Securing multiple color information by optical coherent superpositionbased spiral phase encoding

Muhammad Rafiq Abuturab n

Department of Physics, Maulana Azad College of Engineering and Technology, Patna 801113, Bihar, India

a r t i c l e i n f o

Article history:Received 24 October 2013Received in revised form22 November 2013Accepted 26 December 2013Available online 21 January 2014

Keywords:Optical multiple-color image cryptosystemCoherent superpositionSpiral phase encoding

a b s t r a c t

A new optical multiple-color image cryptosystem using optical coherent superposition based spiral phaseencoding is proposed, which can be applied to achieve a nonlinear multiple-image encryption of the samesize. This multiplexed coding scheme is lensless, non time-consuming and decoding procedure is free fromcross talk and noise effects in real time. In this contribution, a color image is decomposed into threeindependent channels, i.e., red, green and blue. Each channel is then divided into an arbitrarily selectedspiral phase mask (SPM) and a spiral key mask (SKM). The selected SPM is introduced as an encryptedimage for multiple color images. The SKMs are employed as different decryption keys for different images.That means, only need is to send the construction parameters (as the order, the wavelength, the focallength, and the radius) of the SPM independently to multiple-user, but not the key itself, so it enhancesrobustness against existing attacks than double random phase encoding techniques. Moreover, themaximum data can be securely handled with a single parameter variation. The encryption process canbe performed digitally while the decryption process is very simple and can be implemented usingoptoelectronic architecture. A set of numerical simulation results confirm the feasibility and effectivenessof the proposed cryptosystem for multiple-color image encryption.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In order to transmit and receive the digital data, informationsecurity problem has been paid more and more attention with thedevelopment of science and technology. Optical image encryptiontechnology has gained interest as an attractive alternative in infor-mation security system owing to its inherent ability of multipleparameters and parallel operation. The double random phase encod-ing (DRPE) technique has arisen as a promising technique for opticalinformation security system, which uses a phase code key in each ofthe image and Fourier planes to encrypt the data of the originalimage and that employs complex conjugate of the Fourier-planephase code key to decrypt the data [1]. Various optical securityarchitectures have been designed and developed in encryptionapplications [2–6]. Recently, ptychography has been applied inoptically encrypting the complex-amplitude image, which demon-strated that introducing ptychography can not only simplify thearchitecture of an optical encryption system but also greatly enhanceits security by enlarging the key space [7].

The encryption and decryption stages can both be implemen-ted either optically or digitally. In optical security techniques, theencryption algorithm is equivalent to the optical architecture

together with the corresponding procedure. The output plane isconjugate to the input plane in the optical domain, so the linearitybetween the encrypted image at output plane and the originalimage at input plane is evident even though the encrypted datahave become stationary white noise which leads to a security flaw.The asymmetric cryptosystems have been proposed to breakdownthis linearity and enhance the security of the cryptosystem [8,9].Moreover, when a monochromatic light is used to illuminate acolor image during the encryption process, color information islost. It is obvious that a color image provides more informationthan a gray level image. The encryption system of color imagesbased on different methods has been analyzed [10–26]. Morerecently, the asymmetric color cryptosystems using simple tech-niques have also been proposed [22,23].

However, these methods are limited to a single image encryp-tion. Recently, multiple image encryption attracts much attentionin the field of information security system, which is extensivelyapplied in multiple-user authentications and content distributionowing to economic memory occupation and secret informationtransmission. Compared with a single-image encryption, multipleimage encryption encodes two or more images on the input planeinto a single encrypted file on the output plane by optical ornumerical techniques [27–37]. The encrypted file is generallysynthesized by superimposing individual encrypted imagestogether. Digitally, all the images are superimposed in one com-posite charge-coupled device (CCD) frame, and each one of them

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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.2013.12.018

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Optics and Lasers in Engineering 56 (2014) 152–163

can be independently recovered through a digital spatial filtering.Optically, the complexity of the procedure mainly depends on theoptical architecture implemented for recording data. The obviousdrawbacks to these techniques are time-consuming and sensitiveto cross talk and noise effects. Thus, the performance of a multipleimage security system can be improved by depressing cross talksbetween encrypted images and accordingly increase the multi-plexing capacity, and eliminating noise generation due to multiplenon-decrypted images over a single decrypted image.

