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Research on application of image encryption technology based on chaotic of cellular neural network.

1. Introduction

With the development of the Internet, people's life becomes more and more inseparable from the Internet. Everyday people transfer large amounts of information on the Internet, including texts, audios, videos, images and so on. The 21st century is the era of information and technology. We can say that more information you mastered can decide to your success. Enterprises in the market of choice-making are not so much market share rather than more information. Throughout our history, many examples can be often seen. Those countries suffer from disasters since the lost of information. Therefore, security and confidentiality of information is not only related to the privacy of personal communication, but is a key factor for corporate development and national security. Nowadays the digital images have been widely applied to all kinds of fields of our lives. Thus, the importance of digital image and security is particularly important.

Digital image becomes one of the important means for people to express the information because of its features of the large amount of data, the relevant features, and vivid expression. The digital image encryption has been unable to meet the requirements by using the methods of traditional text encryption merely. Hence, a new approach of image encryption technology is of great significance. According to the digital image encryption problems, a lot of literatures propose different encryption methods, yet a new chaotic encryption method rising in recent years is an effective method for the new image encryption.

2. Chaotic of Cellular Neural Network

2.1 The Mathematical Model of Cellular Neural Network

The cellular neural network is a neural network with local interconnection and proposed firstly by Chua in 1988 [1]. The basic unit of CNN called cells, each cell is composed of a linear resistor, a capacitor and a voltage controlled current source. There are M x N cells in cellular neural network and C (i, j) means the i rows and j columns of cells. Figure 1 shows a cellular neural network on a 4 x 4 scale. Figure 2 shows the equivalent circuit of each cell in the network. [2-4]

The cellular neural network is a large-scale nonlinear analog circuit with the high speed and parallel signal processing. And the ultra large scale integrated circuit (VLSI) is easily realized implementation with the structure of its rules and partial connection properties. CNN is a flexible and effective neural network. Researches show that high dimensional CNN can generate more complex, real-time chaotic behavior and more controllable parameters. Therefore, we can use the nonlinear dynamic characteristics and improve the security of communication system. So the high dimensional cellular neural network will play an important role in the field of secure communication. Cellular neural networks can produce a kind of chaos phenomenon which is a complex nonlinear dynamic system and is similar to random process of nonlinear systems with certainties. Chaos is particularly applicable to secure communication technology with its characters of the noise, the initial value sensitivity and the long-term unpredictability.

According to statistics result, 99.13% news story has a news title, which is the simplest and accuratest content summarization. It usually appears in a white rectangular window, whose position is at the height of 1/12 ~ 1/4 screen. Its content covers one or two rows (Figure.2). The time of news title appearing is not fixed, maybe it appears at the position of anchorperson, maybe it appears at the position of interpreter.

Cellular neural networks can produce a kind of chaos phenomenon which is a complex nonlinear dynamic system and is similar to random process of nonlinear systems with certainties. Chaos is particularly applicable to secure communication technology with its characters of the noise, the initial value sensitivity and the long-term unpredictability.

According to statistics result, 99.13% news story has a news title, which is the simplest and accuratest content summarization. It usually appears in a white rectangular window, whose position is at the height of 1/12 ~ 1/4 screen. Its content covers one or two rows (Figure.2). The time of news title appearing is not fixed, maybe it appears at the position of anchorperson, maybe it appears at the position of interpreter.

2.2 State equation of cellular neural network

According to the equivalent circuit diagram of each cell, we can see that each cell C (i, j) has a constant input [u.sub.ij], a threshold [z.sub.ij], a state [x.sub.ij] and one output [y.sub.ij]. State equation of cell can be expressed by first-order nonlinear differential equations.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Output Equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Input equation:

[u.sub.ij] = [E.sub.ij]

Constraint:

[absolute value of [u.sub.ij]] [less than or equal to] 1[x.sub.ij](0) [less than or equal to] 1

Circuit parameters

C > 0, [R.sub.x] > 0

Where: [C.sub.ij] the cell, [x.sub.ij] the initial state;

[E.sub.ij] An independent voltage source, C : a linear capacitor, [R.sub.x] : A linear resistor,

[A.sub.kl] The weights between the cell [C.sub.kl] and output [y.sub.kl] of the adjacent cell [C.sub.kl], and that is also called feedback template;

[B.sub.kl] The weights between the cell [C.sub.kl] and input [u.sub.kl] of the adjacent cell [C.sub.kl], and that is also called control template;

In the template A and B: [A.sub.i,j] (or [B.sub.i,j]) represents the self feedback weight of [C.sub.i,j] and also is the central element of feedback template A (or control template); [A.sub.kl] : (or [B.sub.kl]) represents the feedback weight of In the pro domain of the cell [C.sub.i,j] and also is other elements outside the center element among feedback template (or control template B). The values of k, l follows the definition of B Pro domain.

