Image encryption based on nonseparable fractional Fourier transform and chaotic map

Optics Communications 348 (2015) 43–49

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Image encryption based on nonseparable fractional Fourier transform and chaotic map Qiwen Ran a,b, Lin Yuan a,c,n, Tieyu Zhao a a

State Key Laboratory of Tunable Laser Technology Research, Institute of Optic-Electronics, Harbin Institute of Technology, Harbin 150001, China Nature Science Research Center of Science and Technology, Harbin Institute of Technology, Harbin 150001, China c College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 27 January 2015 Received in revised form 7 March 2015 Accepted 10 March 2015 Available online 11 March 2015

In this paper an image cryptosystem is constructed by using double random phase masks and a chaotic map together with a novel transform which is similar to fractional Fourier transform and gyrator transform to some extent. The new transform is not periodic with respect to the transform order and cannot be expressed as a tensor product of two one-dimensional transforms neither in the space domain nor in the Wigner space-frequency domain. In the cryptosystem, the parameters of Arnold map, transform orders of the proposed transform and phase information serve as the main keys. The numerical simulations have demonstrated the validity and high security level of the image cryptosystem based on the proposed transform. & 2015 Elsevier B.V. All rights reserved.

Keywords: Nonseparable fractional Fourier transform Wigner distribution Inseparability Image encryption

1. Introduction The image processing gains much broader attention and scopes due to the ever growing importance of scientific visualization and the need of acquiring and exchanging efficient information. Image encryption plays an important role in information secure storage and transmission nowadays and has been widely used in various aspects of life. Since the double random phase encoding method (DRPE) was proposed by Refregier and Javidi in 1995 [1], optical encryption methods have been receiving considerable attention [2–6]. In the numerous encryption methods some optical processors or transforms, such as fractional Fourier transform (FRFT) [7–10] and Gyrator transform (GT) [11–13], are employed to enhance the speed and security levels of encryption methods. FRFT and GT both belong to the class of linear canonical transforms (LCT) [14–17] which have been significantly developed and widely used in optical and digital information processing in recent years for their good description of the first order lossless optical systems [18–21]. Despite the wide use of FRFT and GT, their periodicity makes them deficient in image encryption. Furthermore, two-dimensional FRFT can be decomposed into a product of two one-dimensional FRFTs [22]. It means when a given signal is being n Corresponding author at: State Key Laboratory of Tunable Laser Technology Research, Institute of Optic-Electronics, Harbin Institute of Technology, Harbin 150001, China. E-mail address: [email protected] (L. Yuan).

http://dx.doi.org/10.1016/j.optcom.2015.03.016 0030-4018/& 2015 Elsevier B.V. All rights reserved.

processed, its information on two different dimensions is independent of each other. Similarly, regardless of a cross term of two variables in the expression of GT, GT shares the same independency property with FRFT for the reason that GT can be rewritten as two-dimensional FRFT with special angles. According to the relation between FRFT and Wigner distribution function (WDF) that FRFT at parameter α corresponds to an απ /2 rotation of Wigner distribution [23], we can conclude that GT can also be taken as a rotation in Wigner distribution plane. The relation of WDF to FRFT and GT is partly responsible for the information-independency due to which FRFT and GT cannot muddle up the input information along two different dimensions adequately. Thus the encryption effects of schemes based on FRFT or GT are to some extent defective. In this paper we propose an image cryptosystem based on a new optical transform named nonseparable fractional Fourier transform (NFRFT) which, to our knowledge, is firstly proposed and solves the information-independency problem. It highly enhances the security level of the proposed image cryptosystem that NFRFT has information-tangling property and aperiodicity with respect to the transform order unlike FRFT and GT. The capability of information-tangling is resulted from the property that NFRFT cannot be decomposed into a tensor product of two one-dimensional FRFTs no matter in the space domain or in the Wigner distribution space-frequency domain. In addition to NFRFT, Arnold map [24–26] and double random phase masks are also employed to the cryptosystem. The input image is taken as the amplitude and further modified through multiplication with the first random

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phase. The outcome is then transformed into the first NFRFT domain. Subsequently the multiplication with the second random phase and transformation into the second NFRFT domain are performed. In the end Arnold map is employed to scramble the intermediate result to obtain the final output of which the amplitude is taken as the encrypted image. The parameters of Arnold map, two transform orders of NFRFTs and phase masks serve as the main keys with parameters in the coefficients of NFRFT as additional keys. Some numerical simulations have been made to evaluate the validity and security of the cryptosystem. In the latter part of this paper, the definition and properties of NFRFT are presented in section 2. The image cryptosystem is constructed and described in Section 3. Section 4 gives the numerical simulations to show the feasibility of NFRFT to image encryption and to demonstrate the high security level of the proposed image encryption scheme based on NFRFT. Conclusion is presented in the final section.

