Progressive secret image sharing scheme using meaningful shadows
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1 SECURITY AND COMMUNICATION NETWORKS Security Comm. Networks 2016; 9: Published online 30 August 2016 in Wiley Online Library (wileyonlinelibrary.com) RESEARCH ARTICLE Progressive secret image sharing scheme using meaningful shadows Zhi-Hui Wang 1, Ya-Feng Di 1, Jianjun Li 2 *, Chin-Chen Chang 3 and Hui Liu 1 1 School of Software, Dalian University of Technology, Dalian, China 2 School of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou, China 3 Department of Information Engineering and Computer Science, Feng Chia University, Taichung, Taiwan ABSTRACT This paper proposes a novel secret image sharing scheme, which progressively hides a secret image into multiple different meaningful cover (or host) images by utilizing a magic matrix. The produced shadows are high visual quality, meaningful images that differ from each other. As a result, they are not easy to cause the suspicion by the attackers. Moreover, the secret image in the proposed scheme can be recovered progressively via different numbers of shadows. The more shadows used, the better the quality of the secret image. The experimental results demonstrate the aforementioned advantages of the proposed scheme. Copyright 2016 John Wiley & Sons, Ltd. KEYWORDS secret image sharing; progressive secret image sharing; Sudoku; magic matrix *Correspondence Jianjun Li, School of Computer Science and Technology, Hangzhou Dianzi University, Hangzhou, China. lijjcan@gmail.com 1. INTRODUCTION We propose a novel secret image sharing scheme to solve existing problems such as meaningless shadow images, distorted recovery secret image, and stolen shadow image by utilizing the concept of magic matrix. Magic matrix is a number placement puzzle based on logic, and it is represented on a square grid. In this scheme, we can generate the high visual quality, meaningful shadow images that are quite important in medical, military, and artistic fields. In addition, our scheme allows authorized participants to cooperate to recover the secret image reversibly and progressively. The rest of the article is organized as follows. Definitions and the current status of the development of secret image sharing is discussed in Section 2. Section 3 introduces Sudoku and Shamir s(t, n)-threshold scheme briefly. Section 4 elaborates the proposed scheme, consisting of both shadow image generation and secret image recovery, followed by the experimental results in Section 5. Last, Section 6 gives a conclusion of the paper. 2. BACKGROUND With the rapid development of the Internet, a growing number of multimedia works appear in our lives, such as videos, audios, and images. To preserve these media, many methods have been proposed, such as steganography [1 3] and secret sharing scheme [4 7]. Sharing secret data among several involved participants is the purpose of the secret sharing mechanism [8,9]. After dividing the secret data into several parts, distributing them to paticipants, and a certain number of authorized participants can recover the secret data by cooperating. In 1979, Blakely and Shamir [8,9] first proposed the idea of a secret sharing system, the (t, n)-threshold secret sharing system. In this system, the dealer divides the secret data into n shares and then distributes to n participants. Only any t or more participants cooperating with their shares can recover the secret data. Later, Thien and Lin [10] extended the idea into the image field where they regarded the image as the secret data and generated n shadow images. The original secret image can be recovered by any t out of n shadows. Based on the scheme, Wang and Shyu [11] proposed a secret image sharing scheme, which is scalable and has three traits: the multi-secret, the priority, and the progressive. The shares distributed to the participants had different priorities, which determined the quality of the recovered secret image. Nevertheless, the shadow images had no meaning that would attract an intruder s attention. To obtain meaningful shadows, embedding the secret image into a host image was proposed. Lin and Tsai [12] and Wu et al. [13] used the Copyright 2016 John Wiley & Sons, Ltd. 4075
2 Progressive secret image sharing scheme using meaningful shadows Z.-H. Wang et al. (t 1)-degree polynomial to embed the secret image into the host image to acquire meaningful shadow images [14]. However, in this way, the recovered secret image is distorted in that pixel values larger than 250 are truncated. For recovering a lossless secret image, Zhao et al. [15] and Chang et al. [16] proposed using two pixels [17] on behalf of exceeding values, but the visual quality of shadow images was contorted to some degree. To avoid the risk of shadow images being stolen, Chen and Lin [18] proposed a progressive secret image sharing scheme. They divided the secret image into several shares with equal importance to avoid information disclosure in cases in which share is lost, and the secret image is transmitted using Figure 1. Instance of a 9 9 Sudoku grid using digits from 0 to 8. Figure 3. A Sudoku with digits from 0 to 8. Figure 2. Instance of matrix Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd.
