Boolean-Based ( k , n , m ) Multi-Secret Image Sharing

: Multi-secret image sharing technology shares multiple secret images among shared images for participants and the recovery requires enough shared images for fulﬁlling the security property. Simultaneously, the adopted Boolean method preserves high performance. The proposed ( k , n , m ) multi-secret image sharing scheme shares m secret images among n participants and gathering k participants’ shared images perfectly recovers these m secret images. The proposed scheme exhibits high performance because of the usage of Boolean-based processing. Moreover, the usage of a sharing matrix based strategy acquires perfect reconstruction of the secret images under ( k , n ) thresholds. Two proposed schemes, the source-random-mixed ( SRM ) and the source-random-separate ( SRS ) methods, are presented in this study from different strategies of sharing the random image mixed with the secret images or separated to the secret images, respectively. Moreover, the proposed SRS method exhibits the better sharing capacity. Experimental results show that the proposed schemes perform well on sharing efﬁciency and security.


Introduction
The popular usage of multimedia content leads to a security problem on transmission or storage being increasingly more important. Secret image sharing technique, aiming to protect the multimedia content among participants, is one of the protection methods and has become an important research field [1].
The secret image sharing problem was first presented by Thien and Lin [2] with solving by the Shamir-Lagrange method [3]. The secret image sharing technique is then extended to various research areas, such as sharing among host images [4,5], sharing using Boolean operations [6,7], progressive sharing [8][9][10], sharing with visual cryptography property [11], multiple sharing [12,13], and scalable sharing [14]. Among these research topics, sharing multiple secret images efficiently is an important topic in real cases. Moreover, the Booleanbased scheme outperforms the functional-based schemes on computational complexity but is hard to achieve various properties such as (k, n) partial recovery, multiple sharing, or progressive sharing. Therefore, a Boolean-based secret image sharing scheme with (k, n) thresholds on multiple secret images sharing is very important and merits our study.
The Boolean-based approaches can be classified into two categories, the stackingbased and the XOR-based methods. The stacking-based methods [15][16][17][18], which are based on visual cryptography [19], are the least computational Boolean-based methods. However, the stacking-based methods always lead to a rough visual effect. Consequently, the XOR-based methods are the key technique having both efficient computation complexity and perfect recovered results. Therefore, our proposed method is based on XOR-based technique. Moreover, sharing matrix strategy [20] is an efficient method to achieve perfect recovery. Therefore, the proposed scheme is based on XOR operations and the concept of sharing matrix.
Over the past years, several Boolean-based approaches have been presented. Wang et al. [6] adopted simple Boolean operations to present an (n, n) threshold scheme. Chen and Wu [12] 1.
Acquire the random image R with the secret images I i (0 ≤ i ≤ n − 1) by Equation (1): h = H(I 0 I 1 I 2 · · · I n−1 ), SI = image_synthesis(h), R = bit_reverse(I 0 I 1 I 2 · · · I n−1 ) SI, where H() represents the SHA-256 hash function, image_synthesis(h) synthesizes the random image from the seed h, and bit_reverse(I) reverses the image I on the bit plane.

2.
Shift the random image with different pixels to acquire a series of random images R i (0 ≤ i ≤ n − 1), as given in Equation (2): 3. Acquire shared images O i for participant i from Equation (3): The recovery steps are introduced as follows.
1. Collect all n shared images O i and perform XOR operation O 0 ⊕ . . . ⊕ O n−2 ⊕ O n−1 on all shared images, as defined by Equation (4): 2.
Apply the result of Equation (4) to Equations (1) and (2) to obtain a series of random images R i (0 ≤ i ≤ n − 1).

3.
Acquire the recovered secret images I i by applying O i and R i to Equation (5): The recovered secret images I i and original secret images I i are consistent when all the shared images O i are correctly acquired. Moreover, Chen et al.'s scheme [21] only exhibits (n, n) secret image sharing for sharing among n shared images and recovery requirement is all n shared images. For acquiring a Boolean-based and robust (k, n) secret image sharing scheme, Bao et al. [22] presented sharing an important matrix-based method, which is introduced in the next section.

Boolean-Based (k, n) Secret Image Sharing Scheme
The Boolean-based scheme is difficult to achieve with (k, n) secret image sharing when k ≤ n. Bao et al. [20] present a novel (k, n) secret image sharing scheme by using the proposed sharing matrix to achieve the recovery of only acquiring k shared images rather than n shared images. Assume that a secret image with size r × c is shared among n shared images with recovery thresholds (k, n); their sharing algorithm is introduced as follows.