2. Nonseparable fractional Fourier transform Define nonseparable fractional Fourier transform with order α as 3

R αE [f (ri )](ro) ≜ (R αE f )(ro) =

∑ A k (α) fk (ro), k=0

(1)

where f0 (ro) ≜ f (xo, yo ), f1(ro)

≜ F (yo , xo), f2 (ro) ≜ f ( − xo, − yo ), f3 (ro) ≜ F 3(yo , xo), F (yo , xo )is the transposed Fourier transform of f (x i , yi ) denoted by F˜ (f )(xo, yo ) and ri, o = (x i, o, yi, o ) indicates the input/output coordinate here and further. It can be inferred that fk = F˜ (fk − 1 ), k = 1, 2, 3. The symbols Ak (α), k = 0, 1, 2, 3 are coefficients satisfying the additive condition R αE R Eβ = R Eβ R αE = R αE + β . Through mathematical manipulation, the analytical expression of Ak (α), k = 0, 1, 2, 3 is

A k (α ) =

1 4

3

∑ l= 0

⎧ iπ ⎫ exp ⎨ − [(α − k) l] + αirl ⎬ , ⎩ 2 ⎭

(2)

where rl, l= 0, 1, 2, 3 is any real number. Therefore, the NFRFT at parameter α reads

(R αE f )(ro) =

1 4

3

3

∑∑ k = 0 l= 0

⎧ iπ ⎫ exp ⎨ − [(α − k) l] + αirl ⎬ fk (ro). ⎩ 2 ⎭

(3)

It is inherent in the formulas (2) and (3) that NFRFT has the following properties: 1. Linearity: R αE [a1f (ri ) + a2 g (ri )] = a1R αE [f (ri )] + a2 R αE [g (ri )]; 2. Additivity: R αE R Eβ = R Eβ R αE = R αE + β ; 3. Unitarity: R αE ⋅ (R αE ) H = I , where ‘H ’ denotes the complex conjugate transpose operation; 4. Aperiodicity: R αE + T ≠ R αE , for any real or complex number T ≠ 0; 5. Parseval relation: ∫ {(R αE f )(ro)}{(R αE g)(ro)}⁎dro = ∫ f (ri ) g (ri )⁎dri; 6. Coincidence: when α is any integer and rlm = 0for l = 0, 1, 2, 3, m = 1, 2, NFRFT coincides with GT. Eq. (3) implies that NFRFT is a linear combination of four functions which dominate in the space, frequency domain, the symmetrical space and frequency domain respectively. Therefore, NFRFT cannot be rewritten as a tensor product of two one-

dimensional FRFTs and further does not correspond to any rotation of Wigner distribution. Such a property makes the input information sufficiently mixed together not only within each dimension but also between two dimensions. Note that despite their similarity, GT and NFRFT do not totally coincide due to the aperiodicity and inseparability of NFRFT contrasting to the periodicity and separability of GT. The properties of NFRFT make it a suitable tool for information processing.

3. Image encryption scheme In the past two decades, among various optical encryption technologies, double random phases encoding (DRPE) is a classical method and many algorithms for DRPE in different frequency domains have been proposed [27–30]. In these algorithms only with the correct random phase masks and in rightful domains, can the image be recovered. Chaos-based encryption schemes [25,26,31,32] are also widely used in cryptosystem for their ergodicity and pseudorandomness which lead to good performances in image encryption. Chaotic maps are highly sensitive to the control parameters and initial conditions. In this paper, we will employ the DRPE and Arnold map together with NFRFT to construct an encryption scheme. The Arnold map used in this paper is in a generalized form [31,32] and expressed as

