SECRET IMAGE SHARING BASED ON CHAOTIC

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International Journal of Wavelets, Multiresolution and Information Processing Vol. 10, No. 3 (2012) 1250023 (18 pages) c World Scientific Publishing Company DOI: 10.1142/S0219691312500233

SECRET IMAGE SHARING BASED ON CHAOTIC MAP AND CHINESE REMAINDER THEOREM

CHUNQIANG HU∗ , XIAOFENG LIAO† and DI XIAO‡ State Key Laboratory of Power Transmission Equipment System Security and New Technology, College of Computer Science Chongqing University, Chongqing 400030, China ∗[email protected] †[email protected] ‡[email protected] Received 15 September 2010 Revised 19 July 2011 Published 18 May 2012 Secret sharing is an efficient method for transmitting the image securely. This paper proposes an efficient secret sharing scheme for secret image. The protocol allows each participant to share a secret gray image with the rest of participants. In our scheme, a secret digital image is divided into n pieces, which are further distributed into n participants. The secret digital image can be reconstructed if and only if r or more legal participants cooperate together. These schemes have no pixel expansion. It is general in nature and can be applied on any image size. The proposed scheme is based on the chaotic map and the Chinese Remainder theorem. The security of the scheme is analyzed and the protocol is proven to be secure and be able to resist statistic and exhaustive attacks. Keywords: Image sharing; chaotic map; Chinese Remainder theorem (CRT); information security. AMS Subject Classification: 22E46, 53C35, 57S20

1. Introduction With the rapid development of network technology, image information can be distributed and transformed on the internet rapidly and conveniently; image security becomes more and more important. Image security includes Image Hiding, Digital Watermarking and Digital Image Sharing. Due to the characteristic of image, many researchers devote to study on Image Encryption and digital Watermarking, so there are many novel research papers published.6,7,9,11,19 Secret image sharing is just recently discovered.

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Secret image sharing is an important subject of digital image information security. It is based on secret sharing schemes. Recently, the secret image sharing has become a key technology to keep confidentiality in the field of information security and protection. However, there is an important issue that such techniques should deal with accidental or intentional loss/corruption of images when a single party accesses to the data. If the secret image is always protected by only one person, it will be easy to be lost, destroyed, or modified by attacker. Image sharing which protects and distributes a secret image among a group of participants provides solutions to this issue. Blakley3 and Shamir15 firstly proposed a concept, named (r, n)-threshold scheme, to solve this problem independently. A (r, n)-threshold scheme consists of a trusted dealer and n participants. The dealer divides the secret document into n shadows and distributes them to n participants. The secret document cannot be reconstructed unless some specific groups of more than r − 1 out of n shadow holders pool their shadows. Usually a (r, n) threshold scheme has three features: (1) the secret is divided into n shadows; (2) any r or more shadows can be used to reconstruct the secret; (3) any r − 1 or fewer shadows reveal no knowledge about the secret. Recently, several cryptographic protocols for (r, n)-threshold cryptography have been proposed in Refs. 5 and 21. A (r, n)-threshold scheme is called ideal if the size of every share is equal to the size of the shared secret, and a (r, n)-threshold scheme is said to be perfect if the knowledge of any r − 1 or fewer shares provides no information about the original secret in Refs. 17 and 18. Numerous secret sharing schemes have been proposed in the past decades. The first (r, n)-threshold scheme presented to share secret images was credited to Naor and Shamir13 and is called the visual cryptography. It is based on visual threshold schemes r of n, i.e. the secret image is divided into n shares in such a way that each of them is photocopied in a transparency and then, the original image is recovered by superimposing any r transparencies but no less. However, this scheme is perfect but not ideal. This is because the size of the shared images is bigger than the original one. After that, Chang and Hwang4 introduced the codebook idea into the image sharing. Logical operations10,24–26 are the common methods to achieve (r, n)threshold. An elegant method10 was given to prevent hackers from seeing secret images; however, the disadvantage of this method is that each (gray or color) shadow is several times bigger than the original image. In 2005, Yang25 proposed a visual secret sharing scheme, in which the size of shadow image is bigger than the given image. In 2004, Yang24 proposed another one which achieved no pixel expansion, but his scheme only supports black-and-white images. In Yan’s scheme,23 which has small size, fault tolerance and security are presented. The original idea, proposed by Shamir15 and Blakley,3 is the extension of the secret sharing scheme to image. It is well-known that the secret sharing is a reliable method for cryptographic key protection, and has many valuable properties. For example, it is a perfect threshold scheme, in which the size of each share does not exceed the size of the secret, and the 1250023-2

