees with m leaves.

[0395] The difficulty assumption of integer factorization is the basis of the security of many cryptographic algorithms. No polynomial algorithm for calculating the number n(m, k) has been reported.

[0396] Theorem 9. There are infinitely many trees with the same number of leaves.

[0397] A growing algorithm for set-ordered graceful labeling of caterpillar trees

[0398] Step-1. Caterpillar Tree T 1 It's Star Tree K 1，q(1) , whose central vertex is u 1 , Caterpillar Tree T 1 There are q(1)(≥1) leaves. Using the meta-coding matrix in Definition 6, we can give the caterpillar tree T 1 Implement set-ordered linear graceful coloring to obtain the set-ordered linear graceful topological encoding matrix in Definition 7

[0399] Step-2. Give the caterpillar tree T again 1 The full coloring topological encoding matrix T code (T 1 ) implements the dual coloring transformation of the matrix θ 1 , that is, the element encoding matrix (q(1)-0, k, q(1)-k) T =θ 1 [(0, k, k) T ], get the caterpillar tree Caterpillar Tree T 2 The set-ordered linear graceful topological encoding matrix is

[0400]

[0401] Making the Caterpillar Tree T 2 A leaf of 2 The coloring is 0.

[0402] Step-3. Give the caterpillar tree T 2 The leaf u with color 0 2 Connect q(2)(≥1) new leaves to get the caterpillar tree T 3 . Use the meta-coding matrix in Definition 6 to encode the caterpillar tree T 3 Color the q(2) leaves that are not colored, and get the caterpillar tree Making the Caterpillar Tree T 4 The set-ordered linear graceful topological encoding matrix is

[0403]

[0404] Step-4. Give the caterpillar tree T 4 The full coloring topological encoding matrix Tcode (T 4 ) implements the dual coloring transformation of the matrix θ 2 , get the caterpillar tree Making the Caterpillar Tree T 5 The set-ordered linear graceful topological encoding matrix is ​​T code (T 5 )=θ 2 [T code (T 4 )], and the caterpillar tree T 5 A leaf of 5 The coloring is 0, and the full coloring topological encoding matrix T code (T 4 ) each element encoding matrix (a, b, c) T The dual coloring transformation of is

[0405]

[0406] Step-(n). According to the inductive hypothesis, the caterpillar tree T n A leaf of n The color of is 0. Give the caterpillar tree T n Leaves n Connect q(n)(≥1) new leaves to get the caterpillar tree T n+1 . Give the caterpillar tree a T n+1 Color the q(n) leaves that are not colored, and get the caterpillar tree Caterpillar Tree T n+2 The set-ordered linear graceful topological encoding matrix

[0407]

[0408] Step - (n+1). Give the caterpillar tree T n+2 The topological encoding matrix T code (T n+2 ) implements the dual coloring transformation of the matrix θ n+2 , get the caterpillar tree T n+3 , and there is Caterpillar Tree T n+3 The set-ordered linear graceful topological encoding matrix is ​​T code (T n+3 )=θ n+2 [T code (T n+2 )], and the caterpillar tree T n+3 A leaf of n+3 The coloring is 0.

[0409] The topological coding matrix T in the growing algorithm for the set-ordered graceful labeling of caterpillar treescode (T n+2 ) is also called the graph structure type full coloring topological encoding matrix, or the spanning tree full coloring topological encoding matrix, caterpillar tree T n+2 is the topological encoding matrix T code (T n+2 ) (Definition 19). Attached Figure 12 uses the caterpillar tree T in Attached Figure 11 to give the complete process of implementing the "growth algorithm for set-ordered graceful labeling of caterpillar trees".

[0410] Theorem 10. According to Definition 7, Theorem 4 and the growing algorithm for set-ordered graceful labeling of caterpillar trees, every caterpillar tree has every coloring type matrix -k (k∈[3,24]) in Definition 7.

[0411] Since the set-ordered graceful labeling of trees can derive dozens of tree-based labeling and coloring, the graph group generated by the growth algorithm of the set-ordered graceful labeling of caterpillar trees has a higher convenience and practicality in the previous algorithms. Importantly, the problem of allocating the number of leaves of the caterpillar tree (see Figure-11 and Theorem 9) can be reduced to the integer splitting problem, and the number of added leaves is random, which brings unbearable time overhead to the codebreaker.

