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  • 1. IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 7, NO. 1, FEBRUARY 2012 197 A Probabilistic Model of Visual Cryptography Scheme With Dynamic Group Sian-Jheng Lin and Wei-Ho Chung, Member, IEEE Abstract—The visual cryptography (VC) is a secret higher contrast represents the better visual quality, and there-sharing scheme where a secret image is encoded into trans- fore the stacking secret with high contrast is the goal of pur-parencies, and the stacking of any out of transparencies reveals suit in VC designs. Naor and Shamir [1] define a contrast for-the secret image. The stacking of or fewer transparencies isunable to extract any information about the secret. We discuss the mula which has been widely used in many studies. Based onadditions and deletions of users in a dynamic user group. To reduce the definition of contrast, there are studies attempting to achievethe overhead of generating and distributing transparencies in user the contrast bound of VC scheme [4], [14]–[20]. For in-changes, this paper proposes a VC scheme with unlimited stance, Blundo et al. [17] give the optimal contrast of based on the probabilistic model. The proposed scheme allows VC schemes. Hofmeister et al. [19] provide a linear program to change dynamically in order to include new transparencies which is able to compute exactly the optimal contrast forwithout regenerating and redistributing the original transparen-cies. Specifically, an extended VC scheme based on basis matrices VC schemes. Krause and Simon [20] provide the upper boundand a probabilistic model is proposed. An equation is derived from and lower bound of the optimal contrast for VC schemes.the fundamental definitions of the VC scheme, and then the Moreover, there exist VC related researches using differential VC scheme achieving maximal contrast can be designed definitions of contrast [21]–[23]. Another important metric isby using the derived equation. The maximal contrasts with the pixel expansion denoting the number of subpixels in trans-to are explicitly solved in this paper. parency used to encode a secret pixel. The minimization of pixel Index Terms—Contrast, random grids (RGs), secret sharing, vi- expansions has been investigated in previous studies [24], [25].sual cryptography (VC). The probabilistic model of the VC scheme was first intro- duced by Ito et al. [2], where the scheme is based on the basis matrices, but only one column of the matrices is chosen to en- I. INTRODUCTION http://ieeexploreprojects.blogspot.com rather than the traditional VC scheme code a binary secret pixel, utilizing the whole basis matrices. The size of the generatedV ISUAL cryptography (VC) is a branch of secret sharing. transparencies is identical to the secret image. Yang [31] also In the VC scheme, a secret image is encoded into trans- proposed a probabilistic model ofparencies, and the content of each transparency is noise-like so two cases and VC scheme, and the are explicitly constructed to achievethat the secret information cannot be retrieved from any one the optimal contrast. Based on Yang [31], Cimato et al. [32] pro-transparency via human visual observation or signal analysis posed a generalized VC scheme in which the pixel expansion istechniques. In general, a -threshold VC scheme has the fol- between the probabilistic model of VC scheme and the tradi-lowing properties: The stacking of any out of those VC gener- tional VC scheme.ated transparencies can reveal the secret by visual perception, Encrypting an image by random grids (RGs) was first intro-but the stacking of any or fewer number of transparen- duced by Kafri and Keren [26] in 1987. A binary secret imagecies cannot retrieve any information other than the size of the is encoded into two noise-like transparencies with the same sizesecret image. Naor and Shamir [1] proposed a -threshold of the original secret image, and stacking of the two transparen-VC scheme based on basis matrices, and the model had been cies reveals the content of the secret. Comparing RGs with basisfurther studied and extended. The related works include the VC matrices, one of the major advantages is that the size of gener-schemes based on probabilistic models [2]–[4], general access ated transparencies is unexpanded. The RG scheme is similar tostructures [5], [6], VC over halftone images [7], [8], VC for the probabilistic model of the VC scheme, but the RG schemecolor images [9], cheating in VC [10], [11], the general formula is not based on the basis matrices. The recent studies includeof VC schemes [12], and region incrementing VC [13]. the RG for color image [27], RG, and RG schemes Contrast is one of the important performance metrics for VC [28], [29]. We also compare the proposed method withschemes. Generally, the stacking revelation of the secret with RG [29] in Section IV. We consider the scenario of a dynamic user group, where Manuscript received May 09, 2011; revised August 15, 2011; accepted Au- new participants are to join the user group with original par-gust 17, 2011. Date of publication September 06, 2011; date of current version ticipants; and the transparencies need to