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\title{An Empirical Investigation of Using ANN Based N-State Sequential Machine and Chaotic Neural Network in the Field of Cryptography}
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             \author[1]{Abhinav  Tiwari}

             \affil[1]{  Shri Ram Institute of Technology Jabalpur}

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\date{\small \em Received: 13 December 2011 Accepted: 3 January 2012 Published: 15 January 2012}

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\begin{abstract}
        


Cryptography is the exchange of information among the users without leakage of information to others. Many public key cryptography are available which are based on number theory but it has the drawback of requirement of large computational power, complexity and time consumption during generation of key [1]. To overcome these drawbacks, we analyzed neural network is the best way to generate secret key. In this paper we proposed a very new approach in the field of cryptography. We are using two artificial neural networks in the field of cryptography. First One is ANN based n-state sequential machine and Other One is chaotic neural network. For simulation MATLAB software is used. This paper also includes an experimental results and complete demonstration that ANN based n-state sequential machine and chaotic neural network is successfully perform the cryptography.

\end{abstract}


\keywords{ANN, n-state Sequential Machine, Chaotic Neural Network, Cryptography.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
n Artificial Neural Network (ANN) is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain, process An ANN is configured for a specific application, such as pattern recognition or data classification, through a learning process \hyperref[b10]{[12]}. Learning in biological systems involves adjustments to the synaptic connections that exist between the neurons. \hyperref[b10]{[12]}Cryptosystems are commonly used for protecting the integrity, confidentiality, and authenticity of information resources. In addition to meeting standard specifications relating to encryption and decryption, such systems must meet increasingly stringent specifications concerning information security. This is mostly due to the steady demand to protect data and resources from disclosure, to guarantee the authenticity of data, and to protect systems from web based attacks. For these reasons, the development and evaluation of cryptographic algorithms is a challenging task.\par
This paper is an investigation of using ANN based n-state sequential machine and chaotic neural network in the field of cryptography .the rest of the paper is organized as follows: section 2 discusses background and related work in the field of ANN based cryptography, section 3 proposed method related to nstate sequential machine and chaotic neural network section 4 discusses implementation section 5 discusses experimental report and test result and finally section 6 discusses conclusion. 
\section[{II.}]{II.}\par
Background and related work Jason L. Wright , Milos Manic Proposed a research paper on Neural Network Approach to Locating Cryptography in Object Code. In this paper, artificial neural networks are used to classify functional blocks from a disassembled program as being either cryptography related or not. The resulting system, referred to as NNLC (Neural Net for Locating Cryptography) is presented and results of applying this system to various libraries are described \hyperref[b1]{[ 2]}.\par
John Justin M, Manimurugan S introduced A Survey on Various Encryption Techniques. This paper focuses mainly on the different kinds of encryption techniques that are existing, and framing all the techniques together as a literature survey. Aim an extensive experimental study of implementations of various available encryption techniques. Also focuses on image encryption techniques, information encryption techniques, double encryption and Chaos-based encryption techniques. This study extends to the performance parameters used in encryption processes and analysing on their security issues \hyperref[b2]{[3]}.\par
Ilker DALKIRAN, Kenan DANIS¸MAN introduced a research paper on Artificial neural network based chaotic generator for cryptology . In this paper, to overcome disadvantages of chaotic systems, the dynamics of Chua's circuit namely x, y and z were modeled using Artificial Neural Network (ANN). ANNs have some distinctive capabilities like learning from experiences, generalizing from a few data and nonlinear relationship between inputs and outputs. The proposed ANN was trained in diffrent structures using different learning algorithms. To train the ANN, 24 different sets including the initial conditions of Chua's circuit were used and each set consisted of about 1800 input-output data. The experimental results showed that a feed-forward Multi Layer Perceptron (MLP), trained with Bayesian Regulation back propagation algorithm, was found as the suitable network structure. As a case study, a message was first encrypted and then decrypted by the chaotic dynamics obtained from the proposed ANN and a comparison was made between the proposed ANN and the numerical solution of Chua's circuit about encrypted and decrypted messages  {\ref [5]}.\par