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Secret Image Sharing Based on Chaotic Map and Chinese Remainder Theorem

security does not rely on unproven mathematical assumptions. To maintain these advantages, Thien and Lin changed the values of polynomial coefficients aj into a corresponding pixel value of a secret image. According to their design, the size of the shares is very small, and their security has also been analyzed, but the recovery image is lost if the pixel value of the secret image exceeds 251 in Thien and Lin’s scheme.20 To overcome the information loss problem, without extra overhead, Yang et al.26 modified the field to Galois field in mod two8 to obtain lossless recovery of a secret image; however, this approach not only reduces the quality of the stego images but also allows a significant probability that the fake stego image can be authenticated successfully. Asmuth and Bloom2 proposed a (r, n)-threshold secret sharing scheme based on CRT. Some of the following studies can be found in Refs. 12, 16 and 23. All of these researches focused on how to share a number x. In Yan’s scheme,23 one fatal disadvantage of their approaches is that the image is divided into n shadows, which are further distributed to n participants, but we can access to information of the original image. In addition, the approach is lost if the pixel value of the original image exceeds 251. In Meher’s scheme,12 the size of the shadow’s image has expanded to compare with the original image; the proposed scheme cannot be ideal; and the disadvantage of their approach that it is not have fault tolerance. However, in Shyu and Chen’s scheme,16 the shadow’s images are smaller than the original image, then the scheme is not an ideal according to the criterion given in Refs. 17 and 18. In Ref. 22, Huffman code was used to further reduce the shadow size. In Ref. 1, the method does not have fault tolerance; if one shadow image is lost, the original image cannot be recovered. The characteristics of the chaotic maps have attracted the attention of cryptographers to develop new encryption algorithms. As chaotic maps have many fundamental properties such as ergodicity, mixing property and sensitivity to initial condition/system parameter, which can be considered analogous to some cryptographic properties of ideal ciphers such as confusion, diffusion, balance and avalanche property and so on in Ref. 8. Moreover we will introduce the chaotic map into the secret image sharing. In this paper, our aim is to design a secure threshold (r, n)-secret image sharing scheme based on logistic map and CRT. The advantage of our scheme is provided with fault tolerance, ideal and perfect according to the criterion in Refs. 17 and 18. The rest of this paper is organized as follows: preliminaries knowledge is given in Sec. 2. In Sec. 3, our threshold secret image sharing scheme based on chaotic map and CRT is proposed and gives the experimental results of our scheme in Sec. 4, and then analyze the security of the scheme in Sec. 5. Finally, some conclusions of are presented in Sec. 6.

2. Preliminaries In this section, we will introduce Asmuth–Bloom scheme, logistic map and CRT. 1250023-3

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2.1. Asmuth–Bloom scheme In Ref. 2, Charles Asmuth and John Bloom introduced a key safeguarding scheme; which was called Asmuth–Bloom scheme. In this scheme, named a (r, n)-threshold scheme, we always choose a large prime number p. Then, a set of integers p, m1 < m2 < · · · < mn is chosen subject to the following conditions: (i) (mi , mj ) = 1 for i = j, r−1 r (ii) i=1 mi > p i=1 mn−i+1 . Here, as before, n denotes the number of shadows. Any r shadows will suffice for key recovery. Estimates of the density of primes had been also shown that one could easily find primes mi to satisfy (ii). To find composite is still easier. Finally, r let M = i=1 mi . The decomposition process begins with the key x; we assume that 0 < x < p, and let Y = x + Ap,

(2.1)

where A is an arbitrary integer subject to the condition 0 ≤ Y < M . Then the shadows yi : yi ≡ Y

mod mi .

