Efficient Arnold and Singular Value Decomposition based Chaotic Image Encryption

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1 Efficient and Singular Value ecomposition based Chaotic Encryption Ashraf Afifi epartment of Computer Engineering Computers and Information Technology college Taif University, Al-Hawiya 974, Kingdom of Saudi Arabia Abstract This paper proposes an efficient image encryption that is based on transform AT) and the Singular value decomposition SV). The proposed method employs AT on a plain image to transpose all image pixels in the positions, then a diffusion process is applied to the resulted encrypted image via SV decomposing into three segments. The decryption process aims to derive the plain image from the cipher image. Matlab simulation experiments are done to examine the suggested method. The achieved results show the superiority of the suggested approach with respect to encryption quality. Keywords Encryption arnold transform; singular value decomposition; chaotic image encryption I. INTROUCTION Nowadays, the internet and multimedia networks have captured attention in information security researches. Encryption is employed to achieve security. encryption is used increasingly in military, communication networks, medical image applications [-6]. The security of multimedia data which have a high relationship among neighboring pixels has drawn a great attention, recently. The conventional data encryption techniques like AES, IEA, Triple-ES, and other symmetric ciphering techniques are well known but they may be unsuitable for efficient image ciphering [7]. The main characteristics of chaotic techniques are their high sensibility to control parameters and initial conditions, thus their features can be exploited for achieving the required cryptographic characteristics. In 998, Fridrich suggested the primary public framework for chaos-based image ciphering. This framework is made up of diffusion and confusion mechanisms [8]. Firstly, the pixels of image are shuffled by employing a chaotic map such as cataker, and standard maps. After that, the values of pixels are sequentially altered utilizing a specific discretized chaotic map through the diffusion mechanism. The Fridrich s framework has been considered the most common architectures in different proposed chaos-based image ciphering techniques [9-]. This paper is mainly focusing on the communications applications with high security levels using AT based SV security schemes. This scheme is worked by transposing the plain image. Then, the original image will be independently AT then decomposed into three matrices via SV. The remainder of the paper has been organized as follows: Section covers the methodology and the main tools employed in the proposed method, namely the AT and SV. Section 3 is devoted to detail the enciphering and deciphering phases of the suggested AT SV image cipher. Section 4 explores the AT SV of the image cipher detailed security study. Finally, Section 5 presents the conclusions. II. METHOOLOGY This section presents a literature survey on the AT and SV which were used for image encryption method. A. The AT The AT is defined as the Cat s mapping [9]. It aims to shift the pixels positions instead of changing their estimates. Recently, it was employed for image ciphering and watermarking [-5]. The AT of a pixel a ) of an image ' ' f a, b) of size NXN pixels is defined by f a ) and can be expressed mathematically: ' a ' AT a, b), N) b Where a a' and b b' a mod b N) ) represent the initial and the position of pixel of shifted image, respectively. Mod defines the modular arithmetic operation. The parameter N is the target image size, which is used to determine the period of AT. The AT period is determined by [6]: p Period min[ p :[ AT f a, b), N)] f a, b)] Where min defines the minimum value and p defines the number of iterations. The number of times AT is represented fixed at different values along with different N for improving the image ciphering security [0]. B. The SV The SV is one of the best fit and reliable techniques for matrix decomposition employed in linear algebra. This analogous to the Hermite matrix or symmetry matrix employing a background of eigenvectors. Such method is not only efficient but also stable in decomposing the image into a collection of linearly independent segments, each of which has its energy contribution [7]. Regarding mxn matrix, orthogonal matrices U and V exist, each with mxn elements, respectively. ) 45 P a g e