accommodate the newJanuary 13, 2012. This work was supported by the National Science Councilof Taiwan, under Grant NSC 100-2221-E-001-004. The associate editor co- users. If the transparencies are to be generated withordinating the review of this manuscript and approving it for publication was the traditional VC scheme, the original transparenciesDr. Carlo Blundo. need to be discarded, and the new transparencies need The authors are with the Research Center for Information Technology In-novation, Academia Sinica, Nankang, Taipei 115, Taiwan (e-mail: whc@citi. to be generated with the traditional VC scheme. Thesinica.edu.tw). regeneration and redistribution of the whole transparencies con- Digital Object Identifier 10.1109/TIFS.2011.2167229 sume computing and communication resources and may lead to 1556-6013/$26.00 © 2011 IEEE
  • 2. 198 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 7, NO. 1, FEBRUARY 2012the potential security vulnerability. The addition and deletion of The frequency probability of appearing onesusers are further discussed in Section IV-A. in a vector . In this paper, we propose a probabilistic model of VCscheme with unlimited . The major contribution is that the , The discrete probability distribution of allproposed scheme accommodates dynamic changes of users in columns in and .the group sharing a VC secret. The proposed scheme allows A memoryless binary sequence with values ofchanges of users without regeneration and redistribution of VC elements as 0 or 1, where the probability oftransparencies, which reduce the computing and communica- assigning each element to 0 is .tion resources in accommodating user changes. The scheme iscapable of generating an arbitrary number of transparencies and A vector .the explicit algorithms are proposed to generate the transparen- , Two vectors to record the nonzero terms incies. For a group with initial users, the proposed Algorithm .1 explicitly generates the required transparencies. For An optimization problem to find the optimalnewly joining participants, the new transparencies can beexplicitly generated through Algorithm 2, and the new trans- contrast of VC scheme.parencies can be distributed to the new participants without A binary secret image.the need to update the original transparencies. The secondary A ready-to-process pixel taken from .contribution is that this paper designs an implementation of VC based on the probabilistic model, and the proposed The th generated transparency.scheme allows the unlimited number of users. For the conven- A pixel at , and the position corresponds totional VC scheme to implement the case , the the position of .mathematical manipulations of infinite size of basis matricesand variables are often required, which is computationally and is the number of original participants in theprohibitive. Our approach designs an implementation scheme user group initially, and is the number ofwhich is capable of producing a finite subset of the complete new participants to join the user group.infinite transparencies through the proposed Algorithms 1 and An index table where is the index of2, with computationally feasible operations. We also derive an the used memoryless sequence tooptimization problem to solve the maximal contrast of the encode the secret pixel .proposed VC scheme. http://ieeexploreprojects.blogspot.com The probability of the event .Notations A group which contains all columns of the matrix as the elements. The threshold of a VC scheme. The number of generated transparencies of a II. EXTENDED VERSION BASED ON BASIS MATRICES AND VC scheme. PROBABILISTIC MODEL The number of subpixels to encode a secret A. Basis Matrices pixel; i.e., the width of the basis matrices. The basis matrices of VC scheme were first introduced Stacking operation, equivalent to the bitwise by Naor and Shamir [1]. In this paper, a white-and-black secret operation “OR.” image or pixel is also described as a binary image or pixel. In and Two collections of Boolean matrices. the basis matrices, to encode a binary secret image , each se- The number of ones in a vector . cret pixel white black will be turned into blocks at the corresponding position of transparencies , The contrast of the VC scheme based on basis respectively. Each block consists of subpixels and each sub- matrices. pixel is opaque or transparent. Throughout this paper, we use 0 and Two Boolean matrices where to indicate a transparent subpixel and 1 to indicate an opaque and . subpixel. If any two subpixels are stacked with matching posi- tions, the representation of a stacked pixel may be transparent, A Boolean matrix containing all when the two corresponding pixels are both transparent. Oth- possible combinations of zeros and erwise, the stacked pixel is opaque. Let denote the stacking ones as the columns. operation, defined as and Two vectors to represent the multiplicities of in and . The contrast of the probabilistic model of VC Actually, can be treated as the bitwise operation “OR.” It is scheme. noted that we use the notation to indicate the stacking of the two transparencies and , since and can be treated , A column randomly selected from and . as two Boolean matrices.
