An image authentication technique based on cross ...

2014 First International Conference on Computational Systems and Communications (ICCSC) | 17-18 December 2014 | Trivandrum

An image authentication technique based on cross chaotic map Rija M Raju

K. Gopakumar

M.Tech Communication systems, Dept. of ECE TKM College of Engineering Kollam, Kerala, India e-mail: [email protected]

Professor, Dept of ECE, TKM College of Engineering Kollam, Kerala, India e-mail: [email protected]

Abstract— Digital data transmitted through the internet can be copied and can be used in malicious ways. This may cause problems referring to the protection of intellectual rights and the integrity of the transmitted information. Watermarking has emerged as a new technology to resolve these problems, providing both copyright claiming mechanisms and integrity verification tools. A novel image authentication technique is proposed in this paper in which both discrete wavelet transform (DWT) and chaotic principle are combined effectively. The watermark is embedded using Haar wavelet transform and the cross chaotic map is generated by combining the non-periodical and non-convergence properties of Henon map and Arnolds Cat map. The map is run for a couple of iterations to generate a chaotic sequence, and the pixels of the watermarked image are encrypted with these sequences. The method proposed is efficient, fast and highly secure that can be used for digital data transmission. Keywords --- DWT; Watermarking; Chaos; Henon map; Cross Chaos

I. INTRODUCTION Chaos is a noise-like phenomenon existing in the deterministic systems that exhibit extreme sensitivity to initial conditions. Even though chaos was experimentally observed by Van der Pol [1] as early as 1927, the science of chaos is relatively new. One of the first models which were shown to exhibit chaotic behaviour in numerical solution was the fluid convection model introduced in 1963 by E.N.Lorenz [2] in his studies on atmospheric weather. Although Lorenz's discovery was an accident, it planted the seed for the new theory of chaos science. The sensitivity of chaos to the initial conditions is known as “butterfly effect”, discovered by Lorenz while observing the weather forecast. Since the self-synchronization property of chaos is demonstrated it has been widely used for image encryption. Earlier the encryption is done by modulating the orbits of continuous time dynamical systems. These schemes strongly urge for the control and synchronisation [3, 4, 5] of chaos however it is difficult to determine the synchronization time and the information transmitted during the transient time is lost. So new promising approaches are emerged in which the chaotic transformation is directly applied to the plain text or plain image. The sensitivity to the initial conditions and the

parameter mixing characteristics of chaos are beneficial to crypto systems. A. Chaotic maps Chaotic Map is a mathematical model used to understand and describe the phenomenon of chaotic motion. A map is a mathematical transformation that is applied again and again in sequence on a single point, so that the point gets mapped repeatedly to new locations. Each implementation of the mathematical equations is called an iteration of the map. Many standard chaotic maps are available and are used extensively in the field of image encryption. Chaotic maps [6] have attracted the attention of cryptographers as a result of the following fundamental properties: (1) Chaotic maps are deterministic, meaning that their behaviour is predetermined by mathematical equations. (2) Chaotic maps are unpredictable and nonlinear because they are sensitive to initial conditions. Even a very slight change in the starting point can lead to significantly different outcome. (3) Chaotic maps appear to be random and disordered but, in fact, they are not. Beneath the random behaviour there is an order and a pattern. II. WATERMARKING TECHNIQUES In general, a digital watermark is a code that is embedded inside an image. It acts as a digital signature, giving the image a sense of ownership or authenticity. More than 700 years ago, watermarks were used in Italy to indicate the paper brand and the mill that produced it. By the 18th century watermarks began to be used as anti-counterfeiting measures on money and other documents. The name was probably given because the marks resemble the effects of water on paper. Digital watermarking techniques [7] can be classified into two categories: Spatial and Transform domain methods. In spatial domain based watermarking scheme, watermark is inserted into the least significant bits of pixels. However this scheme is less secure and the watermark can be easily destroyed. In the case of transform based schemes, the algorithms modify the DFT magnitude and phase coefficients to embed watermarks. Another watermarking scheme was proposed by Chittaranjan Pradhan et al [8] for image authentication using cross chaotic map. According to this method, the watermarked image is Arnolds Transformed and

