A New Encryption Scheme for Color Images Based on

A New Encryption Scheme for Color Images Based on Quantum Chaotic System in Transform Domain Ahmed A. Abd El-Latif 1,2, Ning Wang1, Jia-Liang Peng1,3, Qiong Li1 and Xiamu Niu 1 1

School of Computer Science and Technology, Harbin Institute of Technology, 150080 Harbin China 2 Department of Mathematics, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt 3 Information and Network Administration Center, Heilongjiang University, Harbin, 150080, China. ABSTRACT

This paper presents an efficient image encryption scheme for color images based on quantum chaotic systems. In this scheme, a new substitution/confusion scheme is achieved based on toral automorphism in integer wavelet transform by scrambling only the Y (Luminance) component of low frequency subband. Then, a chaotic stream encryption scheme is accomplished by generating an intermediate chaotic key stream image with the help of quantum chaotic system. Simulation results justify the feasibility of the proposed scheme in color image encryption purpose. Keywords: Color image encryption; quantum chaotic system; integer wavelet transform.

1. INTRODUCTION How chaotic systems can be described in terms of quantum theory is so-called quantum chaos. It is emerged a new branch of physics from the efforts to understand the features of quantum systems which have chaotic deterministic dynamics in the classical limit. One of the aims of quantum chaos is the study of quantum versions of classical chaotic systems. Quantum chaos can be characterized by the sensitivity to parameters in the Hamiltonian that governs the chaotic dynamics [1]. This is an interesting property, which can be used in image crytosystems [2]. Image encryption is a challenging task due to the significant level of sophistication achieved by forgerers and other cybercriminals [3]. Advanced encryption methods for secure transmission, storage, and retrieval of digital images are increasingly needed for a number of military, medical, homeland security, and other applications. The scope of the present work is to utilize quantum chaotic system in color image encryption. This provides a great flexibility in both performance characteristics and security. In the presenet scheme, a new substitution/confusion scheme is achieved based on toral automorphism in integer wavelet transform by scrambling only the Y (Luminance) component of low frequency subband. Then, chaotic stream encryption scheme is accomplished by generating an intermediate chaotic key stream image with the help of quantum chaotic system. Simulation results justify the feasibility of the proposed scheme in color image encryption purpose. The rest of the paper is organized as follows. Section 2 gives the basics of the proposed scheme and the proposed scheme is presented in Section 3. The experimental results are given in Section 4, whereas the conclusion is shown in Section 5.

2. BASICS OF THE PROPOSED SCHEME 2.1 Quantum chaotic system In 1990, Goggin et al. [1] proposed a dissipative quantum logistic map by coupling the quantum kicked to bath of harmonic oscillators. They introduced a quasicontinuum model to describe the dissipation from the bath, and then studied the resulting expectation-value map. In order to study the effects of quantum correlations they wrote ˆ  ˆ  ˆ , where ˆ represents a quantum fluctuation about  ˆ  and obviously has the property:  ˆ  0 . Considering the following equation, that is similar to classical logistic map



Author’s e-mail: [email protected] ,{ning.wang, jialiang.peng, q.li}@ict.hit.edu.cn Fifth International Conference on Digital Image Processing (ICDIP 2013), edited by Yulin Wang, Xie Yi, Proc. of SPIE Vol. 8878, 88781S · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2031074

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†  ˆ i 1   ( ˆ i    ˆ i ˆ i ), where  is an adjustable parameter.

(1)

† ˆ ˆ  and  ˆ ˆ  , etc., have as the Goggin et al. [1] studied what effects correlations of the form   coupling to the bath is varied. Again, by write ˆ  ˆ  ˆ , as suggested in [1], so that Eq. (2) becomes 2 †  ˆ i 1   ( ˆ i    ˆ i  )    ˆ i ˆ i , (2) Based on Eq. (2), an equation in which third-order quantum corrections appear by derive an equation for †  ˆ ˆ  from the Heisenberg equation of motion for ˆ . † ˆ ˆ  ,  ˆ ˆ  and their Hermitian conjugates are neglected, Now, if the higher-order correlations than   the following set of equations results as follows: 2 (3a ) Qi (1)   (Qi 1 (1)  Qi 1 (1)   Qi 1 (2)), 2   * * * (3b)  e [(2  Qi 1 (1)  Qi 1 (1))Qi 1 (2)  Qi 1 (1)Qi 1 (3)  Qi 1 (1)Qi 1 (3)], Qi (2)  Qi 1 (2)e 2   * (3c )  e [2(1  Qi 1 (1))Qi 1 (3)  2Qi 1 (1)Qi 1 (2)  Qi 1 (1)]. Qi (3)  Qi 1 (3)e † ˆ ˆ  and β is the dissipation parameter. where Q (1)  ˆ  , Q (2)  ˆ ˆ  , Q (3)  

If we iterate Eqs.(3) with initial values Q0 (1) , Q0 (2) and Q0 (3) real, then all the successive values Qi (1) , Qi (2) and Qi (3) are real for all i. We note that, Eqs.(3) reduce to the classical logistic map when the quantum corrections Qi (2), Qi (3)  0 . Also, the dissipation parameter β gives the classical logistic map when it leads to infinite value (  ).

