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Optical color image encryption using position multiplexing technique based on phase truncation operation
Optics & Laser Technology 57 (2014) 110–118
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Optical color image encryption using position multiplexing technique based on phase truncation operation Xiangling Ding a,n, Guangyi Chen b a b
Department of Physics and Information Engineering, Huaihua University, Huaihua 418008, China College of information science and technology, Hunan agricultural University, Changsha 410128, China
ar t ic l e i nf o
a b s t r a c t
Article history: Received 6 July 2013 Received in revised form 8 September 2013 Accepted 2 October 2013 Available online 29 October 2013
We propose an optical color image cryptosystem based on position multiplexing technique and phase truncation operation. Compared with the reported color image encryption method, we employ the position multiplexing technique to encrypt the color image in only one spatial channel. Meanwhile, our proposed method can maintain the nonlinear characteristic of the cryptosystem and avoid various types of the currently existing attacks, especially the iterative attack. Simulation results are presented to demonstrate the security and robustness performance of the proposed method. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Optical color image encryption Phase truncation operation The position multiplexing
1. Introduction In the past several decades, optical information processing has been obtained gigantic achievements because of the characteristics of high-speed and paralleling. Up to now, the hottest method in this field is the double random phase-encoding technique (DRPE) [1] that proposed by Refregier and Javidi in 1995. Then, a lot of optical encryption techniques [2–13] that based on DRPE principle have been proposed such as extended fractional Fourier-transform encryption system [2], fractional Fourier-transform encryption system [3], and Fresnel diffraction encryption system [4], etc. However, the encryption techniques proposed in the above mentioned papers are designed for a gray or binary image. When a monochromatic light is used to illuminate a real color image in all these aforementioned encryption techniques, color information of a deciphered image is lost. So, some encryption techniques [14–23] have been further developed to encode a color image. L. Chen and D. Zhao have employed wavelength multiplexing to encrypt a color image based on lensless Fresnel transform holograms [14]. W. Chen, X. Chen, and Colin J. R. Sheppard have presented an optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain [19]. Z. Liu, Y. Zhang and W. Liu etc. have also proposed an optical color image hiding scheme based on chaotic mapping and Hartley transform [23]. However, the above proposed color cryptosystems belong to the category of linear symmetric cryptosystems, in which the encryption key is identical to the decryption key. From the
n
Corresponding author. Tel./fax: þ 86 745 2851011. E-mail address: [email protected] (X. Ding).
0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.10.002
view of cryptography, the linear symmetric cryptosystem would suffer from several problems under the network environment, such as the safe of distribution and management of keys. In order to solve these problems, some gray asymmetric cryptosystems [24,25] have been proposed based on the phase-truncated Fourier transforms (PTFTs) [24], with which one can produce a ciphertext as real-valued and stationary white noise by using two public random phase keys, while a legal user can retrieve the plaintext using another two private phase keys that generated from the encryption process. Then, W. Chen and X. Chen have further proposed two different methods to encrypt a color image [26]. One is the conventional three-channel color image encryption in the Fresnel domain based on the multiplewavelength. The other is the indexed image encryption, in which the indexed image firstly is converted from a color image and then onestep operation in the asymmetric system. However, the first proposed method enciphers the color image in three different single spatial channels of arrangement at a time. The second method although encrypts in only one spatial channel, the outline of the deciphered images cannot be recognized when the encoded matrix occurs a little change in the process of encryption and decryption. Furthermore, the above two proposed methods are changed into the hybrid encryption system, which contains nonlinear components as well as linear components. Due to the linear components, the proposed methods still suffer from several problems, such as the safe of distribution and management of keys. Therefore, an optical color image cryptosystem, which can encrypt in only one spatial channel at a time, also can maintain the nonlinear properties of encryption structure, and has good robustness, has yet to be developed.
X. Ding, G. Chen / Optics & Laser Technology 57 (2014) 110–118
In this paper, an optical color image cryptosystem, which can encrypt in only one spatial channel at a time and maintain the nonlinear properties of encryption structure, is proposed. The encryption process can be realized optically with the help of the computer and the decryption process can also be implemented optically like the decryption of the PTFTs-based asymmetric system. In this method, the color image to be encrypted is first decomposed into three color components (corresponding to red, green and blue). Then the position technique and the PTFTs are used to transform the three color components into a gray ciphertext, which looks like white noise, in only one spatial channel with the asymmetric cryptosystem [24]. During the decryption process, the three color components can be recovered by two private keys, which generated in the encryption procedure and are different from the encryption keys. Compared with the proposed color image encryption method [20], the obvious advantages of the method are not only the one spatial channel which is used for encryption process instead of three channels for the using of the position multiplexing technique but also the maintained nonlinear characteristic of the cryptosystem because of the nonlinear phase-truncated Fourier transforms (PTFTs) and the position multiplexing technique. Thus, due to maintain the nonlinear characteristic of the cryptosystem, the same disadvantages resulted from the linearity and symmetric cryptosystem could be avoided and therefore high robustness could be achieved. Numerical simulations are demonstrated to show the security and robustness of the proposed methods.
