Multi-secret visual cryptography with deterministic ... - Springer Link

Multimed Tools Appl (2014) 72:1867–1886 DOI 10.1007/s11042-013-1479-8

Multi-secret visual cryptography with deterministic contrast Bin Yu & Gang Shen

Published online: 8 May 2013 # Springer Science+Business Media New York 2013

Abstract The multi-secret visual cryptography scheme (MVCS) allows for the encryption of multiple secret images into a given image area. The previous works on MVCS with probabilistic contrast can not guarantee that every original pixel will be reconstructed correctly. However, MVCS with deterministic contrast can reconstruct every original pixel with simple computation for high-end applications, but they are all simple 2-out-of-2 cases. These drawbacks limit the applicability of MVCSs existed. Based on ringed shares, MVCS with deterministic contrast has been defined in this paper. Furthermore, an (k, n)-MVCS with deterministic contrast, which makes the number of secret images not restricted, is proposed on the basis of the share rotation algorithm and the basis matrices of single secret sharing visual cryptography. Experimental results show that our scheme is the first (k, n)-MVCS with deterministic contrast, which can be applied on any k and n. Keywords Visual cryptography . Multiple secret images . Deterministic contrast . Ringed shares 1 Introduction Secret sharing, proposed independently by Shamir [16] and Blakley [2] in 1979, is an effective method for a group to protect the confidential information. The principle is encoding the secret into some shares to be distributed to a set of participants. Only the B. Yu Department of Computer Science and Information Engineering, Zhengzhou Information Science and Technology Institute, Zhengzhou, People’s Republic of China e-mail: [email protected] G. Shen (*) Zhengzhou Information Science and Technology Institute, Zhengzhou, People’s Republic of China e-mail: [email protected]

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qualified sets of these participants can recover the secret, whereas the unqualified sets gain nothing about the secret. As a new secret sharing technology, visual cryptography scheme (VCS) was introduced by Noar and Shamir [15] in Eurocrypt’94. Since the security of VCS is equal to “one time pad” [15] and the reconstruction is performed without complex computation, many researchers pay much attention to this domain. In recent years, the main results of VCS include the general access structure [1], the optimization of pixel expansion and relative difference [4, 5, 8, 17], grey and chromatic images [3, 9, 13, 14], etc. However, the schemes above can only share one secret image. Many shares have to be managed by each participant when sharing multiple images, which will reduce the work efficiency and limit its applications. The purpose of the multi-secret visual cryptography scheme (MVCS) is incorporating multiple secrets at once, so the construction methodology for MVCS is more complicated than that of VCS. According to the approaches discussed in prior schemes, the multi-secret visual cryptography can generally be categorized as either probabilistic MVCS (PMVCS) or deterministic MVCS (DMVCS). PMVCS uses the appearance probabilities of the blackness in the black and white areas of the recovered images for interpreting black and white pixels by human visual system. The probabilities of the blackness in the black areas of the recovered images will be higher than that in the white areas. The contrast of PMVCS is probabilistic, because they give no absolute guarantee on the correct reconstruction of the original pixel: in some cases, the reconstruction of the original pixel is wrong. Lin et al. [12] proposed a (2, 2)-PMVCS without pixel expansion based on the probabilistic model. Stacking the two shares can recover one secret image. Flipping one of the two shares and then stacking with the other share can recover the second secret image. By this method, only 16.7 % contrast for recovered images can be retrieved. Lee and Chiu [11] proposed a novel hybrid encryption algorithm that includes a VCS-based encryption and a camouflaging process. By this algorithm, about 43 % contrast for recovered images without pixel expansion can be achieved. Yang and Chung [23] discussed a general (k, n)-PMVCS for any k and n, which can be easily implemented on basis of a conventional (k, n)-VCS. Although recovered images of the above schemes display the information of secret images in whole, we can’t reconstruct the original pixel correctly from the recovered images with simple computation, which don’t obey the rule of secret recovering without complex computation. DMVCS utilizes a pixel-expanded codebook when encrypting secret images and enables human visual system distinguish between “black” and “white” via the Hamming weights in the recovered images. The Hamming weights of the black areas should be greater than that of the white areas in the recovered images. The contrast of DMVCS is deterministic, because the reconstruction of the secret pixel is guaranteed and in order to get the recovered images more clearly, we can recover all the pixels perfectly from the recovered images with simple computation according to the Hamming weights. To cater for the above conditions, Wu and Chen [21] developed a (2, 2)-DMVCS, which is capable of hiding two square secret images. According to this scheme, directly stacking two shares upon one another recovers the first secret image. Subsequently rotating the first share clockwise recovers the second secret image. Based upon turning over or flipping around, Shyu et al. [18] proposed visual cryptography schemes that are able to encode two or four secrets into two rectangular shares and up to eight secrets into two square shares. In virtue of the angle restriction during stacking rectangular or square shares, Wu and Chang [20] designed a (2, 2)-DMVCS, in which two secret images are encrypted into two

