A new sharing digital image scheme with clearer shadow images

Author’s Accepted Manuscript A new sharing digital image scheme with clearer shadow images Ching-Nung Yang, Chihi-Han Wu, Zong-Xuan Yeh, Dao-Shun Wang, Cheonshik Kim www.elsevier.com

PII: DOI: Reference:

S0920-5489(16)30198-2 http://dx.doi.org/10.1016/j.csi.2016.11.015 CSI3177

To appear in: Computer Standards & Interfaces Received date: 20 July 2016 Revised date: 29 November 2016 Accepted date: 29 November 2016 Cite this article as: Ching-Nung Yang, Chihi-Han Wu, Zong-Xuan Yeh, DaoShun Wang and Cheonshik Kim, A new sharing digital image scheme with clearer shadow images, Computer Standards & Interfaces, http://dx.doi.org/10.1016/j.csi.2016.11.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A new sharing digital image scheme with clearer shadow images

Ching-Nung Yanga*, Chihi-Han Wua, Zong-Xuan Yeha, Dao-Shun Wangb*, and Cheonshik Kimc a

Department of Computer Science and Information Engineering, National Dong Hwa University, Hualien, Taiwan. b Department of Computer Science and Technology, Tsinghua University, Beijing, China. c Department of Computer Engineering, Sejong University, Seoul, Korea. [email protected] [email protected] * Corresponding author. Prof. Ching-Nung Yang, Prof. Dao-Shun Wang Abstract Recently, Wei et al. propose a 2-out-of-2 sharing digital image scheme (SDIS) that shares a color secret image into two shadow images based on Boolean exclusive-or operation. There are three types of shadow images for Wei et al.’s SDIS: noise-like, black-and-white meaningful, and color meaningful shadow images. However, there exist some weaknesses in Wei et al.’s SDIS: the incorrect assignment of color palette data for the color index 255, the erroneous recovery in secret image, and the partial region in shadow image revealing the cover image. In this paper, we solve the weaknesses and propose a new SDIS. Experimental results demonstrate that our scheme effectively avoids these weaknesses.

Keywords: Secret sharing, digital image, color palette, Boolean exclusive-or operation.

1. Introduction Sharing digital image by secret sharing technology is an important research area combining cryptography and image processing. A secret image is shared to some shadow images (referred to as shadows), which do not reveal any secret information. These shadows may be noise-like or meaningful (revealing a cover image on shadow). When shadows are combined in the prescribed way, the secret image can be recovered. Usually, this secret image sharing (SIS) scheme is implemented as a threshold (k, n)-SIS scheme, where k  n, that divides a secret image into n shadows. In a (k, n)-SIS scheme, we may reconstruct the secret image from any k shadows; but (k1) or fewer shadows do not recover the secret image. There are two major categories of SIS scheme: one is the visual cryptography scheme (VCS) and the other 1

is the polynomial-based SIS (PSIS) scheme. Decoding of VCS requires neither knowledge of cryptography nor computer. Participants can easily photocopy their shadows onto transparencies and stack them to visually decode the secret through human visual sight. We call this novel property the stacking-to-see property. Naor and Shamir’s (k, n)-VCS is the first VCS [1]. Afterwards, size-reduced VCSs were proposed [2-4] to reduce the pixel expansion. Other VCSs with specific functions, such as sharing multiple secrets, cheating prevention, sharing color image, providing region incrementing property, providing progressive recovery, and keeping aspect ratio invariant were subsequently proposed [5-10]. On the contrary, the reconstructed image of PSIS scheme can obtain the distortion-less secret image but it needs the more complex computation (note: using Lagrange interpolation). This PSIS scheme can be accomplished via Shamir’s secret sharing [11]. Up to date, there were many researches on PSIS schemes providing various functions [12-17]. For more details of VCS and PSIS schemes, readers can refer to the book [18]. In fact, there are also some schemes combining VCS and PSIS scheme to highlight their respective advantages [19, 20], on which one can decode secret image for preview by the approach of VCS and can recover the high-quality reconstructed image too via PSIS approach. The stacking operation in VCS is the same as Boolean OR operation. As we know, the monotone property of OR operation will cause that a black pixel in one shadow cannot be undone by the color of another pixel in the other shadow laid over it. Thus, VCS has the poor visual quality of reconstructed image. It is reasonable to adopt exclusive-OR (XOR) operation instead of OR operation to enhance the visual quality. In [21], the authors demonstrate that the OR-based (k, n)-VCS is also the XOR-based (k, n)-VCS and vice versa. Also, the contrast of XOR-based (k, n)-VCS is enhanced 2(k−1) times when compared with OR-based (k, n)-VCS. Actually, the XOR operation in VCS is pixel-level based operation. It can be accomplished by hardware with the helps of projector (OR operation) and device with reversing function, e.g., copy machine. Unlike pixel-based VCS, some bit-wise Boolean operation based SIS schemes were proposed [22-25] to obtain a high-quality secret image, but adopting Boolean operations on bit-level needs computer and image manipulation program. Recently, Wei et al. use the bit-wise XOR operation to design a 2-out-of-2 sharing digital image scheme (SDIS) [26] to share a 256-color (or true color) digital image. Even though bit-wise Boolean operation is more complex than the pixel-wise Boolean operation in VCS, its advantage is the recovery of distortion-less image like PSIS scheme. The rapid development of communication over the Internet makes securely sharing digital data via public channel being a critical issue nowadays. Therefore, the protection of important multimedia data from attacks is

2

becoming a challenge when distributing secret data on networks. The SDIS is a type of SIS scheme, and may be used to securely share digital image. In addition, the SDIS only needs the computation of bit-wise XOR operation, which is cost-effective when compared with PSIS scheme and encryption/decryption. The above implies that the SDIS deserves further studying. However, there are three weaknesses in Wei et al.’s SDIS: the incorrect assignment of color palette data for the color index 255, the erroneous recovery in secret image, and the partial region in shadow revealing the cover image. For the color index 255, Wei et al.’s SDIS has an error for embedding the data of color palette. In addition, Wei et al.’s SDIS with color meaningful shadows cannot extract the block data in a shadow correctly for white cover pixels. This will cause the erroneous recovery in secret image. Moreover, Wei et al.’s SDIS uses five black subpixels in a 9-subpixel block to reveal the content of the cover image. Five black dots in a block degrade the visual quality of meaningful shadows. In this paper, we solve all these three weaknesses and propose a new SDIS. The rest of this paper is organized as follows. Sections 2 and 3 review Wei et al.’s SDIS and briefly describe its weaknesses. The proposed SDIS is presented in Section 4. Experiment, discussion and comparison are given in Section 5. Finally, Section 6 concludes the paper.

2. Wei et al.’s SDIS Notations in this paper and their descriptions are listed in Table 1. These notations are used to describe all the schemes throughout this paper.

Notation CP SI CCI, BCI NS1, NS2 CS1, CS2 BS1, BS2

B1  B9 B1i  B9i n n1, n2 xByW B1, B2 n10 , n01 , n11 , n00

Sx, y

Table 1. Notations and Descriptions Description a 256-color color palette a secret image with the size with the size (MN) pixels binary (black-and-white) over image and color cover image with the size (MN) pixels binary noise-like shadows with the size (3M3N) (respectively, (5M5N)) subpixels for 256-color (respectively, true color) secret image color meaningful shadows with the size (3M3N) (respectively, (5M5N)) subpixels for 256-color (respectively, true color) secret image binary meaningful shadows with the size (3M3N) (respectively, (5M5N)) subpixels for 256-color (respectively, true color) secret image a 33-subpixel block including 8-bit color index B1  B8 and one bit B9 of color palate data a 33-pixel block on shadow i, i=1 and 2 the number of 1s in ( B1  B8 ) n1 and n2 are the numbers of “1” of block in NS1 and NS2 x black subpixels and y white subpixels in a block the blocks B1 and B2 have 6B3W and 5B4W subpixels, respectively the number of positions of (10), (01), (11), and (00) from B1 to B2 the subset, where x=0, 1 and j=0, 255, used to represent B9=x and the color index = y 3

RP(256) , RP(true) (256) W

R

( true ) W

,R

CP , CW

the regions in shadows showing the content of cover image RP(256) (sharing 256-color image) and RP(true ) (sharing true color image) for the proposed SDIS the regions in shadows showing the content of cover image RP(256) (sharing 256-color image) and RP(true ) (sharing true color image) for Wei et al.’s SDIS the contrasts of binary meaningful shadows for the proposed SDIS and Wei et al.’s SDIS

About color digital image, the most common method is to use a separate color palette of 256 colors, where each color plane is red, green, and blue. A color is chosen from a palette of 16,777,216 (=224) colors (24 bits: each color plane has 8 bits). In VGA cards, 256 on-screen colors are chosen from a color palette CP, to be most visible to the human eye and meanwhile conserve a bandwidth. For example, consider the conservation of storage. Each pixel is represented by a single byte, on which is the location (index) of the color in a 256-color color palette CP. The palette should be stored along with an image, and via this way, the file size of a color image can be kept small. Consider that a (512512)-pixel digital image represented by using a 256-color CP. The file size is 2563 bytes (color palette) + 5125121 bytes (indices) = 262,912 bytes total. However, using 24-bit true color format, we require 125123=786,432 bytes for storing this file. Recently, Wei et al. propose a simple 2-out-of-2 SDIS to share a 256-color digital image SI into two binary noise-like shadows (NS1 and NS2). These shadows of Wei et al.’s SDIS can be extended to two color meaningful shadows (CS1, CS2), or two binary meaningful shadows (BS1, BS2), which revel color cover image CCI and binary cover image BCI, respectively. The structure of a block in Wei et al.’s SDIS is shown in Fig. 1. Consider a case such that a pixel (8-bit color index) and one bit of color palate data are B1  B9 . These pixels are 2 2 subdivided into 9 subpixels B11  B91 on shadow 1 and 9 subpixels B1  B9 on shadow 2, respectively. The

XOR-ed results of these two shadows are used to recover the 256-color secret image. Because every pixel in SI is represented by a 9 subpixels, shadow sizes are nine times expanded.

