A hierarchical threshold secret image sharing

A hierarchical threshold secret image sharing

Cheng Guo

a

, Chin-Chen Chang

b,c,⇑

, Chuan Qin

b

a

Department of Computer Science, National Tsing-Hua University, Hsinchu 30013, Taiwan, ROC

b

Department of Information Engineering and Computer Science, Feng Chia University, No. 100 Wenhwa Rd., Seatwen, Taichung 40724, Taiwan, ROC

c

Department of Biomedical Imaging and Radiological Science, Chinese Medical University, Taichung 40402, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 31 March 2011 Available online 1 October 2011 Communicated by S. Sarkar Keywords:

Hierarchical threshold Distortion-free Secret image sharing Access structure

a b s t r a c t

In the traditional secret image sharing schemes, the shadow images are generated by embedding the secret data into the cover image such that a sufficient number of shadow images can cooperate to recon-struct the secret image. In the process of reconrecon-struction, each shadow image plays an equivalent role. However, a general threshold access structure could have other useful properties for the application. In this paper, we consider the problem of secret shadow images with a hierarchical threshold structure, employing Tassa’s hierarchical secret sharing to propose a hierarchical threshold secret image sharing scheme. In our scheme, the shadow images are partitioned into several levels, and the threshold access structure is determined by a sequence of threshold requirements. If and only if the shadow images involved satisfy the threshold requirements, the secret image can be reconstructed without distortion.

Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction

A secret sharing scheme is a technique to share a secret among a group of participants. The secret data can be divided into several pieces, called secret shadows, which are distributed to the partici-pants. If enough participants cooperate to reconstruct the secret data and pool their secret shadows together, the secret data can be reconstructed.

In 1979, the first (t, n) threshold secret sharing schemes were pro-posed byShamir (1979)and Blakley (1979), based on Lagrange interpolating and liner project geometry, respectively. In 1995, based on the concept of threshold secret sharing,Naor and Shamir (1995)introduced visual cryptology, or the visual secret sharing scheme (VSS scheme). In the (t, n) VSS schemes (Yang, 2004; Wang et al., 2007; Chang et al., 2009; Lin and Wang, 2010), the secret data is an image comprised of black and white pixels that is encoded into n shadow images, and the secret image can be reconstructed only by stacking t of the shadow images; no information about the secret im-age can be obtained from t  1 or fewer shadow imim-ages. However, in this kind of visual secret sharing scheme, the shadow images are meaningless, which tends to call attention to them. In 2003,Thien and Lin (2003)utilized the steganography approach to embed the secret image into a cover image to generate the shadow images. In their scheme, the shadow images are meaningful and the distortion between the cover image and the shadow images is imperceptible.

Since then, the secret image sharing schemes (Lin and Tsai, 2004; Wu et al., 2004; Yang et al., 2007; Chang et al., 2008; Zhao et al., 2009; Lin et al., 2009; Eslami et al., 2010; Lin and Chan, 2010) have been extensively developed to meet the requirements of our daily lives. In 2004,Lin and Tsai (2004)proposed a secret image sharing scheme with steganography and authentication, in which the sha-dow images are meaningful, while the reconstructed secret image is distorted slightly. In 2007,Yang et al. (2007)improved Lin and Tsai’s scheme by making it possible to restore a distortion free secret image, but their scheme reduces the visual quality of the shadow images and increases the risk of their being deceived by malicious intruders. In 2009,Lin et al. (2009)employed the modulus operator to embed the secret image into a cover image, where each partici-pant can obtain a meaningful shadow image with high visual qual-ity, the authorized participants can detect intruders, and the secret image and the cover image can be recovered losslessly. In 2010,

Lin and Chan (2010)proposed a new secret sharing scheme that achieves an excellent combination of embedding capacity and the visual quality of shadow images. In addition, their scheme can reconstruct the secret image and the cover image without distortion. However, in all of these schemes, all shadow images are equal in terms of privileges and authority in the process of reconstructing the secret image. In this paper, we consider a special situation in which the shadow images may be not equal. We achieve a hierar-chical threshold access structure by introducing Tassa’s threshold secret sharing scheme, and we present a novel hierarchical thresh-old secret image sharing scheme.

