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International Journal of Intelligent Information Technology Application 1:1 (2008) 1-9 Available at http://www.engineering-press.org/IJIITA.htm

A Probabilistic Visual Secret Sharing Scheme for Grayscale Images with Voting Strategy Chin-Chen Chang Department of Information Engineering and Computer Science, Feng Chia University, Taichung 407, Taiwan, R.O.C. Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 621, Taiwan, R.O.C. E-mail: [email protected]

*Chia-Chen Lin Department of Computer Science and Information Management, Providence University, Taichung 433, Taiwan, R.O.C. E-mail: [email protected]

T.Hoang Ngan Le, Hoai Bac Le Department of Computer Science, Natural Science University, 227 Nguyen Van Cu, District 5, HCMC, Vietnam. E-mail: {lhbac, lthngan}@fit.hcmuns.edu.vn

Abstract—Wang et al. proposed two visual secret sharing schemes based on Boolean operations in 2007. One is a probabilistic (2, n) secret sharing scheme, which is called (2, n) ProbVSS scheme, for binary images and the other is a deterministic (n, n) secret sharing scheme for grayscale images. Although Wang et al. only apply probabilistic concept to design the revealing phase of their (2, n) ProbVSS scheme, their (2, n) ProbVSS and (n, n) VSS schemes solve the problems of computational complexity and pixel expansion at the same time. In Wang et al.’s (n, n) VSS scheme for grayscale images, n generated shadows must be collected in advance to completely reconstruct a secret grayscale image. If Wang et al.’s (2, n) ProbVSS scheme is repeated eight times to deal with grayscale images, the image quality of the reconstructed image is significantly decreased when only any two of n shadows are used to generate the reconstructed grayscale image. To provide a (2, n) ProbVSS scheme that demonstrates better image quality of the reconstructed grayscale image than Wang et al.’s scheme without significantly increasing computational complexity, we apply the voting strategy and the least significant bits abandoning approach in combination with Wang et al.’s (2, n) ProbVSS for binary images to handle grayscale images. Experimental results confirm that the image quality of the reconstructed grayscale image achieved with the proposed scheme is better than one achieved by the pure Wang et al.’s scheme. Index Terms—Probabilistic visual secret sharing, voting strategy, (2, n) ProbVSS scheme, grayscale images

I. INTRODUCTION In 1971, a secret sharing scheme also called a (k, n) threshold scheme was firstly introduced by George Blakley [1] and Adi Shamir [2], respectively. The first objective of secret sharing scheme is to protect secret data by dividing it into n pieces; each piece is called a share or

a shadow. Later, the set of shadows is distributed to n participants and each participant holds a single piece of the set of shadows. The secret data can be reconstructed if and only if there is complete knowledge of the k shadows, and k-1 or fewer shadows will reveal no information about the secret data, where k ≤ n . Based on the (k, n) threshold scheme, in 1995 Noar and Shamir firstly introduced a secret image sharing scheme, which is also called visual secret sharing (VSS), that focused on image data [3]. Based on Noar and Shamir’s idea, several schemes for grayscale images [4, 7] and for color images [6, 8, 9] have been proposed. In essence, instead of the original image the VSS scheme uses several random-like images called shadows to be the data transmitted over the Internet. The shadows can thwart malicious attackers and prevent the secret image from being directly accessed. The secret image sharing schemes also inherit the properties of the (k, n) threshold scheme mentioned above. Generally, four criteria are used to evaluate the performance of a (k, n) VSS scheme. The first criterion is security: whether the scheme guarantees that fewer k shadows offer no information about the secret image, where k ≤ n . The second criterion is accuracy: the similarity between the reconstructed image and the original one. Basically, accuracy is respect to the image quality of the reconstructed image. A higher quality in the reconstructed image implies higher accuracy of the VSS scheme. The third criterion is computational complexity: the number of operators required to generate shadows for a secret image and to reconstruct the reconstructed image by using the collected shadows. The last criterion is the size of a shadow called the pixel expansion problem. Larger shadow size implies higher transmission cost and storage cost.

1999-2459/ Copyright © 2008 Engineering Technology Press,Hong Kong July,2008

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Chin-Chen Chang et al / International Journal of Intelligent Information Technology Application

Many (k, n) VSS schemes have been proposed over the past decade. All these schemes satisfy the security and accuracy criteria previously described. However, some of them suffer from the pixel expansion problem [3, 7, 9, 11] although the reconstructed image can be revealed by simply stacking the collected shadows. Some VSS schemes [5, 14] solve the pixel expansion problem but they have high computational complexity. To reduce the computational cost of generating shadows, a probabilistic visual scheme, which is called ProbVSS, was proposed by Yang et al. in 2004 [10]. In the best ProbVSS case, each pixel in the original secret image is represented by one pixel in each shadow. With ProbVSS, a black original pixel will accurately appear as black one in the recovered image but a white one has a higher probability to appear as white pixel in the recovered image than that to appear as black one. In fact, not every pixel will be exactly reconstructed. However, a reconstructed image looks very similar to its original image. In other words, the ProbVSS scheme solves the computation complexity problem and the pixel expansion problem at the same time with a slight decrease in reconstruction accuracy capacity. In 2006, Cimato et al. [12] proposed a generalized ProbVSS scheme that allows a trade in shadow size for recovered image accuracy by setting the pixel expansion factor in shadows generation phase. Inspired by Cimato et al.’s scheme, Wang et al. [13] proposed a new ProbVSS (2, n) scheme, which not only reduces computational cost but also increases the accuracy of the recovered image by using two Boolean operations: XOR and AND. Wang et al.’s scheme was initially designed for use with binary secret images. Experimental results showed that their scheme is superior to existing ProbVSS schemes with respect to security, accuracy and computational complexity. To extend the application of Wang et al.’s scheme for using with secret grayscale images and to increase accuracy without significantly increasing computational complexity, in this paper, we apply a voting method and the least significant bits abandoning approach in combination with Wang et al.’s scheme. In the following discussions, we implicate repeating Wang et al.’s scheme eight times to deal with grayscale images as the pure Wang et al.’s scheme. Experimental results confirm that our proposed scheme can provide better image quality of the recovered image in comparison with the pure Wang et al.’s scheme. The rest of this paper is organized as follows. Section 2 gives a briefly review of Wang et al.’s (2, n) ProbVSS scheme and voting strategy. Section 3 presents our scheme in detail. Section 4 shows the experimental results of the proposed scheme. Future work and some conclusions are given in Section 5. II. RELATED WORKS To extend the application of Wang et al.’s (2, n) ProbVSS scheme to grayscale images, we apply the voting method in combination with Wang et al.’s scheme as one of our strategies. To give readers sufficient background knowledge of our proposed scheme, we

