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Tamper-proof secret image-sharing scheme for identifying cheated secret keys and shared images

[+] Author Affiliations
Chien-Chang Chen

Tamkang University, Department of Computer Science and Information Engineering, Taipei, Taiwan

Chong-An Liu

Tamkang University, Department of Computer Science and Information Engineering, Taipei, Taiwan

J. Electron. Imaging. 22(1), 013008 (Jan 09, 2013). doi:10.1117/1.JEI.22.1.013008
History: Received July 19, 2012; Revised November 15, 2012; Accepted December 11, 2012
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Abstract.  A (t,n) secret image-sharing scheme shares a secret image to n participants, and the t users recover the image. During the recovery procedure of a conventional secret image-sharing scheme, cheaters may use counterfeit secret keys or modified shared images to cheat other users’ secret keys and shared images. A cheated secret key or shared image leads to an incorrect secret image. Unfortunately, the cheater cannot be identified. We present an exponent and modulus-based scheme to provide a tamper-proof secret image-sharing scheme for identifying cheaters on secret keys or shared images. The proposed scheme allows users to securely select their secret key. This assignment can be performed over networks. Modulus results of each shared image is calculated to recognize cheaters of a shared image. Experimental results indicate that the proposed scheme is excellent at identifying cheated secret keys and shared images.

Figures in this Article

Sharing images secretly is essential to protect important images. Conventional (t, n) secret image-sharing methods share one secret image to n shared images, and gathering t shared images recovers the secret image. Thien and Lin1 presented an efficient secret image-sharing scheme by using Shamir method for image sharing and using the Lagrange interpolation method for reconstruction. Many researchers further present functional image-sharing ideas (e.g., reducing load in sharing multiple images,2 progressive,36 weighted,7 visual cryptography and secret image sharing,8,9 scalable,10 and sharing with hiding).11 In addition to the Shamir-Lagrange method, many other methods such as Blakley,12 Boolean,13 and Chinese Remainder Theorem14 are also adopted to share important images secretly.

Although numerous secret image-sharing methods have been proposed, an efficient method of detecting cheaters both in secret key and shared image has not been presented. Currently, the convenience of computer networks allows users to share and recover a secret image over networks easily. However, hackers may use counterfeit secret keys or modified shared image to misappropriate other participants’ authorized secret keys and shared images. Therefore a structure of applying tamper-proof secret image-sharing techniques over computer networks merits the current study. Research has been presented to introduce a method for identifying cheaters. Wu and Wu15 used hash functions to collect shared messages and then generated a large number for verification. Chang and Hwang16 improved the Wu and Wu scheme to increase security by factoring the product of two large prime numbers. Tan et al.17 presented a quadratic residue-based secret sharing scheme. Other researchers further discussed cheaters’ identification approaches to secret image-sharing problems. Chen and Suen18 adopted the Wu and Wu scheme to verify the authenticity of shared images. Zhao et al.19 presented an exponent computation-based secret key verification scheme. Although some works on detecting cheaters in secret image-sharing problems are present, a complete solution for detecting cheaters both in secret keys and shared images is unavailable. Therefore the current study presents a secret image-sharing scheme to efficiently detect cheaters in secret keys and shared images. A significant aspect of the proposed scheme relies on not needing a one-way hash function because a security issues existed in hash functions.17

The rest of this paper is organized as follows. Section 2 reviews important secret image-sharing schemes on detecting cheaters. Section 3 introduces the proposed tamper-proof secret image-sharing scheme. Algorithms of initial procedure, sharing a secret image to shared images, recovering with verification from secret keys and shared images, and security analysis are presented in Secs. 3.13.4, respectively. Section 4 provides experimental results and comparisons between the proposed scheme and other methods. Section 5 offers a conclusion with suggestions for future research.

This section reviews the literatures on cheater detection of secret image-sharing problems. We review the publications of Chen and Suen18 and Zhao et al.19 in Secs. 2.1 and 2.2, respectively.