2. Encryption and decryption process
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(corresponding to red, green and blue). Then, three decomposed color components that combined with three different random phase encryption keys ER ðx; yÞ, EG ðx; yÞ and EB ðx; yÞ respectively are located in the position ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ of the same plane along the axis x and the axis y to form a gray image using the position multiplexing technique. Thus, the input information can be expressed as f o ðx; yÞ ¼ ½f R ðx; yÞ ER ðx; yÞ δðx a1 ; y b1 Þ þ ½f G ðx; yÞ EG ðx; yÞ δðx a2 ; y b2 Þ þ ½f B ðx; yÞ EB ðx; yÞ δðx a3 ; y b3 Þ
ð3Þ
where a proper choice of the parameters ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ would guarantee that the three color components can be reconstructed without being superimposed [12]. Through Fourier transformation and nonlinear operation of phase-truncation, the encoded image can be achieved as follow: c1 ðu; vÞ ¼ TrfFT½f o ðx; yÞg:
ð4Þ
The final ciphertext cðx; yÞ can be obtained by the following step: cðx; yÞ ¼ TrfIFT½c1 ðu; vÞ UBðu; vÞg
ð5Þ
where Bðu; vÞ is another random phase encryption key in the Fourier transform domain. At the same time, a pair of decryption phase keys P 1 ðx; yÞ and P 2 ðu; vÞ should also be generated by the following additional two steps in encryption process. P 1 ðx; yÞ ¼ RefIFT½c1 ðu; vÞ Bðu; vÞg
ð6Þ
P 2 ðu; vÞ ¼ RefFT½f o ðx; yÞg
ð7Þ
It can be seen from Eqs. (4)–(7) that c1 ðu; vÞ, cðx; yÞ and P 1 ðx; yÞ, P 2 ðu; vÞ have simple relations
2.1. Phase-truncated Fourier transforms Before describing the proposed approach, we first review the phase-truncated Fourier transforms (PTFTs) that proposed in the PTFTs-based asymmetric system by Q. Wan and X. Peng in Ref. [24]. PTFT is a process of Fourier transform but with the nonlinear operation of phase truncation that means only the amplitude part of the Fourier spectrum is retained while the phase part of the spectrum is truncated. Let FTð UÞ, Trð UÞ and Reð U Þ denote the operator of Fourier transform, the operator of phase truncation and the operator of phase reservation, respectively. Assume a Fourier transformation of kðx; yÞ is expressed as Kðu; vÞ ¼ FT½kðx; yÞ ¼ jKðu; vÞjexp½iβðu; vÞ, where ðx; yÞ and ðu; vÞ denote the coordinate of the spatial domain and the frequency domain respectively, the phase truncation and the phase reservation can be respectively expressed as follows: Tr½Kðu; vÞ ¼ jKðu; vÞj
ð1Þ
Re½Kðu; vÞ ¼ exp½iβðu; vÞ
ð2Þ
2.2. Encryption process From now on, let us show the encryption process of the color image based on the position multiplexing technique and the PTFTs. The flowchart of the encryption process is shown in Fig. 1, where the symbol and n represents the multiplication operation and the convolution operation, respectively. Let f ðx; yÞ denotes the original color image to be encrypted, ER ðx; yÞ ¼ exp½jμ1 ðx; yÞ, EG ðx; yÞ ¼ exp½jμ2 ðx; yÞ, EB ðx; yÞ ¼ exp½jμ2 ðx; yÞ and Bðu; vÞ ¼ exp½jμ4 ðu; vÞ are four random phase encryption keys where μ1 ðx; yÞ,μ2 ðx; yÞ,μ3 ðx; yÞ and μ4 ðu; vÞ are white sequences statistically independent in the interval ½0; 2π. Firstly, the original color image f ðx; yÞ is decomposed into three color components f R ðx; yÞ, f G ðx; yÞ and f B ðx; yÞ
c1 ðu; vÞ P 2 ðu; vÞ ¼ FT½f o ðx; yÞ
ð8Þ
cðx; yÞ P 1 ðx; yÞ ¼ IFT½c1 ðu; vÞ Bðu; vÞ
ð9Þ
It must be pointed out that the ciphertext cðx; yÞ generated from Eqs. (4) and (5) is a real-valued function, which is more convenient in the process of record and transmission. 2.3. Decryption process The flowchart of the decryption process, which is similar to the decryption process of the PTFTs-based cryptosystem, is shown in Fig. 2. Firstly, the c1 ðu; vÞ can be obtained by operation of Fourier transform and nonlinear operation of phase truncate with the decryption keys P 1 ðx; yÞ. The result c1 ðu; vÞ can be expressed as: c1 ðu; vÞ ¼ TrfFT½cðx; yÞ P 1 ðx; yÞg
ð10Þ
Secondly, c1 ðu; vÞ multiplied with another private key P 2 ðu; vÞ is inverse Fourier transformed. The result can be expressed as f o ðx; yÞ ¼ IFT½c1 ðu; vÞ P 2 ðu; vÞ
ð11Þ
Thirdly, the deciphered result is obtained by the nonlinear operation of phase-truncation, of which process can be described as f M ðx; yÞ ¼ Tr½f o ðx; yÞ ¼ f R ðx; yÞ δðx a1 ; y b1 Þ þ f G ðx; yÞ δðx a2 ; y b2 Þ þf B ðx; yÞ δðx a3 ; y b3 Þ
ð12Þ
Finally, three original color components (f R ðx; yÞ, f G ðx; yÞ and f B ðx; yÞ) can be extracted from the deciphered result and combined into the original color image. 2.4. Performance analysis and optical setup From the processes of encryption and decryption as described above, it can be seen that the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ), as well as the public keys (ER ðx; yÞ, EG ðx; yÞ,
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Fig. 1. Flowchart of the encryption process.
Fig. 2. Flowchart of the decryption process.