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circular shares. The first secret image is recovered by stacking two circular shares atop one another. When one of the shares is rotated to a specific angle, the second secret image is recovered. Later Shyu et al. [19] also proposed a (2, 2)-DMVCS to encrypt multiple secret images within two circular shares, but shapes of the recovered images are distorted. To overcome the distortion, Hsu et al. [10] used ringed shares to construct a (2, 2)DMVCS. Two secret images can be, respectively, recovered by rotating a share with a fixed angle and stacking them. In order to share more secret images, two (2, 2)-DMVCS were proposed in [6, 7] to make the number of secret images not restricted. The above works on DMVCS are all simple 2-out-of-2 cases. In this paper, based on ringed shares, we give a definition of the DMVCS and design a (k, n)-DMVCS for any k and n. Our (k, n)-DMVCS, which is implemented on the basis of the conventional (k, n)-VCS, can share any number of secret images. For the special case k=n= 2, the pixel expansion and contrast of our DMVCS are the same as the scheme in [7]. Finally, we show the experimental results and comparisons to demonstrate the effectiveness of our scheme. The rest of this paper is organized as follows. In Section 2, we present a review of preliminary studies. In Section 3, we propose the construction of our DMVCS on conventional VCS. Experiment and comparison are given in Section 4 and Section 5 concludes the paper.

2 Preliminary studies In this section, we briefly review the deterministic and probabilistic VCS and then introduce Yang and Chung’s basic model by using the approach as the same as the probabilistic VCS. 2.1 Deterministic VCS In the deterministic VCS, it is guaranteed that for every black and white pixel of the original image the recovered image contains a clear difference. Nair et al.’s VCS in [15] expands the pixel into m sub pixels and use “m−h”B“h”W to represent the white pixel and “m−l”B“l”W to represent the black pixel, where 0  l < h  m . Definition 1 A deterministic (k, n)-VCS can be represented as two collections of n×m Boolean matrices C0 and C1. When sharing a white (resp. black) pixel, the dealer randomly chooses one row of the Boolean matrix C0 (resp. C1) to a relative share. The chosen matrix defines the gray level of the m sub pixels in every one of the n shares. Let V (Ci | r), i=0, 1, denote an OR-ed vector of any r rows in Ci, H(•) be the Hamming weight function. A deterministic scheme is considered valid if the following conditions are met: 1. For any r ( r  k 1 ) rows in C0 (resp. C1), the “OR”-ed V satisfies H ðV ðC1 jrÞÞ ¼ H ðV ðC0 jrÞÞ . 2. For any r (r≥k) rows in C0 (resp. C1), the “OR”-ed V satisfies H ðV ðC1 jrÞÞ > H  ðV ðC0 jrÞÞ . The first condition is called security and the second condition is called contrast. Due to the security condition, we cannot get any information about the shared secret if we do not have more than k shares. The contrast defined in [15] is aD ¼ ðH ðV ðC1 jrÞÞ H ðV ðC 0 jrÞÞÞ=m and it reflects the visual quality of the recovered secret image.