introduce Wang et al.’s (2, n) ProbVSS scheme and voting method in detail, respectively, in the following subsections. 2.1. Wang et al.’s (2, n) ProbVSS for binary images In 2007, Wang et al. proposed two ProbVSS schemes, one of which, the (2, n) ProbVSS scheme [7] is designed for binary images with two Boolean operations: AND (&) and XOR (⊕). The shares construction phase and revealing phase of Wang et al.’s scheme are illustrated in the pseudo-codes as follows: Shares construction phase: Input: An integer n with n≥2 and the secret image A Step 1: Generate (n + 1) random matrices B1, B2, ..., Bn+1. Step 2: Compute n intermediate matrices C1, C2, ..., Cn with Ci = Bi&A, for i =1, 2, ..., n. Step 3: Compute n shadows A′1, A′2, ..., A′n with A′i = Bn+1 ⊕ Ci, for i = 1, 2, ..., n. Revealing phase: Step 1: Compute the reconstructed images A′ij = Ai ⊕ Aj , for i = 1, 2, ..., n and i≠j. Step 2: Calculate the number of bits ‘1’ in each image A′ij, for i = 1, 2, ..., n and i≠j. Step 3: Output the reconstructed image: A′. Wang et al.’s scheme does not use the Boolean OR operation or Hamming weight, thus their scheme cannot be implemented by directly viewing the stacked k shares of n shadow images. Therefore, to generate a recovered secret binary image, they must compute the values of each bit of the reconstructed image. The authors claim the pixel-wise Boolean operations AND (&) and XOR (⊕) involving in their scheme can be easily carried out with common software packages such as Photoshop. However, each pixel in the original secret binary image is not always the same as that in the reconstructed image with Wang et al.’s scheme. In other words, the accuracy of their scheme is decreased. Detailed discussions about this phenomenon are described as follow: Let us simply denote pixel Ai (s, t) as Ai. In their revealing phase, when Ai=0, the following operations are performed: Ci = Bi&A = 0, A′i = Bn+1⊕ Ci = Bn+1, A′j =Bn+1⊕ Cj = Bn+1; thus, A′i = A′i⊕A′j = Bn+1 ⊕ Bn+1 = 0. That is, a pixel “0” in the original binary image will remain the same in the reconstructed image. However, when Ai=1, the following operations are performed: Ci = Bi&A = Bj, A′i = Bn+1⊕ Bi, A′j = Bn+1⊕ Bj; thus, A′i = A′i⊕A′j = Bi ⊕ Bj ⊕ Bn+1 ⊕ Bn+1 = Bi ⊕ Bj. That is, a pixel “1” in the original binary image A may be changed or not in the reconstructed image A′. Because there are n shadows A′1, A′2, …, A′n which are generated in the shares construction phase with Wang et al.’s scheme, there are (n-1)×(n-1)/2 ways to choose a pair of shadows A′i and A′j to form a reconstructed image A′ij. Since a pixel “1” in the original binary image A could be changed in reconstructed image A′, any two shadows cannot generate the same reconstructed images. To decide which A′ij is the best, Wang et al. introduce probabilistic concepts demonstrated in the following


example. Let us assume that the secret binary image is A, where A = ⎡1 0 1⎤ and the number of shadows n is 3. ⎢0 0 1⎥ ⎣ ⎦

Example 1: Input: n=3, A =

Shares construction phase: Step 1: Generate (n+1) random matrices Bi's: B1 = ⎡⎢1 1 1⎤⎥ , B2 = ⎣1 1 0⎦

⎡0 1 0 ⎤ , ⎢1 1 0 ⎥ ⎣ ⎦

B3 = ⎡⎢1 0 1⎤⎥ , B4 = ⎡⎢0 0 1⎤⎥ . ⎣0 1 0 ⎦

⎣0 0 1⎦

Step 2: Compute n intermediate matrices using Ci = Bi&A: C1 = ⎡⎢ 1 0 1 ⎤⎥ , C2 = ⎡⎢0 0 0⎤⎥ , C3 = ⎡⎢1 0 1⎤⎥ . ⎣0 0 0 ⎦

⎣0 0 1⎦

⎣0 0 0 ⎦

Step 3: Compute n shadow images, using Ai = Bn+1⊕Ci : ⎡1 0 0 ⎤ 1 0 0⎤ ⎡0 0 1⎤ ⎥ , A2 = ⎢0 1 0⎥ , A3 = ⎢ 0 1 1 ⎥ . 0 1 0 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

A1 = ⎡⎢

Revealing phase: Step 1: Compute the reconstructed images A'ij = Ai ⊕ Aj : A′12 = ⎡⎢ 1 0 1 ⎤⎥ , A′13 = ⎡⎢0 0 0⎤⎥ , A′23 = ⎡⎢1 0 1⎤⎥ . ⎣0 0 0 ⎦

