Design and implementation of asymmetric linear topology encryption technology

Abstract

The invention provides an asymmetric linear topological encryption technology, a topological signature (graph and matrix) key pair is generated by using an adjacent matrix of a full-coloring topological coding tree (graph), a full-coloring topological coding matrix, graph homomorphic operation and graph structure operation of the technology, and a technical theory and a feasible algorithm of asymmetric linear topological encryption are given. A W-type coloring algorithm of random leaf growth of a topological coding tree and an implementation algorithm of asymmetric linear topological encryption are designed. And proposing a measurement standard and management of asymmetric linear topological key intensity. The asymmetric linear topology encryption technology contains problems belonging to an NP-complete problem class, so that the asymmetric linear topology encryption technology and various algorithms generated by the asymmetric linear topology encryption technology have calculation non-decipherability and provable security. At present, a polynomial quantum algorithm for solving some NP-difficulty problems does not exist, so that the algorithm designed by the asymmetric linear topology encryption technology can resist the attack of quantum calculation.

Description

[0001] 1. Technical field: Information encryption

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

[0003] Graph theory is a branch of mathematics. It is widely used in natural sciences, operations research, computational biology, molecular analysis of chemistry, network structure analysis (Internet of Things, logistics network, biological neural network, telecommunication network, power network), coding theory, control theory, feedback theory, random process, reliability theory, computer programming, artificial intelligence, economics, linguistics, social structure and other fields.

[0004] Graph theory problems are the core mathematical problems of many important applications such as big data processing. The National University of Defense Technology (in collaboration with the Academy of Military Science, Sun Yat-sen University and other domestic and foreign institutions) has successfully developed a new type of programmable silicon-based optical quantum computing chip, which has realized the quantum algorithm solution of multiple graph theory problems.

[0005] The topological coding graph used in the present invention is composed of "topological structure" and "mathematical constraints". The "mathematical constraints" of the topological coding graph are based on the theories and practical techniques of graph theory, number theory, set theory, abstract algebra, linear algebra, computer science and other disciplines. The "topological structure" of the topological coding graph is a tool for semi-structured data. It is not a geometric graph or a picture, but a "soft mathematical expression", which is called a "graph" in the graph theory discipline. The graph is a natural representation of the coding relationship structure, and the calculation defined by the graph structure data is widely used in various fields.

[0006] Example of a topologically encoded graph: Graph H in Figure 1 is a topological structure, and Graph T in Figure 2 is a topologically encoded graph (with a fully colored topological structure). The topological structures of Graph H and Graph T are exactly the same, that is, there is graph isomorphism. The topological coding graph T is composed of two elements: "topological structure" and "mathematical constraints". The "mathematical constraints" of the topological coding graph T are that it recognizes a graceful label h, so that the coloring of any two vertices x and y of the graph T satisfies h(x)≠h(y), the coloring of each edge uv of the graph T satisfies h(uv)=|h(u)-h(v)|, and the coloring of all edges of the graph T forms a continuous set of integers {1, 2, ..., 19}.

[0007] The asymmetric linear topology encryption technology of the present invention is based on a linear topology structure and graph operations. An algebraic matrix inputs the asymmetric linear topology structure into a computer, enabling the technology of the present invention to have a faster implementation efficiency. Since the security of the asymmetric linear topology encryption technology is based on NP-Hard or NP-C problems, and there is currently no quantum polynomial algorithm to solve some NP-hard problems, the asymmetric linear topology encryption technology may become a member of the family of cryptographic systems resistant to quantum computing attacks. IV. Description of the Drawings:

[0008] Appendix Figure 1 The trees H1, H2, H3, H4, and H5 all have 11 leaves and are pairwise non-isomorphic. The trees H1 and H2 are caterpillar trees, and their longest paths are both 16; the longest path of tree H3 is 11; the longest path of tree H4 is 13; the longest path of tree H5 is 14.

[0009] Appendix Figure 2 The trees T1, T2, T3, and T4 all have 8 leaves and are pairwise non-isomorphic. The trees T1 and T2 are caterpillar trees, and their longest paths are both 13; the longest path of tree T3 is 8; the longest path of tree T4 is 9.

[0010] Appendix Figure 3 5 spanning trees S1, S2, S3, S4, and S5 of the Chinese character graph C of the character "ugly".

[0011] Appendix Figure 4 The Chinese character graph C of the character "ugly" (see Figure - 3 in the drawings) admits 6 set-ordered graceful labelings, generating topological coding graphs P1, P2, P3, P4, P5, and P6.

[0012] Appendix Figure 5 After performing vertex tearing operations on the topological coding graph P1 of the character "ugly" (see Figure - 4 in the drawings), non-isomorphic vertex tearing trees Q1, Q2, Q3, Q4, Q5, and Q6 are obtained.

[0013] Appendix Figure 6 Process of adding leaves: (a) The graph G admits a graceful (k, d)-total coloring; (b) Add leaves to the graph G to obtain the graph G1; (c) Color the new edges of the graph G1 using the leaf-adding xSLyLS-graceful algorithm to obtain the graph G2; (d) Color the leaves added to the graph G2 to obtain the graph G3; (e) Explain the leaf-adding xLSySL-graceful algorithm using the graph G2.

[0014] Appendix Figure 7 Using the graph G2 in Figure - 6 in the drawings to explain the trees H1, H2, H3, and H4 of 4 leaf-adding graceful algorithms, where W = xSLySL, xLSyLS, ySLxSLS, yLSxLS. Note that there are vertices with the same vertex coloring.

[0015] Appendix Figure 8 2. Continuing to implement the algorithm of adding leaves xSLyLS - graceful to Figure H1 in Figure - 7, the topological - coding graph that admits a (k, d) - graceful total - coloring has two vertices colored 3d, two vertices colored 9d, two vertices colored k + 20d, and two vertices colored k + 22d.

[0016] Appendix Fig. 9 3. An example to explain the "random - adding - leaves edge - magic total - coloring algorithm": The coloring of each edge uv satisfies f(u)+f(uv)+f(v)=10. Tree EM1 is the root of the edge - magic total - coloring, and the roots of trees EM2 and EM3 are tree EM1.

[0017] Appendix Fig.10 4. An example to explain the "random - adding - leaves edge - difference total - coloring algorithm": The coloring of each edge uv satisfies f(uv)+|f(u)-f(v)| = 8. Tree ED1 is the root of the edge - difference total - coloring, and the roots of trees ED2 and ED3 are tree ED1.

[0018] Appendix Fig.11 5. An example to explain the "random - adding - leaves felicitous - difference total - coloring algorithm": The coloring of each edge uv satisfies |f(u)+f(v)-f(uv)| = 2. Tree FD1 is the root of the felicitous - difference total - coloring, and the roots of trees FD2 and FD3 are tree FD1.

[0019] Appendix Fig.12 6. An example to explain the "random - adding - leaves graceful - difference total - coloring algorithm": The coloring of each edge uv satisfies ||f(u)-f(v)|-f(uv)| = 1. The caterpillar tree GD1 is the root of the graceful - difference total - coloring, and the roots of trees GD2 and GD3 are tree GD1.

[0020] Appendix Fig.13 7. (a) and (b) explain the "random - adding - leaves edge - magic total - labeling algorithm". (c) and (d) explain the "random - adding - leaves edge - difference total - labeling algorithm". (e) and (f) explain the "random - adding - leaves felicitous - difference total - labeling algorithm". (g) and (h) explain the "random - adding - leaves graceful - difference total - labeling algorithm".

[0021] Appendix Fig.14 8. The topological - coding graphs of the Chinese characters "qian, shi, gan, tu".

[0022] Appendix Fig.15 9. (a) The edge - magic total - coloring graph G is the result of the edge - adding operation on EM3 in Figure - 9; (b) The edge - difference total - coloring graph H is the result of the edge - adding operation on DM3 in Figure - 10.

[0023] Appendix Fig.16 10. (a) The edge - magic total - coloring graph G EMThe result of performing the same-color vertex coincidence operation on EM3 in Figure 9; (b) edge difference full coloring graph G DM This is the result of performing the same color vertex overlap operation on DM3 in Figure 10.

[0024] Attached Fig.17 , Explanation Module Algorithm-I.

[0025] Attached Fig.18 , explain ATCC algorithm-I.

[0026] The common mathematical concepts, terms, colors and labels used in this invention are all taken from the following documents:

[0027] [1]JABondy, USRMurty.Graph Theory[M].Springer London, 2008.DOI: 10.1007 / 978-1-84628-970-5.

[0028] [2]Joseph A.Gallian.A Dynamic Survey of Graph Labeling[J].The electronic journal of eombinatorics, #DS6, Twenty-fourth edition, December 9, 2021. (576 pages, 3028 references, more than 200 graph numbers)

[0029] [3] Bing Yao, 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 figure numbers and figure coloring)

[0030] [4] Bing Yao (Yao Bing), Xiaohui Zhang, Hui Sun, Jing Su, Fei Ma, HongyuWang. Parameterized Colorings And Labelings Of Graphs In Topological Coding[J].arXiv:2207.03381[cs.IT], 7Jul 2022.1-149.https: / / doi.org / 10.48550 / arXiv.2207.03381

[0031] [5] Bing Yan, Yarong Mu, Yirong Sun, Hui Sun, Xiaohui Zhang, Hongyu Wang, Jing Su, Mingjun Zhang, Sihua Yang, Meimei Zhao, Xiaomin Wang, Fei Ma, MingYan, Chao Yang, Jianming Xie. Using Chinese Characters To Generate TexT-Based Passwords For Information Security[J]. arXiv:1907.05406v1[es.IT]11 Jul 2019.1-59.https: / / doi.org / 10.48550 / arXiv.1907.05406

[0032] [6] Bing Yao, Meimei Zhao, Xiaohui Zhang, Yarong Mu, Yirong Sun, Mingjun Zhang, Sihua Yang, Fei Ma, Jing Su, Xiaomin Wang, Hongyu Wang, HuiSun. Topological Coding and Topological Matrices Toward Network Overall Security. arXiv:1909.01587v2[cs.IT]15 Sep 2019. V. Specific implementation methods:

[0033] The specific implementation method is divided into five parts: (i) coloring definition and security issues, (ii) linear topological key matching design, (iii) computational topological key matching design, (iv) linear topological coloring algorithm, and (v) implementation of asymmetric linear topological encryption.