  • 3. LIN AND CHUNG: PROBABILISTIC MODEL OF VC SCHEME WITH DYNAMIC GROUP 199 For the basis matrices, two collections of Boolean in which is the multiplicity of inmatrices and are constructed to encode the binary pixel , . The relation between and isrespectively. Each row of the matrix in and correspondsto an encoded block, and the elements represent the subpixels. (1)Before describing the definitions of and , we first explainhow to encode . For being white, the dealer randomly chooses Example 1: An example of a (3, 4) VC scheme isa matrix from with uniform distribution and then sends allthe rows to each , respectively. For being black, the dealerrandomly chooses a matrix from with uniform distributionand then sends all the rows to each , respectively. The and are required to meet the conditions described in Definition1. Let denote the hamming weight of a -vector(i.e., the number of ones in ). Definition 1: A VC scheme with subpixels and con-trast can be represented as two collections ofBoolean matrices and . Let be a constant integer. A The arevalid VC scheme is required to meet the following conditions[1]: 1) For any matrix in , the stacked of any out of the rows in the matrix satisfies . 2) For any matrix in , the stacked of any out of the rows in the matrix satisfies . 3) For any -element subset and , the two collections of matrices ob- tained by restricting each matrix in and , to rows are indistinguishable in the sense that they contain the same matrices with the same frequencies. The first and the second conditions represent the contrast re-quirements. In general, with larger , the stacking result is more http://ieeexploreprojects.blogspot.comvisually distinguishable. The third condition represents the se-curity requirement. A valid VC must be able to prevent the Thus, is represented as , and is rep-secret pixels from being revealed by analyzing the patterns or resented as .probability distributions from transparencies for . B. Probabilistic Model of VC Scheme If we find an Boolean matrix and an Boolean matrix , we can construct and by In the scenario where is very large, i.e., , to con-permuting all columns of and [1], expressed as struct is impractical. Thus, the proposal of a new VC scheme is required in order to overcome the above problem. The proba- bilistic model of VC is first introduced by Ito et al. [2]. Instead of the basis matrices expanding a secret pixel into a block with subpixels in transparencies, the probabilistic model of VC only uses one subpixel to represent one secret pixel. The idea is briefly described as follows. To encode the secret image , the probabilistic model of VC constructs two basis matrices and . For the secret pixel being white, the dealer randomly chooses a column fromIf two Boolean matrices and can be adjusted to with uniform distribution and sends each element to ,become the same matrix with reordering columns, and respectively. For being black, the dealer randomly chooses aare equivalent in terms of generating . Therefore, the orders column from with uniform distribution and sends eachof columns of and are technically insignificant. element to , respectively. To decode , any out of the The previous studies indicated that the “totally symmetric” transparencies are stacked .scheme [19] (also called the canonical form [15]) is only consid- Then the region white is probabilistically moreered for constructing and , since the studies [15] and [19] likely to appear white than the region black is.proved that the “totally symmetric” schemes cover the Thus, we haveVC schemes with optimal contrast. Let denote an Boolean matrix consisting of all possible combinations of zeros and ones as columns of . A matrix (2)is called “totally symmetric” if can be divided into some , with horizontal concatenation. Thus, where is the probability of the event , and is athe and of a “totally symmetric” VC scheme can be pre- binary value taken from (if is white) or (if is black). Thesented as two vectors and notation means that are
  • 4. 200 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 7, NO. 1, FEBRUARY 2012any elements in . As a result, can be interpreted by visual black secret pixels , and by (7), and are homogeneous toperception since the human visual system can be treated as a low and for describing and , respectively. Accordingpass filter. On the other hand, to protect , in the case where the to the probabilistic model, we randomly choose a columnnumber of stacked transparencies is smaller than , the region from ; since consists of many submatrices , supposecorresponding to the white pixels in is probabilistically iden- a column is chosen from . This step can be simulated bytical, in terms of appearing black or white, to the region corre- rearranging a vector consisting of zeros and onessponding to the black secret