Cryptography by Using " Hosoya " Polynomials for " Graphs Groups of Integer Modulen and " Dihedral Groups with ' Immersion " Property

In this paper we used Hosoya polynomial ofgroupgraphs Z1,...,Z26 after representing each group as graph and using Dihedral group to"encrypt the plain texts with the immersion property which provided Hosoya polynomial to immerse the cipher text in another"cipher text to become very"difficult to solve.


Introduction""
" In modern"times, cryptography is an important science in"algebra and graph theory, where"many articles linked in"this side, In"2014,"Natalia"tokareva" [1] linked the graph theory"and"cryptography,"with"conclusion"good"results" [2].Thomas Risse studied this concept and gave some important example, Simon"Richard,"Carlos"Cid and Ciaran Mullan in [3], presented:and studied:of group theoryand cryptography where they get good results.The main goal of cryptography is to secure and save important messages by special methods that cannot be easily identified.This paper includes three parts " The first part is the basic concepts." The second partsuggested algorithm of the method." "The third part is the application method."Let G be a graph, then the Hosoya polynomial of G is Where d(G,k) is the number of vertices pairs at distance k , k≥0 , dim(G) is the diameter of the geaph G and X is the guide of the polynomial.

Example 4
) then the simple graph of this group is of Z Neutral Graph .Figure 2 and the hosoya polynomial of this graph is 5+6X+4X Definition 5 [6] } is called the dihedral "Cryptographyis the scientific and practical activity associated with developing of cryptographic security facilities of information and also with argumentation of their cryptographic resistance".

Definition 7 [7]
"Encryption is the process of disguising a message in such a way as to hide its substance (the process of change the plaintext into cipher text by virtue of cipher)".

Definition 8 [7]
"Decryption is the process of turning a cipher text into the plain text".
hen the adverse of T

The Suggested Algorithm
In this section we introduce two algorithms, the first is algorithm of encryption process and the second is algorithm of decryption process Note 11 ""We consider the blank is character, that is the alphabet is 27 letters and we used the function (mod 28).i-algorithm of encryption process .4-Take positive integer number"n .

-"
Divide the text with length 2n by using dihedral group as: "If the"number"0"appears, then"it"always"takes"the"code"#:

Note 13
After the decryption, we always take the first letter and then we cancel two letters after it and take the fourth letter and cancel two letters after it and so on because the clear text is immersed in another text.

Application Method
Now, we apply the above method in two examples.

Example 14
Take the plain text (college)

1-"Encryption
and representing this groups We converts each letter with corresponding groups Z as graphs and extract hosoya polynomaial for all this graphs the graph of this , and

EFDGILEFD_##
Then the plain text is (college).

Example 15
If we encryption the text (college of computer science and mathematics) by the same technique with choose n=3 then we will get P→"college of computer science and mathematics" C→"IDAPVAMQWPKEVUWTROKGEPVAGHIYX_LGENJMKGEPVAGHIYX_LGENJ M IDAJMYFGENHKXX_VUWGAAOTQEECOIWZ_ _G_ENKSFGENSE_ _ _G_EROC CDAT#E#AAA_ _#_ _#_ _" Now we find a table of ratios of letters and a statistical scheme of plan text and cipher text and try to compare these two texts For plan text  Now to compare these percentages we give some observations 1-Notice that the clear text consists of 38 characters while the encrypted text consists of 126 characters this means that each letter of clear text corresponds to three letters of the encryption text and this is the immersion property we mentioned.
2-Notice in the statistical scheme of the encryption text that almost all alphabets were used as well as the symbols added to the alphabet, whereas in the plan text there are nine non-existent characters.3-Notice that the highest ratio of letters or symbols in the encoded text is the ratio of the symbol _ which has been added to the alphabet which does not represent any letter of the clear text and this indicates that this code added to the alphabet has increased the strength of encryption significantly.

Definition1 [ 4 ]
In this section, we will provide some basic concepts known consists of the elements of n ) be a group of integers module n ,then the graph of Z while the edges for any two distinict vertices a,b would be adjcent if a+ n Z .n and the element 0 associated with all elements of Z .n e is the identity element of the groups Z the form " group with order 2n, a, b the elements of this group.Definition 6[7]

1 . 3 -
each letter with corresponding groups Z -Extraction of Hosoya polynomial for all groups graphs.

-
To improve this method we must encryption the first letter because the first letter by using this method stay the same letter always then we encryption the first letter by this equation +

4 Figure
the hosoya polynomail of this graph is and the graph of this group is , and the hosoya polynomail of this graph is (15+18X+84X ) and for all letters we will get )

Table 1 .
The ratio of plan text characters

Table 2 .
The Ratio of Cipher Text Characters