The motivation of this research is to present a multiple-userverification system, which is advantageous of simplicity, compact-ness, nonlinearity, noniterative encoding, enhanced multiplexingcapacity without any crosstalk and noise, and multiple securityparameters. In this paper, for the first time to my knowledge,a new multiple-color information security system based on opticalcoherent superposition principle and spiral phase mask is pro-posed. First, a color image is split into red, green and bluechannels. Then each channel is directly separated into two phasemasks: one is a spiral phase mask (SPM) and other is a spiral keymask (SKM). The SPM is an arbitrarily selected mask, which isindependent of an original color image. Thus, the same SPM istreated as an encrypted image for different color images. The SKMis a modulation of the SPM by the original color image, which isemployed as a decryption key. Therefore, different SKMs are usedas decryption keys for different color images. The SPM is arbitrarilyselected and SKM is directly related to the color image. So theproposed system forms a nonlinear cryptosystem.

The proposed method follows the process of transmitting a singlechosen SPM as encrypted image for different color images indepen-dently to the different authorized users having different SKMs asdecryption keys. First, a single chosen SPM is needed as an encryptedimage for multiple-color image, so additive cross-talks are completelyremoved and consequently increase multiplexing capacity. Second,different images are separately reconstructed by different authorizedusers by employing different SKMs as decryption keys, thus noiseeffects are entirely eliminated and high quality of decrypted images isobtained. Furthermore, a single encrypted result allows a singlespatial light modulator to display it, resulting in a simple and compactreconfigurable-optical-decryption system for multiple-user. Therefore,the encrypted data can be directly transmitted through communica-tion lines and then the decryption process can be achieved by usingthe correct parameters.

The proposed technique has three significant advantages overprevious studies: First, the encryption process is quite simple andfree from inherent time consuming problem of iterative algo-rithms. Second, the SPM not only provides multiple storing keys asthe order, the wavelength, the focal length, and the radius para-meters to achieve a higher security but its concentric focal ringsalso make easier to position in decoding process. Finally, the keyspace is enlarged and the sensitivities of parameters are increasedsignificantly, without increasing the complexity of optical hardwareand computation cost.

2. Theoretical analysis

2.1. Spiral phase mask

The phase function of a spiral phase mask (SPM) is defined as[38]

Sðr; θÞ ¼ Sðxi; yiÞ ¼ exp imθ� iπλf o

r2o

� �ð1Þ

where m is the order, λ is the wavelength, and fo is the focal lengthand ro is the radius of the SPM. ðr; θÞ represents the polarcoordinates, and ðxi; yiÞ denote the input coordinates.

An SPM Sðxi; yiÞ is decomposed as a vector as follows:

Sðxi; yiÞ ¼ ½Srðxi; yiÞ; Sgðxi; yiÞ; Sbðxi; yiÞ� ð2Þwhere Srðxi; yiÞ; Sgðxi; yiÞ; and Sbðxi; yiÞ are, respectively, red, green,and blue channels of SPM, which are expressed as

Srðxi; yiÞ ¼ exp imrθ�iπλr f o

r2o

� �ð3Þ

Sgðxi; yiÞ ¼ exp imgθ� iπλgf o

r2o

� �ð4Þ

Sbðxi; yiÞ ¼ exp imbθ�iπλbf o

r2o

� �ð5Þ

A single chosen SPM is employed as encrypted image for multiple-color image. That means Srðxi; yiÞ; Sgðxi; yiÞ; and Sbðxi; yiÞ are, respec-tively, used as chosen encrypted red channel, encrypted greenchannel, and encrypted blue channel of all color images.

2.2. Coherent superposition method

An original color image is dissociated into red, green, and bluechannels. Then each channel is directly separated into correspond-ing an SPM and an SKM, which means that each channel can bedecrypted by direct recording the intensity of the coherent super-position of its SPM and SKM [39].

A color image which consists of red, green and blue values withcertain proportions is expressed as a vector as follows:

f ðx; yÞ ¼ ½f rðx; yÞ; f gðx; yÞ; f bðx; yÞ� ð6Þ

where f rðx; yÞ; f gðx; yÞ; and f bðx; yÞ are, respectively, red, green, andblue channels of an original color image. ðx; yÞ denote the outputcoordinates.

