A Novel (2, 3) Reversible Secret Image Sharing Based on Fractal Matrix

This article first proposes two types of three-dimensional (3D) fractal reference matrices. Then, a new reversible (2, 3) secret image sharing scheme based on a fractal matrix is presented. With the help of the fractal reference matrix, the secret data can be embedded in an original image to generate three meaningful shadow images. It is worth mentioning that these two types of fractal matrices are designed for different hiding capacity, meanwhile the characteristics of the fractal matrix can guarantee good visual quality of the generated shadow images. Moreover, we provide two authentication mechanisms according to different situations, one of the authentication mechanisms can achieve 100% ability cheat detection. When the authentication is passed, we only need two of the three shadow images to extract the correct secret data and perfectly restore the original image. Experimental results show that our proposed scheme provides excellent authentication ability and good visual quality.


I. INTRODUCTION
With the increasing importance of the network security, people are paying more and more attention to the protection of data. It is risky to directly transmit the sensitive secret data on the network, and it is easy to be intercepted by the eavesdroppers. A lot of image steganographic methods, also known as data hiding techniques, have been proposed recently to hide the secret data in the cover images. These techniques can be categorized into three major classes, which are the transform domain hiding [1]- [3], the compressed space domain hiding [4], [5], and the direct pixel value hiding [6]- [13]. Typical applied file formats for the compressed domain data hiding are the AMBTC and JPEG compressed images [14], [15]. The purpose of data hiding technology is to embed the secret data into some digital carrier with slight modification and transmit the carrier through the Internet. This greatly improves the security of the secret data. The famous direct pixel value hiding schemes include the exploiting modified direction [16], the turtle shell-based [17], [18], the octagon-shaped shellbased [19], and the mini-Sudoku-based schemes [20].
In 1995, Naor and Shamir [21] proposed a visual cryptography model, in which, a secret image is encrypted into n different modified versions (called shares or shadows) The associate editor coordinating the review of this manuscript and approving it for publication was Kaitai Liang . of similar meaningless-looking transparencies. Each original pixel of the secret image is represented by n shares of m black and white sub-pixels, which are printed in close proximity to each other so that human visual system averages their contributions. When k out of n transparencies are stacked together, the secret image can be apparently interpreted by human eyes. But this scheme has two obvious disadvantages: (1) The generated shadows are meaningless and easily cause suspicion by attackers; (2) this scheme suffers from the problem of pixel expansion. Many different secret image sharing (SIS) schemes [22]- [25] have been proposed to deal with these two shortcomings.
Later, the extended visual cryptography [26] were proposed. Each share of transparency not only includes the information of secret image but also appears meaningful by itself. Different techniques have been proposed to encrypt grayscale or colored cover images into meaningful shares of good visual quality.
In the new era, transparencies are phased out; instead, smartphones and portable devices are very popular. The thin client computing-based frame structure is a much more practical way for secret sharing via cover images. In order to further improve the safety of such schemes, authentication is required before extracting the secret data. In 2004, the first SIS scheme with authentication ability was proposed by Yang et al. [27]. This scheme greatly promoted the VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ development of SIS, but it still has two disadvantages. First, its authentication ability is too weak, that is, even a fake share is highly possible to pass authentication. Secondly, the visual quality of the image shares is too low. Therefore, how to improve the authentication ability and the image quality of the shares has become the main objective of later studies. In 2014, Chang et al. [28] proposed a novel SIS scheme based on the turtle shell (TS) reference matrix. Their scheme uses two different images as carriers of the secret data to generate two high-quality shadow images with good authentication ability. In 2020, Chang et al. [29] proposed a SIS scheme based on the maze matrix. In their scheme, the embedding capacity is improved. In addition, a new authentication mechanism was designed to detect tampered share without help of the other share. But these two schemes of SIS cannot recover the original image.
In 2019, Li et al. [18] proposed a reversible (3, 3) SIS scheme based on the TS matrix. In 2020, Gao et al. [30] proposed a reversible SIS scheme based on the stick insect (SI) reference matrix. Both schemes use a single cover image to generate secret image shares with authentication mechanism. Details of these two schemes will be introduced in the next section.
At present, most of the SIS schemes need to use all shadow images when extracting the secret data. If one of the shadow images is fake, the secret data cannot be extracted. In this article, we will introduce a novel 3D reference matrix called fractal matrix with a shadow image generation scheme to implement the secret sharing application, which does not require all shadow images when extracting the secret data and perfectly recovering the cover image. Moreover, the proposed authentication mechanism can achieve 100% ability to detect tampered image shadow.
The rest of this article is organized as follows. Section 2 introduces several previous SIS methods. Section 3 presents the basic model and the fractal matrix. Section 4 introduces the secret share generation, authentication, data extraction, and cover image recovery in detail. Our experimental results are presented in Section 5. Section 6 presents our conclusions.

