- Email: [email protected]
Image double encryption based on iteration Fourier and chaos system
Optik 124 (2013) 4197–4200
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Image double encryption based on iteration Fourier and chaos system Suping You ∗ , Ling Wu, Benxiao Cai School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China
a r t i c l e
i n f o
Article history: Received 31 July 2012 Accepted 13 December 2012
Keywords: Image encryption Iterative Fourier Chaos replacement Double encryption
a b s t r a c t According to the digital image encryption, a double encryption algorithm based on the iterative Fourier transform and chaotic transform is proposed. It uses iterative Fourier transform to encrypt image for the first time, then processes the encrypted image through the chaos replacement, which can change image pure phase distribution to realize double encryption. The simulated result shows that the algorithm can effectively resist the attacks of statistical analysis, and has higher encryption effect and safety which is very sensitive to secret key. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Along with rapid development of multimedia and network technology, the digital image gradually becomes important carrier of information exchange, and image information safety turns into the attention focus. In recent years, image encryption research based on optical information technology aroused many scholars’ interest at home and abroad [1–5]. Owing to the very high space location precision of element of optical 4f system, the digital encryption based on it can’t be practically used. Virtual optics technology composited with digital signal and optics advantage has promoted optical information safety technology, which uses digital method to simulate optical digital processing in computer space, and utilizes hardware to realize digital encryption based on 4f optical system[6–8]. In the image encryption domain, the traditional encryption algorithms have many disadvantages because of image characteristics such as big inherent digital capacity and high correlation of adjacent pixels. In recent years, with the development of chaos theory, image encryption technology based on chaos has been developing. Chaos is the regular pattern control action of nonlinear dynamics, which shows sensibility, statistics character of white noise and ergodic character of chaos sequence. And its form structure is complicate and has non-forecasting character [8–12]. Such big advantages make chaos theory so attractive to image encryption. Regarding to the characteristic of traditional encryption and chaos encryption, the paper proposes a double encryption
∗ Corresponding author. E-mail address: [email protected] (S. You). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.12.064
algorithm. Firstly, iterative Fourier transform is used to encrypt image. Then the encrypted image is encrypted with chaos replacement again. Finally the secret key sensibility, grayscale histogram and correlation etc. are analyzed. 2. Iterative Fourier algorithm based on the optical 4f system 2.1. The key ideas and steps of the Iterative Fourier algorithm The basic principle of the Iterative Fourier algorithm is iterating back and forth between object plane and spectrum plane to Fourier transform, and then putting the known limits on the object and spectrum planes separately to retrieve the phase distribution of the object plane to the maximum. The algorithm idea of the IFTA procedure is as follows: Firstly give any initial phase distribution of the input phase ϕ(x,y) (ϕ value is between 0 and 2, and all later values keep the ones calculated in the fourth step), which together with the incident light amplitude |f(x,y)| to constitute the incident wave function f(x,y), then do Fourier transform to f(x,y) to get f(U,V), in which (x,y) is the airspace coordinates and (u,v) is the frequency domain coordinates; Put the phase part of F(u,v) and the amplitude distribution |F’(u,v)|? to constitute a complex function F’(u,v); Do the inverse Fourier transform for F’(u,v) to get the wave function f(x,y); Take the phase part of f’(x,y) and the scheduled input light amplitude to form the new wave function which would be the input of the next iteration, and then repeat the above four steps. With the iteration times increasing, the output images would slowly converge to the output of the preset image, and in the input face values,ϕ(x,y) is the phase of the distribution needed.