1.
Generate a series of n binary sharing matrices M i (0 ≤ i ≤ n − 1) with size r × c, in which each matrix should satisfy the following two criteria: A.
Gathering k sharing matrices leads to each entry containing at least a 1 among these sharing matrices. B.
Gathering less than k sharing matrices leads to at least an entry without finding a 1 among these sharing matrices.

2.
Encrypt the secret image I to acquire a randomlike image R.

3.
For a sharing matrix M i , each entry M i (j,k)(0 ≤ j≤ r − 1,0 ≤ k ≤ c − 1) determines the corresponding shared image entry S i (j,k), as defined in Equation (6): where Rand() represents a random number generation function.
Since the encrypted image R in Step 2 is a randomlike image, Step 3 acquires a shared image that is also randomlike because part of the entries of R are S i and other entries are random numbers. According to the novel arrangement of the sharing matrices M i , gathering k shared images S i can acquire all entries S i (j,k)(0 ≤ j ≤ r − 1, 0 ≤ k ≤ c − 1) of R and then decrypt to the secret image I perfectly. However, gathering less than k shared images cannot acquire all entries of R and the secret image cannot then be decrypted.
The recovery algorithm is introduced as follows.

1.
Reconstruct the same series of n binary sharing matrices M i (0 ≤ I ≤ n − 1) as applied in the sharing algorithm.

2.
Acquire the encrypted image R from M i and S i by applying Equation (7) R 3.
Decrypt the secret image from R to acquire the recovered secret image. In Step 2, each entry of the randomlike image R will be correctly recovered because of the property of binary sharing matrices M i , as introduced in Step 1 of the sharing algorithm.

The Proposed Approach
This section introduces our proposed XOR-based (k, n, m) multi-secret image sharing scheme, in which m secret images are shared among n participants with k thresholds for perfectly recovering all secret images. Section 3.1 introduces the generation of the sharing matrix, Sections 3.2 and 3.3 introduce two proposed XOR-based (k, n, m) multi-secret image sharing algorithms: the source-random-mixed method (SRM) and source-random-separate method (SRS), respectively. Examples of the proposed SRS sharing and recovery with thresholds (2, 3, 2) are demonstrated in Section 3.4.

Sharing Matrix Generation
Since the sharing matrix has to acquire the property of (k, n) thresholds, all secret images are consistent in size and the sharing matrices P i (i = 1, 2, . . . , n) have the following properties:

1.
Each sharing matrix P i has the same size with secret images.

2.
Each element in P i are either 0 or 1.

3.
Selecting p matrices of P i , for example, i = 1, 2, . . . , p, the following criteria should be satisfied.

3.1
When p ≥ k, at least one matrix has the value of 1 in all entries among these p matrices.

3.2
When p < k, at least one entry has the value of 0 in all p matrices.
Above three properties construct the sharing matrices P i (i = 1, 2, . . . , n) that will be used in our two proposed methods. Without loss of generality, an example of (k, n) using threshold (3,4) and matrix size 4 × 6 is illustrated in Figure 1. The predetermined selection of n = 4 acquires 4 sharing matrices P 1 to P 4 and k = 3 shows that selecting 3 sharing matrices should fit the above criterion 3.1. For example, the selection of P 1 , P 2 , and P 3 fits the requirement of 3.1 being at least one matrix having the value of 1 in all entries. However, the selection of P 1 , and P 2 fits the criterion of Step 3.2, with entries such as P 1 (1,1) = P 2 (1,1) = 0 or P 1 (2,6) = P 2 (2,6) = 0. Therefore, the (k, n) thresholds determine the collection of k sharing matrices acquires at least one matrix that has the value of 1 in all entries and the collection of k − 1 sharing matrices at least have the value of 0 in one entry among sharing matrices. The sharing matrix generation algorithm, a modification of Bao et al.'s method [20], is illustrated as follows for obtaining sharing matrices Pi(i = 1,2,…,n) for (k, n, m) thresholds with r × c secret image size. (8) to acquire an n × 1 binary vector M:

