A n -out-of- n Sharing Digital Image Scheme by Using Color Palette

: A secret image sharing (SIS) scheme inserts a secret message into shadow images in a way that if shadow images are combined in a speciﬁc way, the secret image can be recovered. A 2-out-of-2 sharing digital image scheme (SDIS) adopts a color palette to share a digital color secret image into two shadow images, and the secret image can be recovered from two shadow images, while any one shadow image has no information about the secret image. This 2-out-of-2 SDIS may keep the shadow size small because by using a color palette, and thus has advantage of reducing storage. However, the previous works on SDIS are just 2-out-of-2 scheme and have limited functions. In this paper, we take the lead to study a general n -out-of- n SDIS which can be applied on more than two shadow. The proposed SDIS is implemented on the basis of 2-out-of-2 SDIS. Our main contribution has the higher contrast of binary meaningful shadow and the larger region in color shadows revealing cover image when compared with previous 2-out-of-2 SDISs. Meanwhile, our SDIS is resistant to colluder attack.


Introduction
A secret image sharing (SIS) scheme inserts a secret message into shadow images in a way that if shadow images are combined in a specific way, the secret image can be recovered.A SIS scheme is usually referred to by a threshold (k, n) SIS, where k ≤ n, and can insert a secret image into n shadow images (referred to as shadows).In a (k, n)-SIS, we may recover the secret image by using any k shadows, but cannot recover the secret image from (k − 1) or fewer shadows.There are various types of SIS.Here, we give a brief survey for three major types of SIS schemes: the visual cryptography scheme (VC), the polynomial-based SIS (PSIS), and the bit-wise Boolean-operation based SIS.
The so-called VC [1][2][3][4][5][6] has a novel stacking-to-see property such that the involved participants can easily stack shadows to visually decode the secret through the human eye.This property makes VC applicable in many scenarios.Although VC has the ease of decoding, it has poor visual quality of reconstructed image.Another SIS adopts (k − 1)-degree polynomial like Shamir's secret sharing [7] to design (k, n)-PSIS [8][9][10][11][12][13][14][15].There are two major differences between VC and PSIS: the quality of recovered image and the decoding method.Unlike VC provided with the poor visual quality, the recovered secret image of PSIS is distortion-less.However, the decoding of VC only needs stacking operation but PSIS uses the computation of Lagrange interpolation to recover secret image.Some SIS schemes are based on Boolean operations [16][17][18][19][20]. Note: the stacking operation of VC, strictly speaking, is also a Boolean OR operation.However, this OR operation of VC is pixel-wise operation, which applied on black-and-white dots.However, Boolean operation in [16][17][18][19][20] is bit-wise operations, and can obtain a high-quality secret image (a distortion-less image like PSIS scheme).Besides, using -wise Boolean has much lower complexity when compared with Lagrange interpolation.
Recently, Wei et al. use the bit-wise XOR operation to design a (2, 2) sharing digital image scheme (SDIS) [17] to share a 256-color (or true color) digital image.Wei et al.'s (2, 2)-SDIS is also a type of (k, n)-SIS where k = n = 2. Wei et al.'s (2, 2)-SDIS is the first SIS scheme using a 256-color palette.This color palette has 256 colors, where each color is composed of red (R), green (G), and blue (B) color planes.Each color and is chosen from a palette of 16,777,216(=2 24 ) colors (24 bits: each color plane has 8 bits).In VGA cards, 256 on-screen colors are chosen from a color palette, and these colors are most visible to the human eye and meanwhile conserve a bandwidth.When using a color palette, each pixel is represented by a color index in a 256-color color palette.Consider an example, a 256 × 256-pixel image.The file size is 256 × 256 × 1 bytes (color indices) +256 × 3 bytes (color palette) = 66,304 bytes, but is 256 × 256 × 3 = 196,608 bytes for using 24-bit true color format.Thus, the file size of a color image can be kept small when represented by a color palette.Because Wei et al.'s (2, 2)-SDIS is based on color palette, and thus it has the advantage of reducing storage.
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.In [19], Yang et (2,2)-SDIS are simple 2-out-of-2 scheme and have limited applications.In this paper, we take the lead to study a general (n, n)-SDIS, which can be applied on any n ≥ 3. The main weakness of Wei et al.'s (2, 2)-SDIS is the incorrect assignment of color palette data for some color indices, and this is tackled by using a complicated approach, partitioned sets, in Yang et al.'s (2, 2)-SDIS.In the proposed (n, n)-SDIS, because of the number of shadows more than two, i.e., n ≥ 3, a simple approach reducing Hamming weigh of a temporary block is adopted to easily solve this weakness.In addition, performance of our (n, n)-SDIS are enhanced when compared with the previous (2, 2)-SDIS.The rest of this paper is organized as follows.Section 2 reviews Wei et al.'s (2, 2)-SDIS and Yang et al.'s (2, 2)-SDIS.The proposed (n, n)-SDIS is presented in Section 3. Also, an approach of enhancing visual quality of color meaningful shadow is introduced.A very extreme attack, the (n − 1)-colluder attack, on the proposed (n, n)-SDIS is discussed in Section 4. The experiment, discussion and comparison are in Section 5. Finally, Section 6 concludes the paper.

