A Novel Perceptual Secret Sharing Scheme

A Novel Perceptual Secret Sharing Scheme Xuehu Yan1 , Shen Wang1, , Ahmed A. Abd El-Latif1,2 , Jianzhi Sang3 , and Xiamu Niu1 1

3

School of Computer Science and Technology, Harbin Institute of Technology, 150080 Harbin, China 2 Mathematics Department, Faculty of Science, Menoufia University, Shebin El-Koom 32511, Egypt Department of Mathematics, Harbin Institute of Technology, 150080 Harbin, China {xuehu.yan,jianzhi.sang,xiamu.niu}@ict.hit.edu.cn, [email protected], ahmed [email protected]

Abstract. In this paper, a novel perceptual secret sharing model namely PSS model is defined. Then, a general (1, k out of n) PSS scheme is proposed based on maximum likelihood estimation (MLE). The proposed model has a large number of valuable features that previous secret sharing schemes fail to have. Furthermore, the proposed PSS scheme satisfies P (l, k, n) threshold mechanism which means that no perceptual information of the secret data will be revealed when the number of shadows is less than l, the degraded perceptual quality will be recovered when the number of shadows is greater than or equal to l and less than k, the more shadows the better recovered perceptual quality will be. Lossless perceptual quality will be recovered when the number of shadows is greater than or equal to k and less than or equal to n that could control access for different owners. This property will be useful for prominent prevalence in real applications like pay-per-view videos, Pay- TV/Music and art-work image vending, video on demand (VOD), etc. Keywords: Perceptual secret sharing, progressive secret sharing, Threshold, maximum likelihood estimation (MLE).

1

Introduction

Nowadays, the wide use of media data such as digital images or videos in various applications attracts the attention to the protection issues. In recent years, various cryptographic schemes have been proposed as a possible solution to the security of digital images, among which cryptography attracts more attention due to its prominent prevalence in real applications. In fact, cryptography provides a very powerful method to protect both confidentiality and robustness. In many applications like pay-per-view videos, Pay- TV/Music and art-work image vending, video on demand (VOD), the following feature namely “perceptual secret sharing (PSS)” is very useful. This feature requires that the quality

Corresponding author.

Y.Q. Shi (Ed.): Transactions on DHMS IX, LNCS 8363, pp. 68–90, 2014. c Springer-Verlag Berlin Heidelberg 2014

A Novel Perceptual Secret Sharing Scheme

69

of visual secret data could be partially degraded after the sharing phase i.e., the secret is still partially perceptible after the sharing phase. Such perceptibility makes it possible for potential users to view low-quality copies of the media data products before buying them. Thus, it is advisable that the visual quality degradation can be continuously controlled by a control factor. This control factor is used a rough measure of the degradation due to the fact that there does not exist a well-accepted objective measure of visual quality of digital images. Fig.1 shows the general view of PSS.

threshold P(l,k,n) Control factor

Secret data

Perceptual secret sharing

Shared secret data

Fig. 1. Perceptual secret sharing

To realize the PSS, the following properties are concerned: (1) Implement the P (l, k, n) threshold mechanism: Less than l paying number cannot reveal any perceptual information of the secret data. Degraded perceptual quality will be recovered when paying number is less than k and greater than or equal tol, the more paying the better recovered perceptual quality will be. Lossless perceptual quality will be recovered when paying number is more than or equal to k and less than or equal to n that could control access for different owners. (2) Simple recovery of the original multimedia: only simple computation for light-weighted receivers or devices. (3) The order of the shadow images (shares) is alternative: There is no need to record the order of the shadow images when recovering original multimedia for convenience. (4) Avoid the pixel expansion: the size of the shadow images is the same as the original multimedia, which could reduce the storage and transmission bandwidth, and avoid the shape distortion and size remembrance. (5) No codebook [1] (that designs the coding method consist of basis matrices, encoding block pairs or code words, and decrypted results of different shares corresponding to different secret pixels) design. Since the codebook is not easily constructed, especially for some specific applications. (6) The color representation method is the same as color representation method of digital multimedia (digital color) [2][3][4][5]: 0 denotes black or opaque and 1

70

X. Yan et al.

denotes white or transparent in digital multimedia and common digital processing software for digital applications, such as BMP, JPEG, Matlab and Photoshop This representation can avoid flipping (that is 0 → 1 or 1 → 0), reversing or complementing operations and will be convenient for applications since digital images are portable, easily distributed and transmitted. Cryptographic algorithms especially multimedia encryption and secret sharing may be potential technologies to realize PSS. Several traditional cryptographic algorithms [6][7][8][9] may be used to protect the images, videos and audios in such applications However, these schemes are not loss-tolerant, not P (l, k, n) threshold mechanism, and needs complex computation. Secret image sharing techniques are also used in protecting secret images in addition to traditional cryptography. It distributes a secret image among some participants through splitting the secret image into noise-like shadow images (also called shares or shadows) and recovering the secret by collecting sufficient authorized participants(shadow images). It has attracted more attention to scientists and engineers. Visual secret sharing (VSS) [10][11][12][13], that is also called visual cryptography scheme (VCS), and Shamir’s polynomial-based scheme are the primary branches in this field. Naor and Shamir [10] firstly propose the threshold-based VCS. In this scheme, a binary secret image is shared by generating corresponding n noise-like shadow images And any k or more noise-like shadow images are superposed to recover the secret image visually based on human visual system (HVS) and probability. However, less than k participants cannot reveal any information of the secret image by inspecting their shares. VCS can realize some properties concerned in PSS. The main advantages of the VCS are simple recovery and alternative order of the shadow images Simple recovery means the decryption of secret image is completely based on HVS without any cryptographic computation. However it also suffers from lossy recovery and pixel expansion [11][12] Based on the pioneer work of Naor and Shamir [10], more researches have been done extensively to overcome the disadvantages. The scheme of Giuseppe et al.’s [14] has meaningful shadow images Wang et al. [3] proposes a secret sharing scheme based on Boolean operation that is lossless. However, these schemes suffer from pixel expansion and codebook design. Since VSS by random grids (RG) can avoid pixel expansion and has no complex codebook, some other researchers have paid more attention to RG-based VSS. RG-based encryption method is firstly presented by Kafri and Keren [15] In RG-based encryption method, the binary secret image is generated into two noise-like RG (shadow images or share images). The shadow images have the same size as the original secret image. The decryption operation is the same as traditional VC. The image encryption by RG [1][15][16] satisfies no pixel expansion but lossy recovery. Besides the above mentioned schemes, some other schemes based on Shamir’s polynomial and Lagrange interpolation, have been proposed [17][18] to realize the secret image sharing These schemes have the advantages of (k, n) threshold, meaningful shadow images, lossless recovery and no pixel expansion However,