⎛ xn+ 1 ⎞ ⎛1 p ⎞ ⎛ xn ⎞ ⎟ ⎜ ⎟ mod 1, ⎜y ⎟ = ⎜ ⎝ n + 1 ⎠ ⎝q 1 + pq ⎠ ⎝ yn ⎠

(4)

where “x mod 1” means taking the fractional part of the real number x. Thus x n, yn are both confined to interval [0, 1) where p, q are any positive integers called control parameters and x0, y0 are called initial conditions. The encryption flowchart is shown in Fig.1. The proposed scheme is a cascade of three steps: 1. The original image is taken as an amplitude and multiplied with the first random phase matrix P1, then transformed into NFRFT domain of order α , i.e., g1 = R αE [P1 ⊗ Io ]. The first random phase matrix is stochastically generated by the computer. The encryption keys group used in this step is denoted by K1 including the first random phase P1 and all the parameters of the first NFRFT which are the first transform order α and coefficient-parameters rl1, l = 0, 1, 2, 3. 2. The output g1 of step 1 is multiplied with the second random phase mask P2 and subsequently transformed by NFRFT of order β , namely, g2 = R Eβ [P2 ⊗ g1]. The keys group K2 of this step consists of the second phase matrix P2, the second transform order β and the corresponding coefficient-parameters rl2, l = 0, 1, 2, 3. 3. The intermediate result g2 is permuted by the chaotic Arnold map, i.e., Ie = P {g2 } , where ‘P’ denotes the permutation operation and the keys group K3 is composed of all the parameters of the Arnold map including (x0, y0 ), p, q and the iteration time N . According to the expression (1), we can implement NFRFT by a photoelectric hybrid setup as shown in Fig.2. The symmetrical setup composed of three generalized lenses that contained in a

Fig. 1. Flowchart of the encryption, Io -the original image, Ie -the encrypted image.

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Fig. 2. (a) Photoelectric hybrid setup of the encryption and decryption system based on NFRFT, SLM-spatial light modulator, L-generalized Lens, BE-beam expander, (b) an assembled set of two cylindrical lenses that forms the generalized lenses L1 and L2.

dashed rectangular in Fig.2(a) can perform the transposed Fourier transform F˜ (f ). Each lens is a generalized lens assembled by two cylindrical lenses as shown in Fig.2(b). The detailed description of the generalized lenses and the configuration of the corresponding optical part of the setup are given in [13]. We start with displaying the original signal f0 at the plain P1. By means of holography, the information of f1 is registered by a CCD located at plain P2. After post processing by the computer, f1 is then displayed at the spatial light modulator (SLM) at plain P1 as the input in the next iteration. Likewise, the other two functions fk , k = 2, 3 can be obtained through the iteration in the hybrid setup. Subsequently, after computation of the coefficients and their multiplication with the corresponding basic functions by the computer, NFRFT is acquired by combining the four weighted functions. The generation of the random phase matrices and the multiplication with the functions of the corresponding NFRFT domains are performed in the computer as well. Finally, the encrypted image is obtained through the Arnold map permutation by the computer. The part of decryption process relating to NFRFT can be implemented simply by substituting the transform angles with their opposite numbers using the computer without altering the photoelectric hybrid setup. The rest parts of decryption are performed by the computer. The three used tools: random phase matrices, NFRFT and the Arnold map play different roles in the cryptosystem. NFRFTs enable the cryptosystem to resist the statistical analysis, noise pollution and shearing operation due to their capability of adequately mixing up the information of input image within and between two dimensions and the further transformation into fractional frequency domain; The randomly generated phase matrices by computer turn input images into white noise-alike images and moreover the randomness of the first phase matrix may result in different output images out of the same input image in the encryption process, so that the cryptosystem achieves probabilistic encryption and therefore can resist the statistical analysis and exhaustive attack; The Arnold map scrambles the sequence of image pixels. The high sensitivity of the Arnold map and cryptosystem to K3 guarantees the good resistance of the cryptosystem to exhaustive attack. The encryption keys of the cryptosystem consist of the main keys P1, P2, α , β , K3 and the additional keys, the coefficient-parameters rlm, m = 1, 2,l = 0, 1, 2, 3 of NFRFTs. The decryption process is achievable by inversing the main process of encryption in Fig. 1 except for the last step in which multiplication with the complex conjugation of P1 is replaced by the operation |P1 ⊗ Io |. The first phase is of no need as a key in the decryption process as in the encryption process. Therefore the decryption keys are composed of keys K1, K2, K3 except for P1( ∈ K1).What needs to be pointed out is that the additional keys for decryption differ a little from the ones for encryption due to the property of rlm , i.e., there are many more values of rlm qualified as decryption keys. The multi-choices of rlm are strongly related to the transform order of NFRFT in the manner that different order

corresponds to different number of qualified rlm . Without knowing the true value of the transform order, the valid values of rlm for decryption are hardly to be confirmed. We may call such a property of rlm as the changeable-periodicity which is completely different from the periodicity. The security level of the proposed image encryption system increases greatly due to the information-tangling property and the aperiodicity of NFRFT. The property that NFRFT is not periodic with respect to the transform order α indicates the value range of α falls in the infinite interval ( − ∞ , + ∞) and the only correct value of α as decryption key is hard to be pinpointed. On the contrary, FRFT and GT are defective in image encryption since the periodicity gives rise to more than one correct value of α as decryption keys and cuts the search-range of correct value of α down to a finite interval.