Review of the Chen and Suen’s Secret Image-Sharing Scheme

Chen and Suen18 adopted the Wu and Wu15 plan—which is based on a one-way hash function h, a selected prime number P, and a calculated large number T—to identify cheaters in a secret sharing scheme. Both sharing and recovering strategies are examined in the study of Chen and Suen (t, n) scheme.

In the sharing procedure, the secret image is shared with n shared images y1, y2,,yn using the Shamir secret sharing method. Then a large number T is calculated and publicly accessed, where T=i=1nh(yi)p2(i1)+i=1n1cp2i1, 1c<p and h() is a hash function.

During recovering procedure, all collected shared images yj* are checked by functions Tj*=h(yj*)p2(j1) and TTj*/p2(j1)(modp)=0 to determine whether yj* is a cheating shared image. Then the Lagrange interpolation method is applied to recover the secret image s, when the number of correct yj* is t.

Review of the Zhao et al.’s Secret Image-Sharing Scheme

Zhao et al.19 applied Thien and Lin’s1 secret image-sharing scheme for sharing a secret image and verified it by modulus calculation. Assume that H is the secret image keeper and Pi (i=1,,n) denotes each participant. Three procedures—initial, sharing, and recovering—are needed in their approach.

During the initial process, the keeper H publishes two parameters {g,n0}. Then each participant Pi selects his or her secret key si and calculates his or her public parameter ri.

When sharing the secret image, the keeper H calculates two parameters r0 and wi, and then chooses two other parameters, {r0,f}, for sharing the image. H then calculates the shared message by hj(wi)=(b0+b1wi++bt1wit1)mod251, where b0, b1,,bt1 are pixel values. Then H publishes {hj(wt)}.

During the recovering with verification procedure, each participant Pi calculates the checked message wi. If wi=wi, H confirms participant Pi by providing a verified secret key, and the secret image can be reconstructed. Without verification, Pi is a confirmed cheater. The correct reconstructed image is then calculated using the Lagrange interpolation method.

In Zhao et al.’s scheme, the accuracy of a shared image relies on its corresponding secret key, rather than checking content of the shared image itself. This creates a gap in the security. Therefore we present a secure secret image-sharing scheme that checks the validity of secret key and the shared image.

This section introduces the proposed (t, n) tamper-proof secret image-sharing scheme. Assume that Pii=1,2,,n, denotes each participant. In Sec. 3.1, the proposed scheme first allows each participant to configure his secret key. Section 3.2 presents a description of the image sharing process. Section 3.3 shows the verification and reconstruction processes for keys and shared images. Section 3.4 analyzes the security of the proposed scheme.

Initial Algorithm

This section uses exponent and modulus computation to determine each participant’s secret key. Steps of initial algorithm are illustrated as follows.

  1. The dealer selects two prime numbers, p0 and q0, and calculates n0=p0×q0.
  2. The dealer selects an integer g0, satisfying gcd (g0,n0)=1 and then publishes {n0,g0}.
  3. Each participant Pi chooses two prime numbers pi and qi, and then calculates their product ni, denoted by ni=pi×qi. Pi chooses another integer gi, satisfying gcd (gi,ni)=1, and then calculates its multiplicative inverse fi, satisfying gi×fi=1mod(pi1)×(qi1).
  4. Pi publishes {gi,ni}.
  5. Participant Pi takes pi as his or her secret key and sends fi and ri to the dealer, where ri=g0pt mod n0.

The dealer should preserve each received ri differently, which means that each Pi possesses different secret key pi, to distinguish participant’s role. This extra step requires participants possessing identical ri, to repeat steps 3 to 5 to obtain new secret key. Furthermore, the prime number pi is the secret key that Pi possesses, whereas the dealer retains ri instead of pi.

Sharing Algorithm

When sharing the image, the dealer should first calculate new key wi for each participant. The following algorithm illustrates the steps taken during the sharing process.

