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

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• ### Topological Graphic Passwords And Their Matchings Towards Cryptography

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• ### Graph Theory Towards New Graphical Passwords In Information Networks

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• ### Graphic Lattices and Matrix Lattices Of Topological Coding

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## I Introduction

Graphical passwords (GPWs) are familiar with people in nowadays, such as 1-dimension code, 2-dimension code, face authentication, finger-print authentication, speaking authentication, and so on, in which 2-dimension code is widely used in everywhere of the world. A 2-dimension code can be considered as a GPW, since it is a picture. Researchers have worked on GPWs for a long time ([5, 6, 7]). Wang et al. propose another type of graphic passwords (Topsnut-gpws) in [20] and [21], which differ from the existing GPWs.

As an example, we have two Topsnut-gpws shown in Fig.1(a) and (b), where is as a public key, is as a private key. The authentication in network communication is given in Fig.1(c). By observing Fig.1 carefully, we can see that the labels of nodes (also, vertices) and edges of two Topsnut-gpws and form a complementary relationship, and the labels of each edge and its two nodes in and satisfies some certain mathematical restraints. Another important character of Topsnut-gpws is the configuration, also, the topological structure (called graph hereafter). Thereby, we say that Topsnut-gpws are natural-inspired from mathematics of view. In general, Topsnut-gpws are easy saved in computer by algebraic matrices, and Topsnut-gpws occupy small space rather than that of the existing GPWs such that Topsnut-gpws can be implemented quickly.

Topsnut-gpw can be as a platform for password, cipher code and encryption of information security. As Topsnut-gpws were made by “topological configurations plus number theory”, we will apply a particular class of matrices to describe Topsnut-gpws for the purpose of writing easily in computer and running quickly by computer. These matrices are called Topsnut-matrices, and can yield randomly text-based passwords (TB-paws for short) for authentication and encryption in communication. For the theoretical base, we will introduce some operations on Topsnut-matrices in order to implement them for building up TB-paws flexibly.

As known, Topsnut-gpws are related with many mathematical conjectures or NP-problems, so Topsnut-gpws are computationally unbreakable or provable security. A Topsnut-gpw has an advantage, that is, it can generate text-based passwords with longer byte such that it is impossible to rebuild the original Topsnut-gpw from the derivative text-based passwords made by . This derives us to explore the area of generating text-based passwords from Topsnut-gpws in this article. We believe this transformation from Topsnut-gpws to text-based passwords is very important for the real application of Topsnut-gpws.

### I-a Examples and problems

We write “text-based passwords” by TB-paws, and “topological graphic passwords” as Topsnut-gpws hereafter, for the purpose of quick statement. We will make some TB-paws from a Topsnut-gpw depicted in Fig.2. Along a path shown in Fig.2, we have a TB-paw

 D1=3163321891571570125125

obtained from the labels of vertices and edges on the path .

The Topsnut-gpw depicted in Fig.2

such that each edge holds to be an odd number, and for any pair of vertices , as well as for any two edges and of . By cryptography of view, the Topsnut-gpw has twelve sub-Topsnut-gpws and with to form a larger authentication, where with are public keys, and with are private keys. Moreover, a sub-Topsnut-gpw pictured in Fig.3 distributes us a TB-paw

 D(T11)=095950979709999101210310199105610910310746111105611310711581171091191012311310121111

Obviously, to reconstruct the sub-Topsnut-gpw from the TB-paw is difficult, and the TB-paw does not rebuild the original Topsnut-gpw at all. It means that the procedure of generating TB-paws from Topsnut-gpws is irreversible. On the other hands, this Topsnut-gpw can distribute us TB-paws shown in the formula (22), such that each TB-paw has at least 380 bytes or more.

