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Multilevel image authentication using shared secretthreshold and phase retrievalXuemei Pana, Xiangfeng Menga, Yurong Wanga, Xiulun Yanga, Xiang Pengb, Wenqi Heb,Guoyan Dongc & Hongyi Chend

a Shandong Provincial Key Laboratory of Laser Technology and Application, Department ofOptics, School of Information Science and Engineering, Shandong University, Jinan, Chinab College of Optoelectronics Engineering, Shenzhen University, Shenzhen, Chinac College of Materials Science and Opto-Electronic Technology, University of ChineseAcademy of Sciences, Beijing, Chinad College of Electronic Science and Technology, Shenzhen University, Shenzhen, ChinaPublished online: 23 Jul 2014.

To cite this article: Xuemei Pan, Xiangfeng Meng, Yurong Wang, Xiulun Yang, Xiang Peng, Wenqi He, Guoyan Dong & HongyiChen (2014) Multilevel image authentication using shared secret threshold and phase retrieval, Journal of Modern Optics,61:18, 1470-1478, DOI: 10.1080/09500340.2014.941430

To link to this article: http://dx.doi.org/10.1080/09500340.2014.941430

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Multilevel image authentication using shared secret threshold and phase retrieval

Xuemei Pana, Xiangfeng Menga*, Yurong Wanga, Xiulun Yanga, Xiang Pengb, Wenqi Heb, Guoyan Dongc andHongyi Chend

aShandong Provincial Key Laboratory of Laser Technology and Application, Department of Optics, School of Information Scienceand Engineering, Shandong University, Jinan, China; bCollege of Optoelectronics Engineering, Shenzhen University, Shenzhen,

China; cCollege of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing, China;dCollege of Electronic Science and Technology, Shenzhen University, Shenzhen, China

(Received 29 May 2014; accepted 30 June 2014)

A new kind of multilevel authentication system based on the (t, n) threshold secret sharing scheme and the iterativephase retrieval algorithm in Fresnel domain is proposed, in which, the first phase distribution iteratively generated isdivided into n parts and delivered to n different participants, during high-level authentication, any t (t ≤ n) or more ofthem can be collected to reconstruct the original meaningful certification image; While in the case of low-level authenti-cation, any t − 1 or fewer will gain no significant information of certification image, however, it can result in a remark-able peak output in the nonlinear correlation coefficient of the recovered image and the standard certification image,which can successfully provide an additional authentication layer for the high-level authentication. Theoretical analysisand numerical simulations both validate the feasibility of our proposed scheme.

Keywords: identity authentication; phase retrieval; (t, n) threshold secret sharing

1. Introduction

The optical information security technology has beenextensively studied in recent years and many predeces-sors help open up a wide vision for information securityresearch, since Réfrégier and Javidi introduced the dou-ble random phase encoding (DRPE) technique in 1995[1], it was successfully combined with many opticalinformation processing techniques or principles, such asfractional Fourier transform [2–5], Fresnel transform[6,7], digital holography [8–10], joint transform corre-lator (JTC) [11], phase-shifting interferometry [12–14],polarization encoding [15], gyrator transform [16], dif-fractive imaging [17], ghost imaging [18], fractionalMellin transform [19], two beam’s interference [20–25],aperture movement [26], and phase reservation and com-pression [27].

Besides DRPE and its related methods, the otherimportant kind of optical information security is basedon phase retrieval algorithm or technique. In 1996, Wanget al. first proposed the method based on a modified pro-jection-onto-constraint-sets (POCS) phase retrieval algo-rithm, in which a secret image was encoded into arandom phase distribution at the Fourier plane relating toa fixed phase mask [28]. Li et al. demonstrated an opti-cal security system based on POCS algorithm imple-mented in a JTC [29]. Situ and Zhang proposed asecurity method implemented in 4-f setup where both thephase distributions can be adjusted simultaneously in

each iteration cycle [30,31]. To realize different authoritylevels, in 2007, we developed a hierarchical security sys-tem based on cascaded multiple-phase retrieval by aniterative Fresnel-transform algorithm [32]. To increasethe encryption capacity while avoiding the crosstalknoise, Huang et al. proposed a lensless multiple-imageoptical encryption method based on the cascading modi-fied Gerchberg–Saxton algorithm in the Fresnel domain[33]. Recently, Chen et al. proposed an informationauthentication scheme based on cascaded iterative phaseretrieval algorithm and sparse representation, in whichthe concept of nonlinear correlation is applied to identifythe decoded image when it cannot be recognized bydirect visual inspection [34].