Secret sharing schemes have been extensively studied, and se-cret image sharing and its many variations form an important re-search direction. In 2007, Tassa (2007) proposed a hierarchical threshold secret sharing scheme. Tassa’s scheme is based on the 0167-8655/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.patrec.2011.09.030

⇑ Corresponding author at: Department of Information Engineering and Com-puter Science, Feng Chia University, No. 100 Wenhwa Rd., Seatwen, Taichung 40724, Taiwan, ROC. Tel.: +886 4 24517250x3790; fax: +886 4 27066495.

E-mail addresses: guo8016@gmail.com (C. Guo), alan3c@gmail.com (C.-C. Chang),qin@usst.edu.cn(C. Qin).

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Birkhoff interpolation. In their scheme, the secret is shared by a set of participants partitioned into several levels, and the secret data can be reconstructed by satisfying a sequence of threshold require-ments (e.g., it has at least t0participants from the highest level, as

well as at least t1>t0participants from the two highest levels and

so forth). There are many real-life examples of hierarchical thresh-old schemes. Consider the following example. According to a grad-uate school’s policy, a gradgrad-uate who wants to apply for a postgraduate position must have letters of recommendation. As-sume that the graduate school’s policy concerning such recom-mendations is that the candidate must have at least two recommendations from professors and at least five recommenda-tions from a combination of professors and associate professors. In this scenario, the professor is the highest level, the associate pro-fessor is the second-highest level, the corresponding threshold val-ues are t0¼ 2 and t1¼ 5, and the shadows of a higher level can

substitute for those of a lower level. In this example, recommenda-tions from two professors and three associate professors, three professors and two associate professors, four professors and one associate professor, or five professors are all acceptable.

The same situation also appears in the sharing of secret images. Inspired by the hierarchical secret sharing scheme, we construct a hierarchical threshold secret image sharing scheme.

The existing secret image sharing schemes (Lin and Tsai, 2004; Wu et al., 2004; Yang et al., 2007; Chang et al., 2008; Zhao et al., 2009; Lin et al., 2009; Eslami et al., 2010; Lin and Chan, 2010) have tried to improve the visual quality of the shadow images so the sus-picions of malicious intruders won’t be aroused. Two of the most popular steganographic methods are the least significant bits (LSB) replacement and the modulus operation. Shamir’s (t, n) threshold scheme is an ingenious method with which to share se-cret data among n participants. Traditional sese-cret image sharing schemes (Lin et al., 2009; Lin and Chan, 2010) usually transformed the secret image pixels into base-m representation, s1;s2; . . . ;sMSNS

dlogm255e, where MS NSdenotes the size of the secret image,

and then constructed a Lagrange interpolation polynomial f(x) by using the s1;s2; . . . ;sMSNSdlogm255eas the polynomial coefficients. In

order to make the shadow images meaningful for the purposes of camouflage—that is, to diminish the distortion of the shadow images—they utilized the modulus operator to embed secret data into the pixel of the cover image.

However, in Tassa’s scheme, the hierarchical threshold access structure can work so long as the modulus p is far greater than the threshold t, so it is difficult to combine the existing secret im-age sharing schemes based on the modulus operation directly with Tassa’s hierarchical threshold access structure. Therefore, how to combine the hierarchical threshold access structure proposed by

Tassa (2007)with steganographic methods is our scheme’s main challenge. We also need to guarantee that the secret image can be reconstructed losslessly.

To the best of our knowledge, no hierarchical secret image shar-ing schemes have been proposed in the literature to date. Since we believe that the application of secret image sharing in groups with hierarchical structure has good prospects, we provide a novel hier-archical threshold secret image sharing scheme.