Use Equation
where M includes (k − 1) number of value 0 and (n − k + 1) number of value 1.
2. Acquire initial matrix N by permuting through M as Equation (9): where the function ℎ ( , ) circularly shifts the vector V to x positions. 4. Exchange rows of to acquire exchanged matrix ′ : where the function ℎ ( , ) randomly permutes matrix A on rows by the seed b. 5. Extend the matrix ′ to fit the sizes of secret image for obtaining sharing matrix : where the function repeatMatrix(A,w) duplicates the A matrix w times for acquiring the required size, which is the same size as the secret images. Figure 2 shows sharing matrices generation step by step. Since the (k, n) thresholds are (3,4), the generated binary vector M, as shown in Figure 2a, is 4 × 1 and 2 of these 4 elements are 1, as denoted in Step 1 of Equation (8). Figure 2b shows the generated initial matrix N, as defined in Step 2 and Equation (9). Figure 2c-e shows a series of extended matrix N1, exchanged matrix 1 ′ , and sharing matrix 1 , respectively. These matrices are acquired from Steps 3-5. In Figure 2c, N1 is acquired from the first row of N with one position circular shift. Moreover, Figure 2f of N2 is acquired from the second row of N with one positioned circular shift. The sharing matrix generation algorithm, a modification of Bao et al.'s method [20], is illustrated as follows for obtaining sharing matrices P i (i = 1, 2, . . . , n) for (k, n, m) thresholds with r × c secret image size.

1.
Use Equation (8) to acquire an n × 1 binary vector M: where M includes (k − 1) number of value 0 and (n − k + 1) number of value 1.

3.
Extend one row vector of N, denoted by N(i,:), to acquire an extended matrix N i with size r × t by using Equation (10): where the function shi f tVector(V, x) circularly shifts the vector V to x positions.

4.
Exchange rows of N i to acquire exchanged matrix N i : where the function exchangeRows(A, b) randomly permutes matrix A on rows by the seed b.

5.
Extend the matrix N i to fit the sizes of secret image for obtaining sharing matrix P i : where the function repeatMatrix(A,w) duplicates the A matrix w times for acquiring the required size, which is the same size as the secret images. Figure 2 shows sharing matrices generation step by step. Since the (k, n) thresholds are (3,4), the generated binary vector M, as shown in Figure 2a, is 4 × 1 and 2 of these 4 elements are 1, as denoted in Step 1 of Equation (8). Figure 2b shows the generated initial matrix N, as defined in Step 2 and Equation (9). Figure 2c-e shows a series of extended matrix N 1 , exchanged matrix N 1 , and sharing matrix P 1 , respectively. These matrices are acquired from Steps 3-5. In Figure 2c, N 1 is acquired from the first row of N with one position circular shift. Moreover, Figure 2f of N 2 is acquired from the second row of N with one positioned circular shift.  Figure 3 shows the sharing strategy of the proposed algorithms. The input images Ii are processed to acquire temporary images Ei. On the other hand, sharing matrices Pi are calculated from thresholds (k, n), as introduced in Equations (8)- (12). At last, shared images Sj are acquired from temporary images Ei and sharing matrices Pi.

The Proposed (k, n, m) Source-Random-Mixed (SRM) Algorithm
This section shows our first proposed (k, n, m) SRM sharing algorithm, in which m secret images with identical size are shared among n shared images and collecting k shared images perfectly recover these n secret images. Assume all the secret images are denoted by Ii(1 ≤ i ≤ m) with the same size r × c, and the generated shared images are Sj(1 ≤ j ≤ n). The sharing matrices introduced in Section 3.1 are needed in the proposed sharing and recovery algorithms. Figure 4 shows the proposed SRM sharing steps and the proposed SRM sharing algorithm is introduced as follows.
1. Apply m secret images Ii (1 ≤ i ≤ m) to XOR operation for acquiring IXOR by Equation (13):  Figure 3 shows the sharing strategy of the proposed algorithms. The input images I i are processed to acquire temporary images E i . On the other hand, sharing matrices P i are calculated from thresholds (k, n), as introduced in Equations (8)- (12). At last, shared images S j are acquired from temporary images E i and sharing matrices P i .   Figure 3 shows the sharing strategy of the proposed algorithms. The input images I are processed to acquire temporary images Ei. On the other hand, sharing matrices Pi are calculated from thresholds (k, n), as introduced in Equations (8)- (12). At last, shared images Sj are acquired from temporary images Ei and sharing matrices Pi.

The Proposed (k, n, m) Source-Random-Mixed (SRM) Algorithm
This section shows our first proposed (k, n, m) SRM sharing algorithm, in which m secret images with identical size are shared among n shared images and collecting k shared images perfectly recover these n secret images. Assume all the secret images are denoted by Ii(1 ≤ i ≤ m) with the same size r × c, and the generated shared images are Sj(1 ≤ j ≤ n). The sharing matrices introduced in Section 3.1 are needed in the proposed sharing and recovery algorithms. Figure 4 shows the proposed SRM sharing steps and the proposed SRM sharing algorithm is introduced as follows.

