Vague Sets Security Measure for Steganographic System ...

Research ArticleVague Sets Security Measure for SteganographicSystem Based on High-Order Markov Model

Chun-Juan Ouyang12 Ming Leng12 Jie-Wu Xia12 and Huan Liu12

1Key Laboratory of Watershed Ecology and Geographical Environment Monitoring of NASG Jinggangshan UniversityJirsquoan 343009 China2School of Electronics and Information Engineering Jinggangshan University Jirsquoan 343009 China

Correspondence should be addressed to Chun-Juan Ouyang oycj001163com

Received 26 April 2017 Accepted 12 June 2017 Published 6 August 2017

Academic Editor Yushu Zhang

Copyright copy 2017 Chun-Juan Ouyang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Security measure is of great importance in both steganography and steganalysis Considering that statistical feature perturbationscaused by steganography in an image are always nondeterministic and that an image is considered nonstationary in this paper thesteganography is regarded as a fuzzy process Here a steganographic security measure is proposedThis security measure evaluatesthe similarity between two vague sets of cover images and stego images in terms of n-order Markov chain to capture the interpixelcorrelation The new security measure has proven to have the properties of boundedness commutativity and unity Furthermorethe security measures of zero order first order second order third order and so forth are obtained by adjusting the order value ofn-order Markov chain Experimental results indicate that the larger 119899 is the better the measuring ability of the proposed securitymeasure will be The proposed security measure is more sensitive than other security measures defined under a deterministicdistribution model when the embedding is low It is expected to provide a helpful guidance for designing secure steganographicalgorithms or reliable steganalytic methods

1 Introduction

Security of the steganographic system is the fundamentalissue in the field of the information hiding Image steganog-raphy is the technique of hiding information in digital imageand trying to conceal the existence of the secret informationThe image with and the image without hidden informationare called stego image and cover image respectively [1]Steganography and steganalysis are in a hide-and-seek game[2] They try to defeat each other and also develop witheach other In recent years steganalysis researches have mademuch head-way [3 4] and many attempts have been madeto build up secure steganographic algorithms [5ndash8] Up untilnow there is no standard securitymeasure for steganographicsystem The security of the steganography always dependson the encryption of the steganography which contradictsKerckhoffsrsquo principle [9] Hence it is very necessary tostudy the security measure which can provide guidance for

designing the high-secure steganography and steganalyticalgorithms with high performance

Now the study of the security measure becomes one ofthe hotspots in the steganography research field Researchershave put forward their views from different viewing angelsFrom the point of view of information theory Cachin [10]proposed a security measure in terms of the relative entropybetween the probability mass functions (PMF) of the coverimages and the stego images Sullivan et al [11] employedthe divergence distance of the empirical matrices to definethe security measure They modeled the sequence of imagepixels as first-order Markov chain which could capture oneadjacent pixel dependency Furthermore Zhang et al [12]models the images pixels as n-order Markov chain to providethe security measure Based on game theory Liu et al[13] presented that the counterwork relationship is modeledbetween steganography side and attack side In [14] Schottleand Bohme studied adaptive steganography while taking the

HindawiSecurity and Communication NetworksVolume 2017 Article ID 1790268 13 pageshttpsdoiorg10115520171790268

2 Security and Communication Networks

knowledge of the steganalyst into account Liu and Tang [15]also provided the security for the adaptive steganography In[16] Chandramouli et al proposed an alternative securitymeasure based on steganalyzerrsquos ROC (Receiver OperatingCharacteristic) performance From the point of feature spacePevny and Fridrich [17] provided the MMD (MaximumMeanDiscrepancy) by employing a high-dimensional featurespace set as the covers models

The security measures mentioned above all assume thataccurate statistical estimations can be obtained from the finitedata samples However an image is a nonstationary processits local statistical correlation will change when image ischanged slightly So the statistical features change is nonde-terministic after steganography processing Meanwhile for asteganographic system thewarden is lack of the knowledge ofthe cover distribution Thus the distribution estimates of thecover and stego image are not stable So the securitymeasuresdefined under the deterministic statistical model are hard toapply due to the lack of the accurate distribution

To address this problem we regard the steganography asa fuzzy and indeterministic process The goal of this paperis to provide a practical security measure in terms of thevague sets similarity measure between cover images and thestego images Particularly the sequence of image pixels ismodeled as an n-orderMarkov chain to capture the interpixelcorrelations The main contributions of this work are asfollows

(1) We derive a security measure for a steganographicsystem which is different from the deterministicones The existing security measures are defined byevaluating the difference between cover images andstego images In contrast the new security measureis defined by evaluating the similarity between coverimages and their stego version

(2) The 119899-order security measure based on vague setssimilarity measure is proven to have the propertiesof the boundedness commutativity and unity Theproperties guarantee the security measure is indeeda real distance which indeed satisfies the symmetryand triangle inequality The boundedness guaranteesthe new benchmark can measure the steganographicsecurity

(3) Simulation results verify the effectiveness of the newsecurity measure by benchmarking several popularsteganographic schemes When embedding rate islow the new security measure is more sensitive toreveal the statistical features change than other secu-rity measures Thus the proposed security measurecan provide a better guidance for the design ofsteganography and steganalysis

The rest of the paper is organized as follows Section 2gives a review of the two security measures with the deter-ministic statistical distribution model and introduces the 119899-orderMarkov chainmodelThe n-order securemeasure basedon vague sets similarity measure is presented in detail inSection 3 Experimental results are provided in Section 4to demonstrate the effectiveness and the superiority of the

proposed security measure We draw our conclusions inSection 5

2 Steganographic Security and Cover Model

21 Security Measure Based on Kullback-Leibler (K-L) Diver-gence Suppose 119862 is the set of all the covers and it is anassumption that the selections of the covers and stegos fromthe set 119862 can be described by the random variables 119888 and119904 on 119862 with the probability mass functions (PMF) 119875119888 and119875119904 respectively Cachin [10] quantified the security of asteganographic system in terms of the Kullback-Leibler (K-L) divergence (sometimes called relative entropy) that is