In our scheme, the n shadow images generated from the secret image and the cover image are partitioned into several levels such that each level has a certain number of shadow images and a cor-responding threshold. The secret image can be reconstructed only by meeting a sequence of threshold requirements.

The novel characteristic of the proposed scheme is not available in the existing mechanisms, so the proposed scheme has the potential to work in many applications. In addition to the unique hierarchical threshold characteristic, our proposed scheme has three key characteristics:

1. The secret image can be retrieved losslessly.

2. The scheme solves the problems of overflow and underflow. 3. Unlike the traditional secret image sharing schemes in which

the embedding capacity is proportional to the increase of t, the capacity of the embedded secret data is stable and large. 2. Review of Tassa’s hierarchical threshold secret sharing scheme

In case of hierarchical threshold secret sharing, the set of participants is partitioned into some levels P0, P1, . . . , Pmand the

access structure is then determined by a sequence of threshold requirements t0, t1, . . . , tm according to their hierarchy. Tassa’s

method (2007)for hierarchical threshold secret sharing is based on Birkhoff interpolation. In this section, we briefly introduce Tassa’s hierarchical threshold secret sharing scheme. Assume that there are n participants and one dealer responsible for generating the secret shadows and distributing them to the participants.

1. The dealer generates the polynomial F(x) of degree at most tm 1 over GFq, FðxÞ ¼ S þ a1x þ a2x2þ    þ atm1xtm1, where S

is the shared secret data.

2. An element i

e

GFq is assigned to the participant i, for all

1 6 i 6 n .

3. For any level j, each participant i from the jth level will receive the secret shadow Ftj1_{ðiÞ, where F}tj1_{ð:Þ is the (t}

j1)th derivative

of F(x) and t0= 0.

4. In the reconstruction phase, the participants can cooperate to

reconstruct the shared secret data by using Birkhoff

interpolation. 3. The proposed scheme

Given a shared secret image S and a cover image O, the dealer can generate n shadow images pi, for i = 1, 2, . . . , n. In the proposed

scheme, the n shadow images do not have equal status; instead, the secret image is shared among n shadow images that are parti-tioned into several levels. Based on Tassa’s definition (Tassa, 2007), we define the hierarchical secret image sharing as follows: Definition 1. Let P be a set of n shadow images and assume that P is composed of levels; that is, P ¼ [m

i¼0Pi, where Pi\ Pj¼ / for all

i – j and i, j

e

[0, m]. Let t = ftigmi¼0 be a monotonically increasing

sequence of integers. Then the (t, n) hierarchical threshold access structure is

C

¼ V  P : V \ [ i j¼0 Pj !          Pti;

8

i 2 f0; 1; . . . ; mg ( ) : 3.1. Initialization procedure

First, the dealer constructs n secret shadow images and divides these n shadow images into (m + 1) levels P = {P0, P1, . . . , Pm}

according to the real-life situation. Then the dealer sets a sequence of threshold values {t0, t1, . . . , tm}, 0 < t0< t1. . .< tm, where t = tmis

the overall number of shadow images that are required for recov-ery of the secret image, and assumes that

p1;p2; . . . ;pl02 P0; pl0þ1;pl0þ2; . . . ;pl12 P1; ..

.

where pi;0 6 i 6 n denotes the ith shadow image and Pi;0 6 i 6 m

denotes the set of shadow images of the ith level. The secret image can be reconstructed by satisfying a sequence of threshold require-ments, such as that it has at least t0secret shadow images from the

highest level as well as at least t1> t0secret shadow images from

the two highest levels and so forth.

Assume that the cover image O has M  N pixels,

O = {oi|i = 1, 2, . . . , (M  N)}, and secret image S has MS NSpixels.

Step 1: The dealer selects a large prime modulus p.

Step 2: The dealer obtains all pixels of the secret image S, denoted as S ¼ fsjjj ¼ 1; 2; . . . ; ðMS NSÞg, where sj

e

[0, 255].