Claims

1. The characteristics of the asymmetric topological matrix overall encryption technology for the local area network are: the full-coloring topological coding matrix is ​​constructed based on the union operation of the meta-coding matrix, the edge-coloring adjacent matrix and the full-coloring adjacent matrix are constructed by the topological matrix coloring transformation, and the topological matrix algebraic operation constitutes the topological matrix matching pair, the parameter full-coloring topological coding matrix and the pan-topological coding matrix; the graph homomorphism, topological matrix homomorphism, graph group and topological matrix group technology are used to implement asymmetric overall encryption for the local area network.

2. The union operation of the element coding matrices according to claim 1 constitutes a fully colored topological coding matrix, whose topological characteristics are: (2.1) The meta-coding matrix is ​​a matrix with 3 rows and 1 column. (2.2) The fully colored topological coding matrix is ​​a matrix with 3 rows and q columns, which is the union of q element coding matrices. It corresponds to a topological coding graph called a cluster point, and also corresponds to a graph set, so that each topological coding graph of this graph set (considered as a public key) is color-preserving and is isomorphic to the cluster point (considered as a private key), thus forming a "one-to-many" topological coding key group.

3. According to the edge-colored adjacent matrix and the full-colored adjacent matrix described in claim 1, the matrix transformation characteristics are: (3.1) Perform matrix coloring on the adjacency matrix of the topological coding graph to obtain the edge-colored adjacency matrix and the fully-colored adjacency matrix. (3.2) The directed topological coding graph has a directed topological coding matrix, a directed edge-colored adjacent matrix, and a directed fully-colored adjacent matrix.

4. The parameter fully colored topological coding matrix and the pan-topological coding matrix according to claim 1, characterized in that: (4.1) Performing matrix linear algebra operations on the full-color topological coding matrix to obtain a parameter full-color topological coding matrix; (4.2) By performing algebraic substitution on the full-coloring topological coding matrix, we obtain the following classes of universal topological coding matrices: string-coloring topological coding matrix, matrix-coloring topological coding matrix (similar to tensor matrix), vector-coloring topological coding matrix, set-coloring topological coding matrix (hypergraph-coloring topological coding matrix), and thing-coloring matrix.

5. According to the graph homomorphism and topological matrix homomorphism described in claim 1, the algebraic homomorphism characteristics are: (5.1) Color-preserving graph homomorphisms and color-preserving topological matrix homomorphisms are. (5.2) Adjacent matrices are homomorphic, color-preserving edge coloring adjacent matrices are homomorphic, and color-preserving total coloring adjacent matrices are homomorphic.

6. The topological matrix matching pair according to claim 1, wherein the topological feature is that the matrix matching pair forms a public key and a private key matching pair, (6.1) Edge-coloring adjacent matrix matching pairs; (6.2) Full-coloring adjacent matrix matching pairs; (6.3) Twin parameter topological encoding matrix matching pairs; (6.4) Parameter and harmonious mirror topological coding matrix matching pairs; (6.5) Mirror topological coding matrix matching pairs; (6.6) String set matching pairs; (6.7) Parameter topological coding matrix and curve matching pairs.

7. According to the graph group and topological matrix group described in claim 1, their algebraic topological characteristics are: using finite module and non-module Abelian additive operations to obtain: (7.1) graph group on graph set; (7.2) edge colored adjacent matrix group on edge colored adjacent matrix set; (7.3) fully colored adjacent matrix group on fully colored adjacent matrix set; (7.4) topological coding matrix group on fully colored topological coding matrix set; (7.5) parameter fully colored topological coding matrix group on parameter fully colored topological coding matrix set; (7.6) hybrid topological coding matrix group and 3-dimensional topological coding matrix group on hybrid topological coding matrix set.

8. The asymmetric overall encryption of the local area network according to claim 1 is technically characterized by using a topological coding graph group and a topological matrix group to color the local area network as a whole, so that the communication between network nodes is doubly protected by "topological structure" and "mathematical constraints". The above items (2.1), (2.2), (3.1), (3.2), (5.1) and (5.2) are innovations, and no similar research reports have been found on the Internet.

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