An Adaptive Neural Network Guided Secret Key based Encryption through Recursive Positional

Modulo-2 Substitution for Online Wireless Communication (ANNRPMS)

J. K. Mandal1, Arindam Sarkar2

Department of Computer Science & Engineering University of Kalyani, Kalyani-741235, Nadia, West Bengal, India.

1jkm.cse@gmail.com 2arindam.vb@gmail.com

Abstract—In this paper, a neural network guided secret key based technique for encryption (ANNRPMS) has been proposed using recursive positional modulo-2 substitution for online transmission of data/information through wireless communication. In this approach both the communicating networks receive an indistinguishable input vector, produce an output bit and are trained based on the output bit. The dynamics of the two networks and their weight vectors are found to a novel experience, where the demonstrate networks synchronize to an identical time dependent weight vectors. This observable fact has been used to form a secured variable length secret-key using a public channel. The length of the key depends on the number of input and output neurons. The original plain text is encrypted using recursive positional modulo-2 substitution technique. The secret key also fabricated through encryption process in a cascaded manner. This intermediate cipher text is again encrypted to form the final cipher text through chaining and cascaded xoring of identical weight vector with the identical length intermediate cipher text block. If size of the last block of intermediate cipher text is less than the size of the key then this block is kept unchanged. Receiver will use identical weight vector for performing deciphering process for getting the recursive positional modulo-2 substitution encrypted cipher text and secret key for decoding. A session key based transmission has also been proposed using 161-bit key format of 14 different segments [7,8]. Parametric tests are done and results are compared in terms of Chi-Square test, response time in transmission with some existing classical techniques, which shows comparable results for the proposed system. Keywords – Adaptive Neural Network guided Recursive Positional Modulo-2 Substitution (ANNRPMS), weight vector, input vector, sub key, session key, chaining, wrapping technique.

I. INTRODUCTION These days a variety of techniques are available to defend

data and information from eavesdroppers [1,2,7,8,10]. Every algorithm has its own advantages and disadvantages. For Example in DES, AES algorithms [9] the cipher block length is inflexible. RSBP [10] and RBCMPCC [1] allow only one cipher block encoding. In RPSP algorithm [10] any intermediate blocks during its cycle taken as the encrypted block and this number of iterations acts as secret key.

The organization of this paper is as follows. Section II of the paper deals with the proposed synchronization and key generation technique and also random block length based cryptographic techniques. Session key based encryption technique has been discussed in section III. Example of this technique is given in section IV. Complexity of the algorithm is presented in section V. Results are presented in section VI. Analysis regarding various aspects of the technique has been presented in section VII. Conclusions are drawn in section VIII and that of references at end.

II. THE TECHNIQUE In proposed technique recursive positional modulo-2

substitution algorithm is used to encrypt the plain text along with secret key padded with the last block of intermediate text generated. This intermediate and padded cipher text is again encrypted to form the final cipher text using chaining of cascaded xoring of intermediate cipher text with the identical neural secret key. Using identical weight vector receiver performs deciphering process to regenerate the plain text. Section II.A deals with key generation technique and that of section II.B and II.C describes the encryption and decryption techniques respectively with the help of recursive positional modulo-2 substitution algorithm that of

II.A. Neural synchronization scheme & secret key generation At the beginning of the transmission, a neural network

based secret key generation is performed between receiver and sender. The same may be done through the private channel also or it may be done in some other time, which is absolutely free from encryption. Figure 1 shows the tree based generation process using random number of nodes (neurons) and the corresponding algorithm is given in two-sub section II.A.1 and II.A.2.

.

Figure 1. A tree parity machine with K=3 and N=4

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II.A.1. Neural synchronization algorithm Input: - Random weights, input vectors for both neural

networks Output: - Secret key through synchronization of input and

output neurons as vectors. Method: - Each party (A and B) uses its own (same) tree

parity machine. Neural network parameters: K, N, L values will be identical for both parties and synchronization of the tree parity machines is achieved.