⎣ 0 0 1⎦

⎣0 0 1⎦

Step 2: Calculate the number of bits “1” in each reconstructed image A′ij, for i = 1, 2, ..., n and i ≠ j. Numbers of bits “1” in A′12, A′13, A′23 are derived as 2, 1, 3, respectively. Hence, A′23 is the best outcome. A′ =A′23. Step 3: Output the reconstructed image: A′= ⎡⎢1 0 1⎤⎥ . ⎣0 0 1⎦

2.2. Voting strategy The voting strategy concept is very simple but very useful to generate the best result from a set of options. Voting strategy means that the option representing the majority is selected as the outcome. In our scheme, we use the voting strategy to deal with the inconsistency phenomenon among bits in the same location of shadows. By applying the voting method, we hope to find the best result to represent the corresponding value in the reconstructed image. Let us assume a set of shadows A1, A2, …, An, and suppose reconstructed image M is the representative result. The value of bits in the reconstructed image M is identified by Equation (1). ⎧ ⎪1, M (i , j ) = ⎨ ⎪⎩0,

if

A2(3,3) = 0; A3(3,3) = 0. The number of bits value “0” is 3, which is the majority, thus M (3,3) = 0. Finally, the representative image M for these three shadows should be M=

⎡1 0 1 ⎤ ⎢0 0 1⎥ ⎣ ⎦

A1 (i, j ) + ... + An (i, j ) ≥

n 2.

(1)

otherwise

Example 2: Suppose we have 3 candidate shadows (A1, A2, A3) with values such as A1=

⎡1 0 1 ⎤ ⎢0 1 0 ⎥ ⎢ ⎥ ⎢⎣1 1 0 ⎥⎦

⎡1 0 1 ⎤

, A2= ⎢0 1 1 ⎥ ⎢ ⎥ ⎢⎣1 1 0 ⎥⎦

⎡1 0 1 ⎤

A3= ⎢0 1 0 ⎥ Let us consider value of bit in each location. ⎢ ⎥ ⎢⎣0 1 0 ⎥⎦

For position (1,1): A1(1,1) = 1; A2(1,1) = 1; A3(1,1) = 1. The number of bits “1” is 3, which is the majority, thus M (1,1) = 1. For position (1,2): A1(1,2) = 0; A2(1,2) = 0; A3(1,2) = 0. The number of bit value “0” is 3, which is the majority, thus M (1,2) = 0. For position (3,3): A1(3,3) = 0;

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⎡1 0 1 ⎤ ⎢0 1 0⎥ . ⎢ ⎥ ⎢⎣1 1 0 ⎥⎦

III. PROPOSED SCHEME This section discusses a new (2, n) ProbVSS scheme for grayscale images based on the techniques mentioned in the previous section. To apply Wang et al.’s (2, n) ProbVSS scheme to grayscale images, we must break a secret grayscale image into a set of binary images first. However, it is not simple to apply Wang et al.’s scheme to each binary image in order to achieve our objective because some problems must be solved. First, if we break a grayscale image sized 8×8 into binary images, the number of generated binary images will be 8. If we cannot propose another strategy to combine with Wang et al.’s scheme, the computation cost of our scheme will be 8 times Wang et al.’s. Second, even though improved computing technology may handle the increased computational complexity of our scheme, the image quality of the reconstructed image still remains unsolved. Wang et al.’s scheme is initially designed for secret binary images. Their scheme cannot guarantee reconstruction of each pixel of the secret binary image if only two of n shadows are collected to generate the reconstructed secret image. If we simply apply their scheme to each binary image in our scheme, the same phenomenon will occur. As we know, the left-most bit of a pixel in a grayscale image contains the largest energy. It causes 27 distortions if the left-most bit of a pixel is changed from 1 to 0 or from 0 to 1. In other words, the distortion of the reconstructed grayscale image will increase if we simply apply Wang et al.’s scheme. To provide higher image quality of the reconstructed grayscale image when only two of n shadows are used to generate the reconstructed grayscale image without significantly increasing computational complexity, we propose two new strategies as improvements to Wang et al.’s scheme. First, we adopt the least significant bits abandoning approach to maintain acceptable computational complexity. This approach means the right-most bits of pixels in the grayscale image will be ignored in our scheme. Take a grayscale image sized 8×8, for example. If three bits from the right side of the image’s pixels are abandoned; each pixel contains only five bits. That is, only five binary images are generated after a grayscale image is broken into binary images. In other words, the computational complexity is five eighth of the computational complexity of the method that does not abandon the three right-most bits of pixels in the grayscale image. Second, in the shares construction phase, we repeat Wang et al.’s scheme (2×m+1) times to generate the corresponding elements of the most significant bits of the pixels in the grayscale image for each shadow. That is to say, each shadow contains (2×m+1) options for a part of significant bits of pixels in the grayscale image. Later, in the revealing phase, the

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voting method is used to find out the best outcomes from (2×m +1) options. The following subsections contain detailed descriptions of the shares construction phase and revealing phase. 3.1. Shares construction phase Some initial parameters must be firstly initialized: the number of shares n in the (2, n) ProbVSS scheme, the number of ignored matrices k, the number of options contained in each shadow for a part of significant bits of pixels in the grayscale image (2×m +1), and the number of matrices adopting the voting method t. The shares construction phase consists of five steps as shown in Figure 1.

Step 3.1: Break the binary block into eight binary matrices. The binary matrix is created according to the following rule: the ith bit of binary matrix xj is the jth bit of the ith pixel that belongs to the binary block x. In other words, binary matrix xj is a set of jth bits of binary block x. Figure 4 shows how to form binary matrix x1. Other binary matrices are generated in a similar manner.

Figure 4. Example of procedure for generating binary matrices. Figure 1. Flowchart of shares construction phase.