(2.2)

To recover x, it clearly suffices to find Y , if r shadows or more shadows are known, then by the CRT, we can recover the Y , or else, it cannot recover the Y . 2.2. Chinese Remainder theorem The Chinese Remainder theorem (CRT) is an important theory in number theory, and plays an important role in cryptography.14 Theorem 2.1. Consider n > 2 positive relatively prime module, namely m1 , r m2 , . . . , mn , where M = i=1 mi . Let ai denote the remainder of x modulo mi for 1 < i < n, the formula are as follows:  a1 ≡ x mod m1 ,      a2 ≡ x mod m2 , (2.3) ..    .    an ≡ x mod mn , where x has a unique solution in ZM , ZM = 0, 1, 2, . . . , M − 1 and the solution x can be obtained by the following formula: x=

n

ai M i N i .

i=1

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(2.4)

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Then Mi and Ni can be obtained by the following Eqs. (2.5)–(2.7): M =

n

mi

(2.5)

Mi = M/mi

(2.6)

i=1

(2.7) Mi Ni = 1 mod (mi ) n Proof. Suppose Mi = M/mi = l=1,l=i ml , i = 1, 2, . . . , n, we note that Mi and mi are coprime, so there exists a unique inverse element Ni ∈ ZM , i.e. Mi Ni = 1 (mod mi ). Now we prove that ∀ i = 1, 2, . . . , n, x satisfies ai = x mod mi . Note that mi | Mj holds. So (Mj × Nj mod mj ) mod mi ≡ ((Mj mod mi ) × ((Nj mod mj ) mod mi )) mod mi ≡ 0. Since (Mi × (Ni mod mi )) mod mi ≡ (Mi × Ni ) mod mi ≡ 1, we obtain that x ( mod mi ) ≡ ai . Now we prove that there is a unique solution of Eq. (2.3). We suppose that x is another solution of Eq. (2.3) such that x ≡ ai (mod mi ), (i = 1, 2, . . . , n). Then from x ≡ ai (mod mi ), we can attain that x − x ≡ 0 mod mi , i.e. mi | (x − x). Because mi are coprime, it holds that M | (x − x), i.e. x − x ≡ 0 (mod M ). Then x (mod M ) = x (mod M ). This completes the proof. r From Theorem 2.1, if the fixed r, 1 < r < n, then min(r) = i=1 mi n and max(r − 1) = m , we choose x, which is a positive integer and i i=n−r+2 max(r − 1) < x < min(r). Let ai denote the remainder of x modulo mi for 1 ≤ i ≤ n, the set a1 , a2 , . . . , an is the (r, n)-threshold scheme of x, which is called Asmuth–Bloom scheme. For example: r = 2, m1 = 15, m2 = 19 and m3 = 23, we denote x = 93, 93 ∈ (23, 285), a (2, 3)-threshold, the algorithm can be described as follows: Step 1. Compute ai ; ai ≡ x mod mi , then a1 = 3, a2 = 17, a3 = 1, M = 3 i=1 mi = 6555; Step 2. Compute Mi ; Mi ≡ M mod mi , then M1 = 437, M2 = 245, M3 = 285; Step 3. Compute Ni ; Mi Ni ≡ 1 (mod mi ); then N1 = 8; N2 = 13 and N3 = 18; Step 4. Assume that we know m1 , m2 , a1 and a2 ; then M = m1 × m2 = 285; because that Mi ≡ M mod mi , M1 = 19, M2 = 15 and N1 = 4, N2 = 14; 2 Step 5. Compute x ; x = i=1 ai Mi Ni = 3798; then x = 3798 mod 285 = 93. So the x can be recovered when r > 2.

2.3. Chaotic logistic map In this section, we introduce a chaotic map. The chaotic system which is deterministic has a noise-like behavior, if we give its parameters and initial values, we can reproduce it. These signals are extremely sensitive to initial conditions. 1250023-5

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Theorem 2.2. One of the most famous chaotic systems is logistic map, which is a nonlinear map given by Xn+1 = λXn (1 − Xn ),

(2.8)

where λ ∈ (0, 4), Xn ∈ [0, 1]. When 3.5699 . . . < λ < 4, the system is Chaotic. Figure 1 shows the plot of x(n) under n for X0 = 0.618 and λ = 3.98 after 700 iterations. 3. The Proposed Scheme In this section, we describe a kind of image sharing algorithm based on logistic map and CRT. 3.1. Shares construction procedure The share construction procedure requires two phases to divide a secret image into n shadows as shown in Fig. 2. Phase 1: Modulation for image pixel Consider a M × N gray image A; we first pretreatment the pixel of the original image A by logistic map. The pixel values are in the range [0, 255]. We set aij as the pixel of the image A. We can deal with the aij by Eq. (3.1), where aij is transformed into aij as follows: aij = 256 × Xk × aij ,

(k = 0, 1, 2, . . . , i × N + j),

(3.1)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

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Chaotic behavior of logistic map with X0 = 0.618 and λ = 3.98. 1250023-6

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Image A

λ , Xi

Logistic Map

Phase 1

+p Phase 2

CRT Prime qj (1≤j≤m)

S1 Fig. 2.