Eva Volna ,Martin Kotyrba ,Vaclav Kocian, Michal Janosek developed a Cryptography Based on Neural Network. This paper deals with using neural network in cryptography, e.g. designing such neural network that would be practically used in the area of cryptography. This paper also includes an experimental demonstration \hyperref[b4]{[6]}.\par
Karam M. Z. Othman , Mohammed H. Al Jammas Introduced Implementation of Neural -Cryptographic System Using Fpga. In this work, a Pseudo Random Number Generator (PRNG) based on artificial Neural Networks (ANN) has been designed. This PRNG has been used to design stream cipher system with high statistical randomness properties of its key sequence using ANN. Software simulation has been build using MATLAB to firstly, ensure passing four wellknown statistical tests that guaranteed randomness characteristics. Secondly, such stream cipher system is required to be implemented using FPGA technology, therefore, minimum hardware requirements has to be considered \hyperref[b5]{[7]}.\par
T. Schmidt , H. Rahnama Developed A Review of Applications of Artificial Neural Networks In Cryptosystems. This paper presents a review of the literature on the use of artificial neutral networks in cryptography. Different neural network based approaches have been categorized based on their applications to different components of cryptosystems such as secret key protocols, visual cryptography, design of random generators, digital watermarking, and steganalysis \hyperref[b6]{[8]}.\par
Wenwu Yu, Jinde Cao introduced Cryptography based on delayed chaotic neural networks. In this Letter, a novel approach of encryption based on chaotic Hopfield neural networks with time varying delay is proposed. We use the chaotic neural network to generate binary sequences which will be used for masking plaintext. The plaintext is masked by switching of chaotic neural network maps and permutation of generated binary sequences. Simulation results were given to show the feasibility and effectiveness in the proposed scheme of this Letter. As a result, chaotic cryptography becomes more practical in the secure transmission of large multi-media files over public data communication network \hyperref[b7]{[9]} III. 
\section[{PROPOSED METHOD}]{PROPOSED METHOD}\par
A number of studies have already investigated different machine learning methodologies, specifically neural networks and their applications in cryptography, but It is uncommon technique to using Artificial Neural network based n-state sequential machine and Chaotic neural network in the field of cryptography . 
\section[{a) Sequential Machine}]{a) Sequential Machine}\par
A sequential Machine output depends on state of the machine as well as the input given to the sequential machine. Therefore Michel I .Jordan Network was designed because in which output are treated as input. We are used these type of input as a state. As a sequential machine can be achieved by using a Michel I. Jordan neural network, therefore data are successfully encrypted and decrypted. In this case the starting state of the n-state sequential machine can act as a key. Data is used to train the neural network as it provides the way the machine moves from one state to another. c) Cryptography Achieved by a chaotic neural network Cryptography scheme was done by a chaotic neural network. A network is called chaotic neural network if its weights and biases are determined by chaotic sequence. Specially encryption of digital signal we used chaotic neural network. The reason for using sequential machine for implementation is that the output and input can have any type of relationship and the output depends on the starting state. The starting state is used as a key for encryption and decryption. If the starting state is not known, it is not possible to retrieve the data by decryption even if the state table or the working of the sequential state is known. For training of the neural network, any type of sequential machine can be used with the key showing the complexity or the level of security obtained. 
\section[{c) Cryptography Achieved Through chaotic neural network}]{c) Cryptography Achieved Through chaotic neural network}\par
A chaotic neural network in which weights and biases are determined by a chaotic sequence. g = digital signal of length M and g (n) 0 ? M-1 , be the one-byte value of the signal g at position n .The decryption procedure is the same as the above one except that the input signal to the decryption Chaotic neural network should be g'(n) and its output signal should be g"(n).\par
V. 