4198
S. You et al. / Optik 124 (2013) 4197–4200
Suppose now it is the Kth iteration: Fk (u, v) = FT {fk (x, y)}
(1)
= |Fk (u, v)| exp[ik (u, v)]
Fk (u, v) = |Fk (u, v)| exp ik (u, v)
(2)
fk (x, y) = IFT {Fk (u, v)
(3)
= |fk (x, y)| exp iϕk+1 (x, y) fk+1 (x, y) = |f (x, y)| exp iϕk+1 (x, y)
(4)
If the final result meets the criteria, the iteration ends; if not, then do the next iteration until the criteria are reached. Usually we take MSE (mean square error) as criteria, defined as follows. In frequency domain:
∞ ∞
3. Image encryption with chaos system
|Fk (u, v) − Fk (u, v)|2 dudv EF2 =
−∞−∞ ∞ ∞
(5)
|Fk (u, v)|2 dudv −∞−∞
In the airspace:
∞ ∞
|fk+1 (x, y) − fk (x, y)|2 dxdy EO2 =
on the output plane after iteration, |g(x,y)| is the amplitude of optical field distribution on the output plane after iteration. IFTA is an algorithm used to work out b(x,y) and H(u,v). Because what we concern is the amplitude on the output plane, it is needed to set limits on the output image’s amplitude after every iteration to get the final result of b(x,y) and H(u,v). Cascade phase recovery algorithm’s principle is changing the phase distributions of the input plane phase plate b(x,y) and Fourier plane phase plate H(u,v) at the same time, then taking the stay encrypted image to synthesize into b(x,y) and H(u,v) . . . Use the plane wave and stay encrypted image to modulate the amplitudes of the input and output planes. After the first time iteration, we can take the input plane’s phase as b(x,y), and the phase difference on the Fourier plane formed between Fourier and inverse Fourier transform as H(u,v). This cycle repeats until they meet the criteria, then the loop ends.
−∞−∞
(6)
∞ ∞
|fk (x, y)|dxdy −∞−∞
When the mean-square deviation is less than certain threshold value, we can assume that the airspace and frequency domain satisfy the constraint conditions at the same time, then the iteration stops. 2.2. The iteration Fourier encryption principle based on 4f system The optical 4f system (shown in Fig. 1) is the typical coherent filtering system. The spectrum of the input function can be changed by setting the different filters on the spectrum. Then inverse Fourier transform through the second filter can make the output function get the hoped transform. As the spatial filters on the spectrum space can regulate amplitude and phase of the input function’s frequency spectrum, the system can be applied to image encryption. Express the stay encrypted image (the preset output image) as g(x,y) in the 4f system, and two random phase plates are placed between the input plane and Fourier plane. One is b(x,y)=exp[i(x,y)] and the other is H(u,v)=exp[i(u,v)] b(x,y) and H(u,v) should meet the following formula at last: |g(x, y)| exp[iϕ(x, y)] = IFT {FT {b(x, y)}H(u, v)}
(7)
FT, IFT signify the Fourier transform and inverse Fourier transform respectively, and ϕ(x,y)is the phase of optical field distribution
Digital image scrambling means making the image disorder which is changing and confusing image matrix by a series of nonlinear transformation. So the object of chaos is to generally change every pixel value. The image is a two dimensional matrix, but we can converse the two-dimensional matrix into one-dimensional matrix through a certain way, then encrypt one-dimensional matrix which is formed by pixels through chaos sequence produced by chaos system. We use one-dimensional Logistic mapping, defined by the following formula: xk+1 = xk (1 − xk )
(8)
where 0≤≤4 is called control parameters, and xk ∈(0,1). When 3.2699456≤≤4, the Logistic mapping is in chaos state, It means that the sequences {xk } is aperiodic, convergent and sensitive to initial value, which is generated by Logistic mapping in the initial condition x0 . Choose a random number as initial iteration values x0 from the open interval (0, 1) to iterate according to the above formula, then throw away the iterative data of the front 500 times, and finally use chaotic sequence for encrypting every pixel values. Assume that the pixels matrix of digital image is Z(M,N), then convert it into one-dimensional array f(k), where k = i × n + j, 0 ≤ i ≤ M, 0 ≤ j ≤ N. Encryption system selects x0 , , s and N0 as secret keys. There into, x0 and are initial value and parameter value of chaotic system respectively, N0 is the initial iteration times of chaos system. s is the initial feedback value. The design of the encryption system is as follows: C(k) = (k) ⊕ {[f (k) + (k)]modN} ⊕ C(k − 1) (k) = xk × N
(9) (10)
Where f(k) is the current pixel value, C(k−1) is output value of previous pixel value encryption. 4. Iterative Fourier algorithm based on chaotic transform 4.1. Iterative Fourier encryption algorithm based on chaotic transform
Fig. 1. 4f system.
