Asymmetric topology encryption technology based on topology coding graph set

Abstract

According to the invention, a topological coding graph set is adopted to make asymmetric key pairs and groups, and a corresponding encryption algorithm is designed; a topology encryption management center is created, asymmetric topology key pairs and groups of network users are manufactured and managed, diversity of topology signatures can be achieved, and the asymmetric topology key pairs and groups similar to'one-time pad 'are provided. The encryption technology has the advantages that public key and private key topological signatures and uniqueness thereof are achieved, a topological coding graph is input into a computer through a topological matrix, topological structure space is huge, a plurality of NP-complete problems are associated, topological coloring types are complex, the number is large, new topological coloring is generated every day, character strings exported through the topological matrix are irreversible, data functions are complete, and the method is suitable for being applied to a computer. The asymmetric multi-topology encryption system can cope with AI attacks of equipment quantum computers. The practical algorithm designed by adopting the patent technology does not need to prove that the algorithm has calculation safety and certifiable safety every time.

Description

[0001] I. Technical Field: Information Encryption Field

[0002] II. Background Art: Graph Theory, Combinatorial Mathematics, Topological Coding Theory

[0003] III. Summary of the Invention: Asymmetric topological encryption is a new technology in the field of information encryption. It has functions such as key establishment, encryption and decryption, identity signature, authentication, and complete data, providing a feasible and secure practical technology for identity verification, confidentiality, integrity, and non-repudiation.

[0004] The topological coding graph in asymmetric topological encryption can achieve asymmetric diversity. Based on graph operations and coloring techniques, we can achieve: one public key topological signature corresponding to multiple private key topological signatures, multiple public key topological signatures corresponding to a single private key topological signature, multiple public key topological signatures corresponding to multiple private key topological signatures, and an approximate "one-time pad" asymmetric topological key.

[0005] The topological coding graph and its topological coding matrix in asymmetric topological encryption are the main technical tools of the present invention. "Topological structure" and "mathematical constraints" constitute the topological coding graph and its topological coding matrix, which are "soft mathematical expressions". "Mathematical constraints" are based on the theories and practical technologies of disciplines such as graph theory, number theory, set theory, abstract algebra, linear algebra, and computer science, while "topological structure" is a tool for semi-structured data, not an ordinary geometric graph or picture. This makes it necessary to cross more than 2 essentially different branches of mathematics and computer science to break the topological coding graph and its topological coding matrix.

[0006] Since a graph is a natural representation of an encoding relationship structure, calculations based on graph-structured data are widely used in various fields.

[0007] To have a preliminary understanding of the technology of the present invention, we give the following example: In Figure - 1, L is the Chinese character graph of "You", and its adjacent matrix A(L) 10×10 is given in formula (1), where O m×n is an m×n zero matrix with all elements being zero, and the transpose matrix of the edge adjacent matrix E adj (V) 6×4 is The adjacent matrix A(C) 10×10 is a symmetric square matrix, that is, the adjacent matrix is equal to its own transpose matrix, i.e., A(C) 10×10 = 4 T (C) 10×10 .

[0008]

[0009] The full-coloring topological coding matrix T of the Chinese character graph L of "You" in Figure - 1code (L) satisfies T code (L)=T code (L*). Perform vertex tearing operation on the Chinese character graph L of "you", and obtain the graph L in Figure - 1 1 and the graph L 2 and the graph L 3 . It is not difficult to observe that the topological coding matrices of these 3 graphs all satisfy T code (L*) = T code (L) = T code (L 1 ) = T code (L 2 ) = T code (L 3 ).

[0010] However, the Chinese character graph L of "you", the graph L 1 and the graph L 2 and the graph L 3 are non - isomorphic to each other, that is, their topological structures are non - isomorphic, and their adjacency matrices A(L) 10×10 , A(L 1 ) 12×12 , A(L 2 ) 13×13 and A(L 3 ) 14×14 are pairwise non - similar. The adjacency matrix is the topological coding graph used in the present invention as a topological signature, which is the theoretical guarantee for the uniqueness of identity authentication.

[0011] The adjacency matrix A(L) 10×10 can derive (100)! (≈2 524 ) non - identical digital strings, and the topological coding matrix T code (L*) 3×13 can derive (39)! (>2 153 ) non - identical text strings. These digital strings and text strings are the basic tools used for encryption and decryption in the present invention.

[0012] The Chinese character graph L of "you" is a bipartite graph, and its vertex set is V(L)=X∪Y, where X = {u 1 , u 2 , u 3 , u 4 , u 5 , u 6} and Y = {v 1 , v 2 , v 3 , v 4} and such that for each edge u i v j ∈E(L) of the bipartite graph L, the 2 endpoints ui and v j satisfies u i ∈ X and v j ∈ Y. Note that, graph L, graph L 1 , graph L 2 and graph L 3 are all bipartite graphs. In the discipline of topological coding, bipartite graphs have many good properties, which can be directly applied to the technical theory and algorithm implementation of the present invention.

[0013] Figure - 2 gives 10 distinct set - ordered W - constrained labelings of the Chinese character graph L of "you" (see Definition 7 and the notes of Definition 7). The following graphs with total coloring (total labeling) are collectively called topological coding graphs. In Figure - 2, there are:

[0014] (a) The topological coding graph G 1 admits a set - ordered graceful labeling f 1 , such that for each edge u i v j ∈ E(G 1 ), the coloring satisfies the graceful constraint equation f 1 (u i v j ) = |f 1 (u i ) - f 1 (v j )|, and the edge - coloring set is f 1 (E(G 1 )) = {f 1 (u i v j ) : u i v j ∈ E(L)} = {1, 2,..., 13}.

[0015] (b) The topological coding graph G 2 admits a set - ordered odd - graceful labeling f 2 , such that for each edge u i v j ∈ E(G 2 ), it satisfies the odd - graceful constraint equation f 2 (u i v j ) = |f 2 (u i ) - f 2 (v j )|, and the edge - coloring set f 2 (E(G 2 )) = {f 1 (u i v j ) : u i vj ∈E(G 2 )} = {1, 3, 5, ..., 25}, the topological coding graph G 2 's total coloring topological coding matrix

[0016]

[0017] Total coloring topological coding matrix T code (G 2 , f 2 ) can derive (39)! (> 2 153 ) digital strings with 65 different bytes.

[0018] (c) The topological coding graph G 3 admits a set-ordered edge-difference labeling f 3 such that for each edge u i v j ∈E(G 3 ) the coloring satisfies the edge-difference constraint equation f 3 (u i v j ) + |f 3 (u i ) - f 3 (v j )| = 14.

[0019] (d) The topological coding graph G 4 admits a set-ordered edge-magic labeling f 4 such that for each edge u i v j ∈E(G 4 ) the coloring satisfies the edge-magic constraint equation f 4 (u i ) + f 4 (u i v j ) + f 4 (v j ) = 21.

[0020] (e) The topological coding graph G 5 admits a set-ordered happy-difference labeling f 5 such that for each edge u i v j ∈E(G 5 ) the coloring satisfies the happy-difference constraint equation |f 5 (u i ) + f 5 (v j ) - f 5 (u i v j )| = 7.

[0021] (f) Topological coding graph G 6 Admitting set ordered graceful labeling h 1 ; (g) Topological coding graph G 7 Admitting set ordered odd graceful labeling h 2 ; (h) Topological coding graph G 8 Admitting set ordered edge - difference labeling h 3 ; (i) Topological coding graph G 9 Admitting set ordered arithmetic - progression labeling h 4 ; (j) Topological coding graph G 10 Admitting set ordered edge - magic labeling h 5 .

[0022] The above facts give the following encryption and decryption processes:

[0023] One - to - one correspondence process: A topological coding graph L corresponds to a unique adjacent matrix A(L) to ensure the uniqueness of identity, and a digital string is derived for encrypting and decrypting files.

[0024] Many - to - one correspondence process: Multiple topological coding graphs correspond to a topological coding matrix T code , and a digital string is derived for encrypting and decrypting files.

[0025] Figure - 2 gives the following facts:

[0026] (1) For k = 1, 2, 3, 4, 5, each coloring f k satisfies the set - ordered constraint max f k (X k ) < min f k (Y k ) holds, each coloring h k satisfies the set - ordered constraint max h k (Y k ) < min h k (X k ) holds, V(L) = V(G k ) = X k ∪Y k . This property leads to a random growth algorithm for randomly adding leaves.

[0027] (2) Since the coloring of each vertex w ∈ V(G k ) satisfies the label transformation h k (w) = max f k (V(G k )) + min f k (V(G k )) - f k (w), f k (V(G k )) = {fk (w): w ∈ V(L)}, we call the labeling f k and the labeling h k (k = 1, 2, 3, 4, 5) is a pair of dual labelings, which lead to the labeling f k being the private - key coloring, and the labeling h k being the public - key coloring.

[0028] (3) There exists a one - to - one mapping θ k such that the fully - colored topological encoding matrix T code (G k , f k ) can be transformed into the fully - colored topological encoding matrix T code (G k , h k ), that is, there is a fully - colored topological encoding matrix transformation T code (G k , h k ) = θ k [T code (G k , f k )](k = 1, 2, 3, 4, 5).

[0029] Figure - 3 gives the parametric W - constrained (k, d) - labelings admitted by the Chinese character graph L of "You" (see Figure - 1), which can be used to generate the keys for "one - time pad". According to Definition 9, we have:

[0030] ① The topological encoding graph P a，1 (as the public key) admits a graceful (k, d) - labeling f a，1 such that for each edge u i v j ∈ E(P a，1 ) satisfies the graceful - difference constraint equation f a，1 (u i v j ) = |f a，1 (u i ) - f a，1 (v j )|, and its parametric fully - colored topological encoding matrix is P code (P a，1 , f a，1 |k, d) = k·I 0 +d·T code (P a，1 , f 1 );

[0031] ② The topological encoding graph P a，2 (as the private key) admits a graceful (k, d) - labeling f a，2 , and for each edge u i vj ∈E(P a，2 ) satisfies the graceful difference constraint equation f a，2 (u i v j ) = |f a，2 (u i ) - f a2 (v j )|, and its parameter all - coloring topological coding matrix is P code (P a，2 , f a，2 |d, k) = d·I 0 + k·T code (P a，2 , f 2 );

[0032] ③ The topological coding graph P a，3 admits an edge - difference (k, d) - total labeling f a，3 , such that for each edge u i v j ∈E(P a，3 ) satisfies the edge - difference constraint equation f a，3 (u i v j ) + |f a，3 (u i ) - f a，3 (v j )| = 2k + 12d, and its parameter all - coloring topological coding matrix is P code (P a，3 , f a，3 |k, d) = k·I 0 + d·T code (P a，3 , f 3 );

[0033] ④ The topological coding graph P a，4 admits an edge - magic (k, d) - total labeling, such that for each edge u i v j ∈E(P a，4 ) satisfies the edge - magic constraint equation f a，4 (u i ) + f a，4 (u i v j ) + f a，4 (v j ) = 2k + 19d, and its parameter all - coloring topological coding matrix is P code (P a，4 , f a，4 |k, d) = k·I 0 + d·T code (P a，4, f 4 )。 IV. Description of the Drawings:

[0034] Appendix Figure 1 , Chinese character graph L of "You", topological coding matrix T code (L*).