II. REVIEW
In this section, we briefly introduce four recent schemes of SIS. The former two schemes generate image shadows based on a pair of different cover images. The latter two schemes generate shares based on a single cover image, but the cover image can be restored after secret data extraction. All these schemes provide authentication mechanism to detect tampered share.
A. A TURTLE SHELL-BASED SECRET IMAGE SHARING SCHEME [28] The SIS scheme proposed by Chang et al. [28] embeds the secret data in two different original images and generates two meaningful shadow images with the help of the turtle shell (TS) matrix (see Fig. 1). The turtle shell matrix is composed of many non-overlapping hexagons. We call the six elements on the edge of a hexagon as edge elements, and the inner two elements are called back elements. The embedding rules are as follows. Combine two pixels p 1i , p 2i at the corresponding positions of two original image into a pixel pair (p 1i , p 2i ), and (p 1i , p 2i ) is used as the coordinates of the TS matrix. As shown in Fig. 1, if the element pointed to by (p 1i , p 2i ) is an edge element, the sender looks for a new back element with the same value as the secret data and closest to this edge element in a range consisting of four turtle shells. After that, the sender uses the coordinates of the new element as the new pixel values instead of p 1i and p 2i to generate two shadow images. As shown in Fig. 1, if the element pointed to by (p 1i , p 2i ) is a back element, the range of finding the new element is composed of six turtle shells, and the remaining steps are the same as above.
In extracting the secret data, a pair of pixels collected from corresponding positions of two image shadows is mapped to the TS matrix. If the mapped element is an edge element, the pixel pair contains tampered pixel; else, the stored value is extracted as the secret data. B. A MAZE MATRIX-BASED SECRET IMAGE SHARING SCHEME [29] Similar to Liu et al.'s scheme, the SIS scheme proposed by Chang et al. [29] also embeds the secret data in two different original images and generated two meaningful shadow images with the help of the maze matrix (see Fig. 2). Then, the secret shares are distributed to two participants. When extracting the secret data, two participants need to work together to obtain the correct secret data. The algorithm for generating image shadows is given as follows.
In extracting the secret data, a pair of pixels collected from corresponding positions of two image shadows is mapped to the maze matrix. If the mapped element is an 'x'-marked element, the pixel pair contains tampered pixel; else, the stored value is extracted as the secret data.
In addition, there are a lot of vertical and horizontal gaps in the maze matrix, which are the designed traps to capture the tampered pixels. Without help of the other shadow, a tampered shadow with trapped pixels can be detected individually.

Algorithm for Generating Image Shadows
Step 1: The data sender establishes the maze matrix.
Step 2: The data sender scans the two original images in sequence, and combines the pixels p α , p β at the same position in the two images into a pixel pair (p α , p β ).
Step 3: The sender uses the pixel pair (p α , p β ) as coordinates to locate a 6 × 6 block in the maze matrix.
Step 4: The sender reads a 16-ary secret data and finds an element (p α , p β ) in the block with the same value as the secret data.
Step 5: The sender records p α and p β to the corresponding positions of p α and p β in two shadow images, respectively.
C. A TURTLE SHELL-BASED REVERSIBLE VISUAL SECRET SHARING SCHEME [18] Li et al. [18] proposed a (3, 3) reversible SIS scheme based on the TS matrix (see Fig. 3). In their scheme, three image shadows are generated to embed secret data based on a single cover image. The embedding rules are as follows. Retrieve a pair of cover pixels to constitute the coordinate (P 1 , P 2 ). Then, map the coordinate to the element TS(P 1 ,P 2 ) in the TS matrix. According to the definition in Section 2. A, if a back element is mapped, two additional edge elements in the same turtle shell are selected; else, two edge elements or two back elements in the same shell are selected. Possible collections are mapped to different secret numbers, therefore, secret data can be embedded by recording its corresponding collection of three elements. Finally, the coordinates of the three elements are separately recorded to three image shadows.
Secret data can be extracted and cover image can be restored according to the special configuration of an element collection retrieved from the three shadows. Note that, all three elements in a collection must be within the same turtle shell. When a violating case is encountered, it indicates that at least one image shadow is tampered.

