Optical double image security using random phase fractional Fourier domain encoding and phase-retrieval algorithm

Optics Communications xx (xxxx) xxxx–xxxx

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Optical double image security using random phase fractional Fourier domain encoding and phase-retrieval algorithm Sudheesh K. Rajputa,b, Naveen K. Nishchala, a b

⁎

Department of Physics, Indian Institute of Technology Patna, Bihta, Patna 801103, Bihar, India Department of Systems Science, Graduate School of System Informatics, Kobe University, Rokkodai 1-1, Nada, Kobe 657-8501, Japan

A R T I C L E I N F O

A BS T RAC T

Keywords: Image encryption Double random phase encoding Fractional Fourier transform Phase retrieval algorithm Phase truncation

We propose a novel security scheme based on the double random phase fractional domain encoding (DRPE) and modified Gerchberg-Saxton (G-S) phase retrieval algorithm for securing two images simultaneously. Any one of the images to be encrypted is converted into a phase-only image using modified G-S algorithm and this function is used as a key for encrypting another image. The original images are retrieved employing the concept of known-plaintext attack and following the DRPE decryption steps with all correct keys. The proposed scheme is also used for encryption of two color images with the help of convolution theorem and phase-truncated fractional Fourier transform. With some modification, the scheme is extended for simultaneous encryption of gray-scale and color images. As a proof-of-concept, simulation results have been presented for securing two gray-scale images, two color images, and simultaneous gray-scale and color images.

1. Introduction Optical image encryption is one of the important applications of optical information processing technologies. In recent years, it has drawn considerable attention of global researchers because it provides higher level of security as compared to the digital counterpart. Various encryption techniques have been reported for information security in past few decades [1–37]. Among them, optical encryption based on double random phase encoding (DRPE) [1] is more popular because it can be easily implemented by using a 4 f optical set-up. Several optical transform domains have been used to implement this scheme [2–6]. However, the DRPE scheme has been found to be vulnerable to the chosen-ciphertext attack, chosen plaintext attack, and known-plaintext attack [7–10]. This vulnerability is because of its inherent linearity in the encryption process. To break the linearity in the process, asymmetric cryptosystem based on phase-truncated Fourier transform (PTFT) approach has been proposed [11]. In this scheme, asymmetric keys are generated during encryption. However, this scheme has also been found vulnerable against specific attack [12]. After publication of this work, several variants of asymmetric cryptosystems have been proposed with improved security levels [13–18]. An asymmetric cryptosystem has also been proposed by Liu et al. [19], which uses mixture retrieval type of Yang-Gu algorithm. Wang and Zhao [20] proposed an amplitude-phase retrieval attack-free encryption scheme based on direct attack to PTFT-

⁎

based encryption using random amplitude mask. In communication systems, there is a requirement for several users to share the secured common information simultaneously. In the multiple image encryption (MIE), two or more images are encrypted into a single image using optical or digital techniques. To secure multiple images, various encryption schemes have been proposed [21– 28]. Situ and Zhang [21] proposed a technique of wavelength-multiplexing into the DRPE system for MIE. Barrera et al. [22] proposed MIE scheme using lateral shifting of RPM. An important issue with MIE is that how to reduce cross-talks and accordingly increase the number of images to be encrypted simultaneously. Hwang et al. [23] proposed MIE and multiplexing using modified Gerchberg-Saxton (GS) algorithm. Deng and Zhao [24] proposed MIE using phase retrieval algorithm and intermodulation. In this scheme, all the original images are extracted from the ciphertext without any cross-talk. Alfalou and Brosseau [25] proposed a scheme to compress and encrypt simultaneously multiple images using spectral multiplexing. Some double image encryption schemes have also been reported [26–28]. All the discussed encryption schemes use monochromatic light to illuminate the input image. Hence, the color information of decrypted image is lost. To overcome this drawback, color image encryption schemes have been proposed [29–36]. There are two ways to encode a color image; three channel and single channel image encryption. In three channel encryption scheme, a color image to be encrypted is separated into three primary color channels and then each channel is

Corresponding author. E-mail address: [email protected] (N.K. Nishchal).

http://dx.doi.org/10.1016/j.optcom.2016.11.002 Received 16 August 2016; Received in revised form 30 September 2016; Accepted 2 November 2016 Available online xxxx 0030-4018/ © 2016 Elsevier B.V. All rights reserved.

