Abstract
Graphical password (GPW) is one of various passwords used in information communication. The QR code, which is widely used in the current world, is one of GPWs. Topsnut-GPWs are new-type GPWs made by topological structures (also, called graphs) and number theory, but the existing GPWs use pictures/images almost. We design new Topsnut-GPWs by means of a graph labelling, called odd-elegant labelling. The new Topsnut-GPWs will be constructed by Topsnut-GPWs having smaller vertex numbers; in other words, they are compound Topsnut-GPWs such that they are more robust to deciphering attacks. Furthermore, the new Topsnut-GPWs can induce some mathematical problems and conjectures.
1. Introduction and Preliminary
1.1. Researching Background
Graphical passwords (GPWs) have been investigated for over 20 years, and many important results can be found in three surveys [1–3]. GPW schemes have been proposed as a possible alternative to text-based schemes. However, the existing GPWs have (i) no mathematical computation; (ii) more storage space; (iii) no individuality; (iv) geometric positions; (v) slow running speed; (vi) vulnerable to attack; and (vii) no transformation from lower safe level to high security. However, QR code is a successful example of GPW’ applications in mobile devices by fast, relatively reliable and other functions [4, 5]. GPWs may be accepted by users having mobile devices with touch screen [6, 7].
Wang et al. show an idea of “topological structures plus number theory” for designing new-type GPWs (abbreviated as Topsnut-GPWs, [8–10]). All topological structures used in Topsnut-GPWs can be stored in a computer through ordinary algebraic matrices. And Topsnut-GPWs have no requirement of geometric positions for users and allow users to make their individual passwords rather than learning more rules they do not like and so on.
How to quickly build up a large scale of Topsnut-GPWs from those Topsnut-GPWs having smaller vertex numbers? How to construct a one-key versus more-locks (one-lock versus more-keys) for some Topsnut-GPWs? And how to compute Topsnut-GPWs’ space by the basic computing unit ? Obviously, we need enough graphs and lots of graph coloring/labellings, and we can turn more things into Topsnut-GPWs. Let be the number of graphs having vertices. From [11], we know
| bits | ||
| 18 | 1787577725145611700547878190848 | 100 |
| 19 | 24637809253125004524383007491432768 | 114 |
| 20 | 645490122795799841856164638490742749440 | 129 |
| 21 | 32220272899808983433502244253755283616097664 | 145 |
| 22 | 3070846483094144300637568517187105410586657814272 | 161 |
| 23 | 559946939699792080597976380819462179812276348458981632 | 179 |
| 24 | 195704906302078447922174862416726256004122075267063365754368 | 197 |
where for . It means that adding various graph labellings enables us to design tremendous Topsnut-GPWs with huge topological structures and vast of graph coloring/labellings, since there are over 150 graph labellings introduced in [12]. As a fact, Topsnut-GPWs can generate alphanumeric passwords with longer units. As an example, we take a path in Figure 6(d) to produce an alphanumeric password by selecting the neighbors of each vertex of these four vertices , and . Clearly, such password may have longer unit in a large scale of Topsnut-GPW for meeting the need of high level security.
In this article, we will apply a graph labelling called odd-elegant labelling [13]. And we will define some construction operations under odd-elegant labelling for designing our compound Topsnut-GPWs.
1.2. Preliminary
We use standard notation and terminology of graph theory. Graphs mentioned are loopless, with no multiple edges, undirected, connected, and finite, unless otherwise specified. Others can be found in [14]. Here, we will use A -graph which is one with vertices and edges; the symbol stands for an integer set for integers and with ; indicating an odd-set , where and both are odd integers with ; and represents an even-set , where and are both even integers with respect to .
Definition 1 (see [13]). Suppose that a -graph admits a mapping such that for distinct vertices , and the label of every edge is defined as and the set of all edge labels is equal to . One considers to be an odd-elegant labelling and to be an odd-elegant.
Definition 2 (see [15]). Suppose that a bipartite graph receives a labelling such that , where is the bipartition of vertex set of . We call a set-ordered labelling (So-labelling for short).
As shown in Figure 1, there are four different examples of Definitions 1 and 2.
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Definition 3. Let be a -graph with . A graph obtained by identifying each vertex of with a vertex of into one vertex with is called an -identification graph and denoted as ; the vertices are called the identification-vertices.
Moreover, the -identification graph defined in Definition 3 has vertices and edges. One can split each identification-vertex into two vertices and (called the splitting-vertices) for , such that is split into two parts and . For the purpose of convenience, the above procedure of producing am -identification graph is called an -identification operation; conversely, the procedure of splitting into two parts and is named as the m-splitting operation.
Definition 4. Let be a connected -graph with , and let . If the -identification -graph has a mapping holding the following: (i) for each pair of vertices , (ii) is an odd-elegant labelling of with , and (iii) and , then one calls a twin odd-elegant graph (a TOE-graph), a TOE-labelling, a TOE-source graph, a TOE-associated graph, and a TOE-matching pair.