Applying coherent superposition method, each channel is decom-posed into two phase masks: an SPM and a SKM.

f rðx; yÞ ¼ jSrðxi; yiÞþkrðx; yÞj ð7Þ

f gðx; yÞ ¼ jSgðxi; yiÞþkgðx; yÞj ð8Þ

f bðx; yÞ ¼ jSbðxi; yiÞþkbðx; yÞj ð9Þwhere | � | represents the modulus operation. krðx; yÞ; kgðx; yÞ; andkbðx; yÞ are, respectively, red, green, and blue channels of SKM, whichare defined as

krðx; yÞ ¼ Srðx; yÞþexp iπ� iarccos 1�jf rðx; yÞj22

� �� �ð10Þ

kgðx; yÞ ¼ Sgðx; yÞþexp iπ� iarccos 1�jf gðx; yÞj22

!" #ð11Þ

kbðx; yÞ ¼ Sbðx; yÞþexp iπ� iarccos 1�jf bðx; yÞj22

� �� �ð12Þ

Therefore, each channel is divided into an SPM and a SKM,whose phase difference is determined by the channel itself.

2.3. Multiple-color image algorithm

The whole process includes three steps.

Step 1. Encryption: First, decompose an SPM into its red, green,and blue components as Eq. (2) and then obtain Eqs. (3)–(5) aschosen encrypted red channel, encrypted green channel, andencrypted blue channel of multiple-color image.

M.R. Abuturab / Optics and Lasers in Engineering 56 (2014) 152–163 153

Step 2. Decryption keys: Generate decryption keys for multiple-color image as follows:Let the function f nðx; yÞ represents the intensity distribution ofthe nth, n¼ 1;…;N; original color image and f rn ðx; yÞ; f gn ðx; yÞ;and f bn ðx; yÞ denote its red, green, and blue channels. Then

decryption keys for multiple-color image can be generated byusing Eqs. (10)–(12) as

krn ðx; yÞ ¼ Srðx; yÞþexp iπ� iarccos 1�jf rn ðx; yÞj22

!" #ð13Þ

Fig. 1. (a) Flow diagram of the proposed red-channel decryption scheme, (b) Optical decryption setup for multiple color images based on superposition. (For interpretationof the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Original color images with 512�512 pixel and 24 bit: (a) Barbara, (b) Ali, (c) Olive fruits, and (d) Date tree. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 56 (2014) 152–163154

Fig. 3. Encrypted results with all right keys: (a) Phase part of SPM as encrypted image for multiple color images, and (b) real part of SPM as encrypted image for multiplecolor images. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig.4. (a) Phase distribution of SKM of Barbara, (b) Phase distribution of SKM of Ali, (c) Phase distribution of SKM of olive fruits, and (d) Phase distribution of SKM of datetree. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 56 (2014) 152–163 155

kgn ðx; yÞ ¼ Sgðx; yÞþexp iπ� iarccos 1�jf gn ðx; yÞj22

!" #ð14Þ

kbn ðx; yÞ ¼ Sbðx; yÞþexp iπ� iarccos 1�jf bn ðx; yÞj22

!" #ð15Þ

Step 3. Decryption: The decrypted images can be obtained byusing coherent superposition method where the modulus ofthe sum of Eqs. (3) and (13), Eqs. (4) and (14), Eqs. (5) and (15)produce Eqs. (16)–(18), respectively.

f rn ðx; yÞ ¼ jSrðx; yÞþkrn ðx; yÞj ð16Þ

f gn ðx; yÞ ¼ jSgðx; yÞþkgn ðx; yÞj ð17Þ

f bn ðx; yÞ ¼ jSbðx; yÞþkbn ðx; yÞj ð18Þ

Eqs. (16)–(18) are decrypted red, green, and blue channels ofnth color image. The flow diagram of the proposed red-channeldecryption scheme is shown as Fig. 1(a).