Parameters: K - The number of hidden neurons. N - The number of input neurons connected to each hidden neuron, total (K*N) input neurons. L - The maximum value for weight {-L..+L} Step 1. Initialize random weight values. Step 2. Repeat step 3 to 6 until the full synchronization is achieved, using hebbian-learning rules (eq. 3). Step 3. Generate random input vector X. Inputs are generated by a third party (say the server) or one of the communicating parties (fig. 2)

Figure 2. Neural network for machine A and machine B with K=4 & N=2

Step 4. Compute the values of the hidden neurons

==

N

jijiji xw

1sgnσ ( )

−=

10

1sgn x

ififif

.0,0,0

>=<

xxx

(1) Step 5. Compute the value of the output neuron

∏= =

=K

i

N

jijij xw

1 1sgnτ

(2) Step 6. Compare the output values of both tree parity machines by exchanging between the networks Outputs are not same: Go to step 3 if Output (A) Output (B) else if Output (A) = Output (B) then one of the suitable learning rule is applied to the weights. Update the weights only if the final output values of the neural machines are equivalent. In this paper we have used hebbian-learning rule for synchronization (eq. 3).

( ) ( )BAiiii xww i ττθτσθσ+=+

(3)

When synchronization is finally occurred, the synaptic weights are same for both the networks. In each step there may be three possibilities: 1. Output (A) Output (B): None of the parties updates its weights. 2. Output (A) = Output (B) = Output (E): All the three parties update weights in their tree parity machines. 3. Output (A) = Output (B) Output (E): Parties A and B update their tree parity machines, but the attacker cannot do that. Because of this circumstance his learning is slower than the synchronization of parties A and B. The synchronization of two parties is faster than learning of an attacker. Complexity: O(N) computational steps are needed to generate a key of length N. The average synchronization time up to N=1000 asymptotically one expects an increase like O (log N). II.A.2. Key generation

The technique considers the synchronized weight vector from the first phase in the form of blocks of bits with different size like 8/ 16/32/ 64/ 128/ 256. The rules to be followed for generating a cycle are as follows:

For each block S= s0 s1 s2 s3 s4 … sL-1 of length L bits, the following technique is followed in a stepwise manner to generate the target block T= t0 t1 t2 t3 t4 … tL-1 of the same length (L). 1. Consider the source block S= s0 s1 s2 s3 s4 … sL-1 (where

N=2n, n =3 to 8) and divide it into two equal parts (1st half & 2nd half).

2. Consider any key value (key= 2n, where n=1 to 7) depends upon the source stream that is, key value is the half of the source stream). Suppose key value is 10 (half of the 1st source block).

3. Apply step 4 and step 5 exactly L number of times, for the values of the variable X ranging from 0 to L-1 increasing by 1 after each execution of the loop.

4. Now divide the key value into two halves and 1st half is placed to the MSB position of the 1st half of the source stream and 2nd half placed to the LSB position of the same source stream then perform the modulo-2 addition (X-OR) with the key value to the first half of the source stream, to get the first intermediate block.

5. Make the modulo-2 addition operation again on the rest of the middle bits with the reversed key value to get the second intermediate block. The same operation is performed for whole stream

number of time with a varying block sizes. k such iteration is done and the final intermediate stream after k iterations generates the cascaded form of the encrypted stream(i.e. neural secret key). All of the different block sizes and k constitute the key for the session. This process is repeated until the source neural weight vector is generated.

For different size of weight sub vector different intermediate blocks may be considered as the corresponding

Weight Weight

TPM of Party A TPM of Party B Hidden layer

Input layer

Output layer

Input layer

Output layer

Hidden layer

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encrypted blocks. For example, the policy may be something like that out of three weight sub vector blocks B1, B2, B3 in a key block of bits, the 4th, the 7th and the 5th intermediate blocks respectively are being considered as the encrypted blocks. In such a case, the key of the scheme will become much more complex, which in turn will ensure better security. In the case of blocks with varying lengths, different blocks will require different numbers of iteration to form the corresponding cycle. So, the LCM value, say, P, of all these numbers will give the actual number of iterations required to form the cycle for the entire stream. If i number of iterations are performed to encrypt the entire stream, then a number of (P – i) more iterations will be required to decrypt the encrypted weight vector stream.

II.B. Encryption Technique During encryption, the decimal equivalent of the block of

bits under consideration is one integral value from which the recursive modulo-2 operation starts. The modulo-2 operation is performed to check if the integral value is even or odd and the position of that integral value in the series of natural even or odd numbers is evaluated. Recursively this process is carried out to a finite number of times, which is exactly the length of the source block. After each modulo-2 operation, 0 or 1 is pushed to the output stream in MSB-to-LSB direction; depending on the fact whether the integral value is even or odd respectively. A stream of bits is considered as the plaintext. In this technique, the plaintext is divided into a finite number of blocks, each having a finite fixed number of bits. The recursive positional modulo-2 substitution is then applied for each of the blocks.