3.2. Secret image sharing procedure

The procedure consists of two phases: (1) the sharing phase, and (2) the embedding phase.

3.2.1. Sharing phase

Without loss of generality, assume that we want to embed s0;s1;s2; . . . ;st1into the cover image to generate n shadow images

using a hierarchical access structure. The dealer performs the fol-lowing steps:

Step 1. Construct a (t  1)th-degree polynomial FðxÞ ¼ s0þ s1x

þ    þ st1xt1mod p, where p is a large prime, t = tm, and

s0, s1, s2, . . . , st1 denote the pixel values of the shared secret

image.

Step 2. For the first level shadow images, utilize the (t  1)th-degree polynomial F(x) to generate the shadow images. The shadow images of other levels are processed in the following manner. The shadow images of the ith level in the hierarchy can be generated by using the polynomial Fti1_{ðxÞ, where}

Fti1_{ðxÞ is the (t}

i1) th derivative of F(x).

For example, there are three levels in the shadow images, P = P0[ P1[ P2. Assume that the threshold sequence requirements

are t0= 2, t1= 4 and t2= 7; that is, the secret image can be

recon-structed if and only if there are at least seven shadow images, of which at least four are from P0[ P1, and at least two are from P0.

In this example, we should construct a 6th-degree polynomial FðxÞ ¼ s0þ s1x þ    þ s6x6mod p: First, the dealer utilizes F(x) to

generate the shadow images that belong to P0. Since t0= 2, the

sec-ond level shadow images are generated by using the polynomial F00

ðxÞ, and since t1= 4, the shadow images of the lowest level can

be computed using the polynomial F(4)_(x).

3.2.2. Embedding phase

In order to diminish the distortion of the shadow images, most existing secret image sharing schemes have utilized the modulus operator to embed the secret image data into the pixels of the cov-er image. Howevcov-er, in the proposed scheme, in keeping with Tas-sa’s scheme, we calculate the shadow images in a finite field of size p, which is a large prime, so we need to develop a new method in order to embed the shadow data into the cover image.

Lin and Chan’s scheme (Lin and Chan, 2010) formed a camou-flaged pixel using

Qi¼ boi=kc  k;

qi¼ Qiþ yi;

ð1Þ

where Qiis the quantized value of oi, and qirepresents the ith

cam-ouflaged pixel. Inspired by Lin and Chan’s scheme, we also use a quantization operation to embed the secret data. However, in Lin and Chan’s scheme, all operations are in a field modulo a small prime number

r

, such as 5, 7, or 11, so yican be directly embedded into a

pixel of the cover image without causing a large distortion. However, in our scheme, the modulus p must be far greater than the threshold t, so we obtain a large integer yi= F(i) by feeding an integer

i, i 2 [1, n] into F(x) and need to use r pixels of shadow images to rep-resent shadow data yi. In the traditional secret image sharing

schemes, the dealer feeds a secret key or a unique IDiinto the

poly-nomial F(x) to obtain yi; and in order to facilitate the embedding of yi

into the shadow image, the polynomial F(x) can modulo a small prime. However, in the proposed scheme, the polynomial F(x) needs to modulo a large prime, so we need more pixels to represent yi.

Obviously, the larger yiis, the more pixels are needed to represent yi.

Therefore, in order to minimize yi, we feed a series of integers i,

for i = 1, 2, . . . , n into F(x) instead of feeding IDi. In order to

guaran-tee that r pixels are sufficient to represent yi, we maximize the

shared secret data siand assume that all si= 255, for i = 1, 2, . . . , n.

We first talk about how to generate the highest level shadow images. Assume that the highest level P0includes l0shadow images

p1;p2; . . . ;pl0, and the selected r camouflage pixels in the cover

im-age O are oi, oi+1, . . . , oi+r1. In the embedding phase, we perform

the following steps:

Step 1. Assume that we want to generate the shadow image pi, i

e

[1, l0]. The dealer first computes yiby feeding i into F(x).