3 Z.-H. Wang et al. Progressive secret image sharing scheme using meaningful shadows several channels to avoid the secret image being stolen. The advantage is that the quality of the recovered secret image depends on the number of participating shadow images, that is, the process of recovery is progressive. However, if the visual quality of shadow images is not good enough, it can attract an intruder s attention in transmission over the Internet. 3. THE RELATED WORK In this section, we introduce the concept of Sudoku, a magic matrix [19 22] adopted for embedding and recovering secret images in the proposed scheme. Subsequently, a concrete magic matrix example and Shamir s (t, n)-threshold sharing mechanism [8,9] are illustrated, respectively. Figure 4. Instance of matrix M. Figure 5. Instance of the embedding result. Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd. 4077
4 Progressive secret image sharing scheme using meaningful shadows Z.-H. Wang et al Sudoku Sudoku, originally called Number Placement, is a logicbased puzzle presented in a square grid. The most common Sudoku has a 9 9 grid with nine 3 3 sub-grids (also called blocks). In addition to 9 9 Sudoku, 4 4 and Sudoku are also significant. Figure 1(a) shows a 9 9 Sudoku that has not been completed, and the goal is to complete the grids under the following rules: digits in each row, column, and 3 3 block must be ranged from 0 to 8 and cannot duplicate. Figure 1(b) gives one solution of the incomplete Sudoku shown in Figure 1(a) Magic matrix Many kinds of magic matrix exist; for instance, Kim s magic matrix is filled with digits 0 to 8 and Zhang and Wang s matrix is filled with digits 0 to 4 [23]. Different magic matrixes can bring different experimental results, such as different numbers of shadow images and different quality of shadow images. In our proposed scheme, the size of the magic matrix M is , and the content is composed using Sudoku, assumed as stu, the size of which is N 2 N 2. The method of magic matrix construction is shown as follows: For i ¼ 0 to 255 For j ¼ 0 to 255 M½Šj i ¼ stu i mod N 2 j mod N 2 ; where N ¼ 2; 3; 4; 16 The magic matrix is a composition involving a large number of Sudokus and, as a result, it has all the characteristics of Sudoku. Figure 2 shows part of a magic matrix of size , which is one of the magic matrixes used in our experiments. The Sudoku size in the magic matrix is 9 9, where N =3. authorized participants can recover all the coefficients s, m 1, m 2,, m t 1 by constructing F(x). So the secret can be obtained through the participants cooperation. In contrast, if the number of participants is less than t, the shadow containing these participants is a forbidden set, that is, it cannot reconstruct the F(x) by LIP. One thing to note is that the values of y i sare limited within [0, p 1], and because the coefficients m 1, m 2,, m t 1 are randomly determined, the shadows appear meaningless. To prevent intruders from being suspicious about the shadows, the proposed scheme camouflages the shadows with host images to generate n meaningful shadow images. 4. OUR PROPOSED METHOD Given a secret image, the dealer is responsible for preparing n cover images and then generates n shadow images with the secret image. The shadow-generating procedure is introduced in Subsection 4.1. Using different number of shadow images, secret images with different visual quality can be reconstructed by the participants, that is, the more shadow images are used, the better quality of the reconstructed secret image has. The process of reconstructing is described in Subsection 4.2. To describe the scheme clearly, an example of the two procedures is introduced in Subsection The process of shadow generation The proposed scheme uses grayscale images as test images. The pixel value is ranged from 0 to 255. The small 3.3. The (t, n)-threshold secret sharing system If a secret s is going to be shared, the dealer sets a prime p and generates a (t 1)-degree polynomialf(x)as Fx ðþ¼ s þ m 1 x þ þ m t 1 x t 1 mod p (1) Here, the coefficients m 1, m 2,, m t 1 are randomly determined within integers [0, p 1]. Then n shadows (y 1, y 2,, y n ) can be derived as y 1 ¼ Fð1Þ; y 2 ¼ Fð2Þ; ; y n ¼ Fn ðþ (2) Then, the shadows are distributed to the participants. In general, if the authorized participants have at least t shadows, we call the shadow set a qualified set, which means that these participants in the qualified set can reconstruct F(x) using the Lagrange interpolation polynomial (LIP) with cooperation. In other words, these Figure 6. Instance of the embedding result Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd.