119863(119875119888 119875119904) = sum119909isin119883

119875119888 (119909) log119875119888 (119909)119875119904 (119909) (1)

where 119883 is the set of possible pixel values A steganographicsystem is called perfectly secure if (1) is zero or 120576-secure if0 le 119863(119875119888 119875119904) le 120576 is satisfied The K-L divergence providesa simple yet convenient method for measuring the differencebetween cover images and stego images

In fact we have little information about the PMF involveddue to the large dimensionality of the set 119862 So the securitymeasure is usually definedwith simplified covermodels suchas independent and identically distributed (iid) ones Thesecurity measure of K-L divergence calculates the differencefrom the view of the first-order statistical features (such asone-dimensional histogram feature)

22 Security Measure Based on Divergence Distance Toaccount for the dependence of the pixels Sullivan et al [11]employed the first-order Markov chain model to capture theinterpixel correlation The divergence distance was used toquantify the statistical feature perturbations introduced by asteganography between the two empirical matrices of coverimages and stego images Suppose 119862 and 119878 are two randomsequences of the cover image pixels and the stego imagepixels respectively obtained by a given scanningmethod Let119872119888 and119872119904 be the empirical matrixes of119862 and S respectivelyThe divergence distance is given by

119863(119872119888119872119904) = sum119894119895isin119877

119872119888119894119895 log( 119872119888119894119895sum119895119872119888119894119895sum119895119872119904119894119895119872119904119894119895 ) (2)

where 119872119888119894119895sum119895119872119888119894119895 and 119872119904119894119895sum119895119872119904119894119895 are the transition prob-abilities of cover images and stego images respectively Thetransition probability is commonly calculated by the ratio ofthe total number to the pixel changes from value 119894 to value 119895over the total number of possible pixel changes (eg for an8-bit image the total possible pixel changes number is 256 times256) The constant 119877 is the range of all possible pixel valuesThus the divergence distance provides the difference betweencover images and their stego version from the view of thesecond-order statistical features (such as two-dimensionalhistogram feature and difference histogram feature)

Security and Communication Networks 3

[[[[[[[[[[[[[[[[[[[[[[[[

165

165

165

164

165

164

164

165

165

]]]]]]]]]]]]]]]]]]]]]]]

[[[[

165 164 164

165 165 165

165 164 165

]]]]

164

165[18 28

28 38]

164

165[ 0 17

17 17]

Empirical matrix of rst-order Markov chain

Empirical matrix of second-order Markov chain

164 165

164 165

164 165

y

ynynminus1

ynynminus1

ynminus2 = 164

ynminus2 = 165

164

165[17 17

17 17]

Figure 1 The generating process of the empirical matrixes of first-order and second-order Markov chain

The two security measures mentioned above are definedbased on the Shannon information theory under the assump-tion that the image data statistical distribution is determin-istic Most of the security measures proposed later are alsodefined under the same assumption However the image datashows the sceneries in the aspects of gray texture shape andso forth There are many a kind of indeterministic factors(such as noise) in a steganography process Therefore thesecurity measures with the deterministic statistical distribu-tion model cannot measure the security accurately

23 n-Order Markov Chain Model The weakness of theabove two security measure lies in the fact that the imagemodel such as iid and first-order Markov are too simple tocapture interpixel dependency Therefore here we model thesequence of image pixels as an n-order Markov chain The n-orderMarkov chain is a random sequence indexing the imagepixels scanned by a given mode For instance when 119899 = 2the second-order Markov chain accounting for two adjacentpixelsrsquo correlation meets the following condition

119875 (119884119898 | 119884119898minus1 119884119898minus2 1198841) = 119875 (119884119898 | 119884119898minus1 119884119898minus2) (3)

There are at least two reasons for us to select n-orderMarkov chain model First the model is flexible When119899 = 0 it turns out to be the iid model in which theimage pixels are assumed to be unrelated When 119899 = 1the first-order Markov chain can capture only one adjacentpixel dependence Furthermore the n-order Markov chaincan capture more interpixel relationships among the pixelswhen 119899 ge 2 Second compared with the Markov randomfield model [9] the Markov chain model though simpleis able to calculate the statistical estimation of the imagesamples For n-order Markov chain it is easy to calculate therealistic statistical estimates using the empirical matrixes Inthe following we construct the empiricalmatrixes of the first-order and second-order Markov chain

Let 119884119899 119899 = 1 2 119871 be an n-order Markov chainon the finite set 120596 where 119884119899 is the 119899-indexed set of pixelsobtained by a row column zigzag or Hilbert scanning

method 120596 is the possible gray scale values When 119899 = 1the first-order Markov chain source is defined by the tran-sition matrixes 1198791198941 1198942 = 119875(119884119899 = 1198941 | 119884119899minus1 = 1198942)and marginal probabilities 1199011198941 = 119875(119884119899 = 1198941) For arealization 119910 = (1199101 1199102 119910119871)119879 Let 1205781198941 1198942 be the number oftransitions fromvalues 1198941 to 1198942 in119910The empiricalmatrixes are1198721(119910) = 1205781198941 1198942(119910)(119871 minus 1) That is the 1198941 1198942 element representthe proportion of spatially adjacent pixel pairs with thegrayscale value of 1198941 followed by 1198942 Thus the empiricalmatrixes provide an estimation of the transitionmatrixes andmarginal probabilities The empirical matrixes are similar tothe concurrencematrixes of the image It can be recognized asa matrix form of the two-dimensional normalized histogramfor estimating the joint probability mass function (PMF) ofa source image Similarly when 119899 = 2 we can get the empir-ical matrixes of the second-order Markov chain denotedby 1198722(119910) = 1205781198941 1198942 1198943(119910)(119871 minus 1) 1205781198941 1198942 1198943(119910) is the number oftransitions from values 1198941 to 1198943 via 1198942 in 119910 For an 8-bit imagethe size of the empiricalmatrixes1198722(119910) is 256times256times256Theelement of the empirical matrixes represents the proportionof spatially adjacent pixel group with a grayscale value of 1198941followed by 1198942 and 1198943 A simple example of generating theempirical matrixes of first-order and second-order Markovchain is shown in Figure 1