[0035] Appendix Figure 2 , 10 different set-ordered W-constrained total labelings of the Chinese character graph L of "You" (see Figure - 1).

[0036] Appendix Figure 3 , Graceful (k, d)-labelings of the Chinese character graph L of "You" (see Figure - 1): (a) Graceful (k, d)-total labeling (as public key); (b) Graceful (d, k)-total labeling (as private key); (c) Edge-difference (k, d)-total labeling; (d) Edge-magic (k, d)-total labeling.

[0037] Appendix Figure 4 , Explanation of "Uniform Fractal Forest Problem" I: (a)-(d) Forest H = {A 1 , A 2 , A 3 , A 4}; (e)-(g) 4 non-isomorphic vertex-coincident graphs.

[0038] Appendix Figure 5 , Explanation of "Uniform Fractal Forest Problem" II: (a) Loopless vertex-coincident graph Q = A 1 [·]A 2 [·]A 3 [·]A 4 . After performing the vertex-ripping operation on the loopless vertex-coincident graph Q, 3 uniform fractal forests are obtained: (b) H 1 = {T 1 , T 2 , T 3}; (c) H 2 = {B 1 , B 2 ,..., B 8}; (d) H 3 = {C 1 , C 2 ,..., C 7}.

[0039] Appendix Figure 6 , Example for explaining Definition 11.

[0040] Appendix Figure 7 , Example I for explaining the "v-set e-normal graceful (k, d)-total coloring algorithm for disconnected graphs".

[0041] AppendixFigure 8 Example 2 for explaining the "v - set e - normal graceful (k, d)-total coloring algorithm for disconnected graphs".

[0042] Appendix Figure 9 、 One An example of a 0 - rotatable graceful labeling.

[0043] Appendix Figure 10 Explanation of the basic algorithm - I.

[0044] Appendix Figure 11 One of the functions of the Topological Encryption Management Center (TEMC).

[0045] Appendix Figure 12 Another function of the Topological Encryption Management Center (TEMC).

[0046] Except for common mathematical concepts and symbols, the mathematical concepts, terms and notations such as coloring and labeling used in this invention are all from the following literature:

[0047] [1] J.A. Bondy, U.S.R. Murty. Graph Theory [M]. Springer London, 2008. DOI: 10.1007 / 978 - 1 - 84628 - 970 - 5.

[0048] [2] Joseph A. Gallian. A Dynamic Survey of Graph Labeling [J]. The electronic journal of combinatorics, #DS6, Twenty - fourth edition, December 9, 2021. (623 pages, 3295 references, more than 200 graph labelings)

[0049] [3] Bing Yao (Yao Bing), Hongyu Wang. Recent Colorings And Labelings In Topological Coding [J]. 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 graph labelings and graph colorings)

[0050] [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[J]. arXiv:1909.01587v2[cs.IT] 15 Sep 2019. pp 1 - 29.

[0051] [5] Bing Yao, Xiaohui Zhang, Hui Sun, Jing Su, Fei Ma, Hongyu Wang. Parameterized Colorings And Labellings Of Graphs In Topological Coding[J]. arXiv:2207.03381[cs.IT], 7 Jul 2022. pp 1 - 149. V. Specific implementation manners:

[0052] The specific implementation manners are introduced in four parts: (I) Basic concepts and terms; (II) Set (group) of colorless topological keys; (III) Set (group) of colored topological keys; (IV) Key production and encryption algorithm for the set (group) of topological keys.

[0053] Hereinafter, research objects and concepts are expressed by "definitions", theoretical techniques are expressed by "theorems", and specific algorithmizable, programmable, and practical techniques are expressed by "algorithms".

[0054] (I) Basic concepts and definitions

[0055] (1.1) Basic terms and notations

[0056] The set of all positive integers is denoted as Z + , and the set of all non - negative integers is denoted as Z 0 ={0} ∪ Z + . The number of elements contained in a set S is denoted as |S|. Unless otherwise stated, the numbers used hereinafter are all non - negative integers. For non - negative integers a, b, α, β ∈ Z 0 , 0 ≤ a < b, the notation [a, b] represents a set of non - negative integers {a, a + 1,..., b}; when α and β are odd and α < β, denote [α, β] o ={α, α + 2,..., β}

[0057] For positive integers \(p,q\in\mathbb{Z}\) + , a graph \(G\) with \(p(\geq2)\) vertices and \(q\) edges is simply called a \((p,q)\)-graph; a vertex of degree 1 in graph \(G\) is called a leaf; \(L\) eaf (G) is the set of all leaves of graph \(G\).

[0058] The vertex set of the bipartite graph \(H\) is \(V(H)=X\cup Y\), and the intersection such that for each edge \(uv\) of the bipartite graph \(H\), the two endpoints \(u\) and \(v\) satisfy \(u\in X\) and \(v\in Y\). In the discipline of graph theory, a necessary and sufficient condition for a graph to be bipartite is that the graph contains no odd cycles. A \((p,q)\)-tree graph is a connected graph and contains no cycles, satisfying \(q = p - 1\). Therefore, a tree graph is bipartite, and a tree graph is also called a linear topology.

[0059] A string is a general term for a digital string (digital encoding), a text string (text encoding), or a digital-text mixed string.

[0060] Inputting a graph into a computer uses the adjacency matrix and the full-coloring topological coding matrix of this graph. The adjacency matrix of a \((p,q)\)-graph \(G\) is a symmetric square matrix \(A(G)\) of order "p×p" p×p =(a i，j ) p×p , where \(V(G)=\{u 1 ,u 2 ,\cdots,u p \}, when the graph \(G\) has an edge \(u i u j , a i，j = 1, otherwise a i，j = 0. Algebraic graph theory proves the following conclusion:

[0061] Theorem 1. Let graph \(G\) have 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 an invertible matrix \(Q\) such that \(A(G)=QB(G)Q -1 holds.

[0062] According to Theorem 1, any two adjacency matrices of a graph are similar. Therefore, under the similarity operation of adjacency matrices, we say that the \((p,q)\)-graph \(G\) and its adjacency matrix \(A(G)\) p×p are one-to-one. In asymmetric topological encryption, due to the one-to-one correspondence between the graph \(G\) and its adjacency matrix, using the graph \(G\) as a "topological signature" has the uniqueness of identity authentication. In addition, similar matrices have the same rank, the same determinant, the same eigenvalues, and the same invertibility, providing rich techniques for asymmetric topological encryption technology.

[0063]

[0064] Definition 1: [4] Let the topological coding graph \(G\) with \(q\) edges admit a \(W -\)constrained total coloring \(f:V(G)\cup E(G)\to[a,b]\). The edge set of the topological coding graph \(G\) is \(E(G)=\{e i =x i y i :i\in[1,q]\}\). The matrix

[0065]

[0066] is called the total - coloring topological coding matrix of the topological coding graph \(G\), where \(X f =(f(x 1 ),f(x 2 ),\cdots,f(x q ))\) and \(Y f =(f(y 1 ),f(y 2 ),\cdots,f(y q ))\), \(E f =(f(e 1 ),f(e 2 ),\cdots,f(e q ))\); the coloring of each edge \(e i \in E(G)\) satisfies the \(W -\)constrained equation \(W[f(x i ),f(e i ),f(y i )]=0(i\in[1,q])\).

[0067] Notes on Definition 1:

[0068] ① Different from the adjacent matrix, a topological coding matrix can correspond to multiple topologically non - isomorphic topological coding graphs. The total - coloring topological coding matrices of the graphs \(L 1 \), \(L 2 \) and \(L 3 \) in Figure - 1 all satisfy \(T code (L^*) = T code (L)=T code (L 1 ) = T code (L 2 ) = T code (L 3 )

[0069] ② Two different colorings \(f\) and \(h\) of a topological coding graph \(G\) result in different total - coloring topological coding matrices \(T code (G,f)\) and \(T code (G,h)\).

[0070] If there exists a coloring transformation \(\theta\) such that \(h=\theta(f)\), then there is a total - coloring topological coding matrix transformation \(T\)code (G, h) = θ[T code (G, f)].

[0071] ③ The mathematical constraint "W-constraint" in Definition 1 is an abbreviated form of the n-dimensional constraint W(k 1 , k 2 ,..., k n )(n≥1), and a topological coding graph with n-dimensional constraints can be constructed, which is a multiple encryption technology to prevent attackers from forging and deciphering topological keys.

[0072] ④ The matrices associated with the graph (adjacency matrix, topological coding matrix) are collectively called topological matrices.

[0073] ⑤ The digital string derived from the topological matrix is also called the topological digital string.

[0074] ⑥ If there is no confusion, in the following narrative, we will omit the matrix orders "n×n" and "3×q" at the lower right corners of the adjacency matrix A(G) n×n and the total coloring topological coding matrix A(G) 3×q .

[0075] (II) Colorless topological key set (group)

[0076] (1.2) Vertex tearing operation and non-multiple-edge vertex coincidence operation

[0077] Definition 2: Tear a vertex w∈V(G) of the graph G into k (≥2) vertices w 1 , w 2 ,..., w k , and the vertex neighbor sets N(w 1 ), N(w 2 ),..., N(w k ) of these k vertices and the vertex neighbor set N(w) of the vertex w satisfy N(w) = N(w 1 ) ∪ N(w 2 ) ∪... ∪ N(w k ), and the intersection of the neighbor sets of any two vertices w i and w j makes the vertex w the result of the coincidence of these k vertices, that is, w = w [·]w 1 [·]...[·]w 2 [·]...[·]w k , and the obtained vertex tearing graph is denoted as H = G[∧ k w. The graphs G and H are called a vertex tearing operation topological signature key pair, that is, the graph homomorphism of the graph H to the graph G: H→G, such that |V(H)|≥1 + |V(G)|.