B1 B2 B3

B11 B21 B31

B12 B22 B32

B4 B5 B6

B41 B51 B61

B42 B52 B62

B7 B8 B9 B71 B81 B91 B72 B82 B92 (a) (b) (c) Fig. 1. A block of Wei et al.’s SDIS: (a) expansion of one pixel to 9 subpixels (b) 9-subpixel block in shadow 1 (c) 9-subpixel block in shadow 2. Sharing and recovery procedures of Wei et al.’s SDIS with two noise-like shadows NS1 and NS2 are briefly described below.

4

Sharing procedure: (S1) Obtain B1  B9 from the secret image SI and the color palate CP. (S2) Let n be the number of 1s in ( B1  B8 ) . (S3) Among the positions in ( B1  B8 ) , we randomly choose (n  B9 ) / 2 locations of 1s to fill with 1s (respectively, 0s) on shadow 1 NS1 (respectively, shadow 2 NS2). On the contrary, the remaining (n  B9 ) / 2 locations of 1s are filled with 0s (respectively, 1s) on NS1 (respectively, NS2). (S4) In the rest 0’s locations in in ( B1  B8 ) , we randomly choose 5  (n  B9 ) / 2 (respectively,

(8  n)  (5  (n  B9 ) / 2) locations to fill with 1s (respectively, 0s) on both shadows. (S5) If (n is odd and B9=1) then B19 =1, B29 =0; if (n is odd and B9=0) then B19 =0, B29 =1. (S6) If (n is even and B9=1) then B19 =1, B29 =1; if (n is even and B9=0) then B19 =0, B29 =0. (S7) Repeat (S1)(S6), until every pixel in SI and the data in CP are processed.

Note: the 1 and 0 denote the black and white subpixels, respectively, on shadows. Finally, two shadows NS1 and NS2 are binary noise-like shadows. From the sharing procedure, it is observed that every block has 5B4W subpixels. From steps (S3)(S6), we may derive Eq. (1), which demonstratess that the first eight bits of XOR-ed result in a block may exactly recover the original ( B1  B8 ) . In addition, we have B9  B91 (see steps (S5) and (S6)). The positions in the first eight bits in Eq. (1) could be randomly permuted, and thus any color index can be represented.

B1  B8  B9 1's positions 0 's positions  1  NS1  1 1 0 0 1 1 0 0 B9  B9     NS2  0 0 1 1 1 1 0 0 B92  1 0 0 *  NS1  NS2  1

(1)

Recovery procedure: (R1) Obtain the XOR-ed result ( NS1  NS2 ) . (R2) Recover the color indices ( B1  B8 ) from the first 8 bits of every 9-subpixel block in ( NS1  NS2 ) .

5

(R3) Recover the data of color palette B9 from B91 on NS1. (R4) Repeat (R2) and (R3) until all blocks in ( NS1  NS2 ) are processed.

There are 5B4W subpixels in a block. Therefore, a simple way to generate two color meaningful shadows is directly put the color pixels in two cover images to the black subpixels in the corresponding blocks on two shadows, respectively, and meantime leave white subpixels unchanged. Finally, we can obtain two color meaningful shadows CS1 and CS2 with 5/9 of the region revealing the cover image. Another scheme with binary meaningful shadows can be implemented by complementing blocks where the corresponding position in cover image is white pixel. Because the complement of 5B4W is 4B5W, we may use the blocks 5B4W and 4B5W to represent a black pixel and a white pixel on binary cover image. Finally, we can construct a SDIS with two binary meaningful shadows BS1 and BS2. Consider the recovery of secret image from two meaningful shadows. One can easily obtain NS1 and NS2 from CS1 and CS2 by checking the color of pixel. For two binary meaningful shadows, we can also obtain NS1 and NS2 from BS1 and BS2 by easily complementing the 4B5W block to the 5B4W block. Afterwards, the decoding for two meaningful shadows can be done by the same recovery procedure using NS1 and NS2. Sharing and recovery procedures of Wei et al.’s SDIS with color and binary meaningful shadows are formally given as follows.

Sharing procedure: (Step 1) Divide the shadows generated from (S1)~(S7) into 3×3 blocks. (Step 2) /* for Wei et al.’s SDIS with color meaningful shadows */ Fill in 1s in the blocks of both shadows with the corresponding color of the cover pixel in CCI at that location, and leave 0s in both shadows blank (i.e., white color). (Step 2) /* for Wei et al.’s SDIS with binary meaningful shadows */ Fill in all 1s in the blocks of both shadows with the corresponding color of the cover pixel in BCI (note: the color is only black or white now) at that location. Fill in all 0s in both shadows with the complementary color of the cover pixel at that location. (Step 3) Repeat Step 2 (respectively, Step 2), until all blocks in shadows are processed.

Recovery procedure: (Step 1) /* for Wei et al.’s SDIS with color meaningful shadows */ Regard the color subpixel and the white 6

subpixel in both shadows as “1” and “0”,respectively, to obtain all 3×3 blocks in both shadows. (Step 1) /* for Wei et al.’s SDIS with binary meaningful shadows */ Regard the black subpixel and the white subpixel in both shadows as “1” and “0”,respectively, to obtain all 3×3 blocks in both shadows. If the number of 1s in each block is 4, change the subpixel to its complement. (Step 2) Do the procedures (R1)~(R4) of Wei et al.’s SDIS with noise-like shadows.

3. Weaknesses in Wei et al.’s SDIS There are three weaknesses in Wei et al.’s SDIS: i) the incorrect assignment of color palette data for the color index 255, ii) the erroneous recovery in secret image if the cover pixel is white in color meaningful shadows, and iii) the partial regions in meaningful shadows showing the content of the cover image.

3.1 Incorrect Assignment of B9 for the Color Index 255 In [26], the authors claimed that the XOR-ed result ( B1 , B2 ,

, B8 ) , where Bi  ( Bi1  Bi2 ), 1  i  8,

may represent 256 values 0255, and meanwhile the value of B91 may represent the color palette data B9. There are 5B4W in ( B1  B9 ) . The following theorem demonstrates that the above statement is true for all the cases except the case n=8 and B9 =0. Note: the value n=8 implies ( B1 , B2 ,

, B8 )  (1, 1,

, 1) , i.e. the color

index is 255. In other words, the above implies that we cannot embed the data of color palette B9 into the B91 when the color index 255.

Theorem 1. All blocks of two shadows in Wei et al.’s SDIS have 5B4W subpixels, except the case that the color index is 255 and the data bit of color palette is 1. Proof: From steps (S3)(S6) in sharing procedure, the patterns of block on two shadows are summarized in Table 2.

Block

B1B8

B9=1

odd n even n

Table 2. The patterns of block on two shadows. NS1 NS2 Number NS1 NS2 1 0 1 (n  B9 ) / 2 0

1

1

1

1

0

0 1 1

0 0 1

0   7

(n  B9 ) / 2 5  (n  B9 ) / 2 (8  n)  (5  (n  B9 ) / 2) 1 1

B9=0

odd n even n

0 0

 

1 0

1 1

Let the numbers of “1” of block in NS1 and NS2 are n1 and n2, respectively. Via Table 2, we can derive n1 and n2 in Eqs. (2) and (3), respectively.  n1 6 (n B9 )/2   (n B9 )/2  (for B9 1)  n1 5 (n B9 )/2   (n B9 )/2  (for B9 0) 

(2)

n2 6 (n B9 )/2 (n B9 )/2 (for ( B9 1  even n) or for ( B9 0  odd n))    n2 5 (n B9 )/2 (n B9 )/2 (for ( B9 0  even n) or for ( B9 1  odd n))

(3)

From Eqs. (2) and (3), the values of n1 and n2 are calculated as follows. (Case 1): for B9=1 and odd n

n1 6 (n B9 )/2   (n B9 )/2   6  n /20.5   n /20.5 6(n1)/2( n1)/2615  n2 5 (n B9 )/2   (n B9 )/2  5 n /20.5  n /20.5 5( n1)/2( n1)/25     

(4)

(Case 2): for B9=1 and even n

n1 6 (n B9 )/2   ( n B9 )/2   6  n /20.5   n /20.5 6( n/2)( n2)/2615  n2 6 (n B9 )/2   (n B9 )/2  6 n /20.5  n /20.5 6( n/2) ( n2)/2615     

(5)

(Case 3): for B9=0 and odd n

n1 5 (n B9 )/2   (n B9 )/2   5  n /2    n /2 5  n2 6 (n B9 )/2   (n B9 )/2  6 n /2  n /2 6(n1)/2(n1)/2615     

(6)

(Case 4): for B9=0 and even n

n1 5 (n B9 )/2   (n B9 )/2   5  n /2    n /2  5  n2 5 (n B9 )/2   (n B9 )/2  5 n /2  n /2 5(n /2)( n /2)5     

(7)

By Eqs. (4)~(7), it seems that all blocks in both shadows have 5B4W subpixels. However, Case (4) is not true

8

for the color index 255 (i.e., ( B1 , B2 , is not true. The index ( B1 , B2 ,

, B8 )  (1, 1,

, B8 )  (1, 1,

, 1) ) and N9=0. The following explains why Case (4)

, 1) implies n=8. By Table 2, for n=8 and N9=0, the numbers

of positions of 10 (the bit on NS1 is 1 and the bit on NS2 at the corresponding position is 0), 01, 11, and 00 are derived as follows.