pixels. In other words, the regions randomly, and all possible combinations are equally likely to white are probabilistically indistinguishable to the be a candidate, since all columns in have equal chancesregions black on the stacking result. The property to be selected. This property reduces the memory requirementis expressed as to store and during encoding. For the case , the generated column becomes a (0-1) memoryless sequence where the probability of assigning each element (3) to 0 is (i.e., ). In terms of computer programming, can be simulated with aFor the contrast condition described in probability model [2], the random Boolean generator where the probability of generatingcontrast between white and black pixels is calculated from the 0 is and generating 1 is . For convenience, in thedifferential “probability” of appearing zero and not the differen- rest of the paper, the abbreviation PrVC scheme denotestial “frequency” used in basis matrices. The contrast is defined the probabilistic model of VC with unlimited .as Lemma 1: For any elements generated from the secret pixel with PrVC scheme, after the stacking , the probability of the stacking result ap- (4) pearing zero isto distinguish used in Definition 1. Let de- andnote the frequency probability of appearing ones in . The defi- .nition of Probabilistic VC scheme based on and is Proof: To encode the secret pixel , a candidate ingiven as follows. is chosen, and the generated Definition 2: A Probabilistic VC scheme with contrast elements are assigned to , respec- can be represented as two Boolean matrices tively. The probability of generating as the candidateand http://ieeexploreprojects.blogspot.com is generated to encode the secret pixel. . Let be a constant integer; a valid Probabilistic VC is . Supposemust satisfy the following conditions: For the case where transparencies are stacked, the probability 1) The stacked of any out of the rows in satisfies of the secret pixel appearing zero is , because the . corresponding elements at the transparencies, 2) The stacked of any out of the rows in satisfies are required to be all zeros, and we have . 3) For any -element subset and , the two matrices obtained by restricting each matrix in and , to rows are the identical.C. Proposed VC Scheme The proposed method is based on the basis matrices and the Therefore, the probability of a pixel appearing zero isidea of probabilistic model. For a VC scheme, the “to- .tally symmetric” form of and are both constructed and As stated in the security condition in Definition 2, afterdescribed as and , respectively. For a column randomly stacking any rows ( ) in and to obtain two vectorsselected from with uniform distribution, let be the prob- and , we have . Based on the results ofability that consists of zeros and ones. The is Lemma 1, the equations maintaining the security conditionexpressed by are described as (5)where outputs a group which contains all columns of thematrix as the elements, and we have (8) (6) (7) where , and . On the other hand, by (6), we haveThus, and represent the probability distribu- (9)tions by which the columns are selected to encode white and
  • 5. LIN AND CHUNG: PROBABILISTIC MODEL OF VC SCHEME WITH DYNAMIC GROUP 201and is described as The result of (14) equal to 0 is obtained by the using result of (12). To verify the contrast condition in Definition 2, substi- tuting in (10), we obtain (10)D. The PrVC Scheme With Optimal Contrast (15) In this section, we focus on the construction of thePrVC scheme with optimal contrast. For convenience, the ab- The contrast is larger than since is largerbreviation OptPrVC scheme is used to denote the than 1. Hence we obtain that has better contrast than ,PrVC scheme with optimal contrast. which incurs contradiction. Lemma 2: For and of OptPrVC scheme, if A similar result of Lemma 2 for finite had been pointed out , then ; and if , then . by previous work [12]. Lemma 2 gives an immediate mapping Proof: Suppose and from to and which is described as are the OptPrVC ifscheme with the optimal contrast but and if (16) . Thus, for , satisfies (6)(8)(10) which ifare described as (11) (17) (12) Lemma 3: For OptPrVC scheme, there is a con- sisting of nonzero values and zeros. Proof: By (8), (9), (10), and (17), a linear program to http://ieeexploreprojects.blogspot.com (13) achieve the optimal contrast is described as (18)Then, a new scheme with contrast is constructed by the following steps. First, calculate (19) ; second, construct and each term is (20) if otherwise.To verify that satisfies (6), substituting in (6), we obtain (21) In order to remove the absolute symbols in (19), we divide the feasible region into subregions according to the sign of each . Then the above linear program is divided into subproblems, expressed as (22)To verify the security condition in Definition 2, for , substituting in (8), we obtain (23) (24) (25) (26) where is the sign of in each subregion. There (14) are one formula (23), one formula (24), and formulas (25)