D. THE STICK INSECT MATRIX-BASED REVERSIBLE
SECRET IMAGE SHARING SCHEME [30] In 2020, Gao et al. [30] proposed a reversible SIS scheme based on the stick-insect (SI) reference matrix (see Fig. 4). Although this scheme also uses a single cover image, the basic processing unit is a single pixel. By duplicating a pixel p i retrieved from the cover image, the coordinate (p i , p i ) is then mapped to an element on the main diagonal line of the SI matrix. The insect structure that contains the mapped element is the candidate region of embedding. The algorithm for generating image shadows is given as follows. To extract secret data, a pair of corresponding pixels from the two image shadows is mapped to an element in the SI matrix. The value of the mapped element is the secret data, while the main diagonal element in the same insect structure helps to restore the cover pixel.
The insect structures are aligned along the main diagonal line of the SI matrix. Any shadow pixel pair that maps to an element outside the main diagonal region indicates the existence of a tampered shadow.

Algorithm for Generating Image Shadows
Step 1: The data sender establishes the SI matrix.
Step 2: The data sender scans the original image in sequence, and duplicates each pixel p i into a pixel pair (p i , p i ).
Step 3: The sender uses the pixel pair (p i , p i ) as coordinate and maps to an insect structure in the SI matrix.
Step 4: The sender reads the secret data and finds the element (p i1 , p i2 ) in this insect structure with the same value as the secret data.
Step 5: The sender records the coordinate (p i1 , p i2 ) of this element to the corresponding position of p i in two shadow images.
All of the above SIS schemes need to authenticate all shadow images before extracting the secret data. If one of the shadow image is fake, it will never be possible to extract the correct secret data. Moreover, none of these schemes can 100% verify the authenticity of the shadow images. In order to solve these two weaknesses, we propose a novel SIS framework in the following section.

III. THE PROPOSED SIS SCHEME
In this section, we propose two novel 3D basic matrix models called the fractal models. Based on these two models, we can generate a new reference matrix called the fractal matrix, which can be used in the proposed (2, 3) -SIS scheme.

A. THE FRACTAL MODEL
Before introducing our (2, 3) SIS scheme, we first introduce the fractal model, which constitutes the fractal matrix and plays a key role in our (2, 3) SIS scheme.
The fractal model is the smallest basic structure of the fractal matrix. According to the different hiding capacity, we propose two models of size 2 × 2 × 2 and 3 × 3 × 3, respectively. The proposed fractal models are shown in   along three axes are shown in Fig. 7. It can be observed that each projection of a fractal model is a perfectly square matrix.

B. NUMBER ASSIGNMENT OF A FRACTAL MODEL
As shown in Fig. 5, the 2 × 2 × 2 fractal model is composed of two layers, which are painted purple and yellow, respectively. Each layer has two colored elements. In the construction of a fractal model, we assign distinct numbers to its elements. For the case of a 2 × 2 × 2 fractal model, as shown in Fig. 8, there are four elements in a basic structure. We assign the four distinct numbers of 0 to 3 to the four elements. Its three projections are shown in Fig. 9. For security concerns, the assignment is completely random. Details will be discussed in a later subsection.
The number assignment of the 3 × 3 × 3 fractal model is similar to the 2 × 2 × 2 fractal model. The distinct numbers 0 to 7 and an undetermined number ''#'' are assigned to the nine elements of a model. The resulting fractal model and its projections are shown in Figs. 10 and 11, respectively.