Please cite this article as: Rajput, S.K., Optics Communications (2016), http://dx.doi.org/10.1016/j.optcom.2016.11.002

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S.K. Rajput, N.K. Nishchal

encrypted and decrypted independently [29–31]. Such scheme increases the complexity and computational cost. On the other hand, single channel encryption scheme significantly reduces the complexity and computational cost [32–34]. Further, simultaneous color and grayscale image encryption schemes have also been proposed [35,36]. Most of the above mentioned security schemes are either suitable for two gray-scale images or only two color images or only for simultaneous gray-scale and color images. In this paper, we propose a scheme, which is suitable for all above combinations. The proposed scheme is based on the DRPE and modified G-S phase retrieval algorithm in fractional Fourier transform (FRT) domain. For two image encryption, any one of the images to be encrypted is converted into a phase-only image using modified G-S algorithm. Now another image is encrypted according to DRPE scheme using phase-only image as a fractional domain key. For successful retrieval of the original images, the concept of known-plaintext attack and decryption steps of DRPE are followed with all correct keys. The proposed scheme has also been extended for encryption of two color images and simultaneous encryption of gray-scale and color images. The computer simulation results have been presented in support of the proposed method.

Fig. 1. Block diagram for double image encryption. MGSA: Modified Gerchberg-Saxton algorithm; and POI: Phase-only image. N −1

MSE =

N×N

(6)

The block diagram for the proposed double image encryption scheme is shown in Fig. 1. The first image to be encrypted, A(x,y) is multiplied with an RPM, exp{i2πr1(x,y)}, whose FRT is obtained.

E1 (u , v ) = FRT β [A (x, y ) × exp{i 2πr1 (x, y )}]

(7)

The obtained spectrum is now multiplied with the generated phaseonly image; exp{iϕn (u , v )}, and its FRT of order β is obtained, which is referred to as the encrypted image. This function contains the information of both the gray-scale images.

2. Optical security scheme for double images In this Section, we discuss the proposed scheme for two gray-scale image encryption, two color images encryption, and simultaneous grayscale and color image encryption, one by one.

E (ξ, η) = FRT γ [E1 (u , v ) × exp{iϕn (u , v )}]

(8)

The schematic diagram of DRPE for implementing the proposed image encryption scheme optically is shown in Fig. 2(a). The first image bonded with RPM1 can be displayed on first spatial light modulator (SLM1) and illuminated with coherent light source. The obtained fractional spectrum multiplied with phase-only image as a fractional domain key can be displayed on the SLM2. The encrypted image can be recorded through a charge-coupled device (CCD) camera. For successful retrieval, the DRPE based decryption process is followed. The schematic diagram is shown in Fig. 2(b). Here, the original image can be obtained by displaying the encrypted image bonded with decryption keys on SLM and obtaining FRT of corresponding orders. In this case, input domain RPM, generated phaseonly image of second gray-scale image, and fractional orders, serve as keys for decryption. The FRT of order γ for the encrypted image, E(ξ,η), is obtained,

2.1. Two gray-scale image encryption For encryption of two gray-scale images, first image to be encrypted is converted into a phase-only image using phase retrieval algorithm. Suppose A(x,y) and B(x,y) denote two gray-scale images. The modified G-S algorithm [15,23,38] is applied for converting the second image, B(x,y) into a phase-only image. During phase-only conversion, the second image to be encrypted, B(x,y) is multiplied with a random phase mask (RPM); exp{i2πr(u,v)} and its FRT is obtained. The amplitude and phase parts of the obtained spectrum are separated numerically. The phase part of the spectrum is now inverse fractional Fourier transformed. These steps are repeated until process is converged. Mathematically, the algorithm can be explained with the following steps: 1. Any complex function Bk′ (u , v )after kth iteration can be written as