[0034] In the following description, we use definitions to express the research objects and concepts of asymmetric linear topological encryption technology, use theorems to express theories, and use algorithms to express specific algorithmic and programmable technologies.

[0035] (I) Coloring definition and safety issues

[0036] (1.1) Basic terms and symbols

[0037] The set of all non-negative integers is denoted by Z 0 ; The set of all positive integers is Z + =Z 0 \{0}. Let a, b, k, d, m, α, β∈Z 0 , for 0≤a<b, the set of non-negative integers {a, a+1, ..., b} is abbreviated as [a, b]; when α and β are odd integers and α<β, the set of odd integers {α, α+2, ..., β} is denoted as [α, β] o . The number of elements of set S is denoted by |S|.

[0038] For integers p ≥ 2 and integers q ≥ 1, a (p, q)-graph G has p vertices and q edges. Vertices with degree 1 are called leaves. The set of all leaves of graph G is denoted by L. eaf (G).

[0039] The vertex set of the bipartite graph G is V(G) = X∪Y, and the intersection Let the two endpoints u and v of each edge uv∈E(G) of the bipartite graph G satisfy u∈X and v∈Y. Suppose the bipartite graph G admits a W-constrained coloring f:S→M, where the set The coloring set of the elements of set S is denoted by f(S) = {h(w): w∈S}. If the set of vertex colorings of graph G satisfies the set-ordered constraint

[0040]

[0041] We call this constraint coloring f a set-ordered constraint coloring of the bipartite graph G. When S = V(G), f is called a vertex coloring of the bipartite graph G; when S = V(G) ∪ E(G), f is called a total coloring of the bipartite graph G. If the coloring of any two vertices x and y of the bipartite graph G satisfies f(x) ≠ f(y), f is called a labeling of the bipartite graph G.

[0042] In graph theory, a graph is an even graph if it does not contain odd cycles. A (p, q)-tree graph is a connected graph that does not contain cycles and satisfies q = p-1. Therefore, the tree graph is an even graph, and the tree graph is also called a linear topology. A string is a general term for a string of numbers (digital encoding) or a string of text (text encoding).

[0043] Definition 1. [3]Suppose a connected (p, q)-graph G admits a total coloring f: V(G)∪E(G)→[a, q+a] (a is a non-negative integer), and allows the coloring of two non-adjacent vertices x and y to satisfy f(x)=f(y). If the coloring f(uv) of each edge uv of graph G satisfies f(uv)=|f(u)-f(v)|>0, so that the edge coloring set of graph G {f(uv):uv∈E(G)}=f(E(G))=[1,q], then f is called a graceful total coloring of graph G.

[0044] Note to Definition 1: When a = 0, the total coloring f of the (p, q)-graph G in Definition 1 satisfies: the coloring f(x) ≠ f(y) of any two vertices x and y, the edge coloring set of the graph G {f(uv) = |f(u) - f(v): uv∈E(G)} = f(E(G)) = [1, q], and the vertex coloring set We call f a graceful labeling of the graph G. Furthermore, when the connected (p, q)-graph G is an even graph, that is, V(G) = X∪Y and If a graceful labeling f satisfies the set-ordered constraint max f(X)<min f(Y), we call f a set-ordered graceful labeling.

[0045] Definition 2: [3] Assume G is a connected (p, q)-pair graph, then V(G) = X∪Y, and So that every edge xy∈E(G) satisfies x∈X and y∈Y. For integer k, d ≥ 1, if the graph G admits a total coloring

[0046] f:X→S m，0，d ={0,d,...,md},f:Y∪E(G)→S q-1，k，d ={k, k+d, ..., k+(q-1)d}

[0047] The edge coloring set of the even graph G is

[0048] f(E(G))={f(uv)=|f(u)-f(v)|:uv∈E(G)}=S q-1，k，d ={k, k+d, ..., k+(q-1)d}

[0049] Then the total coloring f is called a (k, d)-graceful total coloring of the dual graph G. The dual total coloring f* of a (k, d)-graceful total coloring f is defined as

[0050] f*(z)=max{f(y):y∈Y}+min{f(x):x∈X}-f(z)(z∈V(G)), f*(xy)=|f*(x)-f*(y)|=f(xy)(xy∈E(G))

[0051] The edge coloring set f*(E(G))=S of the even graph G q-1，k，d = f(E(G)), then the total coloring f* is called the (k, d)-graceful total coloring of the above (k, d)-graceful total coloring f.

[0052] Notes on Definition 2: ① The coloring in Definition 1 and Definition 2 allows the coloring of two non-adjacent vertices to be equal, and also allows the coloring of an edge xy to be the same as the coloring of one of its endpoints x, that is, f(xy) = f(x).

[0053] ② When (k, d) = (1, 1) in Definition 2, Definition 1 is a special case of Definition 2.

[0054] ③When (k, d) = (1, 2) in Definition 2, the (1, 2)-graceful total coloring of Definition 2 is an odd-graceful total coloring, and its dual total coloring is also an odd-graceful total coloring.

[0055] ④ When the connected (p, q)-graph G of Definition 2 is an even graph, that is, V(G) = X∪Y and the intersection A total coloring f is called a set-ordered (k, d)-graceful total coloring of the graph G if the set of vertex colorings satisfies the set-order constraint max f(X)<min f(Y).

[0056] Definition 3. [6] Suppose the edge set of a graph G with q edges is E(G) = {e i =x i y i : i∈[1,q]}, the graph G admits a W-constrained total coloring f: V(G)∪E(G)→[a,b]. Call the following matrix

[0057]

[0058] is the full coloring topology encoding matrix of graph G, where the vertex coloring vector X f =(f(x1), f(x2), ..., f(x q )) and the vertex shading vector Y f =(f(f(y1),f(y2),...,f(y q )), and the edge coloring vector E f =(f(e1), f(e2), ..., f(e q )); each edge e i ∈E(G) satisfies the constraint equation W[f(x i ), f(e i ), f(y i )]=0(i∈[1,q]).

[0059] Annotation of Definition 3: ① In general, the mathematical constraint "W-constraint" in Definition 3 is an abbreviation of the n-dimensional constraint W(k1, k2,..., k n )(n≥1), which can construct a complex topological coding graph to prevent forgery and deciphering of topological keys.

[0060] ② In the following narrative, a graph with full coloring is called a topological coding graph, also known as a topological signature.

[0061] (1.2) The topological structure graph and the topological coding graph rely on matrices to be input into the computer

[0062] Using the adjacent matrix A(G) of graph G n×n and the full-coloring topological coding matrix A(G) 3×q The topological structure graph and the topological coding graph of the present invention are input into the computer, and the digital string or text string derived from the matrix discussed in the present invention is used to encrypt and decrypt electronic files. For example, formula (2) gives the adjacent matrix A(C) of the Chinese character graph C of the word "ugly" in Figure - 3 10×10 , which is a 10×10 matrix, where O m×n is an m×n zero matrix (a matrix with each element being zero is called a zero matrix), and E adj (C) 5×5 is called the edge adjacent matrix, is the transpose matrix of the edge adjacent matrix E adj (C) 5×5 . It is not difficult to see that the adjacent matrix A(C) 10×10 is symmetric, then the adjacent matrix is equal to its own transpose matrix, that is, A(C) 10×10 = A T (C) 10×10 .

[0063]

[0064] Note that the adjacent matrix A(C) in formula (2) 10×10 can derive 100! (> 2 524 ) distinct digital strings. Formula (3) is the full-coloring topological coding matrix T code (P1) 3×11 of the topological coding graph P1 of the word "ugly" in Figure - 4, which is a 3×11 matrix, and it can derive 33! (> 2 122 ) distinct digital strings. It is reported 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 189The number of atoms. The number of digital strings derived from the adjacency matrix and the total coloring topological coding matrix of a topological coding graph with no less than 30 vertex numbers is far greater than the number of atoms in the solar system.

[0065]

[0066] In the following narrative, if there is no confusion, we will omit the adjacency matrix A(G) n×n and the total coloring topological coding matrix A(G) 3×q The matrix order "n×n" and "3×q" in the lower right corner. Algebraic graph theory tells us the following conclusion:

[0067] Theorem 1. (1) Let the 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.

[0068] (2) The necessary and sufficient condition for the graph G to be non-isomorphic to the graph H is that for any invertible matrix P, the adjacency matrices of the two graphs satisfy A(H) ≠ PA(G)P -1 .

[0069] According to Theorem 1, under the similarity operation of the adjacency matrix, a graph and its adjacency matrix are in one-to-one correspondence. Therefore, we use the adjacency matrix of the graph as the "topological signature" to ensure the uniqueness of identity authentication; we use the total coloring topological coding matrix of the topological coding graph to derive the strings for encrypting and decrypting electronic files.

[0070] Example 1. The topological structure of the Chinese character graph C of the character "ugly" in Figure - 3 is exactly the same as the topological structure of the topological coding graph P1 of the character "ugly" in Figure - 4, that is, the graph C is isomorphic to the topological coding graph P1. After performing the vertex tearing operation on the topological coding graph P1 (see Figure - 4), the obtained topological coding graphs are 6 vertex tearing trees Q1, Q2, Q3, Q4, Q5, and Q6, and there is a color-preserving graph homomorphism between the topological coding graphs. Their total coloring topological coding matrices are equal, that is, T code (Qj) = T code (P1) (j ∈ [1, 6]). Under the transformation of the dual label (see Definition 2), each vertex of the Chinese character graph C of the character "ugly" in Figure - 3 can be colored zero by a set-ordered graceful label of the graph C, which is the 0-rotatable graceful label in graph theory. The Chinese character graph C of the character "ugly" in Figure - 3 admits a set-ordered 0-rotatable graceful label.

[0071] (1.3) Provable security and computational indecipherability of asymmetric linear topological encryption

[0072] (1.3.1) Degree-constrained spanning tree

[0073]

[0074] The degree-constrained spanning tree problem is NP-complete because it can be represented by a weakening of the Hamiltonian path problem, and it is still NP-complete even if m is fixed to a value α ≥ 2. The degree-constrained spanning tree with m = 2 is the Hamiltonian path problem.

[0075] (1.3.2) Vertex Tearing Tree with Leaf Number Constraint

[0076]

[0077] For 2≤m≤n-1, it is difficult to find the number of trees with m leaves and n vertices, see the integer splitting problem below:

[0078]

[0079] There is currently no polynomial-time integer factorization algorithm to factor a large integer, especially one that is the product of two randomly chosen prime numbers of similar size. The difficulty assumption of the integer factorization problem is the basis for the security of many modern cryptographic algorithms.