S2

......

Sm

Flowchart of shares construction phase.

where the symbol · is the floor function meaning “the greatest integer less than or equal to”. This phase modifies the pixel, and ensures the security of the images. Phase 2: Constructing the process by CRT Because our encoding process is essentially based on logistic map and CRT, we choose a set of m relatively prime module q1 < q2 < s < qm in this phase, then compute α = q1 × q2 × · · · × qt , β = qm−t+2 × qm−t+3 × · · · × qm . In order to ensure the (t, m)-threshold property in the Asmuth–Bloom’s scheme, we adjust aij to aij such that β < aij < α, aij by adding an offset p, where β < p < α − 21 6. aij is transformed into aij as follows: aij = aij + p,

(3.2)

the aij is within the range of (β, α) so as to be shared in a (t, m) manner by CRT. The encoding scheme is formally described as follows: Input: a M × N gray image A, a set of participants’ m, threshold t(2 ≤ t ≤ m); Output: shadow Sj , ej , and qj for 1 ≤ j ≤ m. Step 1. The original image A is processed by logistic map by Eqs. (3.1) and (3.2), and then we get the aij ; Step 2. Choose q1 , q2 , . . . , qm |(qi , qj ) = 1, 2 ≤ q1 < q2 < · · · < qm ≤ 256; Step 3. Compute α = q1 × q2 × · · · × qt , qm−t+2 × qm−t+3 × · · · × qm ; Step 4. Choose p(β < p < α − 216 ), the aij is converted aij by using (10); Step 5. For (1 ≤ k ≤ m) {Sk = Φ ek = p mod qk } end 1250023-7

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Step 6. For (1 ≤ i ≤ M ) For (1 ≤ i ≤ N ) For (1 ≤ k ≤ m) {sk = aij mod qk Sk = Sk ||{sk }; } end end end Step 7. Output (S1 , S2 , . . . , Sm ; e1 , e2 , . . . , em ; q1 , q2 , . . . , qm ). Note that the original image A is distributed into S1 , S2 , . . . , Sm , and the p is decomposed as ej according to qj by CRT. Furthermore, we can distribute qj and ej to the participators as the key of the original image, then Sj as shadows to participant j for 1 ≤ j ≤ m. 3.2. Recover procedure The proposed scheme is a (r, m)-threshold scheme, so at least any t(r ≤ t ≤ m) of m shadows must be collected separately in advance to restore the secret image. Figure 3 shows the flowchart of our proposed recover procedure. S1

......

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Sm

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......

em

Prime qj (r≤t≤m) CRT

CRT

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−p

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×256

aij' X  i×256

Phase 2

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Fig. 3.