  • 6. 202 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 7, NO. 1, FEBRUARY 2012as the constraints of the linear program, and those formulas are where are don’t-care terms, since those values will be multi-linearly independent as long as the subregion is not empty, and plied by zeros in the later stage. Then is described asthe maximal lies at the boundary sets (26). Thus, consistsof nonzero values and zeros. (33) Since the nonzero terms in represent the criticalvalues, those nonzero terms are recorded by two vectors Lemma 4: For an OptPrVC scheme, if is even, and , where is the then ; otherwise, if is odd, then .probability of using the memoryless sequence to encode Proof: The result is easily verified by (33).the secret pixel taken from the secret image , with By Lemma 4, all values of are grouped into two setsfor white and for black. Without loss of gen- Even integer and Odd integer ; and the sumserality, is arranged in increasing order (i.e., ). In of all elements in both set are 1 and , respectively. By (33),other words, the th nonzero term is recorded in and their mathematical formulations are expressed assuch that and . The equations of andare described as (34) (27) Example 2: For andobtained from Example 1, the , , , , and are (35) By (34) and (35), is expressed as By (8), (9), and (10), the equations substituted with http://ieeexploreprojects.blogspot.com and are expressed as (36) (28) Lemma 5: For the of a OptPrVC scheme, and . (29) Proof: Suppose is a OptPrVC scheme but or . We generate a new where each term in is (30) if ifThen, (28)–(30) are expressed in the matrix form as otherwise. Then the contrast of is better than by substi- . . .. . . . (31) tuting and in (36). . . . . . . . . . . . Finally, the optimization problem is derived by com- bining (27) and (36) with Lemma 5. The optimization problem is described asThe matrix is the transpose of a Vander-monde matrix. By the property of , we usethe inverse Vandermonde matrix to obtain . In fact, the co-efficients of inverse Vandermonde matrix can be expressed asLagrange polynomials[30]. We fill with those coefficientsto obtain (37) . . (38) . . .. . . . . . . . . . The target function is the reciprocal of . Krause and Simon [20] proved that the contrast of VC scheme is no less (32) than , so the outcome of is expected to be .
  • 7. LIN AND CHUNG: PROBABILISTIC MODEL OF VC SCHEME WITH DYNAMIC GROUP 203 TABLE I OF SOME OPTPRVC SCHEMES FOR Fig. 1. Binary secret image.We use Maple to solve for , and the results arelisted in Table I.E. Encoding Algorithm Fig. 2. Results of OptPrVC scheme. (a) . (b) . (c) . For a given value of , the transparencies can be continuouslygenerated with the OptPrVC scheme. However, prac-tical applications require the algorithm to terminate within fi-nite steps. To meet the requirement, a finite number is used to Algorithm 2. The algorithm of OptPrVC scheme byspecify the number of transparencies in the algorithm. The algo- the index tablerithm requires , obtained by solving or by lookingup Table I. The outputs of Algorithm 1 are transparencies Input: An index table , a positive integer , and a vector . and an index table , where is the index Output: transparencies .of the used memoryless sequence http://ieeexploreprojects.blogspot.com to encode the se-cret pixel . 1 for each in do 2 for to do 3 Assign randomly to 0 or 1 whereAlgorithm 1. The algorithm of OptPrVC scheme . 4 end forInput: A binary secret image , two positive integers , , 5 end forand two vectors .Output: transparencies ; an index table .1 for each pixel in do III. EXPERIMENTAL RESULTS2 if white then Three experiments are performed for and . Fig. 13 Generate an integer shows the binary secret image . Fig. 2 shows the first experi- and . ment for . Fig. 2(a)–(b) shows two generated transparen-4 else cies and , and the stacking result is shown in5 Generate an integer Fig. 2(c). We observe that the characters on the stacking result and are clear. Based on Lemma 1, we verify the security and con- . trast of the three experiments. We use to indi-6 end if cate any generated transparencies with the proposed scheme,7 . and is a pixel at and the position corresponds to the secret8 for to do pixel .9 Assign randomly to 0 or 1 where For a single transparency , the probabilities of a white se- . cret pixel appearing zero and a black secret pixel appearing zero10 end for are11 end for In the first round, we use Algorithm 1 to generate trans-parencies and . If we need not to generate more transparen-cies in the future, is not required and discarded. Otherwise, Therefore, the white pixels and black pixels cannot be distin- has to be stored in a safe place, and we can generate more guished by observing only one transparency. In the case wheretransparencies by utilizing . two transparencies are stacked , the contrast