For the encryption of data and dynamic networks, we propose the following problems:

1. How to generate TB-paws from a given Topsnut-gpw?

2. How many TB-paws with the desired -byte are there in a given Topsnut-gpw?

3. How to encrypt a dynamic network by Topsnut-gpws or TB-paws?

We will try to find some ways for answering partly the above problems in the later sections. In graph theory, Topsnut-gpws are called labelled graphs, so both concepts of Topsnut-gpws and labelled graphs will be used indiscriminately in this article.

### I-B Preliminary

The following terminology, notation, labellings, particular graphs and definitions will be used in the later discussions.

1. The notation indicates a consecutive set with integers holding , denotes an odd-set with odd integers with respect to , and is an even-set with even integers .

2. The number of elements of a set is written as .

3. is the set of vertices adjacent with a vertex , is called the degree of the vertex . If we call a leaf.

4. A lobster is a tree such that the deletion of leaves of the tree results in a caterpillar, where the deletion of leaves of a caterpillar produces just a path.

5. A graph having vertices and edges is called a -graph.

6. A spider is a tree having paths with and , its own vertex set , such that its own edge set , and . Clearly, , and for any vertex . We call as the body, and each path is a leg of length of .

7. A ring-like network has a unique cycle , and each vertex of is coincident with some vertex of a tree with .

8. The set of all subsets of a set is denoted as , but the empty set is not allowed in . For example, for a set , then contains: , , , , , , , , , , , , , , .

###### Definition 1.

[33] A labelling of a graph is a mapping such that for any pair of elements of , and write the label set . A dual labelling of a labelling is defined as: for . Moreover, is called the vertex label set if , the edge label set if , and a universal label set if .

A combinatoric definition of set-labellings is as follows.

###### Definition 2.

[33] Let be a -graph.

(i) A set mapping is called a total set-labelling of if for distinct elements .

(ii) A vertex set mapping is called a vertex set-labelling of if for distinct vertices .

(iii) An edge set mapping is called an edge set-labelling of if for distinct edges .

(iv) A vertex set mapping and a proper edge mapping are called a v-set e-proper labelling of if for distinct vertices and two edge labels for distinct edges .

(v) An edge set mapping and a proper vertex mapping are called an e-set v-proper labelling of if for distinct edges and two vertex labels for distinct vertices .

###### Definition 3.

([4, 34, 41]) Suppose that a connected -graph with admits a mapping . For edges the induced edge labels are defined as . Write , . There are the following restrictions:

1. .

2. .

3. , .

4. , .

5. .

6. .

7. is a bipartite graph with the bipartition such that ( for short).

8. is a tree containing a perfect matching such that for each edge .

9. is a tree having a perfect matching such that for each edge .

A graceful labelling holds (a), (c) and (e) true; a set-ordered graceful labelling satisfies (a), (c), (e) and (g), simultaneously; a strongly graceful labelling holds (a), (c), (e) and (h) true; a strongly set-ordered graceful labelling complies with (a), (c), (e), (g) and (h) meanwhile. An odd-graceful labelling holds (a), (d) and (f) true; a set-ordered odd-graceful labelling obeys (a), (d), (f) and (g), simultaneously; a strongly odd-graceful labelling holds (a), (d), (f) and (i) true at the same time; a strongly set-ordered odd-graceful labelling fulfils (a), (d), (f), (g) and (i), simultaneously.

Another group of definitions is about the sum of end labels of edges, we present it as follows:

###### Definition 4.

([4, 42]) A -graph with admits a labelling , where is an integer set. For edges the induced edge labels are defined as or for every edge . And is the vertex label set, and is the edge label set. There are the following constraints:

1. .

2. .

3. .

4. .

5. .

6. .

7. when is even, and when is odd.

8. .

9. .

10. .

11. .

12. .

13. .

14. .

15. .

16. There exists an integer so that .

17. is bipartite with its bipartition so that .

