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An asymmetric single-channel color image encryption based on Hartley transform and gyrator transform
Optics and Lasers in Engineering 69 (2015) 49–57
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
An asymmetric single-channel color image encryption based on Hartley transform and gyrator transform Muhammad Rafiq Abuturab Department of Physics, Maulana Azad College of Engineering and Technology, Patna, Bihar 801113, India
ar t ic l e i nf o
a b s t r a c t
Article history: Received 28 July 2014 Received in revised form 30 December 2014 Accepted 8 January 2015 Available online 5 March 2015
A novel asymmetric single-channel color image encryption using Hartley transform and gyrator transform is proposed. A color image is segregated into R, G, and B channels and then each channel is independently Hartley transformed. The three transformed channels are multiplied and then phase- and amplitude truncated to obtain first encrypted image and first decryption key. The encoded image is modulated with a conjugate of random phase mask. The modulated image is gyrator transformed and then phase- and amplitude truncated to get second encrypted image and second decryption key. The asymmetric (decryption) keys, random phase mask, and transformation angle of gyrator transform serve as main keys. The optoelectronic encryption and decryption systems are suggested. Numerical simulation results have been demonstrated to verify the performance and security of the proposed security system. & 2015 Elsevier Ltd. All rights reserved.
Key words: Asymmetric single-channel color image encryption Hartley transform Gyrator transform
1. Introduction Optical technology has been widely used in the field of information security owing to its inherent advantages of multi-dimensional operation and inherent parallel procession. Refregier and Javidi first proposed the double-random phase-encoding (DRPE) technique, which is an optical symmetric-key method that encrypts an image by multiplying two random phase masks (RPMs), one in the spatial plane and one in the Fourier plane [1]. A number of modified DRPEbased encryption systems of gray-scale images have been put forwarded to achieve higher security [2–6]. The quantitative techniques for simultaneous compression and encryption of images have also been developed [7–10]. Recently, an optimized method of simultaneous compression and encryption designed to process images with close spectra is presented [11]. This approach is well adapted to the compression and encryption of images of a timevarying video sequences but also to static polarimetric images. The spectral plane (containing the information to send and/or to store) is decomposed in several independent areas which are assigned according a specific way. In addition, each spectrum is shifted in order to minimize their overlap. Zhang and Karim [12] first introduced a single-channel encryption algorithm for color image based on DRPE in the Fourier domain, which has been further extended to the fractional Fourier domain [13,14], Arnold transform and discrete cosine transform [15], Fresnel domain [16], and gyrator domain [17–19].
E-mail address: rafi[email protected] http://dx.doi.org/10.1016/j.optlaseng.2015.01.001 0143-8166/& 2015 Elsevier Ltd. All rights reserved.
The color image encryptions based on Hartley transform (HT), with the random rotation of color vector [20] and gyrator transform (GT) domain [21] have been researched. In the former method, the random shift of the four angles in rotation operation is regarded as the key of the encryption system whereas in the later technique, the period and iterative number of Arnold transform, and transformation angles of GT in each channel are employed as main keys of the algorithm. A color image encryption approach by using chaotic mapping and HT has been reported [22]. In this algorithm, the three components of color image are first scrambled by Baker mapping. The scrambled components are used as the positions in Cartesian coordinates and then converted into spherical coordinates. The data of azimuth angle is separated and normalized for serving as main key of this encryption scheme. However, most of the reported optical encryption techniques have been proven to be lacking in security strength because of their inherent linearity. To overcome this linearity, an optical one-way encryption scheme based on phase truncated Fourier transforms has also been proposed [23]. Although high robustness against existing attacks could be achieved owing to phase truncation in Fourier domain, it has been found to be vulnerable to the specific attack based on the iterative Fourier transforms when the two RPMs are used as public keys to encrypt different plaintexts [24]. In order avoid specific attack; the first way is to extend the PTFT-based cryptosystem into the fractional Fourier transform domain [25], gyrator domain directly [26–28]. The second way is to introduce an undercover amplitude modulator in the PTFT-based security system [25]. In this article, for the first time to author’s knowledge, a new asymmetric single-channel color image security system using HT and GT is investigated. In this scheme, a color image to be encoded is
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divided into R;G; and B channels then each channel is separately Hartley transformed. The three transformed channels are multiplied and then phase- and amplitude truncated to obtain first encrypted image and first decryption key. The encoded image is further multiplied with a conjugate of RPM. The obtained image is gyrator transformed and then phase- and amplitude truncated to get second encrypted image and second decryption key. In decryption process, the second encrypted image and second decryption key are multiplied and then inverse gyrator transformed. The received image is first multiplied with the RPM and then independently multiplied with asymmetric (decryption) keys for color channels. The resulting images are Hartley transformed to recover R;G; and B channels. The original image can not be recovered unless the asymmetric keys, random phase mask, and transformation angle of GT are correct. HT is combined with GT to enhance the security of the proposed cryptosystem with transformation angle as very sensitive additional keys. The optoelectronic encryption and decryption designs are presented. Numerical simulation results have been shown to confirm the efficiency and security of the proposed cryptosystem. The suggested method has three advantages in comparison to the reported HT-based methods. First, owing to the nonlinear operation of phase truncation, a one-way encryption scheme is achieved and thus a high robustness against existing attacks could be obtained. Second, only one channel is used instead of three, and consequently a compact and feasible system is achieved. Finally, transformation angle of GT offers remarkably sensitive key, and thus the security of the system is highly enhanced. In addition, a real-value gray ciphertext can bewilder the unauthorized users.
CCP BS 1
SLM 1
L BS
Laser
M1 f Computer System
SLM2
L1
CCD BS 2
SLM1
z
Laser
CCD 1
z
z
Computer System
1
The gyrator transform at parameter α, which is called as the transformation angle, of a two-dimensional complex field funct ion f i xi ; yi , is calculated as [31]
BS
ð2Þ
3. Gyrator transform
α
M1
L Laser
where cas ¼ cos þ sin . According to the definition of Hartley transform, it can be synthesized from two Fourier transforms: Z þ1 Z þ1 f xi ; yi cas 2π xo xi þyo yi dxi dyi H xo ; yo ¼ 1 1
exp iπ iπ ð3Þ F xo ; yo ¼ pffiffiffi4 F xo ; yo þexp 2 2 where F xo ; yo is the p FT of the real function f i xi ; yi . The ffiffiffi constant factor exp iπ =4 = 2 can be neglected in HT operation, because squared modulus of HT is measured in the output plane. It is noted that the HT can be calculated by two FTs with a phase of exp iπ =2 between them.