Preliminaries
Notations in this paper and their descriptions are listed in Table 1.These notations are used throughout the whole paper to describe all the schemes, Wei et al.'s (2, 2)-SDIS [17], Yang et al.'s (2, 2)-SDIS [19], and the proposed (n, n)-SDIS.
In [17], Wei et al. first proposed a simple (2, 2)-SDIS to insert a 256-color digital image SI into two binary noise-like shadows (NS 1 and NS 2 ).In Wei et al.'s (2, 2)-SDIS, every 9-bit block B, i.e., b 1 − b 9 , is obtained from the 256-color secret image SI and the color palette CP.Afterwards, the block B is subdivided into two blocks B (1) and B (2) on shadow 1 NS 1 and shadow 2 NS 2 , respectively, by using XOR operation.As shown in Figure 1, B = B (1) ⊕ B (2) , where each bit b Both shadow blocks of B (1) and B (2) are Y blocks.Accomplish all blocks until all pixels in SI and the data in CP are processed.Because every pixel in SI is represented as a block, shadow sizes are nine times expanded.The first 8 bits b 1 − b 8 in B represents a color index, and the ninth bit b 9 in every block of NS 1 (i.e., the bit b (1)

9
) is collected to covey the CP information.Therefore, from the XOR-ed results NS 1 ⊕ NS 2 we may obtain color indices and the CP to recover SI.There are other two types of shadows for Wei et al.'s (2, 2)-SDIS.Noise-like shadows (NS 1 , NS 2 ) can be extended to two binary meaningful shadows (BS 1 , BS 2 ) and two color meaningful shadows (CS 1 , CS 2 ), on which binary cover image BCI and color cover image CCI can be, respectively, visually viewed.In addition, Wei et al.'s (2, 2)-SDIS can also be extended to directly insert a true color SI without using CP.
9 ) are 5B4W blocks.For the corresponding position of this secret pixel pi, the cover pixels of BCI 1 and BCI 2 are white and black, respectively.We reverse the shadow B (1) = (b (1) 9 ) = (110001101) block to (001110010) (4W5B) to represent the white color pixel in BCI 1 , and we do not change 9 ) = (010110101) (5B4W) to represent the black color pixel in BCI 2 .In secret recovery, the color index can be easily derived from the exclusive OR result from (b (1) It is obvious that more black subpixels in every block may enhance the visual quality of meaningful shadows BS 1 and BS 2 , and CS 1 and CS 2 .Accordingly, in [19], Yang et al. adopted X block and Y block half and half on blocks B (1) and B (2) , such that the average number of black subpixels in B (1) and B (2) is enhanced from 5 to 5.5.This enhancement improved the visual quality of meaningful shadows.Meanwhile, Yang et al.'s (2, 2)-SDIS also solved the other two weaknesses of Wei et al.'s (2, 2)-SDIS.