  • 8. 204 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 7, NO. 1, FEBRUARY 2012 Fig. 4. Results of OptPrVC scheme. (a) . (b) . (c) . (d) . (e) . the characters on the stacking result are barely visible. The verifications of security and contrast are described as follows.Fig. 3. Results of OptPrVC scheme. (a) . (b) . (c) . (d) . For a single transparency , the security verification is(e) . (f) . (g) .verification is In the case where two transparencies are stacked , the security verification isThus, the contrast is . Fig. 3 shows the second experiment http://ieeexploreprojects.blogspot.com for . Fig. 3(a)–(c)show three generated transparencies , , and , andFig. 3(d)–(f) show the results of stacking any two transparen-cies , , and . Fig. 3(g) is the result ofstacking the three transparencies , and we can seecontours of the secret on stacking result. The verifications ofsecurity and contrast are described as follows. In the case where three transparencies are stacked For a single transparency , the security verification is , the security verification isIn the case where arbitrary two transparencies are stacked , the security verification is In the case where four transparencies are stacked , the contrast verification isIn the case where arbitrary three transparencies are stacked , the contrast verification is Thus, the contrast is .Thus, the contrast is . IV. DISCUSSIONS AND APPLICATIONS Fig. 4 shows the third experiment for . Fig. 4(a)–(d)show four generated transparencies , , , and . It is A. Addition and Deletion of Users in the Dynamic User Groupnoted that we skip the stacking results of arbitrary two or Since the proposed scheme allows dynamic changes of usersthree transparencies due to the large number of combinations. in the user group, the operations to add and delete users are elab-Fig. 4(e) shows the stacking result , where orated in this section. For the addition of new participants to
  • 9. LIN AND CHUNG: PROBABILISTIC MODEL OF VC SCHEME WITH DYNAMIC GROUP 205 TABLE II OPTIMAL CONTRAST VC SCHEMES FORthe user group with original participants, the transparenciesneed to be able to accommodate the new users. Fig. 5. Experimental result of (2, 100) PrVC. (a) . (b) . (c) .In the case where the original transparencies are producedthrough the traditional VC scheme, the original trans-parencies need to be eliminated, and then the new set oftransparencies need to be generated through the advantages over the traditional VC scheme. In this case,VC scheme. The frequent regeneration and redistribution of the the contrast of using the VC scheme is similar to thewhole set of transparencies consume huge computing and com- OptPrVC scheme. The VC schemes with optimalmunication resources for a dynamically changing user group, contrast for and various values of are listed inand may lead to potential security risks if certain original trans- Table II, which is duplicated for reference from [19], except forparencies are not discarded completely. For example in the case . For , the contrast isof , an original participant holding one of original which is very closes to . In addition, ForVC transparencies and obtaining one of the new , the contrast is which is alsoVC transparencies might potentially be able to decode the secret very closes to . Thus, the stacking results ofby stacking the two transparencies. In the applications with a OptPrVC scheme and VC scheme are very closeknown maximal number of participants , the better strategy to each other for large . Fig. 5 is an example of the (2, 100)is to apply the VC scheme to generate trans- VC scheme based on probabilistic model with contrast 25/99.parencies; and each authorized participant is assigned a trans- We can see similar qualities in Figs. 5(c) and 2(c). However, theparency. By applying the VC scheme, the addition traditional VC scheme requires larger amounts of compu- tations and memory space to construct basis matrices. A simple http://ieeexploreprojects.blogspot.comand departure of users can be accommodated without frequentregeneration and redistributions of transparencies. However, in analysis is discussed as follows.many applications, the is unknown and applying a larger For two basis matrices and of the traditionalnumber as is a feasible option to avoid the overflow of VC scheme, the memory space is needed to storeusers over the . The proposed OptPrVC scheme and . To encode a secret pixel, first, the arithmetic sequenceaccommodates the addition and departure of users without the is permutated to obtain withproblem of user overflow. In addition, with very large , operations; second, each