3. Image Encryption Technology Based on Chaotic

of CNN

3.1 Brief Introduction of CNN Chaotic System

This paper study the five dimensional chaotic systemybased on the Chaotic of cellular neural network. [5-7] The above content introduces the mathematical model of cellular neural network detailed. For the sake of convenience, this article introduced the CNN cell model which is simplified and generalized. The nonlinear state equation of dimensionless quantity is

[d.sub.ij]/dt = [-x.sub.j]+[a.sub.j]f([x.sub.j])+[G.sub.o]+[G.sub.s]+[[??].sub.j]) (3) f([x.sub.j]) = 1/2 ([absolute value of [x.sub.j]+1] - [absolute value of [x.sub.j]-1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Where: J is a cell mark. [x.sub.j] is a state variable. [a.sub.j] is a constant. [[??].sub.j] is the threshold. [G.sub.o] is a linear combination of output variable of the connected cell. [G.sub.s] is a linear combination of state variable of the connected cell. f([x.sub.j]) is related to the output of the cell and state of the circuit.

According to the formula (3) and (4), five order generalized CNN dynamic model equations can be obtained.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The corresponding Cell parameters are defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Take the parameters into the formula (5), and equation (6) can be obtained.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The above equation is solved by Using the four order Runge-Kutta algorithm, where

[x.sub.j](0) = (0.2, 0.2, 0.2, 0.2)

Then, chaotic attractors generated by the five order full interconnection CNN system are shown in Figure 3.

3.2 Image encryption based on cellular neural network

* Flow chart of image encryption algorithm desige

* Encryption algorithm process

Step 1: In this paper we select the five-dimensional chaotic sequence [x.sub.i] that is generated by chaotic systems and select the digits of the variables in the fixed position. The case selects number 250th of A and B as the keys of key1 and key2. This method makes up the flaws, whose is the key is easy to lose and not easy to manage.

Step 2: Firstly image named X image is readed into M x N matrix. According to the logistic mapping, two groups of chaotic sequences of a and b can be produced by the key of key1 and key2. A typical logistic chaotic sequences are selected in this paper, where l=4,Method is as follows.

[x.sub.n+1] = [lambda][x.sub.n] (1 - [x.sub.n]) [x.sub.n] [member of] [0, 1] (7)

Step 3: According to a certain weight (such as 4:6), two groups of chaotic sequences are superimposed and rounding and the values are limited between 0 and 256. Finally the scrambling matrix e is obtained. Algorithm is

e = round (256 x (0.4 x a + 0.6 x b)) (8)

Step 4: The original image is encrypted. The encrypted image is generated by the method of superposition of the corresponding elements between parameter matrix and the scrambling matrix e in certain weight (such as 1:99). That is

Yimage = 0.01 x Ximage x (1 - 0.01) x e (9)

Step 5: The encrypted image is decrypted. Decryption is the reverse process of encryption.

3.3 The simulation results

The plaintext image in figure 5 (a), the cipher text image in figure 5 (b) and the encrypted image are obtained by the above algorithm. Images are as follows.

3.4 Safety analysis of experimental results

3.4.1 Correlation Analysis

The image is composed of a plurality of pixels, and the image can be completely or partially revealed because of a relationship between pixels. The pixel position of encrypted image changes, and the correlation between pixels will cease to exist. In order to verify the correlations between the plaintext image and encrypted image pixels, 1000 adjacent pixels in the image are selected for comparison randomly in the horizontal direction, vertical and diagonal directions respectively. The calculation method of the correlation coefficient between the adjacent pixels is given in the formula (10) and the experimental results are shown in Table 1.

[[rho].sub.xy] = E(xy) - E(x) E(y) / [square root of (D(x))] [square root of (D(y))] (10)

Data in Table 1 show that the correlation coefficient is very great between adjacent pixels of the plaintext image, and the correlation coefficient is almost close to 1.

However, the correlation coefficient of the encrypted image almost tends to 0, and the adjacent pixels of the encrypted image is not related basically. Therefore, according to the analysis of correlation coefficient, we can draw the conclusion that the effect of image encryption is much better, because the statistical characteristics between pixels of the plaintext image have ceased to exist in the cipher text image.

