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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 10605–10621
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Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps

Liansheng Sui, Kuaikuai Duan, Junli Liang, and Xinhong Hei  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 10605-10621 (2014)
http://dx.doi.org/10.1364/OE.22.010605


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Abstract

A double-image encryption is proposed based on the discrete fractional random transform and logistic maps. First, an enlarged image is composited from two original images and scrambled in the confusion process which consists of a number of rounds. In each round, the pixel positions of the enlarged image are relocated by using cat maps which are generated based on two logistic maps. Then the scrambled enlarged image is decomposed into two components. Second, one of two components is directly separated into two phase masks and the other component is used to derive the ciphertext image with stationary white noise distribution by using the cascaded discrete fractional random transforms generated based on the logistic map. The cryptosystem is asymmetric and has high resistance against to the potential attacks such as chosen plaintext attack, in which the initial values of logistic maps and the fractional orders are considered as the encryption keys while two decryption keys are produced in the encryption process and directly related to the original images. Simulation results and security analysis verify the feasibility and effectiveness of the proposed encryption scheme.

© 2014 Optical Society of America

1. Introduction

With the rapid popularity of computer networks, issues about illegal image data access on internet being more and more serious and image encryption technique has attracted a growing attention. Due to their capability to ensure the security of data transmission and communication, optical image encryption techniques have become an important research field. Various optical encryption and encoding techniques in different domains such as the Fourier transform (FT), Fresnel transform (FrT), fractional Fourier transform (FrFT), gyrator transform (GT) and fractional Mellin transform domain have been proposed in the past two decades [1

1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef] [PubMed]

15

15. N. Zhou, Y. Wang, and J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011). [CrossRef]

]. Moreover, Alfalou and Brosseau [16

16. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009). [CrossRef]

] analyzed the performance on different methods and pointed out many schemes can be used for compression simultaneously. Though these optical methods have excellent properties such as parallel and multidimensional capability of signal processing, most schemes mainly discuss the single image encryption, which reduce the efficiency when encrypting, storing and transmitting double or multiple images. Additionally, these schemes belong to the category of symmetric cryptosystems, in which the encryption keys are identical to decryption ones. Because of the inherently linear property of mathematical or optical transformation, these schemes are vulnerable to various attacks such as chosen plaintext attack.

In order to ensure security and improve efficiency of image transmission and communication on network, the multiple-image encryption has attracted lots of attentions. Situ and Zhang [17

17. G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005). [CrossRef] [PubMed]

, 18

18. G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A, Pure Appl. Opt. 8(5), 391–397 (2006). [CrossRef]

] proposed the multiple-image encryption schemes based on wavelength multiplexing and position multiplexing. Alfalou and Mansour [19

19. A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48(31), 5933–5947 (2009). [CrossRef] [PubMed]

] multiplexed target images by using FT based on double random phase encoding. Subsequently, Alfalou et al. [20

20. A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express 19(24), 24023–24029 (2011). [CrossRef] [PubMed]

] suggested a multiple-image encryption scheme by using the discrete cosine transform (DCT) and the specific spectral filtering technique, which implemented simultaneous fusion, compression and encryption of multiple images. Liu et al. [21

21. Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, and S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (2011). [CrossRef]

] proposed an optical multi-image encryption based on frequency shift technique, where the lower frequency parts of the plain images are selected, shifted and encrypted by using double phase encoding in FrFT domain. Wang and Zhao [22

22. X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284(1), 148–152 (2011). [CrossRef]

] designed a multiple-image encryption based on the nonlinear phase truncation operations in FT domain, which is vulnerable to various attacks such as chosen plaintext attack. Additionally, Wang and Zhao [23

23. X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21–22), 4280–4284 (2012). [CrossRef]

] proposed a fully phase multiple-image encryption based on superposition principle and digital holographic technique. Hwang et al. [24

24. H. E. Hwang, H. T. Chang, and W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009). [CrossRef] [PubMed]

] proposed a multiple images encryption in FrT domain based on modified Gerchberg-Saxton algorithm (MGSA), which reduces the cross-talks of the decrypted images significantly. Based on MGSA, Chang et al. [25

25. H. T. Chang, H. E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011). [CrossRef]

, 26

26. H. T. Chang, H. E. Hwang, C. L. Lee, and M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011). [CrossRef] [PubMed]

] suggested the position multiplexing encryption schemes by using cascaded phase-only masks and Huang et al. [27

27. J. J. Huang, H. E. Hwang, C. Y. Chen, and C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012). [CrossRef]

] designed the scheme with architecture of two adjacent phase-only functions to increase capacity of the cryptosystem. Deng and Zhao [28

28. X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012). [CrossRef]

] proposed a multiple-image encryption algorithm using phase retrieve algorithm and intermodulation in Fourier domain, which can avoid the cross-talk noise completely. Sui et al. [29

29. S. Liansheng, X. Meiting, and T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013). [CrossRef] [PubMed]

] encrypted multiple images based on phase mask multiplexing technique in FrFT domain, in which the capacity is considerable enhanced.