4. Numerical simulations To testify the validity and demonstrate the security level of the cryptosystem, we will give numerical simulations of the cryptosystem. In the simulations, we apply the cryptosystem to Mena of size 256 × 256 image with 256 Gy levels. The transform orders in step 1 and step 2 areα1 = 0.891, α2 = 1.432, the parameters in the coefficients of NFRFTs are r 1 = (2, 5.973, − 6.3, − 5 ) and r 2 = (1, 0.002, e, − 2 ) respectively. The employed Arnold map is determined by x0 = 0.4537, y0 = 0.6821, p = 11, q = 32 and

N = 105. The original and encrypted images are displayed in Fig.3 (a) and (b) respectively. With all the correct keys, the reconstruction of the original image Fig.3(c) is achieved. The decrypted image with wrong phase matrix P2 is shown in Fig.3(d). Decryption with other correct keys but wrong transform order α2 leads to the unrecognizable image Fig.3(e). Wrong value of control parameter p results in the false decryption in Fig.3(f) and wrong initial condition x0 leads to Fig.3(g). When the iteration time N is wrong, the reconstructed image Fig.3(h) is obtained. The corresponding MSEs are also given. Mean Square Error (MSE) between the original image Io and the decrypted image Id is a measurement criterion of the security level of cryptosystems, and expressed as

MSE =

1 N×M

N

M

∑ ∑ ‖Io (x, y) − Id (x, y) ‖ x=1 y=1

(5)

where N × M is the size of input/output image. Fig. 4(a) illustrates the MSE of the encryption scheme with respect to parameter α1. From Fig.4(a) we can learn that: first, the variation of the MSE curve is in no obvious monotonous pattern. The oscillation of the curve well resists the analysis through which attackers locate the correct value of α1 by using monotony of the curve; Second, the interval displayed in Fig.4(a) is finite, but α1 actually distributes on the infinite interval (−∞ , + ∞) due to the aperiodicity of NFRFT. Therefore non-exhaustive search of the

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Fig. 3. (a) Original image, (b) encrypted image, (c) reconstructed image applying correct keys, decrypted image with wrong phase matrix P2 (d), with wrong transform order α2 (e), with wrong p (f), with wrong x0 (g), with wrong N (h).

correct value of α by attackers is in vain and so is exhaustive search which is nearly unachievable in present technological environment. Fig.4(b) gives the conjoint analysis of MSE with respect to r31 and α1. Here we choose a few test values α′1 of α1 in order to well

demonstrate the behavior of MSE with respect to α1 and r31. From formula (4) we can infer that NFRFT is changeable-periodic with respect to r31 with a period of 2π /α1 which is also indicated in Fig.4 (b). However the true values of r31 as decryption keys are hardly located by analyzing the MSE without knowing the true value of

Fig.4. (a) MSE with respect to α1, (b) conjoint analysis of MSE with respect to r31 and α1, (c) MSE comparison between cryptosystems based on NFRFT and FRFT respectively.

Q. Ran et al. / Optics Communications 348 (2015) 43–49

α1. Because for different test values α′1 there are different numbers of r31 corresponding to minimum of MSE as shown in Fig.4(b). For instance within the chosen range of r31 in Fig.4(b), in the case of α′1 = 1, the minimum MSE = 2.42 × 103 only when r31 = − 1.596 and in the case of α′1 = 2.5, MSE achieves the minimum MSE = 6.7337 × 103 at three values of r31. The number of r31 making MSE achieve minimum depends on the test value α′1 in the manner that the larger α′1 is, the smaller the period of r31 becomes and the more the number of r31 increases. It can be inferred from above that when all the keys are taken into account, MSE will be rather large and complicated which means it is difficult for attackers to break the cryptosystem. In addition, the key space of the cryptosystem with K1, K2, K3 as encryption keys is large enough to ensure the security and firmness of the cryptosystem. In order to manifest the effects of FRFT and NFRFT on the security, we substitute FRFT for NFRFT into the cryptosystem and calculate the MSE with respect to the transform order α1. In Fig.4 (c), the thick curve represents the MSE corresponding to NFRFT and the thin curve represents the MSE corresponding to FRFT. As is shown, the thin MSE curve has periodicity that indicates the cryptosystem based on FRFT is deficient. Due to the periodicity of FRFT used in the cryptosystem, the MSE curve exhibits the periodicity so that there exist infinitely many true values as decryption keys corresponding to α1. Furthermore, the searching interval of α1 is reduced to a finite interval with length of 4. Although there are several minimum values on the thick curve, their bad effect to the security is much less than that of the periodic true values of α1 on the thin curve. Therefore, NFRFT is better than FRFT in the respect