The encryption process is carried out digitally while the decryp-tion process can be implemented using simple optoelectronicarchitecture as illustrated in Fig. 1(b). For the sake of simplicity,only the red channel is illustrated. In decryption process, the laser

beam expanded with a beam expander is divided by a beam splitterBS1 into two beams. One beam is modulated by the red componentof an SPM as an encrypted red channel for multiple color images atthe first Spatial Light Modulator (SLM1). The other beam ismodulated by red component of an SKM at the second SpatialLight Modulator (SLM2). The two modulated beams are combinedby the beam splitter BS2. The intensity of the decrypted red channelis recorded by a charged couple device (CCD) camera and stored inthe computer system. The blue and green channels can alsoretrieved using the same procedure. Finally, they are combineddigitally by using computer systems to obtain decrypted an originalcolor image. Note that an SPM as an encrypted at SLM1 remains thesame for all color images while only different SKMs as decryptionkeys at SLM2 are used to retrieve different color images in theproposed system.

3. Numerical simulation results

A set of numerical simulations have been presented on aMatlab 7.11.0 (R2010b) platform to show the viability of theproposed technique. Four color images to be encrypted with512�512�3 pixel and 24 bit are shown as Fig. 2(a)–(d). The para-meters that produce a SPM are orders (mr ¼ 1; mg ¼ 2; mb ¼ 3),wavelengths (λr ¼ 632:8 nm, λg ¼ 532 nm, λb ¼ 488 nm), focallength f ¼ 2:0 cm, and radius r¼ 0:15 mm. The phase part and

Fig. 5. Decrypted results with all correct keys: (a) Barbara, (b) Ali, (c) Olive fruits, and (d) Date tree. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 56 (2014) 152–163156

real part of SPM as encrypted image for multiple color images areillustrated as Fig. 3(a) and (b). The phase distribution of decryptionkeys as SKMs for Fig. 2(a)–(d) are demonstrated as Fig. 4(a)–(d),respectively. The parameters of SKMs are the same as that of theSPM. All original images have been encrypted completely andcannot be recognized. The recovered four images using decryptionkeys with all correct parameters are depicted as Fig. 5(a)–(d). Theycorrespond to the original images well.

The sensitivity of parameters of SKM of the proposed systemused in decryption procedure for image1 (Barbara) has beeninvestigated. If ΔPc denotes the difference between encryptedvalue and decrypted value of a parameter of SKM P ¼m; λ; f or rofor channel c¼ r; g; b or rgb. Then the value of only one channel(r, g or b) of the parameter is changed from its encrypted valueby keeping the values of other two channels remaining the same.But in the case of rgb, the values of all the channels of theparameter are varied from their encrypted values. First, the orderof SKM is shifted from its encrypted value. For Δmr ¼ 0:2;Δmg ¼ 0:2; Δmb ¼ 0:2, and Δmrgb ¼ 0:2, the correspondingdecrypted images are shown as Fig. 6(a)–(d). Second, the wave-length of SKM is changed from its encrypted value. ForΔλr ¼ 1:0 nm, Δλg ¼ 1:0 nm, Δλb ¼ 1:0 nm, and Δλrgb ¼ 1:0 nm,the corresponding reconstructed images are illustrated as Fig. 7(a)–(d). Third, only focal length of SKM is deviated from its

encrypted value by Δf rgb ¼ 0:005 cm, the retrieved image isdemonstrated as Fig. 8(a). Finally, only radius of SKM is changedfrom its encrypted value by Δrrgb ¼ 0:0002 cm, the recoveredimage is depicted as Fig. 8(b). Similarly multiple images can alsobe sensitive to the same extent. It can be seen that the construc-tion parameters of SPM are extremely sensitive for their very smallshift from their correct values. Therefore, the high sensitivity willcause great difficulty in copying the decryption system.

To evaluate the similarity between the original image anddecrypted image, correlation coefficient (CC) is defined as

CC ¼ Ef½Ii�E½Ii��½Io�E½Io��gEffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif½Ii�E½Ii��2gf½Io�E½Io��2g

q ð19Þ

where Io and Ii are, respectively, output and input images, and E½U �denotes the expected value operator.

The CC values corresponding to Fig. 5(a)–(d) for red, green andblue channels are ð1:0000; 1:0000; 1:0000; 1:0000Þ, whichmeans that the color images are fully extracted. Table 1 showsCC values between three channels of four original images and theircorresponding encrypted images. These values are close to zero;as a result no information about the original images can beobtained.