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Example 1 The (2, 2) deterministic VCS:         10 10 01 01 C0 ¼ ; ; and C 1 ¼ : 10 01 01 10 Obviously, H ðV ðC1 j1ÞÞ ¼ H ðV ðC0 j1ÞÞ ¼ 1 and H ðV ðC1 j2ÞÞ ¼ 2 , H ðV ðC0 j2ÞÞ ¼ 1 . Namely, shares, in which every 2-subpixel block is 1B1W, are noise-like, satisfying the security and contrast conditions. And we can reconstruct all the pixels of the original image with simple computation according to the mapping that an original black pixel is corresponding to 2B0W and an original white pixel is corresponding to 1B1W. 2.2 Probabilistic VCS The deterministic scheme requires every pixel to satisfy the contrast condition. Actually, the secret can be successfully recovered as long as human eyes system can distinguish the difference between black and white areas, not every pixel. Therefore, a probabilistic VCS utilizes the frequency of white pixels in the black and white areas of the recovered image for interpreting black and white pixels by human visual system, but the probabilistic nature will diminish the reliability of the scheme as perceived by the human visual system because there is no absolute guarantee on the correct reconstruction of the original pixel. By selecting every column vector in Ci as a one-column matrix, Yang’s VCS in [22] form 0 the sets Ci , i=0, 1, including n×1 column Boolean matrices. Definition 2 A probabilistic (k, n)-VCS can be represented as two collections of n×1 0 0 column Boolean matrices C0 and C1 .When sharing a white (resp. black) pixel, the 0 0 dealer randomly selects one row of the column matrix C0 (resp. C1 ) to a relative share. The chosen column matrix defines the color level of pixel in every one of the n shares. 0 Let V (Ci | r), i=0, 1, denote an OR-ed vector of any r rows in Ci , P(•) be the appearance probabilities of the blackness. A probabilistic scheme is considered valid if the following conditions are met:

0 0 0 1. For any r (r  k 1 ) rows in C0 (resp. C1 ), the “OR”-ed V satisfies P V C1 jr ¼

0 P V C0 jr .

0 2. For any r (r ≥ k) rows in C0 (resp. C1), the “OR”-ed V satisfies P V C1 jr >

0 P V C0 jr . The first condition is called security and the second condition is called contrast. Its



0 0 contrast defined in [22], which is probabilistic, is aP ¼ P V C1 jr P V C0 jr , r≥k and has the same level as the deterministic VCS but not the same.

Fig. 1 The recovered images of two VCSs: a deterministic b probabilistic

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Fig. 2 The sketch map of subset partition in secret images

Example 2 The (2, 2) probabilistic VCS: 0

C0 ¼

        0 1 1 0 0 ; ; and C 1 ¼ : 1 0 0 1









0 0 0 0 So, P V C1 j1 ¼ P V C0 j1 ¼ 1=2 and P V C1 j2 ¼ 1 , P V C0 j2 ¼ 1=2 . According to the definitions, the contrasts of two different VCSs are both 1/2 and the recovered images of the deterministic (2, 2)-VCS and the probabilistic (2, 2)-VCS are shown in Fig. 1. Due to the probabilistic nature, the detailed edge in Fig. 1(b) is blurred. Moreover, we can get the perfect secret image by scanning every 2-subpixel block of the Fig. 1(a) with judgment according to the Hamming weight. 2.3 Probabilistic MVCS To overcome the number restriction of shares and the shape distortion, Yang and Chung’s scheme in [23] proposed a general (k, n, s)-MVCS for any k, n and s, in which shares s secret images in any k out of n ringed shares with the probabilistic approach. Suppose Tj are the jth share and θi, which is a set of rotation angles to rotate n shares clockwise for revealing the ith secret image, is defined as follows:

Fig. 3 The sketch map of the subpixel block

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θi ¼ fði

1Þðj

1Þθjj ¼ 1; 2; . . . ; ng; for 1  i  s ; where θ ¼ 360 =ðs ðn

 θ A rotation operation ! T1 ; . . . ; Tj ; . . . ; Tn i is defined as follows: n  θ ði ! T1 ; . . . ; Tj ; . . . ; Tn i ¼ Tj

1Þðj 1Þθ

1ÞÞ:

o j j ¼ 1; 2; . . . ; n ; for 1  i  s ; where θ ¼ 360 =ðs ðn

1ÞÞ:

Definition 3 Let [r ½Š denote stacking any r shares and Hmi ðÞ be the Hamming weight function for every m-subpixel block in the stacked result and the corresponding secret pixel for these m subpixels is white (i=0) and black (i=1), respectively. A (k, n, s)-MVCS is considered valid if the following conditions are met: 1:PðHm1 ð[r ½fT1 g81 ; fT2 g82 ; . . . ; fTj g8j ; . . . ; fTn g8n ŠÞÞ ¼ PðHm0 ð[r ½fT1 g81 ; fT2 g82 ; . . . ; fTj g8j ; . . . ; fTn g8n ŠÞÞ; for 0 < r  k

2.