[0080] (1.3.3) Uniform Fractal Forest

[0081] Example 2: Perform a vertex tearing operation (Definition 4) on the connected graph H with 19 edges in Figure 1, and obtain the supporting trees H1, H2, H3, H4 and H5 with 19 edges and 11 leaves. Perform a vertex tearing operation on the topological coding graph T with 19 edges in Figure 2, and obtain the supporting trees T1, T2, T3 and T4 with 19 edges and 8 leaves. Note that the connected graph H in Figure 1 and the topological coding graph T in Figure 2 have the same topological structure, that is, According to Definition 5 and Definition 6, we get the colorless graph homomorphism H k →H(k=1,2,3,4,5) and topologically encoded graph homomorphism T j →T(j=1, 2, 3, 4).

[0082] Applying vertex tearing operation to a (p, q)-connected graph G, we obtain a tree T with q edges. Due to the diversity of vertex tearing operation (Definition 4), the trees with q edges form a set S plit (G), so that for each tree T∈S with q edges plit (G) After implementing the vertex coincidence operation (Definition 5), we obtain the graph homomorphism T→G. The vertex tearing operation allows us to raise the following question:

[0083]

[0084] (1.3.4) Number of linear topological structures

[0085] A non-trivial directed tree is called a rooted tree if there is exactly one vertex with in-degree 0 and all other vertices have in-degree 1. The point with in-degree 0 is called the root, the point with out-degree 0 is called the leaf, and the point with in-degree 1 and out-degree greater than 1 is called the interior point. The interior points and the root are collectively called branch points. Currently, there is no polynomial formula to count the number of trees and rooted trees.

[0086]

[0087]

[0088] Suppose there is a complete graph K with n vertices n A vertex coloring f: V(K n )→{1, 2, ..., n}, so that the complete graph K n The coloring of any two vertices x and y satisfies f(x)≠f(y). Cayley’s theorem in graph theory tells us that this calibrated complete graph K n There are n-2 different calibrated support tree graphs, and put these calibrated support tree graphs into a set S pan (K n ). Since the number of trees with 26 vertices is 279,793,450≈2 18 , however, the spanning tree graph set S pan (K 26 The number of calibrated spanning trees of ) is |S pan (K 26 )|=26 24 (≈2 112 ), indicating the support tree graph set S pan (K 26 ) at least 2 94 Spanning tree diagram and other 2 18 The spanning tree graphs have the same topological structure.

[0089] According to the literature, the number of graphs with 23 vertices is G 23 The number of graphs G with 24 vertices 24 They are as follows:

[0090] G 23 =559946939699792080597976380819462179812276348458981632≈2 179

[0091] G 24=195704906302078447922174862416726256004122075267063365754368≈2 197

[0092] We see that: the number of graphs G with 24 vertices is 24 10 more than the number of atoms in the solar system 57 (≈2 189 ) is even bigger.

[0093] According to the facts of (1.3.1), (1.3.2), (1.3.3) and (1.3.4) above, the difficulty of finding a specific (p, q)-connected graph G and its specific spanning tree (see degree-constrained spanning tree) and a specific vertex-tearing tree from a massive amount of graphs can be reduced to the NP-complete problem of subgraph isomorphism. In addition, the number of graphs and the space of coloring types are huge, and there is no general polynomial algorithm for graph coloring. Therefore, it can be asserted that the encryption algorithm designed using asymmetric linear topology technology is computationally unbreakable and provably secure.

[0094] 2. Linear topological key matching design

[0095] (2.1) T-core leaf tree matching group

[0096] Let the set of all leaves of tree G be L eaf (G), then removing all the leaves of tree G will still leave a tree T, denoted by T = GL eaf (G). The vertex set of the tree T is V(T) = {x i : i∈[1,n]}, each vertex x i The set of leaves connected by ∈V(T) in the tree G is L eaf (x i )={y i，j :j∈[1,a i ]}(a i ≥0, i∈[1,n]), if a vertex x of the tree T j is not adjacent to any leaf of tree G, then a j = 0. For vertex x i ∈V(T) connects to the number of leaves a i ≥2(i∈[1,n]), let a be a positive integer i It can be decomposed into S(a i ) group of positive integers, we can convert the positive integer a i Expressed as and

[0097] a i =m i，k[i]，1 +m i，k[i]，2 +…+m i，k[i]，n(k[i]) , where mi，k[i]，j ≥1,j∈[1,n(k(i])],n(k[i])≥2,i∈[1,n],k[i]∈[1,S(a i )]

[0098] Then the vertex x i ∈V(T) connected leaf set L eaf (x i ) is decomposed into the union of leaf subsets Among them, the leaf set L eaf (x i，k[i]，t )={y i，k[i]，t，j : j∈[1, t]}, t∈[1, n(k[i])]. For M=max{n(k[i]): k[i]∈[1, S(a i )], i∈[1,n]}, take the tree H s (s∈[1,M]) satisfies the remaining tree after removing all leaves V(H s -L eaf (H s ))=V(T), and tree T is called each tree H s The nucleus.

[0099] Tree H s Each vertex x s，i ∈V(H s ) The set of leaves connected is L eaf (x s，i ) is the leaf set L eaf (x i ) is a subset L eaf (x i，k[i]，t ). Tree H s The vertex x s，i The leaf set and tree H t The vertex x t，i The leaf set meets (s≠t, and s, t∈[1,M]).

[0100] Core coincidence operation. M trees H1, H2, ..., H M The core T of the tree is combined into a core T, and we get the tree G. Obviously, each tree H s is a subtree of the tree G. We call the tree set {H s : s∈[1,M]} is a T-core leaf tree matching group of the tree G. When the trees H1, H2, ..., H M When is a key group, the tree G is their topological authentication.

[0101] Suppose that the tree G admits a constrained total coloring F: V(G)∪E(G)→[a, a+|E(G)|](a≥0), and F(E(G))=[a, a+|E(G)|], then every tree H s Admit a constrained full coloring F s :V(H s )∪E(H s )→[a, a+|E(G)|](a≥0), so that when s≠t(s, t∈[1,M]), there is always F s (E(H s ))∩F t (E(H t ))=F(ET))=F(E(GL eaf (G))),

[0102] Where E(s) is the leaf set L eaf (H s ) leaves and trees s The edge set connected to the vertices of L, E(t) is the leaf set L eaf (H t ) leaves and trees t The set of edges connected to the vertices of code (H s )≠T code (H s ), so that the T-core leaf tree matches the group of trees H1, H2, ..., H M can be distinguished from each other.

[0103] (2.2) Special T-core leaf tree matching group

[0104] Matching-3-2-1, regular leaf tree. Each vertex x of tree G i ∈V(T) connected leaf set L eaf (x i ) has b elements, namely |L eaf (x i )|=b. When M=2, there are only trees H1 and H2, and each vertex x i The set of leaves connected by ∈V(T) is

[0105] L eaf (x i )=L eaf (x 1，i )∪L eaf (x 2，i ), x 1，i ∈V(H1-L eaf (H1)), x 2，i ∈V(H2-L eaf(H2)

[0106] Notice that Therefore, there are many trees H1 and H2 that form different matching groups, generating multiple key groups.

[0107] Matching-3-2-2, Caterpillar Tree-Core Leaf Tree Matching Group. Caterpillar tree is one of the simplest tree types. It is easy to form a polynomial algorithm for many colorings and is efficient and practical. For the following 4 caterpillar trees H, T, W and Q, we will establish a caterpillar tree key pair.

[0108] The set of all leaves of the caterpillar tree H is L eaf (H), according to the definition of the caterpillar tree, the residual graph obtained by deleting all the leaves of the caterpillar tree H is the spine path P of the caterpillar tree H, that is, HL eaf (H)=P=u1u2…u n . Let the vertex u on the spine path p be i The set of leaves connected by ∈V(P) is LH eaf (u i )={v i，j :j∈[1,a i ]}, integer a≠0(i∈[1,n]), the set of leaves of the caterpillar tree H is Therefore, the set of all vertices of the caterpillar tree H is V(H)=V(P)∪L eaf (H), caterpillar tree H has Assume that the spines of caterpillar trees T, W, and Q are the same as the spine of caterpillar tree H, that is,

[0109] TL eaf (T) = WL eaf (W) = QL eaf (Q)=HL eaf (H)=P=u1u2…u n

[0110] (i) Let each vertex u on the spine path of the caterpillar tree T be i The leaf set is L Teaf (u i )={y i，j :j∈[1,b i ]}(i∈[1,n]), satisfying If the leaf set of vertices on the spine path of caterpillar tree H and caterpillar tree T satisfies the topological verification formula |L Heaf (u i )|+|L Teaf (u i)|=M(i∈[1,n]), then the caterpillar tree H (public key topological signature) and the caterpillar tree T (private key topological signature) are called ordered leaf M-regular complementary trees.

[0111] (ii) Let each vertex u on the spine path of the caterpillar tree W be i The leaf set is L Weaf (u i )={y i，j :j∈[1,c i ]}(i∈[1,n]), satisfying For vertices u1, u2, ..., u on the spine n A full arrangement of If the leaf set of vertices on the spine path of caterpillar tree H and caterpillar tree w satisfies the topological verification formula The caterpillar tree H (public key topological signature) and the caterpillar tree W (private key topological signature) are called leaf M-regular complementary trees.

[0112] (iii) Let the set of leaves of each vertex on the spine of the caterpillar tree Q be L Qeaf (u i )={z i，j :j[1,d i ]}(i∈[1,n]). If the leaf set of vertices on the spine path of the three caterpillar trees Q, H and T satisfies the topological verification formula L Heaf (u i )∪L Teaf (u i )=L Qeaf (u i )(i∈[1,n]), then the caterpillar tree Q (topological authentication) is the parent graph of the caterpillar tree H and the caterpillar tree T, respectively. The caterpillar tree H (public key topological signature) and the caterpillar tree T (private key topological signature) are Q-complementary trees about the caterpillar tree Q. The spine path of the caterpillar tree H is completely overlapped with the spine path of the caterpillar tree T to form a path, and their parent graph caterpillar tree Q is obtained, which is specially denoted as Q=H[· P ]T.