As a special case, double-image encryption techniques also have attracted more attention than ever before. Li and Wang [30

30. H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008). [CrossRef]

, 31

31. H. Li and Y. Wang, “Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain,” J. Mod. Opt. 55(21), 3601–3609 (2008). [CrossRef]

] proposed the double-image encryption schemes in the GT and FrFT domains, where the iterative process between two plain images has high convergent speed. Liu et al. [32

32. Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010). [CrossRef] [PubMed]

, 33

33. Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, and S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013). [CrossRef]

] suggested the double-image encryption schemes in the GT domain not only by using iterative random binary encoding but also by using random phase encoding and pixel exchanging. Additionally, Liu et al. [34

34. Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, J. Dai, and S. Liu, “Double image encryption by using Arnold transform and discrete fractional angular transform,” Opt. Lasers Eng. 50(2), 248–255 (2012). [CrossRef]

] encrypted two plain images into the amplitude and phase of a complex function, in which the discrete fractional angular transform is used. Zhang and Xiao [35

35. Y. Zhang and D. Xiao, “Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform,” Opt. Lasers Eng. 51(4), 472–480 (2013). [CrossRef]

] designed a double optical image encryption by using the discrete Chirikov standard map which is utilized to scramble the pixel positions and intensity values, respectively. Li and Wang [36

36. H. Li and Y. Wang, “Double-image encryption based on discrete fractional random transform and chaotic maps,” Opt. Lasers Eng. 49(7), 753–757 (2011). [CrossRef]

] proposed a double-image encryption based on discrete fractional random transform and chaotic maps, which can raise the efficiency when encrypting, storing or transmitting. Sui et al. [37

37. L. Sui, H. Lu, Z. Wang, and Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014). [CrossRef]

] proposed a double-image encryption based on discrete fractional random transform, where a chaotic confusion-diffusion process is used to break the correlations between adjacent bit planes efficiently. Wang and Zhao [38

38. X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011). [CrossRef]

] suggested an algorithm to encrypt two covert images into an overt image based on phase retrieval and phase-truncated Fourier transform, which is asymmetric and the encryption keys are different from those in decryption process. Furthermore, Wang and Zhao [39

39. X. Wang and D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express 20(11), 11994–12003 (2012). [CrossRef] [PubMed]

] suggested an asymmetric double-image encryption which has a high level of robustness against the specific attack. Sui et al. [40

40. L. Sui, H. Lu, X. Ning, and Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014). [CrossRef]

] designed the asymmetric encryption between two plain images based on convolution operation in FrFT domain, which make the optical implementation of the decryption process convenient and efficient.

Recently, due to the excellent properties such as ergodicity, pseudo-randomness, sensitivity to initial conditions and control parameters, the chaotic maps are used to encrypt image in different transform domains, which can strengthen the nonlinearity of plain image in spatial and transform domains. Singh and Sinha [41

41. N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46(2), 117–123 (2008). [CrossRef]

, 42

42. N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009). [CrossRef]

] proposed an optical image encryption schemes based on chaos not only in FrFT domain but also in GT domain. Additionally, Singh and Sinha [43

43. N. Singh and A. Sinha, “Chaos based multiple image encryption using multiple canonical transforms,” Opt. Laser Technol. 42(5), 724–731 (2010). [CrossRef]

] encrypted multiple images based on chaos and multiple canonical transforms, where three linear canonical transforms such as FrFT, extended FrFT and FrT are utilized. Li et al. [44

44. H. Li, Y. Wang, H. Yan, L. Li, Q. Li, and X. Zhao, “Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform,” Opt. Lasers Eng. 51(12), 1327–1331 (2013). [CrossRef]

] designed a double-image encryption based on the chaos-based local pixel scrambling technique in GT domain, Arnold transform is used to scramble pixels at the local area. Wu et al. [45

45. J. Wu, X. Luo, and N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013). [CrossRef]

] proposed a four-image encryption method based on spectrum truncation, chaos and the multiple-order discrete fractional Fourier transform (MODFrFT), where the spectrum truncation is employed in discrete FT domain and the resultant performance is better than similar algorithm.

In this paper, an asymmetric double-image encryption algorithm based on the cascaded chaotic discrete fractional random transforms and logistic maps is proposed. First, two original images are combined into an enlarged one which is scrambled in a confusion process. The confusion process consists of a number of rounds, and the pixel positions of the enlarged image are relocated by using the cat maps in each round. Next, the enlarged image is divided into two new components. With this confusion process, the statistical information of original images can be destroyed thoroughly. Second, one component is directly separated into two phase masks and used as for encryption keys. By multiplying one phase mask, another component is transformed to a complex interim by using the discrete fractional random transform and its phase is extracted as one decryption key. By multiplying another phase mask, the amplitude of the interim is transformed to a complex image, in which the amplitude is the final ciphertext with stationary white noise distribution and the phase is used as another decryption key. The private keys include the encryption keys such as the initial values of the logistic maps and the fractional orders of the discrete fractional random transform. Simultaneously, two decryption keys produced in the encryption process also used as the private keys, which makes the encryption scheme is asymmetric and high resistance against to the various attacks such as chosen plaintext attack. Simulation results and security analysis verify the feasibility and effectiveness of the proposed method.