EB ðx; yÞ and Bðu; vÞ), are not required for a correct decryption because of the nonlinear phase-truncated Fourier transform. Meanwhile, because of the position multiplexing technique the proposed method encrypts the original color image in only one spatial channel. Moreover, because the position multiplexing technique is employed in the encryption process, it only changes the input image of the PTFTs-based asymmetric system into the located image of the three color components and any linear component is not introduced. So, the nonlinear characteristic is still maintained in our proposed color image cryptosystem. In addition, our proposed method derives from the double images encryption method [11], which was proposed by X. Wang and D. Zhao. So, our proposed color image cryptosystem can also resist several attacks, such as: (1) brute force attack; (2) chosen plaintext attacks; (3) known public key attack; Similar to the double images encryption method [11], our proposed method can also resist the iterative attack [27]. The analysis is as follows: The iterative attack [27] can be avoided as long as the constraints are broken, because the iterative attack works based on the phase retrieval algorithm. Supposed an attacker, who has intercepted the ciphertext cðx; yÞ, attempts to retrieve f ðx; yÞ with the public keys (ER ðx; yÞ, EG ðx; yÞ, EB ðx; yÞ and Bðu; vÞ) based on the phase retrieval algorithm, he must execute two iterative transformations. The first step is to obtain c1 ðu; vÞ by using the public key Bðu; vÞ and the ciphertext cðx; yÞ. The constraints of this step are the ciphertext cðx; yÞ and the public keys (Bðu; vÞ). Obviously, the c1 ðu; vÞ can correctly be obtained with iterative transformations. The second step is to gain f ðx; yÞ by executing the iterative transformations. The constraints of this step are the public keys (ER ðx; yÞ, EG ðx; yÞ, EB ðx; yÞ), the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ) and c1 ðu; vÞ. The constraints are broken and f ðx; yÞ is incorrectly obtained with iterative transformations if we do not make the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ) public. From above discussions, compared with the reported color image encryption method [20], our proposed color cryptosystem can encipher the original color image in the only one spatial channel instead of three channels because of the position multiplexing technique. Meanwhile, our proposed cryptosystem does not introduce any linear component and maintains the nonlinear characteristic due to using the PTFTs and the position multiplexing technique. It is worth noting that although the encryption process is complicated, the decryption process, which is the same as the decryption of the PTFTs-based asymmetric system [24], is very simple. The complexity of the encryption process depends on the position multiplexing and the operation of phase reservation. Although the position multiplexing can be implemented using a joint transform correlator architecture [28], a more simple digital
Fig. 3. An optical setup for color image cryptosystem. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
method is employed to complete this process in our optical scheme for decreasing the complexity of the position multiplexing implementation by the computer. For the operation of phase reservation, a reference beam should be split from the light source to record the truncated phase by interferometry. An optical setup for the proposed color image cryptosystem with the help of some optoelectronic devices is shown in Fig. 3, where SLMs is a space-light modulator. The SLMs and CCD plane are controlled by a computer. In the optical encryption process, the input original color image f ðx; yÞ is firstly decomposed into three color components. Secondly, the three decomposed color components and the three different random phase encryption keys ER ðx; yÞ, EG ðx; yÞ and EB ðx; yÞ, which are located in the position ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ of the same plane along with the axis x and the axis y, are respectively displayed on the SLM1 and SLM2. Thirdly, the SLMs are illuminated by a monochromatic light. Fourthly, the encoding image g 1 ðu; vÞ is recorded by the CCD camera in the focal plane. In order to obtain the encryption result, we just repeat the above operations with the encoding image g 1 ðu; vÞ and Bðu; vÞ respectively displayed on the SLM1 and SLM2 and also illuminated by the same monochromatic light. Finally, the encryption result is recorded by the CCD camera. For the condition of encryption, a reference beam should be split from the light source to record the truncated phase by interferometry. Similarly in the optical decryption process, the ciphertext cðx; yÞ and the first decryption key P 1 ðx; yÞ are firstly displayed on the SLM1 and SLM2 respectively and illuminated by a uniform plane wave, and then the encoding image c1 ðu; vÞ is recorded by the CCD plane. For the decrypted result, we just repeat the above operations with c1 ðu; vÞ and P 2 ðu; vÞ respectively displayed on the SLM1 and SLM2 and also illuminated by the same plane wave. Finally, the decrypted color components are combined into decrypted color image in the computer. However, owing to the current resource limitation in our laboratory, we just made some numerical simulations to verify the security and robustness of the proposed method.
3. Numerical simulation Computer simulations are performed on a Matlab 7.0.0 (R14) platform to verify the security and robustness of this cryptography. An original color image comprising 128 128 pixels, as shown in Fig. 4(a), is used as the input image to be enciphered. In order to guarantee the three color components can be retrieved without being superimposed, we locate the three color components in the centered and zero-padded images (512 512 pixels) as the input
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Fig. 4. (a) original color image; (b) the ciphertext and (c) the correct decrypted image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Fig. 5. the phase part of decryption keys (a) P 1 ðx; yÞ and (b) P 2 ðu; vÞ.