In Figure 1 the small block is derived from the standardimage ldquoLenardquo Its size is 3 times 3 including pixels 164 and 165The example image pixels are scanned vertically The size ofthe empirical matrixes of first-orderMarkov chain in Figure 1is 2 times 2 The element represents the proportion of spatiallyadjacent pixel pairs with (164 164) (164 165) (165 164) and(165 165) The right-hand side of Figure 1 demonstrates theprocedure of the empirical matrixes of second-order Markovchain Its size is 2 times 2 times 2 in which the element representsthe proportion of spatially adjacent pixel groups with (164164 164) (164 165 164) (165 164 164) (165 165 164) and soforth

Since the cover sources are strongly correlated the prob-abilities of two adjacency samples are equal or nearly equalAs a result in the empirical matrixes the masses are more

4 Security and Communication Networks

50 250

50

100

100 150

150

200

250

100

150

200

25020050 250100 150 200

50

Cover empirical matrix Stego empirical matrix

Cover strongly correlated Stego correlation weakened

(a) The original empirical matrixes

100 110 120 130

100

110

120

130

100 110 120 130

100

110

120

130

Cover empirical matrix (zoomed) Stego empirical matrix (zoomed)

Hiding

(b) The zoomed empirical matrixes

Figure 2 Empirical matrixes of a cover image and its stego image

concentrated near the main diagonal in a correlated sourceIn [18] Harmsen and Pearlman considered that informationhiding can be viewed as adding the additive noise to the coverimageThe secret information (additive noise) is uncorrelatedafter hiding and its empirical matrixes spread evenly overthe main diagonal Thus we see that hiding weakens thedependencies among the cover samples which is illustratedin Figure 2(a) Figure 2(b) is part of the zoomed empiricalmatrixes According to the above analysis the steganographytends to spread the density of the pixels pairs away from themain diagonal of the empirical matrixes This property mayshed some light on designing of the security measure for asteganographic systemThus in Section 3 we will propose ann-order security measure in terms of the vague sets similarity

measure by modeling the sequence of images pixels as an n-order Markov chain

3 Security Measure Based onVague Sets Similarity Measure

The vague sets similarity measure [19 20] describes thematching degree of two vague sets In a practical stegano-graphic system there are many indeterministic factors intro-duced by steganography In this work we regard the respond-ing probability distribution sets of the cover samples and thestego samples as two discrete vague sets Then a new securitymeasure is proposed below in terms of vague sets similarity

Security and Communication Networks 5

measure to measure the similarity between cover images andstego images

31 Vague Sets Roughly speaking a fuzzy set is a class withfuzzy boundariesThe fuzzy set A is a class of objects119883 alongwith a grade ofmembership function 120583119860(119909) 119909 isin 119883 It assignsa single value to each object This single value combines theevidence for x isin X and the evidence against 119909 isin 119883 And it isonly a measure of the proscons evidence However in manypractical applicationswe often require pros and cons evidencesimultaneously Gau and Buehrer [21] advanced the conceptof vague setsThe vague sets theory adopts a truemembershipfunction 119905119860 and a false membership function119891119860 to record thelower bounds on 120583119860 These lower bounds are used to create asubinterval on [0 1] namely [119905119860(119909119894) 1minus119891119860(119909119894)] to generalize120583119860(119909119894) of fuzzy sets where 119905119860(119909119894) le 120583119860(119909119894) le 1 minus 119891119860(119909119894)Vague sets expand the value of the membership function toa subinterval of [0 1] instead of a single value thus it hasstronger ability to reveal the indeterminacy than the fuzzy settheory The related definitions of vague sets are as follows

Definition 1 (vague sets) Let 119883 be the universe of discourse119883 = 1199091 1199092 119909119899 119881(119909) denotes all the vague sets of 119883forall119860 isin 119881(119909) The vague set 119860 is characterized by a truemembership function 119905119860 and a falsemembership function119891119860

119905119860 119883 997888rarr [0 1] 119891119860 119883 997888rarr [0 1] (4)

where 119905119860(119909119894) is the lower bound on the grade of membershipof 119909119894 derived from the evidence for 119909119894119891119860(119909119894) is a lower boundon the negation of 119909119894 derived from the evidence against 119909119894satisfying 119905119860(119909119894) + 119891119860(119909119894) le 1 The grade of membership of 119909119894is bounded to a subinterval [119905119860(119909119894) 1minus119891119860(119909119894)] of [0 1]When119883 is discrete a vague set 119860 can be written as

119860 = 119899sum119894=1

[119905119860 (119909119894) 1 minus 119891119860 (119909119894)]119909119894 119909119894 isin 119883 (5)

Definition 2 Let 119883 be the universe of discourse 119883 =1199091 1199092 119909119899 A and 119861 are two vague sets of119883 The entropyof the vague set 119860 119864(119860) is defined as