Fig. 6. Sensitivity of decrypted image 1 (Barbara): (a) for order of SKM of red channel shifted by 0.2, (b) for order of SKM of green channel shifted by 0.2, (c) for order of SKMof blue channel shifted by 0.2, and (d) for order of SKM of all the channels shifted by 0.2. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 56 (2014) 152–163 157

The sensitivity of decryption keys of SKM as the order, thewavelength, the focal length, and the radius can be analyzed bycalculating CC values when they are shifted from their correct

values. The orders of SKMs of four images are shifted from theircorrect values by 0.2. For only shifted red channel, only shiftedgreen channel and only shifted blue channel of four images,

Fig. 7. Sensitivity of decrypted image 1 (Barbara): (e) for wavelength of SKM of red channel shifted by 1.0 nm, (f) for wavelength of SKM of green channel shifted by 1.0 nm,(g) for wavelength of SKM of blue channel shifted by 1.0 nm, and (h) for wavelength of SKM of all channel shifted by 1.0 nm. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

Fig. 8. Sensitivity of decrypted image1 (Barbara): (a) for focal length of SKM shifted by 0.005 cm, and (b) for radius of SKM shifted by 0.0002 cm. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

M.R. Abuturab / Optics and Lasers in Engineering 56 (2014) 152–163158

Table 1CC values between three channels of an original image and encrypted image.

Image CC value

Red channel Green channel Blue channel

Barbara �9:8595� 10�4 0:0021 4:9738� 10�4

Ali �0:0021 �1:1174� 10�4 �9:6542� 10�4

Olive fruits 2:2799� 10�4 �0:0013 0:0024

Date tree 0:0020 �1:6241� 10�4 9:7892� 10�4

Table 2aCC values of three channels with order of only red channel of SKM of an imageshifted from its correct value by 0.2.

Image CC value

Red channel Green channel Blue channel

Barbara �0:0011 0:0021 4:9738� 10�4

Ali 0:0047 �1:1174� 10�4 �9:6542� 10�4

Olive fruits 8:1860� 10�4 �0:0013 0:0024

Date tree 6:4335� 10�4 1:6241� 10�4 9:7892� 10�4

Table 2bCC values of three channels with order of only green channel of SKM of an imageshifted from its correct value by 0.2.

Image CC value

Red channel Green channel Blue channel

Barbara �9:8595� 10�4 0:0030 4:9738� 10�4

Ali �0:0021 0:0035 �9:6542� 10�4

Olive fruits 2:2799� 10�4 �0:0015 0:0024

Date tree 0:0020 �3:5303� 10�4 9:7892� 10�4

Table 2cCC values of three channels with order of only blue channel of SKM of an imageshifted from its correct value by 0.2.

Image CC value

Red channel Green channel Blue channel

Barbara �9:8595� 10�4 0:0021 0:0016

Ali �0:0021 �1:1174� 10�4 �0:0031

Olive fruits 2:2799� 10�4 �0:0013 5:3267� 10�4

Date tree 0:0020 �1:6241� 10�4 �5:1612� 10�4

Table 2dCC values of three channels with order of all color channels of SKM of an imageshifted from its correct value by 0.2.

Image CC value

Red channel Green channel Blue channel

Barbara �0:0011 0:0030 0:0016Ali 0:0047 0:0035 �0:0031Olive fruits 8:1860� 10�4 �0:0015 5:3267� 10�4

Date tree 6:4335� 10�4 �3:5303� 10�4 �5:1612� 10�4

Table 3aCC values of three channels with wavelength of only red channel of SKM of animage shifted from its correct value by 1.0 nm.

Image CC value

Red channel Green channel Blue channel

Barbara 4:3255� 10�4 0:0021 4:9738� 10�4

Ali 1:6785� 10�4 �1:1174� 10�4 �9:6542� 10�4

Olive fruits 0:0028 �0:0013 0:0024Date tree �0:0031 1:6241� 10�4 9:7892� 10�4

Table 3bCC values of three channels with wavelength of only green channel of SKM of animage shifted from its correct value by 1.0 nm.

Image CC value

Red channel Green channel Blue channel

Barbara �9:8595� 10�4 0:0013 4:9738� 10�4

Ali �0:0021 �0:0031 �9:6542� 10�4

Olive fruits 2:2799� 10�4 �8:3715� 10�4 0:0024

Date tree 0:0020 �0:0011 9:7892� 10�4

Table 3cCC values of three channels with wavelength of only blue channel of SKM of animage shifted from its correct value by 1.0 nm.