10: (n B9 )/2   (80)/2  4  01: (n B9 )/2   (80)/2  4  11: 5 (n B9 )/2  5 (80)/2  1 00: (8n)(5 (n B )/2 )(88)(5 (80)/2 ) 1 9     

(8)

For n=8 and N9=0, there are 4 positions of 10 and 4 positions of 01, respectively. Because N9 is 0 and n is even, we have B19 =B29 =0. Therefore, there are 4B5W in this block. This contradicts that every block should have 5B4W subpixels. The inaccuracy comes from that the number of position for 00 is a negative value “1” (see Eq. (8)).



3.2 Erroneous Recovery in Secret Image By observing the generation of color meaningful shadows CS1 and CS2, we found that the original secret pixel cannot be reconstructed for the incorrect extraction of Bi1 (or Bi2 ), 1  i  9, from shadows. Because we directly put the color pixels of cover images to 5 black subpixels in in a block, and keep the other 4 white subpixels unchanged to generate CS1 and CS2. If the color pixel is white, these 5 black subpixels in a block are replaced by white pixel. Therefore, a block has 9 white subpixels, and we cannot obtain the correct

Bi1 (or Bi2 ), 1  i  9 , to recover the secret pixel.

3.3 Partial Region Revealing the Cover Image Wei et al.’s SDIS directly place the color pixels in cover images to 5 black subpixels in in a block, and other 4 white subpixels are unchanged. Finally, there are 5/9 of the region on a shadow to revealing the content of the cover image. Consider the case that secret image SI has 512512 pixels. The color meaningful shadow has 15361536 pixels, where there are 5125125 subpixels revealing the color pixels in cover image. On the other hand, the binary meaningful shadow uses 5B4W and 4B5W to represent black and white pixels in the cover mage. Therefore, the contrast of shadow is determined as (54)/9=1/9. From the above description, obviously, the visual quality of color meaningful shadow and the contrast of binary meaningful shadow are 9

dependent on the number of black subpixels in a block.

4. The Proposed SDIS In this paper, we solve these weaknesses. We first propose a DSIS with more black subpixels in block and correct embedding for color palate. In addition, in the last two subsections, we describe how to address the weakness of erroneous recovery in the proposed SDIS, and extend the proposed SDIS to share true color secret image.

4.1 The Proposed DSIS with More Black Subpixels in Block and Correct Embedding for Color Palette We use two various blocks (6B3W block and 5B4w block) in shadows to obtain clearer shadows (i.e., the better visual quality for color meaningful shadow and the better contrast for binary meaningful shadow). The block 5B4W has the incorrect assignment of B9 for the color index 255. We first show how to address this weakness by the following approach, on which a similar approach can also be adopted to address the same weakness for 6B3W block. Consider that the block is 5B4W. The XOR-ed result ( B1 , B2 ,

, B8 ) , where Bi  ( Bi1  Bi2 ) , 1i8,

represents the color index. Meanwhile, the value B91 on shadow 1 is used to represent the data of color palette B9. The following equation shows that we may embed B9=1 but cannot embed B9=0 into B91 for the color index 255. As shown in Eq. (9-2), every block has 4B5W subpixels, and this contradicts the block structure.

 CS1   CS   2   SI  

B11 B21 B31 B41 B51 B61 B71 B81 B91 1 1 1 1 0 0 0 0 1 B12 B22 B32 B42 B52 B62 B72 B82 B92 0 0 0 0 1 1 1 1 1 B1 B2 B3 B4 B5 B6 B7 B8 B9  B91 1 1 1 1 1 1 1 1 1

(9-1)

 CS1   CS   2   SI  

B11 B21 B31 B41 B51 B61 B71 B81 B91 1 1 1 1 0 0 0 0 0 B12 B22 B32 B42 B52 B62 B72 B82 B92 0 0 0 0 1 1 1 1 0 B1 B2 B3 B4 B5 B6 B7 B8 B9  B91 1 1 1 1 1 1 1 1 0

(9-2)

A trivial solution is to skip the color index 255, and do not use B91 to embed the color palette data for the 10

color index 255. Consider that the secret image has 512512 pixels, , where have 512512 bits for embedding. A 256-color color palette has the size 25638 bits (each color plane uses 8 bits). This solution is reasonable because of (512  512)

(256  3  8) (note: 512  512  262,144 and 256  3  8  6,144 ).

However, consider an extreme case that a secret image has many pixels of color index 255. There are no enough space for embedding the data of color palette. This trivial approach avoiding the color index 255 is not suitable. Eq. (9-1) only shows one pattern ( B11 , B21 , 255. Actually, there are

84  70

, B81 )  (1, 1, 1, 1, 0, 0, 0, 0) to achieve the color index

combination for these 5-out-of 8 binary vectors. Therefore, we can use 35 out

of 70 patterns for ( B11  B81 ) to represent B9=1, and other 35 patterns for B9=0. For example, we can use 35 4-out-of-8 patterns with B81  1 and B81  0 for B9  1 and B9  0 , respectively. Via this approach, we have to recover the color index first. If the color index is 255, we derive B9 from B81 . Next, we introduce how to use two various blocks (6B3W block and 5B4W block) simultaneously in shadows to enhance the contrasts of meaningful shadows. If the number of black subpixels in a block is b, obviously, there are b/9 of the region in CS1 and CS2 showing the content of cover image, and the contrasts of BS1 and BS2 are (2b9)/9. (Note: for Wei et al.’s SDIS, the value of b is 5). Obviously, by enhancing the value of b, we may improve the visual quality of CS1 and CS2, and the contrast of BS1 and BS2. In fact, Wei et al.’s SDIS with two NS1 and NS2 can be simply described by the concept of constant weight code. Next, we analyze the XOR-ed results for stacking any two constant weight binary vectors, w1-out-of-m vectors and w2-out-of-m vectors, on which we may describe Wei et al.’s SDIS and the proposed SDIS. The following lemma determines the weight distribution of XOR-ing these two m-bit binary vectors.

Lemma 1. Let two m-bit binary vectors v1 and v2 with Hamming weights w1 and w2 (w.l.o.g. w1w2), respectively. For (v1v2), there are

 w m w  binary vectors with Hamming weight (w  w  2i) , where 0iw 1

1

2

2

min

and

wmin=min{mw1, w2}. Proof: Because both vectors are constant weight vectors with Hamming weigh w1 and w2, respectively, the difference of positions of 10 and 01 crossing from vectors v1 to v2 should be (w1  w2 ) . Therefore, the XOR-ed vector (v1v2) has the following form, where 0iwmin and wmin=min{mw1, w2}.

11

w1  w2  i w2  i m  w1  i i  v  1 1 1 1 0 0 0 0 1 v  0  0 1 1 1  1 0 0  2

(10)

Obviously, the Hamming weight of (v1v2) in Eq. (10) is (w1  w2  i)  i  (w1  w2  2i) .



Corollary 1. Cinsider that two w-out-of-m constant weight binary vectors are v1 and v2. For (v1v2), there are

 mi 

binary vectors with Hamming weight 2i , where 0iwmin and wmin=min{mw, w}.

Proof: The proof can be easily derived from Lemma 1 by setting w1=w2.



The following theorem demonstrates that Wei et al.’s SDIS may be explained and described by the XOR-ed results from two 5-out-of-9 constant weight binary vectors.

Theorem 2. Two 5-out-of-9 constant weight binary vectors can be used to implement 256 color indices via the first 8 bits in the XOR-ed result, and meanwhile the last bit can be used to represent the color palette. Proof: By Corollary 1, two XOR-ed 5-out-of-9 vectors may represent binary vectors with Hamming weight 2i, where 0i4, and the pattern is shown below.

i 5i i 4i   v1  1 1 1 1 0 0 0 0     v2  0 0 1 1 1 1 0 0

(11)

For i=0, 1, 2, 3 and 4, i.e., the Hamming weights (i.e., the number of 1s) of XOR-ed result are 0, 2, 4, 6, and 8, respectively. Eq. (11) can be written as Eq. (12).

3 3 4 1 4 1 2 2 2   5   11111 0000 v  1 1111 0 000 v  11 111 00 00    1  1 i  0 :  i =1:  i =2:       11111 0000 v2  00 111 11 00  v2  0 1111 1 000  3 3 2 1 4 1 4    v  111 11 000 0 v  1111 1 0000  1  1  i =4:      i  3 :  v2  000 11 111 0 v2  0000 1 1111 

12

(12)

From the pattern i=0 in Eq. (12), we can use the first eight bits with Hamming weight 0 (i.e.,

H ( B1  B8 )  0 ) to represent the color index 0, and the last bit in shadow 1 B91 to represent B9. As show in Eq. (13), this is exactly the same as step (S6) in Wei et al.’s SDIS: If (n is even and B9=1) then B19 =1, B29 =1; if (n is even and B9=0) then B19 =0, B29 =0. Note: the first eight bits may be randomly permuted.

CP data   B1  B8 ( H ( B  B )  0)  (B  1)  11110000 B1  1 1 8 9 9   2  11110000 B9  1  CP data  B1  B8   ( H ( B1  B8 )  0)  (B9  0)  11110000 B91  0 11110000 B 2  0  9  

(13)

Consider another case i=1 in Eq. (12), we can rewrite it as the patterns in Eqs. (14-1) and (14-2), which can be used to represent the color indices for H ( B1  B8 )  1 and H ( B1  B8 )  2 , respectively. For example, the Hamming weight of the eight first bits 1 (i.e., H ( B1  B8 )  1 ) can represent eight color indices: 1, 2, 4, 8, 16, 32, 64, and 128. Actually, Eq. (14-1) is exactly the same as step (S5) in Wei et al.’s SDIS: If (n is odd and B9=1) then B19 =1, B29 =0; if (n is odd and B9=0) then B19 =0, B29 =1. On the other hand, Eq. (14-2) is the same as (S6).