2.2 Discretized cat map.

The cat map is a 2D map that maps the unit square onto itself in a one-to-one manner. It is defined by Eq. (4) [4, 5], where p and q are positive integers, (ri , si ) and (ri 1 , si 1 ) are the i-th and the (i+1)-th states, respectively:

 ri 1   1 p   ri   s    q pq  1  s  mod( M )  i   i 1  

(4)

2.3 Lifting-Wavelet Transform (LWT)

The lifting scheme has several unique properties in comparison with the traditional wavelet [6]: LWT allows for an in-place implementation of the fast wavelet transform and the construction of wavelets, a feature similar to the FFT, without using the Fourier transform. Hence, LWT can be calculated more efficiently and needs less memory space. Furthermore, It is particularly easy to build non-linear wavelet transforms and LWT has the time–frequency localization capability. What’s more, LWT coefficients are integers and do not have quantization errors unlike the traditional wavelet transform.

3. THE PROPOSED COLOR IMGAE ENCRYPTION/DECRYPTION METHOD The encryption process is performed according to the following steps (1) Reading the original color image as RGB color space P  {Rm,n , Gm,n , Bm,n } where 1  m  M , 1  n  N , M, N are height and width of the plainimage in pixels. (2) Transform the plainimage from RGB color space to YCbCr color space. (3) Transform the Y component to lifting wavelet transform (4) Permute the component of Y component based on 2D cat map, Eq.(4)

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(5) (6) (7) (8) (9)

Perform inverse lifting wavelet transform. Transform the scrambled image from YCbCr color space back to RGB color space. Apply Algorithm 1 to generate chaotic key stream sequences using adopted quantum logistic map. Apply Algorithm 2 to encrypt the scrambled image based on a stream cipher structure. Obtain the final cipherimage.

Algorithm 1: Keys generation using adapted quantum logistic map (CQ) Input : initial values for the quantum logistic map Output: Keys sequences image KCQ for i = 1 to M for j=1 to N

KCQiR, j  ( floor (Qi , j 1 (1) * 2 26 )) mod 256

Algorithm 2: Stream encryption using adapted quantum logistic map (CQ) Input: Scrambled image from the first phase Output: Cipher image C for i = 1 to M for j=1 to N for each   {R, G , B} and



KCQ  {KCQ R , KCQG , KCQ B } do

KCQiG, j  ( floor (Qi , j 1 (2) * 2 26 )) mod 256 KCQiB, j  ( floor (Qi , j 1 (3) * 226 )) mod 256

Ci, j  Ci, j  KCQi, j

end end

end end

Decryption is the converse of encryption. At the receiver side, using the same round transformations and the same keys, the decryption can easily derived from the encryption routine as follows: (1) Reading the cipher colored image as RGB color space C  {Rm,n , Gm,n , Bm,n } where 1  m  M , 1  n  N , M, N are height and width of the cipherimage in pixels. (2) Perform Algorithm 1 to generate the confusion keys using adapted quantum logistic map (CQ) with the correct values as the encryption process. (3) Perform Algorithm 2 by considering the cipherimage and KCQ keys as inputs. (4) Transform the obtained image from RGB color space to YCbCr color space. (5) Transform the Y component to lifting wavelet transform (6) Permute the pixels of Y component based on 2D cat as an inverse way. (7) Perform inverse lifting wavelet transform. (8) Transform the obtained image from YCbCr color space back to RGB color space to obtain the original plainimage. In our experiments, the parameters are chosen as follows: Q0 (1)  0.45234444336; Q0 (2)  0.003453324566;

, Q0 (3)  0.001324523564; , Q 0 (1)  0.00186 , Q0 (3)  0.00398 ,   3.99 and   4.489 . *

*

4. EXPERIMENTAL ANALYSES 4.1 Visual test, sensitivity and histogram analysis

Several images have been adopted in our experiments. For convenience, the color “Sailboat” image is taken as an example for our discussion. The encrypted image is shown in Fig. 1 (b). However, with a tiny change in initial condition or control parameter, we cannot decrypt the cipherimage in Fig 1(b) as shown in Fig. 1(c). This illustrated that the proposed scheme is sensitive to tiny change in the keys. To test the histogram, Fig. 2 (a) and (b) show the histogram of the original image and cipherimage, respectively. The latter figure shows that the histogram of cipherimage is nearly flat implying a good statistical property for the proposed color encryption scheme.