CCP
1
and its inverse relation is Z þ1 Z þ1 H xo ; yo cas 2π xo xi þ yo yi dxo dyo H xi ; yi ¼
z
L3
L2
L1
Thetwo-dimensional Hartley transform (HT) of a real function f i xi ; yi is defined as [29,30] Z þ1 Z þ1 f xi ; yi cas 2π xo xi þyo yi dxi dyi ð1Þ H xo ; yo ¼
1
L3
M2
2. Hartley transform
1
L2
f o xo ; yo ¼ G f i xi ; yi xo ; yo Z Z þ1 xo yo þ xi yi cos α xi yo þ xo yi 1 dxi dyi f i xi ; yi exp i2π ¼ α sin sin α 1
ð4Þ
SLM 2 CCD 2
Fig. 1. (Color online) Optical setup for (a)color image encryption system, (b) color image decryption system.
where Gα ½ denotes operator of GT, xi ; yi and xo ; yo are the input and output coordinates, respectively. The GT can be achieved by using an optimized flexible optical system with plano-convex cylindrical lenses, which are rotated to vary the angle α [32]. The inverse GT corresponds to the GT at angle α. The GT can also be simulated by using phase-only filtering, Fourier transform and inverse Fourier transform [33].
4. Proposed security system The steps of the encryption procedure are described as follows. 1. The original color image represented as f xi ; yi is decom posed into R;G; and B channels denoted as f R xi ; yi , f G xi ; yi
M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
51
Fig. 2. (Color online) Simulation results of the proposal: (a) Original color image with 512 512, pixels, (b) real part of the multiplied image, (c) phase part of the multiplied image, (d) first decryption phase key for red channel, (e) first decryption phase key for green channel, (f) first decryption phase key for blue channel, (g) second decryption phase key, and (h) encoded image with all correct keys.
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M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
Fig. 3. (Color online) Decrypted results: (a) without knowing decryption key for red channel, (b) without knowing decryption key for green channel, (c) without knowing decryption key for blue channel, (d) without knowing first decryption phase keys, (e) without knowing second decryption key, (f) without knowing random phase key, (g) with change in transformation angle by 1 10 12 , and (h) with all correct keys.
M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
and f B xi ; yi , respectively. f xi ; yi ¼ f R xi ; yi ; f G xi ; yi ; f B xi ; yi
Red
ð5Þ
B
i
i
1
0.8 Correlation coefficient
2. Each channel is independently Hartley transformed. The transformed R;G; and B channels are multiplied and then phase- and amplitude truncated to obtain first encrypted image and first decryption key. 9 H R ðx; yÞ ¼ H f R xi ; yi > = H G ðx; yÞ ¼ H f G xi ; yi ð6Þ > H ðx; yÞ ¼ H f x ; y ; B
ð7Þ
E1 ðx; yÞ ¼ PT ½C ðx; yÞ
ð8Þ
P 1 ðx; yÞ ¼ AT ½C ðx; yÞ
-0.2 0
0.1
0.2
0.3
0.4
ð9Þ
ð10Þ
P G ðx; yÞ ¼
P 1 ðx; yÞ H R ðx; yÞH B ðx; yÞ
ð11Þ
P B ðx; yÞ ¼
P 1 ðx; yÞ H G ðx; yÞH B ðx; yÞ
ð12Þ
These color channels are combined in computer system to form color image. The optical system of HT composed of Fourier lens L of focal length f and a Michelson-type interferometer [29,30]. A cube corner prism (CCP) in one arm rotates the field through an angle
1 x 10
-9
0.6
0.4
0
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Transformation angle
1 x 10
-9
Blue 1
Correlation coefficient
0.8
0.6
0.4
0.2
0
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
Transformation angle
ð17Þ
0.9
0.2
The steps of the decryption procedure are given as follows.
ð16Þ
0.8
0.8
ð14Þ
2. The obtained image is first multiplied with a random phase mask and then the product is separately multiplied with P R ðx; yÞ,P G ðx; yÞ, and P G ðx; yÞ. The resulting images are Hartley transformed to retrieve red, green and blue channels.
0.7
1
4. The encoded image is modulated with a conjugate of random phase mask Rn ðx; yÞ. The modulated image is gyrator transformed at transformation angle α and then phase- and amplitude truncated to get second encrypted image and second decryption key. ð13Þ E2 xo ; yo ¼ PT Gα E1 ðx; yÞRn ðx; yÞ
1. The second encrypted image and second decryption key are multiplied and then inverse gyrator transformed. ð15Þ D1 ðx; yÞ ¼ G α E2 xo ; yo P 2 xo ; yo
0.6
Green
Correlation coefficient
P 1 ðx; yÞ P R ðx; yÞ ¼ H G ðx; yÞH B ðx; yÞ
0.5
Transformation angle
where the operators PT½ and AT½ stand for phase- and amplitude truncation, respectively. 3. The first decryption keys for red, green and blue channels are deduced as
f R xi ; yi ¼ H ½D2 ðx; yÞP R ðx; yÞ f G xi ; yi ¼ H ½D2 ðx; yÞP G ðx; yÞ
f B xi ; yi ¼ H ½D2 ðx; yÞP B ðx; yÞ
0.4
0
C ðx; yÞ ¼ H R ðx; yÞH G ðx; yÞH B ðx; yÞ
D2 ðx; yÞ ¼ D1 ðx; yÞRðx; yÞ
0.6
0.2
where H½ represents operator of HT.
P 2 xo ; yo ¼ AT Gα E1 ðx; yÞRn ðx; yÞ
53
0.7
0.8
0.9
1 x 10
-9
Fig. 4. (Color online) Sensitivity of transformation angle: MSE versus variation in transformation angles for red, green, and blue channels.
π to go from F xo ; yo to F xo ; yo , and the plane mirror M 1 in the other arm produces a phase difference π =2 between two optical paths. The optical path length in both arms is equal tof . BS is a beam splitter.
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Fig. 5. (Color online) Attack results: (a) with fake color image (b) with first fake decryption key for red channel, (c) with first fake decryption key for green channel, (d) with first fake decryption key for blue channel, (e) with first fake decryption key, (f) with second fake decryption key.