Fig. 6. Decrypted results with (a) no keys; (b) arbitrarily chosen decryption keys; (c) the encryption keys; (d) the correct key P 1 ðx; yÞ while P 2 ðu; vÞ is wrong and (e) the correct key P 2 ðu; vÞ while P 1 ðx; yÞ is wrong. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
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Fig. 7. (a) Arbitrary fake color image; (b) P 1 ðx; yÞ and (c) P 2 ðu; vÞ generated from (a); Decrypted result (d) with the fake phase keys (b) and (c). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
Fig. 8. (a) The decrypted result with the iteration numbers (the iteration times of the first step is 200, the iteration times of the second step is 100); Relation between iteration times and the MSE (b) in the first step (between cðu; vÞ and its estimate); (c) in the second step (between the original color image and its estimate) and (d) in the second step (between the three color components of the original image and its estimate). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
plaintexts, and the three parameters ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ are set as ( 160, 0), (0, 0) and (160, 0), respectively. The ciphertext encrypted by Eqs. (4) and (5) is shown in Fig. 4(b), which looks like white noise. Fig. 4(c) shows the correct decrypted image. The two decryption keys generated in the encryption process and employed for the correct decryption are represented in Fig. 5. To measure the difference between the original image and the retrieved image quantitatively, the mean-square-error (MSE)
function between two images is defined as MSE ¼
1 L ′ ∑ ðf jf i jÞ2 Li¼1 i ′
ð13Þ
where L and f i denote the number of pixels and the estimate of f i , respectively. The security of the proposed method has been investigated. Firstly, the decrypted images obtained by using no keys, arbitrarily
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Fig. 9. Simulation results (a) collision color image; (b) decrypted key, mðu; vÞ generated through the iterative Fourier transforms; (c) decrypted key nðx; yÞ generated through the modified Gerchberg–Saxton algorithm; (d) retrieved the collision color image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Fig. 10. Recognition results obtained (a) between a retrieved original color image and encrypted image; (b) between a retrieved collision color image and encrypted image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
chosen decryption keys, public keys, only P 1 ðx; yÞ and P 2 ðu; vÞ but the other correct keys, are depicted in Fig. 6(a)–(e), respectively. The MSE values corresponding to Fig. 6(a)–(e) for red, green and blue components are (0.0695, 0.0545, 0.1833), (0.0678, 0.0811, 0.1131), (0.0694, 0.0545, 0.1831), (0.0678, 0.0811, 0.1131) and (0.0834, 0.0782, 0.1741). It can be seen from Fig. 6(a)–(e) that any attempt at the decryption of the ciphertext without all correct decryption keys will fail. Secondly, we test the ability of the proposed method against chosen plaintext attack. Obviously, it is easy to generate two phase decryption keys by using any arbitrary chosen color image and the four public keys. Fig. 7(a) is any arbitrary chosen color image. The two phase decryption keys that generated from the above color images can be shown in Fig. 7(b) and (c) respectively. As can be seen from Fig. 7(d), the decoded color image provides no valuable
information but the arbitrary chosen color image by using the above two fake decryption keys. Thirdly, the ability of the proposed method against the iterative attack is represented in Fig. 8. Fig. 8(a) is illustrated that our proposed color image cryptosystem can resist against the iterative attack effectively without any knowledge of the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ). Fig. 8(b)–(d) illustrate the convergence of the iterative attack through the MSE. The relation between iteration times and the MSE (between c′ðu; vÞ and its estimate) in the first step is shown in Fig. 8(b). Fig. 8(c) shows the relation between iteration times and the MSE (between the original color imagef ðx; yÞ and its estimate) in the second step. Fig. 8 (d) shows the relation between iteration times and the MSE (between the three color components of the original image and its estimate) in the second step. From the above attack results, it can be
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Fig. 11. Simulation of occlusion attack: (a), (c) and (e) are the occlusion encrypted images; (b), (d) and (f) are the corresponding retrieved images from (a), (c) and (d), respectively. (g) Gaussian-noised encrypted image with variance value 0.1, and (h) corresponding decrypted image from (g). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
obtained that any attempt at the decryption of ciphertext without any knowledge of the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ andða3 ; b3 Þ), even with public keys and the iterative attack, will fail.
In this section, the collision attack [29] of our proposed method is simulated. A color image of size 128 128 pixels was taken as a collision image and is shown in Fig. 9(a). Fig. 9(b) and (c) shows
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Fig. 12. The MSE curves from (a) the test of occlusion attack; (b) the test of Gaussian noise attack. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
another two different decryption keys, mðu; vÞ and nðx; yÞ, which are generated by using the iterative Fourier transforms and the modified Gerchberg–Saxton algorithm [30], respectively. Fig. 9(d) shows the decrypted collision image. These results infer that an unauthorized user can generate an arbitrary color image independent of the original color image through the encrypted image. But we can check the authorized user and the unauthorized user by using the joint transform correlator architecture [28], which is used the same as the method in Ref. [29]. Fig. 10(a) and (b) shows the recognition results obtained between a retrieved original color image (shown in Fig. 4(c)) and encrypted image (shown in Fig. 4(b)) and between a retrieved collision color image (shown in Fig. 9(d)) and encrypted image (shown in Fig. 4(b)). So, we can easily validate the authorized user with the aid of a recognition scheme. In the last section, the robustness of the proposed method has been checked by the occlusion and noise attacks on the encrypted images. The occluded encrypted images with 25%, 75%, and 90% occlusion sizes are displayed in Fig. 11(a), (c) and (e), respectively. Their corresponding retrieved images, by using the decryption process shown in Fig. 2 with the correct decryption keys, are shown in Fig. 11(b), (d) and (f), respectively. Corresponding to Fig. 11(b), (d) and (f) for red, green and blue components, the MSE values are (0.0096, 0.0090, and 0.0140), (0.0693, 0.0591, and 0.1167) and (0.1172, 0.0995, 0.2004). For noise attack, the Gaussian-noised encrypted image with variance value 0.1 is presented in Fig. 11(g) and the corresponding retrieved image is depicted in Fig. 11(h). The MSE values corresponding to Fig. 11(h) for red, green and blue components are (0.0613, 0.0547, and 0.0886). It can be gained that the encrypted image has lost up to 90% occlusion size or has been contaminated by Gaussian noise with variance value 0.1, but the outline of the recovered images can still be recognized. The MSE curves of the occlusion attack and the Gaussian-noise attack in the ranges [0.1, 0.9] and [0, 0.5] are represented in Fig. 12(a) and (b), respectively. It can be seen from Fig. 12(a) that the MSE value increases with an increase in occlusion size, where the blue channel increases rapidly in comparison to red and green channels. The MSE versus Gaussian-noise is shown in Fig. 12(b), which indicates that the MSE value increases with an increase in Gaussian noise. The above results fully show that our proposed method can resist the occlusion attack and the Gaussian-noise attack.
4. Conclusion In summary, we propose an optical color image encryption technique that based on an asymmetric algorithm, in which the
decryption keys are different from the encryption keys. The proposed method that combines the position multiplexing technique and the nonlinear phase-truncated Fourier transforms encrypts the three color components of the original color image in only one spatial channel instead of three channels. Meanwhile, it does not introduce any linear component and maintains the nonlinear characteristic because of the nonlinear PTFTs and the position multiplexing technique. Additional, an optical architecture based on the computer and two SLMs is used to encipher or decipher the original color image. In the last section the simulation results show that the method has a very high security and robustness against various types of the currently existing attacks.