71

they suffer from P (l, k, n) threshold mechanism, complex recovery and known order of the shadow images. Additionally, progressive visual secret sharing (PVSS) schemes based on VCS [19][20][21][22], RG [23], Shamir’s polynomial [24] or transformation [25] have the progressive or perceptual property. Progressive property means better perceptual quality will be gained when more shadow images are collected However, they overall suffer from the same disadvantages as VCS, RG and Shamir’s polynomialbased secret sharing. Here we aim to introduce a novel direction in secret sharing namely PSS This work is the first attempt, to the best of our knowledge, of introducing a new model for perceptual secret sharing. In this paper, a general 1 k out of n PSS scheme based on maximum likelihood estimation (MLE) is proposed. The proposed scheme is able to model large number of valuable features in secret sharing. It has the properties of P (l, k, n) threshold sharing, which could gain different perceptual quality, lossless recovery, be loss-tolerant and control access, and keeps the same as color representation method of digital images, which is useful for more and more digital applications. Besides, it can realize other features such as simple computation (needs only adding and comparing), alternative order of shadow images in recovery, avoiding the pixel expansion and no codebook design. Experimental results show the feasibility, effectiveness of the proposed scheme. The rest of the paper is organized as follows. The basic definitions of the proposed PSS model are presented in Section 2. The proposed 1 k out of n PSS scheme is introduced in Section 3. Section 4 is devoted to the experimental results and analysis. Finally, Section 5 concludes this paper.

2

Basic Definitions and Preliminaries

In this section, we firstly give some fundamental definitions and preliminaries for the proposed PSS model. The original binary secret image is denoted as S with pixel value S(i, j) (1 ≤ i ≤ M, 1 ≤ j ≤ N ), size (S) = (M, N ),where function size means size of S. The original binary secret image S is shared among n (2 ≤ n, n ∈ Z + ) shadow images(shares) SC1 , SC2 , · · · SCn by generation function Gf , that is (SC1 , SC2 , · · · SCn ) = Gf (S, k, n)(2 ≤ k ≤ n, k ∈ Z + ); While the recovered secret image S is recovered from t (1 ≤ t ≤ n, t ∈ Z + ) shadow images by recovery function Rf , that is St = Rf (SCi1 , SCi2 , · · · SCit ), where (i1 , i2 , · · · , it )is the subsequence of (1, 2, · · · , n). V Q (St ) denotes the perceptual visual quality of the recovered secret image St . AS0(resp., AS1) is the black (resp., white) area of original secret imageS defined as AS0 = {M, 1 ≤ j ≤ N }. 2.1

Fundamental Definitions

Definition 1 (PSS). A secret sharing scheme is called a P (l, k, n) PSS scheme, if it satisfies the following conditions:

72

X. Yan et al.

1. V Q (St ) = 0whent < l; V Q St2 ≥ V Q St1 > 0 whenk ≥ t2 ≥ t1 ≥ l; St = S when t ≥ k 2. The computation of recovery function Rf is simple, only consists of addition, subtraction, cryptographic computation Boolean operations or no 3. Stm = Rf SCim1 , SCim2 , · · · SCimt = St where (m1 , m2 , · · · , mt )is a permutation of (1, 2, · · · , t) 4. size (S ) = size (SCi ) = size (S) , i = 1, 2, · · · , n 5. There is no codebook design in the computation of Gf and Rf except for the parameters and input images. 6. “1” denotes white pixels, “0” denotes black pixels, which are the same as color representation method of digital multimedia Definition 2 (Contrast denoted as α). In PSS, the visual quality of the recovered secret image St , which will decide how well human eyes could recognize the recovered image, corresponding to the original secret image S, is evaluated by contrast. Contrast is defined in Eq.(1) [1][26][27] , as follows: P (S [AS1] = 1) − P (S [AS0] = 1) P1 − P0 = 1 + P0 1 + P (S [AS0] = 1)

(1)

where P0 (resp.,P1 ) is the appearance probability of white pixels in the recovered imageS in the corresponding black (resp., white) area of original secret imageS, that is P1 is the correctly decrypted probability corresponding to the white area of original secret imageS, and P0 is the wrongly decrypted probability corresponding to the black area of original secret imageS Definition 3 (Security and visually recognizable). The recovered secret image S could be recognized as the content of the corresponding original secret image S ifα > 0. The scheme is secure i.e.α < 1 when l ≤ t < k, which means not all information (including content and details) of S could be recognized throughS ; α = 0 when t < l, which means not any information (including content and details) of S could be recognized throughS The recovered secret image S is lossless if α = 1 when t ≥ k, which means all information (including content and details) of S is recognized throughS 2.2

Maximum Likelihood Estimation (MLE)

MLE is one kind of estimation the parameters of a statistical model. MLE seeks the parameter values that are most likely to have produced the observed distribution. It begins with the mathematical expression known as a likelihood function of the sample data. Clearly, the likelihood of a set of data is the probability of obtaining that particular set of data given the chosen probability model [28]. This expression contains the unknown parameters. Those values of the parameter that maximize the sample likelihood are known as the maximum likelihood estimates. Fig.2 is an example for the MLE. By MLE, we decode 00011 → 0. But if only 4 bits are received, such as 0011, it could not decode by MLE.


73

⎧Receive bits MLE Recover secret bit ⎪ 00011 → 0 ⎪ ⎨ 00111 → 1 ⎪ ⎪⎩ 0011 → error Fig. 2. Example of MLE

3

The Proposed PSS (1 k out of n) Scheme

Here, we present the proposed PSS scheme, based on MLE to realize the model in section 2 and satisfy all the properties and needs of PSS applications 3.1

Shadow Images Generation Phase

In this section, we propose a P (1, k, n) PSS scheme based on MLE, namely MLE-based PSS. The shadow images generation architecture is illustrated in Fig.3. The detailed algorithm is described as follows.