C. THE FRACTAL MATRIX
The fractal matrix is a 256 × 256 × 256 3D matrix constituted by many fractal models based on the principles of fractal theory. Before introducing the construction of the whole matrix, we first need to define the structure of a fractal group. Fig. 12 illustrates the constitution of a fractal group. Four fractal models are arranged in the same way as the constitution of a fractal model to form a fractal group. Each gray box in the figure represents a fractal model. An example box with the number 2 is scaled up to show the details inside. As shown in the figure, a fractal group satisfies the self-similarity property of fractal theory.
The fractal matrix of the size 256 × 256 × 256 is constructed by aligning a series of conjunctive fractal groups along its main diagonal line. The size of a fractal model is 2 × 2 × 2. Thus, the size of a fractal group is 4 × 4 × 4. Therefore, a fractal matrix consists of 256/4 = 64 fractal groups. After arranging the fractal groups in the fractal matrix, we begin to assign numbers to the fractal elements in the whole matrix. We initialize the random number generator with a secret key. Then, we process all the fractal models in the fractal matrix one by one. A random sequence from 0 to 3 is generated and assigned to the elements of the current   fractal model in a predefined order. Repeat this process until all models are processed.
The final step of the fractal matrix construction is numbering the fractal models within a fractal group. The assigned  numbers are labeled on the gray boxes in Fig. 12. The rule of number assignment for a fractal group is fixed, because the performance of the SIS scheme is dependent on this assignment. VOLUME 8, 2020   A fractal group constituted by fractal models of the size 3 × 3 × 3 is illustrated in Fig. 13. The corresponding numbering for the fractal models are also labeled.
The size of the fractal group is 9 × 9 × 9. Since 9 is not a factor of 256, only 256/9 = 28 fractal groups are arranged along the main diagonal line of the fractal matrix  starting from (0, 0, 0), where the · denotes the floor operation.

IV. EMBEDDING AND EXTRACTION PROCESSES
In this article, a (2, 3) -SIS scheme based on the fractal reference matrix is proposed. The proposed (2, 3) -SIS scheme generates three meaningful shadow images with secret data embedded based on a single cover image. The proposed scheme consists of three phases: (1) the image shadow generation phase, (2) the authentication phase, and (3) the secret data extraction and the original image recovery phase.

A. THE IMAGE SHADOW GENERATION PHASE
First, according to the required embedding capacity, the data sender establishes the fractal reference matrix according to the procedures described in Section 3. Throughout this section, we will take the 9 × 9 × 9 fractal model to demonstrate our scheme. Then, process the pixels of the cover image in the raster scan order. For pixel p i , we map the coordinate (p i , p i , p i ) to the fractal matrix. Since the main diagonal of the fractal matrix is aligned with conjunctive fractal groups, (p i ,p i , p i ) can locate a single fractal group. Apply N = p i mod 9 to get the target fractal model in the located group and embed secret data by modifying the coordinate to the matched element. The entries of modified coordinate are recorded separately to three different shadows of the same corresponding image location. Repeat the same procedure until all pixels of the cover image are processed and get three image shadows.
For convenience, we take an example in vicinity of origin. Suppose the cover pixel is p i = 7, the coordinate (p i , p i , p i ) =(7, 7, 7) maps to the first fractal group as shown in Fig. 13. Then, the number 7 = 7 mod 9 is applied to obtain the target fractal model as shown in the right-hand side. The exact submatrix corresponding to this fractal model is M (0 : 2, 6 : 8, 3 : 5) of the fractal matrix M . Suppose the secret digit s j = 3 is to be embedded, the matched element is M (1,8, 5). Then, (p 1i ,p 2i , p 3i ) = (1,8, 5) and the three pixels are recorded separately to the i-th pixel of different shadows.
Notice that, a fractal group of size 9 × 9 × 9 consists of 9 fractal models and has a diagonal line of length 9.

Algorithm Numbering of Fractal Models in a Group
Input: matrix of a fractal group Output: numbering of fractal models 1: Diagonal elements: Q = {q 0 , q 1 , . . . , q 8 }.

4:
For i = 0, 1, . . . , 8, Find j minimize q i − c j , n j = i and remove c j from C.
Algorithm Generation of Data Embedded Image Shadows Input: cover image I C = {p 0 , p 1 , . . . , p W ×H }, W and H are width and height; Output: secret data stream S = {b 0 , b 1 , . . . , b L }.
Algorithm Authentication With Two Shadows Input: Shadow image I 1, Shadow image I 2.

Output:
Number of fail pixel pairs N F 1: N F = 0.

End End End 3:
Print N F .

