Color information verification system based on singular valuedecomposition in gyrator transform domains

Muhammad Rafiq AbuturabDepartment of Physics, Maulana Azad College of Engineering and Technology, Patna 801113, India

a r t i c l e i n f o

Article history:Received 28 October 2013Received in revised form3 December 2013Accepted 2 January 2014Available online 28 January 2014

Keywords:Singular value decompositionRandom phase masksGyrator transform domains

a b s t r a c t

A new color image security system based on singular value decomposition (SVD) in gyrator transform(GT) domains is proposed. In the encryption process, a color image is decomposed into red, green andblue channels. Each channel is independently modulated by random phase masks and then separatelygyrator transformed at different parameters. The three gyrator spectra are joined by multiplication to getone gray ciphertext. The ciphertext is separated into U, S, and V parts by SVD. All the three parts areindividually gyrator transformed at different transformation angles. The three encoded information canbe assigned to different authorized users for highly secure verification. Only when all the authorizedusers place the U, S, and V parts in correct multiplication order in the verification system, the correctinformation can be obtained with all the right keys. In the proposed method, SVD offers one-wayasymmetrical decomposition algorithm and it is an optimal matrix decomposition in a least-squaresense. The transformation angles of GT provide very sensitive additional keys. The pre-generated keys forred, green and blue channels are served as decryption (private) keys. As all the three encrypted parts arethe gray scale ciphertexts with stationary white noise distributions, which have camouflage property tosome extent. These advantages enhance the security and robustness. Numerical simulations arepresented to support the viability of the proposed verification system.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Optical information processing technology has been exten-sively used in field of information security system because of theirmultiple parameters and high-speed parallel processing ability.Optical security techniques mainly consist of encryption, recogni-tion, correlation, identification, verification and watermarking. Thepioneering work in field of optical image encryption is double-random phase-encoding (DRPE) technique based on the 4-f opticalcorrelator to encrypt a primary image into stationary white noise[1]. Other methods based on multiplexing [2], digital holography[3], fractional Fourier domain [4,5], Fresnel domain [6,7], diffrac-tive imaging [8], and polarized light [9] have been proposed. In allthese methods, as monochromatic light is used to illuminate a realcolor image, color information of a retrieved image is lost. Sincethe color information of an image plays an important role inoptical information processing, color image encryption based onan indexed image and DRPE has been proposed [10]. Subsequentlythis technique has been further developed to increase the securityand robustness [11–20].

To the best of my knowledge, most of the color encryptiontechniques are considered as symmetric cryptosystems in which

encryption key is identical to the decryption key. Recently, phase-truncated Fourier transform based asymmetric cryptosystem hasbeen proposed [21]. In this method, encryption cannot be reversedwith the encryption keys (public keys) whereas decryption canonly be achieved with the decryption keys (private keys).Although, a specific attack method based on an iterative Fouriertransform can be applied to break asymmetric cryptosystem whentwo random phase masks are used as public keys to encodedifferent plaintexts [22]. The asymmetric cryptosystem can bemade secure by keeping the encryption keys as private keys orapplying different phase keys for different plaintexts during theencryption to evade the known public key attack as well as thespecific attack [23–26]. A new image encryption algorithm basedon SVD and Arnold transform is proposed [27]. In this scheme, anoriginal image is first transformed in fractional Fourier domain,and then decomposed into three segments by SVD. All the threeparts are Arnold transformed at different number of times toobtain three encrypted images. In the decryption process, all thethree encrypted parts are inverse Arnold transformed at corre-sponding times, multiplied in correct order and inverse fractionalFourier transformed with correct fractional orders to reconstructcorrect information.

In this paper, for the first time to my knowledge, a newasymmetric cryptosystem using SVD in GT domain is proposed.In encryption process, an input color image is converted into red,

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Optics and Lasers in Engineering

0143-8166/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.optlaseng.2014.01.006

E-mail address: rafiq.abuturab@gmail.com

Optics and Lasers in Engineering 57 (2014) 13–19

green and blue channels and then multiplied by correspondingrandom phase masks. The obtained images are gyrator trans-formed at different transformation angles. The resulting imagesare combined by multiplication to obtain a single gray ciphertextand then divided into U, S, and V parts by SVD. Finally, correspond-ing parts are separately gyrator transformed at different transfor-mation angles. The three gray ciphertexts can be allocated todifferent permitted users for highly secure authentication.To retrieve original image, the encoded U, S, and V parts areinverse gyrator transformed at the same encryption parameters,three recovered parts are correctly set to get a gray image, the grayimage is individually multiplied with the decryption keys of red,green, and blue channels and corresponding products are inversegyrator transformed at correct transformation angles to recon-struct red, green, and blue channels. Numerical simulationsdemonstrate the feasibility of the proposed security system.