Algorithm 1. The proposed MLE-based PSS scheme. Input: A M × N original binary secret image S, threshold P (l, k, n) (l = 1, 3 ≤ k ≤ n, n ∈ Z + , k ∈ 2Z + + 1). Output: n shadow imagesSC1 , SC2 , · · · SCn Step 1: For each position (i, j) ∈ {(i, j)|1 ≤ i ≤ M, 1 ≤ j ≤ N }, repeat Steps 2-5. Step 2: Randomly generate bp ∈ {0, 1} , 1 ≤ p ≤ n Step 3: Computenum = |{p|bp = S(i, j), p ∈ {1, 2, · · · n}}| and Ln using Eq.(5). If num < Ln go to step 4; else go to step 5. where {p|bp = S(i, j), p ∈ {1, 2, · · · n}}is a set whose elements are serial numbers of bp that is equal to S(i, j), || denotes the elements number of a set, numis the total number of bp that is equal to S(i, j) Step 4: Randomly select Ln − numunrepeated numberslq ∈ {p|bp = S(i, j), p ∈ {1, 2, · · · n}} Compute bp = bp , p ∈ {1, 2, · · · , n} , p = lq q = 1, 2, · · · Ln − num where bp is a flipping (bit-wise complementary, that is 0 → 1 or 1 → 0) operation. Step 5: ComputeSCp (i, j) = bp , p ∈ {1, 2, · · · , n} Step 6: Output the n shadow imagesSC1 , SC2 , · · · SCn

The recovery algorithm of the MLE-based PSS model is described as follows. In order to clarify the idea of the proposed scheme, an example is shown in Fig.4 in a P (1, 5, 6) PSS scheme assuming current secret bitS(i, j) = 1. Randomly generatebp ∈ {0, 1} , 1 ≤ p ≤ 6, where b1 = 0, b2 = 0, b3 = 1, b4 = 0, b5 = 1, b6 = 0. ComputeLn = n − k2 = 6 − 52 = 4using Eq.(5) and num = |{3, 5}| = 2. Sincenum < Ln , hence randomly select 2 unrepeated numbers from {1, 2, 4, 6}such as lq = 2, 6, then b2 , b6 are flipping. Finally SCp (i, j) is obtained by setting SCp (i, j) = bp , p ∈ {1, 2, · · · , n}

74

X. Yan et al.

source binary secret image

threshold P(1,k,n)

S

Random

b1

b2



bn

S (i, j ) Compute

Ln

num = { p | bp = S (i, j ), p ∈ {1, 2, n}}

num < Ln

Yes

No

Randomly select Ln − num unrepeated numbers

lq ∈ { p | bp ≠ S (i, j ), p ∈ {1, 2, n}} Flipping

SC1 (i, j )

SC2 (i, j )



SCn (i, j )

SC1

SC2



SCn

Fig. 3. Shadow images generation architecture of the proposed scheme

Algorithm 2. Secret image recovery of the proposed scheme. Input: t shadow imagesSCi1 , SCi2 , · · · SCit Output: A M × N recovered binary secret image S Step 1: For each position (i, j) ∈ {(i, j)|1 ≤ i ≤ M, 1 ≤ j ≤ N }, repeat Steps 2-3. Step 2: Compute bp = SCip , 1 ≤ p ≤ t t Step 3: Computenum2 = bp and Lt = 2t . p=1

If num2 < Lt S(i, j) = 0; Else if num2 > Lt S(i, j) = 1; Else S(i, j) = 1 or 0 randomly Step 4: Output the recovered binary secret image S


75

threshold P(1,5,6)

Random

b1 = 0

b2 = 0

b3 = 1

b4 = 0

b5 = 1

b6 = 0

S (i, j ) = 1 Compute

k 5 Ln =  n −  = 6 −  = 4 2  2 

num = {3,5} = 2

num < Ln

Yes

Randomly select Ln − num = 2 unrepeated numbers lq ∈ {1, 2, 4, 6} , lq = 2, 6

Flipping

b1 = 0

b2 = 1

b3 = 1

b4 = 0

b5 = 1

b6 = 1

Fig. 4. An example of the proposed scheme, P (1, 5, 6) PSS scheme

From the generation and recovery phases of the proposed scheme, we remark the main merits of the proposed scheme as follows: 1) The proposed PSS model is the first attempt, to our knowledge, of defining the PSS 2) The proposed scheme satisfies P (l, k, n) threshold mechanism which could be lossy or lossless depending on the number of the shadows that the user have to recover. (i.e., not any perceptual quality if the number of shadows is less than l. lossy if the number of shadows is greater than or equal to l and less than k, the more shadows the better recovered perceptual quality will be. Lossless perceptual quality will be recovered when the number of shadows is greater than or equal to k and

76

X. Yan et al.

less than or equal to n , that could control access for different owners. 3) The computation is simple: The generation phase and recovery phase include only addition and comparison. 4) In the recovery phase the order of the shadow images is alternative: The shadow images are equal to each other. 5) Color representation method is the same as color representation method of digital images: 0 denotes black and 1 denotes white. This property could also be applied in other VSS scheme. 6) No pixel expansion: Since the shadow images have the same size with the binary secret image M × N binary secret image S 7) No codebook design: The generation of the shadow images is based on MLE 3.2

Performance Analyses

In this section, we analyze the performance of the proposed scheme by visual quality and security. Here in the proposed scheme. Let x denote a certain pixel X(i, j) of binary imageX with the size of M × N . The probability of pixel color is transparent (1), say , and the same for the probability of pixel color is opaque M N (0). Besides, X (i, j), 1 ≤ i ≤ M, 1 ≤ j ≤ N i=1 j=1

First, Eq. (2) should be satisfied to perform MLE, and k shadow images could be recovered correctly if Eq. (3) satisfies. In the extreme case, the number of lost shadow images n − k is the change value of the determinant number for correct decoding as in Eq. (4). Based on Eqs. (2-4), we obtain Eq. (5). k ∈ 2Z + + 1, n ∈ Z + , k ≤ n

(2)

k Lk = 2

(3)

n − k = Ln − Lk

(4)

k Ln = n − , 2

Ln , n ∈ Z + , k ∈ 2Z + + 1, k ≤ n

(5)