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Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Optical color image encryption using position multiplexing technique based on phase truncation operation Xiangling Ding a,n, Guangyi Chen b a b
Department of Physics and Information Engineering, Huaihua University, Huaihua 418008, China College of information science and technology, Hunan agricultural University, Changsha 410128, China
ar t ic l e i nf o
a b s t r a c t
Article history: Received 6 July 2013 Received in revised form 8 September 2013 Accepted 2 October 2013 Available online 29 October 2013
We propose an optical color image cryptosystem based on position multiplexing technique and phase truncation operation. Compared with the reported color image encryption method, we employ the position multiplexing technique to encrypt the color image in only one spatial channel. Meanwhile, our proposed method can maintain the nonlinear characteristic of the cryptosystem and avoid various types of the currently existing attacks, especially the iterative attack. Simulation results are presented to demonstrate the security and robustness performance of the proposed method. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Optical color image encryption Phase truncation operation The position multiplexing
1. Introduction In the past several decades, optical information processing has been obtained gigantic achievements because of the characteristics of high-speed and paralleling. Up to now, the hottest method in this field is the double random phase-encoding technique (DRPE) [1] that proposed by Refregier and Javidi in 1995. Then, a lot of optical encryption techniques [2–13] that based on DRPE principle have been proposed such as extended fractional Fourier-transform encryption system [2], fractional Fourier-transform encryption system [3], and Fresnel diffraction encryption system [4], etc. However, the encryption techniques proposed in the above mentioned papers are designed for a gray or binary image. When a monochromatic light is used to illuminate a real color image in all these aforementioned encryption techniques, color information of a deciphered image is lost. So, some encryption techniques [14–23] have been further developed to encode a color image. L. Chen and D. Zhao have employed wavelength multiplexing to encrypt a color image based on lensless Fresnel transform holograms [14]. W. Chen, X. Chen, and Colin J. R. Sheppard have presented an optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain [19]. Z. Liu, Y. Zhang and W. Liu etc. have also proposed an optical color image hiding scheme based on chaotic mapping and Hartley transform [23]. However, the above proposed color cryptosystems belong to the category of linear symmetric cryptosystems, in which the encryption key is identical to the decryption key. From the
n
Corresponding author. Tel./fax: þ 86 745 2851011. E-mail address: [email protected] (X. Ding).
0030-3992/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.optlastec.2013.10.002
view of cryptography, the linear symmetric cryptosystem would suffer from several problems under the network environment, such as the safe of distribution and management of keys. In order to solve these problems, some gray asymmetric cryptosystems [24,25] have been proposed based on the phase-truncated Fourier transforms (PTFTs) [24], with which one can produce a ciphertext as real-valued and stationary white noise by using two public random phase keys, while a legal user can retrieve the plaintext using another two private phase keys that generated from the encryption process. Then, W. Chen and X. Chen have further proposed two different methods to encrypt a color image [26]. One is the conventional three-channel color image encryption in the Fresnel domain based on the multiplewavelength. The other is the indexed image encryption, in which the indexed image firstly is converted from a color image and then onestep operation in the asymmetric system. However, the first proposed method enciphers the color image in three different single spatial channels of arrangement at a time. The second method although encrypts in only one spatial channel, the outline of the deciphered images cannot be recognized when the encoded matrix occurs a little change in the process of encryption and decryption. Furthermore, the above two proposed methods are changed into the hybrid encryption system, which contains nonlinear components as well as linear components. Due to the linear components, the proposed methods still suffer from several problems, such as the safe of distribution and management of keys. Therefore, an optical color image cryptosystem, which can encrypt in only one spatial channel at a time, also can maintain the nonlinear properties of encryption structure, and has good robustness, has yet to be developed.
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In this paper, an optical color image cryptosystem, which can encrypt in only one spatial channel at a time and maintain the nonlinear properties of encryption structure, is proposed. The encryption process can be realized optically with the help of the computer and the decryption process can also be implemented optically like the decryption of the PTFTs-based asymmetric system. In this method, the color image to be encrypted is first decomposed into three color components (corresponding to red, green and blue). Then the position technique and the PTFTs are used to transform the three color components into a gray ciphertext, which looks like white noise, in only one spatial channel with the asymmetric cryptosystem [24]. During the decryption process, the three color components can be recovered by two private keys, which generated in the encryption procedure and are different from the encryption keys. Compared with the proposed color image encryption method [20], the obvious advantages of the method are not only the one spatial channel which is used for encryption process instead of three channels for the using of the position multiplexing technique but also the maintained nonlinear characteristic of the cryptosystem because of the nonlinear phase-truncated Fourier transforms (PTFTs) and the position multiplexing technique. Thus, due to maintain the nonlinear characteristic of the cryptosystem, the same disadvantages resulted from the linearity and symmetric cryptosystem could be avoided and therefore high robustness could be achieved. Numerical simulations are demonstrated to show the security and robustness of the proposed methods.