Compared to Ref. [27], the proposed method has three advan-tages. First, this system consists of three asymmetric decryption(private) keys. Second, it provides very high sensitive parameters asthree transformation angles of GT. Third, it offers five sensitive-multiplication orders of U, S, and V parts of SVD. In addition, theMSE values of all parameters are very high for any one incorrect key.

The motivation of this research is to exploit the features of SVDin the proposed verification system. There are three advantages inSVD-based image encryption system: first, the size of the matricesfrom SVD transformation is not fixed, it is a one-way non-symmetrical decomposition algorithm and is an optimal matrixdecomposition in a least-square sense; second, when a smallperturbation is added to an image, its singular values do notchange significantly; and third, singular values represent intrinsicalgebraic image properties [28].

2. Theory

2.1. Singular value decomposition

The singular value decomposition (SVD) is a numerical techni-que used to diagonalize matrices. SVD decomposes an n�n realmatrix A into a product of three matrices as

A¼ USVT ¼ ½u1;u2;…;…;un� �

s1 0 ⋯ 00 s2 ⋯ 0⋮ 0 ⋱ 00 0 … sn

26664

37775

�½v1; v2;…;…; vn�T ð1Þ

where S is an n�n diagonal matrix with non-negative real valuescalled singular values. U is an orthogonal matrix and VT is thetranspose of an orthogonal matrix. The columns of U are eigen-vectors of AAT while the columns of V are eigenvectors of ATA. Theeigenvalues of AAT or ATA are the squares of the singular values forA. When r¼rank[A] then S¼diag(s1,s,…,…,sn) satisfies s1Zs2Z…ZsrZsrþ1¼srþ2…,…,¼sn¼0 [29].

From the viewpoint of linear algebra, an image can be viewedas a matrix with nonnegative scalar entries. SVD is used to extractalgebraic features from an image. Each singular value specifies theluminous of an image, whereas the corresponding pair of signalvectors specifies the geometry of the image. Let A be a matrixwhose elements are pixel values of an image. The image can beexpressed as

A¼ ∑r

i ¼ 1siuivTi ð2Þ

where ui and vi are the ith column vectors of U and V, respectively.

2.2. Gyrator transform

The gyrator transform (GT) of a two-dimensional complex fieldfunction fi(xi,yi) at parameter α, is defined as [30]

f 0ðx0; y0Þ ¼ Gα½f iðxi; yiÞ�ðx0; y0Þ

¼ 1sin α�� ��

Z Z þ1

�1f iðxi; yiÞ

�exp i2πðx0y0þxiyiÞ cos α�ðxiy0þx0yiÞ

sin α

� �dxidyi ð3Þ

where Gα[] indicates GT operator. (xi,yi) and (x0,y0) are the inputand output coordinates, respectively. The GT for large angles α canbe realized by an optimized flexible optical system which consistsof identical plano-convex cylindrical lenses. The angle α is chan-ged by proper rotation of cylindrical lenses [31]. Gα and G2π�α arereciprocal transforms which can be applied in optical imageencryption. The calculation of discrete GT can be implementedby Fresnel diffraction integral in free space under paraxial approx-imation [32]. In this computational method, larger computationalload is required. Therefore, in this paper, fast algorithm of discreteGT based on convolution operation is used in order to improvecomputational speed [33]. Recently, gyrator transformed basedcryptosystem have been presented for the security of imageinformation [15–20,25,26,34–38].

3. Proposed system

The proposed asymmetric color image security system is basedon SVD in GT domains. fr(xi,yi), fg(xi,yi) and fb(xi,yi) are, respectively,red, green and blue components of an RGB image f(xi,yi).

The encryption algorithm consists of following steps:First, fr(xi,yi), fg(xi,yi), and fb(xi,yi) are multiplied by random

phase masks exp iϕRðxi; yiÞ� �

exp iϕGðxi; yiÞ� �

, and exp iϕBðxi; yiÞ� �

,respectively. The corresponding randomized images are indepen-dently gyrator transformed at transformation angles αR, αG and αB.

gRðx; yÞ ¼GαR f Rðxi; yiÞexp iϕRðxi; yiÞ� �� � ð4Þ

gGðx; yÞ ¼ GαG f Gðxi; yiÞexp iϕGðxi; yiÞ� �� � ð5Þ

gBðx; yÞ ¼GαB f Bðxi; yiÞexp iϕBðxi; yiÞ� �� � ð6Þ

Second, Eqs. (4)–(6) are multiplied to get first encrypted imageas

Egðx; yÞ ¼ gRðx; yÞgGðx; yÞgBðx; yÞ ð7ÞThird, E(xi,yi) is divided into U(x,y), S(x,y) and V(x,y) parts by

using SVD.