\section[{Experiment and test result a) Sequential Machine}]{Experiment and test result a) Sequential Machine}\par
A general n-state Sequential Machine was implemented. As an example, the serial adder was implemented using this machine.   There is initial state is informed to the user for the input bits to be added. The output is the sum and the carry bit. After that the execution of program has been completed it is automatically jumps to the new carry state. This output is considered as previous carry state.\par
Following Graph illustrate Mean Square Error when we Enter input 0,1 and 2 .we apply this input in two feed forward Adder One is of Multilayer single output feed forward Adder and other is Multilayer multiple output feed forward Adder.\par
First we apply Input Number 0,1 and 2 on Multilayer single output feed forward and find MSE on Linear scale . Year and generated encrypted letter similarly is the starting state is 1 the letter is shifted by 2.The state is automatically move to next state .If the next input is again A the output will be C as the current state now is 1. For H, state 0 will flip the letter to A while state 1 will flip the output to B. This method can be used to encrypt a word containing only the letters A to H.  \begin{figure}[htbp]
\noindent\textbf{31}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 3 . 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Fig. 5. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4}\includegraphics[]{image-6.png}
\caption{\label{fig_4}Fig. 5. 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-7.png}
\caption{\label{fig_5}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_0}Table 5 .}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.17990291262135924\textwidth}P{0.19640776699029125\textwidth}P{0.16174757281553395\textwidth}P{0.1666990291262136\textwidth}P{0.14524271844660194\textwidth}}
\tabcellsep \multicolumn{4}{l}{Conditions (Values of x(0) and µ())}\\
\multicolumn{5}{l}{Same Input Encrypted with Different Initial Conditions}\\
\multicolumn{3}{l}{(Values of x(0) and µ())}\tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep Output with\tabcellsep Output with\tabcellsep output with\\
\tabcellsep ASCII\tabcellsep x(0)= 0.75\tabcellsep x(0)= 0.85\tabcellsep x(0)= 0.90\\
INPUT\tabcellsep CODE\tabcellsep µ=3.9\tabcellsep µ=3.5\tabcellsep µ=3.2\\
A\tabcellsep 97\tabcellsep 199\tabcellsep 233\tabcellsep 204\\
B\tabcellsep 98\tabcellsep 195\tabcellsep 112\tabcellsep 98\\
C\tabcellsep 99\tabcellsep 200\tabcellsep 239\tabcellsep 11\\
D\tabcellsep 100\tabcellsep 253\tabcellsep 108\tabcellsep 31\\
E\tabcellsep 101\tabcellsep 220\tabcellsep 226\tabcellsep 1\\
F\tabcellsep 102\tabcellsep 17\tabcellsep 115\tabcellsep 25\\
G\tabcellsep 103\tabcellsep 187\tabcellsep 234\tabcellsep 56\\
H\tabcellsep 104\tabcellsep 101\tabcellsep 109\tabcellsep 235\\
I\tabcellsep 105\tabcellsep 6\tabcellsep 236\tabcellsep 49\\
J\tabcellsep 106\tabcellsep 138\tabcellsep 115\tabcellsep 226\\
K\tabcellsep 107\tabcellsep 107\tabcellsep 229\tabcellsep 36\\
L\tabcellsep 108\tabcellsep 32\tabcellsep 110\tabcellsep 225\\
M\tabcellsep 109\tabcellsep 180\tabcellsep 238\tabcellsep 42\\
N\tabcellsep 110\tabcellsep 119\tabcellsep 112\tabcellsep 254\\