The optical system of GT system consists of three generalized lenses L1 , L2 and L3 (withL3 ¼ L1 ) having fixed interval z between them. Each generalized lens is an assembled set of two identical plano-convex cylindrical lenses of the same power. The first and third identical generalized lenses of the same focal lengthf 1 ¼ z are rotated with respect to each other while the second generalized lens of the focal lengthf 2 ¼ z=2 is fixed. The GT operation for different transformation angles α is carried out by suitable rotation of these lenses [32]. The optoelectronic encryption system is shown in Fig. 1(a). The laser beam is split by a beam splitter BS1 into object and reference beams. The object beam modulated by red, green and blue
channels separately at the first spatial light modulator (SLM 1 ) is optically Hartley transformed. The three obtained channels are digitally multiplied to convert into a single channel. Its phase part is reserved as a first decryption key. Its amplitude part is multiplied by conjugate of RPM to get first encrypted image which is fed into the second spatial light modulator (SLM2 ). The beam splitter BS2 combines the first encrypted image with reference beam and then they are optically gyrator transformed. The phase part of the gyrator spectrum is reserved as a second decryption key. The amplitude part of the gyrator spectrum is recorded as second encrypted image by a charged couple device (CCD) camera and stored in a computer system.
M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
55
Fig. 6. (Color online) Decrypted results of encrypted data: (a) with Guassian noise of variance 0.1, (b) with Guassian noise of variance 0.2, (c) with speckle noise of variance 0.1, (d) with speckle noise of variance 0.2.
The optoelectronic decryption system is demonstrated in Fig. 1(b). The second encrypted image multiplied with the second decryption key is displayed onSLM 1 and illuminated by a laser light, and then optically inverse gyrator transformed. The gyrator spectrum is recorded by a CCD1 camera. The obtained image is multiplied with a RPM and first decryption key. The resulting image is fed into SLM 2 and illuminated by a laser light, and then optically Hartley transformed. To retrieve red, green and blue channels, the first decryption key is separately replaced by P R ðx; yÞ,P G ðx; yÞ, and P G ðx; yÞ. Finally, the recovered color channels are independently recorded by a CCD2 camera, which are combined into original color image in the computer system.
5. Numerical results
decryption, and without knowing RPM are demonstrated in Figs. 3 (d), 3(e) and 3(f), respectively. The retrieved images with change in transformation angle by 1 10 12 and with all correct keys are displayed in Figs. 3(g) and 3(h), respectively. From Figs. 3(a)–3(g), it evident that without correct knowledge of decryption key for red channel, decryption key for green channel, decryption key for blue channels, first decryption key, second decryption, RPM, or transformation angle, the original color image will not be obtained. To evaluate the difference between the original image and decrypted image, correlation coefficient (CC) is measured, and defined as
E ½I i E½I i ½I o E½I o ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CC ¼ rn ð18Þ orffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n offi ½I i E½I i 2 ½I o E½I o 2 E
Numerical simulations have been implemented on a Matlab 7.11.0 (R2010b) platform to confirm the proposed idea. An original color image comprising 512 512 3 pixels, as shown in Fig. 2(a) is chosen as an input image. The transformation angle of the GT is α ¼ 0:2 3 : The real and phase parts of the multiplied image are, respectively, demonstrated in Figs. 2(b) and 2(c). The first decryption keys for red, green and blue channels are, respectively, illustrated in Figs. 2(d), 2(e) and 2(f). The second decryption key and encoded image with all correct keys are, respectively, displayed in Figs. 2 (g) and 2(h). The recovered images without knowing decryption key for red channel, without knowing decryption key for green channel, and without knowing decryption key for blue channel, are shown in Figs. 3(a), 3(b) and 3(c), respectively. The reconstructed images without knowing first decryption key, without knowing second
where I o and I i are, respectively, output and input images, and E½ U denotes the expected value operator. The CCs of Fig. 2(h) for R; G; and B channels of encoded image with all correct keys are ( 0:1039; 0:0902; 0:0943), which indicates that the ciphertext provides no valuable information. The CCs exclusive of, only first decryption key, only second decryption key, and only RPM as shown in Figs. 3(d), 3(e), and 3(f) for R; G; and B channels are, respectively, ( 0:3865; 0:3953; 0:3727), (0:0004;0:0036;0:0056) and (0:0337;0:0489; 0:0924). The decrypted results can not be recognized, which demonstrate that the high performance of the proposed method. If the transformation angle of GT is varied by 1:0 3 10 12 from its correct value, then CCs of R; G; and B channels as illustrated in Fig. 3(g) are 0:0254;0:0448; and 0:0018, respectively. Thus the transformation angle is extremely
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M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
3500
3000
2500
2000
1500
1000
500
0
0
100
200
300
400
500
600
5
3
x 10
2.5
2
1.5
6. Conclusion
1
0.5
0
correct parameters are shown in Figs. 5(e) and 5(f), respectively. The corresponding CCs for R; G; and B channels are (0:0093; 0:0020;0:00008) and (0:0004;0:0029; 0:0007). These results verify that the proposed cryptosystem has the ability to resist known-plaintext attack. The robustness against the contaminations by the Gaussian noise and speckle noise has been investigated. The encrypted result as shown in Fig. 2(h) is contaminated by Gaussian noise of variances 0.1 and 0.2; the recovered images are displayed in Figs. 6(a) and 6(b), respectively. The corresponding CCs for R; G; and B channels are (0:9027;0:8777;0:8960) and (0:9021; 0:8766; 0:8951). Now the encrypted result as depicted in Fig. 2(h) is contaminated by speckle noise of variances 0.1 and 0.2, retrieved images are demonstrated in Figs. 6(c) and 6(d), respectively. The corresponding CCs for R; G; and B channels are (0:7620;0:7946;0:8333) and (0:6297; 0:6759; 0:7290). It is no doubt that the decrypted images can be recognized easily despite of noise interference. Thus the proposed algorithm has high robustness against noise contaminations. The statistical analysis has been studied on the proposed security system. Fig. 7(a) shows the histogram of original color image as demonstrated Fig. 2(a), which contains large spikes corresponding to color level values. Fig. 7(b) displays the histogram of encrypted image as illustrated Fig. 2(h), which is quite uniform and considerably different from the original image. This confirms that the encrypted image does not provide any information regarding the distribution of color values to the attacker. Hence, the proposed algorithm can resist any type of histogram based attacks.