Uðx; yÞ;½ Sðx; yÞ;Vðx; yÞ� ¼ svd Egðx; yÞ� � ð8Þ

where svd represents SVD function.Finally, they are independently gyrator transformed at trans-

formation angles αU, αS and αV to get Eu(x,y), ES(x,y) and Es(x,y),respectively.

EU ðx0; y0Þ ¼ GαU Uðx; yÞ½ � ð9Þ

ESðx0; y0Þ ¼ GαS Sðx; yÞ½ � ð10Þ

EV ðx0; y0Þ ¼ GαV V ðx; yÞ½ � ð11ÞThe decryption algorithm consists of following steps:First, Eu(x,y), ES(x,y), and EV(x,y) are separately gyrator trans-

formed at transformation angles �αU, �αS, and �αV, respectively.

DU ðx; yÞ ¼G�αU EUðx0; y0Þ� � ð12Þ

DSðx; yÞ ¼ G�αS ESðx0; y0Þ� � ð13Þ

M.R. Abuturab / Optics and Lasers in Engineering 57 (2014) 13–1914

DV ðx; yÞ ¼G�αV EV ðx0; y0Þ� � ð14Þ

Second, the obtained Du(x,y), Ds(x,y) and DV(x,y) parts aremultiplied in correct order to recover D(x,y).

Dðx; yÞ ¼DUðx; yÞDSðx; yÞDV ðx; yÞ ð15ÞThe decryption keys of red, green, and blue channels are given by

PRðx; yÞ ¼1

gGðx; yÞgBðx; yÞð16Þ

PGðx; yÞ ¼1

gRðx; yÞgBðx; yÞð17Þ

PBðx; yÞ ¼1

gRðx; yÞgGðx; yÞð18Þ

Third, the decrypted channels are generated as

f Rðxi; yiÞ ¼ G�αR ½Dðx; yÞPRðx; yÞ��� �� ð19Þ

z z

Computer System

CCD

SLM L1 L2 L3

Fig. 1. Opto-electronic decryption system for proposed color image.

Fig. 2. Simulation results: (a) original color image with 512�512 pixels and 24 bits; (b) amplitude part of image obtained after multiplication operation; (c) phase part ofimage, obtained after multiplication operation; (d) encoded image of U part; (e) encoded image of, S part; (f) encoded image of V part; (g) decryption key for red channel;(h) decryption key for, green channel; (i) decryption key for blue channel; (j) decrypted image with all correct keys; (k) image used for attack; and (l) decrypted image withfake decryption keys with all correct, parameters.

M.R. Abuturab / Optics and Lasers in Engineering 57 (2014) 13–19 15

Fig. 3. Sensitivity results: (a) with multiplication order UVS; (b) with multiplication. order SUV; (c) with multiplication order SVU; (d) with multiplication order VUS; (e) withmultiplication order VSU; (f) with an error in transformation angle αU by 0.0011; (g) with an error in transformation angle αS by 0.0011; and (h) with an error intransformation, angle αV by 0.0011. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Decryption results: (a) without red channel key; (b) without green channel key; (c) without blue channel key; and (d) without any channel key. (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this article.)

f Gðxi; yiÞ ¼ G�αG ½Dðx; yÞPGðx; yÞ��� �� ð20Þ

f Bðxi; yiÞ ¼ G�αB ½Dðx; yÞPBðx; yÞ��� �� ð21Þ

Finally, these color channels are combined into original colorimage in computer system.

The encryption process can be performed digitally whereas thedecryption process can be implemented opto-electronically. TheGT can be realized in coherent optical system using three general-ized lenses (denoted as L1, L2 and L3) with fixed equal distance zbetween them. Each generalized lens corresponds to an assembledset of two identical plano-convex cylindrical lenses of the samepower. The first and third identical lenses of focal length f1¼z arerotated to change the transformation angle α. The second lens L2of a focal length f2¼z/2 is fixed [31].