Theorem 1 (Contrast). The contrast of the recovered secret image by stacking any t(1 ≤ t ≤ n) shadow images, which are generated by the proposed PSS scheme, is computed as follows: ⎧ 1, f or k ≤ t ≤ n ⎪ ⎪ ! ⎪ ⎪ 2t t− 2t −C t ⎪ w−1 t−w+1 w ⎪ 2 CL C t−w +CL Cn−Ln +···+CLn Cn−Ln ⎪ n n n−Ln n ⎪ ⎪ ! ⎨ if t ∈ 2Z+ − 1, f or 1 ≤ t < k t t− t 2 2 t−w w−1 t−w+1 w t α (1, k, n, t) = 2Cn − CLn Cn−Ln +CLn Cn−Ln +···+CLn Cn−Ln ⎪ « ⎪ t +1 t− t −1 t t ⎪ „ w t−w ⎪ w−1 t−w+1 t 2 2 ⎪ −Cn 2 CLn Cn−Ln +CL Cn−Ln +···+CL2n Cn−L + 12 CL2n Cn−L ⎪ n n n ⎪ « „ ⎪ if t ∈ 2Z+ , f or 1 ≤ t < k ⎪ ⎩ 2C t − C w C t−w +C w−1 C t−w+1 +···+C 2t +1 C t− 2t −1 + 1 C 2t C 2t n

Ln

n−Ln

Ln

n−Ln

Ln

n−Ln

2

Ln

n−Ln

(6)


77

Where α (1, k, n, t)denotes the contrast of the recovered secret image S with t shadow images in the proposed P (1, k, n) PSS scheme Cab denotes the number of combinations selecting b numbers from a numbers, a, b ∈ Z + , b ≤ aLn = n − k/2.,w = min (Ln , t). Proof (of Contrast) The proof considers two cases: Case 1: k ≤ t ≤ n. Case 2: 1 ≤ t < k For Case 1: k ≤ t ≤ n From algorithms 1 and 2, we know that the secret can be recovered losslessly when k ≤ t ≤ n, hence α = 1for k ≤ t ≤ n For Case 2: 1 ≤ t < k Without loss of generality, we use a secret bit 1 as an example. sc1 , sc2 , · · · , scLn , scLn +1 , scLn +2 , · · · , scn are generated shadow images bits according to the secret pixel s = 1 in the proposed P (1, k, n) PSS scheme the first Ln (computed by Eq.(4)) bits are equal to s = 1, and the last n − Ln bits are complementary to s. In the recovery phase, we assume that, among tshadow image bits, u bits are selected fromsc1 , sc2 , · · · , scLn , v bits are selected formscLn +1 , scLn +2 , · · · , scn , t=u+v where tis the recovered shadow image numbers. Hence, 0 ≤ u ≤ Ln , 0 ≤ u ≤ t, v = t − u Let w = min (Ln , t), then 0 ≤ u ≤ w, v = t − u Since k ≤ n, t ≤ n hence, t/2 ≤ n − k/2 ≤ Ln P (S [AS1] = 1) can be commutated in Eq.(7), as follows:

P (S [AS1] = 1) = ⎧ u v ⎨ CLn ×Cn−Ln =

=

⎧ u v ⎨ CLn ×Ctn−Ln ⎩1 × 2

t Cn u v CL ×Cn−L n n t Cn

Cn u v CL ×Cn−L n n t Cn

if u ≥

t 2 t 2

if u > v or u >

if u = v or u =

⎪ ⎩

t 2

and t ∈ 2Z+ − 1

⎩1 × if u = and t ∈ 2Z+ 2 ⎧ t t− t t−w w−1 t−w+1 ⎪ +CL Cn−Ln +···+CLn2 Cn−L2n ⎨ CLwn Cn−L n n t Cn

t 2

(7) if t ∈ 2Z+ − 1

t +1 t− t −1 t t w−1 t−w+1 w 2 2 CL C t−w +CL Cn−Ln +···+CL2n Cn−L + 12 CL2n Cn−L n n−Ln n n n t Cn

if t ∈ 2Z+

whereCab denotes the number of combinations selecting b numbers from a numbers, a, b ∈ Z + , b ≤ a w = min (Ln , t) Let α (1, k, n, t)denote the contrast of the recovered secret image S with t shadow images in the proposed P (1, k, n) PSS scheme

78

X. Yan et al.

Since P (S [AS0] = 1) = P (S [AS1] = 0) = 1−P (S [AS1] = 1).Hence, based on Definition 2, we have Eq.(8). P S [AS1] = 1 − P S [AS0] = 1 P1 − P0 = 1 + P0 1 + P (S [AS0] = 1) P S [AS1] = 1 − 1 − P S [AS1] = 1 2P S [AS1] = 1 − 1 = = 1 + [1 − P (S [AS1] = 1)] 2 − P (S [AS1] = 1) ⎧

t t− t ⎪ t−w w−1 t−w+1 w t ⎪ Cn−L + CL Cn−Ln + · · · + CL2n Cn−L2n − Cn ⎪ 2 CL ⎪ n n n ⎪ ⎪ ⎪ ⎪

if t ∈ 2Z+ − 1 ⎪ ⎪ t ⎪ t− t ⎪ w−1 t−w+1 2 2 w C t−w t ⎪ 2C − C + C C + · · · + C C ⎨ n Ln n−Ln Ln n−Ln Ln n−Ln = ⎪ ⎪ t t t t ⎪ +1 t− −1 ⎪ w−1 t−w+1 w t 2 2 ⎪ 2 CL C t−w + CL Cn−Ln + · · · + CL2n Cn−L + 12 CL2n Cn−L − Cn ⎪ ⎪ n n−Ln n n n ⎪ + ⎪ ⎪ if t ∈ 2Z ⎪ t +1 t− t −1 t t ⎪ ⎪ w−1 t−w+1 1 w C t−w 2 2 2 C2 ⎩ t − 2Cn CL + C C + · · · + C C + C Ln n−Ln Ln n−Ln 2 Ln n−Ln n n−Ln

α (1, k, n, t) =

(8)

Theorem 2 (Security). The proposed PSS scheme is secure. Proof (of Security) When t (1 ≤ t < k)shadow images are stacked, the next proof considers two cases: Case 1: t ∈ 2Z+ − 1. Case 2: t ∈ 2Z+ For Case 1: t ∈ 2Z+ − 1 if t ∈ 2Z+ − 1,based on Eq.(6), we have Eq.(9).