The rest of this article is organized as follows. In Section 2, the basic principles and the processed of encryption and decryption are introduced in detail. In Section 3, numerical simulation results and security analysis are given. Finally, the conclusion is given in Section 4.

2. Encryption and decryption process

2.1 Two phase masks generated from an image

An image can be separated into two phase masks by using the algorithm proposed in [38

38. X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011). [CrossRef]

], in which two phase masks can be considered as two unit vectors in a two-dimensional Cartesian coordinate system mathematically. For the sake of simplicity, the separation process is explained by using one-dimensional notations. Supposingf(x)denotes a normalized image with maxima as 1, the main steps are described as follows

  • (1) Generating a random distributionexp(iϕ1(x))as one phase mask, in which ϕ1(x)is a phase function distributed in the interval[0,2π].
  • (2) Denoting the angle betweenexp(iϕ1(x))and another phase maskexp(iϕ2(x))asα(x), the relation between α(x) and f(x)can be expressed as
    f(x)=22cos(πα(x)).
    (1)

    So the angle α(x)can be obtained as
    α(x)=πarccos2f2(x)2.
    (2)

  • (3) Another phase maskexp(iϕ2(x))can be given by
    exp(iϕ2(x))=exp(i(ϕ1(x)+α(x))).
    (3)

    Additionally, the normalized imagef(x)is the modulus of the sum of two phase masks and can be calculated by the following equation
    f(x)=|exp(iϕ1(x))+exp(iϕ2(x))|.
    (4)

2.2 Logistic map and chaos-based discrete fractional random transform

Chaos theory is a distinguished theory, which describes that the nonlinear dynamical systems convert from ordered state to disordered state. The dynamical systems are established based on various chaos functions such as logistic map, which are very sensitive to the initial parameters. With a chaotic map, a large number of random iterative values with the desirable properties of non-correlation, pseudo-randomness, ergodicity is generated. The chaotic maps have demonstrated great potential for information security, especially for image encryption.

The logistic map is a one-dimensional nonlinear chaos function and defined as
f(x)=px(1x).
(5)
The function is bounded for0p4, which is the system parameter known as bifurcation parameter. The iterative form of the logistic map is written as
xn+1=pxn(1xn).
(6)
where xn(0,1) is the iterative value and x0is the initial value. Whenp[3.5699456,4], the dynamical system is in chaotic state and slight variations of the initial parameter can yield a totally different random iterative value which is a non-periodic and non-converging sequence over time.

Similar to the fractional Fourier transform, the discrete fractional random transform (DFrRT) has mathematical properties such as linearity, multiplicity, index additivity and Parseval energy conservation and so on, which has widely been used in the implementation of image encryption cryptosystem. According to a one-dimensional signalxwhich size isN, the discrete fractional random transform [46

46. Z. Liu, H. Zhao, and S. Liu, “A discrete fractional random transform,” Opt. Commun. 255(4-6), 357–365 (2005). [CrossRef]

] with orderα can be expressed as follows
Fα(x)=Rαx,
(7)
whereRαis the kernel transform matrix of the DFrRT and can be written as
Rα=VDαVt.
(8)
Here, the matrixVsatisfies the constraintVVt=IandDαis a diagonal matrix as
Dα=diag{1,exp(i2παT),,exp(i2(N1)παT)},
(9)
where the coefficientTis a positive number and usually set to 1. The matrixVis generated with the eigenvectors of a symmetric random matrixSwhich can be constructed by using a N×Nreal random matrixQas follows
S=Q+Qt2.
(10)
For an imagefwhich size isN×Npixels, the two-dimensional DFrRT with two fractional ordersα is expressed as
F(α,α)(f)=R1αfR2α,
(11)
where R1αandR2αdenote the kernel transform matrices, respectively.

2.3 Double-image encryption and decryption processes

The proposed double-image encryption and decryption is based on the cascaded DFrRT and logistic maps. The encryption process is shown in Fig. 1
Fig. 1 Flowchart of encryption process.
. Let fi(i=1,2) denote two plaintext images with size of N×Npixels, the detailed procedures are described as follows

  • (1) In the combination and decomposition module, two original images are firstly normalized with maxima as 1, and combined into an enlarged one fewith size of N×2Npixels by the means of connecting f2withf1successively in the horizontal direction. When the enlarged imagefeis obtained, a confusion process which consists of a number of rounds is performed to scramble pixel positions. In the ith round, a cat map is used to relocate pixel positions, which is expressed by the following equation
    [xy]=[1piqipiqi+1][xy],
    (12)
    x=xmod2N,
    (13)
    y=ymodN.
    (14)

    The coefficient(pi,qi) is obtained by the Eqs. (15) and (16) as follows
    pi=(s1(i)×109)mod2N,
    (15)
    qi=(s2(i)×109)mod2N,
    (16)

    wheres1ands2are two random sequences generated by using Eq. (6) based on two logistic maps with the initial valuesx01andx02, respectively. Finally, the scrambled image feis decomposed into two new components denoted byfi(i=1,2). Thus, the statistical information of original images can be destroyed thoroughly.