5 Z.-H. Wang et al. Progressive secret image sharing scheme using meaningful shadows change of pixel value is hard to observe with the human vision system, so we can change the value of every pixel slightly to embed the secret image. In this paper, we assume that the secret image S has pixels, as do the cover images and shadow images. Then, we choose a Sudoku to generate a magic matrix called M whose size is In the proposed scheme, the dealer first prepares host images and a secret image. Assuming the size of the Sudoku we use is N 2 N 2, dlog N 2256eþ 1host images are used in the proposed scheme. The process of shadow generation is as follows: Step 1: Divide the array [0,255] into N 2 average parts with the last part allowed to contain Figure 7. Three groups of host images. Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd. 4079
6 Progressive secret image sharing scheme using meaningful shadows Z.-H. Wang et al. more numbers; that j is, k each of the first N 2 1parts contains 256 numbers and then 2 N 2 th part contains the rest of the numbers. Then compute the average value of every part A i,as Avg i, where i=1, 2,, N 2. Step 2: Take out the first pixel from the secret image, which we assume is a and a A i ; that is, a belongs to the i th part. Then, in the same way, take out the first pixel in both host 1 and host 2, which we assume are b and c, respectively. Then we regard b and c as the horizontal and vertical coordinates to locate the corresponding N N block in the matrix M. Next, we find number i, which is also the part number of a, in the located block and record the abscissa value and ordinate value of i as b 1 and c 1, respectively. Then b 1 is put into shadow 1 and c 1 is put into shadow 2. Also, the average value of part A i is put into the corresponding position of a new secret imagesecret. Repeat Step 2 until all pixels in the secret image and the two host images are processed. Then, we have three new pictures, shadow 1, shadow 2, and secret. Step 3: Use secret and the new picture host 3 to generate shadow 3. First is to divide the aforementioned calculated part A i in Step 2 into N 2 parts and to calculate the average value Avg ij of every new part A ij, where j=1, 2,, N 2. Assuming that the first secret pixel a belongs to part A ij, put the average value of this part into the first position of a new image secret. Next, take out the first pixel from image host 3 and the first pixel value from secret, which is the average of part A i. Then locate a 1 N 2 block from the matrix M, which contains the unit with the aforementioned two values as ordinate and abscissa values, respectively. We find the part number of A ij in the located 1 N 2 block and put its ordinate value in a new image shadow 3. The reason that we use the 1 N 2 block instead of the N N block is because the abscissa is not supposed to be changed The process of reconstructing secret images using shadow images The process of reconstructing is the reverse of the process of producing shadows but very similar to it. The process is as follows: Step 1: Divide array [0, 255] into N 2 average parts as in Step 1 in the process of shadow generation. Then take out the first pixel in both the shadow 1 and shadow 2, which we assume are p and q, respectively. Then mapping p and q at matrix M, we acquire an integer called x. Put the value Avg x, which is the average of the array with the secret data, into image S 1, and S 1 contains the majority of the secret image. Step 2: Divide the array A x into N 2 parts, the average value of which is Avg xj, where j =1, 2,, N 2. Then take the first pixel of shadow 3, which we assume is m. Mapping Avg x and m at matrix M, we obtain an integer y. Put the value Avg xy, which is the average of the array with the secret data after dividing into the image S 2, and S 2 contains more secret image data than S 1. Step 3: Divide the array A xy into N 2 parts, and the averages of them are Avg xyl, where l = 1,2,, N 2. Then take the first pixel in shadow 4, which we assume is n. Mapping Avg xy and n at matrix M, we obtain a digit z. Put the value Avg xyz, which is the average of the array with the secret data, after dividing into the image S 3, and S 3 contains more secret image data than S 2. Step 4: Do as in Step 3 until the last shadow image is processed An example of shadow generation and secret image reconstruction In the example, we use 9 9 Sudoku, which is shown in Figure 3. A part of the matrix generated by the Sudoku is shown in Figure 4. The detailed description of the embedding procedure and the reconstructing procedure is described in Subsections and 4.3.2, respectively. Repeat Step 3 until all pixels insecret and host 3 are processed. Then, we have two new pictures, shadow 3 and secret. Step 4: Generate shadow 4 with image host 4 and secret in the same way as in Step 3. The rest of the shadow images can also be generated by following the same method until we have dlog N 2256eþ 1 images. Figure 8. An example of secret image Airplane Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd.