[0078]

[0079] In the "uniform fractal forest problem", due to the diversity of vertex tearing operations, a connected graph G may have multiple uniform fractal forests, such that the connected graph G and these uniform fractal forests form a "one-to-many" key matching group. The edge set E(G) of the connected graph G is the trivial uniform fractal forest. If the forest T = {B 1 , B 2 ,..., B m} is a uniform fractal forest of the connected graph G, then |E(B i )| = |E(G)| / m.

[0080] For the branch trees of the uniform fractal forest UF = {H 1 , H 2 ,..., H m}, perform the non-repeated-edge vertex coincidence operation (see Definition 3) to obtain multiple non-repeated-edge vertex coincidence graphs, and gather them in the non-repeated-edge vertex coincidence graph set (see the example in Figure - 3), such that each non-repeated-edge vertex coincidence graph G ∈ C oin ([·]UF) can vertex-tear out each tree H k of UF. We say that the non-repeated-edge vertex coincidence graph set U coin ([·]UF) is the public key set of the private key group UF.

[0081] The difficulty of cracking the topological signature key pair under the graph isomorphism operation can be reduced to the subgraph isomorphism NP-complete problem.

[0082] Definition 3: If the vertex neighborhood sets of two vertices u and v in the graph T have no common vertices, that is, their neighbor vertex sets satisfy then the vertices u and v can be coincided into one vertex w, that is, w = u[·]v. The obtained non-repeated-edge vertex coincidence graph is denoted as L = T(u[·]v), and it is said that the graphs T and L are a non-repeated-edge vertex coincidence operation topological signature key pair, that is, the graph homomorphism of the graph T to the graph L: T → L, such that |V(T)| = 1 + |V(L)| and |E(T)| = |E(L)|.

[0083] To produce the key set (group), the present invention proposes the following uniform fractal forest problem:

[0084]

[0085]

[0086] Obviously, finding the optimal uniform fractal forest and the optimal uniform fractal forest of a connected graph G is a subgraph isomorphism NP-complete problem.

[0087] Example 1. An example to explain the "uniform fractal forest problem". In Figure - 4, (a)-(d) are the uniform fractal forest H = {A 1 , A 2 , A 3 , A 4}; (e)-(g) are 4 non-isomorphic vertex - coincident graphs. In Figure - 5, (a) After performing the non - multiple - edge vertex - coincidence operation on the uniform fractal forest H in Figure - 4, the non - multiple - edge vertex - coincidence graph Q = A 1 [·]A 2 [·]A 3 [·]A 4 is obtained; After performing the vertex - tearing operation on the non - multiple - edge vertex - coincidence graph Q, 3 uniform fractal forests are obtained: (b) H 1 = {T 1 , T 2 , T 3}; (c) H 2 = {B 1 , B 2 ,..., B 8}; (d) H 3 = {C 1 , C 2 ,..., C 7}. The uniform fractal forest H 1 = {T 1 , T 2 , T 3} is the optimal uniform fractal forest of the non - multiple - edge vertex - coincidence graph Q and is also a distinguishable uniform fractal forest.

[0088] Note that, after performing the non - multiple - edge vertex operation on the uniform fractal forest H in Figure - 4, many non - multiple - edge vertex - coincidence graphs can be obtained, thus indicating the topological structure complexity of the "uniform fractal forest problem".

[0089] Regarding the "uniform fractal forest problem", we have the following questions:

[0090] (1) How many optimal (distinguishable) uniform fractal forests does a connected graph have?

[0091] (2) Determine all non - trivial uniform fractal forests of a connected graph.

[0092] (3) Characterize the connected graphs with a unique optimal uniform fractal forest.

[0093] (4) After performing the non - multiple - edge vertex - coincidence operation on a uniform fractal forest EF = {T 1 , T 2 ,..., T m}, the non - multiple - edge vertex - coincidence graph set Determine the graph set Coin ([·]EF) The structure and number of graphs in the figure | C oin ([·]EF)|. If there is a graph Q* ∈ C with no multiple edges and vertex coincidence oin ([·]EF), for any pair of non-adjacent vertices u and w in it, there are common adjacent vertices, that is, the neighbor sets of the two vertices u and w satisfy We call the graph Q* the accumulation point of the graph set C oin ([·]EF). For other properties of the graphs in the graph set C oin ([·]EF), please refer to the "forest vertex coincidence graph problem" later.

[0094] The set of q edges of a connected (p, q)-graph G is a uniform fractal forest, which is a trivial solution to the "uniform fractal forest problem". By performing vertex tearing operations on the connected (p, q)-graph G, it is not difficult to obtain the uniform fractal forest F ore (G) = {A 1 , A 2 ,..., A n}, |E(G)| = |E(F ore (G))|, such that the number of edges of each tree A ore (i ∈ [1, n]) in the uniform fractal forest F i (G) satisfies |E(A i )| ≤ 2. Moreover, there are more than two such uniform fractal forests. Generally speaking, this kind of uniform fractal forest F ore (G) is not the optimal uniform fractal forest of the connected graph G. We have proved the following conclusion:

[0095] Theorem 2. Under vertex tearing operations: (1) Each connected graph has a non-trivial optimal uniform fractal forest.

[0096] (2) Each connected graph is the result of non-multiple-edge vertex coincidence operations of no less than two different uniform fractal forests.

[0097] The inverse problem of the "uniform fractal forest problem" is the following forest vertex coincidence graph problem:

[0098]

[0099] Note that for any non-multiple-edge vertex coincidence connected graph L ∈ C oin ([·]T), the accumulation point G* of the graph set C oin ([·]T) satisfies |V(G*)| ≤ |V(L)|.

[0100] Theorem 3. In the "forest vertex coincidence graph problem", if the forest T = {T 1 , T 2 ,..., Tn each branch tree T of k has a diameter D(T k ) ≥ 3. According to Theorem 6, the branch tree T k admits at least β k graceful (k, d)-total colorings (see Definitions 7 and Theorem 5), where and m k + 1 = [D(T k ) / 2]. Then each loopless vertex-coincident connected graph G in the set C oin ([·]T) admits at least parameter set total colorings.

[0101]

[0102] Regarding the "leaf number constrained vertex tearing tree problem", we have:

[0103] (1) Let n d (T) denote the number of vertices of degree d in the tree T. Then there is a formula for calculating the leaves of the tree T where Δ(T) is the maximum vertex degree of the tree T. The leaf calculation formula can assist in the study of the "degree sequence determined tree problem".

[0104] (2) It involves the following integer partition problem:

[0105]

[0106] Definition 4. 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

[0107] (i) Each edge uv ∈ E(H) if and only if the edge F(u)F(v) ∈ E(G);

[0108] (ii) Each vertex x ∈ V(H) satisfies f(F(x)) = h(x), and each edge uv ∈ E(H) satisfies f(F(u)F(v)) = h(uv);

[0109] (iii) The number of edges is equal, |E(H)| = |E(G)|.

[0110] We say that the graph H is color-preserving graph homomorphic to the graph G. The color-preserving graph homomorphic operation is denoted as and there is a color-preserving total coloring topological coding matrix homomorphism

[0111] Definition 5. Let the set of topological coding graphs G rap (T codeEach topological coding graph in code corresponds to the 3×q order all-coloring topological coding matrix T in Definition 1 rap (T code ). If there is a topological coding graph G ∈ G rap (T code ) such that each topological coding graph H ∈ G (T code ) satisfies color-preserving graph homomorphism

[0112] Note of Definition 5: Given the 3×q order all-coloring topological coding matrix T in Definition 1 code corresponding to a topological coding graph set G rap (T code ), there is no network report on calculating the number of topological coding graphs in the topological coding graph set G rap (T code ). Since determining non-isomorphic topological coding graphs in the topological coding graph set G rap (T code ) encounters NP-complete problems, finding the color-preserving accumulation points of T code involves computational complexity problems such as color-preserving graph homomorphism and color-preserving all-coloring topological coding matrix homomorphism. This means that the all-coloring encryption technology using the topological coding graph set G rap (T code ) of the present invention has computational indecipherability and provable security

[0113]

[0114] (III) Colored Topological Key Set (Group)

[0115] (3.1) Coloring of Topological Coding Graphs

[0116] (3.1.1) Parameter Coloring of Topological Coding Graphs

[0117] Definition 6: Let the bipartite graph G admit a W-constrained coloring f: S → M, where the set The coloring set of the elements of the set S is denoted as f(S) = {h(w): w ∈ S}. If the set of vertex colorings of the graph G satisfies the set-ordered constraint

[0118]

[0119] We call this W-constrained coloring f the set-ordered W-constrained coloring of the bipartite graph G

[0120] Definition 7: Let G be a connected (p, q)-bipartite graph, then there is a vertex set V(G) = X ∪ Y, and the intersection such that for each edge \(xy\in E(G)\), \(x\in X\) and \(y\in Y\). For integers \(k,d\geq1\), if the bipartite graph \(G\) admits a \(W -\)constrained total coloring

[0121] \(f:X\rightarrow S\) m，0，0，d \(=\{0,d,\cdots,md\}\), \(f:Y\cup E(G)\rightarrow S\) q-1，k，d \(=\{k,k + d,k + 2d,\cdots,k+(q - 1)d\}\)

[0122] and satisfies the \(W -\)constrained equation \(W[f(u),f(uv),f(v)] = 0\) (\(uv\in E(G)\)), such that the edge - coloring set of the connected bipartite graph \(G\) is then the total coloring \(f\) is called a \(W -\)constrained \((k,d)-\)total coloring of the bipartite graph \(G\).

[0123] Notes on Definition 7:

[0124] (1) If the \(W -\)constrained \((k,d)-\)total coloring \(f\) in Definition 7 satisfies the set - order constraint \(\max f(X)<\min f(Y)\), we call \(f\) a set - ordered \(W -\)constrained \((k,d)-\)total coloring.

[0125] (2) If for any two vertices \(x\) and \(y\) of the connected bipartite graph \(G\), \(f(x)\neq f(y)\), then the total coloring \(f\) is called a \(W -\)constrained \((k,d)-\)total labeling

[0126] (3) Define the dual total coloring \(f^*\) of the \(W -\)constrained \((k,d)-\)total coloring \(f\) as follows:

[0127] (i) \(f^*(z)=\max\{f(z):z\in V(G)\}+\min\{f(z):z\in V(G)\}-f(z)\), \(z\in V(G)\),

[0128] (ii) \(W[f^*(u),f^*(uv),f^*(v)] = W[f(u),f(uv),f(v)] = 0\), \(uv\in E(G))\)

[0129] then the total coloring \(f^*\) is called the dual \(W -\)constrained \((k,d)-\)total coloring of the graceful \((k,d)-\)total coloring \(f\).