Image CC value

Red channel Green channel Blue channel

Barbara �9:8595� 10�4 0:0021 �9:2096� 10�4

Ali �0:0021 �1:1174� 10�4 0:0055

Olive fruits 2:2799� 10�4 �0:0013 0:0043

Date tree 0:0020 1:6241� 10�4 �0:0011

Table 3dCC values of three channels with wavelength of all color channels of SKM of animage shifted from its correct value by 1.0 nm.

Image CC value

Red channel Green channel Blue channel

Barbara 4:3255� 10�4 0:0013 �9:2096� 10�4

Ali 1:6785� 10�4 �0:0031 0:0055

Olive fruits 0:0028 �8:3715� 10�4 0:0043

Date tree �0:0031 �0:0011 �0:0011

Table 4CC values of three channels with focal length of SKM of an image shifted from itscorrect value by 0.005 cm.

Image CC value

Red channel Green channel Blue channel

Barbara 7:9441� 10�4 0:0033 �0:0033

Ali �6:7223� 10�4 �5:6968� 10�4 �0:0019

Olive fruits �0:0031 0:0028 4:6084� 10�4

Date tree 9:5125� 10�4 0:0013 0:0021

M.R. Abuturab / Optics and Lasers in Engineering 56 (2014) 152–163 159

the calculated CC values for their red, green and blue channels areshown as Tables 2a–2c, respectively. For all the three shiftedchannels of four images, the obtained CC values for their red,green and blue channels are displayed as Table 2d. The wave-lengths of SKMs of four images are altered from their right valuesby 1.0 nm. For only altered red channel, only altered green channeland only altered blue channel of four images, the computed CCvalues for their red, green and blue channels are demonstrated asTables 3a–3c, respectively. For all the three altered channels of fourimages, the resulted CC values for their red, green and bluechannels are exhibited as Table 3d. Note that the value of onlyone channel is changed by keeping the values of other two

channels with correct values. The focal lengths of SKMs of fourimages are changed from their accurate values by 0.005 cm. Thedetermined CC values for their red, green and blue channels areillustrated as Table 4. The radii of SKMs of four images are variedfrom their exact values by 0.0002 cm. The calculated CC values fortheir red, green and blue channels are as depicted in Table 5.As shown in Tables 2–5, the CC values are nearer to zero. Thus, itcan be concluded that the order, the wavelength, the focal length,and the radius are exceptionally sensitive for small variationcompared to their right values, which thereby proves the max-imum data can be securely handled with a single parametervariation.

The influence of change in the order, the wavelength, the focallength, and the radius of SKM from their right values has beenfurther studied. For this purpose, the sampling intervals for order,wavelength, focal length, and radius are fixed at the ranges [0.50,1.50], [460, 660], [1.985, 2.015], and [0.1485, 0.1515], respectively.The corresponding sampling lengths are taken at 0.05, 4�10�9,0.001, and 0.0001. The CCs as a function of the decryption order,wavelength, focal length, and radius deviated from correspondingencryption order, wavelength, focal length, and radius for red,green and blue channels of image 1 (Barbara) are shown as Fig. 9(a)–(d), respectively. In all cases, CCs rapidly decrease for verysmall deviation of decryption parameters from encryption para-meters. So the parameters of key SKM ensure the security of theproposed cryptosystem.

Table 5CC values of three channels with radius of SKM of an image shifted from its correctvalue by 0.0002 cm.

Image CC value

Red channel Green channel Blue channel

Barbara �0:0015 �5:0337� 10�4 0:0018

Ali 0:0016 �0:0018 �0:0016Olive fruits �9:0069� 10�4 4:4561� 10�4 �0:0014

Date tree 0:0021 �0:0043 1:8959� 10�4

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Order of SKM of image 1

Cor

rela

tion

Coe

ffici

ent

RedGreenBlue

460 480 500 520 540 560 580 600 620 640 660-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Cor

rela

tion

Coe

ffici

ent

RedGreenBlue

1.985 1.99 1.995 2 2.005 2.01 2.015 2.02-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Focal length (cm) of SKM of image 1

