New Algebraic Groups Produced By Graphical Passwords Based On Colorings And Labellings

. Safety of plain text passwords has been questioned in current researching information passwords. Graphical passwords are another way for alternative text-based passwords and to improve the user account security. As we are constructing Topsnut-graphical passwords that can be traced to an idea of "Graph structure plus the number theory" proposed first by Hongyu Wang with her colleagues, we find that some of Topsnut-graphical passwords can be composed of algebraic groups under the principle of Abelian additive finite group. We apply the odd-elegant labelling of graph theory to produce Topsnut-graphical passwords, and verify our Topsnut-graphical passwords can form algebraic groups, called labelling graphical groups . Our results can provide those users who have business in two or more banks, and our methods are easily transformed into algorithms with polynomial times.


Introduction
Everyone in today's society has at least two or three passwords. Your password can not only ensure the safety of personal privacy, but also provide a sense of security for the user. The password is stolen and lost it, which is often seen in daily life. Safety of plain text password has been questioned. A graphical password is one of the ways to alternative for text-based passwords and to improve the user account security, because of graphical passwords are easy for users and difficult to be broken by attackers [1]. New graphical passwords can be made by the idea of "Graph structure plus the number theory" due to Wang et al. in [2] and [3], we call such new graphical passwords as Topsnut-graphical passwords for distinguishing with other graphical passwords. Studies have shown that humans remember pictures more than texts, and psychological research supports the hypothesis [4]. In addition, Topsnut-graphical passwords differ from those graphical passwords mentioned in [1], since they need less storage space and can be realize in communication quickly.
We show an example to introduce Topsnut-graphical pass-words simply. Alina goes to a bank for her business.
First of all, she is given six topological structures (also, six graphs) shown in Fig.1, and is asked for selecting a graph for making a key. Secondly, she select the graph shown in Fig.1 (e) and Fig.2 (1) as the base of her key.
Next, she labels each small circle of the graph shown in Fig.2 (2) with a positive integer, continuously, she labels each edge that joins two small circles with a positive integer, these two small circles are called the ends (also, vertices in graph theory) of the edge. Finally, she obtain her key shown in Fig.2 (3), this key is called a opsnutgraphical password. Observe the well-labelled graph Fig.2 (3), we can find that three numbers x,y,z on an edge labelled with number y and its two ends labelled with numbers x,z hold x + z = y (mod 16), that is, Naturally, from the view of theoretical investigation and practice, we want to solve some basic problems: (i) Find all graphical passwords for the graph Fig.2(1)? (ii) Find connections among graphical passwords of the graph if the problem (i) are determined; (iii) Find all of graphs having 7 vertices and 8 edges. As known, many graph labellings can be used to other applications out of graph theory. We will apply operations, colorings and labellings of graph theory ( [5], [6], [7]) to answer partially these three problems. In the following discussion, we will show an application of the oddelegant labelling of graph theory.

Preliminary
The shorthand notation [m,n] to indicate a set {m,m + 1,...,n} with integers m,n respect to 0 ≤ m < n; the symbol [s,t] o denotes an odd number set {s,s + 2,...,t} with odd numbers s,t keeping 1 ≤ s < t; the notation [α,β] e stands up an even number set {α,α + 2,...,β} with even numbers α, β falling into 2 ≤ α < β; |X| is the cardinality of elements of a set X. All graphs mentioned here are undirected, finite, no multiple-edges and no loops, and we call them simple graphs. A (p,q)-graph G has its vertex set V (G) of cardinality p and edge set E(G) of cardinality q. Another two notations will be used in

Definition 1.
Let X be a non-empty set, and let be a 2element operation on X. We call the algebraic structure Γ = (X, ) as a group if it holds: (1) Uniqueness and Closure: x y  X holds for any pair of elements x,y  X; (2) Unit element (zero element): There exists an element e ∈ X such that e x = x e = x for each element x ∈ X; (3) Inverse element: Any element x  X corresponds another element y  X such that x y = y x = x, we call y the inverse element of x, denoted as y = x −1 ; (4) Associative law: Any triple x,y,z  X holds (x y) with i + j − k (mod 2q) ∈ [1,2q] follows the equation (

Examples for illustrating Theorem 1
We show an example in Fig.3 for illustrating Theorem 1.

Fig. 3.
A labelling graphical group for illustrating Theorem 1.

A new graph labelling
We, based on the previous results and examples, define a new labelling of graphs as follows: According to Definition 2 we can define a negativelabelling (see Fig.4), or a mixed-labelling for a graph G (see Fig.3 and Fig.4).

A mixed labelling graphical group
In fact, we can generalize our coloring/ labelling/operation graphical quasi-groups/groups. A labelling graphical group Hroup is formed by two collections A and B shown in Fig.3  This example implies a new graph labelling, called a mixed odd-elegant labelling (see g1 shown in Fig.3 and   h1 shown in Fig.4).   Fig. 4. A negative-labelling graphical group opposite with the group shown in Fig. 3.

Conclusion
Based on the new definition of graph coloring / labeling operations, we found the graph coloring / label constitute a algebraic group. In Theorem 1, we found that set

Odde(H) and 2-element operation F(Hi) ⊕ k F(Hj)
constitute a algebraic group for which we are in the same class label, converted from one label to another label provides a feasible method. Our results show that there are groups in which any element can be defined as the zero element by the 2-element operation ⊕ 0 k in [14].
We already know that there is a label f of graph G, then it must have the corresponding dual label f . That is to say, the original label and its dual label always appear in pairs 2. According to Definition 2 we can define a negative-labelling, and a mixed-labelling for a graph G. In theorem 3, We find a way to extend the two groups into a new group. The mixed odd-elegant labelling, mentioned the above, can leads to some mathematical conjectures , such as: There is a mixed odd-elegant labelling of every tree.
Furthermore, we can transplant this new labelling to other graph labellings, for instance, felicitous labelling in [8], (generalized) edge-magic total labelling in [9], [10] and [12], strongly graceful labelling in [11], even other labellings in [13] and so on, which means that exploring new graphical passwords will bring more new mathematical subjects and problems. Moreover, the mixed odd-graceful labeling has been studied in [14].
Exploring the topological structure of some kind of graphs, finding its more intrinsic properties, and providing theoretical help for new graphical cryptography are indispensable. [15] and [16] explore the equivalent definitions of cactus graphs and euler graphs respectively. [17] and [18] explore odd-graceful labeling and odd-elegant labeling on ring computer networks. Finally, we will explore other new 2-element