[0130] (4) The total coloring in Definition 7 allows two non - adjacent vertices to have the same color, and also allows the color of an edge \(xy\) to be the same as the color of one of its endpoints \(x\), i.e., \(f(xy)=f(x)\).

[0131] (5) When the parameters \((k, d)\) in Definition 7 are \((1, 1)\), \(W(f(u), f(uv), f(v)) = 0=f(uv)-|(f(u)-f(v)|\), and \(f(E(G))=[1, q]\), the total coloring \(f\) is an ordinary graceful total coloring (total labeling). When the set-ordered constraint \(\max f(X)<\min f(Y)\) holds, we call \(f\) a set-ordered graceful total coloring (total labeling). Naturally, the dual of the coloring \(f\) is also a set-ordered graceful total coloring (total labeling).

[0132] (6) When the parameters \((k, d)\) in Definition 7 are \((1, 2)\), \(W(f(u), f(uv), f(v)) = 0=f(uv)-|(f(u)-f(v)|\), and \(f(E(G))=[1, 2q - 1]\) o the graceful total coloring \(f\) is an odd-graceful total coloring, and its dual total coloring is also an odd-graceful total coloring. When the set-ordered constraint \(\max f(X)<\min f(Y)\) holds, we obtain a set-ordered odd-graceful total coloring (total labeling), and the dual of the coloring \(f\) is also a set-ordered odd-graceful total coloring (total labeling).

[0133] (7) When the parameters \(k\geq2\) and \(d\geq2\) in Definition 7, the \(W\)-constraint equation \(W[f(u), f(uv), f(v)] = 0=f(uv)-|(f(u)-f(v)|\), and \(f(E(G)) = S\) q-1，k，d we obtain a graceful \((k, d)\)-total coloring (total labeling). Further, if the set-ordered constraint \(\max f(X)<\min f(Y)\) holds, we obtain a set-ordered graceful \((k, d)\)-total coloring (total labeling).

[0134] (8) When the \(W\)-constraint equation \(W[f(u), f(uv), f(v)] = 0=f(uv)+f(u)+f(v)-k(k\geq1)\) holds for each edge \(uv\) of the dual graph \(G\), the total coloring \(f\) is called an edge-magic \((k, d)\)-total coloring (total labeling).

[0135] (9) When the \(W\)-constraint equation \(W[f(u), f(uv), f(v)] = 0=f(uv)+|f(u)-f(v)|-k(k\geq1)\) holds for each edge \(uv\) of the dual graph \(G\), the total coloring \(f\) is called an edge-difference \((k, d)\)-total coloring (total labeling).

[0136] (10) When the \(W\)-constraint equation \(W[f(u), f(uv), f(v)] = 0=||f(u)-f(v)|-f(uv)|-k(k\geq0)\) holds for each edge \(uv\) of the dual graph \(G\), the total coloring \(f\) is called a graceful-difference \((k, d)\)-total coloring (total labeling).

[0137] (11) When the W - constraint equation \(W[f(u), f(uv), f(v)] = 0=\vert f(u)+f(v)-f(uv)\vert - k(k\geq0)\) holds for each edge \(uv\) of the dual graph \(G\) of the total coloring \(f\), the total coloring \(f\) is called a happy - difference \((k, d)\) - total coloring (total labeling).

[0138] (3.1.2) Twin - matching coloring of the topological coding graph

[0139] Definition 8: For positive integers \(p\), \(q\), \(r\) and \(s\), let the connected \((p,q)\) - graph \(G\) admit a W - constraint total coloring \(f:V(G)\cup E(G)\to[a,a + q + s]\) such that the W - constraint equation \(W[f(u), f(uv), f(v)] = 0(uv\in E(G))\), \(a\in Z\) 0 . Another connected \((r,s)\) - graph \(H\) admits a \(W^*\) - constraint total coloring \(h:V(H)\cup E(H)\to[a,a + q + s]\) such that the \(W^*\) - constraint equation \(W^*[h(x), h(xy), h(y)] = 0(xy\in E(H))\). The colorings of the connected graph \(G\) and the connected graph \(H\) satisfy the equality of the edge - coloring sets \(f(E(G)) = h(E(H)) = M\), and the vertex - coloring sets of the two graphs satisfy \(f(V(G))\neq h(V(H))\).

[0140] (1) If there is then the connected graph \(G\) and the connected graph \(H\) are called an edge - M - twin \((W,W\) ＊ ) - matching pair.

[0141] (2) If but \(\vert f(V(G))\vert+\vert h(V(H))\vert\leq q + s\), then the connected graph \(G\) and the connected graph \(H\) are called a universal - M - twin \((W,W\) ＊ ) - matching pair.

[0142] Example 2: Figure - 6 gives an example to explain Definition 8. The connected \((7,8)\) - graph \(G\) in Figure - 6 admits an odd - graceful (W - constraint) total coloring \(f:V(G)\cup E(G)\to[0,15]\) such that for each edge \(uv\) of the connected graph \(G\), the coloring is \(f(uv)=\vert f(u)-f(v)\vert\), and the edge - coloring set is \(f(E(G))=\{1,3,5,7,9,11,13,15\}=[1,15]\) o .

[0143] When \(k\in[1,11]\), each connected graph \(O\) k admits a \(W^*\) - constraint total coloring \(f\) k : \(V(O\) k )\cup E(O\) k )\to[1,16]\), such that the edge - coloring set of the connected graph \(O\) k is \(f(E(O\) k )) = [1,15]o = f(E(G)), and f(V(G)) ≠ h(V(O k )) and According to Definition 8, the connected graph G and the connected graph O k are the edge [1, 15] o -twin (W, W ＊ ). Among them:

[0144] ① For i ∈ [1, 5], each edge xy of each connected graph O i is colored as f i (xy) = |f i (x) - f i (y)|. f is called the general odd-graceful total coloring, that is, "W*-constraint = general odd-graceful". i

[0145] ② For j = 6, 8, 9, each edge xy of each connected graph O j satisfies f j (x) + f j (xy) + f j (y) = 24. f is called the edge-magic total coloring, that is, "W*-constraint = edge-magic". j

[0146] ③ Each edge xy of the connected graph O 7 (O 11 ) satisfies f 7 (xy) + |f 7 (x) - f 7 (y)| = 16 (f 11 (xy) + |f 11 (x) - f 11 (y)| = 16). f 7 and f 11 are called the edge-difference total coloring, that is, "W*-constraint = edge-difference".

[0147] ④ Each edge xy of the connected graph O 10 satisfies |f 10 (x) + f 10 (y) - f 10 (xy)| = 8. f 10 is called the happy-difference total coloring, that is, "W*-constraint = happy-difference".

[0148] (3.1.3) Parameter total coloring topological coding matrix

[0149] Define the unit total coloring topological coding matrix as such that the v-vector X 0 = (0, 0,..., 0) 1×q and the v-vector Y​​0 =(1, 1, ..., 1) 1×q , and the e-vector E 0 =(1, 1, ..., 1) 1×q .

[0150] Definition 9. Let the bipartite graph G=(X, Y) with q edges admit a set-ordered W-constrained total coloring h: V(G)∪E(G)→[a, b] such that the set-ordered constraint max h(X)<min h(Y) holds. Then the parametric total coloring topological coding matrix of graph G is defined as

[0151]

[0152] where both k and d are non-negative integers, and T code (G, h) 3×q is the total coloring topological coding matrix of the topological coding bipartite graph G.

[0153] (3.2) Mixed total coloring

[0154] Definition 10. A connected (p, q)-graph G admits a mixed total coloring f: V(G)→M, f: E(G)→[1, q], where each element of M is a set. For each edge uv of graph G, there is always e u ∈f(u) and e v ∈f(v) such that the coloring of edge uv is f(uv)=|e u -e v |. If the edge coloring set of graph G is f(E(G))=[1, q], we call the mixed total coloring f a v-set e-normal graceful total coloring of the connected (p, q)-graph G; if the edge coloring set of graph G is f(E(G))=[1, 2q - 1] o , we call the mixed total coloring f a v-set e-normal odd-graceful total coloring of the connected (p, q)-graph G.

[0155] Definition 11. Let the coloring set C olor (G) be all the distinct total colorings of the connected graph G. Taking total colorings h olor , h 1 , h 2 ,..., h n from the coloring set C 1 (G), we define a set total coloring F of the connected graph G as follows: The coloring of each vertex x of the connected graph G is F(x)={h 2 (x), h n (x),..., h 1 (x)}, and the coloring of each edge xy of the connected graph G is F(xy)={h 2 (xy), hn (xy)} to obtain the set total - coloring topological coding matrix S of the connected graph G code (T, F) = (X, E, Y) T , where the v - vector X = (F(x 1 ), F(x 2 ),..., F(x q )), the e - vector E = (F(x 1 y 1 ), F(x 2 y 2 ),..., F(x q y q ))), and the v - vector Y = (F(y 1 ), F(y 2 ),..., F(y q ))).

[0156] Since the elements of the set total - coloring topological coding matrix S code (T, F) are all sets, the strength of the string derived from the set total - coloring topological coding matrix S code (T, F) is stronger, and using them to encrypt and decrypt files has high computational security.

[0157] (3.3) Colored forest (constructing a graph with a coloring atlas, which can be torn into a specific atlas)

[0158] Let F color = {T 1 , T 2 ,..., T n} be a colored forest, and each of its branch trees T k admits a W k - constrained total - coloring h k , and the number of vertices |V(T k )| ≥ 2 (k ∈ [1, n]), and any two trees have no common vertices; it is allowed that two trees are isomorphic, but their colorings are different. After performing the non - multiple - edge vertex - coincidence operation on all the branch trees of the colored forest F color , a non - multiple - edge vertex - coincidence connected graph is obtained such that the number of edges each branch tree of the colored forest F color appears in the non - multiple - edge vertex - coincidence connected graph G and appears only once. Since the connected graph G admits a v - set e - normal total - coloring h, which is derived from the (W 1 , W 2 ,..., W n ) - constrained total - colorings h 1 , h 2 ,..., h n ), it is denoted as

[0159] For each tree T of the colored forest F color add leaves to obtain a new tree H k such that the new tree H k also admits a W k -constrained total coloring f k which is derived from the W k -constrained total coloring h k We obtain a new colored forest H k ={H color , H 1 ,..., H 2}, where H n is the colored forest obtained by adding leaves to the colored forest F color such that the loopless vertex-identified connected graph color grows a lot of hairs (the newly added leaves). The loopless vertex-identified connected graph with hairs is denoted as G[+ . If each tree of the colored forest F leaf admits a set total coloring, then the loopless vertex-identified connected graph color admits a set total coloring based on sets.