Belongs to information security, the authenticationscheme, gradually catching the eyes of the public,mainly emphasizes finding the efficient method to recog-nize one’s identification efficiently which will surely helpguarantee the security of the intercommunication processof both sides. Combined with typical optical informationprocessing techniques, the authentication scheme hasbeen extensively studied in recent years and many prede-cessors help open up a wide vision for information secu-rity research.

The traditional authentication system is based on aone to one principle, that is, every participant of theauthentication system possesses only one pair of secretkeys without extra copies, which means that once lost,

*Corresponding author. Email: xfmeng@sdu.edu.cn

© 2014 Taylor & Francis

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the loss caused to the victims can be enormous. In thissense, it might be necessary for a group of participantsto sharing a certain set of secret data. To solve this prob-lem, Shamir first proposed the idea of (t, n) thresholdsecret sharing [35], by which, the secret data are encodedinto n shares and then distributed to n participants, any t(t ≤ n) or more of the shares can be collected to recoverthe secret, but any t − 1 can or fewer of them cannot. Sofar, many attempts have been made and much significantprogress has been achieved in this field [36–38]. Toincrease the authority levels and achieve a higher dis-crimination capability, here we present a kind of multi-level authentication system based on (t, n) thresholdsecret sharing scheme and phase retrieval algorithm, bywhich it is possible to realize different levels of accessi-bility to the original certification image for differentauthority levels with the same system. We first describethe principles and the procedure of this method, thengive its simulation verification, and finally make theconclusion.

2. Description of the authentication system

2.1. (t, n) threshold secret sharing algorithm

The (t, n) threshold secret sharing scheme is based on aLagrange interpolating polynomial where n representsthe sharing number and t the threshold recovery number,not larger than n. Assuming that a polynomial f about xof degree t − 1 is displayed as follows: [35–38]

f ðxÞ ¼ yþ m1xþ m2x2 þ � � � þ mt�1x

t�1; (1)

where m1, m2, … mt−1 represent t − 1 random numbersand y is a constant. In Equation (1), there are tunknowns, and in order to solve the equation, at least tequations about x are needed. Each xk corresponding to asingle value of f(xk) consists of a single point (xk, f(xk)),where k is a random integer in the range of [1, n], andthese n points (x1, f(x1)), (x2, f(x2)), …, (xn, f(xn)) satisfy-ing Equation (1) above make up a point group G, amongwhich any t or more points will be needed in order toacquire the value of y and m1, m2, … mt−1, while fewerthan t pairs of points will gain no information about theoriginal set of the secret data.

Collecting at least t pairs of points from the n onesin G, we can get the following equations:

f ðx1Þ ¼ yþ m1x1 þ m2x12 þ � � � þ mt�1x1t�1;f ðx2Þ ¼ yþ m1x2 þ m2x22 þ � � � þ mt�1x2t�1;

. . .f ðxtÞ ¼ yþ m1xt þ m2xt2 þ � � � þ mt�1xtt�1;

(2)

Thus, the value of y can be acquired from Equation (2)above

y ¼ð�1Þt�1 f ðx1Þ x2x3 � � � xtðx1 � x2Þðx1 � x3Þ � � � ðx1 � xtÞ

�

þf ðx2Þ x1x3 � � � xtðx2 � x1Þðx2 � x3Þ � � � ðx2 � xtÞ þ � � � :

þf ðxtÞ x1x2 � � � xt�1

ðxt � x1Þðxt � x2Þ � � � ðxt � xt�1Þ� (3)

For a Lagrange interpolating polynomial of degree t − 1,more than t points in G are needed to reconstruct the yvalue [35–38].