0
100
200
300
400
500
600
Fig. 7. (a) (Color online) Histogram analysis: (a) histogram of original color image shown in Fig. 2(a), (b) histogram of encrypted image illustrated in Fig. 2(h).
sensitive key. The CCs of Fig. 3(h) for R; G; and B channels are 1:0000; 1:0000; and1:0000, respectively, which indicates that the original image is completely retrieved. Moreover, the CC curves from R; G; and B channels for various values of transformation angle are calculated and shown in Fig. 4. Hence the transformation angle improves the security of the proposed technique remarkably. The known-plaintext attack is adopted to verify the resistance of the proposed scheme. The fake color image of the same size of original image is shown in Fig. 5(a). First, the fake image is employed instead of original image and then followed the same encryption steps 1–3 to generate first fake decryption keys P 0R ðx; yÞ,P 0G ðx; yÞ, and P 0B ðx; yÞ for red, green and blue channels. The second fake decryption key P 02 xo ; yo is produced by using Eq. (14). The decrypted results by using only the first fake decryption key for red channel P 0R ðx; yÞ, only the first fake decryption key for green channel P 0G ðx; yÞ, and only the first fake decryption key for blue channel P 0B ðx; yÞ are displayed in Figs. 5 (b), 5(c), and 5(d), respectively. Note that the first decryption keys for two channels are correct when first fake decryption key for one channel is selected during the decryption procedure. The first fake decryption key consists of all the first fake decryption keys for red, green and blue channels. The recovered images by using only, first fake decryption key and second fake decryption key with all other
A novel non-linear single-channel color image encryption based on Hartley transform and gyrator transform is proposed. In this scheme, a single-channel is obtained by multiplying the three-Hartley transformed channels and then phase- and amplitude truncated to generate first encrypted image and first decryption key. The encrypted image multiplied with a conjugate of random phase mask (RPM) is gyrator transformed and then phaseand amplitude truncated to produce second encrypted image and second decryption key. Compared with previous studies, this security system is compact and feasible system, asymmetric approach provides a high robustness against existing attacks, and the transformation angle of GT provides extremely sensitive key. The proposed system can be realized digitally or optoelectronically. The validity and feasibility of the proposed technique are illustrated by numerical simulation results.
Acknowledgments The author is indebted to Abdul Aziz RA and Muhammad Sulayman RA for their inspiring supports. References [1] Refregier P, Javidi B. Optical image encryption based on input plane and Fourier plane random encoding. Opt Lett 1995;20:767–9. [2] Unnikrishnan G, Joseph J, Singh K. Optical encryption by double-random phase encoding in the fractional Fourier domain. Opt Lett 2000;25:887–9. [3] Situ G, Zhang J. Double random-phase encoding in the Fresnel domain. Opt Lett 2004;29:1584–6. [4] Chen L, Zhao D. Optical image encryption with Hartley transforms. Opt Lett 2006;20:767–9. [5] Hwang HE. An optical image cryptosystem based on Hartley transform in the Fresnel transform domain. Opt Commun 2011;284:3243–7. [6] Chen J, Zhu Z, Liu Z, Fu C, Zhang L, Yu H. A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains. Opt Express 2014;22:7349–61.
M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
[7] Alfalou A, Elbouz M, Mansour A, Keryer G. New spectral image compression method based on an optimal phase coding and the RMS duration principle. J Opt 2010;12:115403. [8] Alfalou A, Mansour A, Elbouz M, Brosseau C. Optical compression scheme to simultaneously multiplex and encode images. In: Crist´obal G, Schelkens P, Thienpont H, editors. Optical and digital image processing fundamentals and applications. Wiley-VCH Verlag GmbH & Co.; 2011. [9] Alfalou A, Brosseau C, Abdallah N, Jridi M. Simultaneous fusion, compression, and encryption of multiple images. Opt Express 2011;19:24023–9. [10] Alfalou A, Brosseau C, Abdallah N, Jridi M. Assessing the performance of a method of simultaneous compression and encryption of multiple images and its resistance against various attacks. Opt Express 2013;21:8025–43. [11] Aldossari M, Alfalou A, Brosseau C. Simultaneous compression and encryption of closely resembling images: application to video sequences and polarimetric images. Opt Express 2014;22:22349–68. [12] Zhang SQ, Karim MA. Color image encryption using double random phase encoding. Microw Opt Technol Lett 1999;21:318–23. [13] Chen L, Zhao D. Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography. Opt Express 2007;15:16080–9. [14] Joshi M, Shakher C, Singh K. Logarithms-based RGB image encryption in the fractional Fourier domain: a non-linear approach. Opt Lasers Eng 2009;47:721–7. [15] Liu Z, Xu L, Liu T, Chen H, Li P, Lin C, et al. Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains. Opt Commun 2011;284:123–8. [16] Chen W, Chen X, Sheppard CJR. Optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain. Opt Express 2012;20:3853–65. [17] Abuturab MR. Color information verification system based on singular value decomposition in gyrator transform domains. Opt Lasers Eng 2014;57:13–9.
57
[18] Abuturab MR. An asymmetric color image cryptosystem based on Schur decomposition in gyrator transform domains. Opt Lasers Eng 2014;58:39–47. [19] Abuturab MR. Single-channel color information security system using LU decomposition in gyrator transform domains. Opt Commun 2014;323:100–9. [20] Liu Z, Dai J, Sun X, Liu S. Color image encryption by using the rotation of color vector in Hartley transform domains. Opt Laser Eng 2010;48:800–5. [21] Abuturab MR. Color image security system based on discrete Hartley transform in gyrator transform domain. Opt Lasers Eng 2013;51:317–24. [22] Liu Z, Dai J, Sun X, Liu S. Optical color image hiding scheme based on chaotic mapping and Hartley transform. Opt Laser Eng 2013;51:967–72. [23] Qin W, Peng X. Asymmetric cryptosystem based on phase-truncated Fourier transforms. Opt Lett 2010;35:118–20. [24] Wang X, Zhao D. A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms. Opt Commun 2012;285:1078–81. [25] Wang X, Zhao D. Security enhancement of a phase-truncation based image encryption algorithm. Appl Opt 2011;50:6645–51. [26] Abuturab MR. Color information cryptosystem based on optical superposition principle and phase-truncated gyrator transform. Appl Opt 2012;51:7994–8002. [27] Abuturab MR. Information Authentication system using interference of two beams in gyrator transform domain. Appl Opt 2013;52:5133–42. [28] Abuturab MR. Securing multiple color information by optical coherent superposition based spiral phase encoding. Opt Lasers Eng 2014;56:152–63. [29] Bracewell RN, Bartelt H, Lohmann AW, Streibl N. Optical synthesis of the Hartley transform. Appl Opt 1985;24:1401–2. [30] Villasenor JD, Bracewell RN. Optical phase obtained by analogue Hartley transformation. Nature 1987;330:735–7. [31] Rodrigo JA, Alieva T, Calvo ML. Gyrator transform: properties and applications. Opt. Express 2007;15:2190–203. [32] Rodrigo JA, Alieva T, Calvo ML. Experimental implementation of the gyrator transform. J Opt Soc Am A 2007;24:3135–9. [33] Liu Z, Chen D, Ma J, Wei S, Zhang Y, Dai J, et al. Fast algorithm of discrete gyrator transform based on convolution operation. Optik 2011;122:864–7.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
An asymmetric single-channel color image encryption based on Hartley transform and gyrator transform Muhammad Rafiq Abuturab Department of Physics, Maulana Azad College of Engineering and Technology, Patna, Bihar 801113, India
ar t ic l e i nf o
a b s t r a c t
Article history: Received 28 July 2014 Received in revised form 30 December 2014 Accepted 8 January 2015 Available online 5 March 2015
A novel asymmetric single-channel color image encryption using Hartley transform and gyrator transform is proposed. A color image is segregated into R, G, and B channels and then each channel is independently Hartley transformed. The three transformed channels are multiplied and then phase- and amplitude truncated to obtain first encrypted image and first decryption key. The encoded image is modulated with a conjugate of random phase mask. The modulated image is gyrator transformed and then phase- and amplitude truncated to get second encrypted image and second decryption key. The asymmetric (decryption) keys, random phase mask, and transformation angle of gyrator transform serve as main keys. The optoelectronic encryption and decryption systems are suggested. Numerical simulation results have been demonstrated to verify the performance and security of the proposed security system. & 2015 Elsevier Ltd. All rights reserved.