4:
If N F = 0, Print ''authentication passed.'' End Each diagonal element can map to a unique fractal model and get the matched element to embed secret digit. As mentioned in the previous section, the numbering of fractal models in a group is not arbitrary. In shadow generation, (p i , p i , p i ) is modified to (p 1i , p 2i ,p 3i ) according to the numbering of fractal models. The distances of p i to p 1i , p 2i , and p 3i directly influence the similarity between the cover image and the corresponding shadow. To minimize the distortion, we apply a greedy algorithm to number the fractal models.
Given the matrix of a fractal group, the centers of fractal models within the group are calculated first. Then, the  algorithm sequentially processes the elements in the main diagonal line. For the i-th element q i , search within the group to find its nearest model center c j . Number its corresponding model with value i and remove c j from list of model centers. Repeat this process until all models are numbered. The algorithm is summarized as follows.
The resulting numbering of the fractal models in a group are as labeled on the gray boxes of Fig. 14. The mapping between the main diagonal elements and the fractal models is fixed throughout the whole fractal matrix to ensure the good visual quality of the image shadows.
Although the elements of a fractal model are randomly assigned with 0 to 7 and #, as mentioned in the previous section, we always assign ''#'' to the central element, as shown in Fig.14. In the data hiding process, 3 bits can be embedded into the eight states of 0 to 7. To utilize the undetermined state ''#'', we search to find the element that is furthest from the model's corresponding diagonal element. Suppose the element 5 on the right-hand side of Fig. 15 is the furthest element. We switch 5 to the center element and assign a number, 5 + 8 = 13, to its original position. All fractal models are processed in the same way. An additional VOLUME 8, 2020  bit can be embedded when the digit to be embedded meets the special case. The algorithm for shadow images generation phase can be summarized as follows.

B. AUTHENTICATION PHASE
The proposed secret image sharing scheme is a (2,3) secret sharing model. That is, the secret data can be perfectly extracted from any two shares of the three distributed image shadows. Suppose the image shadows with the secret data embedded in them are distributed to three different participants, any two of whom can cooperate to extract the secret by sharing their shadows. To prevent cheating, an authentication mechanism is designed according to the properties of the applied fractal matrix.

Algorithm Authentication With Three Shadows
Input: Shadow image I 1, Shadow image I 2, Shadow image I 3.

Output:
Number of fail triplets N F 1: N F = 0.
During the image shadow generation phase, the pixel values of the image shadows are the recorded coordinates of the fractal model elements. To verify a pixel pair retrieved from the corresponding pixels of the two shadows, we need only to VOLUME 8, 2020  check whether it is mapped to a fractal element or not. For the authentication of two shadows, we project the fractal groups into two dimensional space, since only two of the components of a coordinate are available. The projection of the fractal groups into the x-y plane is shown in Fig. 16. It is composed of many 9×9 square matrices aligned with the main diagonal line. There are 28 projected fractal groups on the x-y plane, and each group contains 81 elements. In other words, only 28 × 81 = 2268 elements of the x-y plane are embeddable; however, the entire x-y plane contains 65536 elements. Thus, when one participant provides a fake image shadow, another participant can easily detect the cheat by employing this property.
Assume that participant 1 is honest and participant 2 is dishonest. Suppose we have a pixel pair (p i1 ,p i2 ), where p i1 and p i2 come from the same position of the shadow images provided by shadow image 1 and shadow image 2, respectively. If the element mapped by the coordinate (p i1 , p i2 ) is not among these 2268 elements, this means that participant 2 provided a fake shadow, so participant 1 can stop cooperating with participant 2 and turn to cooperate with participant 3. The algorithm of the authentication with the two shadows is as follows.
The operator · refers to the floor function. The equation p i1 /9 = p i2 /9 defines all elements within the 28 projected fractal groups. Any pixel pair that violates this equation indicates the existence of a fake shadow. Moreover, when the pixel value is greater than 251, it is not embeddable and the two pixels should have the same value.
Although any two shares of the image shadows can perfectly extract the secret message, when all three shadows are available, the process of authentication can be more precise. The matrix space occupied by the fractal groups can be defined by p i1 /9 = p i2 /9 = p i3 /9 , which is valid for all elements within the whole series of fractal groups. More precisely, there are actually only 81 fractal elements within a 9 × 9 × 9 submatrix occupied by a fractal group.