7 Z.-H. Wang et al. Progressive secret image sharing scheme using meaningful shadows Example of shadow generation. Step 1: Divide the array [0, 255] into nine parts. Each of the first eight arrays contains 28 digits and the last array contains the rest of the digits. Thus, the arrays are A 1 = [0, 27], A 2 = [28, 55],, A 9 = [224, 255], and the averages of all the parts are Avg 1 = 14, Avg 2 = 42,, Avg 9 = 240. Step 2: Take out the first pixel from the secret image and assume its value is 36. We obtain both first pixels in Baboon and Cameraman, which are assumed to be a = 36 and b = 4. Referring to a and b, we can acquire a block, and there must be a number 2, which is the segment to which the secret number 36 belongs. Then put a and b, which are the coordinates of the located number 2, into shadow 1 and shadow 2. Last, put the average of the second part, that is, Avg 2 = 42, into secret. Figure 5 shows the process. Figure 9. The three groups of shadow images using 9 9 Sudoku. Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd. 4081
8 Progressive secret image sharing scheme using meaningful shadows Z.-H. Wang et al. Figure 10. The three groups of shadow images using 4 4 Sudoku Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd.
9 Z.-H. Wang et al. Progressive secret image sharing scheme using meaningful shadows Step 3: Continue to divide the second part into nine parts, which are A 21 = [28, 30], A 22 = [31, 33],, A 29 = [52, 55]. Because we have the average of part A 2, we should use the average as the abscissa, and the ordinate c is from the third host image to locate the block shown in Figure 6(a). Then find the new segment number 3, which is the number of the segment to which the secret 36 belongs, in the located block. Then the ordinate number c of number 3 is recorded into the shadow 3 and the average of A 23 is put into secret. Step 4: Continue to divide part A 33 into nine parts. We cannot divide averagely, so we add the same digits as the last digit in the array; that is, we deal with the nine parts like this: [34] [35] [36] [36] [36] [36] [36] [36] [36]. According to Figure 6(b), we put d = 87 into shadow PSNR ¼ 10 log 10 db MSE The mean square error (MSE) of an image with H W pixels is defined as MSE ¼ 1 HW H W u¼1 v¼1 ðp uv p uv Þ 2 where p uv is the original pixel value and p uv is the pixel value of the shadow image Shadow images generated with secret and host images To test the performance of the proposed scheme, we make a comparison by using 9 9 Sudoku and 4 4 Sudoku, the block size of which is 3 3 and 2 2, respectively Example of secret image reconstruction. Following the example of the shadow generation phase, an example of the reconstruction process is illustrated as follows. Step 1: Acquire the first pixel from shadow 1 and shadow 2, which are a = 37 and b = 3. Referring to the matrix, a and b are treated as the horizontal and vertical coordinates, respectively, and we acquire the integer 2 then put the average of the second part, that is, 42, intosecret, as shown in Figure 5. Step 2: Acquire the first pixel from shadow 3, which is c = 52. Referring to the matrix, treat 42 and c as the horizontal and vertical coordinates, respectively, and we acquire the integer 3. Then put the average of the third part, that is, 35, intosecret, as shown in Figure 6(a). Step 3: Acquire the first pixel from shadow 4, which is d. Referring to the matrix, we acquire the integer 3. Then put the average of the third part, that is, 36, intosecret, as shown in Figure 6(b), and secret is the recovered secret image that we want. Table I. The peak signal-to-noise rate (PSNR) value (db) of the shadow images with Group 1. Shadow images PSNR(dB) (with 9 9 Sudoku) PSNR(dB) (with 4 4 Sudoku) Shadow Shadow Shadow Shadow Shadow Average Table II. The peak signal-to-noise rate (PSNR) value (db) of the shadow images with Group 2. Shadow images PSNR(dB) (with 9 9 Sudoku) PSNR(dB) (with 4 4 Sudoku) Shadow Shadow Shadow Shadow Shadow Average EXPERIMENTAL RESULTS We demonstrate the performance of the proposed scheme using 12 grayscale cover images with pixels, as shown in Figure 7. Figure 7 shows three example groups of host images, and every group contains four images, that is, images (a d) belong to group 1, images (e h) belong to group 2, and images (i l) belong to group 3. An example of a secret image is shown in Figure 8. To estimate the quality of the shadow images, we use the peak signal-tonoise rate (PSNR): Table III. The peak signal-to-noise rate (PSNR) value (db) of the shadow images with Group 3. Shadow images PSNR(dB) (with 9 9 Sudoku) PSNR(dB) (with 4 4 Sudoku) Shadow Shadow Shadow Shadow Shadow Average Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd. 4083
10 Progressive secret image sharing scheme using meaningful shadows Z.-H. Wang et al. Figure 11. The test images in Chen and Lin s scheme. Figure 12. The shadow images in our scheme using 9 9 Sudoku Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd.