Key words: Asymmetric single-channel color image encryption Hartley transform Gyrator transform
1. Introduction Optical technology has been widely used in the field of information security owing to its inherent advantages of multi-dimensional operation and inherent parallel procession. Refregier and Javidi first proposed the double-random phase-encoding (DRPE) technique, which is an optical symmetric-key method that encrypts an image by multiplying two random phase masks (RPMs), one in the spatial plane and one in the Fourier plane [1]. A number of modified DRPEbased encryption systems of gray-scale images have been put forwarded to achieve higher security [2–6]. The quantitative techniques for simultaneous compression and encryption of images have also been developed [7–10]. Recently, an optimized method of simultaneous compression and encryption designed to process images with close spectra is presented [11]. This approach is well adapted to the compression and encryption of images of a timevarying video sequences but also to static polarimetric images. The spectral plane (containing the information to send and/or to store) is decomposed in several independent areas which are assigned according a specific way. In addition, each spectrum is shifted in order to minimize their overlap. Zhang and Karim [12] first introduced a single-channel encryption algorithm for color image based on DRPE in the Fourier domain, which has been further extended to the fractional Fourier domain [13,14], Arnold transform and discrete cosine transform [15], Fresnel domain [16], and gyrator domain [17–19].
E-mail address: rafi[email protected] http://dx.doi.org/10.1016/j.optlaseng.2015.01.001 0143-8166/& 2015 Elsevier Ltd. All rights reserved.
The color image encryptions based on Hartley transform (HT), with the random rotation of color vector [20] and gyrator transform (GT) domain [21] have been researched. In the former method, the random shift of the four angles in rotation operation is regarded as the key of the encryption system whereas in the later technique, the period and iterative number of Arnold transform, and transformation angles of GT in each channel are employed as main keys of the algorithm. A color image encryption approach by using chaotic mapping and HT has been reported [22]. In this algorithm, the three components of color image are first scrambled by Baker mapping. The scrambled components are used as the positions in Cartesian coordinates and then converted into spherical coordinates. The data of azimuth angle is separated and normalized for serving as main key of this encryption scheme. However, most of the reported optical encryption techniques have been proven to be lacking in security strength because of their inherent linearity. To overcome this linearity, an optical one-way encryption scheme based on phase truncated Fourier transforms has also been proposed [23]. Although high robustness against existing attacks could be achieved owing to phase truncation in Fourier domain, it has been found to be vulnerable to the specific attack based on the iterative Fourier transforms when the two RPMs are used as public keys to encrypt different plaintexts [24]. In order avoid specific attack; the first way is to extend the PTFT-based cryptosystem into the fractional Fourier transform domain [25], gyrator domain directly [26–28]. The second way is to introduce an undercover amplitude modulator in the PTFT-based security system [25]. In this article, for the first time to author’s knowledge, a new asymmetric single-channel color image security system using HT and GT is investigated. In this scheme, a color image to be encoded is
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divided into R;G; and B channels then each channel is separately Hartley transformed. The three transformed channels are multiplied and then phase- and amplitude truncated to obtain first encrypted image and first decryption key. The encoded image is further multiplied with a conjugate of RPM. The obtained image is gyrator transformed and then phase- and amplitude truncated to get second encrypted image and second decryption key. In decryption process, the second encrypted image and second decryption key are multiplied and then inverse gyrator transformed. The received image is first multiplied with the RPM and then independently multiplied with asymmetric (decryption) keys for color channels. The resulting images are Hartley transformed to recover R;G; and B channels. The original image can not be recovered unless the asymmetric keys, random phase mask, and transformation angle of GT are correct. HT is combined with GT to enhance the security of the proposed cryptosystem with transformation angle as very sensitive additional keys. The optoelectronic encryption and decryption designs are presented. Numerical simulation results have been shown to confirm the efficiency and security of the proposed cryptosystem. The suggested method has three advantages in comparison to the reported HT-based methods. First, owing to the nonlinear operation of phase truncation, a one-way encryption scheme is achieved and thus a high robustness against existing attacks could be obtained. Second, only one channel is used instead of three, and consequently a compact and feasible system is achieved. Finally, transformation angle of GT offers remarkably sensitive key, and thus the security of the system is highly enhanced. In addition, a real-value gray ciphertext can bewilder the unauthorized users.
CCP BS 1
SLM 1
L BS
Laser
M1 f Computer System
SLM2
L1
CCD BS 2
SLM1
z
Laser
CCD 1
z
z
Computer System
1
The gyrator transform at parameter α, which is called as the transformation angle, of a two-dimensional complex field funct ion f i xi ; yi , is calculated as [31]
BS
ð2Þ
3. Gyrator transform
α
M1
L Laser
where cas ¼ cos þ sin . According to the definition of Hartley transform, it can be synthesized from two Fourier transforms: Z þ1 Z þ1 f xi ; yi cas 2π xo xi þyo yi dxi dyi H xo ; yo ¼ 1 1
exp iπ iπ ð3Þ F xo ; yo ¼ pffiffiffi4 F xo ; yo þexp 2 2 where F xo ; yo is the p FT of the real function f i xi ; yi . The ffiffiffi constant factor exp iπ =4 = 2 can be neglected in HT operation, because squared modulus of HT is measured in the output plane. It is noted that the HT can be calculated by two FTs with a phase of exp iπ =2 between them.