Bk′ (x, y ) = B (x, y ) × exp{i 2πrk (x, y )}

N −1

∑x =0 ∑y =0 { Bk"+1 (x, y ) − B (x, y ) }2

(1)

2. Perform the FRT to the complex function, Bk′ (x, y )

Bk +1 (u , v ) = FRT α [Bk′ (x, y )] ⎡ x 2 + y 2 + u2 + v 2 xyuv ⎤ =K ∬ f (x, y ) × exp ⎢iπ − 2iπ sin α ⎥ dxdy tan α ⎣ ⎦ = Bk +1 (u , v ) × exp{iϕk (u , v )}

(2)

where K is a complex constant and α is the order of FRT. Here (x,y) and (u,v) represent the coordinates of input and FRT domains, respectively. 3. Replace amplitude of previous Eq. with unity

Bk′+1 (u , v ) = 1 × exp{iϕk (u, v )}

(3)

′ (u , v ) as 4. Now perform inverse FRT to Bk+1

Bk″+1 (x, y ) = FRT −α [Bk′+1 (u , v )] = Bk″+1 (x , y ) × exp{iϕk′(x , y )}

(4)

5. Replace amplitude of Eq. (4) with the amplitude of input image to be encrypted

f k′+1 (x, y ) = f (x, y ) × exp(iϕk′(x, y ) = f (x, y ) × exp{irk +1 (x, y )}

(5) Fig. 2. Schematic diagrams for (a) encryption and (b) decryption processes. SLM: spatial light modulator; L: lenses; d: fractional distances; PC: personal computer; and CCD: charge-coupled device camera.

The convergence of the iteration process is completed by computing the mean square error (MSE) between abs[Bk+1′′(x,y)] and abs[B(x,y)], 2

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Fig. 3. Simulation results for two gray-scale images; (a) scene-1, (b) scene-2, (c) phase-only image of scene-1, (d) relation between MSE and number of iterations during generation of phase-only image, and (e) encrypted image.

E1 (u , v ) = FRT −γ [E (ξ, η) ]

(9)

these keys, the phase-only image can be obtained,

The spectrum is multiplied with the conjugate of phase-only image; exp{−iϕn (u , v )}, and its FRT is again obtained. Then obtained function is multiplied with conjugate of RPM to get the first original gray-scale image,

A (x , y ) =

FRT −β [E1 (u,

exp{iϕn (u , v )} =

FRT −γ [E (ξ, η)] FRT −β [ A (x, y ) × exp{i 2πr1 (x , y )}]

(11)

The inverse FRT of the retrieved phase-only image (Eq. (11)) gives the second original gray-scale image as it can be seen from steps of phase retrieval algorithm. Mathematically, the second image is obtained as.

v ) × exp{−iϕn (u , v )}] × exp{−i 2πr1 (u , v )} (10)

For decryption of second image, the concept of known-plaintext attack [9,15] is used for retrieving the phase-only image and then it is converted into a gray-scale image. From phase-only image, second image, B(x,y) is obtained by computing inverse FRT. With the help of

B (x, y ) = FRT −α [exp{iϕn (u, v )}]

3

(12)

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Fig. 4. Decrypted gray-scale images; (a) scene-1 obtained after using all correct keys, (b) scene-2 obtained after using all correct keys, (c) scene-1 obtained after using wrong RPM, and (d) scene-2 obtained after using incorrect FRT order.

where

2.2. Double color image encryption

Fnrm (u , v ) = FRT α [Inrm (x, y )] ⎫ ⎪ Fngm (u , v ) = FRT α [Ingm (x, y )]⎬ Fnbm (u , v ) = FRT α [Inbm (x, y )]⎪ ⎭