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2014 First International Conference on Computational Systems and Communications (ICCSC) | 17-18 December 2014 | Trivandrum

then encrypted using a cross chaotic map. Although this method is robust it does not provide a greater key space to enhance the security. The method proposed here eliminates these drawbacks. It has the following steps: A. Watermark Embedding B. Chaotic Encryption of Watermarked Image C. Chaotic Decryption of Watermarked Image D. Watermark Extraction Algorithm A. Watermark Embedding Here discrete wavelet transform (DWT) is used for embedding watermark. The image is analysed using DWT [9]. After one-level decomposition the image is divided into four corners, upper left corner of the original image (LL), lower left corner of the vertical details (HL), upper right corner of the horizontal details (LH), lower right corner of the component of the original image detail (HH). This is illustrated in Fig. 1. Fig. 2: Block diagram of watermark embedding algorithm

Fig. 1: DWT decomposition model

After wavelet decomposition process, the low frequency component (LL) is equivalent or similar to the original image and this component is used for watermarking. The human eye is more sensitive to low frequency components, thus adding a watermark to high frequency components make it insensitive to human eye. But the watermark added to the high frequency part of the image may be lost during compression. On this basis, watermark image is multiplied by the weighted coefficients to solve the visual distortion, and to enhance the robustness of the watermark. The entire process is detailed in the block diagram shown in Fig. 2.

Fig. 3: Block diagram of chaotic encryption of watermarked image

(1) Selection of Chaotic map - 4D cross chaotic map: The image encryption schemes based on one dimensional chaotic map has less key space and lower security. The security can be enhanced by utilizing higher dimensional maps. In the proposed algorithm 4D cross chaotic map is used which improves security and reduces calculation complexity. The 4D cross chaotic map is implemented using Henon map and Arnold’s Cat map. It is defined [10, 11] in Eq. 1.

B. Chaotic Encryption of Watermarked Image The watermarked image is encrypted using the cross chaotic maps. It involves the following steps: (1) Selection of the chaotic map (2) Pixel encryption The entire process is represented by the block diagram shown in Fig.3

(1)

x, y ,p, q € [-1,1] where a, b and N are the control parameters of the system. When a =1.6 and b= 0.3, the system exhibits a great variety of dynamics behaviour.

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2014 First International Conference on Computational Systems and Communications (ICCSC) | 17-18 December 2014 | Trivandrum

The sequence “d” for Arnolds Transformation is obtained from the iterations of a logistic map. (2) when system parameter μ is 3.925 the logistic map is in the chaotic state and the sequence thus obtained is highly chaotic. This sequence is highly randomized when it is Arnold’s transformed. According to Golomb's three postulates [12] for pseudorandom sequence, ideal chaotic sequences should have following statistical properties: 1. The average value is zero. The computation using MATLAB gives mean x is 0.0273, y is 0.0018, p is 0.0196, q is 0.0250. Since the mean is approximately zero it can be concluded that its equality distribution property is good. 2. The autocorrelation should be a delta function. The autocorrelation computed using MATLAB for Eq. (1) is shown in Fig.4. From the figure it can be seen that the autocorrelation is an approximate delta function.

Fig. 5: Evolution of Largest Lyapunov Exponent

Fig. 6: Bifurcation diagram for sequence x with b=0.3.

Fig. 4: The autocorrelation Characteristics

3.

4.

The Lyapunov Exponent should be positive. The computed [13] value is shown in Fig. 5. The system is chaotic because most of the Lyapunov Exponents are above zero level with the largest value being 1.5510. Bifurcation diagram The limiting behaviour of orbits for values of “a” in the range 0 < a < 2 is given by a diagram which is known as bifurcation diagram. By keeping “b=0.3” the parameter “a” is varied, and the bifurcation diagram is plotted in the MATLAB, from the Fig. 6, it can be observed that the system has a highly chaotic behaviour when a > 1.3. After the onset of chaos, slight variations in the initial condition yields dramatically different results over time.