The opto-electronic decryption system for red channel is shownin Fig. 1. The digitally recovered image D(x,y) is first multiplied withdecryption phase keys for red channel PR(x,y) and displayed onspatial light modulator (SLM), illuminated by a uniform plane wave

and inverse gyrator transformed. The retrieved red channel fr(xi,yi) isthen recorded by a CCD camera. Similarly, D(x,y) is individuallymultiplied by PG(x,y), and PB(x,y), and the above procedures arerepeated to retrieve fG(xi,yi) and fB(xi,yi), respectively. Finally, thereconstructed color channels are combined into decrypted colorimage by using a computer system.

4. Numerical simulation results

Numerical simulations have been performed on a Matlab 7.11.0(R2010b) platform to study the viability of the proposed system.The color image with a size of 512�512�3 pixels and 24 bits isused as test image, which is shown in Fig. 2(a). The transformationangles of the GT, αR, αG, and αB are 0.451, 0.551 and 0.651,respectively. The parameters of the GT corresponding to αU, αS

and αV are 0.101, 0.201, and 0.301. The amplitude and phasedistributions of the image obtained after multiplication operationare, respectively, displayed in Fig. 2(b) and (c). The encoded images

0.59 0.592 0.594 0.596 0.598 0.6 0.602 0.604 0.606 0.608 0.610

0.5

1

1.5

2

2.5

3x 106

x 106

x 106

Transformation angle for U part

MS

E

RedGreenBlue

0.59 0.592 0.594 0.596 0.598 0.6 0.602 0.604 0.606 0.608 0.610

0.5

1

1.5

2

2.5

Transformation angle for S part

MS

ERedGreenBlue

0.59 0.592 0.594 0.596 0.598 0.6 0.602 0.604 0.606 0.608 0.610

0.5

1

1.5

2

2.5

Transformation angle for V part

MS

E

RedGreenBlue

Fig. 5. (a) MSE against variation in transformation angle for each channel of U part, (b) MSE against variation in transformation angle for each channel of S part, and (c) MSEagainst variation in transformation angle for each channel of V part. (For interpretation of the references to color in this figure, the reader is referred to the web version ofthis article.)

M.R. Abuturab / Optics and Lasers in Engineering 57 (2014) 13–19 17

of U, S and V parts are illustrated in Fig. 2(d), (e), and (f),respectively. These three encoded gray images have no informa-tion of all the three channels. Fig. 2(g), (h), and (i) demonstratesthe decryption keys for red, green, and blue channels, respectively.The decryption result using the correct parameters is depicted inFig. 2(j). It is evident that recovered image is very similar to theoriginal image without any distortion. The decryption result ofknown-plaintext attack with the keys generated from a chosenimage, which is treated as fake image, is displayed in Fig. 2(k). Theattack result using fake decryption keys with all correct para-meters is shown in Fig. 2(l). It can be seen that the retrieved imageis noise-like signal. The sensitivity results with incorrect multi-plication orders corresponding to UVS, SUV, SVU, VUS, and VSU aredepicted in Fig. 3(a)–(e). That means any attempt at decryption ofciphertext without correct order of U, S, and Vwill fail. Fig. 3(f), (g),and (h) demonstrates the decryption results with an error intransformation angle, αU by 0.0011, αS by 0.0011, and αV by 0.0011,respectively. It can be observed that the reconstructed images arecompletely noise. Fig. 4(a)–(d) displays the brute force attackwithout using, only PR(x,y), only PG(x,y), only PB(x,y), and anydecryption keys (for red, green, and blue channels), respectively.These results indicate a high robustness against brute force attack.

The mean square error (MSE) values are calculated to evaluate thequality of the decrypted image. The definition of MSE is expressed as

MSE¼ 1M � N

∑M

i ¼ 1∑N

j ¼ 1I0ðm;nÞ� Idðm;nÞ�� ��2 ð22Þ

where I0(m,m) and Id(m,m) represent the amplitude values of originaland retrieved images at pixel position(m,n), respectively. (M�N) is thesize of the two images.

The MSE values corresponding to in Fig. 2(d)–(f) for red, green andblue channels for encrypted U, S, and V parts with all correctparameters are, respectively, (2.0866�104, 1.2910�104, 1.1659�104), (3.4710�1012, 3.4710�1012, 3.4710�1012) and (2.0866�104,1.2910�104, 1.1659�104).

The MSE values of encoded U and V parts are equal and that ofencoded S part is very high. Thus, any useful information cannotbe recovered. As all the three encrypted parts are the gray scaleciphertexts with stationary white noise distributions, which havecamouflage property to some extent.