[0160] (3.4) Set of spanning trees

[0161] Consider the set S pan (G) of all non-isomorphic spanning trees of the connected graph G. For two spanning trees T i , T j ∈S pan (G), there is a vertex mapping F i，j : V(T j ) → V(T i ). Then define the total vertex-identified graph of these two spanning trees as H = T i [o v-all T j such that the vertex x ∈ V(T j ) coincides with the vertex F i，j (x) ∈ V(T i ) to form a single vertex, E(H) = E(T i ) ∪ E(T j ), and the intersection of the edge sets is called the edge spine of the total vertex-identified graph H. Then the intersection of the edge sets of all spanning trees of the connected graph G is called the edge core of G. If the edge core of the connected graph G is not an empty set, then the connected graph G is 1-connected; otherwise, the connected graph G is 2-connected. Since the vertex mapping F i，j ​The fully vertex - coincident graph of all distinct and non - isomorphic spanning trees is not necessarily the original connected graph G. For example, the labeled complete graph K n has n n-2 non - identical labeled spanning - tree graphs, among which there are a large number of spanning trees with the same topological structure.

[0162] (4) Key generation and topological key - set (group) encryption algorithm

[0163] For the (colored) graph set G = {G 1 , G 2 ,..., G n}, after performing the operation of vertex - coincident without multiple edges, a (colored) graph set is obtained. Then, randomly add leaves to the graphs in this (colored) graph set, and then tear it apart, which is equivalent to randomly adding leaves to the elements of the graph set G.

[0164] (4.1) Random leaf - adding W - constrained total - coloring algorithm

[0165] Theorem 4 [5] Let the connected (p, q) - bipartite graph G admit a graceful (k, d) - total - coloring. Randomly add leaves to the bipartite graph G to obtain the leaf - added graph G + L eaf which also admits a graceful (k, d) - total - coloring, where L eaf is the set of leaves added to G.

[0166] Proof: We use the random leaf - adding parameter xSLyLS - type graceful algorithm to prove this theorem.

[0167]

[0168]

[0169] In the above - mentioned random leaf - adding parameter graceful algorithm, we first perform edge - coloring on the edges connecting the newly added leaves of vertex x 1 until the edges connecting the newly added leaves of the last vertex x s are edge - colored (from Small to Large, xSL). Then, we perform edge - coloring on the edges connecting the newly added leaves of vertex y i until the edges connecting the newly added leaves of the last vertex y 1 are edge - colored (from Large to Small, yLS). Therefore, the above algorithm is called the random leaf - adding parameter xSLyLS - type graceful algorithm.

[0170] For \(W = xSLySL, xSLyLS, xLSySL, xLSyLS, ySLxSL, ySLxLS, yLSxSL, yLSxLS\), the algorithmic proof of this theorem can derive 8 random plus-leaf parameter \(W\)-type graceful algorithms.

[0171] Theorem 5. A tree \(T\) with diameter \(D(T)(\geq3)\) admits at least \(\beta\) graceful \((k, d)\)-total colorings, where \(2\) m-1 \(\leq\beta\leq2\) m+2 , \(m + 1=[D(T) / 2]\).

[0172] Proof of the algorithm: Initialization. Let \(T\) 1 be a tree with diameter \(D(T\) 1 )(\geq3)\), and \(L\) eaf (\(T\) 1 ) be the set of all leaves of the tree \(T\) 1 . Remove all the leaves of the tree \(T\) 1 to obtain another tree \(T\) 2 \(=T\) 1 \(-L\) eaf (\(T\) 1 ). Proceeding in this way, we get the tree \(T\) i+1 \(=T\) i \(-L\) eaf (\(T\) i )(\(i\in[1,m]\)), and \(T\) m+1 is a star tree whose vertex set is \(V(T\) m+1 )\(=\{x\}\cup\{x\) j : \(j\in[1,n]\}\), and the edge set is \(E(T\) m+1 )\(=\{xx\) j : \(j\in[1,n]\}\).

[0173] For non-negative integers \(k\) and \(d\), we define a graceful \((k, d)\)-total coloring \(f\) m+1 for the star tree \(T\) m+1 as follows:

[0174] (1) \(f\) m+1 (\(x)=0\), \(f\) m+1 (\(x\) j )\(=k+(j - 1)d\); (2) \(f_{x}\) +1 (\(xx\) j )\(=f\) m+1 (\(x\) j )\(-f\) m+1 (\(x\))(\(j\in[1,n]\)).

[0175] Iterative step. Since each tree \(T\) i is obtained by adding the leaf set \(L\) i+1 to the tree \(T\) eaf (\(T\) i) is obtained from the leaves in, that is, T i = T i+1 + L eaf (T i ). According to Theorem 4, the tree T i admits a graceful (k, d)-total coloring f i , and it is derived from the graceful (k, d)-total coloring f i+1 of the tree T i+1 to obtain the graceful (k, d)-total coloring f i . According to induction, the tree T 1 admits a graceful (k, d)-total coloring f 1 .

[0176] Lower bound of the number of graceful total colorings. Since each tree T i has at least 2 leaves, that is, |L eaf (T i )|≥2 (i ∈ [1, m]). Since the vertex set V(T i+1 = T i - L eaf (T i )(i ∈ [1, m - 1]) = X i+1 ) = X i+1 ∪ Y i+1 satisfies and |X i+1 |≥2 and |Y i+1 |≥2. We claim that the tree T 1 admits at least 2 m-1 pairwise distinct graceful (k, d)-total colorings.

[0177] Upper bound of the number of graceful total colorings. Since when W = xSLySL, xSLyLS, xLSySL, xLSyLS, ySLxSL, ySLxLS, yLSxSL, yLSxLS, the above proof can derive 8 leaf-added parametric graceful algorithms, that is, the tree T 1 admits 2 m+2 pairwise distinct graceful (k, d)-total colorings.

[0178] Theorem 6. For each vertex-tearing tree T of a connected (p, q)-graph G with q edges and diameter D(T)≥3, the connected (p, q)-graph G admits β v-set e-normal graceful (k, d)-total colorings, where 2 m-1 ≤β≤2 m+2 , and m + 1 = [D(T) / 2].[[]END]

[0179]

[0180] Theorem 7. Let C nnec(G) is the set of connected graphs obtained from the disconnected simple graph G by Algorithm-I in the "v-set e-normal graceful (k, d)-total coloring algorithm for disconnected graphs". For the connected graph L without multiple edges and vertex coincidence k ∈C nnec (G) (k ∈ [1, N G ), after implementing the vertex tearing operation, the set of trees S plit (L k ) is obtained. According to Theorem 6, each tree T k，i ∈S plit (L k ) admits β k，i v-set e-normal graceful (k, d)-total colorings, where m k，i + 1 = [D(T k，i ) / 2], Let n k = |S plit (L k )|, then the disconnected simple graph G admits n color (G) v-set e-normal graceful (k, d)-total colorings, where

[0181] It should be noted that determining the set of connected graphs C nnec (G) and the set of trees S plit (L k ) in Theorem 7 is extremely difficult because their complexity can be reduced to the subgraph isomorphism NP-complete problem. Figures - 7 and Figures - 8 give examples to explain the "v-set e-normal graceful (k, d)-total coloring algorithm for disconnected graphs".

[0182] Theorem 8: Let the tree T admit a v-set e-normal graceful (k, d)-total coloring, then the tree T admits a v-set e-normal W-constrained (k, d)-total coloring, where W-constrained ∈ {edge-magic, edge-difference, happy-difference}.

[0183] Proof: Let G be a connected (p, q)-bipartite graph, its vertex set is V(G) = X ∪ Y, and there is such that each edge x i y j of the bipartite graph G satisfies x i ∈ X = {x i : i ∈ [1, s]} and y j ∈ Y = {y j : j ∈ [1, t]}, where s + t = p, and its edge set is E(G) = {e i = x i y j: \(i\in[1,q]\). Suppose further that \(G\) admits a graceful \((k,d)\)-total coloring \(f\) (see Definition 7), then there is

[0184] \(f: X\rightarrow S\) m，0，0，d， \(f: Y\cup E(G)\rightarrow S\) q-1，k，d \(=\{k,k + d,\cdots,k+(q - 1)d\},k\geq1\)

[0185] and \(0 = f(x\) 1 )\(\leq f(x\) i )\(\leq f(x\) i+1 )(\(i\in[1,s - 1]\)), and \(f(y\) j )\(\leq f(y\) j+1 )\(\leq f(y\) i )\(=k+(q - 1)d\)(\(j\in[1,t - 1]\)), such that the set of edge - coloring

[0186] \(f(E(G))=\{f(x\) i y\) j )\(=f(y\) j )-f(x\) i ): \(x\) i y\) j \(\in E(G)\}=S\) q-1，k，d

[0187] 1. Let \(h(x\) i )\(=f(x\) i )\(h(y\) j )\(=f(y\) j ) and \(h(x\) i y\) j )\(=2k+(q - 1)d - f(x\) i y\) j ), then there is the edge - difference constraint equation

[0188] \(h(x\) i y\) j )\(+\vert h(y\) j )-h(x\) i )\vert=2k+(q - 1)d - f(x\) i y\) j )\(+[f(y\) j )-f(x\) i )\vert=2k+(q - 1)d - f(x\) i y\) j )\(+f(x\) i y\) j )\(=2k+(q - 1)d\)

[0189] 2. Let \(h(x\) i )\(=\max f(X)+\min f(X)-f(x\) i ), \(h(y\) j) = f(y j ) and h(x i y j ) = 2k + (q - 1)d - f(x i y j ), then there is an edge - magic constraint equation

[0190] h(x i ) + h(x i y j ) + h(y j ) = maxf(X) + minf(X) - f(x i ) + 2k + (q - 1)d - f(x i y j ) + f(y j ) = maxf(X) + minf(X) + 2k + (q - 1)d

[0191] 3. Let h(x i ) = maxf(X) + minf(X) - f(x i ), h(y j ) = f(y j ) and h(x i y j ) = f(x i y j ), then there is a happiness - difference constraint equation

[0192] |h(x i ) + h(y j ) - h(x i y j )| = |maxf(X) + minf(X) - f(x i ) + f(y j ) - f(x i y j )| = maxf(X) + minf(X)

[0193] According to Definition 7, this proof is completed.

[0194] For the composite - set total - coloring topological - coding matrix of a general graph, we give the following problem:

[0195]

[0196]

[0197] According to the "Digital - String - Composite - Set Total - Coloring Topological - Coding Matrix Problem", we assert that the asymmetric topological key designed by the v - set e - normal composite - set total - coloring topological - coding matrix has computational indestructibility and provable security, and the theoretical basis is as follows:

[0198] ① Digital string splitting problem: There is no polynomial method to split the digital string s = c 1 c 2 ...c n into small digital strings α i , α j , α k in the form of

[0199] ② Digital string splitting problem: Writing the small digital strings α i , α j , α k in the form of a set can be reduced to the digital string splitting problem.