Step 1: Divide an original grayscale image into grayscale blocks. The original grayscale image is broken into nonoverlapping grayscale blocks having the same size, b×b. Step 2: Convert a grayscale block into a binary block. Each pixel value in a grayscale block is transformed to binary representation. Take a grayscale block ⎡111 159 20 ⎤ ⎢245 10 198⎥ for example. Its corresponding binary ⎢ ⎥ ⎢⎣ 40 215 100⎥⎦

block is generated as shown in Figure 2:

Step 3.2: Devote the last k binary matrices. Step 3.3: Split the remaining binary matrices into two groups. The first group containing t binary matrices is called the set of most significant matrices, or MS. The second one containing (8-k-t) binary matrices is called the set of less significant matrices, or LS. Step 4: Transform MS and LS to binary intermediate shadows. This process is performed by applying our proposed scheme to the MS and applying Wang et al.’s (2, n) ProbVSS to the LS. Each matrix in the MS is applied (2×m+1) times the shares construction phase of Wang et al.’s (2, n) ProbVSS scheme to generate (2×m+1) candidates for each matrix. The output, which is called the set of most significant intermediate shadows, is shown in Figure 5: Matrix sMSi11

1st time

Matrix MSi

Figure 2. Example of the transformation of a grayscale block to its binary blocks.

Step 3: Separate the binary block into two groups of matrices. Binary Block Step 3.1 BM1

BM2

...

...

BM8

Step 3.3 Step 3.2 MS1 ... MSt

LS1 ... LS8-k-t

The most significant matrices set (MS)

The less significant matrices set (LS)

Devote

Figure 3. The detail procedure of Step 3 in the shares construction phase.

First, generate eight binary matrices from each binary block. After abandoning the last k matrices, split the retained matrices into two groups as shown in Figure 3. Step 3 can be broken into three substeps as follows:

Prob (2, n) nd 2 time VSS of Wang et al. (2*m+1)th time

... Matrix sMSi1n ... Matrix sMSi(2*m+1)1 ... Matrix sMSi(2*m+1)n

Figure 5. Generating the most significant intermediate shadows sets.

Each matrix in the LS is applied a single time the shares construction phase of Wang et al.’s (2, n) ProbVSS scheme. The output is called the set of less significant intermediate shadows sets Step 5: Translate the sets of the most and the less significant intermediate shadows to the set of grayscale shadows. There are many ways to create grayscale shadows by combining intermediate shadows belonging to the most and the less significant intermediate shadows sets generated in Step 4. To reduce the correlation between each generated grayscale shadow and the original grayscale one, the proposed combination strategy adopts a shuffling approach. In this strategy, the sets of the most


and less significant intermediate shadows are combined to generate grayscale shadows. Figure 6 demonstrates how to generate a pixel in grayscale. Note that this Figure shows only the combined order of the sets of the most and the less significant intermediate shadows of the first pixel in the first grayscale shadow. Later, the binary representation of a pixel in a grayscale shadow is transformed into a decimal representation to generate a grayscale shadow. The remaining grayscale shadows are based on the same combination order. Matrix sMS111 ... Matrix MS1

Matrix sMS11n ... Matrix sMS1(2*m+1)1 ... Matrix sMS1(2*m+1)n

…

Matrix MSt

Matrix sMSt11 ... Matrix sMSt1n

k-t) next matrices. Ignore the rest of the (k- 2×m×t) last matrices.

Matrix sLS11 Matrix LS1

... Matrix sLS1n

... Matrix sLS(8-k-t)1 ...

Matrix LS(8-k-t)

Matrix sLS(8-k-t)n

... Matrix sMSt(2*m+1)1 ... Matrix sMSt(2*m+1)n

5

Padding 0

Figure 6. Example of creating the first pixel of grayscale shadow 1.

3.2. Revealing phase After receiving the set of grayscale shadows, a receiver can reconstruct the original grayscale image by using any two of collected shadows with our revealing phase. Our revealing phase includes five steps, as shown in Figure 7.

Figure 8. Generating the set of MS and LS.

Step 4: Compute the reconstructed binary image. Based on the sets of MS and LS, the reconstructed binary image is built uses the same strategy as in Step 5 of the shares construction phase. To reveal each pixel value in the intermediate image, we combine the t bit from the MS and the (8-k-t) bit from the LS. The other k bits are chosen randomly. Figure 9 is an example describing our processes in detail:

(a) Pattern of a pixel of the reconstructed binary image

Figure 7. Flowchart of our revealing phase.

Let GS = {GS 1, GS 2…, GS n} be a grayscale shadows set distributed to n participants. In this phase, assume the receiver uses any two grayscale shadows called GSi and GSk to generate the reconstructed grayscale image. Step1: Divide grayscale shadows GSi and GSk into nonoverlapping grayscale blocks. These blocks have the same size, b×b, which is the same as in the shares construction phase. Step 2: Convert each grayscale block into eight binary matrices. The step is performed like Step 2.1 in the shares construction phase. From two grayscale blocks, we have two binary matrices groups, and each group has eight matrices. Step 3: Generate the sets of MS and LS. Based on the values of t, m and k parameters, we can compute the set of MS and the set of LS. Figure 8 demonstrates this step. After applying Wang et al.’s (2, n) ProbVSS to each pair of binary matrices that belong to the two binary matrices groups GSi and GSk, we have eight matrices. We separate the first (2×m+1)×t matrices into t subgroups and then use the voting method on each subgroup to decide the set of representative MS. The set of LS consists of (8-

(b) Exampel Figure 9. Example of generating the reconstructed binary image.