11 Z.-H. Wang et al. Progressive secret image sharing scheme using meaningful shadows Figure 9 shows the three groups of shadow images with 9 9 Sudoku. Figure 10 shows the three groups of shadow images with the 4 4 Sudoku. The value of the PSNR of the shadow images is shown in Tables I III. We use the notation shadow s.t to represent shadow t in the group s. From Figures 9 and 10, we can see that the distortion is imperceptible with either the 9 9 Sudoku or the 4 4 Sudoku. That is, our proposed scheme can successfully camouflage shadows from intruders. From Tables I III, we can observe that using 4 4 Sudoku achieves higher PSNR than 9 9 Sudoku because the size of the Sudoku matters. As to the experiments with 4 4 Sudoku, the variation range of shadow 1 and shadow 2 is [ 1, 1] and the variation range of the other shadow images is [ 3, 3]. With respect to the experiments with 9 9 Sudoku, the variation ranges of pixels in shadow 1 and shadow 2 are [ 2, 2] and the variation ranges of the other shadow images are [ 8, 8]. In addition, to evaluate the performance of the proposed scheme, we compare the experimental results with Chen and Lin s [18] scheme. The test images used in Chen and Lin s scheme are shown in Figure 11. We use the same test images to make a fair comparison. In the experiment, we test the proposed scheme with 9 9 Sudoku and 4 4 Sudoku again, and the generated shadow images are shown in Figures 12 and 13. Table IV shows the PSNRs of the shadow images from the proposed scheme and from Chen and Lin s. Obviously, the proposed scheme achieves higher PSNR of the same shadow images. In every table, we can see that shadow images like shadow 1 and shadow 2 have a higher value of PSNR, while shadow 3, shadow 4, and other shadow images have a lower Table IV. The contrast of shadow images peak signal-to-noise rate in two schemes. 9 9 Sudoku (db) 4 4 Sudoku (db) value. The reason is that the variation ranges of the pixel values after the embedding in shadow 1 and shadow 2 are [ (N 1), N 1], while in the rest of the shadows, they are [ (N 2 1), N 2 1] Recovered images using shadow images progressively Chen and Lin s scheme (db) Boat Airplane Man Fish Fruit After the process of reconstruction, we obtain the secret image using the shadow images. Sudoku with a 3 3 block using the images is shown in Figure 7. Figure 14 demonstrates one group of recovered images. Image (a) is recovered with two shadow images and image (b) is recovered with three shadow images, while image (c) is recovered with four shadow images and image (d) is the original secret image. Figure 15 shows the progressively recovered secret images with 4 4 Sudoku using the images in Chen and Lin s scheme, as shown in Figure 11. We can figure that by using more shadow images, the recovered image is closer to the original image. When using all the shadow Figure 13. The shadow images in our scheme using 4 4 Sudoku. Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd. 4085
12 Progressive secret image sharing scheme using meaningful shadows Z.-H. Wang et al. Figure 14. The recovered images and the secret image. Figure 15. The recovered images Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd.