The flowchart of our proposed recovers procedure. 1250023-8

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The dealer uses t collected shadows to reconstruct image A by CRT. Suppose t shadows S1 , S2 , . . . , St are collected and e1 , e2 , . . . , et , q1 , q2 , . . . , qt are attained. We can recover the image A. The decoding scheme is formally described as follows: Input: t participants S1 , S2 , . . . , St ; e1 , e2 , . . . , et ; q1 , q2 , . . . , qt ; λ, X0 ; Output: The original image A. Step 1. Compute p; we know e1 , e2 , . . . , et and q1 , q2 , . . . , qt , the p can be computed by CRT; Step 2. Compute S; we know S1 , S2 , . . . , St and q1 , q2 , . . . , qt , the S can be computed by CRT; Step 3. Compute aij ; the value of S subtract the p; Step 4. Compute aij Xt is recovered from λ and X0 by logistic map and aij = aij /256 × Xt ; Step 5. Repeat the steps from Step 2 to Step 4; and then we can recover image A. 4. Experimental Results We demonstrate the feasibility and applicability of our image sharing scheme by showing the experimental results of sharing a gray image in the (3, 4)-threshold. The experimental image A is chosen as 512 × 512 gray Lena image, as shown in Fig. 3. We choose (q1 , q2 , q3 , q4 ) = (241, 247, 251, 253). Then α = q1 × q2 × q3 = 14941277 and β = 63503. We choose p = 64221. Our scheme is coded in Matlab 7.0 running in a PC Windows environment. Figure 4 shows the encoding results of the four shadows with respect to image Lena. Here Figs. 4(b)–4(e) are the shadows of the image Lena respectively. These shadows which are distributed to the four participants are all seemingly random images. The size of each of the four shadows is 512 × 512. We display each of the shadows as a 512 × 512 image. p = 64221 is also shared among the four participants by CRT. We obtain (e1 , e2 , e3 , e4 ) = (115, 236, 216, 212). A set of participants can reconstruct number p and image Lena by using logistic map and CRT with their keys and shadows only when separately t ≥ 3. The experimental result is shown in Fig. 5. Any group of three shadows of S1 , S2 , S3 , S4 or more than three shadows can recover the original image Lena. From Table 1, we may see that the proposed method has lossless and fault tolerance properties. In Fig. 5, it shows that the size of every shadow image is equal to the size of the original image Lena, and the knowledge of any three or fewer shadow images provides no information about the original image, and then, the scheme is ideal and perfect according to the criterion given in Refs. 17 and 18. 5. Security Analysis In this section, we analyze the security of the scheme. Specifically, we will analyze its resistance of the most important attacks. 1250023-9

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(a) Lena

Fig. 4.

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(c) S2

(d) S3

(e) S4

Experimental result: (a) image Lena; (b)–(e) are the shadow images of Lena.

We propose such a method that secret image can be shared by several shadow images. The size of each shadow image is the same as the secret image in our method, and it gives our method certain benefits, such easier process for storage, transmission, and hiding. The size of shadow images are no expansion by comparing with the original image Lena. 1250023-10

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(a) {S1 , S2 , S3 }

(c) {S1 , S3 , S4 }

(b) {S1 , S2 , S4 }

(d) {S2 , S3 , S4 }

(e) {S1 , S2 , S3 , S4 }

Fig. 5. Experimental result: image Lena is recovered from any group of three shadows or more than three shadows (b)–(e) which are in Fig. 4. Table 1. Comparisons of the proposed method and the Thien and Lin’s scheme20 and Shyu and Chen’s scheme.16

The proposed method Thien and Lin’s Scheme 1 Thien and Lin’s Scheme 2 Shyu and Chen’s scheme

The size of shadow images

The reconstructed image

Fault tolerance

The same size 1/m More than 1/m 1/2

Lossless Lossy Lossless Lossy

Yes Yes Yes Yes

5.1. Statistical analysis The statistical test tries to determine the confusion and diffusion properties of the proposed scheme. This test is performed by a correlation test of adjacent pixels of the original image and its shadows. The correlation test between adjacent pixels in the images has been made by selecting in a random way 2500 pairs of two vertically adjacent pixels, 2500 pairs of two horizontally adjacent pixels, and 2500 pairs of two diagonally adjacent pixels, for each original image as well as its shadows. In each case, the correlation coefficient of each pair has been computed and the results are shown in Table 2. As is well-known, the correlation coefficient is defined by the 1250023-11

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C. Hu, X. Liao & D. Xiao Table 2.

Correlation coefficients of adjacent pixels for the shadows and original image.

Horizontal Vertical Diagonal

Lena

Shadow 1

Shadow 2

Shadow 3

Shadow 4

0.9778 0.9864 0.9632

0.0075 −0.0024 0.0021

0.0029 −0.0008 0.0029

0.0038 0.0033 0.0068

−0.0019 0.0017 −0.0034

following expression N

N

(xj × yj ) − N

j=1

N

xj × N

j=1

N

yj

j=1

Cr =  ,

  2   2 

N N N N

 2   2  

N xj − N xj   × N yj − N yj  j=1

j=1

j=1

(5.1)