Figure 3 shows the algorithm for encryption technique.

Figure 3. Encryption algorithm Thus overall complexity of encryption algorithm is

O(L).The key is padded with the encrypted text block to form the intermediate cipher block. Then the neural secret key is use to xored with the same length first intermediate cipher text block to produce the first final cipher block (neural secret key XOR with same length cipher text). This newly generated block again xored with the immediate next block and so on. This chaining of cascaded xoring mechanism is performed until all the blocks get exhausted. If the last block size of intermediate cipher text is less than the require xoring block size (i.e. weight vector size) then this block is kept unchanged

II.C. Decryption Technique During decryption the encrypted message is converted into

the corresponding stream of bits and then this stream is decomposed into a finite set of blocks, each consisting of a finite set of bits using same rule of encryption. It is to be noted that the receiver will take identical weight vectors and same neural key generation algorithm is use to generate the neural key. Then cascaded xoring operation is performed using identical neural secret key with the cipher text. The technique of performing xoring is same that was in encryption process. Finally the outcomes of this step T1 is to be decomposed into the blocks (t1, t2, …, t7). Then from this T1 intermediate encrypted block (T) and key block is extracted and now key is use to decipher the T to get the plain text. Complexity of this decryption algorithm is also O(L).

Figure 4 shows the algorithm for decryption technique .

Figure 4. Decryption algorithm

III. GENERATION OF SESSION KEY To ensure secured encryption of the proposed technique

with varying size of blocks, a 161 bit key format consisting of 14 different segment has been proposed here [7,8]. For the segment of rank the R, there can exist a maximum of

Figure 5. 161 bit key format N=218-R blocks, each of unique size of S=218-R, R

starting from 1 and moving till 14.Full input is divided into blocks of various sizes based on the session key.

Segment with R=1 formed with the first maximum 131072 blocks, each of size 131072 bits

Segment with R=2 formed with the first maximum 65536 blocks, each of size 65536 bits

1 2 3 4 5 6 7 8 9 10 11 12 13 1402468

101214161820

Segment size in bits

Set: P = 0. LOOP: Evaluate: Temp = Remainder of DL-P / 2. If Temp = 0 Evaluate: DL-P-1 = DL-P / 2. Set: tP = 0. Else If Temp = 1 Evaluate: DL-P-1 = (DL-P + 1) / 2. Set: tP = 1. Set: P = P + 1. If (P > (L – 1)) Exit. END LOOP

Set: P = L – 1 and T = 1. LOOP: If tP = 0

Evaluate: T = Tth even number in the series of natural numbers.

Else If tP = 1 Evaluate: T = Tth odd number in the series of natural

numbers. Set: P = P – 1. If P < 0 Exit. END LOOP

Evaluate: S = s0 s1 s2 s3 s4 … sL-1, which is the binary equivalent of T

Key Segment

Segment size in

bits

An Adaptive Neural Network Guided Secret Key based Encryption through Recursive Positional Modulo-2 Substitution for Online Wireless Communication (ANNRPMS)

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Segment with R=3,4,5,----,14 formed with the next maximum 32768, 16384,8192,----,16 blocks, each of size 32768, 16384,8192,-------,16 bits respectively.

With such a structure, the key space becomes 147 bits long and a stream of the maximum size 2.6 GB can be encrypted using the proposed technique. The key structure is represented in Figure 5.This session key may be used during each session as a random manner to enhance the security of wireless communication.

IV. EXAMPLE In this section, we consider a separate plaintext (P) as: “Local Area Network”. The technique being an asymmetric one, the task of

encryption and that of decryption are being discussed separately. Section IV.A shows how the plaintext P is to be encrypted using this technique and section IV.B describes the process of decryption.

IV.A The Process of Encryption:

To generate the corresponding source stream of bits, we take the reference of table I.