CP data CP data   B1  B8  B1  B8  H ( B1  B8 )  1  01111000 B1  1 ;  H ( B1  B8 )  1  11111000 B1  0   (B9  1)  11111000 B 29  0   (B9  0)  01111000 B92  1 9 9     B1  B8 B1  B8 CP data CP data   H ( B  B )  2   H (B  B )  2   1(B 8 1)   01111000 B91  1;   1(B 8 0)   01111100 B91  0 9 9  10111000 B 2  1   10111100 B 2  0  9 9   

(14-1)

(14-2)

Based on the same argument, the cases i=2, 3, and 4 may be used to represent the color indices with Hamming weights 3, 4, 5, 6, 7, and 8, respectively. However, for the case Hamming weight 8, Wei et al.’s SDIS has the weakness: an incorrect assignment of B9 for the color index 255. From Eq. (15), it is observed that for the color index 255, we cannot represent B9=0 except using 4B5W. The above analysis using constant weight is the same as the analysis for Wei et al.’s SDIS.

13

  B1  B8 ( H ( B  B )  8)  (B  1)  11110000 1 8 9  00001111     B1  B8  ( H ( B1  B8 )  8)  (B9  0)  11110000 00001111   

CP data

block pattern on shadows

B  1  NS1: 5B4W NS2 : 5B4W B 1 1 9 2 9

CP data

block pattern on shadows

(15)

B91  0  NS1: 4B5W NS2 : 4B5W B92  0 

The proposed SDIS is based on using 6-out-of-9 binary vector (i.e., 6B3W in a block) on a shadow and the corresponding block on the other shadow is 5-out-of-9 binary vector (i.e., 5B4W in a block). For achieving a higher average value of b, we choose the blocks with 6B3W and 5B4W alternately like a chessboard in Fig. 2. The average value of b is (6+5)/2=5.5 which is larger than that of Wei et al.’s SDIS.

B1 B2 B1 B2

B2 B1 B2 B1

B2 B1 B2 B1

B1 B2 B1 B2

B1 B2 B1 B2

B2 B1 B2 B1

B2 B1 B2 B1

B1 B2 B1 B2

6B3W block 5B4W B2 : block B1 :

(a) (b) Fig. 2. Block patterns on shadows of the proposed SDIS: (a) NS1 (b) NS2.

The proposed SDIS is based on Lemma 1. Theorem 3 demonstrates that the SDIS may share 256-color image and embed the color palette. The blocks B1 and B2 have 6B3W and 5B4W subpixels, respectively. Also the notations ( B11  B91 ) and ( B12  B92 ) in the proposed SDIS are defined as nine subpixels in B1 and B2, respectively.

Theorem 3. The XOR-ed results from 6-out-of-9 binary vector and 5-out-of-9 binary vector can represent 0~255 color indices and embed the data of color palette. Proof: By Lemma 1, the XOR-ed result of 6-out-of-9 binary vector and 5-out-of-9 binary vector has Hamming weight (1+2i), where 0i3, and the pattern is shown below.

1 i 5i i 3i   v1  1 1 1 1 0 0 0 0    v2  0 0 1 1 1 1 0 0

(16)

14

For i=0, 1, 2, and 3, the Hamming weights of XOR-ed result are 1, 3, 5, and 7, respectively. Eq. (16) can be written as Eq. (17).

5 1     v1  1 11111 i  0 :    v2  0 11111  3 3   v  111 111  1   i  2 :  v2  000 111 

2 4 1 2   v1  11 1111 0 00 i =1:    000 v2  00 1111 1 00 3

000

(17)

3 4 2  00 0  v1  1111 11 000 i  3 :    v2  0000 11 111 11 0 2

1

By the same argument in the proof of Theorem 2, we can use the cases i=0, 1, 2, and 3, and 4 to represent the color indices with Hamming weights 0, 1, 2, 3, 4, 5, 6, and 7 (except Hamming weight 8). The embedding of color palette data B9 for the color indices with Hamming weights 16 is similar to that of Wei et al.’s SDIS. However, the assignment of B91 to embed the color palette bit B9 has to be slightly modified. The step (S5) is modified as “if (n is even and B9=1) then B19 =1, B29 =0, and if (n is even and B9=0) then B19 =0, B29 =1”. By rewriting Eq. (17), the following equation shows how to generate two shadows for Hamming weights

H ( B1  B8 )  1 ~ 6 and the CP data B9 (note: the first eight bits of block may be randomly permuted).

  B1  B8  H ( B1  B8 )  1  11111000   (B9  1)  01111000     B1  B8  H ( B1  B8 )  2   10111100   (B9  1)   01111100   B1  B8  H ( B1  B8 )  3     (B  1)   11011100 9  00111100    B1  B8  H ( B  B )  4     1(B 8 1)   11001110 9  00111110      B1  B8  H ( B1  B8 )  5   11100110   (B9  1)  00011110     B1  B8  H ( B1  B8 )  6   11100011   (B9  1)    00011111

 B1  B8 H ( B1  B8 )  1   B  1;   11111100  (B9  0)   B 1   01111100 CP data  B1  B8 H ( B1  B8 )  2   1  B9  1 ;   11111100  (B9  0)   B92  0  00111100 CP data  B1  B8 H ( B1  B8 )  3   1  B9  1 ;   11011110  (B9  0)   B92  1  00111110 CP data  B1  B8 H ( B1  B8 )  4   1  B9  1 ;   11101110  (B9  0)   B92  0  00011110 CP data  B1  B8 H ( B1  B8 )  5   1  B9  1;   11100111  (B9  0)   B92  1  00011111 CP data  B1  B8 H ( B1  B8 )  6   B91  1 ;   11110011  (B9  0)   B92  0   00001111 CP data 1 9 2 9

CP data

B91  0 B92  0

(18-1)

CP data

B91  0 B92  1

(18-2)

CP data

B91  0 B92  0

(18-3)

CP data

B91  0 B92  1

(18-4)

CP data

B91  0 B92  0

(18-5)

CP data

B91  0 B92  1

(18-6)

For the color indices with Hamming weights 0 and 7, using 6B3W block on one shadow and 5B4W on the

15

other shadow also has the weakness of an incorrect assignment of B9 similar to only using 5B4W blocks in Wei et al.’s SDIS. As shown in Eq. (19), for the color index 0 (i.e., H ( B1  B8 )  0 ), we cannot represent B9=0 because B1: 5B4W and B2: 6B3W (see Eq. (19-2)) contradict the definition in Table 2. To represent B9=0 for the color indices with H ( B1  B8 )  7 , we reach the contradiction because B1 has 5B4W and B2 has 4B5W (see Eq. (19-4)).

 ( H ( B1  B8 )  0)  (B9     ( H ( B1  B8 )  0)  (B9    ( H ( B1  B8 )  7)  (B9    ( H ( B  B )  7)  (B 1 8 9  

block patter on shadows CP data  B1  B8 1   1)  11111000 B9  1  B1: 6B3W 11111000 B 2  0 B2: 5B4W 9 

(19-1)

CP data block patter on shadows  B1  B8 1  0)  11111000 B9  0  B1: 5B4W 11111000 B 2  1 B2: 6B3W 9 

(19-2)

block patter on shadows CP data  B1  B8  1)  11110001 B91  1  B1: 6B3W 00001111 B 2  1 B2: 5B4W 9 

(19-3)

CP data block patter on shadows  B1  B8 1   0)  11110001 B9  0  B1: 5B4W 00001111 B 2  0 B2: 4B5W 9 

(19-4)

By using the similar approach for 5B4W block, for the color index 0 and the color indices with

H ( B1  B8 )  7 , we can use 28 out of 56 patterns from B11  B81 (see Eqs. (19-1) and (19-3)) to represent B9=1, and other 28 patterns for B9=0. The pattern in Eq. (17) cannot represent the color index 255 (i.e., H ( B1  B8 )  8 ). Next, we show how to represent the color index 255. When the XOR-ed result ( B1 , B2 ,

, B8 ) is 0, the above shows that we use 28

from 56 5-out-of-8 patterns ( B11  B81 ) to represent B9=1, and other 28 patterns for B9=0. Actually, this approach can be easily modified to represent the color indices 0 and 255 and the CP data B9 simultaneously. Let all 56 5-out-of-8 patterns of ( B11  B81 ) be a set S (see Eq. (19-1)). We equally partition the set S into four subsets S0,0 , S0,255 , S1,0 , and S1,255 . The subset S x , y , where x=0, 1 and j=0, 255, is then used to represent the CP data B9=x and the color index = y.



Block diagrams of the proposed SDIS is illustrated in are briefly described step by step as follows.