j=1

where x and y are the value of two adjacent pixels in the image and N is total number of pixels selected from the image for the calculation. As it is shown in Table 2, the correlation coefficients are far apart. For example, in the original image, the correlation coefficient for two horizontally adjacent pixels is 0.9778, which is very near to 1, as it was expected. Nevertheless, in the four shadows, these coefficients are 0.0075, 0.0029, 0.0038, and −0.0019, respectively, i.e. these are very close to 0. Similar results for vertical and diagonal pixels are obtained (see Table 2). It is observed that the correlation coefficients of the images and the shadows are different enough, which guarantee the confusion and diffusion of the pixels. In addition, the correlations in the shadows are very small indeed. Finally, Fig. 6 shows the correlation distribution of 2500 couples of horizontal adjacent pixels of the secret images and shadows, respectively. The correlation distributions of two vertically and diagonally adjacent pixels have the same appearance in Figs. 7 and 8. As it was expected, the adjacent pixels in the original image are distributed along the diagonal due to the fact that the gray levels of those pixels have a very near value. One can observe how in the first case, the distributions follow, approximately, the principal diagonal which gives an idea of the strong correlation among the pixels of the secret images; nevertheless, the distribution of the three classes of adjacent pixels for the shadows seems to be random. The clouds of points seem to distribute in a random way in Figs. 6–8, which indicates the weak correlation among the pixels of the shadows. 5.2. Security of the sharing scheme The security of the sharing scheme is based on logistic map and CRT. In this method, the threshold is three, and then any two or fewer shadow images cannot get sufficient information to recover the secret image. Without loss of generality, consider a 512 × 512 image a is the pixel of the original image. Let us inspect how a 1250023-12

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C. Hu, X. Liao & D. Xiao Correlation of vertical adjacent pixels of original image

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Secret Image Sharing Based on Chaotic Map and Chinese Remainder Theorem Correlation of diagonal adjacent pixels of original image

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can be recovered, an offset p is introduced into the scheme, the pixel a is modified through Eqs. (3.1) and (3.2). It is transformed into a , a = 256×Xk ×a+p. From Eqs. (2.3) and (2.4), we need n prime numbers q to recover the a. If we only have two shadow images and fewer prime numbers q, we cannot recover the a , because m r the a is out of the range of ( j=m−r+2 qj , i=1 qi ). If a hacker wants to get all the information on the image by the exhaustive attack method, the possibility of obtaining the original image is only (512×512) 218 1 1 + p = + p . (5.2) 28 × 28 216 The p is 64221 in this scheme, and then we have 262144 (512×512) 218 1 1 1 +p = + 64221 = . 28 × 28 216 129757

(5.3)

From Eq. (5.3), we found that the exhaustive attack method is infeasible. Also note that the original image has been shared by the logistic map before doing any sharing process; therefore, there is no correlation among the shadow images. In other words, although the neighboring pixels are usually similar, the lack of information cannot be supplied from the image property. Even if the hacker has

Fig. 9.

(a) {S1 , S2 }

(b) {S2 , S3 }

(c) {S1 , S4 }

(d) {S2 , S3 }

(e) {S2 , S4 }

(f) {S3 , S4 }

The recovered images are recovered by any groups of two out of the four shadow images. 1250023-16

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the information on the shadow image, it is only a part of the original images, and 1 )262144 is a he cannot get all the information on the original image. Since ( 129757 very small number, the hacker only has little chance to obtain a part of the shadow images, it is impossible to obtain the exact original image. Let us consider a more favorable case for the attacker. Suppose that the attacker knows r − 1 shadow images and the qi . In this situation, the attack cannot recover the original image by CRT. Figure 9 illustrates that any groups of two out of the four shadows image cannot recover the image Lena. Similarly, if the attacker has only a shadow image, he cannot get any information of the original image. 6. Conclusions Based on logistic map and CRT, the secret image can be shared by several shadow images. The size of each shadow image has the same size of the secret image, and the knowledge of any r − 1 or fewer shares provides no information about the original image. The proposed scheme meets the requirements of the criterion given in Refs. 17 and 18, so the scheme is ideal and perfect. In addition, the proposed scheme has the ability of fault tolerance, which means although a part of the shadow images are lost, the original image can still be recovered if the quantity of the collected shadow images are equal to threshold r or more than r. On the other hand, the method is a lossless sharing method, and we can attain a lossless recovered image. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants (Nos. 60973114, 60873201, 61070246) and the Natural Science Foundation Project of CQ CSTC (2009BA2024, 2011jjjq40001) and State Key Laboratory of Power Transmission Equipment and System Security and New Technology, Chongqing University (2007DA10512709207), Fundamental Research Funds for the Central Universities (Grant No. CDJXS10182215). References 1. G. Alvarez, L. Hernandez Encinas and A. Martin del Rey, A multisecret sharing scheme for color images based on cellular automata, Inf. Sci. 178 (2008) 4382–4395. 2. C. Asmuth and J. Bloom, A modular approach to key safeguarding, Inf. Theory IEEE Trans. 29 (1983) 208–210. 3. G. R. Blakley, Safeguarding cryptographic keys, in Proc. NCC, Vol. 48 (AFIPS Press, 1979), pp. 313–317. 4. C. C. Chang and R. J. Hwang, Sharing secret images using shadow codebooks, Inf. Sci. 111 (1998) 335–345. 5. K. Kaya and A. A. Selcuk, Threshold cryptography based on Asmuth–Bloom secret sharing, Inf. Sci. 177 (2007) 4148–4160. 6. H. Y. Leung, L. M. Cheng and L. L. Cheng, A robust watermarking scheme using selective curvelet coefficients, Int. J. Wavelets, Multiresolut. Inf. Process. 7 (2009) 163–181. 1250023-17

