Graphical Cryptography, Summaries of Cryptography and System Security

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Graphical Cryptography

1 Introduction

Data security is one the main concerns these days because of the increas- ing cyber crime. There have been many different algorithms related to encryption and decryption of messages for eg. RSA, Advanced Encryp- tion System(AES) etc. But very few of them use graphs for encryption. I have tried a method which uses graphs for encryption particularly Prime Weighted Graphs.

2 Definitions

• Graph Theory: It is a branch of applied mathematics, which deals with problems with the help of graph. It is very easy to solve a problem graphically, as compared to theoretically
• Graph: A graph G = (V, E) consists of a set of objects V = v 1 , v 2 , v 3 , .... called vertices, and another set E = e 1 , e 2 , e 3 , ...., whose elements are called edges[1].
• Encryption: It is the process of encoding information i.e. it converts plain-text to cipher-text. Only authorized persons can decode the cipher-text back to plain-text and access the original information.
• Decryption: It is the conversion of the encoded text or cipher-text back to plain-text with the use of key either public or private.
• Weighted Graph: It is a special type of graph in which each edge is assigned a specific weight. So Prime Weighted Graph is a graph in which these weights are prime numbers.

3 Encryption Procedure

Step 1: Write the corresponding ASCII value of each character of the message.

Step 2: Find the quotient(Q) and remainder(R) of the ASCII value with respect to the length(L) of the message.

Step 3: Encode the given ASCII value as QR.

Now we need to find the quotient(Q) and remainder(R) with respect to length(L)(14 in this case) for each of the values.

alphabet S e c r e t M e s s a g e ascii 83 101 99 114 101 116 32 77 101 115 115 97 103 101 Q 5 7 7 8 7 8 2 5 7 8 8 6 7 7 R 13 3 1 2 3 4 4 7 3 3 3 13 5 3 QR 513 73 71 82 73 84 24 57 73 83 83 613 75 73

We now convert each of the QR’s to the nearest prime number greater than it.

QR 513 73 71 82 73 84 24 57 73 83 83 613 75 73

Value to be added 8 0 0 1 0 5 5 2 0 0 0 0 4 0 Resultant prime 521 73 71 83 73 89 29 59 73 83 83 613 79 73

So now we have encoded each character as a prime number and so now the next step is to draw a graph with random number of vertices and with the edges having their weights as these prime numbers. I will be using 7 vertices to draw the graph.

a (^) b

g

e (^) d

f c

Now the final step of the encoding procedure is to convert the above graph to its corresponding adjacency matrix.

M =

The value in () represent the weight of the edge.

We send this adjacency matrix M to the receiver.

5 Decryption

Decryption is also important so that the receiver gets the original message that we had sent. In this the case the two main keys needed for decoding the sent message are (a)The order of the edges of the graph.

Resultant prime 521 73 71 83 73 89 29 59 73 83 83 613 79 73

(b)The numbers which were added to get the prime numbers.

Value to be added 8 0 0 1 0 5 5 2 0 0 0 0 4 0

Step 1: Write the weight of each edge corresponding to the sequence of edges shown in the table above.

Step 2: Subtract the elements of the second decryption key from the corresponding prime numbers to get the message encoded in the QR form.

prime 521 73 71 83 73 89 29 59 73 83 83 613 79 73 Subtracted value 8 0 0 1 0 5 5 2 0 0 0 0 4 0 QR 513 73 71 82 73 84 24 57 73 83 83 613 75 73