2t t− 2t t−w w−1 t−w+1 t + C C + · · · + C C − C 2 CLwn Cn−L n Ln n−Ln Ln n−Ln n

t t 2 t− 2 t−w w−1 t−w+1 − 2Cnt − CLwn Cn−L + C C + · · · + C C Ln n−Ln Ln n−Ln n

t t 2 t− 2 t−w w−1 t−w+1 t = 3 CLwn Cn−L + C C + · · · + C C − C n < 0 Ln n−Ln Ln n−Ln n where Ln = n − k/2 Hence, α (k, n, t) =

t

t−

t

w−1 t−w+1 w 2 CL C t−w +CL Cn−Ln +···+CLn2 Cn−L2n n n−Ln n

t

(9)

t−

t −Cn

t

n2 , the number of bit 0 in n bits is n − Ln < n2 , the probability of being 1 is greater than the probability of being 0 for every bit, hence more bits will increase the probability that recovering secret bit to be 1. So P (S [AS1] = 1) increases as t increases 2P (S [AS1]=1)−1 Furthermore, 2−P (S [AS1]=1) increases as P (S [AS1] = 1) increases. Hence increases as t increases. 3.3

Extension for Grayscale/Color Images

The proposed scheme can be extended to share grayscale/color images [1][27][29]. For sharing a color image, color decomposition, and color composition are applied. A color image can be described by color model. Here, RGB (red–green– blue) model will be applied, which is additive color model. Based on the color model, a color image can be processed by three grayscale images(R, G, B) with the same extension methods for grayscale images. To share a grayscale image, there are two extension methods can be applied for the proposed scheme. The two extension methods are described as follows: 1. A grayscale image could be divided into eight bit-plane binary images, since every grayscale pixel value is between [0,255], that can be represented of eight bits attributing to eight bit-plane. 2. In order to share a grayscale image, halftone technologies such as error diffusion [2][4] is applied to convert the grayscale image into binary image, then the proposed scheme can be used for the binary image

4

Experimental Results and Analyses

In this section, we conduct experiments and analyses to evaluate the effectiveness of the proposed scheme. First the images will be illustrated to show the effectiveness of the scheme. Then, the visual quality of recovered secret images in the proposed scheme is shown and the shadow images quality of the proposed scheme will be evaluated by average flipping rate (AFR) [30] and the contrast, which all are the similarity measurements between the original binary secret images and the shadow images. Afterwards, the comparisons with related schemes will be given. The secret images with size 256×256 are used to test the efficiency of the proposed scheme. In the first experiment, P (1, 5, 6) (i.e. l= 1, k = 5, n = 6) threshold with secret image shown in Fig.5 (a) is used to do the test of the proposed scheme. Fig.5 (b−g) show the 6 shadow images SC1 ,SC2 ,SC3 SC4 ,SC5 and SC6 . Fig.5 (hl ) show the recovered binary secret images with t = 1, 2, · · · , 5 shadow images,

80

X. Yan et al.

from which the content of secret image could be recognized. In the secret images recovered from 1 or 2 shadow images, the camera and the tower could not be recognized. In the secret images recovered from 3 or 4 shadow images, the camera could be recognized while the tower could not be recognized. In the secret images recovered from 5 shadow images, the tower could be recognized the same as the original binary secret image. And the visual quality of recovered secret images is better in Fig.5 with more shadow images. Fig.5 (h-k ) show the recovered secret images with less than k shadow images, from which not all the information (including content and details) could be recognized. From the illustrations, we can find that the shadow images have low quality, when t(1 ≤ t < k) shadow images are stacked i.e., the content of secret image could be recognized while the details couldn’t be recognized, and the visual quality of recovered secret image increase as t increases. When less than k shadow images are stacked not all the information of the secret image could be recognized, which shows the security of the proposed scheme. When k or more shadow images are stacked all the information of the secret image could be recognized. The next two examples are given to show the effectiveness of the two extension methods for grayscale/color images described in Section 3.3 In the experiment, P (1, 7, 8) threshold with grayscale secret image shown in Fig.6 (a) is used to do the test of the first extension method Fig.6(b) shows one of the 8 shadow images, Fig.6 (c − h) show the recovered secret images with any t(t = 2, 3, · · · , 7) shadow images. The visual quality of the recovered secret image increases as t increases The secret image recovered by t = k = 7 shadow images is lossless in Fig.6(h). In the experiment, P (1, 5, 5) threshold with color secret image shown in Fig.7 (a) is used to do the test of the second extension method Fig.7(b) shows the halftone color secret image, Fig.7 (c) shows one of the 5 shadow images, Fig.7(d − h) show the recovered secret images with any t(t = 1, 2, · · · , 5) shadow images. The same results can be seen as described above. 4.1

Visual Quality of Recovered Secret Images in the Proposed Scheme

In this section, the visual quality of recovered secret image is evaluated by contrast in Definition 2. The same original binary secret image as shown in Fig.5(a) is used to do the experiments of contrast. Average contrast of the proposed P (l, k, n) (3 ≤ n, 3 ≤ k ≤ n, 1 ≤ t ≤ n, k ∈ 2Z+ + 1, t, n ∈ Z+ ) scheme is shown in Tab.1, where t is the number of shadow images. From Tab.1, we can find that in the proposed scheme, generally speaking (some particular cases are caused by uncertainty of random), the contrast increases as t increases for a certain (k, n) scheme, the contrast changes slowly as k increases for a certain t and n, the contrast decreases as n increases for a certain t and k.