119864 (119860)= minus 1119899 ln 2

119899sum119894=1

[119905119860 (119909119894) ln 119905119860 (119909119894) + 119891119860 (119909119894) ln119891119860 (119909119894)] (6)

Definition 3 Let 119883 be the universe of discourse 119883 =1199091 1199092 119909119899 A and 119861 are two vague sets of 119883 The partialentropy of vague set 119860 against vague set 119861 119864119861(119860) is definedas

119864119861 (119860) = minus 119899sum119894=1

[119905119861 (119909119894) ln 119905119860 (119909119894) + 119891119861 (119909119894) ln119891119860 (119909119894)] (7)

32 The 119899-Order Security Measure Based on Vague SetsSimilarity Measure As discussed in Section 23 the n-order

Markov chain model can capture sufficient inherent correla-tions Additionally the changes in image statistical featuresintroduced by steganography are indeterministic Thereforein the new security measure we model the sequence of theimage pixels as an n-order Markov chain Simultaneouslythe empirical matrixes of the n-order Markov chain of coverimages and stego images are regarded as two vague setsThenthe n-order security measure based on the vague sets similar-ity measure is defined as follows

Suppose 119862 and 119878 are n-order Markov chain sequence ofcover images and stego images respectively and then scanthem by a given mode (such as horizontal vertical zigzagand Hilbert mode) MC and MS represent the correspond-ing empirical matrixes 1198981198941 1198942119894119899+1 the element of empiricalmatrixes denotes the joint probability distribution from pix-els 1198941 to 119894119899+1 via the states of 1198942 1198943 and 119894119899 The 1198941 1198942 119894119899+1is the image pixel value 119894 isin [0 255] 119866 denotes the set of allpossible values of1198981198941 1198942119894119899+1 Let1198721198941 1198942 119894119899+1 be the universe ofdiscourse composed of 1198981198941 1198942119894119899+1 ThenMC andMS are twovague sets on1198721198941 1198942119894119899+1 That is

119872119862 = sum255119894=0 [119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) 1 minus 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)]1198981198941 1198942 sdotsdotsdot119894119899+1 1198981198941 1198942 sdotsdotsdot119894119899+1 isin 1198721198941 1198942 sdotsdotsdot119894119899+1

119872119878 = sum255119894=0 [119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1) 1 minus 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)]1198981198941 1198942sdotsdotsdot119894119899+1 1198981198941 1198942 sdotsdotsdot119894119899+1 isin 1198721198941 1198942 sdotsdotsdot119894119899+1

(8)

Definition 4 Let 1198721198941 1198942 119894119899+1 be the universe of discourseMC and MS are two vague sets of 1198721198941 1198942 119894119899+1 The similaritymeasure 119879119899(119872119862119872119878) between the vague sets MC and MS isdefined as the n-order secure measure for a steganographicsystem that is

119879119899 (119872119862119872119878) = 119898 ln 2 (119864 (119872119862) + 119864 (119872119878))119864119872119862 (119872119878) + 119864119872119878 (119872119862) (9)

where 119864(119872119862) and 119864(119872119878) denote the entropy of the vaguesetMC andMS respectively 119864119872119862(119872119878) stands for the partialentropy of vague set119872119878 against vague set119872119862 119864119872119878(119872119862) isthe partial entropy of vague set MC against vague set MS119864(119872119862) and 119864119872119862(119872119878) can be written as

119864 (119872119862) = minus 1119898 ln 2sdot sum1198941 1198942 sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln 119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)]

119864119872119878 (119872119862)= minus sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln119891119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)]

(10)

6 Security and Communication Networks

Similarly 119864(119872119878) and 119864119872119862(119872119878) can be written as

119864 (119872119878) = minus 1119898 ln 2sdot sum1198941 1198942 sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln119891119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)]

119864119872119862 (119872119878)= minus sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1) ln119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)]

(11)

Moreover a steganographic system is called perfectlysecure if119879119899(119872119862119872119878) = 1 or 120576-secure if119879119899(119872119862119872119878) = 120576 120576 isin(0 1) 119879119899(119872119862119872119878) = 0Theorem5 Let119879119899(119872119862119872119878) be the n-order securemeasure ofa steganographic system based on vague set similarity measureThen 119879119899(119872119862119872119878) satisfies the following(1) Boundedness is

0 le 119879119899 (119872119862119872119878) le 1 (12)

(2) Commutativity is

119879119899 (119872119862119872119878) = 119879119899 (119872119878119872119862) (13)

(3) Unity is

119879119899 (119872119862119872119878) = 1 lArrrArr119872119862 = 119872119878 (14)

119879119899(119872119862119872119878) provides a security measure for a stegano-graphic system in terms of the similarity between cover imagesand stego images 119879119899(119872119862119872119878) is limited in a finite interval of[0 1] where 1 denotes ldquoperfectly securerdquo while 0 denotes ldquodef-initely unsecurerdquo However other security measures under thedeterministic statistical model calculate the difference betweencover images and stego images The values range in an infiniteinterval [0infin) The property of the boundedness guaranteesthe proposed security measure can measure a steganographicalgorithm quantitatively Hence it has stronger ability to revealthe statistical changes of the cover images When 119899 = 0 theimage pixels distribution is said to be iid and 1198790(119872119862119872119878)is called the zero-order security measure When 119899 = 1the sequence of image pixels is considered to be a first-orderMarkov chain and 1198791(119872119862119872119878) is defined as the first-ordersecurity measure Thus a different order security measure canbe obtained by adjusting the value of 1198994 Experimental Results and Discussion

In this section we report experimental results that demon-strate the capability of the new security measure First of all

in Section 41 the image databases used for the experimentare described Afterwards in Section 42 we benchmark sev-eral different steganographic methods with 119899-order securitymeasure based on vague sets with particular attention to theeffectiveness of low embedding rate Finally we compare theproposed security measure with previously used benchmarksdesigned under the deterministic statistical model