Double Image encryption is combined with iterative Fourier transform and image scrambling encryption. Denote image chaos encryption by JC () and inverse of image chaos encryption byJ−C (). Denote iterative Fourier transform for IFT() and the input image for f(x,y).
S. You et al. / Optik 124 (2013) 4197–4200
Fig. 2. Encrypted effect image.
Fig. 3. Decrypted effect image.
Deal the input image with iterative Fourier transform, showed as follows: IFT {f (x, y)}
(11)
Make the transformed image with chaotic transform: JC {IFT {f (x, y)}}
(12)
The finally encrypted result is g(x, y) = JC {IFT {f (x, y)}}
(13)
Image decryption is the anti-process of image encryption. Get the original image by carrying out formula (13), see the following formula: f (x, y) = IFT− {{J−C {g(x, y)}}
4199
(14)
4.2. Security analysis of the double encryption algorithm The security of encryption algorithm depends on the size of secret key space, sensibility of secret key for encrypted image and complexity of algorithm. According to the image encryption
scheme, the secret keys used for encryption include: two random phase matrixes ((x,y) & (u,v)) which are uniform distribution among 0∼2, iterative times n whose secret key space magnitude is 102 , x0 of chaos mapping whose secret key space magnitude is 1015 , whose secret key space magnitude is 1013 , N0 and s whose secret key space magnitude is 103 . Therefore, total secret key space magnitude is 1060 , which means that the algorithm space is huge and can resist unlicensed user to do exhaustive attack in the time. 5. Simulate analysis of calculator 5.1. Encrypted effect images as shown in Fig. 2 5.1.1. The sensibility of initial value of secret key. In the condition of all other constant parameters, assume that user sets encrypted key x0= 0.1234567891234567 in the chaos mapping, and decrypted key x0=0.1234567891234568. The decrypted image is showed as Fig. 3b, while the correct decrypted image is showed as Fig. 3c. The experiment result shows that the slight change will generate grate influence on the decrypted image, which also indicates that the algorithm is sensitive to secret key.
4200
S. You et al. / Optik 124 (2013) 4197–4200
Fig. 5. Correlation distribution mage of horizontal adjacent pixels. Fig. 4. Grayscale histogram of image.
diffused to encrypted message, which can effectively against the attack of the statistics (Fig. 5).
5.2. Grayscal histogram analysis From the grayscale histogram of Fig. 4, it shows that pixel gray value before encryption distributes unevenly, while the pixel gray value after encrypted with the paper algorithm distributes evenly from 0 to 255. It proves that the algorithm has strong ability of resisting statistic analysis attack. 5.3. Correlation analysis In order to test and verify the correlation of original image and encrypted image, use the formula blow to calculate the related coefficient of neighborhood pixels, which are horizontal adjacent pixels pairs, vertical adjacent pixels pairs and some diagonal pixels pairs randomly selected from images. E(x) =
N 1
N
xi
(15)
i=1
1 [xi − E(x)]2 N N
D(x) =
(16)
i=1
1 [xi − E(x)][yi − E(y)] N N
Conv(x, y) =
(17)
i=1
xy =
conv(x, y)
D(x)
(18)
D(y)
xy is the related coefficient of neighboring coupled pixels. The correlation of horizontal adjacent pixels is described in Fig. 4. The result shows that the adjacent pixels of original image are highly relevant and the correlation coefficient is close to 1. Moreover, the correlation coefficient of encrypted image is close to 0, which means that the adjacent pixels are basically uncorrelated. Therefore, the statistical characterization of original image is randomly