13 Z.-H. Wang et al. Progressive secret image sharing scheme using meaningful shadows Table V. Peak signal-to-noise rates using different numbers of shadow images. Numbers of shadow images images, the recovered image has no distortion. The PSNRs of the recovered images are shown in Table V. One can see that by using the same number of shadow images, the results with 3 3 Sudoku are better than with 4 4 Sudoku because using 9 9 Sudoku can embed more secret data than 4 4 Sudoku does, so with the same number of shadow images, the 9 9 Sudoku can reconstruct more secret data. 6. CONCLUSIONS In this paper, a progressive secret image sharing scheme with meaningful shadows is proposed. We embed secret images into host images using a magic matrix generated by different sizes of Sudoku and simultaneously obtain different numbers of shadow images. If the size of the Sudoku is smaller, the quality of the generated shadow images is more satisfactory, which can reduce the malicious intruder s attention when the image is transmitted over the Internet. In addition, the quality of the reconstructed secret image is proportional to the number of shadow images; the lossless secret image can be retrieved when all shadow images work together. Also, compared with Chen and Lin s scheme, the proposed scheme achieves a higher PSNR of the same shadow images in the experiments. ACKNOWLEDGEMENTS This work was supported by the National Nature Science Foundation of China under Grant No REFERENCES Chen and Lin s scheme The proposed scheme 4 4 Sudoku (db) 9 9 Sudoku (db) Lossless 5 Lossless Lossless 1. Qin C, Chang CC, Huang YH, Liao LT. An inpainting-assisted reversible steganographic scheme using a histogram shifting mechanism. IEEE Transactions on Circuits and Systems for Video Technology 2013; 23(7): Qin C, Chang CC, Chen YC. Efficient reversible data hiding for VQ-compressed images based on index mapping mechanism. Signal Processing 2013; 93(9): Qin C, Chang CC, Hsu TJ. Reversible data hiding scheme based on exploiting modification direction with two steganographic images. Multimedia Tools and Applications 2015; 74(15): Drăgan CC, Ţiplea FL. Distributive weighted threshold secret sharing schemes. Information Sciences 2016; 339: Hou YC, Quan ZY, Tsai CF. A privilege-based visual secret sharing model. Journal of Visual Communication and Image Representation 2015; 33: Yang CN, Chen CHand Cai SR. Enhanced Booleanbased multi secret image sharing scheme. Journal of Systems and Software 2016; 116: Chang CC, Hwang RJ. Efficient cheater identification method for threshold schemes. IEE Proceedings- Computers and Digital Techniques 1997; 144(1): Blakley GR. Safeguarding cryptographic keys. AFIPS Conference Proceedings 1979; 48: Shamir A. How to share a secret. Communications of the ACM 1979; 22(11): Thien CC, Lin JC. Secret image sharing. Computers & Graphics 2002; 26(5): Wang RZ, Shyu SJ. Scalable secret image sharing. Signal Processing: Image Communication 2007; 22(4): Lin CC, Tsai WH. Secret image sharing with steganography and authentication. Journal of Systems and Software 2004; 73(3): Wu YS, Thien CC, Lin JC. Sharing and hiding secret images with size constraint. Pattern Recognition 2004; 37(7): Tsai CS, Chang CC, Chen TS. Sharing multiple secrets in digital images. Journal of Systems and Software 2002; 64(2): Zhao R, Zhao JJ, Dai F, Zhao FQ. A new image secret sharing scheme to identify cheaters. Computer Standards & Interfaces 2009; 31(1): Chang CC, Hsieh YP, Lin CH. Sharing secrets in stego images with authentication. Pattern Recognition 2008; 41(10): Yang CN, Chen TS, Yu KH, Wang CC. Improvements of image sharing with steganography and authentication. Journal of Systems and Software 2007; 80(7): Chen SK, Lin JC. Fault-tolerant and progressive transmission of images. Pattern Recognition 2005; 38(12): Aaronson L. Sudoku science. IEEE Spectrum 2006; 43(2): Felgenhauer B, Jarvis F. Mathematics of sudoku I. Mathematical Spectrum 2006; 39(1): Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd. 4087
14 Progressive secret image sharing scheme using meaningful shadows Z.-H. Wang et al. 21. Russell E, Jarvis F. Mathematics of sudoku II. Mathematical Spectrum 2006; 39(2): Chang CC, Chou YC, Kieu TD. An information hiding scheme using sudoku. Proceedings of the Third International Conference on Innovative Computing Information and Control, Dalian, 2008; Chang CC, Chen YH, Wang ZH, Li MC. A data embedding scheme based on a magic matrix and wet paper codes. Proceeding of International Conference on Computational Intelligence and Natural Computing 2009; 2: Security Comm. Networks 2016; 9: John Wiley & Sons, Ltd.
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