The MSE values of red, green and blue channels for decryptedimages, with all exact keys and with fake decryption keys are(4.1348�10�23, 1.0532�10�23, 9.8901�10�24) and (5.2309�105,3.33107�105, 5.2289�105) as shown in Fig. 2(j) and (l), respectively.The MSE values close to zero indicate the correct decryption resultwhereas the MSE values of the order 105 signify a high robustnessagainst known-plaintext attack. In order to demonstrate key sensitiv-ity, the MSE of different multiplication orders and change in transfor-mation angles of the GT have been calculated. Note that the other keysare correct when one incorrect key is used. The MSE values of red,green and blue channels corresponding to incorrect multiplicationorders UVS, SUV, SVU, VUS, and VSU are (3.2188�106, 1.7640�106,1.0691�106), (4.2399�105, 2.2215�105, 1.7071�105), (1.9096�106,7.9931�105, 8.7409�105), (1.4643�106, 1.1191�106, 8.8454�105),

0 100 200 300 400 500 6000

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400 500 6000

2

4

6

8

10

12

14

0 100 200 300 400 500 6000

2

4

6

8

10

12

14

0 100 200 300 400 500 6000

2

4

6

8

10

12

14x 104

x 104

x 104

Fig. 6. (a) Histogram analysis: (a) histogram of original color image shown in Fig. 2(a); (b) histogram of encrypted U part as shown in Fig. 2(d); (c) histogram of encrypted Spart as shown in Fig. 2(e); and (d) histogram of encrypted V part as shown in Fig. 2(f). (For interpretation of the references to color in this figure, the reader is referred to theweb version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 57 (2014) 13–1918

and (1.1853�106, 8.1470�105, 8.4650�105) as illustrated in Fig. 3(a)–(e). The MSE values of red, green and blue channels for change intransformation angle, αU by 0.0011 αS by 0.0011 and αV by 0.0011 are,respectively, (1.1167�105, 8.0090�104, 5.2440�104), (3.4366�105,1.6996�105, 1.0322�105), and (1.8100�105, 7.9994�104,6.3760�104) as demonstrated in Fig. 3(f)–(h). It is clear from theMSE values of decrypted images as shown in Fig. 3(a)–(h), theproposed system is more sensitive to both multiplication orders ofUSV and transformation angles of GT. The MSE values of red, green andblue channels with no decryption keys are 3.4688�1012, 3.4689�1012 and 3.4689�1012, respectively. The MSE values of the order 1012

imply that an extremely high robustness against brute force attack.Moreover, compared with the previous SVD based work [27,33], theproposed method provides additional sensitive parameters with veryhigh MSE values.

The parameters of GT are additional keys of the proposedsystem. Thus, the dependence of MSE on the change in theseparameters are shown in Fig. 5(a)–(c). The correct value of eachparameter is fixed 0.61. It is obvious that the MSE values of αU, αS

and αV reach zero at their right values. However, MSE values αU, αS

and αV become very high for a very small difference (0.0011) fromtheir exact values. The noise-like images will be recovered asillustrated in Fig. 3(f)–(h). The high sensitivity of transformationangles will cause great difficulty in copying the decryption system.Finally, image histograms are used in image analysis. The histo-grams of original color image, encoded images of S, U and V partsare demonstrated in Fig. 6(a), (b), (c), and (d), respectively. Theintensity distribution of the histograms of the encrypted images isfully dissimilar from that of the histogram of the original image,which indicates that an intruder cannot perceive useful informa-tion based on statistical properties. The security analyses effec-tively illustrate the robustness of the proposed method.

5. Conclusion

A new asymmetric security system based on SVD in GTdomains has been presented. The red, green and blue channelsare multiplied by corresponding random phase masks and thengyrator transformed. The transformed images are combined bymultiplication to obtain a single gray image, which is segregatedinto U, S, and V parts by SVD and finally they are individuallygyrator transformed at different transformation angles. Theencoded U, S, and V parts can be allocated to three users of anauthorized group. SVD gives a one-way asymmetrical decomposi-tion algorithm. The parameters of GT supply extremely sensitivekeys. The decryption keys for red, green and blue channels, whichare different from encryption keys, are generated during encryp-tion process. Moreover, the gray scale ciphertexts instead of colorciphertexts can easily confuse an attacker. These propertiesimprove the performance of the proposed algorithm. A set ofnumerical simulations is made to illustrate the feasibility of theproposed authentication system.

Acknowledgments

The author is indebted to Muhammad Waizul Haque and Mehr-un-nisa for their inspiring supports.

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