[0200] ③ NP-complete problem: Digital strings have no connection with the topological structure. Finding a graph H is a subgraph isomorphism NP-complete problem.

[0201] ④ Uncertain constraint equation: Finding each v-set e-normal graceful (k m , d m )-total coloring h m of the composite set total coloring θ admitted by the graph H, such that the W m -constraint equation W m [h m (u m ), h m (u m v m ), h m (v m )] = 0 (m ∈ [1, N]) holds. This is an extremely difficult and unprecedented task.

[0202] ⑤ Randomness: Since the parameter sequence is random, it is difficult to derive the digital string s = c code (H, θ) from the v-set e-normal composite set total coloring topological coding matrix P 1 c 2 ...cn.

[0203] (4.2) Set-ordered 0-rotatable graceful labeling algorithm

[0204] Definition 12. A connected (p, q)-graph G admits a set-ordered 0-rotatable graceful labeling if for any vertex w of the connected graph G, the connected graph G admits a set-ordered graceful labeling h such that h(w) = 0.

[0205] Note of Definition 12: The 0-rotatable graceful labeling was given in [2], but without the constraint of "set-ordered". Definition 12 shows that a connected (p, q)-graph G admits at least p set-ordered graceful labelings. Figure - 9 gives an example of 0-rotatable graceful labeling (see the notes of Definition 12 and Definition 7).

[0206]

[0207]

[0208] Theorem 9: There exist infinitely many trees, and each tree admits a set-ordered 0-rotatable graceful labeling.

[0209] Proof: Since the star tree admits a set-ordered 0-rotatable graceful labeling, according to the set-ordered 0-rotatable graceful labeling algorithm, this theorem is proved.

[0210] Application: Let the tree T admit a set-ordered 0-rotatable graceful labeling. Randomly add m leaves to a vertex of the tree T (private key) to obtain the tree H (public key) such that the tree H admits a set-ordered graceful labeling.

[0211] Definition 13: A connected graph G admits a set-ordered 0-rotatable W-constrained (total coloring) total labeling means that for any specified vertex w ∈ V(G), the connected graph G admits a 0-rotatable W-constrained (total coloring) total labeling h such that h(w) = min h(V(G)).

[0212] Theorem 10: If a connected graph G admits a set-ordered 0-rotatable graceful total coloring (total labeling), then the connected graph G admits a 0-rotatable magic total coloring (total labeling): 0-rotatable edge-magic total coloring (total labeling), 0-rotatable felicitous-difference total coloring (total labeling), 0-rotatable edge-difference total coloring (total labeling), 0-rotatable graceful-difference total coloring (total labeling).

[0213] (4.3) Generating the "one-time pad" key for asymmetric topological keys

[0214] (4.3.1) The parameter topological coding matrix derives infinitely many digital string key pairs

[0215] For a general parameter topological coding matrix (see Definition 9), when k ≠ d, we swap the order of k and d so that the parameter topological coding matrix P code (G, F|k, d) generates the public key digital string s i (k, d), and another parameter topological coding matrix P code (G, F|d, k) generates the private key digital string s i (d, k), forming the topological authentication digital string s i (k + d, k + d).

[0216] (4.3.2) Key pair of the digital string attached to the curve

[0217] Given the parameter topological coding matrix P(H, F|x, y) of the topological coding graph H with q known edges, (3q)! distinct digital strings S containing the parametric variables x and y can be derived i (x, y) = s i，1 (x, y)s i，2 (x, y)...s i，3q (x, y)(i ∈ [1, (3q)!]). We take a sequence of positive integer points (x n , y n )(n ∈ [1, m]) from a plane curve y = f(x) such that y n = f(x n ). We obtain the digital string S attached to the plane curve y = f(x), which is derived from the parameter topological coding matrix P(H, F|x n , y n ) i (x n , y n ) = s i，1 (x n , y n )s i，2 (x n , y n )...s i，3q (x n , y n )(n ∈ [1, m]).

[0218] Note that the inverse function x = f -1 (y) is the dual function of the plane curve y = f(x). Therefore, there is a digital string attached to the plane curve x = f -1 (y), which is derived from the parameter topological coding matrix P(H, F|y n , x n )

[0219] S i (x n , y n ) = s i，1 (x n , y n )s i，2 (x n , y n )...s i，3q (x n , y n )(n ∈ [1, m], i ∈ [1, (3q)!])

[0220] Due to the infinity of plane curves, an infinite number of digital strings attached to the curves are generated, attacking or deciphering the digital string S i (x n , y n ) has to find the plane curve y = f(x). Obviously, this is an extremely difficult and unruly problem.

[0221] (4.3.3) Color-preserving graph homomorphism key pair of asymmetric topological keys

[0222] Topological coding graph set G rap (T code ) Each topological coding graph in it corresponds to a fully colored topological coding matrix T code , such that each topological coding graph H ∈ G rap (T code ) is color-preserving graph homomorphic to the fully colored topological coding matrix T code 's color-preserving accumulation point G, that is The color-preserving accumulation point G is the private key, and each topological coding graph H ∈ G rap (T code ) is the public key, thus forming an "one-to-many" asymmetric topological key pair, group.

[0223] (4.3.4) Set of graphs with non-repeated edges and coincident vertices of asymmetric topological keys

[0224] For the consistent fractal forest T = {H 1 , H 2 ,..., H m}, the set of graphs with non-repeated edges and coincident vertices is the key set. For the uniform fractal forest F = {T 1 , T 2 ,..., T m}, the set of connected graphs with non-repeated edges and coincident vertices is the key set. We obtain a set of key sets of graphs with non-repeated edges and coincident vertices composed of multiple public keys corresponding to one private key. However, determining the complexity of the graph set U coin ([·]T) and the graph set C oin ([·]F) can be reduced to the subgraph isomorphism NP-complete problem.

[0225] (4.3.5) Set full coloring of asymmetric topological keys

[0226] Let the coloring set C olor (T) be the set of all distinct graceful (k, d)-total colorings of trees T with diameter D(T) (≥ 3). Take out the graceful (k, d)-total coloring f olor from the coloring set C 1 , f 2 ,..., f n。Define a set \((k, d)\)-total coloring \(F\) of tree \(T\): The coloring of each vertex \(x\) of tree \(T\) is \(F(x)=\{f 1 (x), f 2 (x), \cdots, f n (x)\}\), and the coloring of each edge \(xy\) of tree \(T\) is \(F(xy)=\{f 1 (xy), f 2 (xy), \cdots, f n (xy)\}\), such that \(f k (xy) = |f k (x)-f k (y)|(k\in[1, n])\). Based on this set total coloring \(F\) of tree \(T\), we obtain the set \((k, d)\)-total coloring topological coding matrix \(S code (T, F)=(X, E, Y) T 。

[0227] Theorem 11. The union of the total coloring topological coding matrices \(T code (T 1 , f 1 ), T code (T 2 , f 2 ), \cdots, T code (T n , f n ) of tree \(T\) derives a string set that is exactly the same as the string set derived from the set total coloring topological coding matrix \(S code (T, F|k, d)\) of tree \(T\).

[0228] Let the connected graph \(G\) be a solution to the uniform fractal forest problem, that is, after performing the vertex tearing operation on the connected graph \(G\), a forest \(H = \{T 1 , T 2 , \cdots, T n \}\) is obtained, such that the number of edges of the connected graph \(G\) and the forest \(H\) are equal, \(|E(G)| = |E(H)|\), and the difference in the number of vertices of any two trees \(T i \) and \(T j \) in the forest \(H\) does not exceed 1, that is, it satisfies \(||V(T i )|-|V(T j )||\leq1\). Each branch tree \(T i \) in the forest \(H\) admits a \(W i \)-constrained total coloring \(h i \), which leads to a set total coloring \(h\) of the connected graph \(G\). Using the set total coloring topological coding matrix \(S codeMaking an asymmetric topological key with (G, h) will result in a stronger asymmetric topological key with computational indecipherability and provable security. The following presents the "digit string - forest vertex coincidence graph problem" in the general case:

[0229]

[0230] (4.3.6) Lattice of graphs with non - overlapping edges and vertex coincidence for asymmetric topological keys

[0231] If the uniform fractal forest EF + ={T 1 , T 2 ,..., T m} satisfies that any two trees are non - isomorphic, we call it a distinguishable uniform fractal tree basis. Based on the non - overlapping edge vertex coincidence operation "[·]" and the distinguishable uniform fractal tree basis EF + , the graph set is called the lattice of graphs with non - overlapping edges and vertex coincidence. The lattice L(Z 0 [·]EF + ) can provide infinitely many topological coding graphs for the technology of the present invention. When each tree of the distinguishable uniform fractal tree basis EF + admits a W - constrained (total - labeling) total coloring, each graph H with non - overlapping edges and vertex coincidence in the lattice L(Z 0 [·]EF + ) admits a v - set e - normal (total - labeling) total coloring h, and the set total - coloring topological coding matrix T code (H, h) can derive a digital - string key with high strength. The notation a k T k means a k trees T k .

[0232] (4.4) Topological key set (group) encryption algorithm

[0233] (4.4.1) Basic algorithm (without the help of a third party, the user makes the key by himself and conducts communication)

[0234] The authentication of the following basic algorithm is all on the communication users themselves.

[0235] (*) Basic algorithm - I is based on the uniform fractal forest UF={H 1 , H 2 ,..., H m}(the number of vertices of any two trees H i and H j is equal), the non - overlapping edge vertex coincidence operation "[·]" and the set of graphs with non - overlapping edges and vertex coincidence

[0236] Step - I - 1. Party A randomly selects a graph G with non - overlapping vertices that coincide ∈ C([·]UF). Using the adjacency matrix A(G) of graph G and an associated algorithm θ (k ∈ [1, (p × p)!]) to derive the private - key topological signature digital string A(k) and the public - key topological signature digital string A(k) (k ∈ [1, t]), where p = |V(G)|. Subsequently, Party A saves the private - key topological signature digital string A(k) by himself and sends the public - key topological signature digital string A(k) to Party B. Party A uses the full - coloring topological encoding matrix T(G, F) and an associated algorithm α (j ∈ [1, (3q)!]) to derive the private - key digital string T(j) and the public - key digital string T(j). Subsequently, Party A saves the private - key digital string T(j) by himself and sends the public - key digital string T(j) to Party B. r ∈C oin ([·]UF), using graph G r 's adjacency matrix A(G r ) and an associated algorithm θ k (k ∈ [1, (p r ×p r )!]) to derive the private - key topological signature digital string A pri (k) and the public - key topological signature digital string A pub (k)(k ∈ [1, t]), where p r =|V(G r )|. Subsequently, Party A saves the private - key topological signature digital string A pri (k) by himself and sends the public - key topological signature digital string A pub (k) to Party B. Party A uses the full - coloring topological encoding matrix T code (G r , F r ) and an associated algorithm α j (j ∈ [1, (3q r )!]) to derive the private - key digital string T pri (j) and the public - key digital string T pub (j). Subsequently, Party A saves the private - key digital string T pri (j) by himself and sends the public - key digital string T pub (j) to Party B.