2.2. Designing process

2.2.1. Two phases iteratively generated by phaseretrieval algorithm in Fresnel domain

In the authentication center, a gray-scale image Lena(shown in Figure 1) acted as the example standard certi-fication image is iteratively encoded into two phasemasks located in input plane and transform plane basedon phase retrieval algorithm in Fresnel domain.

Figure 2 depicts the whole design process, in which,the basic structural framework for iterative phases gener-ation is similar to our previous work [32,39]. Two statis-tical independent random phase masks are placed at theinput plane (x1, y1) and the transform plane (x2, y2),respectively, whose amplitude transmittance areexp½i2pw0

1ðx1; y1Þ� and exp½i2pw02ðx2; y2Þ�, respectively,

where w01 and w0

2 are two random distributions in therange of [0, 1]. The standard certification image Lena islocated at the output plane (x, y). The distance betweenthe input and the transform plane is z1, and that betweenthe transform plane and the output plane is z2. When theinput plane is illuminated by an on-axis plane wave ofwavelength λ, the complex amplitude U at the outputplane under the Fresnel approximation is [32,39]:

Figure 1. Standard certification image ‘Lena’.

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Uðx; yÞ ¼FrTz2 FrTz1 exp½i2pw01ðx1; y1Þ�

n onexp½i2pw0

2ðx2; y2Þ�

o;

(4)

where FrT represents the Fresnel transform.The flow chart of the iteration process is similar to

Figure 2 in our recent work [39], the two phase distribu-tions at the input and the transform plane are simulta-neously updated after each loop which helps keep thecirculation running, and the real amplitude of the stan-dard certification image Lena (as shown in Figure 1) g isserved as the amplitude constraints at the output plane.After the kth iteration (k = 1, 2, 3 …), assuming that thetwo phase distributions are w1

k and w2k , respectively,

then the two phase distributions in the (k + 1)th step canbe described as [32,39]:

wkþ12 ¼ angle

IFrTz2 g exp iangle Uk� �� ��

FrTz1 exp iwk1

� �� � !

; (5)

wkþ11 ¼ angle IFrTz1 IFrTz2 g exp iangle Uk

� �� �� exp �iwkþ1

2

� �� �� �;

(6)

where IFrT denotes the inverse Fresnel transform, andangle (•) is the phase extraction operator.

The correlation coefficient (CC) [32,39] is generallyadopted as the convergent criteria to evaluate the similar-ity between the output real amplitude image gk (assumingafter kth iteration) and the standard certification imageg, which is defined as [32,39]

CC ¼ E g � E gð Þ½ � � gk � E gk� �� ��

rgrgk; (7)

where E{} denotes the expected value operator, and σ isthe standard deviation of the corresponding image. Theiteration cycle process does not stop until the CC is lar-ger than a predefined value.

When iterative cycle stops, two final phase distribu-tions ψ1 and ψ2, located in the input plane and transformplane, respectively, are generated and stored. The latterphase ψ2 is fixed in transform plane; while the formerphase ψ1 is used to processed into n shares based on (t, n)threshold secret sharing algorithm.

2.2.2. Phase information splitting and sharing

Each individual pixel of the iterative generated phase ψ1

is handled as a separate secret integer value y, which issplit into n shares based on the Lagrange interpolatingpolynomial f as mentioned in Section 2.1. Authenticationsystem pre-selected n meaningful images, called camou-flage images, for each secret integer value yk in gener-ated phase ψ1, the pixel in the corresponding position ofeach camouflage images is treated as the unknown vari-ables xk in the polynomial, and the corresponding value f(xk) can be calculated by Equation (1), finally, n matrixescalled secret-key-carrier (SKC) images are generated,which have the same size with phase ψ1 and correspond-ing camouflage images. For the purpose of secret shar-ing, the n SKC images (together with its correspondingcamouflage images) are delivered to n different partici-pants of the authentication system.

2.3. Authentication process

The authentication process is illustrated by the flow chartin Figure 3 and is detailed as follows.