Step 5: Transform the reconstructed binary image into a grayscale image. Group each set of 8 bits into a set, and then convert the 8 bits into an integer value. Output the result as the reconstructed grayscale image after all bits are processed. The following example gives a better explanation of our proposed scheme. Let us assume that the number of shadows is 4, the number of ignoring matrices is 3, the number of MSs is 1, the operation times of Wang et al.’s scheme for MS is set as 3 (=2×m+1 and m=1). Example 3: Input: n=4; k=3; t=1; m=1; grayscale image I= ⎡⎢124 200 ⎤⎥ . ⎣180

61 ⎦

Shares construction phase: Step 1: Divide into one block x = ⎡⎢124 200 ⎤⎥ .

⎣180 61 ⎦ Step 2: Convert to binary block x= ⎡⎢01111100 11001000 ⎤⎥ . ⎣10110100 00111101⎦

Step 3: Separate the binary block into two matrix groups.

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First, break the binary block into eight binary matrices as follows: BM1= ⎡⎢0 ⎣1 BM5= ⎡⎢1 ⎣0

1⎤ , BM2= 0 ⎥⎦

⎡1 1 ⎤ , BM = ⎡1 0⎤ , BM = 3 ⎢ 4 ⎢0 0 ⎥ ⎥ ⎣ ⎦ ⎣1 1 ⎦ 1⎤ , BM6= ⎡⎢1 0⎤⎥ , BM7= ⎡⎢0 0 ⎤⎥ , BM8= ⎡⎢0 1⎥⎦ ⎣1 1 ⎦ ⎣0 ⎣0 0 ⎦

⎡1 0⎤ , ⎢1 1 ⎥ ⎣ ⎦ 0⎤ . 1 ⎥⎦

Next, devote three last matrices (BM6, BM7, BM8) with k=3. Set the first matrix as the most significant matrix because of t=1, and set the remaining four matrices as the least significant matrices. It means that MS1=BM1 LS1=BM2, LS2=BM3, LS3=BM4, and LS4=BM5. Step 4: Transform the set of MS and LS to binary intermediate shadows. Because of m=1, repeat (2×m+1) = 3 times Wang et al.’s scheme for MS1. Apply our shares construction phase to MS1 and Wang et al.’s shares construction phase to LS1, LS2, LS3, and LS4. Finally, the sets of binary intermediate shadows MS1 are shown in Figure 10.

GS 2 = {sMS112, sMS122, sMS132, sLS12, sLS22, sLS32, sLS42, ⎡00101001 01110101⎤ ⎥, ⎣11110111 10011110 ⎦

sLS52} = ⎢

GS 3 = {sMS113, sMS123, sMS133, sLS13, sLS23, sLS33, sLS43, ⎡00110011 00000111⎤ ⎥, ⎣01111111 10010111⎦

sLS53} = ⎢

GS 4 = {sMS114, sMS124, sMS134, sLS14, sLS24, sLS34, sLS44, ⎡00111100 00110101⎤ ⎥. ⎣ 01010011 10010011⎦

sLS54} = ⎢

Finally, four grayscale shadows are obtained and shown as follows after grouping 8 bits as a set and converting them into decimal representations. GS1= ⎡⎢38 165⎤⎥ , GS2= ⎡⎢ 41 117 ⎤⎥ , GS3= ⎡⎢ 51 ⎣50 152 ⎦ GS4= ⎡⎢60 53 ⎤⎥ . ⎣ 83 147 ⎦

7 ⎤ ⎥ , 254 151 ⎣ ⎦

⎣ 247 158 ⎦

Revealing phase: Based on the above four grayscale shadows, we demonstrate our revealing phase as follows: Step 1: Arbitrarily choose two of the four grayscale shadows to generate the reconstructed image. Here, we choose GS1= ⎡⎢38 165⎤⎥ and GS2= ⎡⎢ 41 117 ⎤⎥ to generate ⎣50 152 ⎦

⎣ 247 158 ⎦

our reconstructed grayscale image. Step 2: Convert the two chosen shadows into binary images as follows: GS1= ⎡⎢00100110 10100101⎤⎥ , GS2 = ⎡⎢00101001 01110101⎤⎥ . ⎣ 00110010 10011000 ⎦

Figure 10. Binary intermediate shadows sets of MS1.

The binary intermediate shadows of LS1, LS2, LS3 and LS4 are shown below: sLS11 = ⎡⎢0 ⎣1 sLS21 = ⎡⎢0 ⎣0 sLS31 = ⎡⎢1 ⎣0 ⎡ sLS41 = ⎢1 ⎣1 sLS51 = ⎡⎢0 ⎣0

0 ⎤ , sLS = ⎡0 12 ⎢ 1 ⎥⎦ ⎣1 0 ⎤ , sLS = ⎡1 22 ⎢ 1 ⎥⎦ ⎣0 1 ⎤ , sLS = ⎡0 31 ⎢ 0 ⎥⎦ ⎣1 0 ⎤ , sLS = ⎡0 42 ⎢ 0 ⎥⎦ ⎣1 1 ⎤ , sLS = ⎡1 52 ⎢ 0 ⎥⎦ ⎣1

1⎤ , sLS = ⎡1 13 ⎢ 1⎥⎦ ⎣1 0 ⎤ , sLS = ⎡0 23 ⎢ 1 ⎥⎦ ⎣1 1⎤ , sLS = ⎡0 33 ⎢ 1⎥⎦ ⎣1 0 ⎤ , sLS = ⎡1 43 ⎢ 1 ⎥⎦ ⎣1 1 ⎤ , sLS = ⎡1 53 ⎢ 0 ⎥⎦ ⎣1

0 ⎤ , sLS = ⎡1 14 ⎢ 1 ⎥⎦ ⎣1 0 ⎤ , sLS = ⎡1 24 ⎢ 0 ⎥⎦ ⎣0 1⎤ , sLS = ⎡1 34 ⎢ 1⎥⎦ ⎣0 1⎤ , sLS = ⎡0 44 ⎢ 1⎥⎦ ⎣1 1⎤ , sLS = ⎡0 54 ⎢ 1⎥⎦ ⎣1