O\tabcellsep 111\tabcellsep 225\tabcellsep 224\tabcellsep 45\\
P\tabcellsep 112\tabcellsep 184\tabcellsep 113\tabcellsep 225\\
Q\tabcellsep 113\tabcellsep 63\tabcellsep 243\tabcellsep 49\\
R\tabcellsep 114\tabcellsep 168\tabcellsep 83\tabcellsep 227\\
S\tabcellsep 115\tabcellsep 103\tabcellsep 252\tabcellsep 51\\
T\tabcellsep 116\tabcellsep 245\tabcellsep 116\tabcellsep 229\\
U\tabcellsep 117\tabcellsep 160\tabcellsep 244\tabcellsep 53\\
V\tabcellsep 118\tabcellsep 83\tabcellsep 84\tabcellsep 231\\
W\tabcellsep 119\tabcellsep 209\tabcellsep 248\tabcellsep 55\\
X\tabcellsep 120\tabcellsep 219\tabcellsep 120\tabcellsep 233\\
Y\tabcellsep 121\tabcellsep 209\tabcellsep 248\tabcellsep 57\\
Z\tabcellsep 122\tabcellsep 231\tabcellsep 88\tabcellsep 235\end{longtable} \par
 
\caption{\label{tab_1}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.23466257668711657\textwidth}P{0.15513803680981594\textwidth}P{0.1329754601226994\textwidth}P{0.18381901840490797\textwidth}P{0.12645705521472392\textwidth}P{0.016947852760736196\textwidth}}
\multicolumn{5}{l}{Decrypted Using Same and Different Initial Conditions}\tabcellsep \\
\multicolumn{5}{l}{Encrypted Data of Table 1 (Column 2) Decrypted Using}\tabcellsep \\
\tabcellsep \multicolumn{3}{l}{Same and Different Initial Conditions}\tabcellsep \tabcellsep \\
\multicolumn{3}{l}{Output Obtained Using}\tabcellsep \multicolumn{2}{l}{Output Obtained Using}\tabcellsep \\
\multicolumn{3}{l}{Same Initial Condition}\tabcellsep \multicolumn{2}{l}{Different Initial Condition}\tabcellsep \\
\tabcellsep \tabcellsep output with\tabcellsep output with\tabcellsep output with\tabcellsep \\
\tabcellsep ASCII\tabcellsep x(0)= 0.75\tabcellsep x(0)= 0.85\tabcellsep x(0)= 0.90\tabcellsep \\
INPUT\tabcellsep CODE\tabcellsep µ=3.9\tabcellsep µ=3.5\tabcellsep µ=3.2\tabcellsep \\
A\tabcellsep 199\tabcellsep 97\tabcellsep 79\tabcellsep 106\tabcellsep \\
B\tabcellsep 195\tabcellsep 98\tabcellsep 209\tabcellsep 195\tabcellsep \\
C\tabcellsep 200\tabcellsep 99\tabcellsep 68\tabcellsep 160\tabcellsep \\
D\tabcellsep 253\tabcellsep 100\tabcellsep 245\tabcellsep 134\tabcellsep \\
E\tabcellsep 220\tabcellsep 101\tabcellsep 91\tabcellsep 184\tabcellsep \\
F\tabcellsep 17\tabcellsep 102\tabcellsep 4\tabcellsep 110\tabcellsep \\
G\tabcellsep 187\tabcellsep 103\tabcellsep 54\tabcellsep 228\tabcellsep \\
H\tabcellsep 101\tabcellsep 104\tabcellsep 96\tabcellsep 230\tabcellsep \\
I\tabcellsep 6\tabcellsep 105\tabcellsep 131\tabcellsep 94\tabcellsep \\
J\tabcellsep 138\tabcellsep 106\tabcellsep 147\tabcellsep 2\tabcellsep \\
K\tabcellsep 107\tabcellsep 107\tabcellsep 229\tabcellsep 36\tabcellsep \\
L\tabcellsep 32\tabcellsep 108\tabcellsep 34\tabcellsep 173\tabcellsep \\
M\tabcellsep 180\tabcellsep 109\tabcellsep 55\tabcellsep 243\tabcellsep \\
N\tabcellsep 119\tabcellsep 110\tabcellsep 105\tabcellsep 231\tabcellsep \\
O\tabcellsep 225\tabcellsep 111\tabcellsep 110\tabcellsep 163\tabcellsep \\
P\tabcellsep 184\tabcellsep 112\tabcellsep 185\tabcellsep 41\tabcellsep \\
Q\tabcellsep 63\tabcellsep 113\tabcellsep 189\tabcellsep 127\tabcellsep \\
R\tabcellsep 168\tabcellsep 114\tabcellsep 137\tabcellsep 57\tabcellsep \\
S\tabcellsep 103\tabcellsep 115\tabcellsep 232\tabcellsep 39\tabcellsep \\