(2) Pixel encryption: Since the chaotic sequence generated are real valued, they cannot be directly applied to image pixels. Therefore the sequence is multiplied with adequate weights and the encryption matrix K is obtained [12]. Suppose P denotes an image of the size M x N, and Pi,j is the gray scale level of a pixel at position i and j, where 1 ” i ”M , 1 ” j ” N, 0” Pi,j”255 and M and N resized to 400.The image is encrypted using formulae given in Eq. 3. (3) where 1”x”M and 1”y”N. C. Chaotic Decryption of Watermarked Image The chaotic image decryption process is the inverse of the encryption process. The process is illustrated in Fig. 7 D. Watermark Extraction Algorithm The extraction algorithm process is the inverse of embedding process [Fig. 2]. DWT is applied on watermarked image and original image to generate approximate and detailed coefficients separately. From coefficients of watermarked image and original image, watermark is extracted.

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2014 First International Conference on Computational Systems and Communications (ICCSC) | 17-18 December 2014 | Trivandrum

IV. SECURITY ANALYSIS Securely encrypted image should resist all statistical attacks and known plain text attacks. Security analysis has been performed and the results are compared with the 1D Logistic maps [14] and 2D Cross Chaotic maps [12] based image encryption schemes. The 2D cross chaotic maps are generated by mixing the properties of Logistic Map and Chebyshev map.

Fig. 7: Block diagram of chaotic decryption of watermarked image

A. Statistical analysis 1. Histogram Analysis: Histogram is the pictorial representation of distribution of data. It shows the diffusion properties in the encrypted data. The histogram of the plain image Lena is shown in Fig. 10(a) and the histogram of the encrypted image using 1D Logistic maps, 2D Cross chaotic maps, 4D Cross chaotic maps are shown in Fig. 10(b), 10(c) and 10(d) respectively. On comparison it is found that encrypted image using 4D cross chaotic maps is fairly uniform and is significantly different from that of the original image, and that the encrypted images transmitted do not provide any suspicion to the attacker, which can strongly resist statistical attacks.

Fig. 8: Block diagram of watermark extraction algorithm

III. SIMULATION RESULTS The test image for chaotic encryption is the Lena image selected from the internet. The image encryption is done using a laptop with Intel P5, 3 Ghz processor and with 1 GB RAM. MATLAB 7.0 platform is used to perform the computations. The results are depicted in Fig. 9.

Fig. 10: Histogram Analysis

2. Information Entropy: It is well known that the entropy H [15] of symbol source S is (4)

Fig. 9: Simulation results for watermarking using 4D Cross Chaotic Maps

where p(si) represents the probability of symbol si. Information entropy of image shows the distribution of gray scale value. If the gray scale values of image are uniformly distributed its information entropy will be greater. An image with entropy 8 is said to be ideally encrypted. Lower the value of entropy the more easily image is decrypted. The information entropy of encrypted image is computed in Matlab for 1D logistic maps, 2D cross chaotic maps and 4D cross chaotic maps using Eq. 4 and it found to be 7.9904, 7.4280,7.9924 respectively. Since the value approaches the

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2014 First International Conference on Computational Systems and Communications (ICCSC) | 17-18 December 2014 | Trivandrum

ideal value, it can be concluded that information leakage is minimum for 4D maps. 3. Correlation of adjacent pixels: The correlation between adjacent pixels in plain and encrypted image can be computed using the Eqs. (5-8) for the coefficients in horizontal, vertical and diagonal [15]

However, the correlation coefficient of adjacent encrypted images is approximately equal to 0 and little or no correlation. The watermarked image using the 4D cross chaotic maps shows the correlation compared to the other two.

pixels in they have encrypted minimum

TABLE I. CORRELATION COEFFICIENTS OF ADJACENT PIXELS IN PLAIN AND ENCRYPTED IMAGE

(5) (6) (7) (8) where x and y are gray scale values of adjacent pixels in the image, E(x) is the estimation of expectations of x, D(x) is the estimation of variance of x, and COV(x,y) is the estimation of covariance between x and y. About 2000 pairs of adjacent pixels in vertical, horizontal, and diagonal direction are selected randomly and the correlation coefficients are calculated respectively (TABLE I and Fig. 11).