6. Conclusions The paper adopts iterative Fourier algorithm for image encryption, then deals the encrypted phase distribution image with chaos replacement, which can obtain image double encryption. Only making sure the phase distribution after iterative Fourier transform and chaos sequence at the same time, it can decrypt the encrypted image. The simulation result shows that the algorithm has high effect encryption and safety. References [1] N. Towghi, B. Javidi, Z. Luo, Fully phase encrypted image processor, J. Opt. Soc. Am. A. America 16 (August) (1999) 1915–1927. [2] P.C. Mogensen, J. Gluckstad, Phase-only optical encryption, Opt. Lett. Am. 25 (August) (2000) 566–568. [3] H.T. Chang, Image encryption using separable amplitude based virtual image and iteratively retrieved phase information, Opt. Eng. Am. 40 (October) (2001) 2165–2171. [4] O. Matoba, B. Javidi, Secure holographic memory by double-random polarization encryption, Appl. Opt. America 43 (July) (2004) 2195– 2919. [5] G.H. Situ, J.J. Zhang, Y. Zhang, et al., A cascaded-phases retrieval algorithm for optical image encryption, J. Optoelectronics Laser. China 15 (March) (2004) 341–343. [6] Y.H. Wang, X.D. Chai, C.P. Zhou, et al., An image encryption technology based on chaotic sequences and fractional Fourier fransform, Com. Tech. and Dev., China 16 (September) (2006) 213–216. [7] Y.Y. Wang, Y.R. Wang, Y.B. Yang, Optical hierarchical image encryption by use of iterative Fourier transform algorithm, Chi. Jour. of Lasers, China 33 (October) (2006) 1360–1364. [8] Z. Yang, J.C. Feng, Y. Fang, Information image encryption algorithm based on chaos and fractional Fourier transform, Com. Sci. China 35 (September) (2008) 239–241. [9] T.G. Gao, Z.Q. Chen, A new image encryption algorithm based on hyper-chaos, Phys. Lett. A, Netherlands 372 (April) (2008) 394–400. [10] Z.H. Guang, F.J. Huang, W.J. Guan, Chaos-based image encryption algorithm, Phys. Lett. A, Netherlands 346 (March) (2005) 153–157. [11] C. Cokal, E. Solak, Cryptanalysis of a chaos-based image encryption algorithm, Physics Letters A, Netherlands 373 (August) (2009) 1357–1360. [12] J.X. Jin, S.S. Qiu, Cascaded image encryption systems based on physical chaos, Acta Phys. Sin., China 59 (February) (2010) 780–792.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Image double encryption based on iteration Fourier and chaos system Suping You ∗ , Ling Wu, Benxiao Cai School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China
a r t i c l e
i n f o
Article history: Received 31 July 2012 Accepted 13 December 2012
Keywords: Image encryption Iterative Fourier Chaos replacement Double encryption
a b s t r a c t According to the digital image encryption, a double encryption algorithm based on the iterative Fourier transform and chaotic transform is proposed. It uses iterative Fourier transform to encrypt image for the first time, then processes the encrypted image through the chaos replacement, which can change image pure phase distribution to realize double encryption. The simulated result shows that the algorithm can effectively resist the attacks of statistical analysis, and has higher encryption effect and safety which is very sensitive to secret key. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Along with rapid development of multimedia and network technology, the digital image gradually becomes important carrier of information exchange, and image information safety turns into the attention focus. In recent years, image encryption research based on optical information technology aroused many scholars’ interest at home and abroad [1–5]. Owing to the very high space location precision of element of optical 4f system, the digital encryption based on it can’t be practically used. Virtual optics technology composited with digital signal and optics advantage has promoted optical information safety technology, which uses digital method to simulate optical digital processing in computer space, and utilizes hardware to realize digital encryption based on 4f optical system[6–8]. In the image encryption domain, the traditional encryption algorithms have many disadvantages because of image characteristics such as big inherent digital capacity and high correlation of adjacent pixels. In recent years, with the development of chaos theory, image encryption technology based on chaos has been developing. Chaos is the regular pattern control action of nonlinear dynamics, which shows sensibility, statistics character of white noise and ergodic character of chaos sequence. And its form structure is complicate and has non-forecasting character [8–12]. Such big advantages make chaos theory so attractive to image encryption. Regarding to the characteristic of traditional encryption and chaos encryption, the paper proposes a double encryption