TABLE 1 CHARACTER-TO-BYTE CONVERSION FOR THE TEXT

“LOCAL AREA NETWORK”

Character Byte

L 01001100 O 01101111 C 01100011 A 01100001 L 01101100

<Blank> 00100000 A 01000001 R 01110010 E 01100101 A 01100001

<Blank> 00100000 N 01001110 E 01100101 T 01110100 W 01110111 O 01101111 R 01110010 K 01101011

We get the source stream of bits as the following: S=01001100/01101111/01100011/01100001/01101100/00

100000/01000001/01110010/01100101/01100001/00100000/01001110/01100101/01110100/01110111/01101111/01110010/01101011.

Now, we decompose S into a set of 5 blocks, each of the first four being of size 32 bits and the last one being of 16 bits. During this process of decomposition, we scan S along the MSB-to-LSB direction and extract required number of bits for different block, like the first 32 bits in this direction being for the block S1, the next 32 bits being for the block S2, and so on. In this way we get: S1=01001100011011110110001101100001, S2=01101100001000000100000101110010,

S3=01100101011000010010000001001110, S4=01100101011101000111011101101111, S5=0111001001101011.This way of decomposition is to be intimated as the key by the sender of the message to the receiver of the same through a secret channel. For the block S1, corresponding to which the decimal value is (1282368353)10, the process of encryption is shown below:

1282368353 6411841771 3205920891 1602960451 801480231 400740121 200370060 1000185030 50092521 25046260 12523130 6261571 3130791 1565401 787200 391350 195681 97840 48920 24460 12230 6121 3060

1530 771 391 201 100 50 31 21 10 11. From this we generate the target block T1 corresponding to S1 as: T1=11111001001110010000100111001101.

Applying the similar process, we generate target blocks T1, T2, T3, T4 and T5 as follows corresponding to source blocks S1, S2, S3, S4 and S5 respectively.

T2=01110001011111011111101111001001 T3=01001101111110110111100101011001 T4=10001001000100011101000101011001 T5=1110100110110001. Now, combining target blocks in the same sequence, we

get the target stream of bits T as the following: T=11111001/00111001/00001001/11001101/01110001/01

111101/11111011/11001001/01001101/11111011/01111001/01011001/10001001/00010001/11010001/01011001/11101001/10110001. After padding the key with the above intermediate cipher text and xoring with the neural secret key, final encrypted cipher text is generated. This stream of bits, in the form of a stream of characters, is transmitted as the encrypted message.

IV.B The Process of Decryption:

At the destination point, receiver’s secret neural key is use to xoring the cipher text to get back the key and intermediate cipher text. Now, by that secret key, the receiver gets the information on different block lengths. Using that secret key, all the blocks T1, T2, T3, T4 and T5 are formed as follows:

T1=11111001001110010000100111001101 T2=01110001011111011111101111001001 T3=01001101111110110111100101011001 T4=10001001000100011101000101011001 T5=1110100110110001. Now, applying the process of decryption the

corresponding source blocks Si are generated for all Ti, 1 i 5.Here the corresponding 32-bit stream is

S1=01001100011011110110001101100001.” In this way all the source blocks of bits are regenerated

and combining those blocks in the same sequence, the source stream of bits are obtained to get the source message or the plaintext.

V. RESULTS In this section the results of implementation of the

proposed technique has been presented in terms of encryption decryption time, Chi-Square test, source file size vs.

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En

cryp

tion

& d

ecry

ptio

n tim

e

Source size

encryption time along with source file size vs. encrypted file size. The results are also compared with existing RSA technique.

TABLE II ENCRYPTION / DECRYPTION TIME VS. FILE SIZE

Encryption Time (s) Decryption Time (s)

Source Size (bytes)

Proposed ANNRP

MS RPSP

Encrypted Size

(bytes)

Proposed ANNRP

MS RPSP

18432 5. 32 7.85 18432 4.85 7.81 23044 7. 37 10.32 23040 6.96 9.92 35425 13. 98 15.21 35425 13. 37 14.93 36242 14. 53 15.34 36242 14. 01 15.24 59398 22. 39 25.49 59398 21. 88 24.95 Table II shows encryption and decryption time with

respect to the source and encrypted size respectively. It is also observed the alternation of the size on encryption.