Cor

rela

tion

Coe

ffici

ent

RedGreenBlue

0.1485 0.149 0.1495 0.15 0.1505 0.151 0.1515 0.152 0.1525-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Radius (mm) of SKM of image 1

Cor

rela

tion

Coe

ffici

ent

RedGreenBlue

Wavelength (nm) of SKM of image 1

Fig. 9. (a) Correlation Coefficient versus variation in the order of red, green, and blue channels of SKM of image 1 (Barbara), (b) Correlation Coefficient versus variation in thewavelength of red, green, and blue channels of SKM of image 1(Barbara), (c) Correlation Coefficient versus variation in the focal length of red, green, and blue channels ofSKM of image 1 (Barbara), (d) Correlation Coefficient versus variation in the radius of red, green, and blue channels of SKM of image 1 (Barbara). (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

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It is evident from Eqs. (16)–(18), the difference between theSPMs and SKMs is determined by the original images themselveswhich are unknown to unauthorized users. So, it is impossiblefor an illegitimate user to deduce decryption keys from theencrypted result. Moreover, an arbitrarily selected SPM offersthe order, the wavelength, the focal length, and the radius asmultiple keys in a single mask, an illegal user will simply fail toreconstruct the original color image with a SPM different from theone used during the encryption process. There are still threepossibilities for unauthorized users to be explored to recover theplaintext of known size 512�512�3 pixel and 24 bit. First, anunauthorized user makes an effort to produce a decryption keyusing a fake plaintext but correct arbitrarily selected SPM. Thedecrypted result is the fake plaintext itself as illustrated in Fig. 10(a). Second, an illegitimate user attempts to generate a decryptionkey using a fake plaintext and an arbitrarily selected SPM withknown order, focal length, and radius but unknown wavelength.The reconstructed result is incorrect as demonstrated in Fig. 10(b).When an illegal user tries to recover a decryption key using a SKMand an arbitrarily selected SPM with known order, focal length, andradius but unknownwavelength. The retrieved result is erroneous asshown in Fig. 10(c). It can be concluded that the original plaintext canonly be retrieved using the encrypted SPM with the correct SKM asshown in Fig. 10(d).

Finally, the additive noise impact to the encrypted result byGaussian noise with a 0.2 variance has also been investigated. Theencrypted and noised encrypted images are, respectively, shown inFig. 11(a) and (b). The corresponding decrypted results for Barbara,Ali, Olive fruits, and Date tree are depicted in Fig. 11(c)–(f). It isevident that the all the retrieved images can be identified easily.

4. Conclusion

A new multiple-user verification system using optical coherentsuperposition based spiral phase encryption is proposed. The originalcolor image is separated into red, green and blue channels. Then eachchannel is bifurcated into a chosen SPM and a SKM using coherentsuperposition method. The chosen SPM with multiple parameters isemployed as encrypted mask for multiple color images with thesame size, which is independent of original images. The SKMs areused as different decryption keys for different images, which arerelated to the original images and chosen SPM. Much more informa-tion is required to correctly decrypt the multiples image and thus thesystem security is enhanced against existing attacks. The proposedmethod is advantageous of simplicity, compactness, nonlinearity,noniterative encoding, enhanced multiplexing capacity withoutany crosstalk and noise, and multiple security parameters.

Fig. 10. Decrypted result using: (a) a fake plaintext but correct selected SPM (b) a decryption key using a fake plaintext and an arbitrarily selected SPM with known order,focal length, and radius but unknownwavelength, (c) a decryption key using a SKM and an arbitrarily selected SPM with known order, focal length, and radius but unknownwavelength, and (d) 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.)

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The encryption procedure can be performed digitally while thedecryption procedure can be implemented optically using opto-electronic device. Numerical simulations are carried out to demon-strate the feasibility and high security of the scheme.

Acknowledgements

The author is indebted to Abdul Aziz RA and MuhammadSulayman RA for their inspiring supports.

Fig. 11. Additive noise impact: (a) encrypted result as SPM, (b) encrypted result contaminated by Gaussian noise with a 0.2 variance, (c) decrypted image (Barbara),(d) decrypted image (Ali), (e) decrypted image (Olive fruits), and (f) decrypted image (Date tree). (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

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