81

(a) Original binary se- (b) Shadow image SC1 (c) Shadow image SC2 (d)Shadow image SC3 cret image

(e)Shadow image SC4 (f) Shadow image SC5 (g) Shadow image SC6 (h) Recovered image t=1

(i) Recovered image (j) Recovered image (k) Recovered image (l) Recovered image t=2 t=3 t=4 t=5

Fig. 5. Experimental example of the proposed P(1, 5, 6) threshold scheme

The theoretical contrast of the proposed scheme is shown in Tab.2, the relative difference between the experimental contrast and the theoretical contrast is computed in Eq.(10), as follows: 9

β=

n

n

n=3 k=3,k∈2Z + +1 t=1 9

n

|αexp (1, k, n, t) − αtheory (1, k, n, t)| n

n=3 k=3,k∈2Z + +1 t=1

2

= 0.36%

(10)

|αtheory (1, k, n, t)|2

Where αexp (1, k, n, t)denotes the experimental contrast shown in Tab.1, αtheory (1, k, n, t)denotes the theoretical contrast shown in Tab.2. The result is computed by the corresponding figures in the two Tables for n = 3, 4, . . . , 9k = 3, 5, . . . , n, k ∈ 2Z + + 1,,t = 1, 2, . . . , n

82

X. Yan et al.

(a) Original grayscale (b) Shadow image SC1 (c) Recovered image (d) Recovered image secret image t=2 t=3

(e) Recovered image (f) Recovered image (g) Recovered image (h) Recovered image t=4 t=5 t=6 t=7

Fig. 6. Experimental example of the proposed P(1, 7, 8) scheme for grayscale secret image with extension method 1

(a) Original color se- (b)halftone color secret (c) Shadow image SC1 (d) Recovered image cret image image t=1

(e) Recovered image (f) Recovered image (g) Recovered image (h) Recovered image t=2 t=3 t=4 t=5

Fig. 7. Experimental example of the proposed P(1, 5, 5) scheme for color secret image with extension method 2


9

n

n

83

2

|αexp (1, k, n, t) − αtheory (1, k, n, t)| = 0.222893018

n=3 k=3,k∈2Z + +1 t=1 9

n

n

n=3

k=3,k∈2Z + +1

t=1

2

|αtheory (1, k, n, t)| = 62.05692687

From Eq.(10) we can find that the average contrast of the proposed scheme squares with the theory in section 3.3. The experimental results of the contrast verify the validity of the performance analyses on the contrast in the proposed scheme. When n = 11, k = 9. Average contrast of the recovered secret images stacked by t shadow images as t changes from 1 to n , is illustrated in Fig.8. As can be seen from Fig.8, average contrast of the recovered secret images increases as t increases, when t is less than k, the contrast is less than 1, while t is equal to or more than k, the contrast is equal to 1. 4.2

Visual Quality of Shadow Images in Proposed Scheme

In this section, per shadow image average flipping rate (bits) in sharing one secret bit (AFR) [30] and contrast are used to evaluate the quality of the shadow images. The same original binary secret image as shown in Fig. 5(a) is used to do the experiments.

Table 1. Average experimental contrast (αexp (1, k, n, t)) of the proposed P (1, k, n) PSS scheme (k, n)

t=1

t=2

t=3

t=4

t=5

(3, (3, (3, (3, (3, (3, (3, (5, (5, (5, (5, (5, (7, (7, (7, (9,

0.32214 0.43056 0.51359 0.57699 0.62777 0.66794 0.70051 0.21296 0.28508 0.35179 0.40978 0.45931 0.16395 0.21927 0.27093 0.13648

0.32105 0.43035 0.51492 0.57656 0.62771 0.66833 0.70061 0.21324 0.28522 0.35137 0.40996 0.45822 0.16316 0.21856 0.27059 0.13661

1 1 1 1 1 1 1 0.41314 0.54561 0.64556 0.71926 0.77339 0.28808 0.3887 0.47568 0.22982

1 1 1 1 1 1 0.41464 0.54618 0.64579 0.71913 0.77342 0.28809 0.38878 0.47516 0.23

1 1 1 1 1 1 1 1 1 1 0.45605 0.60267 0.70649 0.33182

3) 4) 5) 6) 7) 8) 9) 5) 6) 7) 8) 9) 7) 8) 9) 9)

t=6

t=7

t=8

t=9

1 1 1 1

1 1 1

1 1

1

1 1 1 1 0.45525 0.60322 0.70657 0.33211

1 1 1 1 1 1 0.48384

1 1

1

1 1 1 0.48475 1

84

X. Yan et al.

Table 2. Theoretical contrast (αtheory (1, k, n, t)) of the proposed P (1, k, n) PSS scheme (k, n)

t=1

t=2

t=3

t=4

t=5

(3, (3, (3, (3, (3, (3, (3, (5, (5, (5, (5, (5, (7, (7, (7, (9,

0.25 0.4 0.5 0.57143 0.625 0.66667 0.7 0.14286 0.25 0.33333 0.4 0.45455 0.1 0.18182 0.25 0.07692

0.25 0.4 0.5 0.57143 0.625 0.66667 0.7 0.14286 0.25 0.33333 0.4 0.45455 0.1 0.18182 0.25 0.07692

1 1 1 1 1 1 1 0.30769 0.5 0.625 0.70968 0.76923 0.1875 0.33333 0.4466 0.13559

1 1 1 1 1 1 0.30769 0.5 0.625 0.70968 0.76923 0.1875 0.33333 0.4466 0.13559

1 1 1 1 1 1 1 1 1 1 0.33333 0.54546 0.68085 0.21053

3) 4) 5) 6) 7) 8) 9) 5) 6) 7) 8) 9) 7) 8) 9) 9)

t=6

t=7

t=8

t=9

1 1 1 1

1 1 1

1 1

1

1 1 1 1 0.33333 0.54546 0.68085 0.21053

1 1 1 1 1 1 0.34783

1 1

1

1 1 1 0.34783 1

1 0.9

Contrast

0.8 0.7 0.6 0.5 0.4 0.3 0.2 1

2

3

4

5

6 t

7

8

9

10

11

Fig. 8. Average contrast as t changes in P(1, 9, 11)

AFR could be used to evaluate the quality of shadow images [30]. From Eq. (5), we obtain Eq. (11) from dividing the left and right equations by n. Since the n shadow images generated by Step2 are random, hence the number of


85

bits 0 and 1 are both n/2, then the flipping number B in the n bits isB = Ln − n2 , thus we use Eq. (5) to obtain Eq. (12). 1−

k k Ln Ln < →1− < n 2n 2n n

(11)

n − k2 Ln − n2 Ln 1 1 k 1 1 k AF R = = − = − >1− − = − (12) n n 2 n 2 2n 2 2 2n As can be seen from Eq. (12), AFR ∈ (0, 1), since higher AFR[30] means per shadow image will flip more bits to be the same as original secret bit, hence higher AFR will indicate higher contrast of shadow images and better visual quality of shadow images. AFR is introduced with lower computation and the close-tolinear relationship with contrast for evaluating the quality of shadow images. Further, AFR can be given before sharing to guide the parameters selection in the PSS scheme. When n = 11, k = 3, 5, . . . , 11. AFR values and average contrast of shadow images as k changes are illustrated in Fig.9. When k = 3, n = 3, 4, . . . , 11. AFR values and average contrast of shadow images as n changes are illustrated in Fig.10. As can be seen from Fig.9 and Fig.10, when n does not change, AFR average contrast of shadow images and the visual quality of shadow images decrease as k increases. And when k does not change, AFR, average contrast of shadow images and the visual quality of shadow images curve increase as n increases.