22.39

14.5313.98

7.375.32

21.88

14.01

6.964.85

13.37

0

5

10

15

20

25

18432 23044 35425 36242 59398

Encryption

Decryption

Figure 6. Source size vs. encryption time & decryption time In figure 6 stream size is represented along X axis and

encryption / decryption time is represented along Y-axis. This graph is not linear, because of different time requirement for finding appropriate random number for different block size. It is observed that the decryption time is almost linear, because there is no random number generation during decryption.

Table III shows Chi-Square value for different source stream size after applying different encryption algorithms. It is seen that the Chi-Square value of ANNRPMS is better compared to the algorithm RBCMPCC and comparable to the Chi-Square value of the RSA algorithm.

TABLE III SOURCE SIZE VS. CHI-SQUARE VALUE

Figure 7 shows graphical representation of table II.

Figure 7. Source size vs. Chi-Square value

VI. COMPLEXITY The complexity of the technique will be O (L), which can be computed using following three steps.

Step 1: To generate a key of length N needs O (N)

Computational steps. The average synchronization time is almost independent of the size N of the networks. Asymptotically it may be expressed as O (log N).

Step 2: Complexity of the encryption technique is O (L). Step 2.1: Corresponding to the source block S = s0 s1 s2 s3 s4… sL-1, evaluate the equivalent decimal integer, DL. It takes O (L) amount of time. Step 2.2: Step 2.3 and step 2.4 executed exactly L number of times for the values of the variable P ranging from 0 to (L-1) increasing by 1 after each execution of the loop where the complexity is O (L) as L no. of iterations are needed. Step 2.3: Apply modulo-2 operation on DL-P to check if DL-P is even or odd, takes constant amount of time i.e. O(1). Step 2.4: If DL-P is found to be even, compute DL-P-1 = DL-P / 2, where DL-P-1 is its position in the series of natural even numbers. Assign tP = 0. If DL-P is found to be odd, compute DL-P-1 = (DL-P + 1) / 2, where DL-P-1 is its position in the series of natural odd numbers. Assign tP=1 which generates the complexity as O (1). Step 2.5: evaluate: DL, the decimal equivalent, corresponding to the source block S = s0 s1 s2 s3 s4 … sL-1 which takes O (L) amount of time.

Step 3: Complexity of the decryption technique is O (L). Step 3.1: To set P=L-1and T=1, time complexity of assignments is O (1). Step 3.2: Complexity of loop is O (L) because L iterations are needed. Step 3.3: If tP = 0,T = Tth even number in the series of natural even numbers. If tP = 1,T = Tth odd number in the series of natural even numbers. Complexity is O (1) as it takes constant amount of times. Step 3.4: To set P=P-1. takes constant amount of time i.e. O(1). Step 3.5: Complexity to convert T into the corresponding stream of bits S = s0 s1 s2 s3 s4…sL-1, which is the source block is O(L) as this step also takes constant amount of time for merging s0 s1 s2 s3 s4…sL-1 .

So, overall time complexity of the entire technique is O(L).

Stream

Size (byte

s)

Chi-Square value

(TDES)

Chi-Square value

(Proposed ANNRPMS)

Chi-Square value

(RBCM CPCC)

Chi-Square value (RSA)

1500 1228.5803 2465.0645 2464.0324 5623.14 2500 2948.2285 5643.4673 5642.5835 22638.99 3000 3679.0432 6757.1533 6714.6741 12800.355 3250 4228.2119 6996.6177 6994.6189 15097.77 3500 4242.9165 10572.6982 10570.4671 15284.728

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VII. ANALYSIS OF RESULT From experimental results it is clear that the proposed

technique may achieve optimal performances. Encryption time and decryption time varies almost linearly with respect to the block size For the proposed algorithm Chi-Square value is very high compared to some existing algorithms. In the RBCMCPCC [1] a user input key (minimum length of eight byte) and a set of arbitrarily generated session keys are used to encrypt a source block. But this key generation technique is not sustainable against attack. Because RBCMCPCC key potency thoroughly depend upon strengthness of user input key. A user input key has to transmit over the public channel all the way to the receiver for performing the decryption procedure. So there is a likelihood of attack at the time of key exchange. To defeat this insecure secret key generation technique a neural network based secret key generation technique has been proposed.