Motivation and Design Concept
As described in Section 2, there are three weaknesses in Wei et al.'s SDIS: (1) the incorrect assignment of the color palette data for the color index 255, (2) the partial regions in meaningful shadows showing the content of the cover image, and (3) the erroneous recovery in secret image if the cover pixel is white in color meaningful shadows.Yang et al.'s (2, 2)-SDIS already tackled these weaknesses.
By delving into these three weaknesses, we can see that the third weakness is a minor weakness caused from an intrinsic nature of color.A trivial approach in [19], using a near white color pixel instead of white pixels in cover image, is very efficient in addressing this weakness.Therefore, the approach can be still adopted in the proposed (n, n)-SDIS for solving this minor weakness.Our contribution is not just the extension from 2-out-of-2 scheme to n-out-of-n scheme.The proposed (n, n)-SDIS, where n ≥ 3, has better solutions for other two major weaknesses.Because the number of shadows is more than two, we can easily solve the first weaknesses (note: the detail will be described in Section 3).However, Yang et al.'s (2, 2)-SDIS uses a very complicated approach by partitioned sets to solve this weakness.For the second weakness, our (n, n)-SDIS uses X blocks in most shadows This approach has large average black subpixels in shadow blocks to enhance visual qualities of meaningful shadows.In addition, the proposed (n, n)-SDIS embeds the CP information in b 9 but both (2, 2)-SDISs [17,19] use b (1) 9 in shadow block B (1) .The bit b 9 obtained from the XOR-ed result B is more securely protected than the bit b (1) 9 in one shadow block B (1) .A secret block B = (b 1 ...b 9 ) has 8 bits (b 1 ...b 8 ) to represent a color index, and one bit b 9 for representing the data of color palette CP.Together with CP, this color index can represent a pixel in secret image SI.All 9-bit blocks are obtained from the secret image SI and the color palate CP.Suppose that T is a 9-bit temporary block.Equations ( 1) and ( 2) are main statements in this paper, on which we can design the proposed (n, n)-SDIS.As shown in Equation ( 1), we may randomly generate (n − 2) X blocks B (i j ) , 1 ≤ j ≤ n − 2, and then determine a temporary block T via these (n − 2) blocks and the block B (see upper equation in Equation ( 1)).The content of T is provisional.Afterwards, T is divided into two blocks {B (j 1 ) , B (j 2 ) } where {j 1 , j 2 } = {1, 2, ..., n} − {i 1 , ..., i n−2 }.Using lower equation in Equation ( 1), we may insert T into two blocks based on Wei et al.'s (2, 2)-SDIS or Yang et al.'s (2, 2)-SDIS, which is dependent on the Hamming weigh of block T. In next subsection, we prove that lower equation in Equation ( 1) can be successfully accomplished.Via Equation (1), we can derive Equation ( 2) implies that the block B can be subdivide into n shadow blocks B (1) , B (2) , ..., B (n) , and meanwhile can be recovered from B = B (1) ⊕ ... ⊕ B (n) .All the n shadows in the proposed (n, n)-SDIS are illustrated in Figure 2. The operation of lower equation in Equation ( 1 Moreover, in [17], the authors claimed that the (2, 2)-SDIS has a novel application to cover the transmission of confidential images.For example, as a supplementary aid to existing symmetric cryptography standards like DES which requires a pre-shared key, the (2, 2)-SDIS remains a safe and less risky means for key distribution.Because the prosed scheme is extended from 2-out-of-2 to n-out-of-n, it implies that our (n, n)-SDIS can be applied on a group key distribution, which includes n members in this group.Besides the application in key distribution, the proposed scheme can be also applied to protection of secret image among multiple users.For instance, the colorful image of traffic or medical information are confidential, and our scheme provides a secure and high efficiency approach to safely keeping such image among n users, only all n users are able to recover the image with high quality.
Finally, in a shadow NS i , 1 ≤ i ≤ n there are X blocks with percentage of n−1.5 n (= The more X blocks have the large number of black subpixels and may enhance visual qualities of meaningful shadows, and these percentages have more effective performance for large n.

Sharing and Recovering Algorithms
A block diagram of the proposed (n, n)-SDIS is illustrated in Figure 3. Shadows NS 1 − NS n are noise-like, which is the same as Boolean-operation based SIS [18].For the proposed (n, n)-SDIS, we can complement the blocks for the corresponding white cover pixels to generate binary meaningful shadows (BS 1 − BS n ) from noise-like shadows (NS 1 − NS n ), i.e., 6B3W (or 5B4W) for black color and 3B6W (or 4B5W) for white color.However, the scheme in [18] does dot has this property.On the other hand, to implement color meaningful shadows (CS 1 , CS n ), the 1s in blocks are replaced with the color of the corresponding cover pixel, and leave 0s blank.Therefore, we only describe how to generate noise-like shadows, and how to recover the secret image and color palette from n noise-like shadows.( ) ) (S-2) Randomly generate (n − 2) X blocks B (i 1 ) , B (i 2 ) , ..., B (i n−2 ) .(S-3) By (n − 2) random blocks and the block B, calculate the temporary block . (S-4) If H(T) is 9, we reduce its Hamming weight to H(T) = 7 via modifying any one shadow block of {B (i 1 ) , ..., B (i n−2 ) }.
/* In Lemma 2, we prove that {B (j 1 ) , B (j 2 ) } can be obtained from Y(T) for odd H(T), and from W(T) for even H(T).*/ (S-6) Process all the blocks, and output shadow blocks B (1) ...B (n) on n noise-like shadows NS 1 − NS n , respectively.
Recovering procedure: (S-1) Obtain B by XOR-ing (B (1) /* Theorem 1, demonstrates that we can obtain the original block from B = (B (1) ⊕ ... ⊕ B (n) ) */ (S-2) Recover the color index (b 1 − b 8 ) and the data of color palette b 9 , respectively, from B. (S-3) Repeat the above until all blocks in NS 1 ⊕ ... ⊕ NS n are processed, and finally SI and CP can be recovered.