3.2.2 Analysis of Scrambling Degree

One of the most important standards to measure the effect of encrypted image is scrambling degree. [8-11] Generally scrambling degree (SM) is defined to evaluate the degree of the scrambling image. its computational formula is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Where X = [{[x.sub.ij]}.sub.mxn] is the original image. [??] = [{[[??].sub.ij]}.sub.mxn] is scrambling image. And R = [{[r.sub.ij]}.sub.mxn] is the noise image. which is the uniform distribution and is the same size with the original image. The MATLAB program is as follows.

sx = sum (sum ((Ximage - 256 x rand (M, N)). [and]2)); sy = sum (sum ((Ximage - Yimage). [and]2)); disp ('the output of scramblmg degree DD') SM = sy / sx

Finally, the calculated scrambling degree is given for SM = 0.8612 in this method. Then, we can draw the conclusion that the encrypted effect is obvious.

3.2.3 Histogram analysis

Histograms of the plaintext image and the ciphertext image are drawn in figure 6. We can see clearly in this figure that the histogram distribution of the encrypted image is more uniform than that of the plaintext image. It shows that the scrambling pixel values in the ciphertext are randomly dispersed, which can resists statistical analysis effectively.

3.2.4 Sensitivity

The algorithm depends on a secret key strongly. As long as it is not the original key, the cipher text image can not be decrypted, even if there is a difference of one fifth. key 1 = 0.7867 and key 2 = 0.5570 are used in this paper. Here we assume that key 1 = 0.7867+[10.sup.-16]. The simulation results are in Figure 7, and the decrypted image will not be the original one. So we can prove that the image is more secure.

4. Conclusion

Due to the wide application of the digital image in the network and the importance of image information security, digital image encryption technology has played an important role in the transmission of digital images safely. By the theory of CNN, the paper put forward a technology to encrypt image based on chaotic system which is generated by CNN. Because of the characteristics of chaotic phenomena of randomness and uncertainty, this paper selected a five dimensional chaos sequence which is produced by CNN as a key for the sake of managing the secret key easily. What's more, we improved the safety of image by using logistic chaotic mapping to generate a secret key. Finally, the encrypted image is obtained by scrambling pixels of the original image. The simulation results show that the scrambling degree of the encrypted image is high and the correlation between adjacent pixels is small, the anti-aggressively is stronger and the security is higher, through the analysis of the correlation between plaintext and cipher text image, we made a further proof that the encryption image has better effect. Therefore, it is better for the image to spread in the network after our encryption algorithm.

Received: 11 November 2012, Revised 13 March 2013, Accepted 19 February 2014

Reference

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[4] Yuming, Liu., Dongming, Zhou et al. (2007). Image encryption based on hyper Chaos of Cellular Neural Network, Journal of Yunnan University, 29 (4) 355-358 (In Chinese).

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[8] Yang, Liu., Yunlong, Ding. (2013). The Modeling of Genetic and Tabu Search Algorithm Based BP Neural Network in the Risk Analysis of Investment, Journal Of Digital Information Management, 12 (11) 391-399.

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[10] Pumputis, Vidmantas., Garbinius, Giedrius., Mironov, Valentin. (2013). Traffic Management Facilities Used At Intersection Of Ukmergs And Gele*inio Vilko Streets In Vilnius. Transport and Telecommunication, 14 (3) 187-195.

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Guo-dong Li, Guo-min Zhao (1), Wen-xia Xu (2), Sheng-zhuo Yao (3)

(1) School of Application Mathematic Xinjiang University of Finance & Economics Urumqi 830012, P.R. China

(2) Xinjiang Weather Modification Office Urumqi 830002, P.R. China

(3) Beijing university of civil engineering and architecture Beijing 100044, China lgdzhy@126.com

Table 1. The correlation coefficient of the plain
and cipher image between adjacent pixels of image

                Correlation     Correlation
Direction       Coefficient     Coefficient
                of the Plain   of the Cipher
                   Image           Image

Horizontal
direction          0.9689         0.0298

Vertical
direction          0.9045         -0.0021

Diagonal
direction          0.9198         0.0176
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Author:Li, Guo-dong; Zhao, Guo-min; Xu, Wen-xia; Yao, Sheng-zhuo
Publication:Journal of Digital Information Management
Article Type:Report
Date:Apr 1, 2014
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