ENCRYPTING NUMBERS USING PAIR LABELING IN PATH GRAPHS

Abstract: Encryption today has grown into such specialized field that involve mathematical, non-linear cryptosystem that even a relatively powerful computers take months or even years to break the cipher text. Now a day’s most of the fields say clients’ credit card numbers, social security numbers, tax file numbers etc needs to be kept secret when they send it electronically. There are various methods used to encrypt the numbers. It has been recognized that encryption and decryption mostly emerges from mathematical disciplines. In this paper, we give a new combinatorial technique to encrypt and decrypt numbers through labeled graphs using the residue modulo class Z3.


Introduction
The concept of graph labeling was introduced by Rosa in 1967 [4].A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions.Labeled graphs serve as useful models for a broad range of applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication network addressing and database management.Detail survey of graph labeling can be seen in A dynamic survey of graph labeling by J.A. Gallian [3].
A graph G(p, q) is said to be (1, 1) edge magic with the common edge count k 0 if there exists a bijection f : V (G) ∪ E(G) → {1, 2, . . ., p + q} such that f (u) + f (v) + f (e) = k 0 for all e = (u, v) ∈ E(G).It is said to be (1, 1) edge antimagic if f (u) + f (v) + f (e) are distinct for all e = (u, v) ∈ E(G).A labeling of the type (1, 1, 1) assign labels from the set {1, 2, . . ., p + q + r} to the vertices, edges and faces of a planar graph G in such a way that each vertex, edge and face receives exactly one label and each number is used exactly once as a label.A Labeling of type (1, 0, 0) and (0, 1, 0) is a bijection from the set {1, 2, . . ., p + q} to the vertices and edges of a graph G.
The weight of a face under a label is the sum of labels carried by that face and edges and vertices on its boundary.A labeling is said to be face magic, if for every numbers, all s side faces have equal weight.Variations of magic and antimagic graphs are studied from [7,8].
Theorem 1.1.Let G be a graph obtained from the path P n by duplication of all the vertex by edge.Then G admits (1, 0, 0) face antimagic labeling.
Let G(p, q, r) be a graph obtained from path P n by duplication of all the vertices by an edge.We have 3n vertices and 4n − 1 edges where vertex set . ., n} be the labels of the vertices of P n .We get the graph G with labels {n + 1, n + 2, . . ., 3n} of the vertices and {3n + 1, . . ., 7n − 1} the labels of edges.We calculate face weight as follows We give the following example for vertex by edge duplication of path P 5 that admits (1, 1, 0) face antimagic labeling with distinct weights {101, 102, 103, 104, 105}.In [2] J. Baskar Babujee and S. Babitha has introduced pair labeling in graphs to encrypt and decrypt numbers using Number Theory technique in complete graph k 9 .In this paper we use the same technique of encryption of numbers over the vertex by edge duplication of path P n .

The Proposed Cryptosystem
Graphs can be effectively used for the encryption.We take vertex by edge duplication of the graph P n , where n ≡ 0 (mod 3).To make our encryption complicated we adopt the following steps: 1. Partition the number into three parts using modulo class of Z 3 .
2. Take a path P n with the vertex v i from 1 ≤ i ≤ n, where n ≡ 0 (mod 3).
Construct a vertex by edge duplication for P n .Now the new graph G has 3n vertices, 4n − 1 edges and n faces.The faces are triangles i.e., C 3 cycles.
3. The labels of the vertex of G is an ordered pair [α, i], 1 ≤ i ≤ 3n and that for edges are [β, j], where the first component of an ordered pair [α, i] and [β, j] will be labeling number obtained by using residue class of Z 3 .The second component of pair [α, i] and [β, j] will be the number {1, 2, . . ., 7n − 1}.
4. G is made (1, 0, 0) face and orientation is given for all its edges.
5. Assign edge magic labeling for the three edges of exactly one face C 3 and make all other edges in G as (1, 1) edge antimagic.