In this module, the two initial values of above logistic maps can be used as encryption keys in the encryption and decryption processes. Figure 2(a)
Fig. 2 (a) Plaintext images “Lena” and “Baboon” and (b) two new components.
shows the two grayscale images “Lena” and “Baboon” with256×256pixels and 256 Gy levels. The original images used in this paper are selected from USC-SIPI image database [47]. The two new components are shown in Fig. 2(b).

  • (2) In the phase masks generation module, one phase mask exp(iϕ1)is produced randomly and another one exp(iϕ2)is generated based on the scrambled component f1by using Eq. (2) and (3).
  • (3) Another componentf2is multiplied by the phase maskexp(iϕ1)and transformed to a complex distribution by using the DFrRT with fractional orderα. The complex distribution can be written as
    hexp(iξ1)=F(α,α)(f2exp(iϕ1))=R1α(f2exp(iϕ1))R2α,
    (17)

    where R1αandR2αdenote the kernel transform matrices, respectively. The amplitudeh and phase functionξ1 of the complex distribution can be extracted with the operators |hexp(iξ1)|andarg{hexp(iξ1)}, respectively. The fractional order αis used as the encryption key and phase functionξ1is used as a decryption key.

  • (4) Similarly, the amplitude distribution his multiplied by the phase maskexp(iϕ2)and transformed to another complex distribution by using the DFrRT with fractional orderβ. The resultant distribution can be written as
    gexp(iξ2)=F(β,β)(hexp(iϕ2))=R1β(hexp(iϕ2))R2β,
    (18)

    where R1βandR2βdenote the kernel transform matrices, respectively. The amplitudegof the resultant distribution can be extracted with the operators|gexp(iξ2)|, which is the final ciphertext image with stationary white noise distribution. The corresponding phase function ξ2 is obtained by usingarg{gexp(iξ2)}, which is used as another decryption key while the fractional order βis used as the encryption key.

  • (1) In the inverse DFrRT modules, two interim complex matrices are obtained by the reverse DFrRTs with fractional ordersαandβ, respectively, which are expressed as
    h2exp(iϕ2)=F(β,β)(gexp(iξ2))=R1β(gexp(iξ2))R2β.
    (19)
    h1exp(iϕ1)=F(α,α)(h2exp(iξ1))=R1α(h2exp(iξ1))R2α.
    (20)
  • (2) In the phase masks combination module, the scrambled component f1can be recovered by using Eq. (4) as
    f1=|exp(iϕ1)+exp(iϕ2)|.
    (21)

    Simultaneously, another scrambled componentf2is derived by using the operator|h1exp(iϕ1)|, namely the amplitudeh1of the complex matrixh1exp(iϕ1) .

  • (3) In the inverse combination and decomposition module, the componentsf1, f2are assembled successively to form an enlarged imagefe, and thenfeis scrambled based on the random sequencess1, s2in accordance with the reversed order of confusion rounds in the encryption process. Finally, the scrambledfeis decomposed into two images, namely the decrypted images.

As pointed out in Ref [48

48. J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010). [CrossRef]

], though the exact optical implementation of DFrRT remains an open problem, the encryption scheme based on DFrRT can be approximately implemented optically in the Fourier domain with the help of a typical 4fsystem as shown in Fig. 4
Fig. 4 Optical implementation of DFrRT.
. Additionally, two gray scale images can be encrypted into one with same image size, which means the compression ratio of the proposed encryption scheme can achieve to 1:2.

3. Numerical simulation and security analysis

To verify the feasibility and effectiveness of the proposed encryption scheme, numerical simulations are performed on two original images “Lena” and “Baboon” shown in Fig. 2(a). The initial valuesx01, x02andx03 of three logistic maps are set to 0.19, 0.73 and 0.56, respectively. The fractional ordersαandβare set to 0.23 and 0.85, respectively. The encryption and decryption results are given in Fig. 5
Fig. 5 (a) Ciphertext image, (b) decrypted “Lena” and (c) decrypted “Baboon”.
. Figure 5(a) shows the ciphertext image with stationary white noise distribution. Figure 5(b) and 5(c) display the decrypted images with all correct keys.