Extension of (n, n)-SDIS to Share True Color Secret Image
Same as (2, 2)-SDIS and VC in [5], the proposed (n, n)-SDIS can be used to share a true color image.To share a true color secret image, we use a 25-subpixel block B r , which the first three 8-tuples, r 1 , ..., r 8 , g 1 , ..., g 8 , and bl 1 , ..., bl 8 , are used to represent R, G and B color planes.The other one bit in B r is p 9 .This 25-subpixel block B r is shown in Figure 4a.Because we share R, G and B colors directly, we do not need to use the bit p 9 to covey any information.Thus, this bit p 9 could be abandoned, or used as authentication bits to provide authentication capability like VC in [6] and PSIS in [10].Collect (x 1 ...x 8 ), where x ∈ {r, g, bl}, and append the bit p 9 to form red, green, and blue shadow blocks B x where x ∈ {r, g, bl} as shown in Figure 4b.Detailed procedures of the proposed (n, n)-SDIS for sharing and recovering true color image are briefly described step by step as follows.

Sharing procedure:
(S -1) Obtain 24-bit true color r 1 , ..., r 8 , g 1 , ..., g 8 , and bl 1 , ..., bl 8 from the secret image SI, and random generate a bit p 9 to form a 25-bit block B r , as shown in Figure 4a./* Parity bit p 9 is not used to covey any information, and thus it can be randomly generated */ (S -2) Subdivide the true color block B T to red, green, and blue shadow blocks B r , B g , B bl .(S -3) Using B r , B g , B bl as 9-bit block B in (S-1), respectively, to generate n shadow blocks bl , and append a black subpixel in the 25-th subpixel to generate a 25-bit shadow block B (i) , where 1 ≤ i ≤ n. /* Because we do not use the 25-th bit p 9 in the XOR-ed result B T to convey any information, we can use black subpixel in 25-th subpixel for all shadow blocks to enhance the number of black subpixels.*/ (S -5) Process all the blocks, and output blocks B (1) − B (n) on n noise-like shadows NS 1 − NS n , respectively.

Enhancing Visual Quality of Color Meaningful Shadow
Consider sharing 256-color (respectively, true color) SI, noise-like shadows NS i , 1 ≤ i ≤ n, are 3M × 3N (respectively, 5M × 5N) times expanded.Based on noise-like shadow NS i , we can fill in 1s in shadow blocks with the color of the corresponding cover pixel in CCI, and leave 0s blank to generate color meaningful shadow CS i .Consider the case sharing 256-color SI.As shown in Figure 5a, there is a pixel with a blue color C in CCI.Suppose that the block B (i) at corresponding position for this 5b), and this block B (i) is a X block with 6B3W sub-pixels (see see Figure 5c).By putting the blue cover pixel C into all black sub-pixels in Figure 5c, we have color meaningful shadow CS (i) in Figure 5d.Noise-like shadow and color meaningful shadow have the same size 3M × 3N subpixels and 9 times expanded when compared with CCI.As shown in Figure 5d, the color at 1s in a block are the same.This is because SI and CCI have the same size with M × N pixels.To enhance visual quality of CS i , we use a large color cover image CCI with 3M × 3N pixels (note: the original CCI has only M × N pixels).Obviously, this larger CCI has the high resolution than CCI.As shown in Figure 6, our new approach uses a large CCI (see Figure 6a).By putting the color pixels in to into all 1s of B (1) in Figure 6b, we have the CS i in Figure 6c.Because the color meaningful shadow CS i has more colors, and will have the high resolution.By the same argument, this approach can also be applied to sharing true color SI. in NS i (c) the corresponding color block in CS i .

Main Theorems and Examples
Lemma 1. Suppose that the block T in Equation ( 1) is all-1 block, i.e., H(T) = 9.We may change any two positions (one is 1 → 0 and the other is 0 → 1) in any one block B (i j ) , 1 ≤ j ≤ n − 2, such that the equation B = T ⊕ B (i 1 ) ⊕ ... ⊕ B (i n−2 ) holds, and H(T) is reduced from 9 to 7. Meanwhile, all (n − 2) blocks B (i j ) , 1 ≤ j ≤ n − 2, are still X blocks.
Proof.As shown in Equation ( 1), all these (n − 2) blocks B (i 1 ) − B (i n−2 ) are X blocks.We choose one block B (i j ) , and modify any two positions of 1 → 0 and 0 → 1.This modification will change the 1 in the block T to 0 at these two chosen modified positions.After that, H(T) is reduced to 9 − 2 = 7.Meanwhile, because we change two positions by 1 → 0 and 0 → 1, respectively, the Hamming weight H(B (i j ) ) is unchanged, and this shadow block B (i j ) is still a X block.