  1. The dealer randomly selects an integer s0[2,n0], satisfying gcd [s0,(p01)]=1 and gcd [s0,(p01)]=1.
  2. The dealer computes r0 and wi, where r0=g0s0modn0, wi=ris0modn0.
  3. The dealer sends r0 to each participant Pi.
  4. The dealer partitions the secret image to blocks of t pixels, where Bkk=1,,r denotes each partitioned block and r is the block number. The dealer then applies to each block Bk the following steps.
    1. Replace Bk by BkRk, where Rk is a random block and denotes Exclusive-OR operation.
    2. It constructs a polynomial function fk(x)=(b0+b1x++bt1xt1)mod251, where b0, b1,,bt1 represent t pixels in one Bk block.
    3. The dealer calculates yi,k=fk(wi), where wi (i=1,2,,n) is obtained from step 2.
    4. The dealer calculates xi,k=yi,kgimodni for the shared image belonging to participant Pi.
    5. The dealer randomly selects a prime number c and computes hi,k=cyi,kmodn, where n is a number defined by larger than number of participants and mod (c,n)=1.
    6. The dealer randomly selects a prime number a and computes Tk=i=1nhi,kai, with a>n.
  5. The dealer sends shared image Xi, which is formed from xi,k (k=1,2,,r), to participants Pi and publishes {Tk}.

Since in conventional Shamir-Lagrange method, a prime number is needed and the number is determined by 251 in the proposed scheme. Therefore all parameters bi in step 4.2 must be restricted between 0 and 250. However, largest pixel value is 255. Consequently, this gap can be solved by Thien and Lin’s method.1 For an image pixel g, g will be partitioned to two numbers 250 and g-250 if 250g255. Two numbers 250 and g-250 represent two parameters bi in step 4.2.

Recovering with Verification Algorithm

This section presents the verification algorithm that accompanies the image recovery process. First the dealer verifies the authenticity of each participant’s possessing key pi and shared image Xi. Then the dealer uses these secret keys and shared images to reconstruct the secret image. The following is an algorithm for recovering with verification.

  1. The dealer acquires the participant’s shared image Xi to calculate the original shared message yi,j by yi,j=xi,jfimodni, where xi,j is the j’th number in Xi.
  2. The dealer employs the participant Pi’s secret key pi and yi,j to verify Pi’s authenticity by checking whether wi is equal to wi (wi=r0pimodn0) and whether hi,j (hi,j=cyi,jmodn) is equal to hi,k (hi,k=Tk/aimoda).
  3. When all participants are authenticated, the following Lagrange interpolation method on each set of secret keys and shared messages (wi,yi,j) is calculated by Display Formula
    fk(x)=i=1tyi,jj=1txwjwiwjmod251=(b0+b1x++bt1xt1)mod251,
    where coefficients b0, b1,,bt1 represent pixels of one secret image block Bk.
  4. Replace Bk by BkRk, where Rk is the random block used in sharing algorithm.
  5. Combine all Bk blocks to acquire the reconstructed secret image.

Note that step 3 is performed when all keys pi and shared images Xi are verified.

Security Analysis

This section analyzes the security of the proposed tamper-proof secret image-sharing scheme. First we will check whether any cheated modification on secret key or shared image can be well detected. Then, since the proposed scheme adopts exponent and modulus computation, we also analyze the common modulus attack in this section.

The secret key is verified by exponential computation. In step 2 of the recovering with verification algorithm, wi=r0pi, where r0=g0s0 is defined in step 2 of the sharing algorithm. Therefore wi=r0pi=(g0s0)pi=(g0pi)s0=ris0=wi, since ri=g0pi, as defined in step 5 of the initial algorithm. Consequently, the accuracy of the proposed verification procedure is proved. For any cheated secret key as defined by replacing pi by pi satisfying pipi, the verification becomes checking whether cheated wi=r0pi=(g0s0)pi=(g0pi)s0 and wi=ris0=(g0pi)s0 are the same. Note that all these computations are calculated under mod n0. This equivalence verification can be described as checking whether (g0pimodn0) is equivalent to (g0pimodn0). Thus a participant Pi can choose another secret key pi satisfying g0pi=g0pimodn0. However, when n0 is a very large number, pi is hard to be found.20 Moreover, an attacker can only cheat ri and acquire g0 and n0 over networks. He has to find pi from ri=g0pimodn0 and it’s also a hard work when n0 is a very large number. Since a pi satisfying the equivalence verification is hard to be found, any cheated secret key will always be detected.