Figure 6(a)
Fig. 6 Decrypted “Lena” with (a) incorrectx01, (b) incorrect x02and (c) incorrectx03.
-6(c) display the decrypted image “Lena” withx01=0.19+1.0×1015, x02=0.73+1.0×1013andx03=0.56+1.0×1015, respectively. Similar results are obtained for the decrypted image “Baboon”.

Figure 7(a)
Fig. 7 (a) Decrypted “Lena” with incorrect α, (b) decrypted “Baboon” with incorrect α, (c) decrypted “Lena” with incorrect β, (d) decrypted “Baboon” with incorrect β.
and 7(b) show the decrypted images with incorrectα=0.231. Figure 7(c) and 7(d) show the decrypted images with incorrectβ=0.8505. Figure 8(a)
Fig. 8 (a) Decrypted “Lena” with incorrect ξ1, (b) decrypted “Baboon” with incorrect ξ1, (c) decrypted “Lena” with incorrect ξ2, (d) decrypted “Baboon” with incorrect ξ2.
and 8(b) show the decrypted images with incorrect ξ1which is generated randomly. Figure 8(c) and 8(d) show the decrypted images with incorrectξ2.

In order to evaluate the quality of the decrypted image, the mean square error (MSE) between the original plaintext image and decrypted one is calculated as follows
MSE=1N2i=1Nj=1N|g(i,j)g(i,j)|2,
(22)
whereg(i,j)denotes the original plaintext image andg(i,j)denotes the corresponding decrypted one.

In order to test the sensitivity of the fractional orders, the decryption processes are performed by fixing one fractional order and varying the other. The relationship curves between the average MSE and the deviation of the fractional order are shown in Fig. 9
Fig. 9 Average MSE versus (a) the deviation of fractional orderαand (b) the deviation of fractional orderβ.
, in which the deviation ranges from −0.1 to 0.1 and the step is 0.005. The average MSE is computed as
MSEave=(MSEfirst+MSEsecond)/2,
(23)
whereMSEfirst andMSEsecondare the MSE values of two decrypted images, respectively. Obviously, the average MSE value approximates to zero whenαor β is correct while the value sharply increases whenαorβ slightly departs from the correct value, which indicate that any tiny fluctuation will lead to false decryption. Actually, the content of decrypted image cannot be recognized totally if the deviation of orderαis larger than 0.001 as illustrated in Fig. 7(a) and 7(b) as well as the content cannot be recognized if the deviation of orderβis larger than 0.0005 as illustrated in Fig. 7(c) and 7(d).

According to an ideal cryptosystem, the size of key space should be large to resist the brute-force attack, namely the total number of different keys used in the decryption process should be large enough. As pointed in [35

35. Y. Zhang and D. Xiao, “Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform,” Opt. Lasers Eng. 51(4), 472–480 (2013). [CrossRef]

], the size of key space should at least be larger than2100in order to obtain a high level of security. From the description of the proposed double-image encryption scheme, it is obvious that the chaotic permutation process in the combination and decomposition module is independent from the DFrRT. In the chaotic permutation process, the key spaces of the initial valuesx01andx02should be analyzed, which are denoted byS1andS2, respectively. In the DFrRT process, the key spaces of the initial valuex03and the fractional ordersαandβshould be analyzed, which are denoted byS3, S4andS5respectively. The entire key space of the cryptosystem isS1×S2×S3×S4×S5. From Fig. 6(a)-6(c), the initial valuex01, x02andx03maintain 15, 13 and 15 digits after decimal point respectively, soS1×S2×S3=1043. Figure 7(a)-7(d), the fractional ordersαandβmaintain 3 digits after decimal point, which meansS4×S5=106. Finally, the entire key space of the cryptosystem almost equals2162, which is enormous enough to resist brute-force attack.

Two aspects of statistical analysis are performed, one is testing on the histograms of the ciphertext and the decryption keys according to two groups of original images and the other is testing on the self-correlation values of adjacent pixels in the plaintext images, the ciphertext and the decryption keys. Figures 11(a)
Fig. 11 (a) Histograms of the ciphertext of “Lena” and “Baboon”, (b) histograms of the decryption keyξ1and (c) histogram of the decryption keyξ2.
-12(c)
Fig. 12 (a) Plaintext image “Aerial” and (b) plaintext image “Peppers”.
display the histograms of the ciphertext image and the decryption keys of “Lena” and “Baboon” shown in Fig. 2(a). Figures 12(a) and12 (b) display another two original images “Aerial” and “Peppers”, which histograms of the ciphertext and the decryption keys are displayed in Fig. 13(a)
Fig. 13 (a) Histograms of the ciphertext of “Aerial” and “Peppers”, (b) histograms of the decryption keyξ1and (c) histogram of the decryption keyξ2.
-13(c). In Fig. 11 and Fig. 13, the histograms not only of the ciphertext images but also of the decryption keys have similar distribution, which mean that an illegal user cannot obtain any useful information from this statistical property.