∗ Corresponding author. E-mail address: [email protected] (S. You). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.12.064
algorithm. Firstly, iterative Fourier transform is used to encrypt image. Then the encrypted image is encrypted with chaos replacement again. Finally the secret key sensibility, grayscale histogram and correlation etc. are analyzed. 2. Iterative Fourier algorithm based on the optical 4f system 2.1. The key ideas and steps of the Iterative Fourier algorithm The basic principle of the Iterative Fourier algorithm is iterating back and forth between object plane and spectrum plane to Fourier transform, and then putting the known limits on the object and spectrum planes separately to retrieve the phase distribution of the object plane to the maximum. The algorithm idea of the IFTA procedure is as follows: Firstly give any initial phase distribution of the input phase ϕ(x,y) (ϕ value is between 0 and 2, and all later values keep the ones calculated in the fourth step), which together with the incident light amplitude |f(x,y)| to constitute the incident wave function f(x,y), then do Fourier transform to f(x,y) to get f(U,V), in which (x,y) is the airspace coordinates and (u,v) is the frequency domain coordinates; Put the phase part of F(u,v) and the amplitude distribution |F’(u,v)|? to constitute a complex function F’(u,v); Do the inverse Fourier transform for F’(u,v) to get the wave function f(x,y); Take the phase part of f’(x,y) and the scheduled input light amplitude to form the new wave function which would be the input of the next iteration, and then repeat the above four steps. With the iteration times increasing, the output images would slowly converge to the output of the preset image, and in the input face values,ϕ(x,y) is the phase of the distribution needed.
4198
S. You et al. / Optik 124 (2013) 4197–4200
Suppose now it is the Kth iteration: Fk (u, v) = FT {fk (x, y)}
(1)
= |Fk (u, v)| exp[ik (u, v)]
Fk (u, v) = |Fk (u, v)| exp ik (u, v)
(2)
fk (x, y) = IFT {Fk (u, v)
(3)
= |fk (x, y)| exp iϕk+1 (x, y) fk+1 (x, y) = |f (x, y)| exp iϕk+1 (x, y)
(4)
If the final result meets the criteria, the iteration ends; if not, then do the next iteration until the criteria are reached. Usually we take MSE (mean square error) as criteria, defined as follows. In frequency domain:
∞ ∞
3. Image encryption with chaos system
|Fk (u, v) − Fk (u, v)|2 dudv EF2 =
−∞−∞ ∞ ∞
(5)
|Fk (u, v)|2 dudv −∞−∞
In the airspace:
∞ ∞
|fk+1 (x, y) − fk (x, y)|2 dxdy EO2 =
on the output plane after iteration, |g(x,y)| is the amplitude of optical field distribution on the output plane after iteration. IFTA is an algorithm used to work out b(x,y) and H(u,v). Because what we concern is the amplitude on the output plane, it is needed to set limits on the output image’s amplitude after every iteration to get the final result of b(x,y) and H(u,v). Cascade phase recovery algorithm’s principle is changing the phase distributions of the input plane phase plate b(x,y) and Fourier plane phase plate H(u,v) at the same time, then taking the stay encrypted image to synthesize into b(x,y) and H(u,v) . . . Use the plane wave and stay encrypted image to modulate the amplitudes of the input and output planes. After the first time iteration, we can take the input plane’s phase as b(x,y), and the phase difference on the Fourier plane formed between Fourier and inverse Fourier transform as H(u,v). This cycle repeats until they meet the criteria, then the loop ends.