The shared image is verified by public data Tk and exponent computation. For an attacker, he cannot find n in step 4.4 of the sharing algorithm. Therefore the calculated hi,j from his cheated shared message xi,j and the following computations, hi,j=cyi,jmodn and yi,j=xi,jfimodni, is very hard to be equal to the original hi,j obtaining from public data Tk. Consequently, the shared image is hard to be replaced by any cheated shared image.

The common modulus attack20 indicates that secret message m can be recovered by two secret keys e1 and e2 corresponding with two shared messages m1=me1 and m2=me2, respectively. Since e1 and e2 are relatively prime, there are two numbers a1, a2 such that a1e1+a2e2=1. Therefore the computation is then obtained as following equations (m1)a1(m2)a2=(me1)a1(me2)a2=ma1e1+a2e2=m. Note that all above computations are calculated under mod N.

In the proposed scheme, the shared messages for participants P1 and P2 are [fk(r1s0)]g1modn1 and [fk(r2s0)]g2modn2, respectively. Since fk(r1s0)fk(r2s0), and n1n2, the common modulus attack cannot be mounted by anyone who has only two secret messages. This property also shows that [fk(r1s0)]a1g1[fk(r2s0)]a2g2 cannot recover the coefficients bi in fk(x)=(b0+b1x++bt1xt1), even though a1g1+a2g2=1. Furthermore, since t shared messages corresponding with secret keys meets the proposed (t, n) thresholds, so we can obtain these bi coefficients.

Experimental Results

This section presents the experimental results obtained from the proposed method. The test image is LENA with a size of 512×512, and the selected thresholds are (2, 5). This threshold assignment shares the secret image with five participants, and collecting any two correct participants’ secret keys with shared images recovers the secret image. Figure 1(a) shows the secret image LENA with a size of 512×512 and Fig. 1(b) to 1(f) shows five shared images corresponding with secret keys, as defined in Table 1. The set thresholds of (2, 5) acquire a shared image with size 512×256.

Graphic Jump LocationF1 :

(a) secret image; (b) to (f) five shared images; (g) reconstructed image from (b) and (c).

Table Grahic Jump Location
Table 1The parameters used in the experiments.

Figure 2 uses a cheated secret key w1=23, instead of correct w1=12, to recover the secret image. Since the secret key is wrong, the cheated secret key will be detected in the proposed scheme. If we ignore the wrong detection in step 2 of recovering with verification algorithm, we acquire the recovered secret image as shown in Fig. 2(c). Another experiment on cheated shared image is illustrated in Fig. 3. Figure 3(b) shows the cheated shared image, where Fig. 3(a) and all secret keys are correct. Ignoring the cheated shared image detection and keeping calculation acquire the reconstructed secret image as shown in Fig. 3(c). In these two figures, we find that any cheated secret key or shared image causes the wrong reconstructed secret image. Note that the proposed scheme can detect any cheated secret key or shared image efficiently. Therefore the wrong reconstructed secret image such as Figs. 2 or 3 will not be acquired in the proposed scheme.

Graphic Jump LocationF2 :

(a) shared image in Fig. 1(b); (b) shared image in Fig. 1(c); (c) reconstructed image from (a) and (b) with cheated w1=23 and w2=33.

Graphic Jump LocationF3 :

(a) shared image in Fig. 1(b); (b) cheated shared image in Fig. 1(c); (c) reconstructed image from (a) and (b) with w1=12 and w2=33.

Comparisons and Discussion

The proposed scheme verifies participants in secret image sharing problem. Two comparisons are provided in this section. First an overall comparison between the proposed scheme and other important works13,8,11,18,19,21 is listed in Table 2. Second, a comparison of the secret image-sharing schemes with cheater identification properties is shown in Table 3.

Table Grahic Jump Location
Table 2Characteristics comparison between the proposed scheme and important literatures.