The Proposed (k, n, m) Source-Random-Mixed (SRM) Algorithm
This section shows our first proposed (k, n, m) SRM sharing algorithm, in which m secret images with identical size are shared among n shared images and collecting k shared images perfectly recover these n secret images. Assume all the secret images are denoted by I i (1 ≤ i ≤ m) with the same size r × c, and the generated shared images are S j (1 ≤ j ≤ n). The sharing matrices introduced in Section 3.1 are needed in the proposed sharing and recovery algorithms. Figure 4 shows the proposed SRM sharing steps and the proposed SRM sharing algorithm is introduced as follows.
7. Obtain shared images Sj (1 ≤ j ≤ n) by concatenating initial shared images Sj i (1 ≤ I ≤ m), as shown in Equation (18): At last, the size of Sj is m × (r × c).

1.
Apply m secret images I i (1 ≤ i ≤ m) to XOR operation for acquiring I XOR by Equation (13): where denotes bit-based XOR operation.

2.
Generate a random image R by Equation (14): where the hash() is performed by SHA-256 hash function and the image_synthesis(h,r,c) synthesizes a randomlike image with size r × c from the seed h.

3.
Generate a series of randomlike images R i (1 ≤ i ≤ m) from Equation (15): where pixel_shift(R,i) right circularly shifts the image R with i pixels.

4.
A series of temporary images E i (1 ≤ i ≤ m) are acquired from I i and R i by Equation (16):
Calculate initial shared images S j i by images E i (1 ≤ i ≤ m) and P j (1 ≤ j ≤ n), which is defined as: where the number and size of S i j are m × n and r × c, respectively.

7.
Obtain shared images S j (1 ≤ j ≤ n) by concatenating initial shared images S j i (1 ≤ I ≤ m), as shown in Equation (18): At last, the size of S j is m × (r × c). Figure 5 shows a shared image example of (3,4,5) thresholds. In the proposed (3, 4, 5) SRM sharing method, 5 secret images are shared to 4 shared images and collecting 3 shared images perfectly recovers these 5 secret images. Figure 5a-e shows 5 secret images and Figure 5f shows one of shared image S 1 , which is a concatenation of 5 initial shared images S 1 1 ∼ S 5 1 as virtually segmented by dotted lines.
Axioms 2022, 11, x FOR PEER REVIEW 8 of 21 Figure 5 shows a shared image example of (3,4,5) thresholds. In the proposed (3, 4, 5) SRM sharing method, 5 secret images are shared to 4 shared images and collecting 3 shared images perfectly recovers these 5 secret images. Figure 5a-e shows 5 secret images and Figure 5f shows one of shared image 1 , which is a concatenation of 5 initial shared images 1 1~1 5 as virtually segmented by dotted lines.
The recovery algorithm requires k shared images with the same size m × r × c. The proposed recovery algorithm of SRM is introduced as follows.

Separate all collected shared images
For the entry (k, l) with one of the pixel ( , ) ≠ 0, assign ( , ) by the value of ( , ). If all values in the entry (k, l) are 0, then assign ( , ) by 0. 4. Acquire the temporary image by Equation (19): 5. Apply to Equation (14) for acquiring the random image . 6. Apply to Equation (15)   The recovery algorithm requires k shared images with the same size m × r × c. The proposed recovery algorithm of SRM is introduced as follows.

1.
Collect k shared images and denote them by For the entry (k, l) with one of the pixel S j i (k, l) = 0, assign S re j (k, l) by the value of S j i (k, l). If all values in the entry (k, l) are 0, then assign S j i (k, l) by 0.

4.
Acquire the temporary image I re XOR by Equation (19):

5.
Apply I re XOR to Equation (14) for acquiring the random image R re .
Apply R re to Equation (15) for acquiring randomlike images R re i (1 ≤ i ≤ m).

7.
Apply S re i and R re i to Equation (20) for obtaining recovered secret images I re i : After applying the above recovery algorithm by k correct shared images, the recovered secret images I re i (1 ≤ i ≤ m) are consistent with the original secret images. For the recovery of less than k correct shared images, all randomlike images are acquired. Moreover, in Step 3, the nonzero values of all S j i (k,l)(1 ≤ i ≤ k) should be consistent when no malicious attacks on shared images are applied.
Furthermore, the proposed SRM method assigns 0 to pixels of shared images when the corresponding coordinates of the sharing matrix are 0, as shown in Equation (17). Therefore, each shared image appears dark, as shown in Figure 5f.