1⎤ , 1⎥⎦ 0⎤ , 0 ⎥⎦ 1⎤ , 0 ⎥⎦ 0⎤ , 1 ⎥⎦ 1⎤ . 1⎥⎦

Step 5: Translate the sets of the most and the less significant intermediate shadows into a set of grayscale shadows. There are four shares generated because n=4. Based on the strategy described in the shares construction phase, we obtain four binary shadows by combining the intermediate shadows belonging to the sets of the most and the less significant intermediate shadows: GS1 = {sMS111, sMS121, sMS131, sLS11, sLS21, sLS31, sLS41, ⎡00100110 10100101⎤ sLS51} = ⎢ ⎥, ⎣00110010 10011000⎦

⎣11110111 10011110 ⎦

Step 3: Generate the MS and LS sets from GS1 and GS2. Break GS1 and GS2 into eight binary matrices following the same rule in Step 3.1 of the shares construction phase. The sets of binary matrices are: s1={ ⎡⎢0 1⎤⎥ ; ⎡⎢0 0 ⎤⎥ ; ⎡⎢1 1 ⎤⎥ ; ⎡⎢0 0 ⎤⎥ ; ⎡⎢0 0 ⎤⎥ ; ⎡⎢1 1 ⎤⎥ ; ⎡⎢1 0⎤⎥ ; ⎣ 0 1⎦ ⎣ 0 0 ⎦ ⎣1 0⎦ ⎣1 1 ⎦ ⎣ 0 1 ⎦ ⎣ 0 0 ⎦ ⎣1 0 ⎦ ⎡0 ⎢0 ⎣ ⎡0 ⎢1 ⎣

1⎤ }, s2 = { ⎡⎢0 0 ⎤⎥ ; ⎡⎢0 1 ⎤⎥ ; ⎡⎢1 1 ⎤⎥ ; ⎡⎢0 1⎤⎥ ; ⎡⎢1 0 ⎤⎥ ; ⎥ 0⎦ ⎣1 1 ⎦ ⎣1 0 ⎦ ⎣1 0⎦ ⎣1 1⎦ ⎣ 0 1 ⎦ 1⎤ ⎡ 0 0 ⎤ ⎡1 1 ⎤ ; ; }. 1⎥⎦ ⎢⎣1 1 ⎥⎦ ⎢⎣1 0 ⎥⎦

Based on the values of 2×m+1 = 3 and t = 1, we form three sets for the reconstructed intermediate shadows of sMS′ and, because we use (2, n) ProbVSS, each set is derived by using two matrices. The generated intermediate shadows of sMS′ are shown below: sMS′1 = {s11⊕s21}= ⎡⎢0 1⎤⎥ ⊕ ⎡⎢0 0 ⎤⎥ = ⎡⎢0 1 ⎤⎥ , ⎣ 0 1⎦ ⎣1 1 ⎦ ⎣1 0 ⎦ sMS′2 = {s12⊕s22}= ⎡⎢0 0 ⎤⎥ ⊕ ⎡⎢0 1 ⎤⎥ = ⎡⎢0 1 ⎤⎥ , ⎣0 0⎦ ⎣1 0 ⎦ ⎣1 0 ⎦ sMS′3 = {s13⊕s23}= ⎡⎢1 1 ⎤⎥ ⊕ ⎡⎢1 1 ⎤⎥ = ⎡⎢0 0 ⎤⎥ . ⎣1 0⎦ ⎣1 0⎦ ⎣ 0 0 ⎦ Apply the voting method to these the set of three sMS′ so that the representative of MS appears as ⎡⎢0 1 ⎤⎥ . ⎣1 0 ⎦

Because of t = 1 and k = 3, the number of matrices in the LS is four (=8-k-t). Perform the XOR operation for


7

next four binary matrices. Five reconstructed LSs are generated as follows: LS1= ⎡⎢0 0 ⎤⎥ ⊕ ⎡⎢0 1⎤⎥ = ⎡⎢0 1 ⎤⎥ , ⎣1 1 ⎦ ⎣1 1⎦ ⎣ 0 0 ⎦ LS2= ⎡⎢0 0 ⎤⎥ ⊕ ⎡⎢1 0 ⎤⎥ = ⎡⎢1 0 ⎤⎥ , ⎣0 1 ⎦ ⎣0 1 ⎦ ⎣0 0⎦ LS3= ⎡⎢1 1 ⎤⎥ ⊕ ⎡⎢0 1⎤⎥ = ⎡⎢1 0⎤⎥ , ⎣0 0⎦ ⎣1 1⎦ ⎣1 1 ⎦ LS4= ⎡⎢1 0⎤⎥ ⊕ ⎡⎢0 0 ⎤⎥ = ⎡⎢1 0 ⎤⎥ . ⎣1 0⎦ ⎣1 1 ⎦ ⎣ 0 1 ⎦

(a) Peppers

(b) Baboon

Step 4: Compute the reconstructed binary image. By combining the set of MS and the set of LS, we can obtain a binary image in the same manner as in Steps 3.2 and 3.3 in the shares construction phase, the rest of the last bit that can be chosen randomly when k>0. ⎡001111 110000 ⎤ . 00110 ⎥⎦ ⎣

x={MS}{LS}= ⎢ 100111

It means we must randomly add 2 bits into the end of each pixel as follows: ⎡00111101 11000000 ⎤ x= ⎢ ⎥. ⎣10011100 00011000 ⎦

Step 5: Transform the reconstructed binary image into a grayscale image. Group each set of 8 bits into a set and convert into decimal presentations. The reconstructed grayscale image is shown in reI= ⎡⎢ 61 192 ⎤⎥ . Note that the pixel values in 156 24 ⎣

⎦

reI are very close to those in the original grayscale image I = ⎡⎢124 200 ⎤⎥ . ⎣160