Table 2 shows a comparison of characteristics between these propositions. These characteristics include sharing multiple images,2 image verification,18 progressive,3 visual cryptography and secret image sharing,8 perfect secret image sharing,1 scalability,21 size constraints,11 secret key verification,19 and secret key and shared-image verification proposed in this paper.

Table 3 shows a comparison of the results between the proposed scheme and other secure secret image-sharing schemes.18,19 Four conclusions are drawn from this table. First, the proposed scheme verifies both secret keys and shared images, which perform better than previous studies18,19 that verify only either the shared image or secret key. Second, the required parameters loads, including dealer processing and public sharing, are few more than required by Refs. 18 and 19. Third, the extra load is cause by the free hash function, and the extra load is limited. At last, secret key selection is determined by participant and this process can be done over networks. Therefore, Tables 2 and 3 show that the proposed scheme has significant property of detecting cheaters both in secret key and shared image.

Table Grahic Jump Location
Table 3Comparisons with other secure secret image-sharing schemes.

This paper presents a secret image-sharing scheme with the properties of detecting cheaters both in secret key and shared image. The proposed scheme presents three algorithms: initial, sharing, and recovering with verification. The strategy for key validation is different from previous works. We allow each participant to select his or her secret key, and the dealer checks the validity of each key. Verification during image recovery is also based on the participant’s selected secret key. This property of determining secret key from a participant fits the network requirement well. Security analysis and experimental results demonstrate that the proposed scheme behaves strong security coverage. Future work will focus on combining other characteristics such as multiple image sharing to enhance the benefits of the proposed scheme.

The authors gratefully acknowledge the helpful comments and suggestions of the reviewers. This work was supported in part by the National Science Council project under Grant NSC 100-2221-E-032-056.

Thien  C. C., Lin  J. C., “Secret image sharing,” Comput. Graph.. 26, (5 ), 765 –770 (2002). 0097-8493 CrossRef
Chen  C. C., Chien  Y. W., “Sharing numerous images secretly with reduced possessing load,” Fundamenta Inform.. 86, (4 ), 447 –458 (2008). 0169-2968 
Chen  S. K., Lin  J. C., “Fault-tolerance and progressive transmission of images,” Pattern Recognit.. 38, (12 ), 2466 –2471 (2005). 0031-3203 CrossRef
Fang  W. P., “Friendly progressive visual secret sharing,” Pattern Recognit.. 41, (4 ), 1410 –1414 (2008). 0031-3203 CrossRef
Huang  C. P., Hsieh  C. H., Huang  P. S., “Progressive sharing for a secret image,” J. Syst. Software. 83, (3 ), 517 –527 (2010). 0164-1212 CrossRef
Hung  K. H., Chang  Y. J., Lin  J. C., “Progressive sharing of an image,” Opt. Eng.. 47, (4 ), 047006  (2008). 0091-3286 CrossRef
Lin  S. J., Chen  L. S., Lin  J. C., “Fast-weighted secret image sharing,” Opt. Eng.. 48, (7 ), 077008  (2009). 0091-3286 CrossRef
Lin  S. J., Lin  J. C., “VCPSS: A two-in-one two-decoding-options image sharing method combining visual cryptography (VC) and polynomial-style sharing (PSS) approaches,” Pattern Recognit.. 40, (12 ), 3652 –3666 (2007). 0031-3203 CrossRef
Yang  C. N., Ciou  C. B., “Image secret sharing method with two-decoding-options: lossless recovery and previewing capability,” Image Vis. Comput.. 28, (12 ), 1600 –1610 (2010). 0262-8856 CrossRef
Thien  C. C., Lin  J. C., “An image-sharing method with user-friendly shadow images,” IEEE Trans. Circ. Syst. Video Technol.. 13, (12 ), 1161 –1169 (2003). 1051-8215 CrossRef
Wu  Y. S., Thien  C. C., Lin  J. C., “Sharing and hiding secret images with size constraint,” Pattern Recognit.. 37, (7 ), 1377 –1385 (2004). 0031-3203 CrossRef
Chen  C. C., Fu  W. Y., “A geometry-based secret image sharing approach,” J. Inform. Sci. Eng.. 24, (5 ), 1567 –1577 (2008).
Chen  T. H., Wu  C. S., “Efficient multi-secret image sharing based on Boolean operations,” Signal Process.. 91, (1 ), 90 –97 (2011). 0165-1684 CrossRef
Shyu  S. J., Chen  Y. R., “Threshold secret image sharing by Chinese remainder theorem,” in  Proc. IEEE Asia-Pacific Services Computing Conf. , pp. 1332 –1337,  IEEE ,  Yilan, Taiwan  (2008).
Wu  T.-C., Wu  T.-S., “Cheating detection and cheater identification in secret sharing schemes,” IEE Proc. Comput. Dig. Tech.. 142, (5 ), 367 –369 (1995).CrossRef
Chang  C.-C., Hwang  R.-J., “Efficient cheater identification method for threshold schemes,” IEE Proc. Comput. Dig. Tech.. , 144, (1 ), 23 –27 (1997).CrossRef
Tan  K. J., Zhu  H. W., Gu  S. J., “Cheater identification in (t, n) threshold scheme,” Comput. Commun.. 22, (8 ), 762 –765 (1999). 0140-3664 CrossRef
Chen  C. C., Suen  G. Y., “Sharing an image with cheater identification,” Int. J. Innovat. Comput. Inform. Control. 6, (2 ), 677 –685 (2010).
Zhao  R. et al., “A new image secret sharing scheme to identify cheaters,” Comput. Stand. Interfac.. 31, (1 ), 252 –257 (2009). 0920-5489 CrossRef
Hinek  M. J., Cryptanalysis of RSA and Its Variants. ,  Chapman and Hall/CRC, Taylor & Francis Group ,  Boca Raton, Florida  (2010).
Wang  R. Z., Chien  Y. F., Lin  Y. Y., “Scalable user-friendly image sharing,” J. Vis. Comm. Image Represent.. 21, (7 ), 751 –761 (2010). 1047-3203 CrossRef