The security issue of RBCMCPCC [1] and RPSP [10] algorithm can be improved by using neural secret sub key generation technique. In this case, the two partners A and B do not have to share a common secret but use their indistinguishable weights as a secret key needed for encryption. The fundamental conception of neural network based key exchange protocol [3,4,5,6] focuses mostly on two key attributes of neural networks. Firstly, two nodes coupled over a public channel will synchronize even though each individual network exhibits disorganized behavior. Secondly, an outside network, even if identical to the two communicating networks, will find it exceptionally difficult to synchronize with those parties, those parties are communicating over a public network. An attacker E who knows all the particulars of the algorithm and records through this channel finds it thorny to synchronize with the parties, and hence to calculate the common secret key. Synchronization by mutual learning (A and B) is much quicker than learning by listening (E) [3].

For usual cryptographic systems, we can improve the safety of the protocol by increasing of the key length. In the case of neural cryptography, we improve it by increasing the synaptic depth L of the neural networks. For a brute force attack using K hidden neurons, K*N input neurons and boundary of weights L, gives (2L+1)KN possibilities. For example, the configuration K = 3,L = 3 and N = 100 gives us 3*10253 key possibilities, making the attack unfeasible with today’s computer power. E could start from all of the (2L+1)3N initial weight vectors and calculate the ones which are consistent with the input/output sequence. It has been shown, that all of these initial states move towards the same final weight vector, the key is unique. This is not true for simple perceptron the most unbeaten cryptanalysis has two supplementary ingredients first; a group of attacker is used. Second, E makes extra training steps when A and B are quiet [3, 4, 5]. So increasing synaptic depth L of the neural networks we can make our neural network safe.

VIII. CONCLUSIONS This paper presents a novel approach for generation of

secret key proposed algorithm using neural cryptography. This technique enhances the security features of the RBCMCCC and RPSP algorithm by increasing of the synaptic depth L of the neural networks. In this case, the two partners A and B do not have to exchange a common secret key over a public channel but use their indistinguishable weights as a secret key needed for encryption or decryption. So likelihood of attack proposed technique is much lesser than the simple RBCMCCC algorithm.

ACKNOWLEDGMENT The author expresses deep sense of gratitude to the IIPC

Project of AICTE, of the department of CSE, University of Kalyani where the computational work has been carried out.

REFERENCES [1] Jayanta Kumar Pal, J. K. Mandal “A Random Block Length Based

Cryptosystem through Multiple Cascaded Permutation Combinations and Chaining of Blocks”Fourth International Conference on Industrial and Information Systems, ICIIS 2009, 28 – 31December2009, SriLanka.

[2] J. K. Mandal, et al, “A 256-bit Recursive Pair Parity Encoder for Encryption”, Advances in Modeling, D; Computer Science & Statistics (AMSE), vol. 9, No. 1, pp. 1-14, France, 2004.

[3] T. Godhavari, N. R. Alainelu and R. Soundararajan “Cryptography Using Neural Network ” IEEE Indicon 2005 Conference, Chennai, India, 11-13 Dec. 2005.ggggjgggggggggggg

[4] Wolfgang Kinzel and ldo Kanter, "Interacting neural networks and cryptography", Advances in Solid State Physics, Ed. by B. Kramer (Springer, Berlin. 2002), Vol. 42, p. 383 arXiv- cond-mat/0203011.

[5] Wolfgang Kinzel and ldo Kanter, "Neural cryptography" proceedings of the 9th international conference on Neural Information processing(ICONIP 02).h

[6] Dong Hu "A new service based computing security model with neural cryptography"IEEE07/2009.J

[7] Jha P. K., Mandal, J. K., “Encryption Through Cascaded Recursive Key Rotation of a Session Key withTransposition and Addition of Blocks (CRKRTAB)”Proceedings of National conference on Recent Trends in Information Systems 2006, organized by IEEE Kolkata section, Jadavpur University, Kolkata held on July 14-15th, 2006. pp 69-72.

[8] Jha P. K., Nayak, A.K., “An Encryption Technique through Bitwise Operation of Blocks (BOB)” accepted tothe 97th Indian Science Congress, to be held in Kerala, Jan 2010, India.

[9] Atul Kahate, Cryptography and Network Security, 2003, Tata McGraw-Hill publishing Company Limited, Eighth reprint 2006.

[10] Saurabh Dutta, Ph. D. Thesis “An Approach Towards Development of Efficient encryption Techniques” The University of North Bengal, India, 2004.

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