16

Fig.3 Detailed procedures of sharing and recovery

(2, 2)SDIS

B1 B2 B1 B2 B2 B1 B2 B1

B2 B1 B2 B1 B1 B2 B1 B2 B2 B1 B2 B1

NS2

B1 B2 B1 B2

binary cover image

CS2

color meaningful shadows

SI

NS2

CP

CS1 CS2

NS1

NS2

BS1 BS2

binary meaningful shadows

BS1 BS2

binary meaningful shadows

NS1

NS2

SI NS1  NS 2

CP

NS1

:6B3W B2 :5B4W B1

CS1

NS1

regard the color pixel and the white pixels in both shadows as “1” and “0”,respectively

SI

B1 B2 B1 B2 B2 B1 B2 B1

noise-like shadows

regard the black pixel and the white pixels in both shadows as “1” and “0”,respectively; if the number of 1s in block is 4, change the value to its complement

noise-like shadows

fill in all 1s in the blocks of both shadows with the corresponding color of the cover pixel; fill in all 0s in both shadows with the complementary color of the cover pixel

color cover image

fill in 1s in the blocks with the corresponding color of the cover pixel, and leave 0s in both shadows blank

color meaningful shadows

CP

SI CP

(a) (b) Fig.3. Block diagram of the proposed SDIS: (a) sharing procedure (b) recovery procedure

Sharing procedure: (S1) Obtain B1  B9 from the secret image SI and the color palate CP. (S2) Let w be the Hamming weight of ( B1  B8 ) , i.e., w  H ( B1  B8 ) . And, the number of positions of (10), (01), (11), and (00) from B1 to B2 are n10 , n01 , n11 , and n00 , respectively. The blocks of B1 and B2 are alternately used on NS1 and NS2 in Fig. 2. (S3) For 1 w 6: (S3-1) For odd w: among the positions in ( B1  B8 ) , from B1 to B2, we randomly choose the locations of

n10   w / 2 , n01   w / 2 , n11  6  n10  B9 , and n00  2  n01  B9 . (S3-2) For even w: among the positions in ( B1  B8 ) , from B1 to B2, we randomly choose the locations of

n10  w / 2  1  B9 , n01  w / 2  1  B9 , n11  6  n10  B9 , and n00  2  n01  B9 . /* Note: steps (S3-1) and (S3-2) consist with Eq. (18). For example, for w=3 and B9=1, we have

n10  3 / 2  2 , n01  3 / 2  1 , n11  6  2  1  3 , and n00  2  1  1  2 , which is the same as Eq. (18-3). For another case w=4 and B9=0, we have n10  4 / 2  1  0  3 , n01  4 / 2  1  0  1 , n11  6  3  0  3 , and

n00  2  1  0  1 , which is the same as Eq. (18-4). */ (S3-3) If (w is even and B9=1) then B19 =1, B29 =0; if (n is even and B9=0) then B19 =0, B29 =1. (S3-4) If (n is odd and B9=1) then B19 =1, B29 =1; if (n is odd and B9=0) then B19 =0, B29 =0. (S4) For w =0 and 8:

17

(S4-1) Among the positions in ( B1  B8 ) , from B1 to B2, we randomly choose the locations of n11  5 , and

n00  3 . (S4-2) For w=0 (the color index 0) and B9=0: choose one ( B11  B81 ) from S0,0 and then determine ( B12  B82 ) from Eq. (19-1); for w=0 and B9=1: choose one ( B11  B81 ) from S1,0 and then determine ( B12  B82 ) from Eq. (19-1); for w=8 (the color index 255) and B9=0: choose one ( B11  B81 ) from S0,255 and then determine

( B12  B82 ) from Eq. (19-1); for w=8 (the color index 255) and B9=1: choose one ( B11  B81 ) from S1,255 and then determine ( B12  B82 ) from Eq. (19-1). (S4-3) Set B91  1 and B92  0 . (S5) For w =7: (S5-1) Among the positions in ( B1  B8 ) , from B1 to B2, we randomly choose the locations of n10  4 ,

n01  3 and n11  1 . (S5-2) Partition the set for a color index with w=7 into the sets S1 and S0 of 18 elements and 17 elements. /* There are

74  35 combinations for a color index with w=7. Consider the case such that the color index 127 4-out-of-7 vector

is (B1, B2, …, B8)=(1,1,1,1,1,1,1,0). There are 35 combinations ( B11 ,

, B71 , B81  1) for this color index (see Eq.

(19-3)) */ (S5-3) For B9=1 (respectively, B9=0), choose one ( B11  B81 ) from S1 (respectively, S0), and then determine

( B12  B82 ) from Eq. (19-3). (S5-4) Set B91  1 and B92  1 . (S6) Repeat (S1)(S5), until every pixel in SI and the data in CP are processed.

Recovery procedure: (R1) Obtain the XOR-ed result ( NS1  NS2 ) . (R2) For 1 w 6: (R2-1) Recover the color indices ( B1  B8 ) from the first 8 bits of every 9-subpixel block in ( NS1  NS2 ) . 1 (R2-2) Recover the data of color palette B9 from B9 on shadow 1.

18

(R3) For w=0: (R3-1) Check which sets S x , y the pattern ( B11  B81 ) belongs. (R3-2) Determine the data of color palette B9=x and the color index = y. (R4) For w=7: (R4-1) Recover the color indices ( B1  B8 ) from the first 8 bits of every 9-subpixel block (R4-2) Check which sets (S1 or S0) the pattern ( B11  B81 ) belongs, and determine the data of color palette B9. (R5) Repeat (R2) and (R4) until all blocks in ( NS1  NS2 ) are processed.

Let the values of c and d be the color index and the bit of color palette. An illustrative example gives a quick understanding for the proposed SDIS.

Example 1. Share and recover the following information (c, d) = (135, 1), (0, 1), (255, 0), (191, 1) by the proposed SDIS. We intentionally use these color indices to show various sharing procedures in (S3) (for c=135), (S4) (for c=0 and 255), and (S5) (for c=191), respectively. Given (c, d) = (135, 1), because of (135)10  ( B1 , B2 ,

, B8 )  (11100001)2 , we have w=4 and B9=1. From

B1 to B2, we randomly choose the locations of n10  w / 2  1  B9  4 / 2  1  1  2 , n01  w / 2  1  B9 

4 / 2  1  1  2 , n11  6  n10  B9  6  2 1  3 , and n00  2  n01  B9  2  2  1  1 . Two possible patterns of

( B11  B81 ) and ( B12  B82 ) satisfying the above requirements are given in Eqs. (20) and (21). In fact, there are many patterns can be used to represent (c, d) = (135, 1).

B11  B81

CP data

10110110 B91  1 01010111 B92  0

(20)

B12  B 282

B11  B81

CP data

00101111 B91  1 11001110 B92  0

(21)

B12  B 282

19

Given (c, d) = (0, 1), because of (0)10  (00000000)2 we have w=0 and B9=1. By (S4-2), we can select one element from S1,0 (see Table 3) to represent ( B11  B81 ) . Accordingly, the pattern ( B12  B82 ) may be chosen. By the same argument, we select one element from S0,255 (see Table 3) for (c, d) = (255, 0). Also, via (S4-2), Eqs. (22) and (23) demonstrates one pattern to represent (c, d) = (0, 1) and (c, d) = (255, 0), respectively.

B11  B81

CP data

11110100 B91  1 11110100 B92  0

(22)

B12  B 282

B11  B81

CP data

11101100 B91  1 11101100 B92  0

(23)

B12  B 282

For (c, d) = (191, 1), because of (191)10  (11111101)2 we have w=7. Via step (S5-2) and (S5-3), we select one element for ( B11 , B21 , B31 , B41 , B51 , B61 , B81 ) (i.e., ( B11 , B21 ,

, B81 )  Bi17 ) from S1 (see Table 4) for (c, d) =

(191, 1). After setting B91 and B92 via step (S5-4), one pattern of ( B11  B81 ) and ( B12  B82 ) is given in Eq. (24).

B11  B81

CP data

11101010 B91  1 00010111 B92  1

(24)

B12  B82

For recovery, consider the case ( B11 , B21 , (20). We can obtain ( B1 , B2 ,

, B81 )  (10110100) and ( B12 , B22 ,

, B82 )  (01010111) in Eq.

, B8 )  (11100001) from the XOR-ed result ( B11  B12 , B21  B22 ,

and B9  B91  1 . Therefore, we have (c, d)=(135, 1). For the case

( B11 , B21 ,

( B12 , B22 ,

, B82 )  (11110100) in Eq. (22), we have ( B1 , B2 ,

( B11 , B21 ,

, B81 )  (11110100)  S1,0 , we recover (c, d)=(0, 1). For the case ( B11 , B21 ,

( B12 , B22 ,

, B82 )  (11101100) , we have

( B1 , B2 ,

20

, B81  B82 ) ,

, B81 )  (11110100) and

, B8 ) = (00000000). By checking the pattern , B81 )  (11101100) and

, B8 ) = (00000000). By checking the pattern

( B11 , B21 ,

, B81 )  (11101100)  S0,255 , we recover (c, d)=(255, 0). For recovering

(11101010) and ( B12 , B22 ,

H ( B1 , B2 ,

, B82 )  (00010111) , we have ( B1 , B2 ,

( B11 , B21 ,

, B81 ) 

, B8 ) = (11111101)=191. Because of

, B8 )  7 and the pattern ( B11 , B21 , B31 , B41 , B51 , B61 , B81 )  (1110100)  S1 , this implies B9=1. Finally, we

obtain (c, d) = (191, 1).



Table 3. The 8-tuples in the partitioned sets S0,0 , S1,0 , S0,255 , and S1,255 for w=0 and 8. S0,0 ( B11 , B21 , , B81 )

S1,0 ( B11 , B21 , , B81 )

S0,255 ( B11 , B21 , , B81 )

S1,255 ( B11 , B21 , , B81 )

(11111000) (10111100) (11011010) (11010110) (10101110) (00111110) (10111001) (10110101) (01101101) (11100011) (11001011) (01011011) (01100111) (10001111)

(11110100) (01111100) (10111010) (10110110) (01101110) (11110001) (01111001) (01110101) (10011101) (11010011) (10101011) (00111011) (10010111) (01001111)

(11101100) (11110010) (01111010) (01110110) (10011110) (11101001) (11100101) (11001101) (01011101) (10110011) (01101011) (11000111) (01010111) (00101111)

(11011100) (11101010) (11100110) (11001110) (01011110) (11011001) (11010101) (10101101) (00111101) (01110011) (10011011) (10100111) (00110111) (00011111)

Table 4. The 7-tuples in the partitioned sets S0 and S1 for the color index with w=7. ( B11 , B21 ,

S0 , B81 )  Bi1

( B11 , B21 ,

(1111000) (1101100) (0111100) (1101010) (0111010) (1010110) (1001110) (0011110) (1101001) (0111001) (1010101) (1001101) (0011101) (1010011) (1001011) (0011011) (0100111) (0001111)