41 ImageDatabase For the experimental validationwe usedtwo image databases The first one is BOWS2 [22] imagedatabase including 10000 grayscale images with fixed size512 times 512 The other one is NRCS Photo Gallery [23] Weselected 1500 images from NRCS Photo Gallery All imageswere converted into grayscale and central cropped to a sizeof 512 times 512 for experimental purposes The images in ourexperiments show a wide range of scenarios including housemanmade objects and animal Some images are shown inFigure 3

42 Verification of the Effectiveness of the Proposed Secu-rity Measure To evaluate the performance of the proposedmethod for measuring the security of the steganographicalgorithms the new security measure with different orders isused tomeasure the security of different steganographic algo-rithms with different embedding rates First we select somespatial-domain steganographic algorithms including LSBM(least significant bit matching) [24] LSB plusmn 2 HUGO [25](highly undetectable steganography) We use 2000 imagesfrom BOWS2 image database all the images are grayscalewith the fixed size 512 times 512As discussed in Section 23 first-order and second-order Markov chain models have capturedsufficient interpixel correlations Additionally consideringthe computation complexity we use the zero-order first-order and second-order securitymeasure based on vague setsto measure the LSBM LSBM2 and HUGO steganographicmethods with the embedding rate ranging from 01 bpp (bitsper pixel) to 1 bpp in a step size of 01 bpp The averagemeasure results for zero-order first-order and second-ordersecurity measure of 2000 images with different embeddingrates are depicted in Figure 4

In Figure 4 all curves indicate that the value of secu-rity measure gradually decreases with an increase in theembedding rate for the same steganographic algorithm It isconsistent with the definition of the security measure basedon vague sets Its value is limited in an interval of [0 1]where 1 denotes ldquoperfectly securerdquo for the steganographicsystem Hence the value of the n-order security measuresatisfies monotonic decreasing property that is the higherthe security of the stego images the larger the value of thesecurity measure Furthermore as is evident in Figure 4 thevalues of the same order security measure are different fordifferent stego schemes with the same embedding rate Notethat LSB plusmn 2 obtains the lowest value in Figure 4 implyingthat it is most unsecure among the three hiding methodsunder the same condition On the contrary HUGO gains thehighest value All the measure results are coincident with thetheoretical analysis of the three embedding schemes

Furthermore in order to evaluate the measuring abilityof different order security measures we compare the security

Security and Communication Networks 7

(a) Some images of BOWS2

(b) Some images of NRCS

Figure 3 Some images of image database

for the same steganographic algorithm using different ordersecuritymeasures Figure 6 shows the averagemeasure resultsof zero-order first-order and second-order security measurefor LSBM LSB plusmn 2 and HUGO respectively In fact allthe data in Figure 5 is derived from Figure 4 As demon-strated in Figure 5 for the same steganographic methodthe values of the zero-order first-order and second-ordersecurity measure are different at the same embedding rateIt is demonstrated that the value of the first-order securitymeasure is smaller than that of the zero-order measure butlarger than that of the second-order measure for the samesteganographic method with the same embedding rate Theexperiments show that the second-order security measureprovides the largest measure interval to reveal the securitychange of the cover images with the embedding rate rangingfrom 01 bpp to 1 bpp So we can conclude that second-order security measure can provide more obvious statisticaldistributed changes caused by steganography

To further verify the effectiveness of the proposed secu-rity measure We used it to benchmark JPEG steganographicalgorithms schemes on different database And we focus onlow payloads to see if any of the test steganographic schemesbecomes distinguishable by using the vague sets securitymeasure with finite image sample

We selected 1500 images from NRCS Photo Gallery Allimages were converted into grayscale and central cropped toa size of 512 times 512 for experimental purposes The imageswere embedded with pseudorandom payloads with 5 1015 and 20 bpac (bits per nonzero AC coefficient) Thetested stego schemes include F3 F5 without shrinkage (nsF5)[26] Model Based Steganography without deblocking (MB1)[27] andModel Based Steganographywith deblocking (MB2)

[28] The cover images were single-compressed JPEGs withquality factor 70 The measure results using zero-order first-order and second-order securitymeasure based on vague setsare showed in Table 1 The data in Table 1 indicates that forthe same steganography the larger the embedding rate thelower the value of the same securitymeasure It also exhibitedthat for the same steganography the higher the order ofthe security measure the smaller the value of the securitymeasure suggesting that second-order security measure canget a value lower than the other two security measures underthe same condition

The data in Table 1 also shows according to the sameorder security measure the MB2 is the least statisticallydetectable followed by MB1 and nsF5 while F3 is the mostdetectable All the measure results are coincident with thetheoretical security among adopted stego algorithms In aword the experimental results indicate that the proposedsecurity measure is effective for measuring the securityfor different steganographic methods on different imagedatabase Meanwhile the greater the order the stronger themeasure ability of the security measure

43 Comparison with Security Measure under DeterministicStatistical Model To show the superiority of the proposedsecurity measure 119879119899(119872119862119872119878) we compare it with twosecurity measures under the deterministic statistical modelOne is the Kullback-Leibler (K-L) divergence between theprobabilitymass functions (PMF) proposed by Anderson [9]denoted by 119863(119875119888 119875119904) The other denoted as 119863(119872119888119872119904) isthe divergence distance between the two empirical matricesproposed by Cachin [10] To be unbiased the zero-ordermeasure 1198790(119872119862119872119878) is compared with 119863(119875119888 119875119904) when