CCP
1
and its inverse relation is Z þ1 Z þ1 H xo ; yo cas 2π xo xi þ yo yi dxo dyo H xi ; yi ¼
z
L3
L2
L1
Thetwo-dimensional Hartley transform (HT) of a real function f i xi ; yi is defined as [29,30] Z þ1 Z þ1 f xi ; yi cas 2π xo xi þyo yi dxi dyi ð1Þ H xo ; yo ¼
1
L3
M2
2. Hartley transform
1
L2
f o xo ; yo ¼ G f i xi ; yi xo ; yo Z Z þ1 xo yo þ xi yi cos α xi yo þ xo yi 1 dxi dyi f i xi ; yi exp i2π ¼ α sin sin α 1
ð4Þ
SLM 2 CCD 2
Fig. 1. (Color online) Optical setup for (a)color image encryption system, (b) color image decryption system.
where Gα ½ denotes operator of GT, xi ; yi and xo ; yo are the input and output coordinates, respectively. The GT can be achieved by using an optimized flexible optical system with plano-convex cylindrical lenses, which are rotated to vary the angle α [32]. The inverse GT corresponds to the GT at angle α. The GT can also be simulated by using phase-only filtering, Fourier transform and inverse Fourier transform [33].
4. Proposed security system The steps of the encryption procedure are described as follows. 1. The original color image represented as f xi ; yi is decom posed into R;G; and B channels denoted as f R xi ; yi , f G xi ; yi
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Fig. 2. (Color online) Simulation results of the proposal: (a) Original color image with 512 512, pixels, (b) real part of the multiplied image, (c) phase part of the multiplied image, (d) first decryption phase key for red channel, (e) first decryption phase key for green channel, (f) first decryption phase key for blue channel, (g) second decryption phase key, and (h) encoded image with all correct keys.
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Fig. 3. (Color online) Decrypted results: (a) without knowing decryption key for red channel, (b) without knowing decryption key for green channel, (c) without knowing decryption key for blue channel, (d) without knowing first decryption phase keys, (e) without knowing second decryption key, (f) without knowing random phase key, (g) with change in transformation angle by 1 10 12 , and (h) with all correct keys.
M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
and f B xi ; yi , respectively. f xi ; yi ¼ f R xi ; yi ; f G xi ; yi ; f B xi ; yi
Red
ð5Þ
B
i
i
1
0.8 Correlation coefficient
2. Each channel is independently Hartley transformed. The transformed R;G; and B channels are multiplied and then phase- and amplitude truncated to obtain first encrypted image and first decryption key. 9 H R ðx; yÞ ¼ H f R xi ; yi > = H G ðx; yÞ ¼ H f G xi ; yi ð6Þ > H ðx; yÞ ¼ H f x ; y ; B
ð7Þ
E1 ðx; yÞ ¼ PT ½C ðx; yÞ
ð8Þ
P 1 ðx; yÞ ¼ AT ½C ðx; yÞ
-0.2 0
0.1
0.2
0.3
0.4
ð9Þ
ð10Þ
P G ðx; yÞ ¼
P 1 ðx; yÞ H R ðx; yÞH B ðx; yÞ
ð11Þ
P B ðx; yÞ ¼
P 1 ðx; yÞ H G ðx; yÞH B ðx; yÞ
ð12Þ
These color channels are combined in computer system to form color image. The optical system of HT composed of Fourier lens L of focal length f and a Michelson-type interferometer [29,30]. A cube corner prism (CCP) in one arm rotates the field through an angle
1 x 10
-9
0.6
0.4
0
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Transformation angle
1 x 10
-9
Blue 1
Correlation coefficient
0.8
0.6
0.4
0.2
0
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
Transformation angle
ð17Þ
0.9
0.2
The steps of the decryption procedure are given as follows.
ð16Þ
0.8
0.8
ð14Þ
2. The obtained image is first multiplied with a random phase mask and then the product is separately multiplied with P R ðx; yÞ,P G ðx; yÞ, and P G ðx; yÞ. The resulting images are Hartley transformed to retrieve red, green and blue channels.
0.7
1
4. The encoded image is modulated with a conjugate of random phase mask Rn ðx; yÞ. The modulated image is gyrator transformed at transformation angle α and then phase- and amplitude truncated to get second encrypted image and second decryption key. ð13Þ E2 xo ; yo ¼ PT Gα E1 ðx; yÞRn ðx; yÞ
1. The second encrypted image and second decryption key are multiplied and then inverse gyrator transformed. ð15Þ D1 ðx; yÞ ¼ G α E2 xo ; yo P 2 xo ; yo
0.6
Green
Correlation coefficient
P 1 ðx; yÞ P R ðx; yÞ ¼ H G ðx; yÞH B ðx; yÞ
0.5
Transformation angle
where the operators PT½ and AT½ stand for phase- and amplitude truncation, respectively. 3. The first decryption keys for red, green and blue channels are deduced as
f R xi ; yi ¼ H ½D2 ðx; yÞP R ðx; yÞ f G xi ; yi ¼ H ½D2 ðx; yÞP G ðx; yÞ
f B xi ; yi ¼ H ½D2 ðx; yÞP B ðx; yÞ
0.4
0
C ðx; yÞ ¼ H R ðx; yÞH G ðx; yÞH B ðx; yÞ
D2 ðx; yÞ ¼ D1 ðx; yÞRðx; yÞ
0.6
0.2
where H½ represents operator of HT.
P 2 xo ; yo ¼ AT Gα E1 ðx; yÞRn ðx; yÞ
53
0.7
0.8
0.9
1 x 10
-9
Fig. 4. (Color online) Sensitivity of transformation angle: MSE versus variation in transformation angles for red, green, and blue channels.
π to go from F xo ; yo to F xo ; yo , and the plane mirror M 1 in the other arm produces a phase difference π =2 between two optical paths. The optical path length in both arms is equal tof . BS is a beam splitter.
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Fig. 5. (Color online) Attack results: (a) with fake color image (b) with first fake decryption key for red channel, (c) with first fake decryption key for green channel, (d) with first fake decryption key for blue channel, (e) with first fake decryption key, (f) with second fake decryption key.