The Proposed (k, n, m) Source-Random-Separate (SRS) Algorithm
This section introduces our second proposed (k, n, m) source-random-separate (SRS) algorithm. The sharing matrices introduced in Section 3.1 are also needed in the proposed sharing and recovery algorithms. Figure 6 shows the proposed SRS sharing steps, and the proposed SRS sharing algorithm is introduced as follows.

1.
Apply m secret images I i (1 ≤ i ≤ m) to XOR operation for acquiring I XOR by Equation (13).

2.
Generate a randomlike image Z with size r × c by the function image_synthesis(b,r,c) defined in Equation (14). The seed b is the same seed as used to generate the exchanged matrix in Equation (11).
Use Equation (22) to calculate images SR j (1 ≤ j ≤ n) by sharing matrix P j .
where j ranges between 1 and n, and randi(255) acquires a random integer between 0 and 255.

6.
Apply I XOR and Equation (14) to generate the random image R.

7.
Generate a series of randomlike images R i (1 ≤ i ≤ m) from the random image R by Equation (15). 8.
Apply I i and R i by Equation (16) to obtain encrypted images E i (1 ≤ i ≤ m).

9.
Randomly select m columns from the initial matrix N in Equation (9) to acquire matrix N part , i.e., where h is acquired from Equation (14) and matrix N is acquired from Equations (8) and (9) with size n × m. (24) to acquire share E i j by the matrix value of N part :

Use Equation
where two coordinates i and j are ranged as 1 ≤ i ≤ m and 1 ≤ j ≤ n.
11. Concatenate SR j and E i j (1 ≤ i ≤ m) to acquire shared images S j , i.e., Axioms 2022, 11, x FOR PEER REVIEW 10 of 21 (1   4) acquires all (1   5) for the further recovery processing. Furthermore, size of the shared images is determined by the number of elements containing 1 in each row of the matrix. Therefore, size of shared image is t × r × c, in which t is equal to 1 + | | and | | denoted number of elements with 1 in the j rows of matrix . The recovery algorithm of the proposed (k, n, m) SRS is introduced as follows.  image_synthesis(b,r,c) defined in Equation (14), in which the seed b is the same seed as used to generate the exchanged matrix in sharing matrix generation step of Equation (11). 5. Use Equation (26) to acquire 0 : where 1 ≤ ≤ , 1 ≤ ≤ and 1 ≤ ≤ . 6. Apply XOR operation between the 0 and the randomlike image Z to acquire random image IXOR, i.e., for the further recovery processing. Furthermore, size of the shared images is determined by the number of elements containing 1 in each row of the N part matrix.
Therefore, size of shared image S j is t × r × c, in which t is equal to 1 + N j part and N j part denoted number of elements with 1 in the j rows of matrix N part .
The recovery algorithm of the proposed (k, n, m) SRS is introduced as follows.

1.
Collect k shared images and denote them by S j (1 ≤ j ≤ k).

2.
Extract each shared image S j with size r × c to acquire SR j (1 ≤ j ≤ k).

4.
Generate a randomlike image Z with size r × c by the function image_synthesis(b,r,c) defined in Equation (14), in which the seed b is the same seed as used to generate the exchanged matrix in sharing matrix generation step of Equation (11).

6.
Apply XOR operation between the SR 0 and the randomlike image Z to acquire random image I XOR , i.e.,

8.
Apply the initial matrix N and coefficient h with Equation (23) to acquire the matrix N part . 9.
Split the remaining shared image S j in Step 2 to groups of r × c matrices and assign each r × c share to E i j (1 ≤ i ≤ m) by Equation (28): 10. Replace E i by extracting the nonempty r × c matrix to share E i j (1 ≤ j ≤ k). Moreover, two nonempty elements among r × c share E i j1 , and E i j2 (j1 = j2) should be identical. 11. Apply E i and R i to Equation (29) for covering the secret images I re i (1 ≤ i ≤ m), i.e., In Step 10, if two nonempty r × c sharing E i j1 and E i j2 (j1 = j2) are not identical, at least one of the r × c shares E i j1 or E i j2 is maliciously modified.

Examples of Sharing and Recovery of the Proposed SRS with Thresholds (2, 3, 2)
In this section, examples of the (2, 3, 2) SRS sharing and recovery processes are introduced. The first way is to generate sharing matrix by the predefined (2, 3, 2) thresholds as follows.