For double color image encryption, both color images to be encrypted are separated into red (r), green (g), and blue (b) color components. Suppose In(x,y) (n=1, 2) denote two color images and Inr(x,y), Ing(x,y), and Inb(x,y) are corresponding r, g, and b components, respectively. Now individual color components are modulated by multiplying with different RPMs. The modulated color components of both color images can be written as [33,34]

Inrm (x, y ) = Inr (x, y ) Rnr (x, y ) ⎫ ⎪ Ingm (x, y ) = Ing (x, y ) Rng (x, y ) ⎬ Inbm (x, y ) = Inb (x, y ) Rnb (x, y )⎪ ⎭

(17)

The obtained spectrum, Cn(u,v), is amplitude- and phase-truncated. The phase truncation and amplitude truncation of the fractional spectrum as obtained in Eq. (17), are defined as.

(13)

k n (u , v ) = AT [Cn (u , v )]

(18)

en (u , v ) = PT [Cn (u, v )]

(19)

The modulated color components of both the images can be expressed as,

The phase-truncated value en(u,v), as obtained in Eq. (16), with the help of Eq. (19), can be rewritten as,

cn (x, y ) = Inrm (x, y ) ⊗ Ingm (x, y ) ⊗ Inbm (x , y )

en (u , v ) = Fnrm (u , v ) ⋅ Fngm (u , v ) ⋅ Fnbm (u , v )

(14)

where the symbol ⊗ denotes the convolution operation. Now the functions corresponding to both the images, cn(x,y) are fractional Fourier transformed with an order parameter α,

Cn (u , v ) = FRT α [cn (x, y )]

This amplitude-truncated value helps generate asymmetric keys for individual color components. Here |.| denotes the modulus. The decryption keys for each component; knj(u,v), (for j= r, g, b and n=1,2) are generated as

(15)

With the help of convolution theorem [39], it can be written as,

Cn (u , v ) = Fnrm (u , v ) × Fngm (u , v ) × Fnbm (u , v )

(20)

k nr =

(16) 4

AT [Fnrm (u , v )] Fngm (u , v ) × Fnbm (u , v )

(21)

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Fig. 5. Simulation results of color image. (a) flower-1, (b) flower-2, (c) convolved gray-scale image of flower-1, and (d) convolved gray-scale of image of flower-2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

k ng =

k nb

AT [Fngm (u , v )] Fnrm (u , v )) × Fnbm (u , v )

AT [Fnbm (u , v )] = Fnrm (u , v ) × Fngm (u , v )

2.3. Simultaneous gray-scale and color image encryption (22) For simultaneous encryption of color and gray-scale images, the color image to be encrypted is convolved into a gray-scale image and individual keys are generated using Eqs. (13–23). Then gray-scale image to be encrypted is converted into a phase-only image. Now convolved image is encrypted according to Eqs. (12) and (13) using phase-only image of gray-scale image. The decryption of convolved image is performed using Eqs. (9) and (10) and then decrypted color image can be obtained by procedure explained through Eqs. (21–23). Finally, the decrypted gray-scale image is retrieved with the help of Eqs. (11) and (12).

(23)

The second convolved gray-scale image, e2(u,v) is converted into phase-only image using phase retrieval algorithm. The first convolved gray-scale image is encrypted according to Eqs. (7) and (8) using phase-only image of second convolved gray-scale image as a FRT domain key. The proposed encryption scheme as shown in Fig. 2(a) can be followed. The first convolved gray-scale image is decrypted according to Eqs. (9) and (10). For decryption of second convolved gray-scale image, the Eqs. (11) and (12) can be used. Now obtained both the convolved grayscale images are multiplied with the individual decryption keys, knj(u,v), obtained in Eqs. (21–23) and is fractional Fourier transformed with order −α.

Inj (x, y ) = I−α [en (u , v ) × k nj (u , v )]

3. Simulation results and discussion Computer simulations using MATLAB 7.10 platform has been performed to check the validity of the proposed scheme. Two grayscale images; scene-1 and scene-2, each of size 256×256 pixels, as shown in Fig. 3(a) and (b), respectively have been used. The second gray-scale image (scene-2) is converted into a phase-only image, as shown in Fig. 3(c). Fig. 3(d) shows the relation between MSE and number of iterations during generation of phase-only image. With this plot, it can be seen that MSE is converged to zero, which infers the complete formation of phase-only image. Fig. 3(e) shows the encrypted

(24)

All the color components of both the images are combined to retrieve the original color images.