2 The SV of X is defined as: T X U * S * V 3) S diag,... ),,,... ) Where where i i are the singular values of the matrix X with min m, n) and satisfying.... The right singular vectors are the first columns of V and the left singular vectors are the first columns of U. The SV technique can be applied in digital image cipher and watermarking. The image can be split into three segments then secure them in a variety of ways so that only at the time all the three image segments come together and are multiplied with the right order the information could be retrieved [8-0] III. THE SUGGESTE AT-SV CIPHERING METHO In this section, the suggested AT- SV image cryptosystem will be defined in terms of two basic processes namely the encryption and decryption. A. Encryption Process This process can be listed in the following steps: The clear image is independently shuffled employing the AT. The scrambled plain image is decomposed into three encrypted segments with SV into USV as illustrated in Fig. a). B. ecryption Process This process can be listed in the following steps: The three encrypted images are firstly multiplied by applying the SV. The inverse AT is implemented to the resulted ciphered image to retrieve the decrypted image as illustrated in Fig. b). IV. SIMULATION RESULTS AN PERFORMANCE ANALYSIS Several measuring tests are carried out to examine the proposed AT- SV image cryptosystem. Also, its performance is compared with the AT. Tests are performed using 5x5- sized Girl, Peppers and Baboon images as illustrated in Fig.. The ciphering outcomes after employing the proposed AT- SV image cryptosystem and conventional AT are shown in Fig. 3 Girl, Peppers and Baboon images respectively. It is shown that encrypting with the suggested AT- SV image cipher succeeded in all images concealment in details. A. Information Entropy The information entropy determines the expected entropy value included for the ciphered image. The information entropy is defined as [-3]: N H K) P K ) logp K ) i i i where P K i ) represents the probability of symbol. It is shown that the image is good if it has a high estimation of entropy. The entropy information estimations for the cipher images uses the suggested AT- SV image cipher and AT as depicted in Table I. The information entropy results, illustrate the efficiency of the suggested AT- SV image cipher when compared to AT cipher. Finally, the entropy outcomes of ciphered image channels resulted by the proposed AT-SV image cipher is the same with their comparing values in AT plain images. a) Girl b) Peppers c) Baboon Fig.. Test s - Girl, Peppers and Baboon. 4) Encryption Scheme a). AT-SV Encryption Block iagram. Fig. 3. Ciphered Results of Girl, Peppers and Baboon s Employing and the Proposed Cryptosystem. Fig.. AT-SV ecryption Block iagram. 453 P a g e

3 TABLE I. ENTROPY OUTCOMES OF CIPHERE GIRL, PEPPERS AN BABOON IMAGES EMPLOYING ARNOL AN THE PROPOSE ARNOL SV IMAGE CRYPTOSYSTEM Encrypted image with and Girl Peppers Baboon B. Histogram Test The histogram test is performed to ensure that the suggested AT- SV image cipher. For better encryption, the ciphered image s histograms should be completely different from the plain image s histograms. Fig. 4 illustrates the histograms of Girl, Peppers and Baboon images and the histogram of their ciphered versions. The histograms of ciphered Girl, Peppers and Baboon plain images are completely distinguishable from the Girl, Peppers and Baboon plain images histograms. C. Encryption Quality Results The correlation coefficient Cc), the histogram deviation H ) and irregular deviation I ) and, are calculated for comparing the quality of different ciphered images. The correlation coefficient r I mg, E ) is estimated between the nt original and ciphered image that arranged like - sequences as [-3]: r I mg, E nt ) cov I mg, E nt ), I mg ) E nt ) L cov I mg, E nt ) I mg l) Mean I )) E nt l) Mean E nt )), L i l L I mg ) I mg l) Mean I mg )), L i l L E nt ) E nt l) Mean E nt )), L i l where L is the pixel numbers within the source image. The aim is to get small Cc estimations between the original image I x, y ) and cipher E x, y ) image. Table II shows mg i nt i the Cc estimates among the sourse image and encrypted image for AT and the suggested AT- SV image cipher. The outcomes proof that the suggested AT-SV image cipher achieve Cc values that are close to ones achieved by AT in Girl, Peppers and Baboon image. This archives the success of the proposed AT- SV image cipher with respect to Cc experiment. 5) 6) 7) 8) The H calculates the encryption quality between the source and the cipher images. The H can be computed as [-3]: H I, E) 55 d i) i0 MxN, where d i) is the absolute difference amplitude value among the enciphered and the source image histograms of at level i. The estimates M and N resemble the plain image dimensions. The main obect is to verify higher H value confirming the encrypted images are deviated from their corresponding image. Table III illustrates the H experiment calculations for the original and ciphered image using AT and the proposed AT- SV image cipher. The results of the proposed AT- SV image cipher give larger H values compared with. Girl Pepper s Baboo n Original image Encrypted image with and Fig. 4. Histogram Results of the Original s and Encrypted using Proposed and. TABLE II. CC BETWEEN PLAIN AN ENCRYPTE IMAGES EMPLOYING SUGGESTE ARNOL AN ARNOL SV Encrypted image with and Girl Peppers Baboon TABLE III. HISTOGRAM EVIATION BETWEEN SOURCE AN CIPHER IMAGES USING IMAGES USING PROPOSE ARNOL AN ARNOL SV Encrypted image with and Girl Peppers Baboon ) 454 P a g e