5.
Acquire sharing matrices P 1 , P 2 , and P 3 by applying matrices N i (1 ≤ i ≤ 3) to Equation (12):  Figure 7 shows an example of the proposed SRS method with thresholds (2, 3, 2). In the proposed (2, 3, 2) method, 2 secret images are shared among 3 shared images with XOR operations and gathering 2 of these 3 shared images perfectly recover these 2 secret images. Figure 7a The sharing processes of the proposed (2, 3, 2) SRS method is introduced as follows.

1.
Apply secret images I 1 and I 2 to Equation (13) for obtaining I XOR .
Apply I XOR and Equation (14) to generate the random image R.

7.
Generate randomlike images R 1 and R 2 from the random image R by Equation (15). 8.
Apply I 1 , I 2 and R 1 , R 2 by Equation (16) to obtain encrypted images E 1 , E 2 , as shown in Figure 7c,d, respectively.

9.
Randomly select two columns of N = 10. Apply N part and E 1 , E 2 to Equation (24) to generate E i shown in Figure 7e-g, respectively.
XOR operations and gathering 2 of these 3 shared images perfectly recover these 2 secret images. Figure 7a,b shows two secret images with size 256 × 256 and Figure 7c,d shows two encrypted images in Step 8 of sharing algorithm. Figure 7e-g shows three generated shared images with size 256 × 512, 256 × 512, 256 × 768, respectively. The recovery of acquiring two secret images should collect 2 shared images. Applying Figure 7e,f to the proposed recovery algorithm acquires 2 secret images, as shown in Figure 7a,b. The sharing processes of the proposed (2, 3, 2) SRS method is introduced as follows.

1.
Collect two shared images S 1 and S 2 .

4.
Generate a randomlike image Z with size r × c by Equation (14).
Use Equation (27) with SR 0 and Z to acquire I XOR . 7.
Apply Equations (14) and (15) with I XOR to acquire R 1 and R 2 .

8.
Acquire the matrix N part =

9.
Assign the remaining part of shared images S 1 and S 2 to E 1 and E 2 , respectively. 10. Apply Equation (29) with R 1 , R 2 and E 1 , E 2 to recover secret images I 1 , I 2 .