S1 , B81 )  Bi1

(1110100) (1011100) (1110010) (1011010) (1100110) (0110110) (0101110) (1110001) (1011001) (1100101) (0110101) (0101101) (1100011) (0110011) (0101011) (1000111) (0010111)

4.2 Resolving Erroneous Recovery in Secret Image The erroneous recovery comes from the confusion between the color subpixels and the white subpixels in a block on CS1 and CS2 when the color pixels from cover images are white pixels. As shown in Fig. 4 (a), a secret pixel with color index 13 and a bit of color palette “1” is represented by ( B1 , B2 ,

, B9 ) =(1, 0, 1, 0, 0, 0 , 0, 0,

1), which is subdivided into two blocks on NS1 and NS2. The blocks in NS1 and NS2 have 6B3W and 5B4W subpixels (see Figs. 4(b) and (c)). When extending the noise-like shadows to color meaningful shadows, a shadow CS1 with the color pixel C from cover image is shown in Fig. 4(d). For decoding the binary values of

( B11 , B21 ,

, B91 ) , one can easily check the subpixel is a color pixel or white pixel. Suppose that the pixel from

cover image is a white pixel, the shadow CS1 will have 9 white subpixels in a block, as shown in Fig.4(e). Therefore, the original ( B11 , B21 ,

, B91 ) cannot be obtained. To avoid the confusion when the pixel from

cover image is white, we use a near white color pixel W' to represent the white subpixel in a block. Finally, the visual quality is quite near when compared with nine white subpixels in a block, and meanwhile the

( B11 , B21 ,

, B91 ) is correctly decoded (see Fig. 4(f)). 21

B1 B2 B3

1

0

B11 B21 B31

1

1 4

1 5

1 6

1

1

0

B12 B22 B32

0

1

2 6

1

0

1

1

0

0

2 4

2 5

B4 B5 B6

0

0

0

B

B

1

0

1

B

B7 B8 B9

0

0

1

B71 B81 B91

1

0

1

B72 B82 B92

B

(a) C C

B

(b) 6B3W

B

1

(c) 5B4W

1

1

0

C

C

1

0

1

0

C

C

1

0

1

0

0

0

W'

0

0

W'

0

0

W'

0

1

1

0

W'

1

0

1

W'

1

0

1

W'

(d) (e) (f) Fig. 2 4. Block patterns: (a) the color index 5 and the color palette B9=1 (b) NS1 (c) NS2 (d) CS1 and the recovered binary patterns for the color pixel C (e) colorful CS1 and the recovered binary patterns for the white-color pixel (f) CS1 and the recovered binary patterns for the near-white color pixel W'.

4.3 Extension of the Proposed DSIS Sams as the Wei et al.’s SDIS, the noise-like shadows of the proposed SDIS may be extended to color meaningful shadows (CS1 and CS2), and binary meaningful shadows BS1 and BS2. In addition, it can be used to share true color image. To implement the scheme with CS1 and CS2, six black subpixels in B1 and five black subpixels in B2 are replaced with the color pixel of the corresponding cover pixel. For the proposed SDIS with BS1 and BS2, we reverse the color of subpixel to represent the white color, i.e., 6B3W (or 5B4W) for black color and 3B6W (or 4B5W) for white color. To share the true color, we use a 25-subpixel block, which three 8-tuples are used to represent R, G and B color planes. Because we share R, G and B color bits directly, we do not need color palette and thus one additional bit in 25-subpixel block is abandoned. Based on the average number of subpixels in a block is enhanced from 5 to 5.5, our schemes with meaningful shadows and sharing true color image have the better performance than Wei et al.’s SDIS. Let the regions in shadows showing the content of cover image be RP(256) (sharing 256-color image) and RP(true ) (sharing true color image) for the proposed SDIS, and

RW(256) (sharing 256-color image) and RW(true ) (sharing

true color image) for Wei et al.’s SDIS. In addition, let the contrasts of binary meaningful shadows for the proposed SDIS and Wei et al.’s SDIS be CP and CW , respectively. In the following theorem, we demonstrate that RW(256) < RP(256) , RW(true ) < RP(true ) , and CW < CP .

Theorem 4. When sharing 256-color or true color images, our regions in shadows showing the content of cover images are greater than those of Wei et al.’s SDIS. Also, our contrast of binary meaningful shadow is better than that of Wei et al.’s SDIS. Proof: The proposed SDIS alternately uses 6B3W-block B1 and 5B4W-block B2 in shadows, so that the average 22

number of black subpixels in a block is (5+6)/18=5.5/9, and we obtain RP(256)  5.5 / 9 . On the other hand, Wei et al.’s SDIS only uses 5B4W block in shadows, and thus RW(256)  5 / 9 . In Wei et al.’s SDIS, the last bit of a block ( B91 or B92 ) is 1 and 0 half and half (see steps (S5) and (S6)). Moreover, the last bit ( B91 or B92 ) is discarded when sharing true color image. The 25-subpixel block for sharing true clor image is shown in Fig. 5, where the red, green, and blue colors show 8 bits for R, G, and B color planes, respectively. The average number of black subpixels in every eight subpixels (for each color plane) is 50.5=4.5. In [26], the authors claimed that the 25th subpixel B25 can be abandoned, or used to hide the secret information and digital watermark. Therefore, to enhance the more regions showing the content of cover image, we may 1 and B252 . Finally, the region showing the content of cover image is RW(true )  choose the black subpixel for B25

(4.5  3  1) / 25  14.5 / 25 .

B1 B2 B3 B4 B5

B11 B21 B31 B41 B51

B12 B22 B32 B42 B52

B6 B7 B8 B9 B10

B61 B71 B81 B91 B101

B62 B72 B82 B92 B102

B11 B12 B13 B14 B15

B111 B121 B131 B141 B151

B112 B122 B132 B142 B152

B16 B17 B18 B19 B20

1 B161 B171 B181 B191 B20

B162 B172 B182 B192 B202

1 1 1 1 1 B21 B22 B23 B24 B25 B21 B212 B222 B232 B242 B252 B25 B23 B24 B22 (a) (b) (c) Fig. 5. The 25-subpixel block for sharing true clor image: (a) expansion of one pixel to 25 subpixels (b) 25-subpixel block in shadow 1 (c) 25-subpixel block in shadow 2.

In the proposed SDIS, the blocks B1 and B2 are alternately used in shadows. From steps (S3-3) and (S3-4) (for 1w6), and steps (S4-3) (for w=0, 8), the average number of black subpixels in every eight subpixels (every color plane) is 5 (see Eq. (18) and Eq. (19-1)). In addition, from step (S5-4) (set B91  1 and B92  1 for w=7), the average number of black subpixes in every eight subpixels (for each color plane) is 4.5 (see Eq. (19-3)). Therefore, the total average number of black subpixels in in every eight subpixels (for each color plane) is 5  248 / 256 (for w  0 ~ 6, 8)  4.5  8 / 256 =4.984. Finally, the value of

RP(true )

is determined as

(4.984  3  1) / 25  15.952 / 25 . The above implies RW(256) < RP(256) and RW(true ) < RP(true ) . Next, we calculate the contrasts of binary meaningful shadows for the proposed scheme and Wei et al.’s scheme. Suppose that the cover image is black-and-white. In the proposed SDIS, we use 6B3W in B1 and 5B4W in B2 to represent the black color in cover image.

23

Meanwhile, 3B6W in B1 and 4B5W in B2 are used to represent the white color. Finally, the contrast CP is ((63)/9+(54)/9)/2=2/9. In Wei et al.’s SDIS, the black color and white color in cover image are represented as 

5B4W and 4B5W. The contrast CW is (54)/9=1/9. Therefore, we have CW < CP .

5. Experiment and Comparison 5.1 Experimental Results Four experiments (Experiments AD) are conducted to evaluate the proposed SDIS: (A) noise-like shadows NS1 and NS2 sharing 256-color image (B) color meaningful shadows CS1 and CS2 sharing 256-color image (C) binary meaningful shadows BS1 and BS2 sharing 256-color image (D) color meaningful shadows CS1 and CS2 sharing true color image. Experiments A, B and C share 256-color image, while Experiment D shares true color image. Experiment A has noise-like shadows. Other three experiments have meaningful shadows: Experiments B and D have color shadows, and Experiment C has binary shadows. In these schemes, two 256-color images C1 (Pelican) and C2 (Anhinga), and two black-and-white images C3 and C4 with printed-text A and B are used as cover images (see Figs. 6 (a-d)). Also, two secret images S1 (Lena: 256-color image) and S2 (Kaleidoscope: true color image) are shown in Figs. 6(e) and (f), respectively. All the images in Fig. 6 are 512512 pixels.

(a)

(b)

(c)

(d) (e) (f) Fig. 6. Cover images and secret images used in all experiments: (a) C1: 256-color Pelican (b) C2: 256-color 24

Anhinga (c) C3: black-and-white A (d) C4: black-and-white B (e) S1: 256-color Lena (f) S2: true color Kaleidoscope.

In these four experiments, both schemes adopt the approaches using partitioned sets to represent the color indices for some cases. The 256-color image in experiments is converted from the RGB color image by using rgb2ind() function in Matlab. The 256-color secret image Lena (S1) in Fig. 6(e) has the number for color indices: 0, 127, 191, 223, 239, 247, 251, 253, 254 , and 255 are 1452, 847, 436, 1362, 411, 1023, 397, 1616, 961, and 148, respectively. For the proposed SDSI, for w=7 (i.e., the color indices: 127, 191, 223, 239, 247, 251, 253, 254), there are 7053 positions cannot be used for embedding data. We can avoid using these positions for embedding color palette because of (512  512)

(256  3  8) . For the cases the color index 0 and the color

index 255, we use the color index 0 (w=0) to represent the color index 255 by four partitioned sets (see Table 3). The cover images Pelican (C1) and Anhinga (C2) have 70,957 and 5,357 white pixels, respectively. Therefore, we replace the white color with the near-white color for both schemes to address erroneous recovery.