0.8 Average contrast of shadow images AFR

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3

5

7 k

9

11

Fig. 9. AFR values and average contrast of shadow images as k changes under n = 11

86

X. Yan et al.

0.8 0.7

Average contrast of shadow images AFR

0.6 0.5 0.4 0.3 0.2 0.1 3

4

5

6

7 n

8

9

10

11

Fig. 10. AFR values and average contrast of shadow images as n changes under k = 3

4.3

Comparisons with Related Schemes

In this section, we compare the proposed scheme with related schemes, especially a PVSS [21]. Indeed, PVSS has similar property, that is better perceptual quality will be recovered with more shadow images. In addition, PVSS in [21] has no pixel expansion, simple recovery (stacking) and good visual quality. In [21] Hou and Quan design two n × n matrices denoted by C and C 1 , which represent the sharing basic matrix for white and black pixels of the secret image, respectively. In Hou and Quan’s scheme 0 represents white, 1 represents black Since the shadow images have the same size as the original secret image, hence, there is no pixel expansion for the shadow images. The recovery method of Hou and Quan’s scheme is based on stacking (Boolean OR operation) two or more shadow images, hence the scheme is a (2, n) threshold scheme. To make an explicit comparison, we have used the same secret binary image (shown in Fig.11 (a1)) as well as the same shadow images number (i.e., n =7). In the proposed scheme, k = 5. Fig.11 (b1-h1) show the recovered secret images with any t (taking the first t shadow images as an example) shadow images, and the contrast is marked at the same time. From Fig.11, we can summarize the following points: -The two schemes both have progressive or perceptual visual quality of the recovered secret images. -There is no pixel expansion in the two schemes The proposed scheme could recovered the secret image losslessly, while Hou and Quan’s scheme [21] not -The visual quality of the proposed scheme is better than Hou and Quan’s scheme [21]


87

Moreover, the color representation of the proposed scheme is the same as the digital images, while Hou and Quan’s scheme [21] not. The codebook is needed in Hou and Quan’s scheme [21], while the proposed scheme not.

The proposed scheme

(a1) Original binary (b1) secret image SC1

Hou and Quan’s scheme [21]

Shadow

image (a2) Original binary (b2) secret image SC1

Shadow

image

(c1) Recovered image (d1) Recovered image (c2) Recovered image (d2) Recovered image t=2, contrast=0.352 t=3, contrast= 0.644 t=2, contrast= 0.081 t=3, contrast= 0.179

(e1) Recovered image (f1) Recovered image (e2) Recovered image (f2) Recovered image t=4, contrast= 0.644 t=5, contrast= 1 t=4, contrast= 0.297 t=5, contrast= 0.442

(g1) Recovered image (h1) Recovered image (g2) Recovered image (h2) Recovered image t =6, contrast= 1 t =7, contrast= 1 t =6, contrast= 0.627 t =7 contrast= 0.858

Fig. 11. Comparisons of visual quality of the recovered secret images (n = 7) between the proposed scheme and Hou and Quan’s scheme [21]

88

X. Yan et al.

In order to highlight the merits of the proposed PSS scheme, Tab.3 summarizes the valuable advantages of the proposed scheme and its comparison with other schemes. The properties of the proposed scheme are summarized as follows: 1. P (l, k, n)threshold The proposed model is able to implement P (l, k, n) threshold. This property is a significant in real applications 2. No pixel expansion The size of shadow images is the same as the original secret image, there is no pixel expansion occurred in the proposed scheme. 3. No complex or extra codebook design The proposed scheme avoids complex codebook (sometimes considerably difficult for different applications) design 4. Digital color representation In the proposed scheme, the color representation is the same as color representation method of digital images: 0 denotes black or opaque and 1 denotes white or transparent in digital images and common digital processing software for digital applications, such as BMP, JPEG, Matlab and Photoshop, which can avoid flipping (that is 0 → 1 or 1 → 0), reversing or complementing operations. 5. In addition, the idea applied in the proposed scheme, that is the color representation method is the same as color representation method of digital images can be extended to traditional VSS schemes. 6. In the recovery phase the order of the shadow images is alternative The computation of the generation phase and recovery phase is simple only addition and comparison.

Table 3. Properties comparison with relative schemes Scheme

P (l, k, n)

Recovering measure

Ref. [1] Ref. [2] Ref. [3] Ref. [10] Ref. [12] Ref. [14] Ref.[20] Ref.[21] Ref.[22] Ref. [29] Ref. [31] Ref. [32] Proposed Scheme

× × × × × × × × × × × × √

Stacking Stacking Boolean Stacking Boolean Stacking Stacking Stacking Stacking Stacking Stacking Stacking Addition and comparison

No pixel ex- No codebook Alternative pansion design order √ √ √ √ × × √ √ √ √ × × √ √ √ √ × × √ √ × √ √ × √ × × √ √ √ √ √ × √ √ √ √ √ √

Digital color × × × × × × × × × × × × √


5

89

Conclusion

A novel perceptual secret sharing model is defined in this paper. Then, we proposed a simple and efficient perceptual secret sharing scheme based on maximum likelihood estimation. The proposed scheme satisfies P (l, k, n) threshold sharing, which can gain different perceptual quality, lossless recovery, loss-tolerant and control access. In addition, it has as the same color representation method of digital images, which is useful for more and more digital applications. Furthermore, the proposed scheme inherits conventional benefits of VSS, such as no pixel expansion and design of complex codebook. Extensive experiments are conducted to evaluate the security and efficiency of the scheme. The results show effectiveness of the novel scheme. The defined model and the novel scheme will be useful for the prominent prevalence in real applications such as Pay- TV/Music and art-work image vending, etc. Acknowledgement. The authors wish to thank the anonymous reviewers for their suggestions to improve this paper. This work is supported by the National Natural Science Foundation of China (Grant Number: 61100187) and the Fundamental Research Funds for the Central Universities (Grant Number: HIT. NSRIF. 2010046, HIT. NSRIF. 2013061).