2.3.1. High-level authentication

Based on the (t, n) threshold secret-sharing algorithmmentioned in Section 2.1, any t (t < n) or more partici-pants of the system with their SKC images will get anexplicit knowledge of the information of ψ1 and pass theauthentication system with the retrieved phase key w0

1,locating in the input plane, together with original phaseψ2 fixed in the transform plane, when the system is

Figure 2. System designing process.

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illuminated by a plane wave with the correct wavelengthλ, a recovered image gʹ is obtained in the output plane,which is mathematically expressed as

g0 ¼ abs FrTz2 FrTz1 exp iw01

� �� �exp iw2ð Þ� � �

; (8)

where ‘abs’ denotes the operator of taking the realamplitude.

Then the authentication center calculates the CCbetween the recovered image gʹ and the standard certifi-cation image g, if the CC is higher than the predeter-mined threshold (e.g. 0.90), that means the quality of therecovered image is so good that the authentication is suc-cessful, while lower means a failure, and this authentica-tion process can be called the high-level authentication.

2.3.2. Low-level authentication

If t − 1, or fewer participants, even only one of the sys-tem attempt to pass the authentication system, any oneof the t − 1 (or fewer) SKC images is regarded as thefirst retrieved phase key w00

1, locating in the input plane,as shown in Figure 3, together with original phase ψ2

fixed in the transform plane, according to the similarprocess in Section 2.3.1, a noise-like image g″, is recov-ered in the output plane, whose CC is too low to identifyany useful information by direct visual inspection, thatis, in this case, it cannot successfully pass through thehigh-level authentication.

To provide an additional authentication layer for thehigh-level authentication and thus achieve a higher dis-crimination capability, nonlinear correlation coefficient(NCC) distribution [34,40] is applied and calculated tocompare the recovered noise-like image g″ with the stan-dard certification image g, which is defined as [34,40]:

NCC n; gð Þ ¼ IFT FT gðn; gÞ½ �f g FT g00ðn; gÞ½ �f gj jx�1���

FT gðn; gÞ½ �f g FT g00ðn; gÞ½ �f g����2 ð9Þ

where FT and IFT represent the Fourier and inverseFourier transform, respectively, (ξ, η) is the coordinate ofthe spectrum transverse plane. The parameter ω definesthe strength of the applied nonlinearity whose range is in[0.2, 0.4] in which ω best suits to the verificationapplication.

The recovered image g″ is a noise-like image withtoo low CC, however, by calculating and displaying the3D NCC distributions, one remarkable peak is generatedin the NCC distributions outputs, which can help authen-ticate the information without the direct visualization ofhidden information.

If some attempters possess respective false SKCimages, in final, a noise-like image without any usefulinformation is obtained, furthermore, only noisy NCCdistributions without a remarkable peak are obtained, asthe false SKC images contain not any useful and authen-tic data.

In this case, the authentication process can be calledthe low-level authentication, for the recovered noise-likeimage with low CC, if there exists a remarkable peak inits NCC distributions, which means the low-level authen-tication is successful, while not any remarkable peakmeans a failure.

3. Computer simulations

A series of computer simulations have been made to ver-ify the feasibility of our proposed method and investigateits performance. All the images shown in the followingsimulations are 256 × 256 pixels in size. All the resolu-tions of the digital images in the input, transform and out-put planes are 15 μm, λ = 0.532 μm, z1 = z2 = 108.3 mm,and ω = 0.4.

Figure 4(a) and (b) depict the two phase distributionsψ1 and ψ2 iteratively generated by the phase retrievalalgorithm after 2000 times of iterations. Here, we take

Figure 3. The authentication process. (The colour version of this figure is included in the online version of the journal.)

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(3, 5) threshold secret sharing algorithm as an example,five meaningful camouflage images pre-selected areshown in Figure 5(a)–(e), based on the idea of phaseinformation splitting and sharing mentioned inSection 2.2.2, their corresponding SKC images to deliver

to five different participants are shown in Figure 6(a)–(e),respectively.