Grahic Jump LocationImage not available.

Chien-Chang Chen received a BS from the Department of Computer and Information Science at Tunghai University, Taiwan, in 1991, and a PhD from the Department of Computer Science at National Tsing Hua University, Taiwan, in 1999. He is currently an associate professor at the Department of Computer Science and Information Engineering, Tamkang University, Taiwan. His research interests include secret image sharing, watermarking, and texture analysis.

Grahic Jump LocationImage not available.

Chong-An Liu received an MS from the Department of Computer Science and Information Engineering, Tamkang University, in 2012. His research interests include secret image sharing.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation

Chien-Chang Chen and Chong-An Liu
"Tamper-proof secret image-sharing scheme for identifying cheated secret keys and shared images", J. Electron. Imaging. 22(1), 013008 (Jan 09, 2013). ; http://dx.doi.org/10.1117/1.JEI.22.1.013008


Figures

Graphic Jump LocationF1 :

(a) secret image; (b) to (f) five shared images; (g) reconstructed image from (b) and (c).

Graphic Jump LocationF2 :

(a) shared image in Fig. 1(b); (b) shared image in Fig. 1(c); (c) reconstructed image from (a) and (b) with cheated w1=23 and w2=33.

Graphic Jump LocationF3 :

(a) shared image in Fig. 1(b); (b) cheated shared image in Fig. 1(c); (c) reconstructed image from (a) and (b) with w1=12 and w2=33.

Tables

Table Grahic Jump Location
Table 1The parameters used in the experiments.
Table Grahic Jump Location
Table 2Characteristics comparison between the proposed scheme and important literatures.
Table Grahic Jump Location
Table 3Comparisons with other secure secret image-sharing schemes.