Step 2. We utilizeLin and Chan’s method (2010)to generate and camouflage the shadow images. Firstly, we convert the secret data yiinto the

r

-ary notational system. In Lin and Chan’s

scheme (2010), Eq.(1)may lead to an overflow situation. There-fore, we must ensure that boi=kc  k þ

r

6255. Meanwhile, the

parameters (k,

r

) can also affect the quality of the shadow images and the embedding capacity. The greater the value

r

is, the larger the embedding capacity is. However, the value

r

may increase the gap between adjacent pixel values, especially for the smooth image. Therefore, the greater the value

r

is, the less smooth the shadow image is. In regard to the different cov-er images, if the covcov-er image is smooth, we need to select a small

r

. On the contrary, if the cover image is rich, we can select a great

r

aiming at improving the embedding capacity. In order to simplify the proposed method, in this paper, we let k = 10 and transform the yi into base-5 representation. For instance, if

yi= 1304, we obtain yi= (2, 0, 2, 0, 4)5. This pair (10, 5) can

effec-tively avoid the overflow problem since b255=10c  10 þ 5 6_{255. And, we need r ¼ dlog}

5Fðl0Þe pixels of the shadow image

to represent yi. As to the ith level, ri can be computed by

ri¼ dlog5Fti1ðliÞe; where Fti1ðliÞ denotes the (ti1)th derivative

of the function F(x).

Step 3. Without loss of generality, assume that we use r pixels pij;j ¼ 1; 2; . . . ; r of the shadow image pi to represent yi as

follows: pi1¼ boi=10c  10 þ yi1; pi2¼ boiþ1=10c  10 þ yi2; .. . pir¼ boiþr110c  10 þ yir; ð2Þ

where each yij;j ¼ 1; 2; . . . ; r, denotes yi’s base-5 representation.

Step 4. By repeating Steps 1–3, the dealer can camouflage all se-cret data yi into the cover pixels, and by feeding i, for

i = 1, 2, . . . , l0into F(x), the dealer can obtain the first level

sha-dow images.

As to the shadow images at the second highest level, since the threshold values are {t0, t1, . . . , tm}, the dealer uses the polynomial Ft0_{ðxÞ to generate the shadow data yi, and so forth, so the shadow}

images of the ith level in the hierarchy are computed using the polynomial Fti1_{ðxÞ .}

The generation process for the other levels of shadow images is the same as that of the highest level except that different polyno-mials are used. Repeat the above steps until all shadow images of various levels are generated.Fig. 1displays the flowchart of the se-cret image sharing scheme.

3.3. Secret image retrieving procedure

In the traditional secret image sharing schemes, given any t sha-dow images, the shared secret image can be reconstructed. In our scheme, according to Tassa’s threshold access structure, given sha-dow images must satisfy a sequence of threshold requirements. In order to extract the secret digits, the polynomial F(x) must be reconstructed by retrieving the shadow data yifrom the shadow

images pi’s. The same method is used for different levels of shadow

images. The details are as follows: Step 1. Compute yiby

yi¼ yi1jjyi2jj . . . jjyir; ð3Þ

where yij¼ pijmod 5, for j = 1, 2, . . . , r, and pijdenotes the jth pixel

value of the ith shadow image, for i = 1, 2, . . . , tm.

Step 2. Collect enough t pairs (i, yi) ’s to satisfy the hierarchical threshold access structure and employ the Birkhoff interpola-tion to reconstruct the (t  1) degree polynomial F(x).

Step 3. Extract the corresponding t coefficients s0, s1, s2, . . . , st1.

Step 4. Repeat Steps 1–3 until all secret data is extracted. Step 5. Reconstruct the secret image.

4. Experimental results and analysis

This section describes some experimental results in order to demonstrate the characteristics of the proposed scheme.