. In order to test the self-correlation values of adjacent pixels, the 2000 pairs of adjacent pixels are randomly selected in vertical, horizontal and diagonal directions from the plaintext images, the decryption keys of “Lena” and “Baboon”, respectively. LetNdenotes the total number of pixels selected from the image, the self-correlation coefficients of two adjacent pixels is calculated as follows
Cor=i=1N(xix¯)(yiy¯)(i=1N(xix¯)2)(i=1N(yiy¯)2),
(24)
where x¯=1/Ni=1Nxi and y¯=1/Ni=1Nyi.Table 1

Table 1. Correlation Results in the Plaintext Images, Ciphertext and Phase Distribution

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shows the results of self-correlation of the plaintext images, the ciphertext and the decryption keys, which indicates that the correlations of two adjacent pixels of the original image is significant while the results of ciphertext and the decryption keys are very low. Therefore, the proposed encryption scheme has strong capability of resisting this statistical data, which means that an illegal user cannot obtain any valid information from this analysis.

As an important requirement in the decryption process, the robustness against noise such as Gaussian random noise should be analyzed. A Gaussian random noise is added to the ciphertext image of “Lena” and “Baboon”, in which the noise interferes with the ciphertext in the following way
C=C(1+kG),
(25)
where C andC are the ciphertext image and the noise-affected ciphertext, respectively, k is a coefficient which denotes the noise strength andGis a Gaussian random noise with zero-mean and identity standard deviation. Figure 14
Fig. 14 Decrypted images with coefficientk: (a) k = 0.4, (b) k = 0.6, (c) k = 0.8 and (d) k = 1.0.
shows the decrypted images of “Lena” when k is set to 0.4, 0.6, 0.8 and 1.0, from which it can be seen that the content of the decrypted images can be recognized despite of noise interference and the quality of the decrypted image decreases as the noise level increases. Similar results are obtained for image “Baboon”. So, the proposed double-images encryption scheme has high robustness against noise attack.

The robustness against occlusion is mainly carried out to evaluate the performance of the encryption scheme against loss of data in the ciphertext image. To do so, the decryption process is performed on the occluded ciphertext image of “Lena” and “Baboon” with all correct private keys, where the ciphertext image is occluded partly. Figure 15(a)
Fig. 15 (a) Ciphertext with 50% occlusion from the left side, (b) decrypted “Lena” and (c) decrypted “Baboon”.
shows the occluded ciphertext whose pixel values at the left-top corner are replaced with 0 in simulation, namely the ciphertext is cropped by 50% from the left side. Figure 15(b) and 15(c) displays the corresponding decrypted images “Lena” and “Baboon” from Fig. 15(a). Apparently, the main information of the original plaintext images can be recognized visually from Fig. 15(b) and 15(c). In other words, the proposed encryption scheme shows enough robustness against loss of ciphertext data. Similar results can be obtained when the ciphertext is cropped by 50% from the right side.

Usually, there are four potential attacks including cipher only attack, known plaintext attack, chosen plaintext attack and chosen ciphertext attack, in which chosen plaintext attack is the most powerful attack. As pointed out in [35

35. Y. Zhang and D. Xiao, “Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform,” Opt. Lasers Eng. 51(4), 472–480 (2013). [CrossRef]

], the cryptosystem should resist other attacks if it can resist chosen plaintext attack. For the proposed double-image encryption scheme, supposing an illegal user has known the encryption keys such as the initial valuesx01, x02,x03and the fractional ordersα,β, he can encrypt two fake plaintext images and produce the decryption keysξ1andξ2. Then, he can use obtained decryption keys to decrypt the original ciphertext. The images “Aerial” and “Peppers” shown in Fig. 12(a) and 12(b) are chosen as fake plaintext images, the decryption keysξ1andξ2are obtained after encryption and used to decrypt the ciphertext of “Lena” and “Baboon”. Figure 16(a)
Fig. 16 (a) Decrypted “Lena” and (b) decrypted “Baboon”.
and 16(b) display the decrypted images of “Lena” and “Baboon”, from which any valuable information cannot be obtained though the content of the fake images “Aerial” and “Peppers” can be seen faintly. As pointed out in [49

49. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005). [CrossRef] [PubMed]

51

51. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010). [CrossRef] [PubMed]

], the encryption algorithms based on double random phase encryption has been demonstrated vulnerable to chosen plaintext attacks due to the linearity of the cryptosystem. However, the chaotic DFrRT of the proposed encryption scheme not only makes the encryption process more complex but also strengthens the nonlinearity both in spatial domain and DFrRT domain. Thus, the proposed encryption scheme can resist the potential types of attacks.

4. Conclusion

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant number U1334211.

References and links

1.

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef] [PubMed]

2.

B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997). [CrossRef]

3.

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004). [CrossRef] [PubMed]

4.

L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14(19), 8552–8560 (2006). [CrossRef] [PubMed]

5.

W. Chen and D. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284(16–17), 3913–3917 (2011). [CrossRef]

6.