[0113] Matching-3-2-3, prawn tree-core leaf tree matching group. A tree L is called a prawn tree. If all leaves of tree L are deleted, the remaining graph LL is obtained. eaf (L) is exactly a caterpillar tree. The definitions of ordered leaf M-regular complementary tree, unordered leaf M-regular complementary tree and Q-complementary tree based on the prawn tree are the same as the definition of the caterpillar tree-core leaf tree matching group.

[0114] Matching-3-2-4. T-core leaf tree matching group based on vertex symmetric tree and edge symmetric tree. Let a copy of tree H be H*, take a vertex w of tree H, and its image vertex in tree H* be w*. We have the following two types of symmetric trees:

[0115] (i) Use a new edge ww* to connect the two trees H and H*. The resulting new tree is denoted as Also called an edge-symmetric tree.

[0116] (ii) The vertex w of the tree H and its image vertex w* in the tree H* are combined into one vertex. The resulting new tree is denoted as T = H(w[·]w*)H*, which is also called a vertex-symmetric tree.

[0117] Edge Symmetric Tree The vertex-symmetric tree T = H(w[·]w*)H* can help us prove the conclusion of Theorem 2. Importantly, the construction algorithm for edge-symmetric trees and vertex-symmetric trees is polynomial.

[0118] Theorem 2: When the tree H admits graceful labeling, the edge-symmetric tree *Accept a set-ordered graceful labeling that can derive dozens of other labelings and colorings of trees in a polynomial way (see reference [3]).

[0119] The main reasons why the T-core leaf tree matching group is computationally unbreakable and provably secure are as follows:

[0120] (1) The difficulty of finding all matching sets of T-core leaf trees of tree G can be reduced to the integer splitting problem.

[0121] (2) If the tree G admits a total coloring, finding its entire set of T-core leaf tree matchings can be reduced to the color-preserving integer splitting problem.

[0122] (3) The difficulty of cracking topological signature key pairs under graph isomorphism operations can be reduced to the subgraph isomorphism NP-complete problem.

[0123] 3. Computational topological key matching design

[0124] (3.1) Vertex tearing operation, vertex coincidence operation topological signature key pair

[0125] Definition 4: Split a vertex w∈V(G) of graph G with a vertex degree not less than k into k vertices w1, w2, ..., w k , the vertex neighbor sets of these k vertices are N(w1), N(w2), ..., N(w k ) and the vertex neighbor set of vertex w satisfy N(w)=N(w1)∪N(w2)∪...∪N(w k ),as well as So that vertex w is the result of the coincidence of these k vertices, that is, w = w1[·]w2[·]...[·]w k The obtained vertex tearing graph is denoted as H = G[∧ k ]w, we call graph G and graph H a vertex tearing operation topological signature key pair, that is, there is a graph isomorphism from graph H to graph G: H→G, such that |V(H)|≥1+|V(G)|.

[0126] Definition 5: If the vertex neighbor sets of two vertices u and vertex v in graph T have no common vertices, that is, their neighbor vertex sets satisfy Then we can overlap the vertex u and the vertex v into a vertex w, that is, w = u[·]v. The resulting non-overlapping vertex overlap graph is denoted as L = T(u[·]v), so that |E(L)| = |E(T)|, then graph T and graph L are called non-overlapping vertex overlap computational topological signature key pairs, that is, there is a graph isomorphism from graph T to graph L: T→L, so that |V(T)| = 1+|V(L)|.

[0127] Note: ① Figure 1 shows an example of graph homomorphism: H i →H,|V(H i )|≥1+|V(H)|(i∈[1,5]). ②Vertex tearing operation and non-duplicate vertex overlap operation are a pair of inverse operations. ③The security proofs of vertex tearing operation topological signature key pairs and non-duplicate vertex overlap operation topological signature key pairs can be reduced to integer splitting problem and vertex tearing tree problem with leaf number constraint. ④Graph homomorphic encryption technology can be applied to cloud computing security.

[0128] (3.2) Vertex splitting computational topology signature key pair. Assume that graph H and vertex tearing graph H* satisfy H*=H[∧2]w, where vertex w∈V(H) is torn into two vertices w′ and w″, and then these two vertices w′ and w″ of vertex tearing graph H* are connected together by an edge. The final graph is denoted as T=H*(w′[°-°]w″). Graphs T and H are called vertex splitting computational topology signature key pairs. Since a vertex w with degree d(w) is torn into two vertices w′ and w″, so that the degrees of the three vertices satisfy d(w)=d(w′)+d(w″)=k+[d(w)-k](k∈[1,d(w)-1]), there are multiple vertex splitting computational topology signature key pairs.

[0129] (3.3) Color-preserving graph homomorphism

[0130] Definition 6: Let the topological coding graph G admit a set-ordered constraint total coloring f, and the topological coding graph H admit a set-ordered constraint total coloring h, if there exists a mapping F: V(H) → V(G) such that

[0131] (i) Every edge ux∈E(H) if and only if the edge F(u)F(v)∈E(G);

[0132] (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);

[0133] (iii) The number of edges is equal |E(H)| = |E(G)|.

[0134] We call graph H a color-preserving graph isomorphic to graph G, and the color-preserving graph isomorphism operation is denoted as Graphs H and G are called color-preserving graph homomorphic topological signature key pairs, and color-preserving total coloring topological encoding matrix homomorphisms.

[0135] Example 3, Figure 1 is an example of a colorless graph isomorphism H i →H(i∈[1,5]), Figure 2 is an example of color-preserving graph homomorphism In encryption applications, H and T are private keys. i and T j They are all public keys.

[0136] Definition 7: Let the topological coding graph set G be rap (T code ) corresponds to the 3×q-order full-coloring topological coding matrix T in Definition 3. 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 fully colored topological coding matrix T code The color-preserving focus.

[0137] Note to Definition 7: The 3×q-order full coloring topological encoding matrix T in Definition 3 code Corresponding to a topological coding graph set G rap (T code ), no network reports have been found to determine the topological coding graph set G rap (T code ) and their number. Since the topological coding graph set G rap (T code ) is an NP-complete problem. Finding T code The color-preserving clustering points involve color-preserving graph homomorphism, color-preserving full-coloring topological coding matrix homomorphism, etc. This means that using the topological coding graph set G rap (T code) is computationally unbreakable and provably secure. Like graph isomorphism, color-preserving graph isomorphism can be applied to cloud computing security.

[0138] (3.4) Add and subtract edge operations topological signature key pair

[0139] If a graph T and a graph H satisfy H = T + xy-uv, where edge uv∈E(T), edge Then graphs T and H are called the addition and subtraction edge operation topological signature key pair, and the inverse operation graph is T=H+uv-xy.

[0140] (3.5) Graph Isomorphism Operation Topology Signature Key Pair

[0141] The Kelly-Ulam reconstruction conjecture in graph theory is still unsolved: Graph G and Graph H are both n-vertex graphs. If there is a one-to-one mapping f: V(G) → V(H) such that for every vertex u∈V(G), there is always a vertex-free graph Then graph G is isomorphic to graph H.

[0142] Suppose that graph G has two different vertices u and w. If the vertex-removed graph Gu and the vertex-removed graph Gw satisfy graph isomorphism Then the graph Gu and the graph Gw are said to be a pair of graph isomorphism operation topology signature key pairs.

[0143] We construct two special graphs: Let the connected graph H* be a copy of the connected graph H, and the image vertex u∈V(H) in the copy graph H* is u 0 . Connect the vertex u of the connected graph H to the image vertex u in the copy graph H* 0 Recombining into a vertex, we get the vertex symmetric graph H(u[·]u 0 )H*; use an edge to connect vertex u in graph H with its image vertex u in the copy graph H* 0 Connect them together to get an edge-symmetric graph

[0144] Vertex symmetric graphs and edge symmetric graphs can help us prove the following conclusions:

[0145] Theorem 3. There are infinitely many trees T such that the following graph isomorphism operation topology signature key pair holds:

[0146] ① Reduce leaves. Each tree T has 2 different leaves u and v, satisfying

[0147] ②Add leaves. Each tree T has two different vertices u and x, satisfying where v and y are the leaves newly added by 2H to the tree T.

[0148] ③Vertex decomposition operation. Each tree T has two different vertices w and z.w =T[∧2]w and T z =T[∧2]z, so that the vertex partitioning operation graph

[0149] ④ Add and subtract leaves. Each tree T has two different edges cd and uv, so that the two addition and subtraction edge operation graphs satisfy The side and edge

[0150] Theorem 4. There exist infinitely many connected graphs G such that the following graph isomorphism operation topology signature key pair holds:

[0151] ① Each connected graph G has two different non-leaf vertices w and z, so that the two vertex tear graphs satisfy

[0152] ② Each connected graph G has two different edges cd and uv, so that the two addition and subtraction edge operation graphs satisfy The side and edge

[0153] Theorems 3 and 4 above can help us design graph isomorphism operation topology signature key pairs.

[0154] (IV) Linear topological coloring algorithm

[0155] (4.1) Randomly adding leaf parameters graceful algorithm

[0156] Theorem 5. [4] Suppose a connected (p, q)-bipartite graph G admits a (k, d)-graceful total coloring, randomly add leaves to the bipartite graph G, and the resulting leaf-added graph G+L eaf We also admit a (k, d)-graceful total coloring, L eaf is the set of leaves added to G.

[0157] Proof: We use the xSLyLS-type graceful algorithm with leaf parameters to prove this theorem.

[0158]

[0159]

[0160] In the above random leaf addition parameter graceful algorithm, we first color the edges of the newly added leaves connected to vertex x1, and continue until the last vertex x s The edges connecting the newly added leaves are colored (from Small to Large, xSL). Then, first give vertex y tThe edges of the newly added leaves are connected and colored until the edges of the newly added leaves connected to the last vertex y1 are colored (from Large to Small, yLS). Therefore, the above algorithm is called a random leaf parameter xSLyLS-type graceful algorithm.

[0161] For W = xSLySL, xSLyLS, xLSySL, xLSyLS, ySLxSL, ySLxLS, yLSxSL, yLSxLS, the proof of this theorem can lead to 8 random leaf-adding parameter type graceful algorithms. Figures 6, 7 and 8 give examples to explain the leaf-adding parameter type graceful algorithms.

[0162] The proof of this theorem is complete.