We perform experiments for n = 10. A secret image can be gen-erated in ten shadow images, and the ten shadow images are par-titioned into three levels. Assume that the first (highest) level has three shadow images, the second level has three shadow images, and the third (lowest) level has four shadow images. Assume a se-quence of threshold requirements t ¼ ðt0;t1;t2Þ ¼ ð2; 4; 7Þ; that is,

the secret image can be reconstructed if and only if a subset of sha-dow images has at least seven shasha-dow images, of which at least four are from the first two levels and at least two are from the first level.

4.1. Simulation results

The peak signal-to-noise ratio (PSNR), defined in Eq.(4), can be used to measure the distortion of the shadow images after the se-cret data have been embedded into the cover image.

PSNR ¼ 10log10

2552 MSE !

dB: ð4Þ

The mean square error (MSE) between the cover image with M  N pixels and the shadow image is defined as

MSE ¼ 1 M  N X MN j¼1 ðpj p0jÞ 2 ; ð5Þ

where pjis the original pixel value and p0jis the pixel value of

sha-dow image.

We use fifteen grayscale images with size 512  512 pixels as the test images, as shown inFig. 2, and the secret sharing image Airplane is set to 256  256 pixels, as shown inFig. 3.

Table 1displays the PSNR value of the shadow images achieved using the proposed scheme. Although the pixel values of the shadow images of the proposed scheme are slightly lower than those of the existing secret image sharing methods, it is clear that, regardless of what we use as the cover image, the PSNR values of the shadow images always maintain a steady level and are within [36.00,41.00]. Further-more, we obtained a new access structure that may have many appli-cations, and the distortion between the shadow images and the cover image is imperceptible by visual perception.

In order to demonstrate the visual perception of the shadow images, we use Peppers as the cover image with size 512  512 pix-els and Airplane as the secret image with size 256  256 pixpix-els. If the secret images involved meet the hierarchical threshold access structure, our method can reconstruct them without distortion.

Fig. 4(a) and (b) shows the cover image and the extracted secret image, respectively.

Fig. 5(a)–(j) display ten shadow images of Peppers that are partitioned into three levels. Since the distortion between the cover image and the shadow images is slight, we can success-fully conceal the embedded the secret image data’s existence from intruders.

(a) Bird (b) Woman (c) Lake (d) Man (e) Tiffany

(f) Peppers (g) Lena (h) Fruits (i) Baboon (j) Airplane

(k) Couple (l) Crowd (m) Cameraman (n) Boat (o) House

Fig. 2. The test images.

5. Discussions

Progressive visual secret sharing mechanism (Fang, 2008; Huang et al., 2010) has the similar characteristics with our pro-posed scheme. Progressive visual secret sharing can be utilized to reconstruct the shared secret image gradually by superimposing more and more shadow images. By increasing the number of the shadow images being stacked, the details of the shared secret im-age can be revealed progressively. In our scheme, if we use the con-stant term and all coefficients of the polynomial F(x) to hide the secret data, our proposed scheme also can achieve a progressive ef-fect. Meanwhile, our scheme has a hierarchical threshold feature. That is, when the shadow images of a level involved meet a corre-sponding level threshold, the recovery of the shared secret image will be clearer. Further more, if we just use t0coefficients of the

F(x) to hide the secret data, the proposed scheme can achieve an ideal hierarchical threshold access structure. If the shadow images involved can not satisfy the hierarchical threshold requirement, they can not obtain anything about the secret image. Progressive visual secret sharing is an important mechanism for application in transmission while hierarchical threshold secret image sharing provides a hierarchical threshold access structure for secret image sharing. To the best of our knowledge, our proposed scheme has a

unique hierarchical threshold characteristic as compared with the existing secret image sharing schemes.