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7. H. Y. Leung, L. M. Cheng and L. L. Cheng, Robust watermarking schemes using selective curvelet coefficients based on a hvs model, Int. J. Wavelets, Multiresolut. Inf. Process. 8 (2010) 941–959. 8. X. F. Liao, D. Xiao, Y. Chen and T. Xiang, Theory and Application of Chaotic Cryptography (Science Press, China, 2009). 9. O. J. Lou, Z. X. Wang and X. H. Wang, A new contourlet domain based image watermarking scheme resilient to geometrical attacks, Int. J. Wavelets, Multiresolut. Inf. Process. 7 (2009) 115–130. 10. R. Lukac and K. N. Plataniotis, Bit-level based secret sharing for image encryption, Pattern Recognition 38 (2005) 767–772. 11. S. P. Maity and M. K. Kundu, Performance improvement in spread spectrum image watermarking using wavelets, Int. J. Wavelets, Multiresolut. Inf. Process. 9 (2011) 1–33. 12. P. K. Meher and J. C. Patra, A new approach to secure distributed storage, sharing and dissemination of digital image, in Circuits and Systems, 2006. ISCAS 2006. Proceedings 2006 IEEE International Symposium on (IEEE, 2006 ), pp. 373–376. 13. M. Naor and A. Shamir, Visual cryptography, in Advances in Cryptology/EUROCRYPT’94 (Springer, 1995), pp. 1–12. 14. B. Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in C (A1bazaar, 2007). 15. A. Shamir, How to share a secret, Communications of the ACM 22 (1979) 612–613. 16. S. J. Shyu and Y. R. Chen, Threshold secret image sharing by Chinese Remainder theorem, in Asia-Pacific Services Computing Conference, 2008. APSCC’08. IEEE (IEEE, 2008), pp. 1332–1337. 17. D. R. Stinson, An explication of secret sharing schemes, Designs, Codes and Cryptography 2 (1992) 357–390. 18. D. R. Stinson, Cryptography: Theory and Practice (CRC press, 2006). 19. N. Taneja, B. Raman and I. Gupta, Chaos partial encryption of spiht compressed images, Int. J. Wavelets, Multiresolut. Inf. Process. 9 (2011) 317–331. 20. C. C. Thien and J. C. Lin, Secret image sharing, Computers & Graphics 26 (2002) 765–770. 21. L. Wang, Z. Cao, X. Li and H. Qian, Simulatability and security of certificateless threshold signatures, Inf. Sci. 177 (2007) 1382–1394. 22. R. Z. Wang and C. H. Su, Secret image sharing with smaller shadow images, Pattern Recogn. Lett. 27 (2006) 551–555. 23. W. Q. Yan and Wei Ding, Image sharing based on Chinese Remainder theorem, J. North China Univ. Tech. 12 (2000) 6–9. 24. C. N. Yang, New visual secret sharing schemes using probabilistic method, Pattern Recogn. Lett. 25 (2004) 481–494. 25. C. N. Yang and T. S. Chen, Aspect ratio invariant visual secret sharing schemes with minimum pixel expansion, Pattern Recogn. Lett. 26 (2005) 193–206. 26. C. N. Yang, T. S. Chen, K. H. Yu and C. C. Wang, Improvements of image sharing with steganography and authentication, J. Syst. Software 80 (2007) 1070–1076.

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