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000). [CrossRef] [PubMed]

7.

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000). [CrossRef] [PubMed]

8.

B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003). [CrossRef] [PubMed]

9.

Y. Sheng, Z. Xin, M. S. Alam, L. Xi, and L. Xiao-Feng, “Information hiding based on double random-phase encoding and public-key cryptography,” Opt. Express 17(5), 3270–3284 (2009). [CrossRef] [PubMed]

10.

M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009). [CrossRef] [PubMed]

11.

W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18(26), 27095–27104 (2010). [CrossRef] [PubMed]

12.

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain,” Opt. Express 20(4), 3853–3865 (2012). [CrossRef] [PubMed]

13.

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51(3), 317–324 (2013). [CrossRef]

14.

M. R. Abuturab, “Noise-free recovery of color information using a joint-extended gyrator transform correlator,” Opt. Lasers Eng. 51(3), 230–239 (2013). [CrossRef]

15.

N. Zhou, Y. Wang, and J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011). [CrossRef]

16.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009). [CrossRef]

17.

G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005). [CrossRef] [PubMed]

18.

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A, Pure Appl. Opt. 8(5), 391–397 (2006). [CrossRef]

19.

A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48(31), 5933–5947 (2009). [CrossRef] [PubMed]

20.

A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express 19(24), 24023–24029 (2011). [CrossRef] [PubMed]

21.

Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, and S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (2011). [CrossRef]

22.

X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284(1), 148–152 (2011). [CrossRef]

23.

X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21–22), 4280–4284 (2012). [CrossRef]

24.

H. E. Hwang, H. T. Chang, and W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009). [CrossRef] [PubMed]

25.

H. T. Chang, H. E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011). [CrossRef]

26.

H. T. Chang, H. E. Hwang, C. L. Lee, and M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011). [CrossRef] [PubMed]

27.

J. J. Huang, H. E. Hwang, C. Y. Chen, and C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012). [CrossRef]

28.

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012). [CrossRef]

29.

S. Liansheng, X. Meiting, and T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013). [CrossRef] [PubMed]

30.

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008). [CrossRef]

31.

H. Li and Y. Wang, “Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain,” J. Mod. Opt. 55(21), 3601–3609 (2008). [CrossRef]

32.

Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010). [CrossRef] [PubMed]

33.

Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, and S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013). [CrossRef]

34.

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, J. Dai, and S. Liu, “Double image encryption by using Arnold transform and discrete fractional angular transform,” Opt. Lasers Eng. 50(2), 248–255 (2012). [CrossRef]

35.

Y. Zhang and D. Xiao, “Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform,” Opt. Lasers Eng. 51(4), 472–480 (2013). [CrossRef]

36.

H. Li and Y. Wang, “Double-image encryption based on discrete fractional random transform and chaotic maps,” Opt. Lasers Eng. 49(7), 753–757 (2011). [CrossRef]

37.

L. Sui, H. Lu, Z. Wang, and Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014). [CrossRef]

38.

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011). [CrossRef]

39.

X. Wang and D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express 20(11), 11994–12003 (2012). [CrossRef] [PubMed]

40.

L. Sui, H. Lu, X. Ning, and Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014). [CrossRef]

41.

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46(2), 117–123 (2008). [CrossRef]

42.

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009). [CrossRef]

43.

N. Singh and A. Sinha, “Chaos based multiple image encryption using multiple canonical transforms,” Opt. Laser Technol. 42(5), 724–731 (2010). [CrossRef]

44.

H. Li, Y. Wang, H. Yan, L. Li, Q. Li, and X. Zhao, “Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform,” Opt. Lasers Eng. 51(12), 1327–1331 (2013). [CrossRef]

45.

J. Wu, X. Luo, and N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013). [CrossRef]

46.

Z. Liu, H. Zhao, and S. Liu, “A discrete fractional random transform,” Opt. Commun. 255(4-6), 357–365 (2005). [CrossRef]

47.

Original images: http://sipi.use.edu/database/database.php.

48.

J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010). [CrossRef]

49.

A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005). [CrossRef] [PubMed]

50.

X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006). [CrossRef] [PubMed]

51.

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010). [CrossRef] [PubMed]

OCIS Codes
(060.4510) Fiber optics and optical communications : Optical communications
(100.2000) Image processing : Digital image processing

ToC Category:
Image Processing

History
Original Manuscript: March 26, 2014
Revised Manuscript: April 21, 2014
Manuscript Accepted: April 21, 2014
Published: April 24, 2014

Citation
Liansheng Sui, Kuaikuai Duan, Junli Liang, and Xinhong Hei, "Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps," Opt. Express 22, 10605-10621 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10605