5

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Fig. 6. (a) Relation between MSE and number of iterations during generation of phase-only image, (b) phase-only image of convolved image of flower-2, and (c) encrypted image.

Fig. 7. Decrypted color images obtained after using all correct keys; (a) flower-1 and (b) flower-2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6

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Fig. 8. Decrypted color images; (a) flower-1 obtained after using encryption keys, (b) flower-2 obtained after using encryption keys, (c) flower-1 obtained after using incorrect convolved image, (d) flower-2 obtained after using incorrect convolved image, (e) flower-1 obtained after using wrong fractional orders, and (f) flower-2 obtained after using wrong fractional orders. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

image. The arbitrarily chosen orders of FRT are α=0.40, β=0.50, and γ=0.60. Fig. 4(a) and (b) show the decrypted gray-scale images; scene1 and scene-2, respectively obtained after using all correct keys. Decrypted images of scene-1 and scene-2 obtained after using wrong

RPM and incorrect FRT order are shown in Fig. 4(c) and (d), respectively. For color image encryption, two color images; flower-1 and flower2, each of size 256×256×3 pixels, as shown in Fig. 5(a) and (b), 7

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Fig. 9. Simulation results for simultaneous color and gray-scale image (a) encrypted image, (b) decrypted color image of flower-1 obtained after using all correct keys, and (c) decrypted gray-scale image of scene-1 obtained after using all correct keys. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the encrypted image and Fig. 9(b) shows the decrypted color image of flower-1 obtained after using all correct keys. Fig. 9(c) shows the decrypted second gray-scale image of scene-1 obtained after using all correct keys. The scheme has been checked with wrong keys and the output remains same, as shown in Fig. 5(c–d) and Fig. 8(a–d), respectively. The MSE values are calculated to check the quality of decrypted images. The calculated values of MSE for Fig. 4(a) and (b) are 1.6×10– 19 and 4.7×10–18, respectively. The calculated values of MSE for Fig. 4(c) and (d) are 1.5×103 and 1.4×103, respectively. The MSE values are quite high and hence no useful information is retrieved. The MSE value has been calculated using the following expression,

respectively have been used. The primary color components of both the images are convolved into a single gray-scale image. The convolved gray-scale images of flower-1 and flower-2 are shown in Fig. 5(c) and (d), respectively. The plot between MSE and number of iterations during generation of phase-only image is shown in Fig. 6(a). The convolved gray-scale image of flower-2 converted into a phase-only image is shown in Fig. 6(b) and the encrypted image is shown in Fig. 6(c). The decrypted color images of flower-1 and flower-2 obtained after using all correct keys are shown in Fig. 7(a) and (b), respectively. The performance of this double color image encryption scheme is also checked by using incorrect keys. The decrypted color images of flower-1 and flower-2 obtained after using incorrect encryption keys are shown in Fig. 8(a) and (b), respectively. The decrypted color images of flower-1 and flower-2 obtained after using incorrect convolved images are shown in Fig. 8(c) and (d), respectively. The decrypted color images of flower-1 and flower-2 obtained after using wrong fractional orders are shown in Fig. 8(e) and (f), respectively. The wrong orders of FRT used are α=0.30, β=0.70, and γ=0.80. These results infer that the scheme is valid for color information security and the use of any one of wrong keys in decryption fails successful retrieval of original color images. Fig. 9(a–c) show the simulation results for simultaneous encryption of color and gray-scale images. The gray-scale image (Fig. 3(a)) and color image (Fig. 5(a)) have been used as input images. Fig. 9(a) shows

MSErgb =

MSEr + MSEg + MSEb 3

(25)