−∞−∞
(6)
∞ ∞
|fk (x, y)|dxdy −∞−∞
When the mean-square deviation is less than certain threshold value, we can assume that the airspace and frequency domain satisfy the constraint conditions at the same time, then the iteration stops. 2.2. The iteration Fourier encryption principle based on 4f system The optical 4f system (shown in Fig. 1) is the typical coherent filtering system. The spectrum of the input function can be changed by setting the different filters on the spectrum. Then inverse Fourier transform through the second filter can make the output function get the hoped transform. As the spatial filters on the spectrum space can regulate amplitude and phase of the input function’s frequency spectrum, the system can be applied to image encryption. Express the stay encrypted image (the preset output image) as g(x,y) in the 4f system, and two random phase plates are placed between the input plane and Fourier plane. One is b(x,y)=exp[i(x,y)] and the other is H(u,v)=exp[i(u,v)] b(x,y) and H(u,v) should meet the following formula at last: |g(x, y)| exp[iϕ(x, y)] = IFT {FT {b(x, y)}H(u, v)}
(7)
FT, IFT signify the Fourier transform and inverse Fourier transform respectively, and ϕ(x,y)is the phase of optical field distribution
Digital image scrambling means making the image disorder which is changing and confusing image matrix by a series of nonlinear transformation. So the object of chaos is to generally change every pixel value. The image is a two dimensional matrix, but we can converse the two-dimensional matrix into one-dimensional matrix through a certain way, then encrypt one-dimensional matrix which is formed by pixels through chaos sequence produced by chaos system. We use one-dimensional Logistic mapping, defined by the following formula: xk+1 = xk (1 − xk )
(8)
where 0≤≤4 is called control parameters, and xk ∈(0,1). When 3.2699456≤≤4, the Logistic mapping is in chaos state, It means that the sequences {xk } is aperiodic, convergent and sensitive to initial value, which is generated by Logistic mapping in the initial condition x0 . Choose a random number as initial iteration values x0 from the open interval (0, 1) to iterate according to the above formula, then throw away the iterative data of the front 500 times, and finally use chaotic sequence for encrypting every pixel values. Assume that the pixels matrix of digital image is Z(M,N), then convert it into one-dimensional array f(k), where k = i × n + j, 0 ≤ i ≤ M, 0 ≤ j ≤ N. Encryption system selects x0 , , s and N0 as secret keys. There into, x0 and are initial value and parameter value of chaotic system respectively, N0 is the initial iteration times of chaos system. s is the initial feedback value. The design of the encryption system is as follows: C(k) = (k) ⊕ {[f (k) + (k)]modN} ⊕ C(k − 1) (k) = xk × N
(9) (10)
Where f(k) is the current pixel value, C(k−1) is output value of previous pixel value encryption. 4. Iterative Fourier algorithm based on chaotic transform 4.1. Iterative Fourier encryption algorithm based on chaotic transform
Fig. 1. 4f system.
Double Image encryption is combined with iterative Fourier transform and image scrambling encryption. Denote image chaos encryption by JC () and inverse of image chaos encryption byJ−C (). Denote iterative Fourier transform for IFT() and the input image for f(x,y).
S. You et al. / Optik 124 (2013) 4197–4200
Fig. 2. Encrypted effect image.
Fig. 3. Decrypted effect image.
Deal the input image with iterative Fourier transform, showed as follows: IFT {f (x, y)}
(11)
Make the transformed image with chaotic transform: JC {IFT {f (x, y)}}
(12)
The finally encrypted result is g(x, y) = JC {IFT {f (x, y)}}
(13)
Image decryption is the anti-process of image encryption. Get the original image by carrying out formula (13), see the following formula: f (x, y) = IFT− {{J−C {g(x, y)}}
4199
(14)
4.2. Security analysis of the double encryption algorithm The security of encryption algorithm depends on the size of secret key space, sensibility of secret key for encrypted image and complexity of algorithm. According to the image encryption
scheme, the secret keys used for encryption include: two random phase matrixes ((x,y) & (u,v)) which are uniform distribution among 0∼2, iterative times n whose secret key space magnitude is 102 , x0 of chaos mapping whose secret key space magnitude is 1015 , whose secret key space magnitude is 1013 , N0 and s whose secret key space magnitude is 103 . Therefore, total secret key space magnitude is 1060 , which means that the algorithm space is huge and can resist unlicensed user to do exhaustive attack in the time. 5. Simulate analysis of calculator 5.1. Encrypted effect images as shown in Fig. 2 5.1.1. The sensibility of initial value of secret key. In the condition of all other constant parameters, assume that user sets encrypted key x0= 0.1234567891234567 in the chaos mapping, and decrypted key x0=0.1234567891234568. The decrypted image is showed as Fig. 3b, while the correct decrypted image is showed as Fig. 3c. The experiment result shows that the slight change will generate grate influence on the decrypted image, which also indicates that the algorithm is sensitive to secret key.