Experimental Results and Discussions
This section demonstrates the experimental results of our proposed (k, n, m) approach. Section 4.1 presents experimental results of the proposed approach with different thresholds and recovery of attacked shared images. All experiments are performed on a PC with an Intel i7-4720HQ CPU, and 16 G RAM, using MATLAB 9.10 software. The size of all secret images is 256 × 256. Section 4.2 presents the performance of the proposed approach with comparing other multiple secret image sharing methods. Figures 8 and 9 show the experimental results of two proposed schemes, the SRM and SRS methods, by sharing with thresholds (k, n, m) = (3, 4, 5) for sharing 5 secret images among 4 shared images and collecting 3 shared images that perfectly recover these 5 secret images. Figure 8 shows the results of our SRM method presented first. Figure 8a-e shows the 5 secret images: Lenna, Baboon, House, Cameraman, and Barbara with size 256 × 256. Figure 8f-i shows 4 shared images with size 256 × 1280, in which each shared image is the combination of five parts for every secret image with size 256 × 256. Therefore, the size of the shared images is proportional to the number of secret images as 5 times 256 × 256 in this example. Two kinds of recovered secret images are provided. Figure 8j Figure 8f-h fits the recovery threshold k = 3 for perfectly acquiring the images in Figure 8j-n that are identical to the original secret images. However, the collection shown in Figure 8f,g cannot satisfy the recovery threshold k = 3 and the recovered secret images all appear as noise images because of the failed acquisition of I re XOR in Equation (19). Moreover, the proposed SRM method uses a sharing matrix to assign part of pixels in shared images to 0, as defined in Equation (17). Therefore, the shared images all appear darker than the random images. Moreover, because of the large usage of 0 pixels, some kinds of regularity can also be found. Figure 9 shows the results of our second proposed SRS method. Figure 9a-e shows the same 5 secret images with size 256 × 256.  (23). The first and third elements in the first row of the matrix N part exhibit the element 1 and it leads the shares of E 1 and E 3 as defined in Equation (24). These two shares with the sharing of SR 1 lead to the shared image with a size that is three times that of the secret image. Therefore, the first shared image is shown in Figure 9f with the size 256 × 768, which is three times the secret image size 256 × 256. Moreover, the third row of N part leads to the share of E 3 , E 4 , and E 5 that are needed. Consequently, four times the secret image size is obtained, as shown in Figure 9h, with the size 256 × 1024. Therefore, sizes of shared images are according to the number of elements with 1, as shown in Figure 9f-i, being 256 × 768, 256 × 768, 256 × 1024, and 256 × 1024, respectively. In the recovery process, collecting k = 3 number of shared images is needed for perfect recovery. Collecting shared images shown in Figure 9f-h recovers the reconstructed images, as shown in Figure 9j-n. The collected shared images less than k, as shown in Figure 9g,i, recover the reconstructed images, as shown in Figure 9o-s, in which all recovered images appear as noise images. Table 1 shows the computation time of the two proposed schemes, SRM and SRS, among three different thresholds. The two proposed XOR-based schemes require limited CPU computation time, and the performances of the proposed schemes are exhibited. Moreover, the computation complexities between the two proposed schemes, SRM and SRS, are different. In our first proposed SRM method, the complexity of the sharing and recovery algorithms are O(nm) and O(km), respectively. Furthermore, the sharing and recovery complexities of our second SRS method are O(n) and O(k), respectively.   number of elements with 1, as shown in Figure 9f-i, being 256 × 768, 256 × 768, 256 × 1024, and 256 × 1024, respectively. In the recovery process, collecting k = 3 number of shared images is needed for perfect recovery. Collecting shared images shown in Figure 9f-h recovers the reconstructed images, as shown in Figure 9j-n. The collected shared images less than k, as shown in Figure 9g,i, recover the reconstructed images, as shown in Figure  9o-s, in which all recovered images appear as noise images.   Since our second proposed SRS method outperforms our first proposed SRM method in both less computation time and smaller size of shared images, more experimental results for our second proposed SRS method are illustrated. Figure 10 shows experimental results of the proposed (3, 5, 3) SRS method by the shared images with homogeneous regions.  × m under thresholds (k, n, m). From definition of the first row in N part , the first shared image, shown in Figure 10d, includes the random image SR 1 and one share of the third secret image from Figure 10c. Therefore, the size of the first shared image is twice the secret images at 256 × 512. Moreover, Figure 10e shows the second shared image of the combination of the random image SR 2 and two shares of the first and the second secret images from Figure 10a,b, respectively. Therefore, the size of the second shared image is three times that of the secret images at 256 × 768. At last, the fourth shared image, shown in Figure 10g with size 256 × 1024, includes the random image SR 4 and three shares of the first, second, and third secret images from Figure 10a-c, respectively. Therefore, three different kind of sizes are found in the proposed (3,5,3) SRS method, as shown in Figure 10d-h, and these shared images all exhibit randomlike images. Two kinds of recovery results are also provided. perimental results of the proposed (3,5,3) SRS method by the shared images with homogeneous regions. Figure 10a-c are three generated secret images, which are constructed by simple color regions with size 256 × 256. The generated is defined by From definition of the first row in , the first shared image, shown in Figure 10d, includes the random image 1 and one share of the third secret image from Figure 10c. Therefore, the size of the first shared image is twice the secret images at 256 × 512. Moreover, Figure 10e shows the second shared image of the combination of the random image 2 and two shares of the first and the second secret images from Figure 10a,b, respectively. Therefore, the size of the second shared image is three times that of the secret images at 256 × 768. At last, the fourth shared image, shown in Figure 10g with size 256 × 1024, includes the random image 4 and three shares of the first, second, and third secret images from Figure 10a-c, respectively. Therefore, three different kind of sizes are found in the proposed (3, 5, 3) SRS method, as shown in Figure 10d-h, and these shared images all exhibit randomlike images. Two kinds of recovery results are also provided. Figure 10i-k are the recovered secret images from Figure 10d-f. Figure  10l-n are the recovered secret images from Figure 10g,h. Since the collected number of shared images is less than threshold k, the recovered secret images are all noiselike images that exhibit the threshold recovery property of our proposed scheme.  Figure 11 shows the recovery results of applying attacks on shared images of homogeneous secret images. Figure 11a-e exhibit five shared images, which are the same as  Figure 11b,c,f is shown in Figure 11j-l, which shows that any attack leads to false recovery because of the incorrect random image generated. Therefore, security of the proposed scheme is identified.

Experimental Results
256; (d): the first shared image with size 256 × 512; (e,f): the second and third shared images with size 256 × 768; (g): the fourth shared image with size 256 × 1024; (h): the fifth shared image with size 256 × 512; (i-k): recovered secret images from (d-f); (l-n): recovered secret images from (g,h). Figure 11 shows the recovery results of applying attacks on shared images of homogeneous secret images. Figure 11a-e exhibit five shared images, which are the same as Figure 10d-h as the shared images of Figure 11a-c. Figure 11f shows a malicious attack result on Figure 11a by replacing one small black block in the left area. Figure 11g-i shows the recovered shared image from Figure 11a-c. The recovery of Figure 11b,c,f is shown in Figure 11j-l, which shows that any attack leads to false recovery because of the incorrect random image generated. Therefore, security of the proposed scheme is identified.