2. Encryption and decryption process
111
(corresponding to red, green and blue). Then, three decomposed color components that combined with three different random phase encryption keys ER ðx; yÞ, EG ðx; yÞ and EB ðx; yÞ respectively are located in the position ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ of the same plane along the axis x and the axis y to form a gray image using the position multiplexing technique. Thus, the input information can be expressed as f o ðx; yÞ ¼ ½f R ðx; yÞ ER ðx; yÞ δðx a1 ; y b1 Þ þ ½f G ðx; yÞ EG ðx; yÞ δðx a2 ; y b2 Þ þ ½f B ðx; yÞ EB ðx; yÞ δðx a3 ; y b3 Þ
ð3Þ
where a proper choice of the parameters ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ would guarantee that the three color components can be reconstructed without being superimposed [12]. Through Fourier transformation and nonlinear operation of phase-truncation, the encoded image can be achieved as follow: c1 ðu; vÞ ¼ TrfFT½f o ðx; yÞg:
ð4Þ
The final ciphertext cðx; yÞ can be obtained by the following step: cðx; yÞ ¼ TrfIFT½c1 ðu; vÞ UBðu; vÞg
ð5Þ
where Bðu; vÞ is another random phase encryption key in the Fourier transform domain. At the same time, a pair of decryption phase keys P 1 ðx; yÞ and P 2 ðu; vÞ should also be generated by the following additional two steps in encryption process. P 1 ðx; yÞ ¼ RefIFT½c1 ðu; vÞ Bðu; vÞg
ð6Þ
P 2 ðu; vÞ ¼ RefFT½f o ðx; yÞg
ð7Þ
It can be seen from Eqs. (4)–(7) that c1 ðu; vÞ, cðx; yÞ and P 1 ðx; yÞ, P 2 ðu; vÞ have simple relations
2.1. Phase-truncated Fourier transforms Before describing the proposed approach, we first review the phase-truncated Fourier transforms (PTFTs) that proposed in the PTFTs-based asymmetric system by Q. Wan and X. Peng in Ref. [24]. PTFT is a process of Fourier transform but with the nonlinear operation of phase truncation that means only the amplitude part of the Fourier spectrum is retained while the phase part of the spectrum is truncated. Let FTð UÞ, Trð UÞ and Reð U Þ denote the operator of Fourier transform, the operator of phase truncation and the operator of phase reservation, respectively. Assume a Fourier transformation of kðx; yÞ is expressed as Kðu; vÞ ¼ FT½kðx; yÞ ¼ jKðu; vÞjexp½iβðu; vÞ, where ðx; yÞ and ðu; vÞ denote the coordinate of the spatial domain and the frequency domain respectively, the phase truncation and the phase reservation can be respectively expressed as follows: Tr½Kðu; vÞ ¼ jKðu; vÞj
ð1Þ
Re½Kðu; vÞ ¼ exp½iβðu; vÞ
ð2Þ
2.2. Encryption process From now on, let us show the encryption process of the color image based on the position multiplexing technique and the PTFTs. The flowchart of the encryption process is shown in Fig. 1, where the symbol and n represents the multiplication operation and the convolution operation, respectively. Let f ðx; yÞ denotes the original color image to be encrypted, ER ðx; yÞ ¼ exp½jμ1 ðx; yÞ, EG ðx; yÞ ¼ exp½jμ2 ðx; yÞ, EB ðx; yÞ ¼ exp½jμ2 ðx; yÞ and Bðu; vÞ ¼ exp½jμ4 ðu; vÞ are four random phase encryption keys where μ1 ðx; yÞ,μ2 ðx; yÞ,μ3 ðx; yÞ and μ4 ðu; vÞ are white sequences statistically independent in the interval ½0; 2π. Firstly, the original color image f ðx; yÞ is decomposed into three color components f R ðx; yÞ, f G ðx; yÞ and f B ðx; yÞ
c1 ðu; vÞ P 2 ðu; vÞ ¼ FT½f o ðx; yÞ
ð8Þ
cðx; yÞ P 1 ðx; yÞ ¼ IFT½c1 ðu; vÞ Bðu; vÞ
ð9Þ
It must be pointed out that the ciphertext cðx; yÞ generated from Eqs. (4) and (5) is a real-valued function, which is more convenient in the process of record and transmission. 2.3. Decryption process The flowchart of the decryption process, which is similar to the decryption process of the PTFTs-based cryptosystem, is shown in Fig. 2. Firstly, the c1 ðu; vÞ can be obtained by operation of Fourier transform and nonlinear operation of phase truncate with the decryption keys P 1 ðx; yÞ. The result c1 ðu; vÞ can be expressed as: c1 ðu; vÞ ¼ TrfFT½cðx; yÞ P 1 ðx; yÞg
ð10Þ
Secondly, c1 ðu; vÞ multiplied with another private key P 2 ðu; vÞ is inverse Fourier transformed. The result can be expressed as f o ðx; yÞ ¼ IFT½c1 ðu; vÞ P 2 ðu; vÞ
ð11Þ
Thirdly, the deciphered result is obtained by the nonlinear operation of phase-truncation, of which process can be described as f M ðx; yÞ ¼ Tr½f o ðx; yÞ ¼ f R ðx; yÞ δðx a1 ; y b1 Þ þ f G ðx; yÞ δðx a2 ; y b2 Þ þf B ðx; yÞ δðx a3 ; y b3 Þ
ð12Þ
Finally, three original color components (f R ðx; yÞ, f G ðx; yÞ and f B ðx; yÞ) can be extracted from the deciphered result and combined into the original color image. 2.4. Performance analysis and optical setup From the processes of encryption and decryption as described above, it can be seen that the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ), as well as the public keys (ER ðx; yÞ, EG ðx; yÞ,
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Fig. 1. Flowchart of the encryption process.
Fig. 2. Flowchart of the decryption process.