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of the capability of avoiding α1 being ascertained. Moreover, on a larger distribution interval of α1, MSE curve of NFRFT better shows the property of no-obvious-pattern. The gray level histograms of image of Mena and Lena are given in Fig.5(a) and (b) respectively with corresponding encrypted images histograms shown in Fig.5(c) and (d). As it is can be seen, the histograms in Fig.5(c) and (d) are very similar to each other so that for the attackers it is difficult to obtain the encryption keys by analyzing the histograms of encrypted images. Therefore the cryptosystem is independent of the encryption keys and capable of resisting statistical analysis. To testify the robustness of such an image encryption scheme against occlusion attack, the encrypted image Mena is cut off by 1/4 and 1/16 respectively and decrypted with correct decryption keys respectively. Fig. 6(b) and (d) are the retrieved images respectively corresponding to the 1/4 -cut-off image in Fig.6(a) and 1/16-cut-off image in Fig. 6(c). Fig. 6 shows us the encrypted image under a certain degree of occlusion attack can be recovered recognizable. The noise attack is simulated by distorting the encrypted image with additive white Gaussian noise in the manner of In = Ie ⋅ (1 + c⋅NGau ) where In and Ie are Noisy encrypted image and the original encrypted image respectively, NGau is the Gaussian noise with zero-mean and the standard deviation c is strengthcoefficient of Gaussian noise. Fig. 7(a), (b), (c) are obtained by applying c = 0.1, c = 0.3, c = 0.6 respectively to distort the encrypted image. It is easy to see that although the encrypted images are polluted by the noises, the cryptosystem can recover the noisy images using correct keys and the retrieved images are

Fig. 5. Histogram of image Mena (a), Lena (b) and histogram of the corresponding encrypted image Mena (c), Lena (d).

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Q. Ran et al. / Optics Communications 348 (2015) 43–49

Fig. 6. Encrypted image cut off by 1/4 (a) and the corresponding decrypted image (b), encrypted image cut off by 1/16 (c) and the corresponding decrypted image (d).

Fig. 7. The recovered images of noise attack (a) c¼ 0.1, (b) c ¼0.3, (c) c¼ 0.6.

recognizable. Therefore the cryptosystem has the capability against the noise attack.

5. Conclusion We proposed a cryptosystem based on nonseparable fractional Fourier transform (NFRFT) along with Arnold map and random phase masks. NFRFT is a newly defined transform of which the primary advantage of NFRFT is the capability of tangling the information along and across both directions together. Such a property lies in the inseparability not only in space domain but also in other fractional frequency domains. The proposed cryptosystem fully takes the advantage of NFRFT that NFRFT at parameter α has only one correct transform order value −α as the corresponding decryption key due to the aperiodicity. On the contrary, the cryptosystems based on GT or FRFT have many more values corresponding to the transform order as decryption keys because of the periodicity with respect to the transform orders. Hence NFRFT, without the deficiency of FRFT and GT, highly enhances the security level of the cryptosystem. Moreover, the Arnold map also contributes to the security due to the high sensitivity of the cryptosystem to the parameters of Arnold map. The simulation results testify the validity of the cryptosystem. In the end, we demonstrate the security level of the cryptosystem by using several common attacks from which, we conclude that the cryptosystem is secure and suitable for image encryption.

References [1] P. Refregier, B. Javidi, Optical encryption based on input plane and Fourier plane ran dom encoding, Opt. Lett. 20 (1995) 767–769. [2] A. Alfalou, C. Brosseau, Optical image compression and encryption methods, Adv. Opt. Photon 1 (2009) 589–636. [3] D. Weber, J. Trolinger, Novel implementation of nonlinear joint transform correlators in optical security and validation, Opt. Eng. 38 (1999) 62–68. [4] I. Mehra, N.K. Nishchal, Asymmetric cryptosystem for securing multiple images using two beam interference phenomenon, Opt Laser Technol. 60 (2014) 1–7.