Proof. Let
The following theorem shows that the proposed (n, n)-SDIS is a n-out-of-n sharing scheme that we can recover SI and CP from n noise-like shadows (NS 1 − NS n ), and cannot obtain SI and CP from (n − 1) or fewer shadows.
Theorem 1.The proposed (n, n)-SDIS is n-out-of-n sharing scheme that the XOR-ed result of n shadow blocks can represent 0 255 color indices and the data of color palette.
Next, we consider the recovery.As shown in Equation ( 2), we can recover the original block B = (b 1 ...b 9 ) from B = B (1) ⊕ ... ⊕ B (n) .Therefore, we can determine the color index (b 1 ...b 8 ) and the data of color palette b 9 .After obtaining all blocks, we can recover SI and CP.Because of B = B (1) ⊕ ... ⊕ B (n) , it is obvious that we cannot recover the original block B via (n − 1) or fewer shadow blocks.
Let the ratio of average number of black subpixels in a block (i.
It is obvious that R P ≥ 5.5 9 with equality for n = 3.From these values R W = 5 9 , R Y = 5.5 9 , R P = 6−1.5/n9 , we have R W < R Y < R P The contrast is the difference of blackness for black block and white block.In binary meaningful shadows BS 1 − BS n , we complement the blocks for the corresponding white cover pixels to generate white shadow blocks.Thus, if the number of black subpixels in a black shadow block is n B , then the he number black subpixels in a white shadow block is 9 − n B .Thus, we have C W = 5−(9−5) B (1) = (101110110) and B (2) = (111101001) with H(B (1) ) = H(B (2) ) = 6, and then we obtain the temporary block T via the following equation.
Let R P be the ratio of average numbers of black subpixels in a 25-bit shadow block for the proposed (n, n)-SDIS sharing true color image.The following theorem demonstrates R P > R P , i.e., the meaningful shadows of sharing true color secret image have the better visual quality than those of sharing 256-color secret image.Theorem 3. The ratio of average numbers of black subpixels in a 25-bit block for the proposed (n, n)-SDIS sharing true color image is R P = 17 25 − 0.16 n , where R P > R P .

Proof. If the blocks
bl are X (respectively, Y ) blocks, then the first 8 bits in g , B bl is 6 × = 40 3 for X blocks.Therefore, the average number of black subpixels in every 8 bits in the first 24 bits of B (i) is 16n−4 3n , as derived below.
Because the 25-th bit in shadow block is always 1, and thus the value of R P is determined as The following equation implies R P > R P .