The calculated values of MSE for decrypted color image obtained after using all correct keys as shown in Fig. 7(a) and (b) are 4.3×10–17 and 6.5×10–17, respectively. The calculated values of MSE for decrypted color image obtained after using wrong keys as shown in Fig. 8(a–d) are 12.3, 9.7, 14.2 and 16.5, respectively. The calculated values of MSE for decrypted color image obtained after using wrong keys as shown in Fig. 8(e, f) are 15.3 and 17.4, respectively. The quality of decrypted images for simultaneous color and gray-scale encryption has also been checked by computing MSE values. From these calcu8

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lated values of MSE, it is inferred that the error is very small when correct keys are used for decryption but the error becomes very high when incorrect keys are used for decryption. 4. Conclusion A security scheme under the framework of DRPE and modified G-S phase retrieval algorithm for double image encryption has been proposed. For two image encryption, the first image to be encrypted is converted into a phase-only image using modified G-S algorithm. The phase-only image is used as a fractional domain key for encrypting the first image according to DRPE scheme. Both the original images are retrieved if all correct keys are used applying the concept of knownplaintext attack. The scheme has also been used for double color image encryption with the help of convolution theorem in FRT domain. The scheme is also useful for performing simultaneous encryption of grayscale and color images. The proposed scheme can be implemented optically as well as digitally. For optical implementation, optical set-up has been suggested. Simulation results have been presented for proposed double image security scheme. Acknowledgment The authors acknowledge the funding from the Council of Scientific and Industrial Research (CSIR), Government of India, under Grant no. 03/(1351)/16/EMR-II. S.K. Rajput acknowledges the funding from the Science and Engineering Research Board (SERB), Government of India, through award No. SB/OS/PDF-117/2015-16. References [1] P. Refregier, B. Javidi, Optical image encryption based on input plane encoding and Fourier plane random encoding, Opt. Lett. 20 (1995) 767–769. [2] G. Unnikrishnan, J. Joseph, K. Singh, Optical encryption by double-random phase encoding in the fractional Fourier domain, Opt. Lett. 25 (2000) 887–889. [3] B. Hennelly, J.T. Sheridan, Optical image encryption by random shifting in fractional Fourier domains, Opt. Lett. 28 (2003) 269–271. [4] O. Matoba, B. Javidi, Encrypted optical memory system using three-dimensional keys in the Fresnel domain, Opt. Lett. 24 (1999) 762–764. [5] G. Situ, J. Zhang, Double random-phase encoding in the Fresnel domain, Opt. Lett. 29 (2004) 1584–1586. [6] J.A. Rodrigo, T. Alieva, M.L. Calvo, Applications of gyrator transform for image processing, Opt. Commun. 278 (2007) 279–284. [7] A. Carnicer, M.M. Usategui, S. Arcos, I. Juvells, Vulnerability to chosen-cyphertext attacks of the optical encryption schemes based on double random phase keys, Opt. Lett. 30 (2005) 1644–1646. [8] X. Peng, H. Wei, P. Zhang, Chosen-plaintext attack on lensless double random phase encoding in Fresnel domain, Opt. Lett. 31 (2006) 3261–3263. [9] X. Peng, P. Zhang, H. Wei, B. Yu, Known plaintext attack on optical encryption based on double random phase keys, Opt. Lett. 31 (2006) 1044–1046. [10] Y. Frauel, A. Castro, T.J. Naughton, B. Javidi, Resistance of the double random phase encryption against various attacks, Opt. Express 15 (2007) 10253–10265. [11] W. Qin, X. Peng, Asymmetric cryptosystem based on phase-truncated Fourier transforms, Opt. Lett. 35 (2010) 118–120. [12] X. Wang, D. Zhao, A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms, Opt. Commun. 285 (2012) 1078–1081. [13] S.K. Rajput, N.K. Nishchal, Image encryption based on interference that uses fractional Fourier domains asymmetric keys, Appl. Opt. 51 (2012) 1446–1452. [14] X. Ding, X. Deng, K. Song, G. Chen, Security improvement for asymmetric

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