4200
S. You et al. / Optik 124 (2013) 4197–4200
Fig. 5. Correlation distribution mage of horizontal adjacent pixels. Fig. 4. Grayscale histogram of image.
diffused to encrypted message, which can effectively against the attack of the statistics (Fig. 5).
5.2. Grayscal histogram analysis From the grayscale histogram of Fig. 4, it shows that pixel gray value before encryption distributes unevenly, while the pixel gray value after encrypted with the paper algorithm distributes evenly from 0 to 255. It proves that the algorithm has strong ability of resisting statistic analysis attack. 5.3. Correlation analysis In order to test and verify the correlation of original image and encrypted image, use the formula blow to calculate the related coefficient of neighborhood pixels, which are horizontal adjacent pixels pairs, vertical adjacent pixels pairs and some diagonal pixels pairs randomly selected from images. E(x) =
N 1
N
xi
(15)
i=1
1 [xi − E(x)]2 N N
D(x) =
(16)
i=1
1 [xi − E(x)][yi − E(y)] N N
Conv(x, y) =
(17)
i=1
xy =
conv(x, y)
D(x)
(18)
D(y)
xy is the related coefficient of neighboring coupled pixels. The correlation of horizontal adjacent pixels is described in Fig. 4. The result shows that the adjacent pixels of original image are highly relevant and the correlation coefficient is close to 1. Moreover, the correlation coefficient of encrypted image is close to 0, which means that the adjacent pixels are basically uncorrelated. Therefore, the statistical characterization of original image is randomly
6. Conclusions The paper adopts iterative Fourier algorithm for image encryption, then deals the encrypted phase distribution image with chaos replacement, which can obtain image double encryption. Only making sure the phase distribution after iterative Fourier transform and chaos sequence at the same time, it can decrypt the encrypted image. The simulation result shows that the algorithm has high effect encryption and safety. References [1] N. Towghi, B. Javidi, Z. Luo, Fully phase encrypted image processor, J. Opt. Soc. Am. A. America 16 (August) (1999) 1915–1927. [2] P.C. Mogensen, J. Gluckstad, Phase-only optical encryption, Opt. Lett. Am. 25 (August) (2000) 566–568. [3] H.T. Chang, Image encryption using separable amplitude based virtual image and iteratively retrieved phase information, Opt. Eng. Am. 40 (October) (2001) 2165–2171. [4] O. Matoba, B. Javidi, Secure holographic memory by double-random polarization encryption, Appl. Opt. America 43 (July) (2004) 2195– 2919. [5] G.H. Situ, J.J. Zhang, Y. Zhang, et al., A cascaded-phases retrieval algorithm for optical image encryption, J. Optoelectronics Laser. China 15 (March) (2004) 341–343. [6] Y.H. Wang, X.D. Chai, C.P. Zhou, et al., An image encryption technology based on chaotic sequences and fractional Fourier fransform, Com. Tech. and Dev., China 16 (September) (2006) 213–216. [7] Y.Y. Wang, Y.R. Wang, Y.B. Yang, Optical hierarchical image encryption by use of iterative Fourier transform algorithm, Chi. Jour. of Lasers, China 33 (October) (2006) 1360–1364. [8] Z. Yang, J.C. Feng, Y. Fang, Information image encryption algorithm based on chaos and fractional Fourier transform, Com. Sci. China 35 (September) (2008) 239–241. [9] T.G. Gao, Z.Q. Chen, A new image encryption algorithm based on hyper-chaos, Phys. Lett. A, Netherlands 372 (April) (2008) 394–400. [10] Z.H. Guang, F.J. Huang, W.J. Guan, Chaos-based image encryption algorithm, Phys. Lett. A, Netherlands 346 (March) (2005) 153–157. [11] C. Cokal, E. Solak, Cryptanalysis of a chaos-based image encryption algorithm, Physics Letters A, Netherlands 373 (August) (2009) 1357–1360. [12] J.X. Jin, S.S. Qiu, Cascaded image encryption systems based on physical chaos, Acta Phys. Sin., China 59 (February) (2010) 780–792.