We illustrate Definition 4 with Figure 2. In other words, a twin odd-elegant graph with its TOE-source graph and TOE-associated graph , where is a TOE-matching pair.
Furthermore, if each with is a connected graph in Definition 4, and the TOE-source is a bipartite connected graph having its own bipartition and a labelling satisfying Definition 2, we call the 2-identification graph a set-ordered twin odd-elegant graph (So-TOE-graph) and a set-ordered twin odd-elegant labelling (So-TOE-labelling). Notice that the source graph is a set-ordered odd-elegant graph by Definitions 1 and 2. In vivid speaking, a source graph and its associated graph defined in Definition 4 can be called a TOE-lock-model and a TOE-key-model ([10]), respectively.
1.3. Techniques for Constructing 2-Identification Graphs
The following three operations, CA-operation, edge-series operation, and base-pasted operation, will be used in this article.
(O-1) CA-Operation. Suppose each graph has an odd-elegant labelling and with . Clearly, for with , there are vertices and such that . For example, some has a vertex such that the label with . We can combine those vertices that have the same labels into one vertex, which gives us a new graph, denoted by . This process is called a CA-operation on .
(O-2) Edge-Series Operation. Given two groups of disjoint trees with there are vertices with . Joining the vertex with the vertex by an edge for produces a tree (denoted by ) with ; next we let one vertex coincide with one vertex into one vertex with . The resulting graph is just a 2-identification graph.
(O-3) Base-Pasted Operation. Given two disjoint trees (called base-trees) having vertices and two groups of disjoint trees with , we let a vertex coincide with the vertex into one vertex for such that the resulting tree (i.e., ) has , for . We overlap one vertex with one vertex into one vertex with to build up a -identification graph holding and
In the following, we give the diagrams with for edge-series operation and base-pasted operation, shown in Figures 3 and 4, respectively.
2. Main Results and Their Proofs
Lemma 5. Each star is a TOE-source tree of a So-TOE-tree.
Proof of Lemma 5 is shown in Figure 5. It describes the construction process of the So-TOE-tree by any TOE-source tree .
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Theorem 6. Every set-ordered odd-elegant graph being not a star is a So-TOE-source graph of at least two So-TOE graphs.
Proof. Suppose that -graph having vertex bipartition is , where , , , and . By the hypothesis of the theorem, has a set-ordered odd-elegant labelling defined by , ; , , ; . Hence, . It is not difficult to observe that ; that is, and .
Case . We construct a labelling of a new tree having vertices by the labelling such that , such that , where for . This tree can be built up in the following way: a bipartition with and , where , such that , ; , . Any edge satisfies with and . We construct the edge set of as such that the edge labels are , for and . Observe that , , and .
Now, we can combine the vertex and of with the vertex and of into one (two identification-vertices) and , respectively, so we obtain the desired graph . And has a labelling defined as , ; , ; , , , , , and . Clearly, any pair of two vertices of are assigned different numbers. According to Definition 4, is an So-TOE-graph having the source graph . Examples that illustrate Case 1 of Theorem 6 are shown by Figures 6(a), 6(b), and 6(d).
Case . Similarly to Case 1, we can get the following results: let , , and furthermore for . This tree can be built up in the following way: a bipartition with and , such that , ; , , . Any edge satisfies with and . We construct the edge set of as such that the edge labels are , for , and , for . Observe that , , and .
Now, we can combine the vertex and of with the vertex and of into one (the identified vertex) and , so we obtain the desired tree . And has a labelling defined as , ; , ; , , , , , and . Clearly, any pair of two vertices of are assigned different numbers. According to Definition 4, is a So-TOE-graph having the source graph . An example for illustrating Case 2 of Theorem 6 is given by Figures 6(a), 6(c), and 6(e).
Theorem 7. Suppose that is a So-TOE-graph, where each is a source tree for . Then obtained by the edge-series operation has a So-TOE-labelling.
Proof. By the hypothesis of the theorem, every -graph has a set-ordered odd-elegant source--graph and an associated--graph for . Let ; the vertex set of each graph can be partitioned into with , where , , and for and . By Definition 4, every has a So-TOE-labelling with such that ; with ; ; for ; and .
Therefore, , where for distinct vertices , which means and . Clearly, the labels of other vertices of differ from each other.
Firstly, we split into two parts and , that is, doing a 2-splitting operation on every with . Secondly, our discussion focuses on and with . We construct a graph by joining the vertex with the vertex by an edge, where and , called . For the purpose of convenience, we set , , , , and . For , , and , we define a new labelling as follows:(T-1);(T-2);(T-3);(T-4).By the labelling forms (T-1) and (T-2) above, we can verify with and have the following properties: (i) ; (ii) ; and (iii) .
Computing the labelling forms (T-3) and (T-4) enables us to obtain for . Now, we combine the vertex with the vertex into one vertex and then combine the vertex with the vertex into one vertex. (i.e., do the 2-identification operation). Thus the labelling is a So-TOE-labelling of ; therefore, is a So-TOE-graph too.