[0163] Theorem 6 [4] A tree T with diameter D(T) (≥3) admits at least β (k, d)-graceful total colorings, 2 m-1 ≤β≤2 m +2 , m+1=[D(T) / 2].

[0164] Proof: Let D(T1) be a tree with diameter D(T1) ≥ 3, L eaf (T1) is the set of leaves of tree T1. Remove all leaves of tree T1 and get another tree T2 = T1-L eaf (T1). In this way, we get tree T i+1 =T i -L eaf (T i )(i∈[1,m]), and T m+1 is a star tree whose vertex set is V(T m+1 )={x}∪{x j :j∈[1,n]}, the edge set is E(T m+1 )={xx j : j∈[1,n]}.

[0165] For non-negative integers k and d, we define the star tree T m+1 Define a (k, d)-graceful total coloring f m+1 as follows:

[0166] (i)f m+1 (x) = 0, f m+1 (x j ) = k + (j-1) d; (ii) f m+1 (xx j )=f m+1 (x j )-f m+1 (x)(j∈[1,n]).

[0167] Since each tree T i It is for the tree i+1 Add leaf set L eaf (T i ) in the leaves, namely T i =T i+1 +L eaf (T i ). According to Theorem 5, the tree T i Admit a (k, d)-graceful total coloring f i , and is from the tree T i+1 The (k, d)-graceful total coloring f i+1 To derive the (k, d)-graceful full coloring f i By induction, the tree T1 admits a (k, d)-graceful total coloring f1.

[0168] Because each tree T i There are at least 2 leaves, that is |L eaf (T i )|≥2(i∈[1,m]). When each tree T i+1 =T i -L eaf (T i )(i∈[1,m-1]) of the vertex set V(T i+1 )=X i+1 ∪Y i+1 satisfy |X i+1 |≥2 and |Y i+1 |≥2, we assert that tree T1 admits at least 2 m-1 mutually distinct (k, d)-graceful total colorings.

[0169] Because, for W = xSLySL, xSLyLS, xLSySL, xLSyLS, ySLxSL, ySLxLS, yLSxSL, yLSxLS, the proof of Theorem 5 can lead to 8 random leaf-plus parameterized graceful algorithms. In other words, the tree T1 admits 2 m+2 mutually distinct (k, d)-graceful total colorings.

[0170] The proof of this theorem is complete.

[0171] Definition 8: A connected (p, q)-graph G admits a mixed total coloring f:V(G)→M, f:E(G)→[1,q], where the elements of M are sets such that for each edge uv of the graph G, there are always a number e u ∈f(u) and the number e v ∈f(v), so that the coloring of edge uv is f(uv)=|e u -ev |, and the set of edge colorings of the graph G is f(E(G)) = [1, q]. We say that a connected (p, q)-graph G admits a v-set e-normal (k, d)-graceful total coloring.

[0172] Theorem 7. For every vertex-tearing tree T with q edges and diameter D(T) ≥ 3 in a connected (p, q)-graph G, the connected (p, q)-graph G admits β v-set e-normal (k, d)-graceful total colorings, where 2 m-1 ≤β≤2 m+2 And m+1=[D(T) / 2].

[0173] (4.2) Leaf-Growing Algorithm for Set-Ordered Graceful Trees

[0174]

[0175] Theorem 8. According to the "leaf growth algorithm for set-ordered graceful labeling of trees", if the connected bigraph G (as the private key) recognizes the set-ordered graceful labeling h, color the vertex w0 of the connected bigraph G as 0 and connect the leaves u1, u2, ..., u m , get the connected dual graph H; then connect two non-adjacent vertices of the dual graph H, remove an edge of the connected dual graph H, get the connected dual graph L (as the public key) that recognizes the set-ordered graceful labeling, so that a leaf of the connected dual graph L is colored 0 under the set-ordered graceful labeling.

[0176] Since the set-ordered graceful labeling of trees can derive dozens of tree-based labeling and coloring, the graph group generated by the "leaf-growing algorithm for set-ordered graceful labeling of trees" has higher convenience and practicality than the previous algorithms.

[0177] (4.3) Graceful Tree (Graph) Nested Growth Algorithm

[0178]

[0179]

[0180] Theorem 9: In the graceful tree nesting growth algorithm, if the nested tree L (as the private key) and the embedded tree H (as the public key) both recognize set-ordered graceful labeling, then the vertex coincidence tree L = T[o]H (as the authentication of the private key and the public key) recognizes set-ordered graceful labeling.

[0181] Theorem 10. According to the "graceful tree nested growth algorithm", if the connected dual graph G (as the private key) recognizes the set-ordered graceful labeling, and the connected graph H (as the public key) recognizes a graceful labeling, then a vertex of the connected dual graph G can be overlapped with a vertex of the connected graph H to obtain a connected vertex overlap graph L = G[o]H that recognizes the graceful labeling (as the authentication of the private key and the public key).

[0182] Theorem 11. If a connected (p, q)-bipartite graph G admits set-ordered graceful labeling, then G admits set-ordered (k, d)-graceful labeling.

[0183] Proof: Suppose a connected (p, q)-bipartite graph G admits a set-ordered graceful labeling h, so that the set-ordered constraint max h(X)<minh(Y) holds, where the vertex set of the bipartite graph G is V(G)=X∪Y; the number of elements in the vertex coloring set of the bipartite graph G is |h(V(G))|=p, and the edge coloring set of the bipartite graph G is h(E(G))=[1,q]. For integers k, d>0, a full label h* of the (p, q)-diptych G is defined as: h*(x)=h(x)·d(x∈X), h*(y)=k+[h(y)-1]d(y∈Y); the coloring of each edge xy∈E(G) is h*(xy)=h*(y)-h*(x)=k+[h(y)-1]dh(x)·d=k+[h(xy)-1]d, and the edge coloring set of the p, q-diptych G is h*(E(G))={k,k+d,k+2d,…,k+(q-1)d}, the number of elements of the vertex coloring set is |h*(V(G))|=p, and the set ordered constraint max h*(X)<min h*(Y) is satisfied.

[0184] According to the note of Definition 2, the global labeling h* is a set-ordered (k, d)-graceful labeling of a connected (p, q)-dual graph G.

[0185] (4.4) Magic Constraints Full Coloring and Full Labeling

[0186] Definition 9: Let a connected (p, q)-graph G admit a total coloring f: V(G)∪E(G)→[1,M], such that the coloring of the two endpoints of every edge xy∈E(G) satisfies f(x)≠f(y), and the coloring of any two adjacent edges uv and uw satisfies f(uv)≠f(uw).

[0187] M-1. If the edge magic constraint M(f(x), f(xy), f(y)) = f(x) + f(xy) + f(y) - k = 0 (k ≥ 1) holds, f is called edge magic total coloring.

[0188] M-2. If the graceful difference constraint M(f(x), f(xy), f(y)) = ||f(x)-f(y)|-f(xy)|-k = 0 (k≥≥0) holds, f is called a graceful difference total coloring.

[0189] M-3. If the edge difference constraint Mf(f(x), f(xy), f(y)) = f(xy) + |f(x) - f(y) | - k = 0 (k ≥ 1) holds, f is called edge difference total coloring.

[0190] M-4. If the happiness difference constraint M(f(x), f(xy), f(y)) = |f(x)+f(y)-f(xy)|-k = 0 (k≥≥0) holds, f is called a happiness difference total coloring.

[0191] Further, according to the above full coloring f and magic constraint M(f(x), f(xy), f(y)) = k(xy∈E(G)), and W∈{edge magic, graceful difference, edge difference, happiness difference}:

[0192] M-5. If the vertex and edge coloring sets of a graph G satisfy f(V(G)∪E(G))=[1,p+q], then f is called W-universal labeling.

[0193] M-6. If the number of elements in the coloring set of vertices and edges of the graph G |f(V(G)∪E(G))|=p+q, then f is called sub-W-full labeling.

[0194] M-7. If a graph G is a tree with p vertices and f(V(G)∪E(G)) = [1, p], then f is called the minimum W total coloring of the tree G.

[0195] M-8. If G is a connected (p, q)-pair graph, that is, V(G) = X∪Y and When a total coloring (full labeling) f satisfies the set-ordered constraint max f(X)<min f(Y), then f is called a set-ordered W-total coloring (set-ordered W-full labeling) of a connected (p, q)-dual graph G.

[0196] Theorem 12. If a connected (p, q)-bipartite graph G admits set-ordered graceful labeling, then G admits W-total coloring (W-total labeling), where W∈{edge magic, graceful difference, edge difference, happiness difference}.

[0197] Algorithmic proof: Initialization. Assume that a connected (p, q)-bipartite graph G admits a set-ordered graceful label h, such that the set-ordered constraint max h(X)<min h(Y) holds, where the vertex set of graph G is V(G)=X∪Y and The number of elements in the vertex coloring set of graph G is |h(V(G))|=p, and the edge coloring set of graph G is {h(xy)=h(y)-h(x):xy∈E(G)}=h(E(G))=[1,q].

[0198] Algorithm-1, edge magic full coloring. Define a full coloring h of the even graph G em As follows: em (x)=max h(X)+minh(X)-h(x)(x∈X),h em (y) = h(y)(y∈Y); h em(xy) = q + 1 - h (xy) = q + 1 - [h (y) - h (x)] (xy ∈ E (G)), then there is the following edge magic constraint

[0199] h em (x)+h em (y)+h em (xy)=[max h(X)+min h(X)-h(x)]+h(y)+(q+1-[h(y)-h(x)])=max h(X)+min h(X))+q+1

[0200] and satisfy the set order constraint max h em (X)<min h em (Y) and |h em (V(G))|=p. Note that under the set-ordered graceful labeling h, "max h(X)+min h(X)+q+1" is a constant. According to Definition 9, the total coloring h em It is a magic total coloring of the even graph G with set-ordered edges.

[0201] Algorithm-2, edge difference total coloring. Define a total coloring h of the even graph G ed As follows: ed (x) = h(x)(x∈X), h ed (y) = h(y)(y∈Y); h ed (xy) = q + 1 - h (xy) (xy ∈ E (G)), then there is a side difference constraint h em (xy)+|h ed (y)-h ed (x)|=[q+1-h(xy)]+h(y)-h(x)=q+1, and satisfies the set-ordered constraint max h ed (X)<min h ed (Y), and |h em (V(G))|=p. So the total coloring h ed is a set-ordered edge-difference total coloring of the even graph G.