EB ðx; yÞ and Bðu; vÞ), are not required for a correct decryption because of the nonlinear phase-truncated Fourier transform. Meanwhile, because of the position multiplexing technique the proposed method encrypts the original color image in only one spatial channel. Moreover, because the position multiplexing technique is employed in the encryption process, it only changes the input image of the PTFTs-based asymmetric system into the located image of the three color components and any linear component is not introduced. So, the nonlinear characteristic is still maintained in our proposed color image cryptosystem. In addition, our proposed method derives from the double images encryption method [11], which was proposed by X. Wang and D. Zhao. So, our proposed color image cryptosystem can also resist several attacks, such as: (1) brute force attack; (2) chosen plaintext attacks; (3) known public key attack; Similar to the double images encryption method [11], our proposed method can also resist the iterative attack [27]. The analysis is as follows: The iterative attack [27] can be avoided as long as the constraints are broken, because the iterative attack works based on the phase retrieval algorithm. Supposed an attacker, who has intercepted the ciphertext cðx; yÞ, attempts to retrieve f ðx; yÞ with the public keys (ER ðx; yÞ, EG ðx; yÞ, EB ðx; yÞ and Bðu; vÞ) based on the phase retrieval algorithm, he must execute two iterative transformations. The first step is to obtain c1 ðu; vÞ by using the public key Bðu; vÞ and the ciphertext cðx; yÞ. The constraints of this step are the ciphertext cðx; yÞ and the public keys (Bðu; vÞ). Obviously, the c1 ðu; vÞ can correctly be obtained with iterative transformations. The second step is to gain f ðx; yÞ by executing the iterative transformations. The constraints of this step are the public keys (ER ðx; yÞ, EG ðx; yÞ, EB ðx; yÞ), the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ) and c1 ðu; vÞ. The constraints are broken and f ðx; yÞ is incorrectly obtained with iterative transformations if we do not make the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ) public. From above discussions, compared with the reported color image encryption method [20], our proposed color cryptosystem can encipher the original color image in the only one spatial channel instead of three channels because of the position multiplexing technique. Meanwhile, our proposed cryptosystem does not introduce any linear component and maintains the nonlinear characteristic due to using the PTFTs and the position multiplexing technique. It is worth noting that although the encryption process is complicated, the decryption process, which is the same as the decryption of the PTFTs-based asymmetric system [24], is very simple. The complexity of the encryption process depends on the position multiplexing and the operation of phase reservation. Although the position multiplexing can be implemented using a joint transform correlator architecture [28], a more simple digital
Fig. 3. An optical setup for color image cryptosystem. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
method is employed to complete this process in our optical scheme for decreasing the complexity of the position multiplexing implementation by the computer. For the operation of phase reservation, a reference beam should be split from the light source to record the truncated phase by interferometry. An optical setup for the proposed color image cryptosystem with the help of some optoelectronic devices is shown in Fig. 3, where SLMs is a space-light modulator. The SLMs and CCD plane are controlled by a computer. In the optical encryption process, the input original color image f ðx; yÞ is firstly decomposed into three color components. Secondly, the three decomposed color components and the three different random phase encryption keys ER ðx; yÞ, EG ðx; yÞ and EB ðx; yÞ, which are located in the position ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ of the same plane along with the axis x and the axis y, are respectively displayed on the SLM1 and SLM2. Thirdly, the SLMs are illuminated by a monochromatic light. Fourthly, the encoding image g 1 ðu; vÞ is recorded by the CCD camera in the focal plane. In order to obtain the encryption result, we just repeat the above operations with the encoding image g 1 ðu; vÞ and Bðu; vÞ respectively displayed on the SLM1 and SLM2 and also illuminated by the same monochromatic light. Finally, the encryption result is recorded by the CCD camera. For the condition of encryption, a reference beam should be split from the light source to record the truncated phase by interferometry. Similarly in the optical decryption process, the ciphertext cðx; yÞ and the first decryption key P 1 ðx; yÞ are firstly displayed on the SLM1 and SLM2 respectively and illuminated by a uniform plane wave, and then the encoding image c1 ðu; vÞ is recorded by the CCD plane. For the decrypted result, we just repeat the above operations with c1 ðu; vÞ and P 2 ðu; vÞ respectively displayed on the SLM1 and SLM2 and also illuminated by the same plane wave. Finally, the decrypted color components are combined into decrypted color image in the computer. However, owing to the current resource limitation in our laboratory, we just made some numerical simulations to verify the security and robustness of the proposed method.
3. Numerical simulation Computer simulations are performed on a Matlab 7.0.0 (R14) platform to verify the security and robustness of this cryptography. An original color image comprising 128 128 pixels, as shown in Fig. 4(a), is used as the input image to be enciphered. In order to guarantee the three color components can be retrieved without being superimposed, we locate the three color components in the centered and zero-padded images (512 512 pixels) as the input
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Fig. 4. (a) original color image; (b) the ciphertext and (c) the correct decrypted image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Fig. 5. the phase part of decryption keys (a) P 1 ðx; yÞ and (b) P 2 ðu; vÞ.
Fig. 6. Decrypted results with (a) no keys; (b) arbitrarily chosen decryption keys; (c) the encryption keys; (d) the correct key P 1 ðx; yÞ while P 2 ðu; vÞ is wrong and (e) the correct key P 2 ðu; vÞ while P 1 ðx; yÞ is wrong. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
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Fig. 7. (a) Arbitrary fake color image; (b) P 1 ðx; yÞ and (c) P 2 ðu; vÞ generated from (a); Decrypted result (d) with the fake phase keys (b) and (c). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
Fig. 8. (a) The decrypted result with the iteration numbers (the iteration times of the first step is 200, the iteration times of the second step is 100); Relation between iteration times and the MSE (b) in the first step (between cðu; vÞ and its estimate); (c) in the second step (between the original color image and its estimate) and (d) in the second step (between the three color components of the original image and its estimate). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
plaintexts, and the three parameters ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ are set as ( 160, 0), (0, 0) and (160, 0), respectively. The ciphertext encrypted by Eqs. (4) and (5) is shown in Fig. 4(b), which looks like white noise. Fig. 4(c) shows the correct decrypted image. The two decryption keys generated in the encryption process and employed for the correct decryption are represented in Fig. 5. To measure the difference between the original image and the retrieved image quantitatively, the mean-square-error (MSE)