References 1. Chen, T.H., Tsao, K.H.: Threshold visual secret sharing by random grids. J. Syst. Softw. 84(7), 1197–1208 (2011) 2. Zhou, Z., Arce, G.R., Di Crescenzo, G.: Halftone visual cryptography. IEEE Trans. Image Process. 15(8), 2441–2453 (2006) 3. Wang, D., Zhang, L., Ma, N., Li, X.: Two secret sharing schemes based on boolean operations. Pattern Recognit. 40(10), 2776–2785 (2007) 4. Wang, Z., Arce, G.R., Di Crescenzo, G.: Halftone visual cryptography via error diffusion. IEEE Trans. Inf. Forensics Security 4(3), 383–396 (2009) 5. Yi, F., Wang, D., Li, S., Dai, Y.: Probabilistic visual cryptography scheme with reversing. J. Tsinghua Univ. (Sci. & Tech.) 1, 121–123 (2008) 6. Li, L., El-Latif, A.A.A., Shi, Z., Niu, X.: A new loss-tolerant image encryption scheme based on secret sharing and two chaotic systems. Research Journal of Applied Sciences Engineering and Technology 4, 877–883 (2012) 7. Yang, B., Busch, C., Niu, X.: Perceptual image encryption via reversible histogram spreading. In: Proceedings of 6th International Symposium on Image and Signal Processing and Analysis, ISPA 2009, pp. 471–476. IEEE (2009) 8. Torrubia, A., Mora, F.: Perceptual cryptography on mpeg layer iii bit-streams. IEEE Trans. Consumer Electronics 48(4), 1046–1050 (2002) 9. Li, S., Chen, G., Cheung, A., Bhargava, B., Lo, K.T.: On the design of perceptual mpeg-video encryption algorithms. IEEE Trans. Circuits Syst. Video Tech. 17(2), 214–223 (2007) 10. Naor, M., Shamir, A.: Visual cryptography. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 1–12. Springer, Heidelberg (1995)

90

X. Yan et al.

11. Weir, J., Yan, W.: A comprehensive study of visual cryptography. In: Shi, Y.Q. (ed.) Transactions on DHMS V. LNCS, vol. 6010, pp. 70–105. Springer, Heidelberg (2010) 12. Wu, X., Sun, W.: Image sharing scheme based on error diffusion. J. Comput. Appl. 31(1), 74–81 (2011) 13. Li, L., Yan, X., Wang, N., El-Latif, A.A.A., Niu, X.: Meaningful image sharing threshold scheme based on error diffusion. Int. J. Digit. Content Technol. Appl. (JDCTA) 6(13), 275–284 (2012) 14. Ateniese, G., Blundo, C., Santis, A.D., Stinson, D.R.: Extended capabilities for visual cryptography. Theor. Comput. Sci. 250(1), 143–161 (2001) 15. Kafri, O., Keren, E.: Encryption of pictures and shapes by random grids. Optics Letters 12(6), 377–379 (1987) 16. Shyu, S.J.: Image encryption by random grids. Pattern Recognition 40(3), 1014–1031 (2007) 17. Thien, C.C., Lin, J.C.: An image-sharing method with user-friendly shadow images. IEEE Trans. Circuits Syst. Video Technol. 13(12), 1161–1169 (2003) 18. Zhao, R., Zhao, J.J., Dai, F., Zhao, F.Q.: A new image secret sharing scheme to identify cheaters. Comput. Stand. Interfaces 31(1), 252–257 (2009) 19. Jin, D., Yan, W.Q., Kankanhalli, M.S.: Progressive color visual cryptography. Journal of Electronic Imaging 14(3), 033019–033019 (2005) 20. Hou, Y.C., Quan, Z.Y., Tsai, C.F., Tseng, A.Y.: Block-based progressive visual secret sharing. Information Sciences (2013) 21. Hou, Y.C., Quan, Z.Y.: Progressive visual cryptography with unexpanded shares. IEEE Trans. on Circ. and Sys. for Video Tech. 21(11), 1760–1764 (2011) 22. Fang, W.P., Lin, J.C.: Progressive viewing and sharing of sensitive images. Pattern Recognition and Image Analysis 16(4), 632–636 (2006) 23. Chen, S.K.: Friendly progressive visual secret sharing using generalized random grids. Optical Engineering 48(11), 117001–117001 (2009) 24. Chen, S.K., Lin, J.C.: Fault-tolerant and progressive transmission of images. Pattern Recognition 38(12), 2466–2471 (2005) 25. Huang, C.P., Hsieh, C.H., Huang, P.S.: Progressive sharing for a secret image. J. Syst. Softw. 83(3), 517–527 (2010) 26. Yan, X., Wang, S., El-Latif, A.A.A., Sang, J., Niu, X.: Threshold visual secret sharing based on boolean operations (2013) (to be appear) 27. Wu, X., Sun, W.: Random grid-based visual secret sharing with abilities of or and xor decryptions. J. Vis. Commun. Image R (2012) 28. Nist/sematech e-handbook of statistical methods, http://www.itl.nist.gov/div898/handbook/eda/section3/eda3652.htm 29. Wu, X., Sun, W.: Improving the visual quality of random grid-based visual secret sharing. Signal Processing (2012) 30. Yan, X., Wang, S., Li, L., El-Latif, A.A.A., Wei, Z., Niu, X.: A new assessment measure of shadow image quality based on error diffusion techniques. J. Inf. Hiding Multimedia Signal Process (JIHMSP) 4(2), 118–126 (2013) 31. Yang, C.N.: New visual secret sharing schemes using probabilistic method. Pattern Recognit. Lett. 25(4), 481–494 (2004) 32. Chen, T.H., Tsao, K.H.: User-friendly random-grid-based visual secret sharing. IEEE Trans. on Circ. and Sys. for Video Tech. 21(11), 1693–1703 (2011)