[5] H. Li, Y. Wang, Double-image encryption based on iterative gyrator transform, Opt. Commun. 281 (2008) 5745–5749. [6] T. Zhao, Q. Ran, Y. Chi, Image encryption based on nonlinear encryption system and public-key cryptography 338 (2015) 64–72. [7] V. Namias, The fractional Fourier transform and its application in quantum mechanics, Inst. Math 25 (1980) 241–265. [8] H.M. Ozaktas, D. Mendlovic, Fractional Fourier transform and their optical implementation-I, J. Opt. Soc. Am. A 10 (1993) 1875–1881. [9] H.M. Ozaktas, D. Mendlovic, Fractional Fourier transform and their optical implementation-II, J. Opt. Soc. Am. A 10 (1993) 2522–2531. [10] H.M. Ozaktas, Z. Zalevsky, M. Alper Kutay, The fractional Fourier transform with applications in optics and signal processing, John Wiley & Sons, 2001. [11] J.A. Rodrigo, T. Alieva, M.L. Calvo, Gyrator transfrom: properties and applications, Opt. Express 15 (2007) 2190–2203. [12] J.A. Rodrigo, T. Alieva, M.L. Calvo, Applications of gyrator transform for image processing, Opt. Commun. 278 (2007) 279–284. [13] J.A. Rodrigo, T. Alieva, M.L. Calvo, Experimental implementation of the gyrator transform, J. Opt. Soc. Am. A 24 (2007) 3135–3139. [14] M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations, J. Math. Phys 12 (1971) 1772–1780. [15] T. Alieva, M.J. Bastiaans, Properties of the linear canonical integral transformation, J. Opt. Soc. Am. A 24 (2007) 3658–3665. [16] S. Pei, J. Ding, Eigenfunctions of linear canonical transform, Signal Processing, IEEE Transactions on 50 (2002) 11–26. [17] S.A. Collins, Lens-System diffraction integral written in terms of matrix optics, J. Opt. Soc. Am. 60 (1970) 1168–1177. [18] M. Nazarathy, J. Shamir, First-order optics—a canonical operator representation: lossless systems, J. Opt. Soc. Am. 72 (1982) 356–364. [19] M.J. Bastiaans, Wigner distribution function and its application to first-order optics, J. Opt. Soc. Am. 69 (1979) 1710–1716. [20] M. Nazarathy, J. Shamir, First-order optics: operator representation for systems with loss or gain, J. Opt. Soc. Am. 72 (1982) 1398–1408. [21] M.J. Bastiaans, T. Alieva, Classification of lossless first-order optical systems and the linear canonical transformation, J. Opt. Soc. Am. A 24 (2007) 1053–1062. [22] A. Sahin, H.M. Ozaktas, D. Mendlovic, Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions, Opt. Commun. 120 (1995) 134–138. [23] A.W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform, J. Opt. Soc. Am. A 10 (1993) 2181–2186. [24] V. Arnold, A. Avez, Ergodic problem of classical mechanics, Benjamin, New York, 1968. [25] Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, S. Liu, Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains, Opt. Commun. 284 (2011) 123–128. [26] L. Chen, D. Zhao, F. Ge, Image encryption based on singular value decomposition and Arnold transform in fractional domain, Opt. Commun. 291 (2013) 98–103. [27] G. Unnikrishnan, J. Joseph, K. Singh, Optical encryption by double random phase encoding in the fractional Fourier domain, Opt. Lett. 25 (2000) 887–889.

Q. Ran et al. / Optics Communications 348 (2015) 43–49

[28] G. Situ, J. Zhang, Double random-phase encoding in the Fresnel domain, Opt. Lett. 29 (2004) 1584–1586. [29] M. He, Q. Tan, L. Cao, Q. He, G. Jin, Security enhanced optical encryption system by random phase key and permutation key, Opt. Express 17 (2009) 22462–22473. [30] Z. Zhong, J. Chang, M. Shan, B. Hao, Double image encryption using double pixel scrambling and random phase encoding, Opt. Commun. 285 (2012)

49

584–588. [31] R. Ye, A novel chaos-based image encryption scheme with an efficient permutation-diffusion mechanism, Opt. Commun. 284 (2011) 5290–5298. [32] G. Ye, K. Wong, An efficient chaotic image encryption algorithm based on a generalized Arnold map, Nonlinear dynamics 69 (2012) 2079–2087.