8 Security and Communication Networks

0097

0975

098

0985

099

0995

1

Embedding rate

Zero

-ord

er se

curit

y m

easu

re

HUGOLSBMLSB2

01 02 03 04 05 06 07 08 09 1

base

d on

vag

ue se

t

(a) Zero-order security measure

HUGOLSBMLSB2

07

075

08

085

09

095

1

Embedding rate

Firs

t-ord

er se

curit

y m

easu

re

01 02 03 04 05 06 07 08 09 1

base

d on

vag

ue se

t

(b) First-order security measure

HUGOLSBMLSB2

01 02 03 04 05 06

06

07

07

08

08

09

09

1055

065

075

085

095

1

Embedding rate

Seco

nd-o

rder

secu

rity

mea

sure

base

d on

vag

ue se

t

(c) Second-order security measure

Figure 4 The same order security measure for different steganographic methods with different embedding rate

119863(119875119888 119875119904) is used under the assumption that the imagemodelis iid Similarly the first-order measure 1198791(119872119862119872119878) iscompared with 119863(119872119888119872119904) since their image pixel sequencesare all modeled as the first-order Markov chain In the exper-iments the same 2000 images from BOWS2 are adopted1198790(119872119862119872119878) 119863(119875119888 119875119904) 1198791(119872119862119872119878) and 119863(119872119888119872119904)are used to measure the security of the HUGO with theembedding rate ranging from 005 bpp to 1 bpp in a stepsize of 005 bpp Figures 6(a) and 6(b) show the averagemeasure of 1198790(119872119862119872119878) and 119863(119875119888 119875119904) with differentembedding rates respectively The average measure values of1198791(119872119862119872119878) and 119863(119872119888119872119904) are also illustrated in Figures7(a) and 7(b) respectively

Looking at Figures 6 and 7 we see that the value ofsecuritymeasure based on vague sets decreases as the embed-ding rate increases whereas the value of security measureunder the deterministic distribution model increases as theembedding rate increases All the curves in Figures 6 and 7indicate that both the security measure models are effectivein measuring the security of the steganography In orderto show the superiority of the proposed security measurewe define 120575 = Δ119910119910 as the sensitivity of where Δ119910 isthe security measure variation of a given embedding ratechange range andyis the total security measure variation ofthe embedding rate change Obviously Figures 6(b) and 7(b)demonstrate that 120575 of security measure is very small whenembedding rate is lower than 05 bpp So its corresponding

Security and Communication Networks 9

082

084

086

088

09

092

094

096

098

1

Embedding rate for HUGO

Di

eren

t ord

er m

easu

re b

ased

on

Zero-orderFirst-orderSecond-order

01 02 03 04 05 06 07 08 09 108

vagu

e set

sim

ilarit

y

(a) For HUGO

Zero-orderFirst-orderSecond-order

075

085

095

1

Embedding rate for LSBM

Di

eren

t ord

er m

easu

re b

ased

on

01 02 03 04 05 06 0707

08

08

09

09

1

vagu

e set

sim

ilarit

y

(b) For LSBM

Zero-orderFirst-orderSecond-order

055

065

075

085

095

1

Embedding rate for LSB2

Di

eren

t ord

er m

easu

re b

ased

on

01 02 03 04 05 06

06

07

07

08

08

09

09

1

vagu

e set

sim

ilarit

y

(c) For LSB2

Figure 5 The different order security measures for the same steganography with different embedding rate

security measure is not sensitive to the statistical distributionchange Hence the new security measure can reveal moreobvious statistical change than the security measures underdeterministic statistical distribution model when embeddingrate is low

5 Conclusions

Vague sets similarity measure is a simple yet effective toolfor measuring the similarity between two vague sets Inthis work a novel security measure for a steganographicsystem in terms of the vague sets similarity measure isproposed to measure the similarity between cover imagesand stego images Particularly in the new security measurethe sequence of image pixels is modeled as an n-order

Markov chain to capture sufficient interpixel dependenciesThe proposed security measure is proven to have such prop-erties as boundedness commutativity and unity Variousorder security measures can be obtained by adjusting thevalue of 119899 Experimental results confirm the effectivenessof the proposed security measure for evaluating differentsteganographic algorithms Meanwhile the security measurewith a higher order always has a better measure abilityAdditionally when the embedding rate is low the n-ordersecurity measure based on vague sets is more sensitive thanother security measures under the deterministic distributionmodel Considering the computational complexity and ste-ganalytic ability two issues should be tackled in our furtherresearch One is how to use the n-order security measureto design reliable steganalytic methods by extracting thestatistical feature from the empirical matrixes The other is

10 Security and Communication Networks

0 01 02 03 04 05 06 07 08 09 10975

098

0985

099

0995

1

Embedding rate for HUGO

Zero

-ord

er m

easu

re b

ased

on

vagu

e set

sim

ilarit

y

(a) 1198790(119872119862119872119878)

01 02 03 04 05 06 07 08 09 10

1

2

3

4

5

6

Embedding rate for HUGO

K-L

dist

ance

bet

wee

n co

ver a

nd st

ego

imag

e

times10minus6

(b) 119863(119875119888 119875119904)

Figure 6 1198790(119872119862119872119878) and119863(119875119888 119875119904) for HUGO with different embedding rates

0 01 02 03 04 05 06 07 08 09 1

09

092

094

096

098

1

Embedding rate for HUGO

Firs

t-ord

er m

easu

re b

ased

on

vagu

e set

sim

ilarit

y

(a) 1198791(119872119862119872119878)

01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

35

4

45

Embedding rate for HUGO

Div

erge

nce d

istan

ce b

etw

een

times10minus5

cove

r and

steg

o im

age

(b) 119863(119872119888119872119904)