row of or is duplicated to thethe contrast of the OptPrVC scheme is similar to the corresponding transparency, and the elements of each row are VC scheme addressed in Section IV-B. arranged by following the orders with For the deletion of users, we suppose that out of ( I/O operations. Thus, the encoding includes computa- ) participants are to leave the user group. If the contrast is to be tions and I/O operations, and the dominates theoptimized, the original transparencies need to be eliminated, traditional VC scheme due to the essentiality of the dupli-and the new transparencies need to be generated through cating operations during the production of transparencies. It isthe VC scheme; the process consumes computing noted that the encoding complexity is if the I/O operationand communication resources for a frequently changing user is excluded from the complexity measure. On the other hand, thegroup. Another strategy is to retrieve the transparencies from proposed probabilistic VC scheme requires the values ofthe departing participants; those retrieved transparencies can be two vectors and , whose lengths are both . To encodeassigned to the new participants. For the proposed Opt- a secret pixel, an -element memoryless sequence is generated,PrVC scheme to accommodate the scenarios of departing users, which takes operations. The value of depends on thethe departing users can simply discard the transparencies, and construction method of and . Blundo et al. [17] provedthe new transparencies for new users can simply be generated that a traditional VC scheme with holds thethrough Algorithm 2; the retrieval of the transparencies from property . Thus, the space complexity and the timethe departing users is not required. The authentication and se- complexity of the VC scheme are both . On thecurity of the retrieved transparencies present a more challenging other hand, the proposed scheme takes spaceproblem and relevant studies can be found in [11] and [33]. and operations. Thus, the proposed method substantially reduces the computations and memory space for large .B. Contrast Comparisons of the Proposed Scheme WithTraditional VC Schemes C. Comparisons of With Two Linear Programs In general, the contrast of the OptPrVC scheme is Hofmeister et al. [19] provide an elegant linear program tolower than certain VC schemes with a finite value of . calculate the optimal contrast of VC scheme. In addition,Nevertheless, if is very large, the OptPrVC has some the basis matrices can be constructed by the solution of the linear
  • 10. 206 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 7, NO. 1, FEBRUARY 2012 TABLE III COMPARISONS OF WITH OTHER OPTIMIZATION EQUATIONS TABLE IVprogram. We restate the linear program as follows: COMPARISONS OF THE OPTPRVC SCHEME WITH RGS (39) (40) (41) (42) TABLE V ENCODING ALGORITHM OF RG SCHEME PROPOSED BY requires variables to cal- CHEN AND TSAO [29]culate the optimal contrast of VC schemes. Obviously, for , (39)–(42) cannot be explicitly formulated due to the in-finite terms of the sigma summation, so is not compu-tationally feasible. The proposed http://ieeexploreprojects.blogspot.com is a specialized optimiza-tion problem under , and (37) and (38) are well formu-lated capable of solving the variables .On the other hand, for a finite integer of , is more suit-able than . Moreover, Bose and Mukerjee [14] also provide the -normoptimization equation to calculate the optimal contrast of the VC scheme; the equation is restated as follows: (43) and , ; and forwhere , and the contrast is given black, we have , andas . For , (43) also cannot be explicitly , . The probabilitiesformulated due to the infinite terms of the sigma summation. are identical to Algorithm 1 for . Thus, the RG [29]Table III lists the comparisons. performs the same distribution in generating transparencies through the Algorithm 1 for ; and based on the observa-D. Comparisons of the Proposed Method With RG Scheme tion, the proposed OptPrVC scheme is also a generalized Table IV shows the comparisons of the OptPrVC version of RG for general and unlimited .scheme with RGs [26]–[29]. Rather than the basis matrices,the size of RG’s transparencies is identical to the secret image, V. CONCLUSIONand this property is similar to the proposed OptPrVC We have proposed a VC scheme with flexible valuescheme. It is noted that certain studies [26]–[29] propose of . From the practical perspective, the proposed scheme ac-various encoding algorithms to achieve various stacking results commodates the dynamic changes of users without regeneratingand contrasts, and therefore, the contrast is not unique. The and redistributing the transparencies, which reduces computa-definition of the contrast follows (4). The proposed Opt- tion and communication resources required in managing the dy-PrVC scheme is a generalized scheme for arbitrary , and the namically changing user group. From the theoretical perspec-contrast is optimal under the definition of Ito et al. [2]. In order tive, the scheme can be considered as the probabilistic modelto give a detailed comparison, Table V shows the RG of VC with unlimited . Initially, the proposed scheme isproposed by Chen and Tsao [29]. As stated in [29], for a white based on basis matrices, but the basis matrices with infinite sizesecret pixel white, we have , cannot be constructed practically. Therefore, the probabilistic