1 , where 0  8 j  360 .   h i   h i > P Hm0 [r !fT 1 ; T 2 ; . . . ; T n gθi P Hm1 [r !fT 1 ; T 2 ; . . . ; T n gθi , for r≥k, where 1  i  s .

The first condition is security, which implies that the MVCS has the same probability of blackness in black and white areas for stacking any less than k shares rotated by any angles. The second condition is contrast, which shows that when  θ stacking k shares with the specific corresponding angles in ! T1 ; . . . ; Tj ; . . . ; Tn i , one can recover the secret Si through the different probabilities of blackness. Hence, Yang and Chung’s MVCS has the similar probabilistic performance like the probabilistic VCS, which can not obtain perfect secret images with simple computation. 3 The proposed deterministic MVCS Although DMVCS can reconstruct secret images perfectly with simple computation, all previous DMVCSs are the simple 2-out-of-2 cases. In this paper, we consider a deterministic (k, n)-MVCS for any k and n, which shares t secret images. Our (k, n)MVCS is constructed based on ringed shares and with the deterministic approach. For the special case k =n=2, the proposed (2, 2)-DMVCS has the same contrast and pixel expansion as Fu et al.’s (2, 2)-DMVCS.

Fig. 4 An example of random permutation

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Fig. 5 The procedure of the secrets sharing

3.1 Basic definitions In this section, we give some definitions and notations about DMVCS based on ringed shares that will be used throughout the paper. Consider a set of n shares (T1, T2, …, Tn) and t secret images (S1, S2, …, St) in a DMVCS. 0 Suppose Si , i=1, 2, …, t, be the recovered images and m be the pixel expansion. Let [r ½Š , 0 < r  n 1 , denote the stacking operation of any r shares in (T1, T2, …, Tn) and H1(H0)

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Fig. 6 The sketch map of the secrets recovering

is the Hamming weight function for every subpixel block in the stacked result and the corresponding secret pixel for these blocks is black (white). Definition 4 A set of rotation  angles to rotate n shares clockwise for recovering the ith secret image is defined by Θi, Θ i ¼ θij j j ¼ 1; 2; . . . ; n , 1  i  t , 1  j  n , θij 2 ½0 ; 360 Š . Every set of rotation angles is not the same and the corresponding secret images are different. θij denotes the rotation angle to rotate jth share for recovering the ith secret image.  Θ Definition 5 A rotation operation ! T1 ; . . . ; Tj ; . . . ; Tn i is defined as follows: n o  Θ θ ! T1 ; . . . ; Tj ; . . . ; Tn i ¼ Tj ij jj ¼ 1; 2; . . . ; n ; for 1  i  t; 1  j  n:  θij Tj denotes rotating the share Tj by the angle θij clockwise. Definition 6 A (k, n)-DMVCS is considered valid if the following conditions are met:  8   
1. H 1 [r fT1 g81 fT2 g82 ; . . . ; Tj 8j ; . . . ; fTn g8n ¼ H 0 ð[r ½fT1 g81 fT2 g82 ; . . . ; Tj j ; . . . ; fTn g8n ŠÞ; 0 H 0 [r !fT 1 ; T 2 ; . . . ; T n gΘi ; r  k



The first condition is security, which implies that the MVCS has the same Hamming weight in black and white areas for stacking any less than k shares rotated by any angles. The second condition is contrast, which shows that when stacking k shares with the specific  Θ corresponding angles in ! T1 ; . . . ; Tj ; . . . ; Tn i , one can recover the secret Si through the different Hamming weight. Hence, the DMVCS has the different performance compared to the PMVCS and according to the definition in [15], the contrast αD is   h i  h i H 0 [r !fT 1 ; T 2 ; . . . ; T n gΘ i =m; H 1 [r !fT 1 ; T 2 ; . . . ; T n gΘ i which is of deterministic nature.