References

Thien  C. C., Lin  J. C., “Secret image sharing,” Comput. Graph.. 26, (5 ), 765 –770 (2002). 0097-8493 CrossRef
Chen  C. C., Chien  Y. W., “Sharing numerous images secretly with reduced possessing load,” Fundamenta Inform.. 86, (4 ), 447 –458 (2008). 0169-2968 
Chen  S. K., Lin  J. C., “Fault-tolerance and progressive transmission of images,” Pattern Recognit.. 38, (12 ), 2466 –2471 (2005). 0031-3203 CrossRef
Fang  W. P., “Friendly progressive visual secret sharing,” Pattern Recognit.. 41, (4 ), 1410 –1414 (2008). 0031-3203 CrossRef
Huang  C. P., Hsieh  C. H., Huang  P. S., “Progressive sharing for a secret image,” J. Syst. Software. 83, (3 ), 517 –527 (2010). 0164-1212 CrossRef
Hung  K. H., Chang  Y. J., Lin  J. C., “Progressive sharing of an image,” Opt. Eng.. 47, (4 ), 047006  (2008). 0091-3286 CrossRef
Lin  S. J., Chen  L. S., Lin  J. C., “Fast-weighted secret image sharing,” Opt. Eng.. 48, (7 ), 077008  (2009). 0091-3286 CrossRef
Lin  S. J., Lin  J. C., “VCPSS: A two-in-one two-decoding-options image sharing method combining visual cryptography (VC) and polynomial-style sharing (PSS) approaches,” Pattern Recognit.. 40, (12 ), 3652 –3666 (2007). 0031-3203 CrossRef
Yang  C. N., Ciou  C. B., “Image secret sharing method with two-decoding-options: lossless recovery and previewing capability,” Image Vis. Comput.. 28, (12 ), 1600 –1610 (2010). 0262-8856 CrossRef
Thien  C. C., Lin  J. C., “An image-sharing method with user-friendly shadow images,” IEEE Trans. Circ. Syst. Video Technol.. 13, (12 ), 1161 –1169 (2003). 1051-8215 CrossRef
Wu  Y. S., Thien  C. C., Lin  J. C., “Sharing and hiding secret images with size constraint,” Pattern Recognit.. 37, (7 ), 1377 –1385 (2004). 0031-3203 CrossRef
Chen  C. C., Fu  W. Y., “A geometry-based secret image sharing approach,” J. Inform. Sci. Eng.. 24, (5 ), 1567 –1577 (2008).
Chen  T. H., Wu  C. S., “Efficient multi-secret image sharing based on Boolean operations,” Signal Process.. 91, (1 ), 90 –97 (2011). 0165-1684 CrossRef
Shyu  S. J., Chen  Y. R., “Threshold secret image sharing by Chinese remainder theorem,” in  Proc. IEEE Asia-Pacific Services Computing Conf. , pp. 1332 –1337,  IEEE ,  Yilan, Taiwan  (2008).
Wu  T.-C., Wu  T.-S., “Cheating detection and cheater identification in secret sharing schemes,” IEE Proc. Comput. Dig. Tech.. 142, (5 ), 367 –369 (1995).CrossRef
Chang  C.-C., Hwang  R.-J., “Efficient cheater identification method for threshold schemes,” IEE Proc. Comput. Dig. Tech.. , 144, (1 ), 23 –27 (1997).CrossRef
Tan  K. J., Zhu  H. W., Gu  S. J., “Cheater identification in (t, n) threshold scheme,” Comput. Commun.. 22, (8 ), 762 –765 (1999). 0140-3664 CrossRef
Chen  C. C., Suen  G. Y., “Sharing an image with cheater identification,” Int. J. Innovat. Comput. Inform. Control. 6, (2 ), 677 –685 (2010).
Zhao  R. et al., “A new image secret sharing scheme to identify cheaters,” Comput. Stand. Interfac.. 31, (1 ), 252 –257 (2009). 0920-5489 CrossRef
Hinek  M. J., Cryptanalysis of RSA and Its Variants. ,  Chapman and Hall/CRC, Taylor & Francis Group ,  Boca Raton, Florida  (2010).
Wang  R. Z., Chien  Y. F., Lin  Y. Y., “Scalable user-friendly image sharing,” J. Vis. Comm. Image Represent.. 21, (7 ), 751 –761 (2010). 1047-3203 CrossRef

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