We call to be: (1) a felicitous labelling if c-3, c-8 and c-10 hold true; (2) an odd-elegant labelling if c-4, c-9 and c-13 hold true; (3) a harmonious labelling if c-2, c-8 and c-10 hold true, when is a tree, exactly one edge label may be used on two vertices; (4) a properly even harmonious labeling if c-5, c-9 and c-11 hold true; (5) a -harmonious labeling if c-2, c-6 and c-15 hold true; (6) an even sequential harmonious labeling if c-5, c-7 and c-12 hold true; (7) a -harmonious harmonious labeling if c-1, c-6 and c-14 hold true; (8) a strongly harmonious labeling if c-3, c-8, 16 and c-10 hold true; (9) a set-ordered harmonious labeling if c-3, c-8, c-17 and c-10 hold true; (10) an set-ordered odd-elegant labelling if c-4, c-9, c-17 and c-13 hold true;

## Ii Techniques for generating TB-paws from Topsnut-gpws

Our methods for generating TB-paws from Topsnut-gpws are mainly based on the following disciplines: Topsnut-configurations, graph-labellings, Topsnut-matrices, Topsnut-matchings and graphic groups, these are two invariable quantities of Topsnut-gpws.

### Ii-a Topsnut-configurations

By simple and clear statements, we utilize the odd-graceful/odd-graceful labellings and Topsnut-configuration to show several methods for creating TB-paws.

#### Ii-A1 Path-neighbor-method

As known, each of caterpillars (see Fig.4) and lobsters admits an odd-graceful labelling [41].

is a caterpillar of a -graph admitting a set-ordered odd-graceful labelling . So, the deletion of leaves of is just a path in the caterpillar , such that each has its own leaf set with and , and the vertex set is

 V(H)=V(P)∪L(u1)∪L(u2)∪⋯∪L(un)

See a caterpillar depicted in Fig.4. Thereby, we can get a vv-type TB-paw and a vev-type TB-paw

 Dvev(P)=f(u1)f(u1u2)f(u2)f(u2u3)⋯f(un−1un)f(un)

by the path-method for deriving two types of TB-paws from Topsnut-gpws.

From a path revealed in Fig.5, we can get a vv-type TB-paw and a vev-type TB-paw by the path-method.

Next, we introduce the path-neighbor-method.

Let a vertex have its neighbor set with , we have a vv-type TB-paw

 Dvv(u)=f(u)f(v1)f(v2)⋯f(vmu)f(u)

and

 Dvev(u)=f(u)f(uv1)f(v1)f(v1v2)f(v2)⋯f(vmu−1vmu)f(vmu)f(vmuu)f(u)

by the mini-principle, and moreover we get a vv-type TB-paw

 D′vv(u)=f(u)f(vmu)f(vmu−1)⋯f(v2)f(v1)f(u)

and another vev-type TB-paw

 D′vev(u)=f(u)f(uvmu)f(vmu)f(vmuvmu−1)f(vmu−1)⋯f(v3v2)f(v2)f(v2v1)f(v1)f(v1u)f(u)

by the maxi-principle. Let with , where . By the mini-principle, for the edge , we write a vv-type TB-paw

 Dvv(uv1)=f(u)f(v1)f(v2)⋯f(vmu)f(u)f(v1)f(v1,1)f(v1,2)⋯f(v1,mv1)f(v1) (1)

by the mini-principle, denoted as

 Dvv(uv1)=Dvv(u)⊎Dvv(v1), (2)

and moreover we can write a vev-type TB-paw

 Dvev(uv1)=f(u)f(uv1)f(v1)f(v1v2)f(v2)⋯f(vmu−1)f(vmu−1vmu)f(vmu)f(vmuu)f(u)f(v1)f(v1v1,1)f(v1,1)f(v1,1v1,2)f(v1,2)⋯f(v1,mv1−1v1,mv1)f(v1,mv1)f(v1,mv1v1)f(v1) (3)

by the mini-principle, denoted as

 Dvev(uv1)=Dvev(u)⊎Dvev(v1). (4)

Similarly with (2) and (4), we can write and by the maxi-principle.