3.2 Share rotation algorithm Because the stacking position is a relative position, the first share is not rotated and is taken for a reference. To simplify the algorithm, we define the column vectorΘ ¼ ðΘ 1 ;Θ 2 ;


;Θ t ÞT and the  corresponding column vector W ¼ ðW1 ; W2 ;


; Wt ÞT , Wi ¼ wij j j ¼ 1; 2; . . . ; n , wij 2 N . The specific share rotation algorithm is listed as follows. share rotation algorithm Input: the threshold structure (k, n), 2  k  n , and the number of secret images t, t≥2. Output: the column vector W and a variable h, h ∈ N. Step 1: For all i ∈ [1,t] and j ∈ [1,n], initially set all elements in W to “0”. Step 2: For all 1  i0 < i and j 2 ½1; nŠ , Dj, a set of distance between wi′j and wij, is defined as follows:

Table 3 Pixel expansions of our scheme for t=3

n

2

3

4

5

6

6

21

40

70

108

12

42

80

140

24

105

240

k 2 3 4 5 6

48

210 96

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Table 4 Pixel expansions of our scheme for t=4

n

2

3

4

5

6

8

36

84

145

222

16

72

168

290

32

180

504

64

360

k 2 3 4 5 6

n Dj ¼ d d ¼ wij

128

wi0 j ; i0 ¼ 1; 2; . . . ; i

o 1 :

Step 3: Initially set i=1. Step 4: For i=i+1 and j ¼ 2; 3; . . . ; k , make wi 1;n plus 1 and assign it to wij. Step 5: If k=n, go to Step6; else for j ¼ k þ 1; . . . ; n , assign wij a minimum value to satisfy the condition that there is no intersection between Dj and any k−1 of all Dj′, j′ H 0 [r !fT1 ; T2 ; . . . ; Tn gΘ i : qpððwij þl 1Þmodhþ1Þ

Proof When Si(q, p, l)=1(0), T1qpl ðl; :Þ; . . . ; Tj

ðl; :Þ; . . . ; Tnqpððwin þl

1Þmodhþ1Þ

ðl; :Þ constitute the basic matrix C1(C0) of a (k, n)-VCS. Because r≥k, we have    qpððwij þl 1Þmodhþ1Þ ðl; :Þ; . . . ; Tnqpððwin þl 1Þmodhþ1Þ ðl; :Þ H 1 [r T1qpl ðl; :Þ; . . . ; Tj    qpððwij þl 1Þmodhþ1Þ > H 0 [r T1qpl ðl; :Þ; . . . ; Tj ðl; :Þ; . . . ; Tnqpððwin þl 1Þmodhþ1Þ ðl; :Þ :

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By shares rotation algorithm we then have    qpððwij þl 1Þmodhþ1Þ H 1 [r T1qpl ; . . . ; Tj ; . . . ; Tnqpððwin þl 1Þmodhþ1Þ    qpððwij þl 1Þmodhþ1Þ > H 0 [r T1qpl ; . . . ; Tj ; . . . ; Tnqpððwin þl 1Þmodhþ1Þ : qpððwij þl 1Þmodhþ1Þ In addition, Tj denotes rotating Tj by the angle θij, so,  h i  h i H 1 [r !fT1 ; T2 ; . . . ; Tn gΘ i > H 0 [r !fT1 ; T2 ; . . . ; Tn gΘ i :

4 Experiment and comparison 4.1 Experimental results Table 1 shows the pixel expansions of (k, n)-VCS in the scheme [5]. Based on the above pixel expansion mc, the pixel expansion of our scheme is h×mc, and according to the proposed approach, Tables 2, 3 and 4 summarize the pixel expansions of our (k, n)-DMVCS for t=2, 3, 4. It is seen that our scheme is of the generality (any k, n and t). To evaluate the performance of the proposed scheme, we conduct a series of experiments for t=3. Experiment 6: (2, 2, 3)-DMVCS is the scheme like (2, 2, 3)-MVCS in [7]. Experiment 7: (2, 3, 3)-DMVCS demonstrates the generality of our construction, which is not only for the simple 2-out-of-2 scheme. Experiment 8: (3, 3, 3)-DMVCS shows the effectiveness of our construction for k>2. In these three experiments, we take the same three secret images, as shown in Fig. 7. Moreover, all images in experiments are displayed in a planar way. Example 6 Construct the proposed (2, 2, 3)-DMVCS 2 3based on a (2, 2)-VCS of C0 ¼     0 0 1 0 1 0 . Since W ¼ 4 0 1 5 and h=3, so we could recover and C1 ¼ 0 1 1 0 0 2 h i 0 h i   0 0 three secrets by S1 ¼ [2 ½T1 ; T2 Š , S2 ¼ [2 T1 ; fT2 g120 , S3 ¼ [2 T1 ; fT2 g240 . Figure 8 shows the shares and the recovered images. We can gain nothing from any share, whereas we can visually reveal the secrets by stacking two ringed shares, respectively. The pixel expansion and the contrast of the proposed DMVCS are 6 and 1/6. Example 7 Construct the proposed (2, 3, 3)-DMVCS based on a (2, 3)-VCS of 2 3 3 0 0 0 0 1 5 . Since W ¼ 4 0 1 2 5 and h=7, 1 0 3 6 h i 1  2  0 0 so we could recover three secrets by S1 ¼ [2 ½T1 ; T2 ; T3 Š , S2 ¼ [2 T1 ; fT2 g7360 ; fT3 g7360 , h i 3  6  0 S3 ¼ [2 T1 ; fT2 g7360 ; fT3 g7360 . 2