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References

  1. P. Refregier, B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef] [PubMed]
  2. B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997). [CrossRef]
  3. G. Situ, J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004). [CrossRef] [PubMed]
  4. L. Chen, D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14(19), 8552–8560 (2006). [CrossRef] [PubMed]
  5. W. Chen, D. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284(16–17), 3913–3917 (2011). [CrossRef]
  6. G. Unnikrishnan, J. Joseph, K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000). [CrossRef] [PubMed]
  7. B. Zhu, S. Liu, Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000). [CrossRef] [PubMed]
  8. B. Hennelly, J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003). [CrossRef] [PubMed]
  9. Y. Sheng, Z. Xin, M. S. Alam, L. Xi, L. Xiao-Feng, “Information hiding based on double random-phase encoding and public-key cryptography,” Opt. Express 17(5), 3270–3284 (2009). [CrossRef] [PubMed]
  10. 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(25), 22462–22473 (2009). [CrossRef] [PubMed]
  11. W. Chen, X. Chen, “Space-based optical image encryption,” Opt. Express 18(26), 27095–27104 (2010). [CrossRef] [PubMed]
  12. W. Chen, X. Chen, C. J. R. Sheppard, “Optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain,” Opt. Express 20(4), 3853–3865 (2012). [CrossRef] [PubMed]
  13. M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51(3), 317–324 (2013). [CrossRef]
  14. M. R. Abuturab, “Noise-free recovery of color information using a joint-extended gyrator transform correlator,” Opt. Lasers Eng. 51(3), 230–239 (2013). [CrossRef]
  15. N. Zhou, Y. Wang, J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011). [CrossRef]
  16. A. Alfalou, C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009). [CrossRef]
  17. G. Situ, J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005). [CrossRef] [PubMed]
  18. G. Situ, J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A, Pure Appl. Opt. 8(5), 391–397 (2006). [CrossRef]
  19. A. Alfalou, A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48(31), 5933–5947 (2009). [CrossRef] [PubMed]
  20. A. Alfalou, C. Brosseau, N. Abdallah, M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express 19(24), 24023–24029 (2011). [CrossRef] [PubMed]
  21. Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (2011). [CrossRef]
  22. X. Wang, D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284(1), 148–152 (2011). [CrossRef]
  23. X. Wang, D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21–22), 4280–4284 (2012). [CrossRef]
  24. H. E. Hwang, H. T. Chang, W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009). [CrossRef] [PubMed]
  25. H. T. Chang, H. E. Hwang, C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011). [CrossRef]
  26. H. T. Chang, H. E. Hwang, C. L. Lee, M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011). [CrossRef] [PubMed]
  27. J. J. Huang, H. E. Hwang, C. Y. Chen, C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012). [CrossRef]
  28. X. Deng, D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012). [CrossRef]
  29. S. Liansheng, X. Meiting, T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013). [CrossRef] [PubMed]
  30. H. Li, Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008). [CrossRef]
  31. H. Li, Y. Wang, “Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain,” J. Mod. Opt. 55(21), 3601–3609 (2008). [CrossRef]
  32. Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010). [CrossRef] [PubMed]
  33. Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013). [CrossRef]
  34. Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, J. Dai, S. Liu, “Double image encryption by using Arnold transform and discrete fractional angular transform,” Opt. Lasers Eng. 50(2), 248–255 (2012). [CrossRef]
  35. Y. Zhang, D. Xiao, “Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform,” Opt. Lasers Eng. 51(4), 472–480 (2013). [CrossRef]
  36. H. Li, Y. Wang, “Double-image encryption based on discrete fractional random transform and chaotic maps,” Opt. Lasers Eng. 49(7), 753–757 (2011). [CrossRef]
  37. L. Sui, H. Lu, Z. Wang, Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014). [CrossRef]
  38. X. Wang, D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011). [CrossRef]
  39. X. Wang, D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express 20(11), 11994–12003 (2012). [CrossRef] [PubMed]
  40. L. Sui, H. Lu, X. Ning, Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014). [CrossRef]
  41. N. Singh, A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46(2), 117–123 (2008). [CrossRef]
  42. N. Singh, A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009). [CrossRef]
  43. N. Singh, A. Sinha, “Chaos based multiple image encryption using multiple canonical transforms,” Opt. Laser Technol. 42(5), 724–731 (2010). [CrossRef]
  44. H. Li, Y. Wang, H. Yan, L. Li, Q. Li, X. Zhao, “Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform,” Opt. Lasers Eng. 51(12), 1327–1331 (2013). [CrossRef]
  45. J. Wu, X. Luo, N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013). [CrossRef]
  46. Z. Liu, H. Zhao, S. Liu, “A discrete fractional random transform,” Opt. Commun. 255(4-6), 357–365 (2005). [CrossRef]
  47. Original images: http://sipi.use.edu/database/database.php .
  48. J. Lang, R. Tao, Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010). [CrossRef]
  49. A. Carnicer, M. Montes-Usategui, S. Arcos, I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005). [CrossRef] [PubMed]
  50. X. Peng, H. Wei, P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006). [CrossRef] [PubMed]
  51. W. Qin, X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010). [CrossRef] [PubMed]

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