[0237] Step - I - 2. Party B randomly selects a graph G with non - overlapping vertices that coincide ∈ C([·]UF). Using the adjacency matrix A(G) of graph G and an associated algorithm θ (i ∈ [1, (p × p)!]) to derive the private - key topological signature digital string B(k) and the public - key topological signature digital string B(k) (k ∈ [1, t]), where p = |V(G)|. Subsequently, Party B saves the private - key topological signature digital string B(k) by himself and sends the public - key topological signature digital string B(k) to Party A. Party B uses the full - coloring topological encoding matrix T(G, F) and an associated algorithm α (j ∈ [1, (3q)!]) to derive the private - key digital string T(j) and the public - key digital string T(j). Subsequently, Party B saves the private - key digital string T(j) by himself and sends the public - key digital string T(j) to Party A. s ∈C oin ([·]UF), using graph G s 's adjacency matrix A(G s ) and an associated algorithm θ i (i ∈ [1, (p s ×p s )!]) to derive the private - key topological signature digital string B pri (k) and the public - key topological signature digital string B pub (k)(k ∈ [1, t]), where p s =|V(G s )|. Subsequently, Party B saves the private - key topological signature digital string B pri (k) by himself and sends the public - key topological signature digital string B pub (k) to Party A. Party B uses the full - coloring topological encoding matrix T code (G s , F s ) and an associated algorithm α j (j ∈ [1, (3q s)!) to derive the private key digital string U pri (j) and the public key digital string U pub (j). Subsequently, B saves the private key digital string U pri (j) by himself and sends the public key digital string U pub (j) to A.

[0238] Step - I - 3. B first uses the public key topological signature digital string A pub (k) and the public key digital string T pub (j) given by A to encrypt a plaintext F, and then uses his own private key topological signature digital string B pri (k) for encryption, and finally obtains a ciphertext D with triple protection, and then sends it to A.

[0239] Step - I - 4. After receiving the ciphertext D, A first uses the public key topological signature digital string B pub (k) given by B to decrypt the ciphertext D. When the adjacent matrix A(G s ) passes the topological authentication A uth [B pri (k), B pub (k)], and the non - multiple - edge vertex - coincident graph G s passes the authentication of the consistent fractal forest UF = {H 1 , H 2 ,..., H m}, it is also confirmed that the non - multiple - edge vertex - coincident graph G s ∈C oin ([·]UF) (to prevent an attacker from forging B's topological signature G s ). A knows that this ciphertext is sent by B, that is, B's identity is confirmed.

[0240] Step - I - 5. For the ciphertext with double protection, A uses his own private key topological signature digital string A pri (k) and the private key digital string T pri (j) for decryption. When the adjacent matrix A(G r ) passes the topological authentication A uth [A pri (k), A pub (k)], and the fully - colored topological coding matrix T code (G r , F r ) passes the topological authentication A uth [T pri (j), T pub (j)], A obtains the plaintext F.

[0241] The basic algorithm - I gives the key sequence: ① The private key topological signature sequence derived from the adjacent matrix A(G k ) and the public key topological signature sequence , where 2 ≤ M(A, 1) ≤ (p k × p k )!. ② The total coloring topological coding matrix T code (G k , F k )-derived private key digital string sequence and the public key digital string sequence , where 2 ≤ M(T, 1) ≤ (3q r )!, see Figure - 10. Thus, an approximate "one - time pad" topological key pair and group are provided for users.

[0242] (**) The basic algorithm - II is based on the uniform fractal forest EF = {T 1 , T 2 …, T m}(the number of vertices of any two trees T i and T j satisfies ||V(T i )| - |V(T j )|| ≤ 1), the non - multiple - edge vertex coincidence operation "[·]" and the non - multiple - edge vertex coincidence graph set

[0243] Step - II - 1. Party A arbitrarily takes a non - multiple - edge vertex coincidence graph G k ∈ C oin ([·]EF), and uses the adjacency matrix A(G k ) of the graph G k and an associated algorithm θ i (i ∈ [1, (p k × p k )!]) to derive the private key topological signature digital string A pri (i) and the public key topological signature digital string A pub (i), where p k = |V(G k )|. Subsequently, Party A saves the private key topological signature digital string A pri (i) by himself / herself and sends the public key topological signature digital string A pub (i) to Party B.

[0244] Step - II - 2. Party A uses the total coloring topological coding matrix T code (G k , F k ) and an associated algorithm α r (r ∈ [1, (3q r )!]) to derive the private key digital string T pri (r) and the public key digital string T pub(r). Subsequently, Party A saves the private key digital string T by itself. pri (r), and sends the public key digital string T to Party B. pub (r).

[0245] Step - II - 3. Party B uses the public key topological signature digital string A pub (i) and the public key digital string T pub (r) to encrypt a plaintext F, obtaining a ciphertext D with double - layer protection, and then sends it to Party A.

[0246] Step - II - 4. After receiving the ciphertext D, Party A first uses the private key topological signature digital string A pri (i) to decrypt the ciphertext D. When the adjacent matrix A(G k ) passes the topological authentication A uth [A pri (i), A pub (i)], and the graph G without multiple - edge vertex coincidence passes the authentication of the uniform fractal forest EF = {T k , T 1 , …, T 2 , …, T m}, it also confirms that the graph G without multiple - edge vertex coincidence k ∈C oin ([·]EF). At this point, Party A knows that this ciphertext is sent by Party B and confirms Party B's identity.

[0247] Step - II - 5. For the ciphertext with only single - layer protection, Party A uses the private key digital string T pri (r) to decrypt it. When the fully - colored topological coding matrix T code (G k , F k ) passes the topological authentication A uth [T pri (r), T pub (r)] (to prevent an attacker from forging Party B's topological public key digital string T pub (r)), Party A obtains the plaintext F.

[0248] Note that the basic algorithm - II gives the key sequence: ① The private key topological signature sequence k and the public key topological signature sequence derived from the adjacent matrix A(G ), where 2 ≤ M(A, 2) ≤ (p k ×p k )!. ② The private key digital string sequence code (G k , F k ) and the public key digital string sequence and the public key digital string sequence , where \(2\leq M(T, 2)\leq(3q r )!\). Thus, an approximate "one-time pad" topological key pair and group are provided for the user.

[0249] (*) The basic algorithm - III is based on the Chinese character graph group \(HZ = \{G 1 , G 2 , \cdots, G M \}, where each \(G k \) is a Chinese character graph.

[0250] From the Chinese character graph group \(HZ\), we obtain \(M!\) Chinese sentences. Let the Chinese sentence \(H k = G k，1 G k，2 \cdots G k，M (k\in[1, M!])\) be a full permutation of the Chinese character graphs \(G 1 , G 2 , \cdots, G M \) in the Chinese character graph group \(HZ\). The set \(A rran (HZ)\) contains these full permutations. According to Theorem 7, \(H k \) admits a \(v\)-set \(e\)-normal graceful \((k, d)\)-total coloring \(h i (i\in[1, M(H k )])\), such that the edge coloring set of the Chinese sentence \(H k \) is

[0251]

[0252] and the Chinese sentence \(H k \) has a total coloring topological coding matrix \(T code (H k , h i ) (i\in[1, M(H k )])\).

[0253] Step - III - 1. Party A arbitrarily selects a Chinese sentence \(H rran \) from the set \(A k (HZ)\) and its adjacent matrix \(A(H k )\) to make public key and private key topological signature digital strings. For the detailed steps, refer to the basic algorithm - II.

[0254] Step - III - 2. Party A uses the total coloring topological coding matrix \(T k \) of the Chinese sentence \(H code (H k , h i ) (i\in[1, M(H k )])\) to make public key and private key digital strings. For the detailed steps, refer to the basic algorithm - II.

[0255] Step - III - 3. The remaining steps are the same as those of the basic algorithm - II. Refer to Figures - 7 and - 8 for examples of the "v - set e - normal graceful (k, d) - total coloring algorithm for disconnected graphs".

[0256] (4.4.2) Topological Encryption Management Center, abbreviated as TEMC.

[0257] The authentication of the TEMC - algorithm must all pass through TEMC. Refer to Figures - 11 and - 12.

[0258] (*) TEMC - algorithm - I. Generate an asymmetric topological key pair for the user and manage the user's asymmetric topological keys.

[0259] · The main techniques for generating an asymmetric topological key pair for the user using the previously introduced techniques are as follows: The parameter topological coding matrix derives a digital string key pair, the key pair of the digital string attached to the curve, the key pair of the parameter - type W - constrained total - colored digital string, the color - preserving graph homomorphism key pair of the asymmetric topological key, the set of graphs with non - overlapping edges and vertex coincidences of the asymmetric topological key, the set total - coloring of the asymmetric topological key, the lattice of graphs with non - overlapping edges and vertex coincidences of the asymmetric topological key.

[0260] · The topological signature key pair derives a digital string key pair, group.

[0261] · Use Chinese character graphs to generate an easy - to - remember topological signature key pair for the user. Based on Chinese character literary works, TEMC can generate complex topological signature key pairs and long - byte - string key pairs, groups for the user.

[0262] · Generate asymmetric topological signature key pairs and string key pairs with special requirements for users in special groups. For example, "one - to - many" asymmetric topological key pairs, groups, "many - to - many" asymmetric topological key pairs, and "one - time - one - key" asymmetric topological key pairs, groups, etc.

[0263] (*) TEMC - algorithm - II. Users in the network communicate through TEMC, and TEMC manages the asymmetric topological keys for the users.

[0264] TEMC - algorithm - II - 1. TEMC sends user A's private - key topological signature G Apri ∈C oin ([·]EF) and the private - key string S Apri . TEMC sends user B's private - key topological signature G Bpri ∈C oin ([·]EF) and the private - key string S Bpri

[0265] TEMC - algorithm - II - 2. User A uses his own private - key topological signature G Apriand its own private key string s Apri to encrypt a plaintext F, and the resulting ciphertext is denoted as F 2 , which has two - layer protection. Party A sends the ciphertext F 2 to TEMC, and requests TEMC to perform technical processing and then send it to Party B.