61 ⎦

IV. EXPERIMENTAL RESULTS Generally, four criteria were used to evaluate the proposed scheme: security, accuracy, computation complexity and pixel expansion. To conduct our experiments, the four 128×128 grayscale images shown in Figure 11, “Lena”, “Peppers”, “Jet”, and “Baboon” served as the test images. The proposed scheme and Wang et al.’s scheme were programmed by Matlab 7.0. The experiments were conducted on a PC with an Intel(R) Core™2 CPU with 1.83 GHz and 1-GB RAM. The operating system is Windows XP Professional. The image quality of the reconstructed grayscale image in our proposed scheme depends on four parameters: k, t, m and n. Because each pixel in the reconstructed image is generated by combining (2×m+1) bits of the MS and (8-kt) bits of the LS, we can establish the relationship among these parameters as 2×m×t≤k and k≤4. To conduct our experiments, a set of possible values for these parameters is defined as: (k, m, t) = {(2, 1, 1); (4, 1, 1); (4, 2, 1); (4, 1, 2)}.

(c) Jet

(d) Lena

Figure 11. Four 128×128 grayscale test images.

Computation complexity evaluation Let O be the number of operators performed in Wang et al.’s (2, n) ProbVSS scheme. Instead of executing 8×O operators when we simply apply Wang et al.’s scheme to grayscale images, our scheme requires only 6×O≤ (8+2×m×t-k)×O≤8×O operators. Thus, the computational complexity of our scheme is less than that of pure Wang et al.’s scheme when the secret image is a grayscale image. Pixel expansion Basically, because our scheme is probabilistic VSS, each generated shadow is a random-like grayscale image. In addition, the size of each generated shadow is the same as for the original grayscale image. Therefore, our scheme does not have the pixel expansion problem. Accuracy Peak signal-to-noise ratio (PSNR) is used in the following experiments to evaluate the image quality of the reconstructed secret grayscale image using the proposed secret sharing scheme. PSNR is defined as follows: PSNR = 10 log 10

255 2 dB , MSE

(2)

where MSE is the mean square error between the original image and the reconstructed one. For an original image size of w × h, the formula for MSE is as follows:

1 ∑ ∑(x − y ) , w× h w

MSE =

h

i =1 j =1

2

ij

ij

(3)

where xij and yij are the pixel values of the original image and the reconstructed one, respectively. A higher PSNR means that the quality of the reconstructed secret grayscale image is similar to that of the original image. Table 1 shows the image quality and computation times for the proposed scheme with different parameters and Table 2 compares our proposed scheme and pure Wang et al.’s scheme with respect to the image quality of the reconstructed image.

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TABLE I. IMAGE QUALITY AND COMPUTATION TIME OF RECONSTRUCTED IMAGES WITH DIFFERENT PARAMETERS

Images

n

b

k

t

2×m +1

3 3 3 10 10 10 15 15 15 3 3 3 10 10 10 15 15 15 3 3 3 3 3 3 10 10 10 3 3 3 3 3 3 15 15 15

4 4 4 4 4 2 2 2 2 2 2 2 4 4 2 2 2 2 4 4 4 2 2 2 4 4 2 4 4 4 2 2 2 2 2 2

2 4 4 2 4 4 2 4 4 2 4 4 2 4 4 2 4 4 2 4 4 2 4 4 2 4 4 2 4 4 2 4 4 2 4 4

1 2 1 1 2 2 1 1 2 1 2 1 1 2 2 1 1 2 1 2 1 1 2 1 1 2 2 1 2 1 1 2 1 1 1 2

3 3 5 3 3 3 3 5 3 3 3 3 3 3 3 3 5 3 3 3 5 3 3 3 3 3 3 3 3 5 3 3 3 3 5 3

PSNR (dB)

Computation

25.32 3 26.25 6 26.48 5 25.19 5 26.82 3 28.13 9 25.83 7 26.83 4 30.43 0 25.37 5 27.35 6 25.78 5 25.04 5 26.20 2 27.82 1 25.43 2 26.89 8 28.38 7 25.12 4 26.53 9 25.88 7 25.21 9 26.56 4 26.32 9 25.49 5 27.67 7 28.71 0 25.40 9 25.53 9 26.14 2 25.24 3 25.47 7 25.43 6 26.57 1 27.38 3 30.19 3

4.5780 4.6840 4.5950 10.3290 11.0260 14.4530 25.7530 26.6560 33.7850 8.6250 9.6140 7.1250 8.9650 11.3410 14.1290 27.5320 26.7150 33.3280 5.102 6.587 5.632 7.948 9.328 8.142 9.875 11.457 16.046 5.129 6.436 6.045 8.129 8.437 9.527 26.656 26.715 31.374

time (s)

From Table I, we can see that the quality of the reconstructed image is better when the value of the following parameters is increased: number of shadows n, number of matrices applying the voting method t and the number of times to repeatedly generate shadows (2×m+1). However, the increased n, t and (2×m+1) will increase computation time for generating shadows. Therefore, there is a tradeoff between the image quality of the reconstructed grayscale image and the required computation time. From Table II, we can see that the quality of the reconstructed image using our scheme is better than that using Wang et al.’s scheme. The difference between PSNRs generated with our scheme and one with pure Wang et al.’s scheme is ranged from 1 dB to 1.5 dB.