1 C0 ¼ 4 1 1

1 1 1

3 2 1 0 0 5 and C1 ¼ 4 1 0 0

1 0 1

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Figure 9 shows the shares and the recovered images. We can gain nothing from any share, whereas we can visually reveal the secrets by stacking two or three ringed shares, respectively. The pixel expansion and the contrast of the proposed DMVCS are 21 and 1/21. Example 2 8 0 C0 ¼ 4 0 0

Construct 3 the proposed 2 (3, 3, 3)-DMVCS 2 3)-VCS of3 3 based on a (3, 0 0 0 1 1 0 1 0 0 1 1 0 1 5 and C1 ¼ 4 0 1 0 1 5 . Since W ¼ 4 0 1 1 5 and h=3, 0 2 2 0 1 1 0 0 1 1 h i   0 0 so we could recover three secrets by S1 ¼ [3 ½T1 ; T2 ; T3 Š , S2 ¼ [3 T1 ; fT2 g120 ; fT3 g120 , h i   0 S3 ¼ [3 T1 ; fT2 g240 ; fT3 g240 .

Figure 10 shows the shares and the recovered images. We can gain nothing from any share by stacking two shares, whereas we can visually reveal the secrets by stacking three ringed shares, respectively. The pixel expansion and the contrast of the proposed DMVCS are 12 and 1/12.

4.2 Comparison Table 5 shows a comparison between our proposed scheme and those from previous works on MVCS for the following items—the contrast, the access structure, the number of secret images, the pixel expansion and the share shape. The proposed (k, n)-MVCS performs like a deterministic VCS compared to the scheme in [23] and the perfect reconstruction of every original pixel is guaranteed. Compared to the MVCS in [19], our scheme and the DMVCSs in [6, 7, 21] are not based on circular shares, so that the shapes of recovered images won’t be distorted. All previous DMVCSs are simple 2-out-of-2 schemes, while our scheme can be applied on any k and n. The MVCS in [21] are only designed for the particular number of secret images and our scheme could work on t secret images, where t≥2. A comparison between our scheme and the existing (2, 2)-DMVCSs [6, 7, 10, 18–21] is also given in Table 6 to examine the pixel expansion and the contrast.

5 Conclusion In this paper, we discuss the general (k, n)-DMVCS, which can be applied on any k, n and any number of secrets. Also we define the formal security and contrast conditions of (k, n)-DMVCS and have developed the share rotation algorithm and the subset sharing algorithm for sharing multiple secret images with deterministic contrast. Besides, on the condition that the intelligent terminals with computing power, which offers convenient conditions to the implementation of simple computation, are universal and available, the proposed scheme can reconstruct every original pixel correctly and retrieve all secret images perfectly. However, our scheme is concentrated on the nature of contrast. Reducing the pixel expansion in DMVCS is encouraged and deserves further studying.

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Bin Yu received the B.S. degree in Dept. of Electronic Engineering from the University of Shanghai Jiaotong in 1986, the M.S. degree in Dept. of Automatic Engineering from South China University of Technology in 1991 and the Ph.D. degree in 1999. From 1997 to 1999, he worked as a research assistant at Hong Kong University of Science and Technology. From 2003 to 2004, he worked as a vice professor at in the University of Waterloo, ON, Canada. Currently, he is a professor of the Department of Computer Science and Information Engineering at Zhengzhou Information Science and Technology Institute, P. R. China. His research interests include the design and analysis of algorithms, visual secret sharing and network security.

Gang Shen was born in Henan province, P. R. China, in December 1986. He received the B.S. degree in the Zhengzhou Information Science and Technology Institute in 2010. He is studying the M.S. in the Department of Computer Science and Information Engineering at Zhengzhou Information Science and Technology Institute. His research interest is visual secret sharing.