4 The I characterizes the encryption quality in terms of deviation outcomes [-4]. The irregular deviation can be estimated as [-4]: I h d 55 h l) l0 d, MxN i) h l) M, 0) ) where the histogram of ciphered image at level i is h i), and M is the calculated average value for ciphered image an ideal consistent histogram. The main aim is to achieve lower I values that show a good quality of ciphered. Table IV show I estimates for AT and the suggested AT- SV image cipher. The suggested AT- SV image cipher has tiny Imd values against AT which conforms their achieved encrypted quality.. ifferential Result The differential test measures how the cipher images with AT and our AT- SV image cipher are affected by one-pixel modification. Two common measurements are used; unified average changing Intensity UACI) and number-of pixels changing rate NPCR). Assume two cipher images E and E having their source images with only one-pixel difference. In E and E, the pixel estimates at index a i ) are E a i ) and E a i ), respectively. In bipolar array a i ) with equal ciphered image sizes, the coefficients a i ) are calculated with E a i ) and E a i ) values. If E a i ) = E a i ), then a i ) ; otherwise a i ) 0. The NPCR can be calculated as [4-5]: i, a i ) NPCR E, E ) 00%, W H ) Where, W and H correspond to the width and the height of the ciphered image. The UACI can be calculated as [-4]: E a i ) E a i ) UACI E, E ) x 00%, i y M N 55 3) TABLE IV. IRREGULAR EVIATION BETWEEN SOURCE AN ENCRYPTE IMAGES USING IMAGES USING PROPOSE ARNOL AN ARNOL SV Encrypted image with and Girl Peppers Baboon The UACI computes the mean intensity of difference among the two ciphered images. The outcomes of the NCPR and the UACI values are shown in Table V. The outcomes ensure that the proposed AT-SV image cipher is too sensitive to little modificationsut the AT ciphering is less sensitive to small modifications in image. E. Noise Immunity Measure The robustness of the proposed AT-SV image cipher in the existence is measured in the process of deciphering. The PSNR The PSNR estimates the encryption image components. The PSNR is expressed as [4-5]: 55) PSNR I, ) 0log W H [ I x, y ) x, y )] i0 0 i i 4) where, I x i, y ) and x i, y ) are the gray level of the intensity value at location x i, y ) the original and of decipher image, respectively. High PSNR estimates show good resistance to noise. The noise resistance values are given in Tables VI and VII. The outcomes prove that the suggested AT- SV image cipher has good resistance to noise thus it can be a best choice for ideal telecommunication applications. The SSIM The SSIM can be computed as [5]: ) ) ) ) ) 5) TABLE V. ESTIMATIONS OF NPCR AN UACI FOR TWO ENCRYPTION IMAGES FOR TESTE IMAGES USING SUGGESTE ARNOL AN ARNOL SV Girl Peppers Baboon Encrypted image with and NCPR UACI NCPR UACI NCPR UACI TABLE VI. PSNR ESTIMATES OF IMAGES IN THE EXISTENCE OF WITH IFFERENT SNR SNR IN B) USING ARNOL SV PSNR 0 db 0 db 30 db 40 db Girl Peppers Baboon P a g e

5 TABLE VII. PSNR IMAGES ESTIMATES IN THE EXISTENCE OF WITH IFFERENT SNR SNR IN B) USING ARNOL SV Girl Peppers Baboon PSNR 0 db 0 db 30 db 40 db where, V, V are minor constants, and are the mean of and regions, respectively. is the variance of region and is covariance among two regions. The SSIM outcomes are given in Table VIII. The values prove that the suggested AT- SV image cryptosystem has better resistance to noise. The FSIM The FSIM permits to evaluate the deciphered image and can be definedas [5]: ) ) ) 6) where Ω is the special domain of image, ) corresponds the overall equivalance between ) and two images is phase congruency value. Large FSIM values make the noise immunity better. The FSIM measures outcomes are given in Table IX. The results prove that the AT- SV image cryptosystem is noise resistant. TABLE VIII. SSIM ESTIMATES IN THE EXISTENCE OF WITH IFFERENT SNR SNR IN B) USING ARNOL SV Structure Similarity Index SSIM) 0 db 0 db 30 db 40 db Girl 9.085* * Peppers.058* * * Baboon 9.77* * * TABLE IX. FSIM ESTIMATES IN THE EXISTENCE OF