See Figures 7, 8 and 9 for understanding Theorem 7.
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In experiments, for each arrangement of , there are many possible constructions of for holding Theorem 7 (as shown in Figure 9).
Theorem 8. Suppose that is a So-TOE-graph, where each is a source graph for . Then obtained by the base-pasted operation has a So-TOE-labelling if two base-trees and are set-ordered.
Proof. By the hypothesis of the theorem, every -graph has a set-ordered odd-elegant source--graph and an associated--graph for . Let ; the vertex set of each graph can be partitioned into with , where , , , and for and . Every , by Definition 4, has a So-TOE-labelling with , and has the following properties: ; for ; ; ; with ; ; and .
Thus, the properties of each So-TOE-labelling induce , and alsowhere if and . In other words, we have and . The labels of other vertices of differ from each other.
Let , such that there exists a set-ordered odd-elegant labelling , satisfying with , and the bipartition of vertex set of satisfies for and .
Next, we discuss all graphs and with by the parity of positive integer in the following two cases.
Case . For considering the case and , we define a new labelling with and in the following way:(C-1) with ;(C-2) with ;(C-3) with ;(C-4) with ;(C-5).Based upon the labelling forms (C-1)–(C-4), we computeThereby, we have shown that and and furthermore the labels of vertices, except and , differ from each other, and the labels of edges differ from each other.
Next, after computing the labelling forms (C-5) with , we obtainwhere and . By the above deduction, we can know thatwhere . Next, for each vertex with and , we setAccording to formula (7), we obtain with , , and , which meansDoing a CA-operation on and having labelling for produces a new graph with . Now, we combine the vertex with the vertex into one vertex and moreover identify the vertex with the vertex into one vertex (i.e., do a 2-identification operation).
By Definitions 2 and 4 and formulae (3)–(8), the labelling is a So-TOE-labelling of . Hence, is a So-TOE-graph. Here, we have proven Case 1. For understanding Case 1, see Figures 10 and 11.
Case . We, for the case and , define a new labelling for and in the following way:(L-1) with ;(L-2) with ;(L-3) with ;(L-4) with ;(L-5).From the above labelling forms (L-1)–(L-4), we can computeThereby, we conclude that and in which the labels of vertices and edges, except and , differ from each other, respectively.
Again, by computing the labelling form (L-5) for each , we obtainwhere and .
Synthesizing the above argument, we get , where the set . For each vertex with and , we setThe above formula (12) enables us to obtain with , , and . Thereby, we have shownAfter performing a CA-operation on and having labelling for , then we obtain a new graph with . Now, we overlap the vertex with the vertex into one vertex and overlap the vertex with the vertex into one vertex (i.e., do a 2-identification operation) in order to obtain . Furthermore, by Definitions 2 and 4 and formulae (9)–(13), the labelling is a So-TOE-labelling of , which implies that is a So-TOE-graph.
Hence, the proof of Case 2 is finished, and illustrating this case is given in Figures 12 and 13.
The proof of the theorem is complete.
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3. Conclusion and Further Research
There are new Topsnut-GPWs having twin odd-elegant labellings introduced here. We define the twin odd-elegant labelling and investigate the -identification graph , called twin odd-elegant graph. We think that finding all possible TOE-matching pairs defined in Definition 4 may be interesting for a given TOE-graph with an odd-elegant labelling . Let be the set of all TOE-associated graphs , so, we have a TOE-book with book-back and book-pages .
We should pay attention to the following problems:
(i) Since for any pair of book-pages , does ?
(ii) Suppose that has pairwise different odd-elegant labellings . Find some possible relationships among TOE-books with .
For the future researching work on Topsnut-GPWs, we propose the following.
Conjecture 9. Let each be a TOE-graph for with . The -identification graph obtained by the edge-series operation (resp., the base-pasted operation) admits a TOE-labelling. r
Conjecture 10. Every simple and connected TOE-graph admits an odd-elegant labelling.
Conjecture 11. Each connected graph is the TOE-source graph of a certain TOE-graph.
A more interesting problem is to design super Topsnut-GPWs such that each super Topsnut-GPW will not be deciphered by attacks of nonquantum computers, since (i) our methods introduced here can construct quickly large scale of Topsnut-GPWs having hundreds vertices; (ii) the space of the Topsnut-GPWs given in Theorem 8 is quite tremendous; (iii) the 2-identification graphs of Theorem 7 and of Theorem 8 are the compound type of Topsnut-GPWs based on smaller scale of Topsnut-GPWs with , and they induce the TOE-books and ; it may be guessed that there is no polynomial algorithm for determining the TOE-books; and (iv) no polynomial algorithm was reported for finding all odd-elegant labellings of a given graph.
Thereby, we hope to discover such super Topsnut-GPWs which can be used in the era of quantum information.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Key R&D Program of China (no. 2016YFB0800700) and the National Natural Science Foundation of China (nos. 61572046, 61502012, 61672050, 61672052, 61363060, and 61662066).