[0202] Algorithm-3, happiness difference total coloring. Define a total coloring h of the even graph G fd As follows: fd (x)=max h(X)+minh(X)-h(x)(x∈X),h fd (y) = h(y)(y∈Y); h fd (xy) = q + 1 - h (xy) (xy ∈ E (G)), then there is a happiness difference constraint

[0203] |h fd (x)+h fd (y)-hfd (xy)|=|max h(X)+min h(X)-h(x)+h(y)-[q+1-h(xy)]|=q+1-[max h(X)+min h(X)]

[0204] and satisfy the set order constraint max h fd (X)<min h fd (Y) and |h em (V(G))|=p. Note that under the set-ordered graceful labeling h, "q+1-[max h(X)+min h(X)]" is a constant. According to Definition 9, the total coloring h fd is a set-ordered happy difference total coloring of the even graph G.

[0205] Algorithm-4, graceful difference total coloring. Define a total coloring h of the even graph G gd As follows: gd (x) = h(x)(x∈X), h gd (y) = h(y)(y∈Y); h gd (xy) = q + 1 + h (xy) (xy ∈ E (G)), then there is a graceful difference constraint || h gd (y)-h gd (x)|-h gd (xy)|=|h(xy)-[q+1+h(xy)]|=q+1, and satisfy the set order constraint max h gd (X)<min h gd (Y) and |h gd (V(G))|=p. According to Definition 9, the total coloring h gd is a set-ordered graceful difference total coloring of the even graph G.

[0206] The theorem is proved.

[0207] (4.5) Magic Constrained Full Coloring, Full Label Random Leaf Addition Algorithm

[0208] According to Theorem 12, we have

[0209] Theorem 13. If a graph G is a tree of p vertices and admits set-ordered graceful labeling, then G admits set-ordered W-full labeling, where W∈{edge magic, graceful difference, edge difference, happiness difference}.

[0210]

[0211] Note that the "random leaf-edge magic full coloring algorithm" is also the "random leaf-edge magic full labeling algorithm". An example of explaining the "random leaf-edge magic full coloring algorithm" is given in Figure 9. (a) and (b) in Figure 13 are examples of explaining the "random leaf-edge magic full labeling algorithm".

[0212]

[0213] Figure 10 gives an example to explain the "random leaf-edge difference full coloring algorithm", and (c) and (d) in Figure 13 explain the "random leaf-edge difference full labeling algorithm".

[0214]

[0215] The example in Figure 11 is an explanation of the "random addition of leaves - happiness difference full coloring algorithm". (e) and (f) in Figure 13 are examples of the "random addition of leaves - happiness difference full labeling algorithm".

[0216]

[0217] Figure 12 is an example of explaining the “random addition of leaves-graceful difference full coloring algorithm”, and (g) and (h) in Figure 13 are examples of explaining the “random addition of leaves-graceful difference full labeling algorithm”.

[0218] (V) Implementation of asymmetric linear topological encryption

[0219] (5.1) Asymmetric linear topology key pair (group)

[0220] Asymmetric linear topology keys (ALT-keys) consist of tree graphs, including a private key tree graph and a public key tree graph.

[0221] (5.1.1) T-core leaf tree matching group: ① fully colored caterpillar tree-core leaf tree matching group; ② fully colored prawn tree-core leaf tree matching group; ③ fully colored T-core regular leaf tree matching group; ④ fully colored vertex symmetric tree-core leaf tree matching group; ⑤ fully colored edge symmetric tree T-core leaf tree matching group.

[0222] (5.1.2) Finitely many tree matching groups. For a tree graph T, tree graph H = T + uv-ab, where ab∈E(T) and H = public key tree, T = private key tree.

[0223] (5.1.3) Infinite random leaf tree matching groups. Tree H k =T+L eaf (k)(1≤k), where L eaf (k) is the set of leaves that randomly add new leaves to the tree T, where H k= set ordered graceful public key tree, T = set ordered graceful private key tree. For the specific implementation algorithm, see (IV) Algorithm design of linear topological coloring.

[0224] (5.1.4) Infinitely many random nested trees. G = T[o]H, where G = authentication tree, nested tree T = set-ordered graceful private key tree, and embedded tree H is a graceful public key tree or a set-ordered graceful public key tree. For the specific implementation algorithm, see (IV) Algorithm design of linear topological coloring.

[0225] (5.1.5) "One to many". One private key tree corresponds to multiple public key trees. A company votes, A sends a voting document F to B-k, and B-k returns an encrypted vote AF k (k∈[1,n]) (see the uniform fractal forest problem).

[0226] (5.1.6) Trees with special topological structures and colors. ① Trees or forests composed of Chinese character graphs.

[0227] ② Calibrated complete graph K n There are n-2 There are different calibrated support tree graphs. We can define full coloring for these calibrated support tree graphs:

[0228] Let the calibrated spanning tree T admit vertex coloring h T (By calibrating the complete graph K n derive the vertex coloring of the tree T), we define the coloring h of the edge uv of the calibrated support tree T T (uv) satisfies the constraint equation F(h T (u), h T (uv),h T (v)) = 0, for example h T (uv)=|h T (u)-h T (v)|.

[0229] (5.2) One-time-one-pad keys in asymmetric linear topology

[0230] 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 .

[0231] Definition 10. Let the bipartite graph \(G=(X,Y)\) with \(q\) edges admit a set-ordered constrained total coloring \(h: V(G)\cup E(G)\to[a,b]\) such that the two vertex coloring sets satisfy the set-ordered constraint \(\max h(X)<\min h(Y)\). Then the parametric total coloring topological coding matrix of graph \(G\) is defined as

[0232]

[0233] 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 \(G\) (Definition 3).

[0234] Example 5. The topological coding graph \(P1\) of the character 'ugly' in Figure - 4 admits a graceful labeling \(h1\). According to Definition 10, we obtain the parametric total coloring topological coding matrix of the topological coding graph \(P1\) of the character 'ugly' as where

[0235]

[0236] and

[0237]

[0238] Obviously, the parametric total coloring topological coding matrix \(P\) code (P1, F1|k, d) can provide infinitely many parametric digital strings for the 'one-time pad' encryption technology.

[0239] 'One-time pad' encryption method:

[0240] (5.2.1) Private key and public key matching pair. Use two parametric total coloring topological coding matrices \(P\) code (G, F|k, d) and \(P\) code (G, F*|d, k) to derive the private key digital string \(s(k, d)\) and the public key \(s*(d, k)\) to form the key matching pair \([s(k, d), s*(d, k)]\).

[0241] (5.2.2) 'One-to-many' key. For positive integers \(a0\) and \(b0\), use a (private key) parametric total coloring topological coding matrix \(P(G, F|k0, d0)\) to correspond to \(n\) (public key) parametric total coloring topological coding matrices \(P(H, F|k\) r , d r ), where \(k\) r =r·a0·k0, d r =r·b0·d0 (1≤r≤n). Each digital string \(s(k0, d0)=c1(k0, d0)c2(k0, d0)...c\) 3q(k0, d0) all correspond to another n digital strings s(k r , d r )=c1(k r , d r )c2(k r , d r )...c 3q (k r , d r )(1≤r≤n), where each digital string s(k r , d r ) is encoded by the parameterized full coloring topological matrix P(G, F|k r , d r ) is derived. We get a "one-to-many" string key pair, where s(k0, d0) is the private key string, and each digital string s(k r , d r ) is the public key string. There are n·(3q)! such "one-to-many" string key pairs.

[0242] (5.2.3) Parameters of the curve constraint. The curve function y = f(x) generates a sequence of integer pairs (k n , d n ), so that d n =f(k n )(1≤n≤N). Parameterized full coloring topological encoding matrix P code (G,F|k n , d n ) When encrypting and decrypting the digital string derived from the cipher, not only the parameter full coloring topological coding matrix authentication is required, but also the verification curve function d n =f(k n ), which makes different curves authenticate differently, bringing greater complexity overhead to the decryptor.

[0243] (5.2.4) Multiple encryption. Use the integer pair sequence (k n , d n )(1≤n≤N) is a digital string generated by the parameterized full coloring topological encoding matrix {s n} to encrypt a plaintext D, that is, use the digital string s1 to encrypt the plaintext D to obtain the ciphertext F1; then use the digital string s2 to encrypt the plaintext ciphertext F1 to obtain the ciphertext F2; continue in this way, and finally obtain the N-fold encrypted ciphertext F N As computers become faster and faster, the time taken for encryption and decryption can be ignored. The multiple encryption technology here has the potential to resist quantum computing.

[0244] The basis of “one-time one-key”: for the adjacent matrix A(G) n×n and the full coloring topological encoding matrix T code(G, h) 3×q and the parameter full-coloring topological coding matrix P code (G, F|k, d) 3×q The derived string. Since the factorial numbers (n 2 )! and (3q)! are quite large, it makes the communicators unable to use up (n 2 )! and (3q)! topological signature key pairs and digital string key pairs during their lifetimes of communication. Therefore, each time the two communicators encrypt and decrypt a file, they use unused topological signature key pairs and digital string key pairs, thus achieving the effect of "one-time pad".

[0245] (5.3) Composite technology of asymmetric linear topology

[0246] Full-coloring topological coding matrix T code (G1, f1), T code (G2, f2),, T code (G n , f n )'s union operation is defined as

[0247]

[0248] Example 6: Figure - 14 shows the topological coding graphs G1, G2, G3, and G4 of 4 Chinese characters "qian, shi, gan, tu". It is not difficult to see that for any two topological coding graphs G k and G j , their full-coloring topological coding matrices satisfy T code (G k , f k ) 3×6 ≠T code (G j , f j ) 3×6 (k ≠ j). However, for any two topological coding graphs G k and G j , their topological structures are the same, that is, there is graph isomorphism Note that the Chinese word meanings of the 4 Chinese character topological coding graphs in Figure - 14 are different from each other and vary greatly. The Chinese character phrase "qian shi gan tu" forms a topological coding graph G, which is composed of 4 topological coding graphs G1, G2, G3, and G4, that is, there is a union graph operation The full-coloring topological coding matrix of the topological coding graph G is

[0249]

[0250] and

[0251]

[0252] It should be noted that the union matrix T in formula (5) code The string s* derived from (G, f) is more complex. It is impossible to find each fully colored topological coding matrix T from the string s* code (G k ,f k )(k ∈ [1, n]), because:

[0253] ①Tearing a fully colored topological coding matrix T code (G, f) into several fully colored topological coding matrices T code (G1, f1), T code (G2, f2),..., T code (G n ,f n ) is difficult.