Experiment A. Consider the case such that the proposed SDIS has two noise-like shadows NS1 and NS2 for sharing a 256-color secret image. Fig. 7 reveals experimental results. By sharing procedure, we obtain two noise-like NS1 and NS2. The 6B3W blocks and 5B4W blocks are alternately arranged in both shadows. If the block on NS1 is 6B3W, the corresponding block on NS2 is 5B4W, and vice versa. Figs. 7(a) and (b) are two noise-like NS1 and NS2 for the proposed scheme which the sizes are expanded to 15361536 pixels. As shown in Fig. 7(c), via recovering procedure, we can recover the original 256-color secret image Lena. Note: because shadows are nine times expanded, for demonstrating all the images in a single page, these images are not proportional.

(a) (b) (c) Fig. 7. Noise-like shadows and the recovered 256-color secret image: (a, b) NS1 and NS2 for the proposed SDIS (c) the recovered 256-color image Lena. 25

Experiment B. Consider the case such that the proposed SDIS has two color meaningful shadows CS1 and CS2 for sharing a 256-color secret image. By adopting the color pixels in C1 and C2 into black subpixels in a block of NS1 and NS2, respectively, we generate two color shadows CS1 and CS2 with the size of 15361536 pixels. In the proposed SDIS, each color shadow has the 5.5/9 of region to show the content of cover image. As shown in Figs. 8(a) and (b), it is observed that the images of Pelican and Anhinga are revealed on CS1 and CS2. Via recovering procedure, we may recover the 256-color secret image Lena (see Fig. 8(c)).

(a) (b) (c) Fig. 8. Color meaningful shadows and the recovered 256-color secret image: (a, b) CS1 and CS2 for the proposed SDIS (c) the recovered 256-color image Lena.

Experiment C. Consider the case such that the proposed SDIS has two binary meaningful shadows BS1 and BS2 for sharing the 256-color secret images S1. By revering (respectively, unchanging) the color of subpixels in a block of NS1 and NS2 to represent the white (respectively, black) color in C3 and C4, respectively. Our scheme has the contrast 2/9 (see Theorem 4) in two binary meaningful shadows. It is observed that the printed-texts A and B are revealed indeed from BS1 and BS2 with the size of 15361536 pixels (see Figs. 9(a) and (b)). Via recovering procedure, we may recover the 256-color secret image Lena (see Fig. 9(c)).

26

(a) (b) (c) Fig. 9. Binary meaningful shadows and the recovered 256-color secret image: (a, b) BS1 and BS2 for the proposed SDIS (c) the recovered 256-color image Lena.

Experiment D. Consider the case such that the proposed SDIS has two color meaningful shadows CS1 and CS2 for sharing a true color secret image. For each color plane, by adopting the color pixels in C1 and C2 into black subpixels in a 25-subixle block, we can generate two color shadows CS1 and CS2 with the size of 25602560 pixels (25 times expanded). There are 15.925/25 of region to show the content of cover image (see Theorem 4). As shown in Figs. 10(a) and (b), it is observed that the images of Pelican and Anhinga are revealed on CS1 and CS2. Via recovering procedure, we may recover the true color secret image Kaleidoscope (see Fig. 10(c)).

(a) (b) (c) Fig. 10. Color meaningful shadows and the recovered true color secret image: (a, b) CS1 and CS2 for the proposed SDIS (c) the recovered true color image Kaleidoscope.

5.2 Discussion and Comparison 5.2.1 Security Analysis The proposed SDIS and Wei et al.’s SDIS share a secret image into two noise-like (or meaningful) shadows. Both schemes are 2-out-of-2 (denoted as (2, 2)) secret sharing schemes. In fact, a (2, 2) secret sharing 27

scheme should provide a perfect threshold property, i.e., one cannot get any information about the secret image from one shadow. Although Wei et al.’s SDIS is not a (2, 2) secret sharing scheme with perfect security, it is secure enough for practical use. We explain why it is secure enough below. From one single shadow

85  56

84  70

( B11 , B21 ,

, B81 , B91 )  (111100001) , there are

( B12 , B22 ,

, B82 ) . Thus, attackers know that the recovered color indices could be only one of

color indices ( B1 , B2 ,

5-out-of-8 and

4-out-of-8 patterns for

, B8 ) (see Eq. (25)). For example, the XOR-ed result ( B1 , B2 ,

index 127 (i.e., ( B1 , B2 ,

, B8 )  (11111110) ) if ( B11 , B21 ,

85  84  126

, B8 ) is not the color

, B81 , B91 ) is (111100001) . For a (2, 2) secret

sharing scheme, basically, we should not obtain any secret information from one shadow. However, for this case, attackers know some information. Even though Wei et al.’s SDIS does not provide perfect security like the (2, 2) secret sharing scheme, attackers still do not know the exact color index from the shadow

( B11 , B21 ,

, B81 , B91 ) = (111100001). Therefore, Wei et al.’s SDIS is secure enough for practical use.

 ( B11 B21 B31 B41 B51 B61 B71 B81 B91 )  (11110000 1)  2 2 2 2 2 2 2 2 2 5-out-of-8 0  ( B1 B2 B3 B4 B5 B6 B7 B8 B9 )      4-out-of-8 1   representations of     ( B1 B2 B3 B4 B5 B6 B7 B8 ) 126 colo indices   

(25)

Consider the case such that we use 35 4-out-of-8 patterns with B81  1 and B81  0 to represent the data of color palette B9  1 and B9  0 , respectively. Wei et al.’s SDIS solves the incorrect assignment of B9 for the color index 255. If an attacker obtains a block ( B11 , B21 ,

, B81 , B91 )  (111100001) from a shadow like Eq. (25),

he just has the same information like the above analysis. Because attackers do not have the corresponding block

( B12 , B22 , is ( B1 , B2 ,

, B82 , B92 )  (000011111) from the other shadow. Therefore, attackers cannot know the XOR-ed result , B8 )  (11111111) , i.e., the color index 255, and then further derive the data of color palette.

Therefore, using the partitioned sets of 4-out-of-8 patterns for ( B11 , B21 ,

, B81 ) to represent the data of color

palette is secure. Consider the proposed SDIS. For the cases 1w6 (see steps (S3)), we have the similar analysis like the analysis in Eq. (25). For example, if an attacker has a block ( B11 , B21 ,

28

, B81 , B91 )  (111110001) (note: 6B3W),

he still only has the information that the XOR-ed result ( B1 , B2 ,

, B8 ) could be one of 126 color indices.

These 126 possible values in Eq. (26) may be different to those in Eq. (25) because of different ( B11 , B21 ,

 ( B11 B21 B31 B41 B51 B61 B71 B81 B91 )  (11111000 1)  2 2 2 2 2 2 2 2 2 5-out-of-8 0  ( B1 B2 B3 B4 B5 B6 B7 B8 B9 )      4-out-of-8 1   representations of      ( B1 B2 B3 B4 B5 B6 B7 B8 )  126 colo indices  

, B81 ) .

(26)

For the color index 255, we may use the approach of partitioned sets in Section 4.1 to solve the problem of Wei et al.’s SDIS. In the proposed SDIS, we may use four partitioned sets ( S0,0 , S1,0 , S0,255 , and S1,255 ) for the cases w=0, 8, and two partitioned sets ( S0 and S1 ) for w=7. By the above same security analysis, our scheme is also secure for the cases w=0, 7, and 8.

5.2.2 Complexity Analysis of Encoding and Decoding Consider the cases that using partitioned sets to address the problem for Wei et al.’s SDIS (i.e., embedding the color index 255) and the proposed SDIS (i.e., embedding the color index for w=0, 7, and 8). They all use the approach of partitioned set, and we should use lookup table (LUT) for encoding and decoding. For example, two small LUTs with the size of 35 bytes are necessary in Wei et al.’s SDIS for encoding/decoding the color index 255. In the proposed SDIS, for w=0 and w=8, we need four small LUTs with the size of 14 bytes; for w=7, we need two small LUTs with the size of 18 bytes and 17 bytes, respectively. Finally, the sizes of total storage are 70 bytes and 91 bytes for Wei et al.’s SDIS and the proposed SDIS, respectively. Our scheme is slightly greater than Wei et al.’s scheme. On the other hand, Wei et al.’ SDIS only needs LUT-based encoding/decoding for the color index 255. However, the proposed SDIS needs this LUT strategy for 10 color indices (i.e., the color index 0 (w=0), the color index 255 (w=8), and the color indices 127, 191, 223, 239, 247, 251, 253 and 254 (w=7)). Notice that the above comparison is based on the premise that we also use our approach (partitioned sets) to solve the incorrect assignment of color palette data. However, even though we use partitioned sets to solve the problem of Wei et al.’s SDIS, it cannot enhance the visual quality of meaningful shadows lie the proposed scheme.

5.2.3 Test Erroneous Recovery in Secret image 29

Errors in recovered secret image come from the confusion between the white subpixels in cover images and the original white subpixels in a block. If the cover pixel is white, we have nine white subpixels in a block. This causes the wrong XOR-ed result ( B1 , B1 ,

, B8 ) . In experiments, we intentionally choose the cover image

Pelican (C1) with a very large number of 70,957 white pixels. Another cover image Anhinga (C2) has 5,357 white pixels in the cover image. Between these two cover images, there are 38 overlapped white pixels. Thus, 76,276 (=70,957+ 5,35738) erroneous pixels occur in the recovered image. Figs. 11(a) and (b) shows a cropped shadows CS1 and CS2 without using a near white color pixel to represent the white subpixel in a block. Obviously, from the recovered image Lena (see Fig. 11(c)), it is observed that the recovered Lena has many speckled pixels in the areas, which are the white sky area in Pelican image and the white rock area in Anhinga image. In Experiment B, by our approach in Section 4.2 , the shadow also has almost the same color of white color (see Figs. 8(a) and (b)), and meantime we can recover the original Lena without distortion (see Fig. 8(c)) to solve the weakness of erroneous recovery.