Security Analysis: The (n − 1)-Colluder Attack
Here, we consider an attack way that (n − 1) participants collude together and want to figure out SI and CP.The (n − 1)-colluder attack is a very extreme attack for the proposed (n, n)-SDIS, because it needs (n − 1) participants jointly providing their shadows for guessing SI and CP.We first discuss the (n − 9 of every block on NS 1 .Therefore, the CP can be completely obtained from NS 1 .Even though Participant 1 has the color palette CP, but he cannot obtain the information about color index.An attacker has 1 256 ≈ 0.004 probability to figure out the correct color index (b 1 ...b 8 ) of block B without any shadow.This value of 1  256 is a brute-force probability, which tries all possible 256 colors in the color palette.However, for the (n − 1)-colluder attack, Participant 1 has B (1) .By cryptanalytic attacks relying on knowing one shadow (the first eight bit of B (1) ), Participant 1 may guess the color index.= 1/84+1/126 2 ≈ 0.01.Both probabilities 0.08 and 0.01 are higher than the brute-force probability 0.004.However, these probabilities 0.08 and 0.01 are still practically secure for guessing 256 colors.
Let the successful probability to recover the block B for (n − 1)-colluder attack, for the proposed (n, n)-SDIS, be P P .In the following theorem, we theoretically prove ).
Theorem 4. The successful probability to recover the block B in the proposed (n, n)-SDIS for (n − 1)-colluder attack is P P = 1 ), where P W ≤ P Y ≤ P P .
Proof.Suppose that there are (n − 1) shadows (say B (1) − B (n−1) ) for reconstruction, on which we may guess the type of shadow block in the corresponding position of B (n) .The block B (n) has X block and Y block with 2n−3 2n probability and 3 2n probability, respectively, which are derived below.
) probability to guess the correct color index (b 1 ...b 8 ), which is better than brute-force probability 1  256 .Thus, P P is calculated as follows.
Since P W = 1 For n = 3, the value of P P is P P = = 1 84 ≈ 0.012 is almost the same as P y ≈ 0.01.For this extreme case, the (n − 1)-colluder attack, the security of the proposed (n, n)-SDIS is close to that of Yang et al.'s (2, 2)-SDIS.By the same argument, for other cases collecting (n − 2) or shadows, the possible combination of collected shadows is more difficult to analyze compared with collecting (n − 1) shadows, and even less than the brute-force probability.
In the proposed (n, n)-SDIS, the color palette information is conveyed by b 9 (the ninth bit in B), but not the ninth bit b (1) 9 of the block B (1) in NS 1 .Therefore, the color palette CP may be obtained from only one shadow for Wei et al.'s (2, 2)-SDIS and Yang et al.'s (2, 2)-SDIS.Even though an attacker has the CP information, he still cannot obtain the secret image SI.For the proposed (n, n)-SDIS, the color palette information in B is securely protected and only can be determined from XOR-ing n blocks B (1) ⊕ ... ⊕ B (n) .This makes the cryptanalysis is more difficult for the proposed (n, n)-SDIS.The following theorem demonstrates the successful probability P C to recover a correct color in CP for the proposed (n, n)-SDIS when collecting (n − 1) shadows.
Theorem 5.The successful probability to recover a correct color in CP for the proposed (n, n)-SDIS when collecting (n − 1) shadows is P C = ( 23 − 1/6 n ) 24 .
Proof.Each color information in CP is encapsulated in 24 blocks, which every block should be derived from B = B (1) ⊕ ... ⊕ B (n) .If colluders have (n − 1) shadows (say NS 1 − NS n−1 ), for a block B, they have the XOR-ed result B = B (1) ⊕ ... ⊕ B (n−1) , and can guess that the shadow block B (n) is X block and Y block with 2n−3 2n probability and 3 2n probability, respectively.For X block, it implies that we have 6  9 probability that the bit b 9 is the complementary bit b 9 in B .On the other hand, we have 5 9 probability that the bit b 9 is the complementary bit b 9 in B for Y block.Therefore, the average probability of guessing b 9 is derived as n Note: every block has one-bit color palette information, and a true color is represented by 24-bit R, G, and B color planes.Because colluders can guess the bit b 9 with 2  3 − 1/6 n probability, Therefore, the value P C = ( 2 3 − 1/6 n ) 24 is less than ( 2 3 ) 24 ≈ 5.94 × 10 −5 , and this implies that the color palette cannot be recovered under (n − 1)-colluder attack.Experiments E and F demonstrate binary and color meaningful shadows for (4, 4)-SDIS and (5, 5)-SDIS, respectively.In Experiment G, we redo Experiment C to enhance the visual quality of color meaningful shadows by using the approach in Figure 6.