T U\tabcellsep 245 160\tabcellsep 116 117\tabcellsep 245 33\tabcellsep 100 224\tabcellsep ( D D D D ) D\\
V\tabcellsep 83\tabcellsep 118\tabcellsep 113\tabcellsep 194\tabcellsep \\
W\tabcellsep 209\tabcellsep 119\tabcellsep 94\tabcellsep 145\tabcellsep \\
X\tabcellsep 219\tabcellsep 120\tabcellsep 219\tabcellsep 74\tabcellsep \\
Y\tabcellsep 209\tabcellsep 121\tabcellsep 80\tabcellsep 145\tabcellsep \\
Z\tabcellsep 231\tabcellsep 122\tabcellsep 197\tabcellsep 118\tabcellsep \end{longtable} \par
 
\caption{\label{tab_2}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.23699376947040496\textwidth}P{0.160202492211838\textwidth}P{0.1218068535825545\textwidth}P{0.19727414330218068\textwidth}P{0.1337227414330218\textwidth}}
\multicolumn{5}{l}{Decrypted Using Same and Different Initial Conditions}\\
\multicolumn{5}{l}{Encrypted Data of Table 1 (Column 3) Decrypted Using}\\
\tabcellsep \multicolumn{4}{l}{Same and Different Initial Conditions}\\
\multicolumn{3}{l}{Output Obtained Using}\tabcellsep \multicolumn{2}{l}{Output Obtained Using}\\
\multicolumn{3}{l}{Same Initial Condition}\tabcellsep \multicolumn{2}{l}{Different Initial Condition}\\
\tabcellsep \tabcellsep output with\tabcellsep output with\tabcellsep output with\\
\tabcellsep ASCII\tabcellsep x(0)= 0.75\tabcellsep x(0)= 0.85\tabcellsep x(0)= 0.90\\
INPUT\tabcellsep CODE\tabcellsep µ=3.9\tabcellsep µ=3.5\tabcellsep µ=3.2\\
A\tabcellsep 233\tabcellsep 79\tabcellsep 97\tabcellsep 68\\
B\tabcellsep 112\tabcellsep 209\tabcellsep 98\tabcellsep 112\\
C\tabcellsep 239\tabcellsep 68\tabcellsep 99\tabcellsep 135\\
D\tabcellsep 108\tabcellsep 245\tabcellsep 100\tabcellsep 23\\
E\tabcellsep 226\tabcellsep 91\tabcellsep 101\tabcellsep 134\\
F\tabcellsep 115\tabcellsep 4\tabcellsep 102\tabcellsep 12\\
G\tabcellsep 234\tabcellsep 54\tabcellsep 103\tabcellsep 181\\
H\tabcellsep 109\tabcellsep 96\tabcellsep 104\tabcellsep 238\\
I\tabcellsep 236\tabcellsep 131\tabcellsep 105\tabcellsep 180\\
J\tabcellsep 115\tabcellsep 147\tabcellsep 106\tabcellsep 251\\
K\tabcellsep 229\tabcellsep 229\tabcellsep 107\tabcellsep 170\\
L\tabcellsep 110\tabcellsep 34\tabcellsep 108\tabcellsep 227\\
M\tabcellsep 238\tabcellsep 55\tabcellsep 109\tabcellsep 169\\
N\tabcellsep 112\tabcellsep 105\tabcellsep 110\tabcellsep 224\\
O\tabcellsep 224\tabcellsep 110\tabcellsep 111\tabcellsep 162\\
P\tabcellsep 113\tabcellsep 185\tabcellsep 112\tabcellsep 224\\
Q\tabcellsep 243\tabcellsep 189\tabcellsep 113\tabcellsep 179\\
R\tabcellsep 83\tabcellsep 137\tabcellsep 114\tabcellsep 194\\
S\tabcellsep 252\tabcellsep 232\tabcellsep 115\tabcellsep 188\\
T\tabcellsep 116\tabcellsep 245\tabcellsep 116\tabcellsep 229\\
U\tabcellsep 244\tabcellsep 33\tabcellsep 117\tabcellsep 180\\
V\tabcellsep 84\tabcellsep 113\tabcellsep 118\tabcellsep 197\\
W\tabcellsep 248\tabcellsep 94\tabcellsep 119\tabcellsep 184\\
X\tabcellsep 120\tabcellsep 219\tabcellsep 120\tabcellsep 233\\
Y\tabcellsep 248\tabcellsep 80\tabcellsep 121\tabcellsep 184\\
Z\tabcellsep 88\tabcellsep 197\tabcellsep 122\tabcellsep 201\end{longtable} \par
 
\caption{\label{tab_3}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.2905193236714976\textwidth}P{0.16014492753623188\textwidth}P{0.021557971014492753\textwidth}P{0.1201086956521739\textwidth}P{0.15295893719806763\textwidth}P{0.10471014492753623\textwidth}}