WITH IFFERENT SNR SNR IN B) USING ARNOL SV Structure Similarity Index FSIM) 0 db 0 db 30 db 40 db Girl * Peppers 8.76* Baboon 8.858* V. CONCLUSION An efficient chaotic encryption of images based on AT and SV is the main finding of this paper. In the proposed technique, the plain image is submitted to AT confusion and SV diffusion. For the decryption phase, the cipher image segments are composed by the SV diffuser and then inversely transformed by AT in order to derive the original image. A set of experiments have been performed in order to test the AT- SV image cryptosystem. The obtained outcomes prove the efficiency of the suggested AT- SV image cipher. REFERENCES [] R. Tao, J. Lang, Y. Wang, "Optical image encryption based on the multiple parameter fractional Fourier transform," Opt. Lett. 33, pp , 008. [] H.M. Ozaktas, A. Koc, I. Sari, M.A. Kutay, "Efficient computation of quadratic phase integrals in optics," Opt. Lett. 3, pp , 006. [3] Z. Liu, S. Liu, "Randomization of the Fourier transform," Opt. Lett. 3, pp , 007. [4] Z. Liu, S. Liu, "ouble image encryption based on iterative fractional Fourier transform," Opt. Commun. vol. 75, pp , 007. [5] Chengqing L, Shuun L, Asim M, Nunez J, Alvarez G, On the security defects of an image encryption scheme, Vis comp.; , 009. [6] Chengqing L, Lo K., Optimal quantitative encryptanalysis of permutation-only multimedia ciphers against plaintext attacks, Signal Process, 9:949-54, 0. [7] Li, S., Chen, G., Zheng, X.: Chaos-based encryption for digital images and videos. In: Furht, B., Kirovski,. eds.) Multimedia Security Handbook. CRC Press, Florida 004). [8] J. Fridrich, Symmetric ciphers based on two-dimensional chaotic maps, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 8, no. 6, pp , 998. [9] R. Ye, H. Huang, Application of the Chaotic Ergodicity of Standard Map in Encryption and Watermarking, I. J., Graphics and Signal Processing, 9 9, 00. [0] H. Liu, X. Wang, Color image encryption using spatial bit-level permutation and high-dimension chaotic system, Optics Communications, 84, , 0. [] R. Ye, A novel chaos-based image encryption scheme with an efficient permutation-diffusion mechanism, Optics Communications, 84, , 0. [] ZJ, Liu, I, Xu, T, Liu, H, Chen, P.F, Li, C, Lin, S.T., Liu. Optics Communications 84,3, 0. [3] W, Chen, C, Quan, C.J, Tay, Optics Communications, pp , 009. [4] X.P,eng,.M, Zhao, Optics Communications, pp , 0. [5] Z.J, Liu,M, Gong Y.K. ou, F, Liu, S, Lin, M.A. Ahmed, J.M. ai, S.T. S.T. Liu, Optics and Lasers in Engineering, pp , 0. [6] Vilardy JM, Torres CO, Jumenez, C ouble image encryption method using the transform in the fractional Hartley domain Proc SIE 8785, [8785R-8785/SR], 03. [7] Bhatnagar G, Wu QMJ, Raman B., SV based robust watermarking using fractional cosine transform, PIE, 7708, -, 00. [8] Gaurav B, Bhatnagar, Q.M, Jonathan WU, Selective image encryption based on pixels of interest and singular value decomposition SP ournal, vol., , 0. [9] Linfei Chen, aomu Zhao, Fan Ge, encryption based on singular value decomposition and transform in fraction domain Optics communication ournal, vol. 9, 98-03, 03. [0] Phool Singh, A.K.. Yadav, Kehar Singh, Phase image encryption in the fractional Hartley domain using transform and singular value decomposition Optics and Laser in Engineering ournal, vol. 9, 87-95, P a g e

6 [] Andrew B. Watson, Compression Using the iscrete Cosine Transform, Technical Report, NASA Ames Research Center, Mathematica Journal, 4), 994, p. 8-88, 994. [] J. Fridrich, "Symmetric Ciphers Based on Two-dimensional Chaotic Maps," International Journal of Bifurcation and Chaos, vol. 86), pp , 998. [3] Osama S. Faragallah, "igital Encryption Based on the RC5 Block Cipher Algorithm," Sensing and Imaging: An International Journal, vol. 3), pp , Springer, 0. [4] Osama S. Faragallah, "An Enhanced Chaotic Key-Based RC5 Block Cipher Adapted to Encryption," International Journal of Electronics, vol. 997), pp , Taylor & Francis, 0. [5] Ensherah A. Naeem, Mustafa M. Abd Elnaby, Hala S. El-sayed, Fathi E.Abd El-Samie, and Osama S. Faragallah, Wavelet Fusion for Encrypting s with Few etails, Computers and Electrical Engineering, vol. 60, pp , 06 AUTHOR S PROFILE Ashraf Afifi received the B.Sc.Hons.), M.Sc., and Ph.. degrees in Electronic and Communication Engineering from Zagazig University, Egypt, in 987, 995, and 00, respectively. He is currently Associate Professor with the epartment of Computer Engineering, Faculty of Computers and Information Technology, Taif University, Saudi Arabia. He is a coauthor of about 35 papers in international ournals and conference proceedings. His research interests cover communication security, image processing, and image encryption. 457 P a g e