  • 11. LIN AND CHUNG: PROBABILISTIC MODEL OF VC SCHEME WITH DYNAMIC GROUP 207model is adopted in the scheme. As the results listed in Table I, [21] E. R. Verheul and H. C. A. Van Tilborg, “Constructions and propertiesthe proposed scheme also provides the alternate verification for of out of visual secret sharing schemes,” Designs, Codes, Cryp- tography, vol. 11, no. 2, pp. 179–196, May 1997.the lower bound proved by Krause and Simon [20]. For , [22] P. A. Eisen and D. R. Stinson, “Threshold visual cryptographythe contrast is very low so that the secret is visually insignif- schemes with specified whiteness levels of reconstructed pixels,”icant. Therefore, in practical applications, the values of 2 or 3 Designs, Codes, Cryptography, vol. 25, no. 1, pp. 15–61, 2002. [23] F. Liu, C. K. Wu, and X. J. Lin, “A new definition of the contrast offor are empirically suggested for the proposed scheme. visual cryptography scheme,” Inf. Process. Lett., vol. 110, no. 7, pp. 241–246, Mar. 2010. 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Tsai, “Cheating in visual cryptography,” Sian-Jheng Lin was born in Taiwan. He received the Designs, Codes, Cryptography, vol. 38, no. 2, pp. 219–236, Feb. 2006. http://ieeexploreprojects.blogspot.com M.S., and Ph.D. degrees in computer science [11] C. M. Hu and W. G. Tzeng, “Cheating prevention in visual cryptog- B.S., from National Chiao Tung University, in 2004, 2006, raphy,” IEEE Trans. Image Process., vol. 16, no. 1, pp. 36–45, Jan. and 2010, respectively. 2007. He is currently a postdoctoral fellow with the Re- [12] H. Koga, “A general formula of the -threshold visual secret search Center for Information Technology Innova- sharing scheme,” in Proc. 8th Int. Conf. Theory and Application of tion, Academia Sinica. His recent research interests Cryptology and Information Security: Advances in Cryptology, Dec. include data hiding, error control coding, and secret 2002, pp. 328–345. sharing. [13] R. Z. Wang, “Region incrementing visual cryptography,” IEEE Signal Process. Lett., vol. 16, no. 8, pp. 659–662, Aug. 2009. [14] M. Bose and R. Mukerjee, “Optimal visual cryptographic schemes for general ,” Designs, Codes, Cryptography, vol. 55, no. 1, pp. 19–35, Apr. 2010. [15] C. Blundo, P. D’Arco, A. De Santis, and D. R. Stinson, “Contrast op- Wei-Ho Chung (M’11) was born in Kaohsiung, timal threshold visual cryptography schemes,” SIAM J. Discrete Math., Taiwan, in 1978. He received the B.Sc. and M.Sc. vol. 16, no. 2, pp. 224–261, Feb. 2003. degrees in electrical engineering from National [16] M. Bose and R. Mukerjee, “Optimal visual cryptographic Taiwan University, Taipei City, Taiwan, in 2000 schemes,” Designs, Codes, Cryptography, vol. 40, no. 3, pp. 255–267, and 2002, respectively, and the Ph.D. degree in the Sep. 2006. Electrical Engineering Department, University of [17] C. Blundo, A. De Santis, and D. R. Stinson, “On the contrast in visual California, Los Angeles (UCLA), in 2009. cryptography schemes,” J. Cryptology, vol. 12, no. 4, pp. 261–289, From 2002 to 2005, he was a system engineer at 1999. ChungHwa TeleCommunications Company. From [18] S. Cimato, R. De Prisco, and A. De Santis, “Optimal colored threshold 2007 to 2009, he was a Teaching Assistant at UCLA. visual cryptography schemes,” Designs, Codes, Cryptography, vol. 35, In 2008, he was a research intern working on CDMA no. 3, pp. 311–335, Jun. 2005. systems in Qualcomm Inc. Since January 2010, he has been a faculty member [19] T. Hofmeister, M. Krause, and H. U. Simon, “Contrast-optimal out of holding the position of assistant research fellow in the Research Center for secret sharing schemes in visual cryptography,” Theoretical Comput. Information Technology Innovation, Academia Sinica, Taiwan. His research Sci., vol. 240, no. 2, pp. 471–485, Jun. 2000. interests include wireless communications, signal processing, statistical detec- [20] M. Krause and H. U. Simon, “Determining the optimal contrast for tion and estimation theory, and networks. secret sharing schemes in visual cryptography,” Combinatorics, Prob- Dr. Chung received the Taiwan Merit Scholarship, sponsored by the National ability, Comput., vol. 12, no. 3, pp. 285–299, May 2003. Science Council of Taiwan, from 2005 to 2009.