function between two images is defined as MSE ¼
1 L ′ ∑ ðf jf i jÞ2 Li¼1 i ′
ð13Þ
where L and f i denote the number of pixels and the estimate of f i , respectively. The security of the proposed method has been investigated. Firstly, the decrypted images obtained by using no keys, arbitrarily
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Fig. 9. Simulation results (a) collision color image; (b) decrypted key, mðu; vÞ generated through the iterative Fourier transforms; (c) decrypted key nðx; yÞ generated through the modified Gerchberg–Saxton algorithm; (d) retrieved the collision color image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Fig. 10. Recognition results obtained (a) between a retrieved original color image and encrypted image; (b) between a retrieved collision color image and encrypted image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
chosen decryption keys, public keys, only P 1 ðx; yÞ and P 2 ðu; vÞ but the other correct keys, are depicted in Fig. 6(a)–(e), respectively. The MSE values corresponding to Fig. 6(a)–(e) for red, green and blue components are (0.0695, 0.0545, 0.1833), (0.0678, 0.0811, 0.1131), (0.0694, 0.0545, 0.1831), (0.0678, 0.0811, 0.1131) and (0.0834, 0.0782, 0.1741). It can be seen from Fig. 6(a)–(e) that any attempt at the decryption of the ciphertext without all correct decryption keys will fail. Secondly, we test the ability of the proposed method against chosen plaintext attack. Obviously, it is easy to generate two phase decryption keys by using any arbitrary chosen color image and the four public keys. Fig. 7(a) is any arbitrary chosen color image. The two phase decryption keys that generated from the above color images can be shown in Fig. 7(b) and (c) respectively. As can be seen from Fig. 7(d), the decoded color image provides no valuable
information but the arbitrary chosen color image by using the above two fake decryption keys. Thirdly, the ability of the proposed method against the iterative attack is represented in Fig. 8. Fig. 8(a) is illustrated that our proposed color image cryptosystem can resist against the iterative attack effectively without any knowledge of the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ and ða3 ; b3 Þ). Fig. 8(b)–(d) illustrate the convergence of the iterative attack through the MSE. The relation between iteration times and the MSE (between c′ðu; vÞ and its estimate) in the first step is shown in Fig. 8(b). Fig. 8(c) shows the relation between iteration times and the MSE (between the original color imagef ðx; yÞ and its estimate) in the second step. Fig. 8 (d) shows the relation between iteration times and the MSE (between the three color components of the original image and its estimate) in the second step. From the above attack results, it can be
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Fig. 11. Simulation of occlusion attack: (a), (c) and (e) are the occlusion encrypted images; (b), (d) and (f) are the corresponding retrieved images from (a), (c) and (d), respectively. (g) Gaussian-noised encrypted image with variance value 0.1, and (h) corresponding decrypted image from (g). (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
obtained that any attempt at the decryption of ciphertext without any knowledge of the parameters of the position (ða1 ; b1 Þ, ða2 ; b2 Þ andða3 ; b3 Þ), even with public keys and the iterative attack, will fail.
In this section, the collision attack [29] of our proposed method is simulated. A color image of size 128 128 pixels was taken as a collision image and is shown in Fig. 9(a). Fig. 9(b) and (c) shows
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Fig. 12. The MSE curves from (a) the test of occlusion attack; (b) the test of Gaussian noise attack. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
another two different decryption keys, mðu; vÞ and nðx; yÞ, which are generated by using the iterative Fourier transforms and the modified Gerchberg–Saxton algorithm [30], respectively. Fig. 9(d) shows the decrypted collision image. These results infer that an unauthorized user can generate an arbitrary color image independent of the original color image through the encrypted image. But we can check the authorized user and the unauthorized user by using the joint transform correlator architecture [28], which is used the same as the method in Ref. [29]. Fig. 10(a) and (b) shows the recognition results obtained between a retrieved original color image (shown in Fig. 4(c)) and encrypted image (shown in Fig. 4(b)) and between a retrieved collision color image (shown in Fig. 9(d)) and encrypted image (shown in Fig. 4(b)). So, we can easily validate the authorized user with the aid of a recognition scheme. In the last section, the robustness of the proposed method has been checked by the occlusion and noise attacks on the encrypted images. The occluded encrypted images with 25%, 75%, and 90% occlusion sizes are displayed in Fig. 11(a), (c) and (e), respectively. Their corresponding retrieved images, by using the decryption process shown in Fig. 2 with the correct decryption keys, are shown in Fig. 11(b), (d) and (f), respectively. Corresponding to Fig. 11(b), (d) and (f) for red, green and blue components, the MSE values are (0.0096, 0.0090, and 0.0140), (0.0693, 0.0591, and 0.1167) and (0.1172, 0.0995, 0.2004). For noise attack, the Gaussian-noised encrypted image with variance value 0.1 is presented in Fig. 11(g) and the corresponding retrieved image is depicted in Fig. 11(h). The MSE values corresponding to Fig. 11(h) for red, green and blue components are (0.0613, 0.0547, and 0.0886). It can be gained that the encrypted image has lost up to 90% occlusion size or has been contaminated by Gaussian noise with variance value 0.1, but the outline of the recovered images can still be recognized. The MSE curves of the occlusion attack and the Gaussian-noise attack in the ranges [0.1, 0.9] and [0, 0.5] are represented in Fig. 12(a) and (b), respectively. It can be seen from Fig. 12(a) that the MSE value increases with an increase in occlusion size, where the blue channel increases rapidly in comparison to red and green channels. The MSE versus Gaussian-noise is shown in Fig. 12(b), which indicates that the MSE value increases with an increase in Gaussian noise. The above results fully show that our proposed method can resist the occlusion attack and the Gaussian-noise attack.
4. Conclusion In summary, we propose an optical color image encryption technique that based on an asymmetric algorithm, in which the
decryption keys are different from the encryption keys. The proposed method that combines the position multiplexing technique and the nonlinear phase-truncated Fourier transforms encrypts the three color components of the original color image in only one spatial channel instead of three channels. Meanwhile, it does not introduce any linear component and maintains the nonlinear characteristic because of the nonlinear PTFTs and the position multiplexing technique. Additional, an optical architecture based on the computer and two SLMs is used to encipher or decipher the original color image. In the last section the simulation results show that the method has a very high security and robustness against various types of the currently existing attacks.
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