For example, by means of a caterpillar exhibited in Fig.5 and two formulae (1) and (3), we have two vv-type TB-paws

 Dvv(H)=03739414345470372468103710252729313335102510122512131517192123
 D′vv(H)=047454341393703710864237103531292725102512102512232119171513

according to the mini-principle and the maxi-principle. Similarly,

 Dvev(H)=03737393941414343454547470373523343162982710371015251727192921312333253510251510131225121133155177199211123

is a vev-type TB-paw by the mini-principle, and moreover,

 D′vev(H)=047474545434341413939373703727102983163343523710273725352333213119291727152510251312151025121123921719517315113

is obtained by the maxi-principle.

It is easy to see that there are many ways to generate vv-type/vev-type TB-paws from a Topsnut-gpw made by a labelled caterpillar, except the mini-principle and the maxi-principle. In a vv-type/vev-type TB-paw , we say , , and in the vv-type/vev-type TB-paw . So, we have permutations for writing , and a caterpillar with the path distributes us vv-type/vev-type TB-paws at least.

#### Ii-A2 Cycle-neighbor-method

By a caterpillar depicted in Fig.4, we add an edge to for joining the vertex with , the resulting graph is denoted as , in which there is a cycle . So, we have a vv-type TB-paw

 Dvv(T′)=Dvv(u1)⊎Dvv(u2)⊎⋯⊎Dvv(un)⊎Dvv(u1)=[⊎nk=1Dvv(uk)]⊎Dvv(u1) (5)

along the cycle , and a vev-type TB-paw

 Dvev(T′)=Dvev(u1)⊎Dvev(u2)⊎⋯⊎Dvev(un)⊎Dvev(u1)=[⊎nk=1Dvev(uk)]⊎Dvev(u1). (6)

Since we have initial vertices of the cycle , so the number of vv-type/vev-type TB-paws distributed from is equal to

 Ntbp(C)=n⋅(m1+1)!⋅(mn+1)!⋅n−1∏i=2(mi)!. (7)

#### Ii-A3 Lobster-neighbor-method

In [41] and [42], the authors have proven: Each lobster admits one of odd-graceful labelling and odd-elegant labelling. Thereby, we can apply lobsters to make Topsnut-gpws, or we select sub-Topsnut-gpws being lobsters of Topsnut-gpws to derive vv-type/vev-type TB-paws. Another advantage about lobsters is helpful for us to produce random Topsnut-gpws that generate random vv-type/vev-type TB-paws.

Recall, a lobster is defined as a tree such that the deletion of leaves of results in a caterpillar, that is, the remainder is just a caterpillar, where is the set of all leaves of . In other words, each lobster can be constructed by adding leaves to some caterpillar. The results in [41] and [42] enable us to build up lobsters admitting odd-graceful/odd-elegant labellings by caterpillars admitting set-ordered odd-graceful/odd-elegant labellings through adding leaves.

We show an example for illustrating “adding leaves to a caterpillar admitting a set-ordered odd-graceful labelling produces a lobster admitting an odd-graceful labelling”. Based on a caterpillar , as revealed in Fig.5, we can see that Fig.6 gives the procedure of “adding randomly leaves to ”, and the labelling new edges is shown in Fig.7, and moreover the procedure of “labelling new vertices and relabelling old vertices” presents the desired odd-graceful lobster (see Fig.8).

We, now, come to introduce the lobster-neighbor-method for getting vv-type/vev-type TB-paws from a Topsnut-gpw made by an odd-graceful lobster in the following algorithm.

###### Theorem 1.

There exists an efficient and polynomial algorithm (LOBSTER-algorithm) for generating vv-type/vev-type TB-paws from Topsnut-gpws made by odd-graceful lobsters.

###### Proof.

We, directly, use an algorithmic proof here for generating vv-type/vev-type TB-paws from Topsnut-gpws.