Figure 7 1198791(119872119862119872119878) and119863(119872119888119872119904) for HUGO with different embedding rates

how to use the new security measure to design highly securesteganographic algorithms

Appendix

Proof of Theorem 5 (1)

119864119872119862 (119872119878) + 119864119872119878 (119872119862) minus 119898 ln 2 (119864 (119872119862) + 119864 (119872119878))= minus sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119862 (119898119894119895) ln119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)]minus sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)

+ 119891119872119878 (119898119894119895) ln119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)]+ sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln 119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)+ 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)]+ sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)]= sum1198941 1198942 sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)+ 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)119891119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1) ]

Security and Communication Networks 11

Table 1 Different order vague sets security measure for different steganography methods

Steganographymethod

Embedding rate(bpac) Zero-order First-order Second-order

F3

5 09768 09662 0956910 09755 09647 0956915 09714 09608 0949820 09683 09584 09477

nsF5

5 09865 09736 0959310 09847 09711 0954215 09840 09687 0953120 09818 09656 09515

MB1

5 09994 09879 0978510 09991 09866 0969915 09965 09849 0967320 09959 09837 09656

MB2

5 09999 09868 0968710 09996 09842 0962415 09987 09922 0961720 09982 09818 09609

+ sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)119891119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)]

(A1)

Since the inequality satisfies ln119909 ge (1 minus 1119909) we have119864119872119862 (119872119878) + 119864119872119878 (119872119862) minus 119898 ln 2 (119864 (119872119862) + 119864 (119872119878))

ge sum1198941 1198942 sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)

sdot (1 minus 119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)) + 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)

sdot (1 minus 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1))]

+ sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)

sdot (1 minus 119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ) + 119891119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)

sdot (1 minus 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) )]

ge sum1198941 1198942 sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) minus 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) minus 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)]+ sum1198941 1198942 sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) minus 119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1) minus 119891119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)] = 0

(A2)

Hence 119864119872119862(119872119878) + 119864119872119878(119872119862) minus 119898 ln 2(119864(119872119862) + 119864(119872119878)) ge0 such that 119864119872119862(119872119878) + 119864119872119878(119872119862) ge 119898 ln 2(119864(119872119862)+119864(119872119878))Since 119905119872119862(1198981198941 1198942 sdotsdotsdot119894119899+1) 119905119872119878(1198981198941 1198942 sdotsdotsdot119894119899+1) 119891119872119862(1198981198941 1198942 sdotsdotsdot119894119899+1) and119891119872119878(1198981198941 1198942 sdotsdotsdot119894119899+1) are all in the range of [0 1] and 0 times ln 0 = 0119864(119872119862) 119864(119872119878) 119864119872119862(119872119878) and 119864119872119878(119872119862) are all positiveHence 0 le 119879119899(119872119862119872119878) le 1(2) According to the definition of the n-order security

measure 119879119899(119872119862119872119878) is described as

119879119899 (119872119862119872119878) = 119898 ln 2 (119864 (119872119862) + 119864 (119872119878))119864119872119862 (119872119878) + 119864119872119878 (119872119862) (A3)

And it can also be described as

119879119899 (119872119878119872119862) = 119898 ln 2 (119864 (119872119878) + 119864 (119872119862))119864119872119878 (119872119862) + 119864119872119862 (119872119878) (A4)

Hence 119879119899(119872119862119872119878) = 119879119899(119872119878119872119862)

12 Security and Communication Networks

(3) From the proving procedure of property (1) we have

119864119872119862 (119872119878) + 119864119872119878 (119872119862) minus 119898 ln 2 (119864 (119872119862) + 119864 (119872119878))ge 0 (A5)

119864119872119862 (119872119878) + 119864119872119878 (119872119862) minus 119898 ln 2 (119864 (119872119862) + 119864 (119872119878))= sum1198941 1198942 sdotsdotsdot119894119899+1isin119866

[119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln 119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)+ 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1) ln 119891119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ]

+ sum1198941 1198942sdotsdotsdot119894119899+1isin119866

[119905119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)119905119872119862 (1198981198941 1198942 sdotsdotsdot119894119899+1)+ 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1) ln 119891119872119878 (1198981198941 1198942sdotsdotsdot119894119899+1)119891119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1)]

(A6)

If and only if

119905119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1) = 119905119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1) 119891119872119862 (1198981198941 1198942sdotsdotsdot119894119899+1) = 119891119872119878 (1198981198941 1198942 sdotsdotsdot119894119899+1)

(A7)

Namely when 119872119862 = 119872119878 and 119872119862 = 119872119878 such that119864119872119862(119872119878) + 119864119872119878(119872119862) minus 119898 ln 2(119864(119872119862) + 119864(119872119878)) = 0Hence 119879119899(119872119862119872119878) = 1 hArr 119872119862 = 119872119878

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural Foundationof China (nos 61462046 61363014) the Science andTechnology Research Projects of Jiangxi Province EducationDepartment (nos GJJ16079 GJJ160750) the Natural ScienceFoundation of Jiangxi Province (nos 20151BAB20702620161BAB202050 and 20161BAB202049) JinggangshanUniversity Doctoral Scientific Research Foundation (nosJZB1311 JZB15016) and Key Laboratory of WatershedEcology and Geographical Environment Monitoring ofNASG (nos WE2015012 WE2016013)

References

[1] B Li J He J Huang and Y Q Shi ldquoA survey on imagesteganography and steganalysisrdquo Journal of Information HidingandMultimedia Signal Processing vol 2 no 2 pp 142ndash172 2011

[2] X-P Zang Z-X Qian and S Li ldquoProspect of digital steganog-raphy researchrdquo Journal of Applied Sciences-Electronics andInformation Engineering vol 34 no 5 pp 475ndash489 2016