4. MMSE and PSNR: The performance of the watermarked image can also be evaluated on the basis of Peak Signal to Noise Ratio (PSNR) and Minimum Mean Squared Error (MMSE). PSNR greater than 30 db is considered to have an acceptable quality in which watermark cause no alteration to the quality of image [13]. Higher the PSNR, better is the quality of watermarked image. The Eqs [9-10] can be used for finding PSNR and MMSE [16] (9) (10) The results are tabulated in TABLE.II. From the results it can be concluded that for the 4D cross chaotic map encryption scheme has the maximum PSNR thus the quality of the image is better. TABLE II. PSNR OF ORIGINAL AND WATERMARKED IMAGE

Fig.11: Correlation Analysis of adjacent pixels in plain and encrypted image

On analysing data from TABLE I, it can be seen that adjacent pixels from plain images have correlation coefficients approximately equal to unity and they are highly correlated.

B. Key Space Analysis The key space is large enough to resist all kinds of brute force attacks. Figure 12(c) corresponds to the decryption obtained with a=1.6 and b=0.3. The same decryption is carried out with a=1.6 and b=0.300000000001 the result of which is shown in Figure 12(d). There is considerable difference between results shown in Fig. 12(c) and 12(d). It demonstrates that the proposed scheme is very sensitive to the secret key. If the decryption key is slightly changed, the decrypted image will be greatly different from the original plain image. The parameters μ, a, b, N or initialization vector {d0, q0, y0} can

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2014 First International Conference on Computational Systems and Communications (ICCSC) | 17-18 December 2014 | Trivandrum

be key values. Therefore, the key space can reach to (1016)7 with 16 bit precision [17]. This encryption scheme provides a large keyspace. An image encrypted with such a large key space is sufficient for resisting various brute-force attacks.

Fig.12: Key Sensitivity analysis

V. CONCLUSIONS In the proposed watermarking method, a gray-scale visual watermark image is inserted into the host image using the Haar Wavelet Transform. The watermarked image is then encrypted using 4D cross chaotic sequences. It utilized the randomness of the chaotic maps to encrypt the image. The pixel position was changed according to the randomness of the chaotic element generated from the chaotic map. The slightest difference in the parameter and initial value made decryption impossible. The experimental results for encrypting the image using 4D cross chaotic maps have been obtained and it confirmed that this technique is robust with high fidelity.

Wavelet Domain” IEEE Conference on Image and Graphics, 2007 (ICIG), pp. 329 – 336, 2007. [8] Chittaranjan Pradhana, Shibani Rathb, Ajay Kumar Bisoi, “Non blind digital watermarking technique using DWT and Cross chaos”, Procedia Technology, vol. 6, pp. 897 – 904, 2012. [9] Stephane G. Mallat, “Multifrequency channel decompositions of images and wavelet models”, IEEE Transactions on Acoustics. Speech. and Signal Processing, vol. 37 , pp. 2091 – 2110, 1989. [10] S. V.Gonchenko , J. D.Meiss , I. I. O vsyannikov, “Chaotic dynamics of three-dimensional Henon maps that originate from a homoclinic bifurcation” Regular And Chaotic Dynamics, vol 11, pp. 191–212, 2006. [11] Fei Chen, Kwok-Wo Wong, Xiaofeng Liao, Tao Xiang, “Period Distribution of the Generalized Discrete Arnold Cat Map for N = 2e ”, IEEE Transactions on Information Theory, pp. vol. 59, pp. 3249 – 3255, 2012. [12] Ling Wang, Qun Ye, Yaoqiang Xiao, Yongxing Zou, Bo Zhang, “An Image Encryption Scheme Based on Cross Chaotic Map”, IEEE Congress on Image and Signal Processing, pp. 22 – 26, 2008. [13] Michael T. Rosenstein, James J. Collins, and Carlo J. De Luca “A practical method for calculating Largest Lyapunov Exponent from small data sets”. [14] K. Gopakumar, Rija M Raju "A Novel Watermarking Technique Based on Chaos Theory", International Journal of Applied Engineering Research, vol-8, pp. 1081 - 1089, 2013. [15] Gelli M B, S. S. Kumar, V. Chandrasekaran, “A novel image encryption scheme using Lorenz attractor”, IEEE Conference on Industrial Electronics and Applications, pp.3662 – 3666, 2009. [16] Welstead, Stephen T., “Fractal and wavelet image compression techniques”, SPIE Publication, pp. 155–156, 1999. [17] G R Chen, Y B Mao, K Charles, “A symmetric image encryption scheme based on 3D chaotic cat maps”, Chaos, Solitons and Fractals, vol.21, pp.749-761, 2004.

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