Comparison and Discussion
This section compares the proposed method with other stacking or XOR-based secret image sharing methods. The comparison metrics include lossless recovery or not, recovery strategy, sharing threshold, multiple sharing or not, and sharing capacity. The stacking-based schemes generate visual similarity on recovered secret images. Therefore, only stacking-based schemes are not lossless recovery and other XOR-based methods can recover without loss all secret images. In sharing thresholds, the method of Kabirirad and Eslami [24] satisfies restrictively consecutive k shared images and only adopt sharing matrix strategies that satisfy the general (k, n) threshold property with ≤ . Other methods, such as stacking-based or XOR-based, only have (2, 2) or (n, n) thresholds. It should be noted that many related methods cannot solve the way to construct multi-secret image sharing with general (k, n) thresholds; however, the proposed multiple sharing property exhibits the structure of sharing several secret images simultaneously. In a Boolean-based (n, n) secret image sharing method, all the secret and shared

Comparison and Discussion
This section compares the proposed method with other stacking or XOR-based secret image sharing methods. The comparison metrics include lossless recovery or not, recovery strategy, sharing threshold, multiple sharing or not, and sharing capacity. The stacking-based schemes generate visual similarity on recovered secret images. Therefore, only stacking-based schemes are not lossless recovery and other XOR-based methods can recover without loss all secret images. In sharing thresholds, the method of Kabirirad and Eslami [24] satisfies restrictively consecutive k shared images and only adopt sharing matrix strategies that satisfy the general (k, n) threshold property with k ≤ n. Other methods, such as stacking-based or XOR-based, only have (2, 2) or (n, n) thresholds. It should be noted that many related methods cannot solve the way to construct multi-secret image sharing with general (k, n) thresholds; however, the proposed multiple sharing property exhibits the structure of sharing several secret images simultaneously. In a Boolean-based (n, n) secret image sharing method, all the secret and shared images are the same size. The sharing capacity is the measurement of dividing the number of secret images by the number of shared images required to recover secret images. Therefore, in a Boolean-based (k, n) secret image sharing method, the sharing capacity is extended to dividing the size of secret images by the size of shared images required to recover secret images. Large sharing capacity means smaller shared image acquired. The sharing capacities of the proposed SRM method and SRS method are 1 k and n ∑ k i=1 u i +1 with u i < n, respectively. Furthermore, the value u i is determined by the matrix N part as the number of elements with 1 in the corresponding row. Table 2 shows that the proposed SRM and SRS methods are the only XOR and sharing matrix-based methods that can perform (k, n) thresholds on multiple sharing with good sharing capacity.

Conclusions
In this study, our contribution is a novel framework that can share m secret images among n participants, and gathering k participants' shared images perfectly recover these m secret images. We propose two XOR-based (k, n, m) multi-secret image sharing schemes. The first is the source-random-mixed (SRM) method, which mixes the random image and secret images to generate shares for each secret image. The second is the sourcerandom-separate (SRS) method, which separates the random image and secret images for constructing the shares for each secret image. Both of the proposed schemes reach the general (k, n) threshold by the usage of a sharing matrix, which is a popular property in mathematical-calculation-based methods such as the Shamir-Lagrange-based method. The proposed schemes have the properties of lossless recovery on collecting k shared images and no useful information acquired on collecting less than k shared images. The proposed schemes outperform other multiple sharing methods on the sharing rate of larger than or equal to 1. Therefore, these two proposed schemes exhibit good improvements on sharing multiple secret images through (k, n) thresholds by using XOR and a sharing matrix based Boolean operations.
In this study, we propose two efficient XOR-based (k, n, m) multi-secret image sharing schemes; however, there are several challenges and limitations to the proposed approach. First, the size of all the secret images used in the proposed approach must be equal, the multiple shared images generated are affected by the randomly generated matrix, and the sizes of each shared image are likely to be different. Second, the performance is relatively low compared to the secret image sharing method that simply uses Boolean operations. Based on these challenges and limitations, how to improve our proposed SRS method to identical size and how to apply the proposed approach in various actual applications, such as share authentication [28,29] and e-voting [30], merit our future study.