[0266] TEMC - Algorithm - II - 2. TEMC makes a package with the ciphertext F 2 and Party A's public - key topological signature G Apub and Party A's public - key string S Apub , and then encrypts this package again with Party B's public - key topological signature G Bpub and Party B's public - key string S Bpub to obtain a ciphertext F with four - layer protection 4 , and sends it to Party B.

[0267] TEMC - Algorithm - II - 3. After receiving the ciphertext F 4 , Party B first uses its own private - key topological signature G Bpri (to confirm that this ciphertext is sent to itself) and its own private - key string S Bpri to decrypt the ciphertext F 4 , and obtains the ciphertext F 2 and Party A's public - key topological signature G Apub and Party A's public - key string s Apub . Here, Party B's topological signature authentication and string authentication are both at Party B.

[0268] TEMC - Algorithm - II - 4. Party B uses Party A's public - key topological signature G Apub (to confirm that the ciphertext is sent by Party A) and the public - key string S Apub to decrypt the ciphertext F 2 , and finally obtains the plaintext F. Note that Party A's topological signature authentication and Party A's string authentication both need to pass through TEMC.

[0269] (4.5) Theoretical basis for the provable security and computational indecipherability of asymmetric topological encryption

[0270] (4.5.1) Topological coding graph input into the computer. Using the adjacency matrix A(G) of the topological coding graph G n×n and the total - coloring topological coding matrix A(G) 3×q input the topological structure graph and topological coding graph of the present invention into the computer, and the calculation is fast and accurate. The digital string or text string derived from the topological matrix (adjacency matrix, total - coloring topological coding matrix) can be used for file encryption and decryption. The larger the order of the topological matrix, the longer the byte length of the derived string and the higher the computational security. Algebraic graph theory has proved that the computational complexity of the "similarity judgment" of the adjacency matrix is O(n 3) It can quickly verify the identity uniqueness of the topological signature.

[0271] (4.5.2) Multiple Asymmetric Topological Encryption System. The first layer is the topological coding graph (topological signature) with identity authentication function, the second layer is the topological string encryption, and the third layer is the topological coding graph with Chinese meaning.

[0272] (4.5.3) The topological structure space is huge. In the field of graph theory, there are a large number of graphs with complex topological structures. There is no polynomial algorithm to draw all non-isomorphic graphs with a specified number of vertices because it has been proven that subgraph isomorphism is an NP-complete problem.

[0273] Let there be a vertex coloring f: V(K n ) → {1, 2,..., n} for a complete graph K n ) with n vertices, such that the coloring of any two vertices x and y of the complete graph K n ) satisfies f(x) ≠ f(y). Cayley's theorem in graph theory tells us that this labeled complete graph K n has n n-2 non-identical labeled spanning tree graphs. Put these labeled spanning tree graphs into the set S pan (K n ). Since the number of trees with 26 vertices is 279,793,450 ≈ 2 18 , however, the number of elements in the set S pan (K 26 ) is |S pan (K 26 )| = 26 24 (≈ 2 112 ), indicating that there are at least 2 pan spanning tree graphs in the set S 26 (K 94 ) that have the same topological structure as 2 18 other spanning tree graphs.

[0274] Network reports: It is estimated that the number of stars in the universe is 3·10 22 ≈ 2 74 , and the number of atoms in the solar system is approximately 10 57 ≈ 2 189 . There are literature reports that the number G 24 of graphs with 24 vertices is:

[0275] G 24 = 195704906302078447922174862416726256004122075267063365754368 ≈ 2 197 So, the number G of graphs with 24 vertices24 is much larger than the number of atoms in the solar system, which is 10 57 (≈2 189 ).

[0276] In practical applications, a topological coding graph with no less than 30 vertices is used, so that the number G of graphs with 30 vertices 30 > 2e = 2 200 , ensuring complete resistance to AI attacks and codebreaking by equipped quantum computers in a short time.

[0277] (4.5.4) The number and variety of topological colorings are numerous and uncertain. There are thousands of W-constrained colorings and W-constrained labelings in graph theory. The variety of colorings of topological coding graphs is uncertain, and new colorings and labelings are generated every day. That is, for a graph G and a specific W-constrained coloring, there is no polynomial algorithm that can give all different W-constrained coloring schemes of graph G. For a positive integer q, Sheppard proved in 1976 that there are q! graceful graphs with q edges, and q! / 2 of them correspond to different graceful labelings of the same graceful graph.

[0278] (4.5.5) Irreversibility of the strings and number strings derived from topological matrices. The number string derived from the topological matrix has no association with the topological structure, and the inverse process of backtracking the string and reconstructing the original topological coding graph and its topological matrix does not exist. It is extremely difficult to find a specific (p, q)-connected graph G, its specific spanning tree (see degree-constrained spanning tree), and specific vertex tearing tree from a large number of graphs, and they all involve the subgraph isomorphism NP-complete problem. Since a graph admits multiple colorings and labelings, a graph corresponds to multiple topological coding matrices, and each topological coding matrix associated with the NP-complete problem derives a huge number of strings. In addition, there is a situation where a number string is derived from more than 2 topological coding graphs. These difficult problems are the cornerstone of the computational security of the present invention.

[0279] (4.5.6) Complete data functions. Based on coloring transformation and universal coloring, the asymmetric topological encryption of the present invention can achieve data compressibility, data anti-modification, data irreversibility, extremely long-bit string keys, and has the potential to resist quantum computing.

[0280] (4.5.7)Countermeasures against AI attacks with quantum computers: Some research reports indicate that even AI attacks with quantum computers cannot effectively cause damage by spanning back and forth across two or more essentially different mathematical domains. Since Chinese characters are natural topological graphs, finding the topological coding matrices for deriving string and their corresponding topological coding graphs from two or more essentially different mathematical domains requires the support of multiple technical theories and time overhead. In addition, problems such as clique, feedback vertex set, vertex cover, subgraph isomorphism, Hamiltonian path problem, and graph planarity in graph theory are all NP-complete problems. The topological structure of the topological matrix is closely related to these graph structures, which is the cornerstone of the provable security of the technology of this invention. Graph theory contains open problems such as the graceful tree conjecture, edge-magic total coloring conjecture, odd-graceful tree conjecture, total coloring conjecture, vertex coloring conjecture, and equitable total coloring conjecture that have remained unsolved for a long time. These conjectures are all related to NP-class problems, which are also the cornerstone of the computational security of the technology of this invention.

[0281] Based on the above facts, the asymmetric topological encryption of this invention not only has practicality and convenience, but also has computational security and provable security, enabling the practical algorithms designed with the technology of this invention to avoid the complicated work of proving computational security and provable security every time, greatly facilitating various practical application scenarios.

Claims

1. The asymmetric topological encryption technology of the present invention is: based on a set of (colorless and colored) topological coding graphs, asymmetric key pairs and groups are produced, and the production technology of asymmetric topological key pairs and groups is given according to the consistent fractal forest, uniform fractal forest and Chinese character graph group, and the corresponding encryption algorithm is designed; a topological encryption management center (TEMC) is created to produce and manage asymmetric topological key pairs and groups for network users. The asymmetric topological encryption technology of the present invention is not only practical and convenient, but also has computational security and provable security. One purpose: the practical algorithm designed by the technology of the present invention does not need to prove the computational security and provable security every time. The second purpose: to achieve the diversity of topological signatures and design approximate "one-time one-key" asymmetric topological key pairs and groups.

2. According to claim 1, "making asymmetric key pairs and groups based on (colorless, colored) topological coding graph sets", it is characterized by: After performing vertex tearing operations and non-duplicate edge vertex overlap operations on the connected graph, a consistent fractal forest and a uniform fractal forest are obtained.

3. The method of “making asymmetric key pairs or groups based on a colored topological coding graph set” according to claim 1 is characterized in that: Coloring of topologically coded graphs, parametric coloring of topologically coded graphs, twin matching coloring of topologically coded graphs, parametric full coloring of topologically coded matrices, set of colored forest group construction colored atlases, and set of colored spanning trees.

4. According to the "technology for making asymmetric topological key pairs and groups and designing corresponding encryption algorithms" described in claim 1: parameter topological coding matrix derives infinite digital string key pairs, digital string key pairs attached to curves, color-preserving graph homomorphic key pairs of asymmetric topological keys, and asymmetric topological keys with no overlapped edges and vertices. The encryption algorithms include: random leaf-added W-constrained full coloring algorithm, set ordered 0-rotatable graceful labeling algorithm, and three basic algorithms based on consistent fractal forests, uniform fractal forests, and Chinese character graph groups.

5. According to the "creation of a topology encryption management center (TEMC)" described in claim 1, its functions are: TEMC creates asymmetric topology key pairs and groups for local area network users, and manages the asymmetric topology keys (groups) of network users; users in the network can encrypt and decrypt files through TEMC to achieve secure communication.

6. According to claim 1, "asymmetric topological encryption technology is not only practical and convenient, but also computationally secure and provably secure", and its basis is: the topological coding graph relies on the topological matrix to input the computer, has multiple asymmetric topological encryption systems, the topological structure space is huge, the topological coloring is complicated and the number is uncertain, the character strings and number strings derived from the topological matrix are irreversible, and the complete data function can cope with the attack of AI equipped with quantum computers; associated NP-complete problems or NP-like problems, such as finding the optimal consistent fractal forest and the optimal uniform fractal forest of a connected graph, are subgraph isomorphism NP-complete problems.

7. According to "one of the purposes" described in claim 1: since the adjacent matrix of the topological coding graph G of the W-constrained full coloring f is recognized as a topological signature (identity uniqueness), the full coloring topological coding matrix of the topological coding graph G is a production machine for encrypting and decrypting digital strings. Due to the irreversibility of digital strings and the multitude of topological coding graphs corresponding to the full coloring topological coding matrix (subgraph NP-complete problem), as well as the randomness of multi-dimensional constrained coloring, the practical algorithm designed based on the technology of the present invention has natural computational security and provable security.

8. According to the "second purpose" described in claim 1: In order to deal with AI attacks equipped with quantum computers, the above-mentioned technologies 2, 3, 4 and 5 can achieve: (1) topological signatures and their uniqueness; (2) one public key topological signature corresponds to multiple private key topological signatures, multiple public key topological signatures correspond to a single private key topological signature, and multiple public key topological signatures correspond to multiple private key topological signatures; (3) random topological signatures; (4) approximate "one-time one-pad" asymmetric topological key pairs and groups.

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