The optical system of GT system consists of three generalized lenses L1 , L2 and L3 (withL3 ¼ L1 ) having fixed interval z between them. Each generalized lens is an assembled set of two identical plano-convex cylindrical lenses of the same power. The first and third identical generalized lenses of the same focal lengthf 1 ¼ z are rotated with respect to each other while the second generalized lens of the focal lengthf 2 ¼ z=2 is fixed. The GT operation for different transformation angles α is carried out by suitable rotation of these lenses [32]. The optoelectronic encryption system is shown in Fig. 1(a). The laser beam is split by a beam splitter BS1 into object and reference beams. The object beam modulated by red, green and blue
channels separately at the first spatial light modulator (SLM 1 ) is optically Hartley transformed. The three obtained channels are digitally multiplied to convert into a single channel. Its phase part is reserved as a first decryption key. Its amplitude part is multiplied by conjugate of RPM to get first encrypted image which is fed into the second spatial light modulator (SLM2 ). The beam splitter BS2 combines the first encrypted image with reference beam and then they are optically gyrator transformed. The phase part of the gyrator spectrum is reserved as a second decryption key. The amplitude part of the gyrator spectrum is recorded as second encrypted image by a charged couple device (CCD) camera and stored in a computer system.
M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
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Fig. 6. (Color online) Decrypted results of encrypted data: (a) with Guassian noise of variance 0.1, (b) with Guassian noise of variance 0.2, (c) with speckle noise of variance 0.1, (d) with speckle noise of variance 0.2.
The optoelectronic decryption system is demonstrated in Fig. 1(b). The second encrypted image multiplied with the second decryption key is displayed onSLM 1 and illuminated by a laser light, and then optically inverse gyrator transformed. The gyrator spectrum is recorded by a CCD1 camera. The obtained image is multiplied with a RPM and first decryption key. The resulting image is fed into SLM 2 and illuminated by a laser light, and then optically Hartley transformed. To retrieve red, green and blue channels, the first decryption key is separately replaced by P R ðx; yÞ,P G ðx; yÞ, and P G ðx; yÞ. Finally, the recovered color channels are independently recorded by a CCD2 camera, which are combined into original color image in the computer system.
5. Numerical results
decryption, and without knowing RPM are demonstrated in Figs. 3 (d), 3(e) and 3(f), respectively. The retrieved images with change in transformation angle by 1 10 12 and with all correct keys are displayed in Figs. 3(g) and 3(h), respectively. From Figs. 3(a)–3(g), it evident that without correct knowledge of decryption key for red channel, decryption key for green channel, decryption key for blue channels, first decryption key, second decryption, RPM, or transformation angle, the original color image will not be obtained. To evaluate the difference between the original image and decrypted image, correlation coefficient (CC) is measured, and defined as
E ½I i E½I i ½I o E½I o ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CC ¼ rn ð18Þ orffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n offi ½I i E½I i 2 ½I o E½I o 2 E
Numerical simulations have been implemented on a Matlab 7.11.0 (R2010b) platform to confirm the proposed idea. An original color image comprising 512 512 3 pixels, as shown in Fig. 2(a) is chosen as an input image. The transformation angle of the GT is α ¼ 0:2 3 : The real and phase parts of the multiplied image are, respectively, demonstrated in Figs. 2(b) and 2(c). The first decryption keys for red, green and blue channels are, respectively, illustrated in Figs. 2(d), 2(e) and 2(f). The second decryption key and encoded image with all correct keys are, respectively, displayed in Figs. 2 (g) and 2(h). The recovered images without knowing decryption key for red channel, without knowing decryption key for green channel, and without knowing decryption key for blue channel, are shown in Figs. 3(a), 3(b) and 3(c), respectively. The reconstructed images without knowing first decryption key, without knowing second
where I o and I i are, respectively, output and input images, and E½ U denotes the expected value operator. The CCs of Fig. 2(h) for R; G; and B channels of encoded image with all correct keys are ( 0:1039; 0:0902; 0:0943), which indicates that the ciphertext provides no valuable information. The CCs exclusive of, only first decryption key, only second decryption key, and only RPM as shown in Figs. 3(d), 3(e), and 3(f) for R; G; and B channels are, respectively, ( 0:3865; 0:3953; 0:3727), (0:0004;0:0036;0:0056) and (0:0337;0:0489; 0:0924). The decrypted results can not be recognized, which demonstrate that the high performance of the proposed method. If the transformation angle of GT is varied by 1:0 3 10 12 from its correct value, then CCs of R; G; and B channels as illustrated in Fig. 3(g) are 0:0254;0:0448; and 0:0018, respectively. Thus the transformation angle is extremely
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3500
3000
2500
2000
1500
1000
500
0
0
100
200
300
400
500
600
5
3
x 10
2.5
2
1.5
6. Conclusion
1
0.5
0
correct parameters are shown in Figs. 5(e) and 5(f), respectively. The corresponding CCs for R; G; and B channels are (0:0093; 0:0020;0:00008) and (0:0004;0:0029; 0:0007). These results verify that the proposed cryptosystem has the ability to resist known-plaintext attack. The robustness against the contaminations by the Gaussian noise and speckle noise has been investigated. The encrypted result as shown in Fig. 2(h) is contaminated by Gaussian noise of variances 0.1 and 0.2; the recovered images are displayed in Figs. 6(a) and 6(b), respectively. The corresponding CCs for R; G; and B channels are (0:9027;0:8777;0:8960) and (0:9021; 0:8766; 0:8951). Now the encrypted result as depicted in Fig. 2(h) is contaminated by speckle noise of variances 0.1 and 0.2, retrieved images are demonstrated in Figs. 6(c) and 6(d), respectively. The corresponding CCs for R; G; and B channels are (0:7620;0:7946;0:8333) and (0:6297; 0:6759; 0:7290). It is no doubt that the decrypted images can be recognized easily despite of noise interference. Thus the proposed algorithm has high robustness against noise contaminations. The statistical analysis has been studied on the proposed security system. Fig. 7(a) shows the histogram of original color image as demonstrated Fig. 2(a), which contains large spikes corresponding to color level values. Fig. 7(b) displays the histogram of encrypted image as illustrated Fig. 2(h), which is quite uniform and considerably different from the original image. This confirms that the encrypted image does not provide any information regarding the distribution of color values to the attacker. Hence, the proposed algorithm can resist any type of histogram based attacks.