(a) (b) (c) Fig. 11. Cropped areas of colo meaningful shadows and the recovered secret image without using a near white color pixel: (a, b) two shadows CS1 and CS2 (c) the recovered Lena with speckled pixels.

5.2.4 Comparison The proposed SDIS solves weaknesses in Wei et al.’s SDIS: the incorrect assignment of B9 for the color index 255, and the erroneous recovery in secret image for white cover pixels. Meantime, we also enhance the average number of black subpixels in a block to obtain better meaningful shadows. Table 5 illustrates the comparison of the structure of block, the region in color meaningful shadows showing the content of cover image, the contrast of binary meaningful shadows, the embedding of color palette data, the erroneous recovery in secret image, the encoding/decoding complexity, and the security between the proposed SDIS and Wei et al.’s SDIS.

30

The above compared items are all objective measurements. We also add a subjective measurement to evaluate the real visual qualities of meaningful shadows between the proposed SDIS and Wei et al.’s SDIS. For simplicity, we perform an experiment on binary meaningful shadows and colorful shadows. These images are viewed and evaluated by nine people in our lab. As we know, this measurement is very personally subjective and bias because it is based on human visual sight. Finally, for binary meaningful shadows, all people can easily recognize the printed-text for these two schemes. To further observing the detail of color meaningful shadows, we enlarge the color meaningful shadows on screen for a better assessment. Except for one person, eight people think that our SDIS has the better visual quality than Wei et al.’s SDIS.

structure of block region in color shadows showing the content of cover image contrast of binary meaningful shadows evaluation of visual quality for meaningful shadows by human visual sight embedding the data of color palette data

erroneous recovery in secret image

encoding/decoding complexity

security

Table 5. Comparison of the proposed SDIS and Wei et al.’s SDIS. Wei et al.’s SDIS the proposed SDIS commentary all blocks of Wei et al.’s SDIS are 5B4W; our SDIS 5B4W 6B4W and 5B4W has 6B3W and 5B4W blocks half and half we have 6/9 region and 5/9 region for B1 and B2, and 5/9 5.5/9 thus the average is (6/9+ 5/9)/2= 5.5/9 the contrasts are (63)/9 and (54)/9 for B1 and B2, 1/9 2/9 and thus the average is (3/9+1/9)/2=2/9 all people can easily for color meaningful we perform an experiment recognize the printed-text shadows, our SDIS has on meaningful shadows; on binary meaningful the better visual quality these images are viewed shadows for Wei et al.’s than Wei et al.’s SDIS. and evaluated by 9 people SDIS and our SDIS our approach by using Embedding the data of partitioned sets can also having a weakness for the color palette for some address the weakness of color index 255 color indices by using Wei et al.’s SDIS when partitioned sets embedding color palette data for color index 255 YES the approach using a NO (the incorrect extraction replacement of near white (replacing the white cover from 9-tuple white subpixels can address the pixel by a near white subpixels causes erroneous weakness of both schemes subpixel) recovery in secret image) for the white cover pixel using XOR operation and if Wei et al.’s SDIS wants LUT; note: four LUTs with to solve the weakness of the size of 14 bytes for embedding color palette using XOR operation w=0 and 8, and two LUTs data for the color index with the size of 18 bytes 255; two LUTs with 35 and 17 bytes for w=7 are bytes are required required Wei et al.’s SDIS is not a (2, 2) secret sharing see the analysis in Section Same as Wei et al.’s SDIS scheme; however it is 5.2.1 practically secure enough

31

5. Conclusion We discuss three weaknesses in Wei et al.’s SDIS. The incorrect assignment of color palette, and the erroneous recovery may bring about errors when recovering the secret image, and the third weakness is that the contrast color meaningful shadow is not good enough to reveal the cover image. In this paper, we prose a SDIS, which can recover the correct secret image. Meanwhile, the visual quality of meaningful shadows is enhanced. Additionally, theoretical analyses and experiments are also given to demonstrate the effectiveness.

Acknowledgement This

research

was

supported

in

part

by Ministry

of

Science

and

Technology,

under

grant

105-2221-E-259-015-MY2, in part by the National Natural Science Foundation of China under Grant Nos. 61373020, 61272435, U1536102 and U1536116.

References [1] M. Naor and A. Shamir, “Visual cryptography,” Advances in Cryptology-EUROCRYPT’ 94, LNCS 950, pp. 1-12, 1995. [2] H. Kuwakado and H. Tanaka, “Size-reduced visual secret sharing scheme,” IEICE Trans. on Funda. of Elect., Comm. and Comp. Sci., Vol. E87-A, pp. 1193-1197, May, 2004. [3] C. N. Yang, “New visual secret sharing schemes using probabilistic method,” Pattern Recognition Letters, Vol. 25, pp. 481-494, 2004. [4] S. Cimato , R. De Prisco, and A. De Santis, “Probabilistic visual cryptography schemes,” The Computer Journal, Vol. 49, pp. 97-107, 2006. [5] S.J. Shyu and H.W. Jiang, “General constructions for threshold multiple-secret visual cryptography Schemes,” IEEE Transactions on Information Forensics Security, Vol. 8, pp. 733-743, 2013. [6]

Y.C. Chen, G.B. Horng, and D.S. Tsai, “Comment on cheating prevention in visual cryptography,” IEEE Transactions on Image Processing, Vol. 21, pp. 3319-3323, 2012.

[7]

Y.C. Hou, “Visual cryptography for color images,” Pattern Recognition, Vol. 36, pp.1619-1629, 2003.

[8]

C.N. Yang, H.W. Shih, C.C. Wu, and L. Harn, “k out of n region incrementing scheme in visual cryptography,” IEEE Transactions on Circuits and Systems for Video Technology, Vol. 22, pp. 779-810, 2012. 32

[9]

Y.C. Hou, Z.Y. Quan, C.F. Tsai, and A.Y. Tseng, “Block-based progressive visual secret sharing,” Information Sciences, Vol. 233, pp. 290-304, 2013.

[10] C.N. Yang and C.Y. Lin, "Almost-aspect-ratio-invariant visual cryptography without adding extra subpixels," Information Sciences, Vol 312, pp. 131–151, 2015. [11] A. Shamir, “How to share a secret,” Communications of the Association for Computing Machinery, Vol. 22 pp. 612-613, 1979. [12] C.C. Thien and J.C. Lin, “Secret image sharing,” Computer & Graphics, Vol. 26, pp. 765-770, 2002. [13] C.C. Chang, Y.P. Hsieh, and C.H. Lin, “Sharing secrets in stego images with authentication,” Pattern Recognition, Vol. 41, pp. 3130-3137, 2008. [14] Y.Y. Lin and R.Z. Wang, “Scalable secret image sharing with smaller shadow images,” IEEE Signal Processing Letters, Vol. 17, pp. 316-319, 2010. [15] C.C. Wu, S.J. Kao, and M.S. Hwang, “A high quality image sharing with steganography and adaptive authentication scheme,” Journal of Systems and Software, Vol. 84, pp. 2196-2207, 2011. [16] C.N. Yang, J.F. Ouyang, and L. Harn, “Steganography and authentication in image sharing without parity bits,” Optics Communications, Vol. 285, pp. 1725-1735, 2012. [17] P. Li, C.N. Yang, C.C. Wu, Q. Kong, and Y. Ma, “Essential secret image sharing scheme with different importance of shadows,” Journal of Visual Communication and Image Representation, Vol. 24, pp. 1106-1114, 2013. [18] S. Cimato and C.N. Yang, Visual cryptography and secret image sharing, CRC Press, Taylor & Francis, 2011. [19] S.J. Lin and J.C. Lin, “VCPSS: A two-in-one two-decoding-options image sharing method combining visual cryptography (VC) and polynomial-style sharing (PSS) approaches,” Pattern Recognition, Vol. 40, pp. 3652-3666, 2007. [20] P. Li, P.J. Ma, X.H. Su, and C.N. Yang, “Improvements of a two-in-one image secret sharing scheme based on gray mixing model,” Journal of Visual Communication and Image Representation, Vol. 23, pp. 441-453, 2012. [21] C.N. Yang and D. Wang, “Property analysis of XOR based visual cryptography,” IEEE Transactions on Circuits and Systems for Video Technology, Vol. 24, pp. 189-197, 2014. [22] R. Lukac and K. N. Plataniotis, “Bit-level based secret sharing for image encryption,” Pattern Recognition, Vol. 38, pp. 767-772, 2005.

33

[23] C.N. Yang, G.C. Ye, and C. Kim, “Data hiding in halftone images by XOR block-wise operation with difference minimization,” KSII Transactions on Internet and Information Systems, Vol. 5, pp. 457-476, 2011. [24] C.C. Chen and W.J. Wu, “A secure Boolean-based multi-secret image sharing scheme,” Journal of Systems and Software, Vol. 92, pp. 107-114, 2014. [25] C.N. Yang, C.H. Chen, and S.R. Cai, "Enhanced Boolean-based Multi Secret Image Sharing Scheme," Journal of Systems and software, Vol. 116, pp. 22-34, 2016. [26] S.C. Wei, Y.C. Hou, and Y.C. Lu, “A technique for sharing a digital image,” Computers Standards & Interfaces, Vol. 40, pp. 53-61, 2015.

Highlights



We show the problem of incorrect assignment for color palette data in Wei et al.’s SDIS.



We show the problem of erroneous recovery problem for secret image in Wei et al.’s SDIS.



We propose a new SDIS with better visual quality of shadow images and also solve the problems of Wei et al.’s SDIS.

34