Seven experiments (Experiments
In all experiments, five binary cover images BCI 1 − BCI 5 with black-and-white printed texts A , B , C , D , E , and five color cover images CCI 1 − CCI 5 with photos of birds are used.In addition, two secret images SI 1 (Lena: 256-color image), SI 2 (Kaleidoscope: true color image) are used.All these images BCI 1 − BCI 5 (see Figure 7), CCI 1 − CCI 5 (see Figure 8), and SI 1 , SI 2 (see Figure 9) are 256 × 256 pixels.has high resolution.These three images CCI 1 − CCI 3 are omitted here for brevity.By using the approach in Figure 6, we use color pixels in CCI 1 − CCI 3 into black subpixels in blocks B (1) , B (2) , and B (3) on NS 1 , NS 2 and NS 3 , respectively to generate three color meaningful shadows CS 1 − CS 3 with the size of 768 × 768 pixels.As shown in Figure 16a-c, it is observed that Figure 16 has better visual quality than Figure 12.However, the photos CCI 1 − CCI 3 used in this experiment may not clearly demonstrate the enhancement.Here, we use a cover image, a colorful centered fractal, for testing.Figure 17( For fairer comparison, we adopt visual quality assessment, the structural similarity (SSIM) index, and the feature similarity (FSIM) index to compare Figure 17(a-1) and Figure 17 (b-1).Let the original image be a colorful centered fractal with the size 768 * 768 pixels.According to the image quality assessment from Laboratory for Computational Vision in New York University (please refer to https://www.cns.nyu.edu/~lcv/ssim/#usage), to calculate SSIM and FSIM for color images, it would be better to convert the color image to gray image with the formula 0.2989R + 0.5870G + 0.1140B, and then calculate its SSIM and FSIM.Finally, SSIM and FSIM of Figure 17(a-1) are 0.2532 and 0.8400, and SSIM and FSIM of Figure 17(b-1) are 0.3300 and 0.8498, respectively.These values of SSIM and FSIM demonstrate a consistency with the performance in Figure 17    In fact, we may further enhance R P by using W block instead of X block to generate (n − 2) The value of P P is 1 ) for using W block in step (S-2), it is still practically secure for applications.This is because our CP information is protected in the XOR-ed result, but not conveyed on b (1) 9 in B (1) like (22)-SDIS [17,19].For example, when using 8B1W as W block.If colluders have (n − 1) shadows (say NS 1 − NS n−1 ), for a block B, they have the XOR-ed result B = B (1) ⊕ ... ⊕ B (n−1) , and can guess the shadow block B (n) is W block with a very high probability for large n (note: 2n−4 2n → 1 for large n).It implies that there is about 8  9 probability that the bit b 9 in B is the complementary bit b 9 of B .By using the same argument in proof of Theorem 5, for this case, the successful probability to recover a correct color in CP is P C = ( 89 ) 24 0.059.Therefore, we cannot get the correct CP back.Although colluders may recover the first 8 bits (b 1 − b 8 ) in B, i.e., a color index by complementing the first 8 bits (b 1 − b 8 ) in B with 1  9 probability.This probability of guessing a color index is larger than the brute-force probability 1 256 .However, colluders do not have the correct CP, and thus they cannot recover the original SI.
Obviously, it is more difficult to apply (n − 1)-colluder attack on using 7B2W as W block, because P C is ( 7 9 ) 24 0.0024.This is why we claim that using W block is still practically secure for applications.To demonstrate the above phenomenon, we use 8B1W as W block in the proposed (5, 5)-SDIS.Five color meaning shadows using color cover images CCI 1 − CCI 5 in Figure 8a-e are illustrated in Figure 18a, where the approach of enhancing visual quality in Section 4.3 is also adopted.  .This problem comes from from all-1 9-bit vector.In [19], Yang et al. adopted a complicated approach using partitioned sets to address this problem.In the proposed (n, n)-SDIS, the number of shadows of (n, n)-SDIS is more than two, i.e., n ≥ 3. Thus, we can easily adopt a simple approach by reducing H(T) to H(T) = 7 in step (S-4) via modifying any one shadow block to solve this problem.Meantime, as described in Section 5.1, we may enhance R P and simultaneously retain the practical security by using W block.
As shown in

Figure 3 .
Figure 3. Block diagram of the proposed (n, n)-SDISFor noise-like shadows (NS 1 , NS n ), detailed procedures of sharing and recovering procedures are briefly described step by step as follows.

Figure 5 .
Figure 5. Block patterns: (a) a pixel with a color in CCI (b) the corresponding block B (i) in NS i (c) the corresponding 6B3W block in NS i (d) the corresponding block in CS i

Figure 6 .
Figure 6.Block patterns: (a) 9 color pixels with color C 1 − C 9 in CCI (b) the corresponding block B (i) in NS i (c) the corresponding color block in CS i .
1)-colluder attack on Wei et al.'s (2, 2)-SDIS and Yang et al.'s (2, 2)-SDIS.Suppose that Participant 1 wants to predict SI and CP from his own shadow NS 1 .Because the color palette CP information is conveyed by the ninth bit b (1) Let the successful probability to recover the block B for Wei et al.'s (2, 2)-SDIS and Yang et al.'s (2, 2)-SDIS be P W and P Y , respectively, when collecting one shadow.Because both shadow blocks of Wei et al.'s (2, 2)-SDIS are all Y blocks (5B4W), obviously P W is 1 On the other hand, shadow blocks of Yang et al.'s (2, 2)-SDIS are evenly composed of X blocks and Y blocks.Thus, P Y =

2 ,
we have P W < P Y .About P Y and P P , the relation is derived as follows.
a-1,b-1) shows two color meaningful shadows using the original one and new enhancement, respectively.For clear observation, cropped image areas of Figure 17(a-1,b-1) are shown in Figure 17(a-2,b-2).Visual inspection of cropped image areas in Figure 17(a-2,b-2) reveals that the original method spoils some edges and fine details in shadow images.Our enhancement has clear color sharpness, especially the clearness of edges. (a-2,b-2).
, for large n.Even though these values are larger than P P = 1

9 =
Figure 18b are the 256-color SI (Lena), and its corresponding CP.The recovered 256-color secret image SI and the color palette CP are shown in Figure 18c.It is observed that these five color meaning shadows in Figure 18a have high resolutions with R P = 8−5.5/n0.767 for n = 5, which have better visual qualities than those in Figure 15b.From, Figure 18c, there is not any secret information of CP and SI leaked for (n − 1)-colluder attack.