Table 2compares the functionality of the proposed scheme with that of related schemes. As presented inTable 2, the new method satisfied the camouflage purpose and provided the satisfactory quality of shadow images. Meanwhile, the secret image can be reconstructed lossless. And the proposed secret image sharing scheme can provide a hierarchical threshold access structure. The new mechanism allows the participants to be partitioned into sev-eral levels, and the access structure is then determined by a se-quence of threshold requirements. In comparison with the traditional secret image sharing schemes (Yang et al., 2007; Chang et al., 2008; Lin et al., 2009; Lin and Chan, 2010), the proposed hier-archical threshold secret image sharing scheme can not recover the cover image. And, quality of shadow images needs to be improved. In the proposed experiment, the ten shadow images are parti-tioned into three levels, and the corresponding thresholds are t0= 2, t1= 4, t2= 7. We can compute the secret image data yi

embedded into shadow images by using a (t2 1)th-degree

poly-nomial F(x). Since yiis a large integer, we need to use r pixels of

the shadow image to represent yi, so parameter r is important in

order to maximize the secret capacity. In our experiment, three dif-ferent r values correspond with the three levels of shadow images. Table 1

The PSNR value (dB) of the shadow images for test images, n = 10, t0= 2, t1= 4, t2= 7.

Test images The first level The second level The third level PSNR(dB)

1 2 3 4 5 6 7 8 9 10 Bird 36.87 37.31 37.58 37.94 38.02 38.17 37.99 38.06 38.10 38.14 Woman 38.29 38.72 38.99 39.33 39.42 39.57 39.40 39.45 39.51 39.54 Lake 37.94 38.39 38.67 39.01 39.10 39.26 39.08 39.14 39.19 39.22 Man 37.73 38.17 38.42 38.76 38.86 39.01 38.83 38.87 38.94 38.97 Tiffany 36.37 36.81 37.08 37.43 37.52 37.67 37.49 37.55 37.59 37.63 Peppers 37.32 37.76 38.03 38.36 38.45 38.60 38.42 38.49 38.53 38.56 Lena 37.78 38.22 38.50 38.83 38.94 39.08 38.91 38.96 39.02 39.05 Fruits 38.42 38.86 39.14 39.49 39.58 39.74 39.55 39.59 39.65 39.69 Baboon 37.25 37.68 37.96 38.31 38.39 38.55 38.38 38.42 38.47 38.51 Airplane 37.48 37.92 38.18 38.53 38.61 38.77 38.59 38.64 38.69 38.72 Couple 38.38 38.84 39.10 39.45 39.55 39.69 39.51 39.56 39.63 39.66 Crowd 38.49 38.92 39.19 39.55 39.64 39.79 39.61 39.66 39.22 39.75 Cameraman 38.32 38.77 39.03 39.38 39.47 39.62 39.45 39.50 39.54 39.59 Boat 37.67 38.11 38.37 38.70 38.78 38.93 38.76 38.81 38.87 38.90 House 39.02 39.57 39.92 40.37 40.48 40.68 40.45 40.52 40.58 40.63 Average 37.82 38.27 38.54 38.90 38.99 39.14 38.96 39.01 39.04 39.10

r0¼ dlog5Fðl0Þe; r1¼ dlog5F00ðl1Þe; r2¼ dlog5F ð4Þ ðl2Þe: :

In this example, the embedding capacity (the number of pixels) can be computed as

Capacity ¼ 512  512 maxfr0;r1;r2g

 

 t2:

(a) The shadow from the first level, PSNR=37.32 dB

(b) The shadow from the first level, PSNR=37.76 dB

(c) The shadow from the first level, PSNR=38.03 dB

(d) The shadow from the second level, PSNR=38.36 dB

(e) The shadow from the second level, PSNR=38.45 dB

(f) The shadow from the second level, PSNR=38.60 dB

(g) The shadow from the third level, PSNR=38.42 dB

(h) The shadow from the third level, PSNR=38.49 dB

(i) The shadow from the third level, PSNR=38.53 dB

(j) The shadow from the third level, PSNR=38.56 dB

Fig. 5. The results of Peppers, n = 10, t0= 2, t1= 4, t2= 7.

Table 2

Comparisons of the related secret image sharing schemes.