Step 1. Suppose that a lobster corresponds to a caterpillar obtained by deleting some leaves from . Write as the set of deleted leaves, so . Conversely, is obtained by adding the leaves of to . Let be the path as the remainder after the deletion of leaves of the caterpillar , and let be a set-ordered odd-graceful labelling of . Thereby, we have

 Dvv(ui)=g(ui)g(vi,1)g(vi,2)⋯g(vi,mi)g(ui)

with and

 Dvev(ui)=g(ui)g(uivi,1)g(vi,1)g(vi,1vi,2)g(vi,2)⋯g(vi,mi−1vi,mi)g(vi,mi)g(vi,miui)g(ui)

with . Thereby, we have a vv-type TB-paw

 Dvv(H)=Dvv(u1)⊎Dvv(u2)⊎⋯⊎Dvv(un)=⊎nk=1Dvv(uk) (8)

and a vev-type TB-paw

 Dvev(H)=Dvev(u1)⊎Dvev(u2)⊎⋯⊎Dvev(un)=⊎nk=1Dvev(uk). (9)

Step 2. Adding randomly leaves to for forming a lobster . Since is a set-ordered odd-graceful labelling of the caterpillar , so with , and any edge of holds and such that . By the hypothesis above, we can write with for , and with for . Suppose that each vertex is added leaves from the set with , and each vertex is added leaves from the set with . Here, it is allowed some or . The resulting tree is just . Therefore,

 |E(T)|=|E(H)|+s∑i=1ai+t∑j=1bj,

and write .

We define a labelling for in the following steps.

Substep 2.1. We label the edges of in the increasing order: for , for , and

 f(xkuk,j)=2j+k−1∑i=1f(xiui,ai), j∈[1,ak]

with . Thus, .

Substep 2.2. For the edges , we set in the decreasing order: with , with , and

 f(yt−jwt−j,k)=2k+j∑i=1f(yt−i+1wt−i+1,bt−i+1),

with and .

Substep 2.3. We come to label the vertices of in the following way: for ; for with ; for ; and for with

Step 3. Producing a vv-type TB-paw and a vev-type TB-paw from the lobster . We use the notation to denote the set of new leaves added to hereafter. We set for with , and get a vv-type sub-TB-paw

 D′vv(ui)=f(ui)⊎Dvv(L∗(ui))⊎f(vi,1)⊎Dvv(L∗(vi,1))⊎f(vi,2)⊎Dvv(L∗(vi,2))⊎⋯⊎Dvv(L∗(vi,mi−1))⊎f(vi,mi)⊎Dvv(L∗(vi,mi))⊎f(ui)

with , where is the set of new leaves added to and for . Hence, we get the desired vv-type TB-paw

 Dvv(T)=D′vv(u1)⊎D′vv(u2)⊎⋯⊎D′vv(un)=⊎nk=1D′vv(uk) (10)

Next, for getting a vev-type TB-paw from the lobster , we take

for , and moreover

 Dvev(L∗(vi,j))=f(vi,jβi,1)f(βi,1)f(βi,1βi,2)f(βi,2)⋯f(βi,ci−1βi,ci)f(βi,ci)f(βi,civi,j)f(vi,j)

for . So,

 D′vev(ui)=f(ui)⊎Dvev(L∗(ui))⊎f(uivi,1)f(vi,1)⊎Dvev(L∗(vi,1))⊎f(vi,1vi,2)f(vi,2)⊎Dvev(L∗(vi,2))⊎f(vi,2vi,3)f(vi,3)⊎⋯⊎f(vi,mi−1vi,mi)f(vi,mi)⊎Dvev(L∗(vi,mi))⊎f(vi,miui)f(ui)

with . Thereby, the lobster distributes a vev-type TB-paw as follows

 Dvev(T)=D′vev(u1)⊎D′vev(u2)⊎⋯⊎D′vev(un)