Algorithm for Encryption
Input: The secret number N , the number of vertices of path P n , [n ≡ 0 (mod 3)] and the magic constant K.
Output: Encrypted labeled digraph G. begin Step 1: Define the graph G obtained from the path P n by duplication of all the vertices by edges, n ≡ 0 (mod 3) with vertex set Step 2: The vertex labeling of V is given by the bijective function Step 3: If N ≡ 0 (mod 3) goto step 4 else goto step 5.
Step 4: Split N as N = 2 i=0 N i such that N i ≡ i (mod 3) for i = 0, 1, 2 and Assign the labels for the edges of Step 5: Split N as N = 2 i=0 N i such that any two of N i has same residue value over Z 3 and N i > 3. Create a directed cycle C 3 , by taking v p ∈ V 1 , v q ∈ V 2 and v r ∈ V 3 which does not have uni-direction but depends upon the residue value of N i for i = 0, 1, 2. The orientation of this cycle C 3 will be Assign the labels for the edges C 3 as follows.If two of the N i (say N a , N b ) have residue value 0 when it is divisible by 3, then Step 6: The orientation of E is defined as, Step 7: Edge labeling of E is given by the bijective function Labeling of E 1 is given in Step 4 and Step 5. Now we have to define the function of the set E.
The first component of edge labeling (i.e.,) β value given as follows.
If i (mod 3) = (i − 1) mod 3 then The second component of the edge labeling j value given in Step 8.
Step 8: We assign the j value by ignoring the orientation of edges in E.
(i) Store the elements of S in a array of size (4n − 1) − 3.
(ii) Start with a[0], assign labels to the edges f ( − − → v i v j ) consecutively by ignoring the orientation of 1 ≤ i, j ≤ n in E.
(iii) Do the process until the array will be empty.
Step 9: Except one cycle C 3 (v p , v q , v r ) all the cycles receive (1, 1) edge antimagic labeling is obtained for the digraph by duplication of vertex by edge of P n , where n ≡ 0 (mod 3).end.

Rigor of the Encryption and Decryption Algorithm with Illustration
Consider the graph P n where n ≡ 0 (mod 3).Obtain the graph G by the vertex duplication of the path P n .The vertex set and edge set are defined as Each vertex of G is labeled as [α, i] where α takes the values of residue class of Z 3 (i.e.,) 0, 1 and 2 repeatedly to all the vertices of graph G and i takes the values of {1, 2, . . ., 3n}.The given number N is split into three numbers in such a way that each number N i ; 0 ≤ i ≤ 2 should be greater than 3 and N i ≡ i (mod 3).If the given number N is divisible by 3 then each partition number N i ; 0 ≤ i ≤ 2 will give residues [0], [1] and [2] respectively when it is divisible by 3. If the number N is not divisible by 3 then also it split into three but two of the partition number N i ; 0 ≤ i ≤ 2 will give some residue value when it is divisible by 3.Each edge of G is labeled as [β, j].Consider any one of the cycle C 3 with vertex v p ∈ v 1 ; v q ∈ v 2 and v r ∈ v 3 whose vertex labels are [0, p], [1, q] and [2, r] respectively.The orientation of this } in unidirectional when N is divisible by 3 and directed cycle C 3 when N is not divisible by 3. Next we define edge label [β, j] of the cycle C 3 is as follows.First we define β value then define the j value of the cycle C 3 .Consider the number N which is going to be encrypted.Here N i be the partition number which has residue value α when it is divisible by 3, then the number β = N i 3 is placed to any one of the edges which is incident to the vertices having the label α and the direction of edge will be towards the same vertex having label α.Label j value to the edges as The orientation of all edges is given as, for i < j, The edge labeling [β, j] for remaining edges other than the cycle C 3 is given as follows: For β value, the numbers N i ; 0 ≤ i ≤ 2 be the partition numbers which has residue value α when it is divisible by 3, then the number β = N i 3 is placed to the edges which is incident to the vertex having the label α.For j value, ignoring the orientation of the remaining edges other than the cycle The remaining (4n−1)−3 edges labeled using the set of elements of S.There are many ways to place the j values to the edges without forming magic cycle C 3 .First we write the numbers in the set S in ascending order.Starting with the first element of S, store the elements in the edges f (v i v j ) for 1 ≤ i, j ≤ n in E consecutively by ignoring the orientation.Continue this until the set S will be empty.i.e., until all the edges will be labeled.Finally we get the encrypted digraph.

Illustration-1
The secret number to be encrypted is 360 which is divisible by 3. Encryption: Given secrete number N = 360.This secret number can be encrypted as follows: Since n = 9, we have n faces i.e., 9 triangles and let total magic constant k = 71.
• The vertex set and edge set are as defined earlier.
• Each vertex of graph G is labeled as [α, i] where α takes the values of residue class Z 3 (i.e.,) 0, 1 and 2 repeatedly to all the vertices by edge duplication of path P n i.e., G and i takes the values of {1, 2, . . ., 3n}.Each edge of G is labeled as [β, j].
• Next we calculate the j value by ignoring the orientation of edges in E.
• Start with 28 and label the edges of E 1 in the set S − {28, 29, . . ., 35}.Then for edges of E 2 from 37 by remaining numbers and continue the process until S will be empty.

Conclusion
We have exhibited face antimagic labeling for duplication of all the vertices by edge in path graph.By using pair labeling technique we have done encrypted numbers using number theory as a tool.In future we have planned to work in encryption and decryption of numbers which leads to several application of graph labeling.

Figure 2 :
Figure 2: Encrypted directed labeled graph for G-vertex by edge duplication of P 9

Figure 4 :vFigure 5 :
Figure 4: Encrypted labeled for directed graph G-vertex by edge duplication of P 9