TABLE II. IMAGE QUALITY OF RECONSTRUCTED IMAGES WITH THE PROPOSED SCHEME AND PURE WANG ET AL.’S SCHEME WHEN N=15, T=2, K=4, M=1 Our reconstructed image Pure Wang et al.’s reconstructed image Reconstructed image

PSNR

Reconstructed image

PSNR

(dB)

(dB)

27.365

28.415

27.716

29.067

27.476

28.138

27.783

29.212

Security To prove that each shadow reveals no information about the secret grayscale image even when each shadow is grayscale image, a set of shadows generated by our scheme for test images “Baboon”, “Jet”, “Peppers” and “Lena” when the number of shadows is set as 5 is demonstrated below. TABLE III. CORRESPONDING SHADOWS FOR EACH TEST IMAGE WITH OUR SCHEME WHEN N=5 Images

Shadow1

Shadow2

Shadow3

Shadow4

Shadow5


Note that each shadow is a random-like grayscale image; therefore, the security of the proposed scheme is guaranteed. V. CONCLUSION In this paper, we extend the application of Wang et al.’s (2, n) ProbVSS scheme, which was designed for binary images, into the domain of grayscale images. To maintain the advantages of Wang et al.’s scheme: low computation cost and no pixel expansion problem, we propose two strategies in combination with Wang et al.’s scheme: the voting method and the least significant bits abandoning approach. Experimental results confirm that our proposed scheme guarantees the four requirements of the secret sharing problem. The computational complexity of our proposed scheme can be successfully reduced about half of that required with the pure Wang et al. scheme on grayscale images when setting parameters (k, m, t) = (4, 1, 2). Moreover, on average, the PSNR of the reconstructed image in our scheme is greater by about 1 dB than that in Wang et al.’s scheme. In short, our scheme provides higher quality reconstructed images with less complexity than that can be achieved with the pure Wang et al.’s scheme. ACKNOWLEDGMENT This work was supported in part by a grant from National Science Council of Taiwan (96-2416-H-126008-MY1.) REFERENCES [1] G. R. Blakley, “Safeguarding Cryptographic Keys,” Proceedings of the National Computer Conference, American Federation of Information Processing Societies Proceedings, New York, USA, pp. 313-317, June, 1979. [2] A. Shamir, “How to Share a Secret,” Communications of ACM, Vol. 22, No. 11, pp. 612-613, 1979. [3] M.Naor and A.Shamir, “Visual Cryptography,” Advances in Cryptology - Eurocrypt’94, LNCS, Vol. 950, pp. 1-12, 1995. [4] E. R. Verheu and H. C. A Van Tilbor, “Constructions and Properties of k out of n Visual Secret Sharing Schemes, “ Designs, Codes and Cryptography, Vol. 11, pp. 179-196, 1997. [5] R. Ito, H. Kuwakado and H. Tanaka, “Image Size Invariant Visual Cryptography,” IEICE Trans. Fundamentals, E82A, No. 10, pp. 2172-2177, 1999. [6] Innes Muecke, “Greyscale and Colour Visual Cryptography,” Thesis of degree of Master of Computer Science, Dalhouse University – Daltech, 1999. [7] C. Blude, A. De Santis and M.Naor, “Visual Cryptography for Grey Level Images,” Information Processing Letters, Vol. 27, pp. 255-259, 2000. [8] C. N. Yang and C. S. Laih, “New Colored Visual Secret Sharing Schemes,” Designs, Codes and Cryptography, Vol. 20, pp. 325-335, 2000. [9] A. Adhikari and S. Sikdar, “A New (2, n)-Visual Threshold Scheme for Color Images,” INDOCRYPT 2003, Lecture Notes in Computer Science, Vol. 2904, Springer, Berlin, pp. 148-161, 2003.

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[10] C. N. Yang, “New Visual Secret Sharing Schemes Using Probabilistic Method,” Pattern Recognition Letters, Vol. 25, Issue 4, pp. 481-494, 2004. [11] S. J. Shyu, “Efficient Visual Secret Sharing Scheme for Color Images,” Pattern Recognition, Vol. 39, Issue 5, pp. 866-880, May 2006. [12] S. Cimato, R. De Prisco and A. De Santis, “Probabilistic Visual Cryptography Schemes,” The Compututer Journal, Vol. 49, No. 1, pp. 97-107, 2006. [13] DaoshunWang, Lei Zhang, Ning Ma and Xiaobo Li, “Two Secret Sharing Schemes Based on Boolean Operations,” Pattern Recognition, Vol. 40, pp. 2776-2785, 2007. [14] Y. F. Chen, Y. K. Chan, C. C. Huang, C. C. Tsai, M. H. and Y. P. Chu, “A Multiple-Level Visual Secret-Sharing Scheme Without Image Size Expansion,” Information Sciences, Vol. 177, No. 21, pp. 4696-4710, Nov. 2007.

CHIN-CHEN CHANG received his B.S. degree in Applied Mathematics in 1977 and his M.S. degree in Computer and Decision Sciences in 1979 from the National Tsing Hua University, Hsinchu, Taiwan. He received his Ph.d. in Computer Engineering in 1982 from National Chiao Tung University, Hsinchu, Taiwan. Since February 2005, he has been a Chair Professor of Feng Chia University. In addition, Dr. Chang is a Fellow of IEEE and a Fellow of IEE. He is also a member of the Chinese Language Computer Society, the Chinese Institute of Engineers of the Republic of China, and the Phi Tau Phi Society of the Republic of China. His research interests include database design, computer cryptography, and data compression.

CHIA-CHEN LIN: received her B.S. degree in information management in 1992 from the Tamkang University, Taipei, Taiwan. She received both her M.S. degree in information management in 1994 and Ph.D. degree in information management in 1998 from the National Chiao Tung University, Hsinchu, Taiwan. Dr. Lin is currently a professor of the Department of Computer Science and Information Management, Providence University, Sha-Lu, Taiwan. Her research interests include image and signal processing, image hiding, mobile agent, and electronic commerce.

T.HOANG NGAN LE: received her Bachelor degree in Information Technology in 2005. From Sep.2006 to Jan.2008, she was studying as Master student at University of Natural Sciences, HCM City, Vietnam. From Jan.2008 up to now, she is studying and researching in MSN Lab, FengChia University, Taichung, Taiwan. Her current research interests include image processing, multimedia security.