0
100
200
300
400
500
600
Fig. 7. (a) (Color online) Histogram analysis: (a) histogram of original color image shown in Fig. 2(a), (b) histogram of encrypted image illustrated in Fig. 2(h).
sensitive key. The CCs of Fig. 3(h) for R; G; and B channels are 1:0000; 1:0000; and1:0000, respectively, which indicates that the original image is completely retrieved. Moreover, the CC curves from R; G; and B channels for various values of transformation angle are calculated and shown in Fig. 4. Hence the transformation angle improves the security of the proposed technique remarkably. The known-plaintext attack is adopted to verify the resistance of the proposed scheme. The fake color image of the same size of original image is shown in Fig. 5(a). First, the fake image is employed instead of original image and then followed the same encryption steps 1–3 to generate first fake decryption keys P 0R ðx; yÞ,P 0G ðx; yÞ, and P 0B ðx; yÞ for red, green and blue channels. The second fake decryption key P 02 xo ; yo is produced by using Eq. (14). The decrypted results by using only the first fake decryption key for red channel P 0R ðx; yÞ, only the first fake decryption key for green channel P 0G ðx; yÞ, and only the first fake decryption key for blue channel P 0B ðx; yÞ are displayed in Figs. 5 (b), 5(c), and 5(d), respectively. Note that the first decryption keys for two channels are correct when first fake decryption key for one channel is selected during the decryption procedure. The first fake decryption key consists of all the first fake decryption keys for red, green and blue channels. The recovered images by using only, first fake decryption key and second fake decryption key with all other
A novel non-linear single-channel color image encryption based on Hartley transform and gyrator transform is proposed. In this scheme, a single-channel is obtained by multiplying the three-Hartley transformed channels and then phase- and amplitude truncated to generate first encrypted image and first decryption key. The encrypted image multiplied with a conjugate of random phase mask (RPM) is gyrator transformed and then phaseand amplitude truncated to produce second encrypted image and second decryption key. Compared with previous studies, this security system is compact and feasible system, asymmetric approach provides a high robustness against existing attacks, and the transformation angle of GT provides extremely sensitive key. The proposed system can be realized digitally or optoelectronically. The validity and feasibility of the proposed technique are illustrated by numerical simulation results.
Acknowledgments The author is indebted to Abdul Aziz RA and Muhammad Sulayman RA for their inspiring supports. References [1] Refregier P, Javidi B. Optical image encryption based on input plane and Fourier plane random encoding. Opt Lett 1995;20:767–9. [2] Unnikrishnan G, Joseph J, Singh K. Optical encryption by double-random phase encoding in the fractional Fourier domain. Opt Lett 2000;25:887–9. [3] Situ G, Zhang J. Double random-phase encoding in the Fresnel domain. Opt Lett 2004;29:1584–6. [4] Chen L, Zhao D. Optical image encryption with Hartley transforms. Opt Lett 2006;20:767–9. [5] Hwang HE. An optical image cryptosystem based on Hartley transform in the Fresnel transform domain. Opt Commun 2011;284:3243–7. [6] Chen J, Zhu Z, Liu Z, Fu C, Zhang L, Yu H. A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains. Opt Express 2014;22:7349–61.
M.R. Abuturab / Optics and Lasers in Engineering 69 (2015) 49–57
[7] Alfalou A, Elbouz M, Mansour A, Keryer G. New spectral image compression method based on an optimal phase coding and the RMS duration principle. J Opt 2010;12:115403. [8] Alfalou A, Mansour A, Elbouz M, Brosseau C. Optical compression scheme to simultaneously multiplex and encode images. In: Crist´obal G, Schelkens P, Thienpont H, editors. Optical and digital image processing fundamentals and applications. Wiley-VCH Verlag GmbH & Co.; 2011. [9] Alfalou A, Brosseau C, Abdallah N, Jridi M. Simultaneous fusion, compression, and encryption of multiple images. Opt Express 2011;19:24023–9. [10] Alfalou A, Brosseau C, Abdallah N, Jridi M. Assessing the performance of a method of simultaneous compression and encryption of multiple images and its resistance against various attacks. Opt Express 2013;21:8025–43. [11] Aldossari M, Alfalou A, Brosseau C. Simultaneous compression and encryption of closely resembling images: application to video sequences and polarimetric images. Opt Express 2014;22:22349–68. [12] Zhang SQ, Karim MA. Color image encryption using double random phase encoding. Microw Opt Technol Lett 1999;21:318–23. [13] Chen L, Zhao D. Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography. Opt Express 2007;15:16080–9. [14] Joshi M, Shakher C, Singh K. Logarithms-based RGB image encryption in the fractional Fourier domain: a non-linear approach. Opt Lasers Eng 2009;47:721–7. [15] Liu Z, Xu L, Liu T, Chen H, Li P, Lin C, et al. Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains. Opt Commun 2011;284:123–8. [16] Chen W, Chen X, Sheppard CJR. Optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain. Opt Express 2012;20:3853–65. [17] Abuturab MR. Color information verification system based on singular value decomposition in gyrator transform domains. Opt Lasers Eng 2014;57:13–9.
57
[18] Abuturab MR. An asymmetric color image cryptosystem based on Schur decomposition in gyrator transform domains. Opt Lasers Eng 2014;58:39–47. [19] Abuturab MR. Single-channel color information security system using LU decomposition in gyrator transform domains. Opt Commun 2014;323:100–9. [20] Liu Z, Dai J, Sun X, Liu S. Color image encryption by using the rotation of color vector in Hartley transform domains. Opt Laser Eng 2010;48:800–5. [21] Abuturab MR. Color image security system based on discrete Hartley transform in gyrator transform domain. Opt Lasers Eng 2013;51:317–24. [22] Liu Z, Dai J, Sun X, Liu S. Optical color image hiding scheme based on chaotic mapping and Hartley transform. Opt Laser Eng 2013;51:967–72. [23] Qin W, Peng X. Asymmetric cryptosystem based on phase-truncated Fourier transforms. Opt Lett 2010;35:118–20. [24] Wang X, Zhao D. A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms. Opt Commun 2012;285:1078–81. [25] Wang X, Zhao D. Security enhancement of a phase-truncation based image encryption algorithm. Appl Opt 2011;50:6645–51. [26] Abuturab MR. Color information cryptosystem based on optical superposition principle and phase-truncated gyrator transform. Appl Opt 2012;51:7994–8002. [27] Abuturab MR. Information Authentication system using interference of two beams in gyrator transform domain. Appl Opt 2013;52:5133–42. [28] Abuturab MR. Securing multiple color information by optical coherent superposition based spiral phase encoding. Opt Lasers Eng 2014;56:152–63. [29] Bracewell RN, Bartelt H, Lohmann AW, Streibl N. Optical synthesis of the Hartley transform. Appl Opt 1985;24:1401–2. [30] Villasenor JD, Bracewell RN. Optical phase obtained by analogue Hartley transformation. Nature 1987;330:735–7. [31] Rodrigo JA, Alieva T, Calvo ML. Gyrator transform: properties and applications. Opt. Express 2007;15:2190–203. [32] Rodrigo JA, Alieva T, Calvo ML. Experimental implementation of the gyrator transform. J Opt Soc Am A 2007;24:3135–9. [33] Liu Z, Chen D, Ma J, Wei S, Zhang Y, Dai J, et al. Fast algorithm of discrete gyrator transform based on convolution operation. Optik 2011;122:864–7.