\multicolumn{6}{l}{Decrypted Using Same and Different Initial Conditions}\\
\multicolumn{6}{l}{Encrypted Data of Table 1 (Column 4) Decrypted Using}\\
\tabcellsep \multicolumn{5}{l}{Same and Different Initial Conditions}\\
\multicolumn{4}{l}{Output Obtained Using}\tabcellsep \multicolumn{2}{l}{Output Obtained Using}\\
\multicolumn{4}{l}{Same Initial Condition}\tabcellsep \multicolumn{2}{l}{Different Initial Condition}\\
\tabcellsep \tabcellsep \multicolumn{2}{l}{output with}\tabcellsep output with\\
\tabcellsep ASCII\tabcellsep \multicolumn{2}{l}{x(0)= 0.75}\tabcellsep x(0)= 0.85\tabcellsep output with x(0)=\\
INPUT\tabcellsep CODE\tabcellsep \tabcellsep µ=3.9\tabcellsep µ=3.5\tabcellsep 0.90 µ=3.2\\
A\tabcellsep 204\tabcellsep \tabcellsep 106\tabcellsep 68\tabcellsep 97\\
B\tabcellsep 98\tabcellsep \tabcellsep 195\tabcellsep 112\tabcellsep 98\\
C\tabcellsep 11\tabcellsep \tabcellsep 160\tabcellsep 135\tabcellsep 99\\
D\tabcellsep 31\tabcellsep \tabcellsep 134\tabcellsep 23\tabcellsep 100\\
E\tabcellsep 1\tabcellsep \tabcellsep 184\tabcellsep 134\tabcellsep 101\\
F\tabcellsep 25\tabcellsep \tabcellsep 110\tabcellsep 12\tabcellsep 102\\
G\tabcellsep 56\tabcellsep \tabcellsep 228\tabcellsep 181\tabcellsep 103\\
H\tabcellsep 235\tabcellsep \tabcellsep 230\tabcellsep 238\tabcellsep 104\\
I\tabcellsep 49\tabcellsep \tabcellsep 94\tabcellsep 180\tabcellsep 105\\
J\tabcellsep 226\tabcellsep \tabcellsep 2\tabcellsep 251\tabcellsep 106\\
K\tabcellsep 36\tabcellsep \tabcellsep 36\tabcellsep 170\tabcellsep 107\\
L\tabcellsep 225\tabcellsep \tabcellsep 173\tabcellsep 227\tabcellsep 108\\
M\tabcellsep 42\tabcellsep \tabcellsep 243\tabcellsep 169\tabcellsep 109\\
N\tabcellsep 254\tabcellsep \tabcellsep 231\tabcellsep 224\tabcellsep 110\\
O\tabcellsep 45\tabcellsep \tabcellsep 163\tabcellsep 162\tabcellsep 111\\
P\tabcellsep 225\tabcellsep \tabcellsep 41\tabcellsep 224\tabcellsep 112\\
Q\tabcellsep 49\tabcellsep \tabcellsep 127\tabcellsep 179\tabcellsep 113\\
R\tabcellsep 227\tabcellsep \tabcellsep 57\tabcellsep 194\tabcellsep 114\\
S\tabcellsep 51\tabcellsep \tabcellsep 39\tabcellsep 188\tabcellsep 115\\
T\tabcellsep 229\tabcellsep \tabcellsep 100\tabcellsep 229\tabcellsep 116\\
U\tabcellsep 53\tabcellsep \tabcellsep 224\tabcellsep 180\tabcellsep 117\\
V\tabcellsep 231\tabcellsep \tabcellsep 194\tabcellsep 197\tabcellsep 118\\
W\tabcellsep 55\tabcellsep \tabcellsep 145\tabcellsep 184\tabcellsep 119\\
X\tabcellsep 233\tabcellsep \tabcellsep 74\tabcellsep 233\tabcellsep 120\\
Y\tabcellsep 57\tabcellsep \tabcellsep 145\tabcellsep 184\tabcellsep 121\\
Z\tabcellsep 235\tabcellsep \tabcellsep 118\tabcellsep 201\tabcellsep 122\\
\tabcellsep \multicolumn{5}{l}{It is clear from table 2, 3 and 4 that we can}\\
\multicolumn{6}{l}{decrypt an encrypted data correctly by knowing the}\\
\multicolumn{2}{l}{exact values of}\tabcellsep \tabcellsep \multicolumn{3}{l}{x (0) and µ otherwise we get the}\\
\multicolumn{5}{l}{wrong data as shown in table 2,3 and 4.}\\
\tabcellsep \multicolumn{2}{l}{VI.}\tabcellsep \multicolumn{2}{l}{Conclusion}\end{longtable} \par
 
\caption{\label{tab_4}Table 4 :}\end{figure}
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