Figure 18 .
Figure 18.The proposed (5, 5)-SDIS using 8B1W block (a) five color meaningful shadows (b) 256-color SI and its corresponding CP (c) the recovered 256-color SI 1 and color palette CP under (n − 1)-colluder attack.6.2.2.Comparison We extend (2, 2)-SDIS to the proposed (n, n)-SDIS.Because the percentage of X block is greater than 50%, the resolution of binary and color meaningful shadows are enhanced.Note: Yang et al.'s (2, 2)-SDIS uses X block and Y block half and half, while Wei et al.'s (2, 2)-SDIS only uses Y blocks.On the other hands, Wei et al.'s (2, 2)-SDIS has the incorrect assignment of color palette data for the color index 255.This problem comes from from all-1 9-bit vector.In[19], Yang et al. adopted a complicated approach using partitioned sets to address this problem.In the proposed (n, n)-SDIS, the number of shadows of (n, n)-SDIS is more than two, i.e., n ≥ 3. Thus, we can easily adopt a simple approach by reducing H(T) to H(T) = 7 in step (S-4) via modifying any one shadow block to solve this problem.Meantime, as described in Section 5.1, we may enhance R P and simultaneously retain the practical security by using W block.As shown in Table 2, a complete comparison is given among Wei et al.'s (2, 2)-SDIS, Yang et al.'s (2, 2)-SDIS, and the proposed (n, n)-SDIS.The comparison includes the structure of block, percentages al. address these weaknesses and propose a new (2, 2)-SDIS.Both Wei et al.'s (2, 2)-SDIS and Yang et al.'s

Table 1 .
Notations and Descriptions.× 3-subpixel block B including 8-bit color index b 1 − b 8 and one bit b 9 (Note: the bit b 9 in B is collected to covey the CP information for the proposed (n, n)-SDIS) B ) of NS 1 is collected to covey the CP information for Wei et al.'s (2, 2)-SDIS and Yang et al.'s (2, 2)-SDIS) r a 3 × 3-subpixel block B r including the first three 8-tuples, (r 1 − r 8 ), (g 1 − g 8 ), and (bl 1 − bl 8 ), are used to represent R, G and B color planes, and the other one bit in B r is p 9 .B (i) a 3 × 3-pixel block on shadow i, where i = 1, 2, ..., n, including 8-bit b i 1 − b i 8 and one bit b i 9 .(Note: the ninth bit in every block B (1) (i.e., b 9 xByW x black subpixels and y white subpixels in a block X , Y X and Y blocks have 6B3W and 5B4W subpixels, respectively H(•) Hamming weight function, the number of 1 in a binary vector W(•) Operation of Wei et al.'s (2, 2)-SDIS, i.e., W(B) = B (1) ⊕ B (2) where both are Y blocks Y(•) Operation of Yang et al.'s (2, 2)-SDIS, i.e., Y(B) = B (1) ⊕ B (2) where one is X block and the other is Y block Wei et al's (2, 2)-SDIS has some weaknesses.For the color index 255, it has a problem with embedding the data of color palette.In addition, Wei et al.'s (2, 2)-SDIS with color meaningful shadows cannot correctly extract the block data for white cover pixels, and this will cause erroneous recovery in the secret image.Moreover, Wei et al.'s SDIS uses Y blocks on both shadows.Five black dots in a block B may not sufficiently demonstrate the visual quality of meaningful shadows.
X 1 be X block, and both Y 1 and Y 2 be Y blocks.We first prove that the possible Hamming weights of (Y 1 , Y 2 ) are 0, 2, 4, 6, 8, and the possible Hamming weights of (X 1 , Y 2 ) are 1, 3, 5, 7.Because both blocks Y 1 and Y 2 have the same Hamming weight 5, the number of positions of 1 → 0 and 0 → 1 crossing from vectors Y 1 to Y 2 should be the same.Suppose that this number is y.Therefore, the (Y 1 , Y 2 ) has the following form (see Equation (3)), where 0 ≤ y ≤ 4. Obviously, the Hamming weight of (Y 1 Y 2 ) in Equation (3) is 2y, and thus H(Y 1 Y 2 ) may be 0, 2, 4, 6, 8.