First, we verify the validity of the high-level authen-tication system. Any three or more participants with theircorrect SKC images can attempt to pass through the

Figure 4. (a)–(b) Two retrieved phase distributions (ψ1 and ψ2) iteratively generated after 2000 times of iterations.

Figure 5. (a)–(e) Five meaningful camouflage images pre-selected.

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high-level authentication system. Based on the high-levelauthentication method mentioned in Section 2.3.1,Figure 7 shows the final recovered image obtained fromthe three SKC images (Figure 6(a), (b), and (d)); it isclear that the recovered image with good quality is verysimilar to the standard certification image, and the CCreaches 0.9616, if the CC’s predetermined threshold is

set as 0.90, that means the high-level authentication issuccessful.

Next, we analyze and test the results of low-levelauthentication system. When only one participant withhis correct SKC image (such as Figure 6(a)) tries to testthe high-level authentication, a noise-like image withlow CC is retrieved in the output plane, as shown inFigure 8(a), from which we can see, any meaningfulinformation cannot be identified by direct visual inspec-tion, that means only one participant with one SKCimage cannot successfully pass through the high-levelauthentication, as the CC is far less than the thresholdvalue set beforehand. However, we can turn attention tocalculating the NCC distributions of the final retrievedimage and the standard certification image, Figure 8(b)displays the 3D NCC distributions of Figure 8(a), it isobviously one remarkable peak is generated. We extendthis simulation to the other four different SKC images,Figure 9(a)–(d) show the corresponding the NCC distri-butions, when the only one SKC image is chosen asFigure 6(b)–(e), respectively, the results of which arevery similar to Figure 8(b), a remarkable peak can all beobviously observed in the corresponding NCC distribu-tions. That means, the low-level authentication is suc-cessful, which provides an additional authentication layerfor the high-level authentication.

Figure 6. (a)–(e) Five SKC images.

Figure 7. The final recovered image obtained from three SKCimages (Figure 6(a), (b), and (d)).

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Figure 8. When only one correct SKC image (taking Figure 6(a) as the example) attempts to test the authentication system, (a) thefinal recovered image; (b) The 3D NCC distributions of (a). (The colour version of this figure is included in the online version of thejournal.)

Figure 9. (a)–(d) The 3D NCC distributions, when the only one SKC image is chosen as Figure 6(b)–(e), respectively. (The colourversion of this figure is included in the online version of the journal.)

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If the SKC image, one participant possessed is falseor incorrect, it will result in a failure of low-level authen-tication. Figure 10(a) shows the recovered image in theoutput plane, when a random noise image of one partici-pant, rather than his/her correct SKC image, is adoptedin the authentication system, and its corresponding NCCdistributions are given in Figure 10(b). Obviously, theretrieved output image is too fuzzy or meaningless todistinguish the original authentication information, andonly the noisy NCC distributions are generated, that is,both the high-level authentication and the low-levelauthentication are failed.

4. Conclusions

We proposed a new kind of multilevel authenticationsystem based on the (t, n) threshold secret-sharingscheme and the iterative phase retrieval algorithm inFresnel domain, in which multiple participants areinvolved in order to gain the accurate information aboutthe system that can overcome the weakness of the tradi-tional system based on one to one principle. With thisapproach, it is possible to realize different levels ofaccessibility to the original certification image for differ-ent authority levels with the same system. CC and NCCdistributions are utilized as the judgment criterion ofhigh-level authentication and low-level authentication,respectively. The feasibility of this method has been con-vincingly verified by theoretical analysis and numericalsimulations.

AcknowledgementWe thank the reviewers for some useful suggestions.

Funding

This work is supported by the National Natural Science Foun-dation of China [grant number 61275014], [grant number60907005], [grant number 61171073], [grant number51102148], [grant number 11104188]; the National NaturalScience Foundation of Shandong province [grant numberZR2011FQ011]; the National science and Technology programsof Shandong province [grant number 2011GGH20119]; theResearch Award Fund for Outstanding Young Scientists ofShandong Province [grant number BS2011DX023].

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