Functionality Yang et al. (2007) Chang et al. (2008) Lin et al. (2009) Lin and Chan (2010) Ours

Hierarchical threshold No No No No Yes

Meaningful shadow image Yes Yes Yes Yes Yes

Quality of shadow images 40 dB 40 dB 43 dB 42 dB 38 dB

Lossless secret image Yes Yes Yes Yes Yes

Lossless cover image No No Yes Yes No

Maximum capacity MN

4 MN4 ðt3ÞMN3 ðt  1Þ  M  N=dlogr255e bM  N= maxfrigc  tm

Table 3

The maximum capacity under different n and ti.

n Level Threshold value Capacity (pixels)

1 2 3 t0 t1 t2 7 2 2 3 1 2 4 174762 8 2 2 4 1 2 4 174762 10 3 3 4 2 4 7 166818 10 2 4 4 1 3 5 163940 12 2 4 6 1 3 6 157286 14 2 6 6 1 4 8 161318

Assume also that the shadow images are partitioned into m levels, and the threshold of the mth level is tm. We can compute the

embedding capacity of our scheme using the following formula:

Capacity ¼ 512  512 maxfrig16i6m $ %  tm; ð6Þ where ri¼ dlog5F ti1_ðl iÞe.

In the secret image sharing scheme, the embedding capacity is an important measurement. In the traditional secret image sharing schemes, the embedding capacity is proportional to the threshold t. However, in our scheme, yimay be increased with the rise of the

threshold, so yishould be represented using more pixels in the

sha-dow image.Table 3describes the maximum size of the secret im-age embedded into the 512  512 cover imim-age under different n and different hierarchical threshold values ti.

In the proposed scheme, the shadow images that must cooper-ate to reconstruct the secret image must satisfy the hierarchical threshold access structure; if the shadow images don’t meet some hierarchical threshold value, the secret image cannot be fully reconstructed.Fig. 6(a) and (b) display the extracted secret images without satisfying the two threshold requirements, respectively.

Fig. 6(a) and (b) show that the secret image cannot be fully reconstructed. However, even this may not be secure enough for some secret images with highly sensitive information since the vi-sual perception ofFig. 6(a) and (b) may leak out some information about the secret image. Our scheme uses the constant term and all coefficients of the polynomial F(x) to hide the secret data, aiming to increase the embedding capacity. Therefore, some shadow images that satisfy only the third level threshold requirement or that sat-isfy only the second and third level requirements can also recover the corresponding coefficients of the F(x). If we relax the restric-tions for the embedding capacity, we can use just t0coefficients

of the polynomial that includes the constant term to hide the se-cret data. In that case, if the shadow images do not meet the hier-archical threshold requirements, they cannot obtain anything about the secret image. (Of course, the changes that decrease the embedding capacity will not affect the visual perception of the sha-dow images.) The embedding capacity can be computed as follows:

Capacity ¼ 512  512 maxfrig16i6m

$ %

 t0; ð7Þ

where ri¼ dlog5Fti1ðliÞe .

6. Conclusions and future work

In this paper, based on Tassa’s hierarchical threshold secret sharing scheme, we propose a novel secret image sharing scheme with a hierarchical threshold access structure. In our scheme, the dealer can generate shadow images by embedding secret data into the cover images. The shadow images are partitioned into several levels, and the dealer sets a sequence of threshold values such that, if and only if the shadow images involved satisfy the threshold requirements, the secret image can be retrieved without distortion. The experimental results show that the shadow images generated by the proposed scheme have the hierarchical threshold character-istic, and the visual quality of the shadow images and the embed-ding capacity are both satisfactory.

However, there are also a number of technical problems that merit attention but that were not fully addressed in this paper. Foremost among these is the distortion-free reconstruction of the cover image. Another problem has to do with how to improve the visual quality of the shadow images and the embedding capac-ity. We believe that some new and interesting approaches will be found by investigating and studying this problem.

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