[3] J Fridrich and J Kodovsky ldquoRich models for steganalysis ofdigital imagesrdquo IEEE Transactions on Information Forensics andSecurity vol 7 no 3 pp 868ndash882 2012

[4] V Holub and J Fridrich ldquoLow-complexity features for JPEGsteganalysis using undecimated DCTrdquo IEEE Transactions onInformation Forensics and Security vol 10 no 2 article A1 pp219ndash228 2015

[5] V Sedighi R Cogranne and J Fridrich ldquoContent-adaptivesteganography by minimizing statistical detectabilityrdquo IEEETransactions on Information Forensics and Security vol 11 no2 pp 221ndash234 2016

[6] N Zhou H Li DWang S Pan and Z Zhou ldquoImage compres-sion and encryption scheme based on 2D compressive sensingand fractional Mellin transformrdquo Optics Communications vol343 pp 10ndash21 2015

[7] L Guo J Ni and Y Q Shi ldquoUniform embedding for efficientJPEG steganographyrdquo IEEE Transactions on Information Foren-sics and Security vol 9 no 5 pp 814ndash825 2014

[8] N Zhou S Pan S Cheng and Z Zhou ldquoImage compression-encryption scheme based on hyper-chaotic system and 2Dcompressive sensingrdquo Optics and Laser Technology vol 82 pp121ndash133 2016

[9] R Anderson ldquoWhy information security is hard - An economicperspectiverdquo in Proceedings of the 17th Annual Computer Secu-rity Applications Conference ACSAC 2001 pp 358ndash365 usaDecember 2001

[10] C Cachin ldquoAn information-theoretic model for steganogra-phyrdquo Information and Computation vol 192 no 1 pp 41ndash562004

[11] K Sullivan U Madhow S Chandrasekaran and B S Man-junath ldquoSteganalysis for Markov cover data with applicationsto imagesrdquo IEEE Transactions on Information Forensics andSecurity vol 1 no 2 pp 275ndash287 2006

[12] Z Zhang G J Wang W Jun et al ldquoSteganalysis of spreadspectrum image steganography based on high-order markovchain moderdquo ACTA Electronica Sinica vol 38 no 11 pp 2578ndash2584 2010

[13] G-J Liu Y-W Dai Y-X Zhao and Z-Q Wang ldquoModelingsteganographic counterwork by game theoryrdquo Journal of Nan-jing University of Science and Technology vol 32 no 2 pp 199ndash204 2008

[14] P Schottle and R Bohme ldquoGame theory and adaptive steganog-raphyrdquo IEEETransactions on Information Forensics and Securityvol 11 no 4 pp 760ndash773 2016

[15] J Liu and G-M Tang ldquoGame research on large-payload andadaptive steganographic counterworkrdquo Acta Electronica Sinicavol 42 no 10 pp 1963ndash1969 2014

[16] R Chandramouli M Kharrazi and N Memon ldquoImagesteganography and steganalysis concepts and practicerdquo inProceedings of the IWDWrsquo03 vol 2939 pp 35ndash49 2003

[17] T Pevny and J Fridrich ldquoBenchmarking for steganographyrdquo inInformation Hiding 10th International Workshop pp 251ndash267Santa Barbara Calif USA 2008

[18] J J Harmsen and W A Pearlman ldquoSteganalysis of additivenoise modelable information hidingrdquo in Proceedings of theISTSPIE 15th Annu Symp Electronic Imaging Science Technol-ogy pp 21ndash24 San Jose Calif USA January 2003

[19] F Li and Z-Y Xu ldquoMeasures of similarity between vague setsrdquoJournal of Software vol 12 no 6 pp 922ndash927 2001

[20] S Y Quan ldquoThe vague set similaritymeasure based onMeaningof InformationrdquoComputer Engineering andApplications vol 43no 25 pp 87ndash89 2007

[21] W L Gau and D J Buehrer ldquoVague setsrdquo IEEE Transactions onSystems Man and Cybernetics vol 23 no 2 pp 610ndash614 1993

Security and Communication Networks 13

[22] P Bas T Filler and T Pevny ldquoBreak our steganographicsystemmdashthe ins and outs of organizing BOSSrdquo in Proceedingsof the 13th International Workshop on Information Hiding pp59ndash70 Berlin Germany 2011

[23] United States Department of Agriculture Natural resourcesconservation service photo gallery [DBOL] httpphotogallerynrcsusdagov 2002

[24] T Sharp ldquoAn implementation of key-based digital signalsteganographyrdquo in Proceedings of the Information Hiding Work-shop vol 2137 pp 13ndash26 2001

[25] T Pevny T Filler and P Bas ldquoUsing high-dimensional imagemodels to perform highly undetectable steganographyrdquo LectureNotes in Computer Science (including subseries Lecture Notes inArtificial Intelligence and Lecture Notes in Bioinformatics) vol6387 pp 161ndash177 2010

[26] J Fridrich D Soukal and M Goljan ldquoMaximum likelihoodestimation of length of secret message embedded using plusmnKsteganography in spatial domainrdquo in Proceedings of SPIE-IS and T Electronic Imaging - Security Steganography andWatermarking of Multimedia Contents VII vol 5681 pp 595ndash606 January 2005

[27] P Sallee ldquoModel-Based Steganographyrdquo in Digital Watermark-ing T Kalker Ed vol 2939 of Lecture Notes in ComputerScience pp 154ndash167 Springer Berlin Heidelberg 2004

[28] P Sallee ldquoModel-based methods for steganography and ste-ganalysisrdquo International Journal of Image and Graphics vol 5no 1 pp 167ndash189 2005

RoboticsJournal of

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Submit your manuscripts athttpswwwhindawicom

VLSI Design

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