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(a) (b) (c) (d) Fig. 1. (a) Original image (b) encrypted image with key 1 (c) decrypted image with wrong key (d) decrypted image correct key 1.

Aii, (a) Red

(b) Green

(c) Blue

'LlialliLhaLL1111

(d) Red

(e) Green

(f) Blue

Fig. 2: Histogram of the original and encrypted Sailboat image 4.2 Information Entropy

To measure the entropy H (s) of a source s, we have: 8

2 1

H  (s)  

 P ( ) log i

2

P ( i )

(5)

i 0

where   {R , G , B} are the color components of the pixel si ∈ s; P ( i ) is the probability of occurrence of symbol si and log denotes the base 2 logarithm so that the entropy is expressed in bits. For a random source emitting 256 symbols, its entropy is H(s) = 8 bits. For the cipherimage of Sailboat, the corresponding entropy is 7.9992. This implies that the information leakage in the proposed encryption process is negligible and the encryption algorithm is secure against the entropy attack. 4.3 Differential attack (NPCR and UACI)

The expected values of NPCR and UACI for a good encryption scheme are 99.6094% and 33.4635%, respectively [7, 8]. We test NPCRR , G , B and UACI R , G , B values for large number of images by the proposed cryptosystem approach to asses the influence of changing a single pixel in the plainimage on the cipherimage. The percentage of pixels changed in encrypted image is greater than 99.6 % for NPCRR , G , B and UACI R , G , B is greater than 33.2 % for even with one-bit difference in the plainimages. This result shows that the proposed scheme is very sensitive with respect to the tiny changes in the plainimage. It is therefore has a strong ability to resist the differential attack. 4.4 Performance speed

The proposed color image encryption only needs some simple operations, so, it just takes us 0.23 s to complete the encryption process for an image of 256  256. The proposed encryption scheme is fast compared to other schemes such as in Refs. [4], [7] and [9], which takes more than 0.5 s for the same image.

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5. CONCLUSION A new color image encryption scheme based on quantum chaotic system is proposed in this paper. In this scheme, a new substitution/confusion scheme is achieved based on 2D toral automorphism in integer wavelet transform by scrambling only the Y (Luminance) component of low frequency subband to save the computation. Then, a chaotic stream encryption scheme is accomplished by generating an intermediate chaotic key stream image with the help of quantum chaotic system. Experimental results show that the proposed encryption scheme has good performance for the protection of color images over transmission channels.

6. ACKNOWLEDGMENT This work is supported by the National Natural Science Foundation of China (Grant Number: 60832010, 61100187), the Fundamental Research Funds for the Central Universities (Grant Number: HIT. NSRIF. 2010046, HIT. NSRIF. 2013061), and the Higher Education Commission of Egypt.

7. REFERENCES [1]. Goggin, M., Sundaram, E B., Milonni, P. W., “Quantum logistic map”, Physical Review A (Atomic, Molecular, and Optical Physics), Volume 41, Issue 10, pp.5705-5708 (1990). [2]. Abd El-Latif, A. A., Li, L., Wang, N., Han, Q., Niu, X., “A new approach to chaotic image encryption based on quantum chaotic system, exploiting color spaces”, Signal Processing, http://dx.doi.org/10.1016/j.sigpro.2013.03.031. [3]. Abd El-Latif, A. A., Li, L., Zhang, T., Wang, N., Song, X., Niu, X., ” Digital Image Encryption Scheme Based on Multiple Chaotic Systems” Sensing and Imaging: An International Journal , Volume 13, Issue 2, pp 67-88 (2012). [4]. Chen, G., Mao, Y., Chui, C.K., ”A symmetric image encryption scheme based on 3D chaotic cat maps”, Chaos, Solitions and Fractals (21),749–761 (2004). [5]. Abd El-Latif A. A., Li L., Wang, N., Li, Q., Niu, X.,”A new image encryption based on chaotic systems and singular value decomposition”, Proc. SPIE. 8334, Fourth International Conference on Digital Image Processing (ICDIP 2012) 83343F (May 1, 2012) doi: 10.1117/12.964281. [6]. Lei, B., Soon, I.Y., Zhou, F., Li, Z., Lei, H. , A robust audio watermarking scheme based on lifting wavelet transform and singular value decomposition, Signal Processing, (92)_1985–2001 (2012). [7]. Rhouma, R., Meherzi, S., Belghith,, S., ”OCML-based colour image encryption,” Chaos, Solitons & Fractals (40) 309-318 (2009). [8]. Amin, M., Abd El-Latif, A. A.,” Efficient Modified RC5 Based on Chaos Adapted to Image Encryption” Journal of Electronic Imaging 19(1), 013012 (Jan–Mar 2010). [9]. Huang, X., “Image encryption algorithm using chaotic chebyshev generator” Nonlinear Dynamics, Volume 67, Issue 4, pp 2411-2417 (2012).

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