[0254] ②Each fully colored topological coding matrix T code (G k ,f k )(k ∈ [1, n]) corresponds to a set of topological coding graphs. Finding the topological coding graph G corresponding to the key in this set of topological coding graphs k will encounter NP-complete problems.

[0255] ③Finding each topological coding graph G from thousands of graph colorings and labelings k and the admitted coloring f k is non-polynomial work.

[0256] ④There are other operations for the algebraic operations of topological matrices: intersection operation, linear operation, tearing, coincidence operation, splitting operation, coloring transformation, homomorphism operation, etc.

[0257] ⑤The topological coding graph corresponding to the fully colored topological coding matrix T code (G, f) may be a Chinese sentence. For example, the Chinese character topological coding graph in Figure - 14 can form Chinese phrases: qianshigan tu, qianshitugan, tuganqianshi, tuqianshigan, etc.

[0258] (5.4) Packing the asymmetric linear topological cipher into a "box"

[0259] Purpose: Let the attacker find a specific asymmetric linear topological key from the "box" to increase the cost of decryption. There are the following techniques:

[0260] (5.4.1) The edge addition operation generates a "box". The asymmetric linear topology key matching is "packed" into the connected graph. Specifically, new edges are connected between non-adjacent vertices of the asymmetric linear topology key matching to obtain a connected graph equivalent to the "box", and the "box" is sent to the communicating party. Figure 15 gives an example of a "box": the edge magic full coloring tree EM3 of Figure 9 is the support tree of the edge magic full coloring graph G, where f(u)+f(uv)+f(v)=10 and the cardinality of the vertex set is equal: |V(G)|=|V(EM3)|; the edge difference full coloring tree DM3 of Figure 10 is the support tree of the edge difference full coloring graph H, and has h(uv)+|h(u)-h(v)|=8, and the cardinality of the vertex set satisfies |V(H)|=|V(DM3)|. Computational unbreakability can be reduced to the previous NP-complete problem "degree-constrained support tree problem".

[0261] (5.4.2) The vertex overlap operation generates a "box". The topological coding graph key is subjected to a vertex overlap operation without common adjacent vertices to obtain a "box". The user uses a vertex tearing operation to disassemble the "box" and obtain the required topological coding graph key. Figure 16 gives an example of a "box": the edge magic full coloring graph G EM The vertex tearing operation is performed to obtain the edge magic full shading tree EM3 of Figure 9, where f(u)+f(uv)+f(v)=10 and the cardinality of the vertex set is unequal: |V(G EM )|<|V(EM3)|;edge difference total coloring graph G DM The vertex tearing operation is performed to obtain the edge difference full coloring tree DM3 in Figure 10, and h(uv)+|h(u)-h(v)|=8, and the cardinality of the vertex set is unequal |V(G DM )|<|V(DM3)|. The computational unbreakability can be reduced to the previous NP-complete problem "Vertex Tearing Tree Problem with Leaf Number Constraint".

[0262] (5.4.3) T-core leaf tree matching groups produces "boxes".

[0263] (5.5) Total coloring of sets of asymmetric linear topology

[0264] Theorem 6 tells us that a tree T with diameter D(T) not less than 3 admits at least β (k, d)-graceful total colorings, of which 2 m-1 ≤β≤2 m+2 and m+1=[D(T) / 2]. All trees T with diameter D(T)(≥3) that are all different (k, d)-graceful colorings are grouped into the coloring set C. olor (T), we start from the coloring set C olor (T) Take a (k, d)-graceful total coloring f1, f2, ..., fn , to construct the set total coloring F of the tree T as follows: The coloring of each vertex x of the tree T is F(x) = {f1(x), f2(x), ..., f n (x)}, the coloring of each edge xy in the tree T is F(xy) = {f1(xy), f2(xy), ..., f n (xy)}. We get the set total coloring topological encoding matrix S of the tree T code (T, F) = (X, E, Y) T , where v-vector X = (F(x1), F(x2), ..., F(x q )), e-vector E = (F(x1y1), F(x2y2), ..., F(x q y q )), v-vector Y = (F(y1), F(y2), ..., F(y q )), the number of edges q = |E(T)| = |V(T)|-1. Since the component F(x i ) and F(y j ) are all sets, so that the set of trees T is fully colored with the topological encoding matrix S code The strings derived from (T, F) are stronger, and encryption and decryption using them are computationally unbreakable.

[0265] Note that the full coloring topology of the tree T encodes the matrix T code (T1, f1), T code (T2, f2), ..., T code (T n , f n ) The derived set of strings and the set of trees T are fully colored topological encoding matrices S code The set of strings derived from (T, F) is exactly the same. The difficulty of cracking the topological signature key pair under graph isomorphism operation can be reduced to the subgraph isomorphism NP-complete problem.

[0266] Theorem 14. Every simple graph admits a v-set e-normal (odd) graceful total coloring.

[0267] Proof: This is the case when (k, d) = (1, 1) for a v-set e-normal (k, d)-graceful total coloring of Definition 8.

[0268] Case-1: If the simple graph G is a connected (p, q)-graph, use the vertex tearing operation to tear the graph G into a tree T with q edges. Using Theorem 6, we get a graceful total coloring h of the tree T, and then perform the vertex coincidence operation on the tree T to get the original graph G. At this time, the graph G admits a v-set e-normal graceful total coloring h* derived from the graceful total coloring h (see Definition 8).

[0269] Case-2: If the simple graph G is a disconnected (p, q)-graph, it has branches G1, G2, ..., G m . Use vertex tearing operation to connect the branches G i Torn into a tree i (i∈[1,m]), so that the number of edges q i =|E(G i )|=|E(T i )|. Then the tree T j A vertex of the tree T j+1 One vertex of coincides with each other, and we get the tree T j [o]T j+1 (j∈[1,m-1]), tree T=T1[o]T2[o]...[o]T m The number of edges |E(G)|=|E(T)| is satisfied. Using the method of case 1, we can obtain a graceful full coloring f of the tree T. Then, we can perform vertex tearing operation on the tree T to obtain the forest {T1, T2, ..., T m}, each tree T i Admit a full coloring f i , so that Finally, do graph homomorphism operation T i →G i (i∈[1,m]), we get the original disconnected graph G, which admits a v-set e-normal graceful total coloring f*, so that the edge coloring set f*(E(G)) of the disconnected (p,q)-graph G = [1,q].

[0270] The theorem is proved.

[0271] Theorem 14 solves conjecture 15 in [3]. Regarding the set total coloring topological encoding matrix induced digital string, we propose the following question:

[0272]

[0273] The main difficulties in solving the "digital string-set full coloring topological coding matrix problem" are as follows:

[0274] (1) There is no polynomial method to convert the number string s = c1c2...c n Written as a small string of numbers α i , α j , α k in the form of.

[0275] (2) There is no polynomial method to convert the small number string α i , α j , α k Written in the form of a collection.

[0276] (3) Finding a graph G that admits a total coloring of a set F is an NP-complete problem involving subgraph isomorphism.

[0277] (4) Find every W that forms the full coloring of the set F m - Constrained full coloring f1,f2,...,f N , so that they satisfy W m - Constraint equation W m [f m (u m ), f m (u m v m ), f m (v m )] = 0(m∈[1,N]), which is extremely difficult because the number of graph colorings in graph theory is not fixed and new colorings are generated every day.

[0278] We assert that the topological key designed using the set-to-full-coloring topological encoding matrix is ​​computationally unbreakable and provably secure.

[0279] (5.6) Modular algorithm, ATCC algorithm

[0280] "Asymmetric linear topology key" is referred to as "ALT-key". We have

[0281] Strength measurement of ALT-key: ① The length of ALT-key; ② The mathematical constraints of ALT-key W(k1, k2, ..., k n ) dimension n; ③ the topological structure of the ALT-key; ④ the topological structure of the ALT-key is a Chinese sentence; ⑤ the cardinality of the space for constructing the ALT-key is at least 2e (=2 200 ), that is, the number of vertices of the graph is not less than 30.

[0282] The topological structure of ALT-key satisfies: strong constraints, strong randomness, "one-time one-key" capability, and infinity.

[0283] The quantity form of ALT-key satisfies: "one-to-many", "many-to-many", "infinitely many".

[0284] ALT-key management tasks: key registration, key production, key distribution, key use, key authentication, key revocation, key update, acceptance of communication partner's key, etc.

[0285] Assuming that A and B are communication users of the network, we designed the following algorithm:

[0286] (5.6.1) Module Algorithm-I (see Figure-17), A sends a key package to B, which contains B's identity public key I pub(i) and the file encryption public key D pub (i)(i∈[1,n]), A retains and does not disclose his private key I pri (i) and file encryption private key D pri (i)(i∈[1,n]). In which, A uses the topological coding graph G i The adjacent matrix A(G i )Export the identity private key I pri (i) and identity public key I pub (i) Using the topological coding graph G i The total coloring topological encoding matrix T of (i∈[1,n]) code (G i )Export file encryption private key D pri (i) and the file encryption public key D pub (i).

[0287] Step-I-1, in the i-th communication, B uses the public key I given by A pub (i) and the file encryption public key D pub (i) Encrypt the plaintext F(i) to obtain the ciphertext S(i) with double protection, and then send it to A.

[0288] Step-I-2: In the i-th communication, after receiving the ciphertext S(i) from B, A first uses the identity private key I pri (i) Decrypt the ciphertext S(i), A uses the topological coding graph G i The adjacent matrix A(G i ) to authenticate identity A dja [I pri (i) I pub After (i)] is passed, it is confirmed that the ciphertext S(i) is sent by B (topological signature authentication to ensure unique identity), and the ciphertext's 1-fold protection is removed. Then the file is decrypted with the private key D pri (i) To continue decrypting the ciphertext with only 1-fold protection, A uses the topological coding graph G i The full coloring topological encoding matrix T code (G i ) for authentication code [D pri (i), D pub (i)] After passing, A sees the plaintext F(i).

[0289] Characteristics of modular algorithm-I: When the positive integer n is large, only key I is used in the i-th communication pri (i) and (I) pub (i) D pri (i) and D pub(i) The previously used key is not used, so that the modular algorithm-1 has the effect of "one-time one-key". i+1 is the topological encoding graph G i The topological coding graph G is obtained by randomly adding leaves to make j, i∈[1,n] i Different from the topological coding graph G j (i≠j), even if the decryptor uses the key used in the previous i-th communication, these methods are invalid for the subsequent keys, forcing the decryptor to find a new decryption method, thereby increasing the decryption cost.

[0290] (5.6.2) Module Algorithm-II, A sends toolkit B to B tool [I pub , D pub ], so that B can create his own public key I pub (i) and the file encryption public key D pub (i)(i∈[1,n]). Toolkit A tool [I pri , D pri ] is a topologically encoded graph G i The adjacent matrix A(G i ) derive A's identity private key IA respectively pri (i) and B's public key IB pub (i) Using the topological coding graph G i The total coloring topological encoding matrix T of (i∈[1,n]) code (G i ) Export A's file encryption private key DA respectively pri (i) and B's file encryption public key DB pub (i).

[0291] B sends tool kit B to A tool [I pub , D pub ] is a topological coding graph H i The adjacent matrix A(H i ) derive B's identity private key IB pri (i) and A's public key IA pub (i), using the topological coding graph H i The total coloring topological encoding matrix T of (i∈[1,n]) code (H i ) Export B's file encryption private key DB respectively pri (i) and A's file encryption public key DA pub (i). Wherein, the topological coding graph G i And the topological coding graph H i(i∈[1,n]) forms a match.

[0292] Step-II-1, in the i-th communication, B uses toolkit B tool [I pub , D pub ] Make your own public key pub (i) and the file encryption public key D pub (i), and then encrypt the plaintext F(i) to obtain the ciphertext S(i) with double protection, and then send the ciphertext S(i) to A.

[0293] Step-II-2, in the i-th communication, after receiving the ciphertext S(i)(i∈[1,n]) sent by B, A uses his own toolkit A tool [I pri , D pri ]Create identity private key IA pri (i) and the file decryption private key DA pri (i), and then perform identity authentication of the adjacent matrix A on the ciphertext S(i). dja [LA pri (i), IB pub (i)] and the digital encryption authentication of the fully colored topological coding matrix T code [DA pri (i), DB pub (i)] to obtain the plaintext F(i).

[0294] The characteristics of modular algorithm-II: Before the ith communication between parties A and B, parties A and B do not make a key, and the attacker cannot obtain the key information to be destroyed. Using the techniques introduced in the previous (IV) and here (5.1) to (5.5), the topological coding graph G can be i And the topological coding graph H i (i∈[1,n]) forms a matching M atch [G i , H i ], and make it match M atch [G i , H i ] is not equal to match M atch [G j , H j ](i≠j); put the ALT-key into a "box" and use the fully colored topological coding matrix to generate complex and infinite keys; let the decryptor often run in different fields of learning and spend more time.

[0295] (5.6.3) ATCC Algorithm-I (see Figure-18), Party A and Party B rely on a third party for communication, the third party is called the Asymmetric Topology Cryptography Center, referred to as ATCC. Party A designs the key pair by itself.

[0296] Step-I-1, A sends the encrypted package P to ATCC ack (A→B), which contains B’s identity public key I pub (i) and the file encryption public key D pub (i)(i∈[1,n]).

[0297] Step-I-2, ATCC will encrypt the package P ack (A→B) then performs security processing and creates an encrypted package C[P ack (A→B)] and then send it to B.

[0298] Step-I-3, B receives the encrypted packet C[P ack (A→B)], unzip it and get the encrypted package P sent by A. ack (A→B), and then unwrap it. In the i-th communication, B uses the unwrapped package P ack The public key I in (A→B) pub (i) and the file encryption public key D pub (i) Encrypt the plaintext F(i) to obtain the ciphertext S(i) with double protection, then send the ciphertext S(i) to ATCC and request ATCC to send it to A.

[0299] Step-I-4: After ATCC performs security processing on the ciphertext S(i), it obtains the encrypted package C[S(i)] and sends it to A. After A decrypts the encrypted package C[S(i)], it obtains the ciphertext S(i). The remaining steps are the same as those of the module algorithm-1.

[0300] Features of ATCC Algorithm-I: Every communication between Party A and Party B is protected by ATCC, so that decryption requires attacking the key under the topological structure. The attacker must switch to search in different mathematical fields, which cannot form an effective attack, and its benefits are far less than its cost. Most importantly, users do not need to put their public keys and secret keys on a public platform, or transmit public keys and secret keys to the communication party through public channels.

[0301] (5.6.4) ATCC Algorithm-II. Party A and Party B completely rely on ATCC to make keys, and ATCC uses all the technologies of the present invention to make keys.

[0302] Step-II-1, ATCC sends key package P to A ack (A), which contains A's identity private key IA pri(i) and the file decryption private key DA pri (i)(i∈[1,n]), and the public key IA to encrypt the file sent to B pub (i) and the file encryption public key DA pub (i)(i∈[1,n]).

[0303] Step-II-2, ATCC sends key package P to Party B ack (B), which contains B's identity private key IB pri (i) and file decryption private key DB pri (i)(i∈[1,n]), and the public key IB to encrypt the file sent to B pub (i) and file encryption public key DB pub (i)(i∈[1,n]).

[0304] Step-II-3, the steps of communication between Party A and Party B are the same as those of Module Algorithm-1.

[0305] Features of ATCC Algorithm-II: ① It is a great help to network communication users who cannot make keys. Users do not need to make keys, distribute keys, update keys and other high-tech and tedious work by themselves, so that users can obtain non-public topological signature keys and non-public encryption keys in time. After Party A and Party B obtain the keys sent by ATCC, they can communicate directly and securely without the help of a third party.

[0306] ②ATCC can use graph groups and grids to perform overall encryption of communities and local area networks, and perform overall key management tasks such as key registration, key production, key distribution, key use, key authentication, key revocation, and key update for users of communities and local area networks.

[0307] ③ATCC can be miniaturized and become private key management software for individual network users.

Claims

1. The design and implementation of asymmetric linear topology (ALT) encryption technology are characterized by: using the adjacency matrix of the fully colored topological coding tree (graph) and the fully colored topological coding matrix to design the technical theory and feasible algorithm of ALT-encryption. The technical theory includes problems belonging to the NP-complete problem class, which makes ALT-encryption technology and various algorithms generated by it computationally unbreakable and provably secure. Use graph homomorphic operations and graph structure operations to generate topological signatures and topological key pairs. Give a W-type coloring algorithm for random leaf growth of the topological coding tree, design the metric and management of ALT-key strength, and implement the algorithm of ALT-encryption.

2. The fully colored topological coding tree (Figure) according to claim 1, characterized in that: The topological coding graph is composed of "topological structure" and "mathematical constraints". "Mathematical constraints" are theories and practical techniques based on graph theory, number theory, set theory, abstract algebra, linear algebra, computer science and other disciplines; "topological structure" is a tool for semi-structured data, not an ordinary geometric graph or picture.

3. According to the adjacency matrix and the fully colored topological coding matrix described in claim 1, its topological characteristics are: (3.1) The adjacency matrix determines the uniqueness of the topological signature; (3.2) The fully colored topological coding matrix derives the encrypted and decrypted strings, but a fully colored topological coding matrix can correspond to multiple mutually incompatible topological coding graphs; (3.3) ALT-encryption technology is a linear topological structure and linear topological operation. Various matrices are tools of ALT-encryption technology with faster implementation efficiency.

4. The technical theory described in claim 1 includes problems belonging to the NP-complete problem class: degree-constrained bracing tree problem, vertex-tearing tree problem with leaf number constraints, and digital string-set topological coding matrix problem.

5. The computational unbreakability and provable security of claim 1, characterized by: The number of linear topological structures is based on the cardinality 2e(=2 200 ) in the form of; there are no polynomial algorithms for the integer splitting problem, the uniform fractal forest problem, and the digital string-set full coloring topological coding matrix problem; finding a specific graph and its specific supporting tree and specific vertex tearing tree from a massive graph involves the NP-complete problem of subgraph isomorphism, and there are no polynomial algorithms for the structural construction of the graph and its coloring method.

6. According to claim 1, graph homomorphic operations and graph structure operations are used to generate topological signatures and topological key pairs, and the operation characteristics are: color-preserving graph homomorphic operations for topological signature key pairs, vertex tearing operations for topological signature key pairs, vertex coincidence operations for topological signature key pairs, color-preserving full-coloring topological encoding matrix homomorphic key pairs, and putting the ALT-password into a "box" to construct a "one-time-one-pad" ALT-key.

7. The W-type coloring algorithm for random leaf growth of the topological coding tree according to claim 1 is: a W-type graceful algorithm with random leaf addition parameters designed to achieve "one-time one-key"; a set-ordered graceful labeling-leaf growth algorithm for trees; a graceful tree nested growth algorithm; a random leaf addition W-magic (full labeling) full coloring algorithm, where W-magic = edge magic, edge difference, happiness difference, and graceful difference.

8. The method of designing a metric and management of ALT-key strength and an implementation algorithm of ALT-encryption according to claim 1, characterized in that: (8.1) Strength measurement: ① The length of the ALT-key; ② The mathematical constraints W(k1, k2, ..., k n ) dimension n; ③ the topological structure of the ALT-key; ④ the topological structure of the ALT-key is a Chinese sentence; ⑤ the cardinality of the space for constructing the ALT-key is at least 2e (=2 200 ), that is, the number of vertices of the graph is not less than 30. ⑥ Strong constraints, strong randomness, "one-time one-key" and infinite number; ⑦ Quantity forms of "one-to-many", "many-to-many" and "infinite". (8.2) ALT-key management: key registration, key creation, key distribution, key use, key authentication, key revocation, key update, receiving the key of the other party, etc. This is achieved by using the Asymmetric Topology Cryptography Center (ATCC) algorithm. (8.3) The implementation algorithm of ALT-encryption is the ATCC (Asymmetric Topology Cryptography Center) algorithm, which has the following characteristics: every communication of the user is protected by ATCC, and the user does not need to put his public key on a public platform or transmit the public key to the communication party through a public channel. The user does not need to make keys, distribute keys, update keys, and other high-tech and tedious tasks by himself. The ATCC algorithm can use graph groups and grids to encrypt the community and local area network as a whole, and manage the keys of users in the community and local area network.

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