C. SECRET DATA EXTRACTION AND IMAGE RECOVERY PHASE
Any two verified image shadows can be applied to perfectly extract the secret message and reconstruct the cover image. According to the versions of the participated image shadows, the fractal matrix is projected to their corresponding plane first. Then, the pixels of each shadow is rearranged in raster scan order. Corresponding pixels of two shadows are paired to extract secret data and recall the cover pixel. For each pixel pair, apply its value as coordinate and map to an element in the projected fractal matrix. According to the value of mapped element and the assigned number of its mother fractal model, the secret data and cover pixel can be recovered. The details are given as follows.

V. EXPERIMENTAL RESULTS
In our experiment, twelve standard grayscale test images of size 512 × 512 as shown in Fig. 18 are applied. All programs are implemented with MATLAB R2018a. The experiments can be divided into two major parts. The first part demonstrates the applicability of our proposed secret sharing scheme. Visual quality of shadow images is also assessed. The second part proves the function of authentication mechanism to detect tampered or noise corrupted shadows.

A. VISUAL QUALITY
To demonstrate the applicability of the proposed secret image sharing scheme, our first experiment uses the cover image '' Fig. 18(l) Lake'' to embed the secret image '' Fig. 18 (h) Airplane.'' In our scheme, the fractal reference matrix based on the 4 × 4 × 4 fractal groups can hide 2 bits in a cover pixel to generate a pixel triplet. The triplet is then distributed to the three image shadows. That is, a cover pixel can  To know the details of the difference, we apply the visual quality index of Peak-Signal-to-Noise Ratio (PSNR), as defined by where W and H represent the width and height of the cover image; p i and p j represent the corresponding pixel values in the cover image and the shadow image, respectively. The PSNR values of the three shadows range from 36 to 38 dB. The human visual system is incapable of discriminating their difference. By applying any two image shadows, the cover image and secret image can be perfectly recovered as shown in Figs. 19 (f) and (g). Since the proposed scheme is based on a uniform embedding approach, the visual quality of the image shadows is independent of the features of the cover image. To verify this, 12 test images are examined, and the PSNR values are listed in Table 1, where random numbers are applied as secret data. In addition, the reference matrix (RM) based on the fractal groups of size 9 × 9 × 9 are also implemented and the experimental values are listed in the table. In this case, the PSNR values range from 31 to 34 dB, which is also acceptable in practical applications. Although the visual quality of the image shadows is slightly degraded, the embedding capacity is increased to 3.125 bits per cover pixel.

B. AUTHENTICATION ABILITY
To test the authentication ability of our scheme, we assume that the three image shadows 1, 2, and 3 are distributed to participants 1, 2, and 3, respectively. While cooperating to extract the secret data, participant 1 is dishonest, and participants 2 and 3 are honest. Our authentication mechanism includes two-shadow and three-shadow versions as described in Section 4.2. Here, we use the detection rate (DR) as an index to measure the authentication ability, which is defined as where N d represents the number of detected pixels, and N t represents the total number of tampered pixels. Two examples are applied to demonstrate the authentication mechanism. Referring to Fig. 20, the image ''Peppers'' shown in (a) is used as the hostile image. A small region of image shadow 1 generated using ''Airplane'' and shown in (b) is replaced by the hostile image. With the help of faithful shadow 2, as shown in (c), the authentication result is given in (d). The rare white pixels represent the ones that passed the authentication test.
The second example is shown in Fig. 21, where the hostile image is shown in (a), and a tampered shadow is shown in (b). With the help of (c) and (d), the authentication result is given in (e). We can see that the tampered region is entirely black. This means that all tampered pixels have been detected.
To investigate the ability of authentication for the different cover images, the detection rate of the tampered pixels for 12 test images is listed in Table 2 for the two-shadow version and in Table 3 for the three-shadow version. In Table 2, the DR values are above 95% for the fractal matrix with fractal groups sized 4 × 4 × 4, while they are above 91% for the 9 × 9 × 9 version. In Table 3, the DR values are all 100%. That means a pixel value in the third shadow is entirely determined by the corresponding pixels in the first and second shadows. This characteristic makes it possible to uniquely identify secret data and cover the pixel value with two shadows.
When an image shadow is corrupted by noise, we need to detect such events and reject the noisy shadow. The authentication ability when there is noise corruption is provided in Table 4 for the two-shadow version and in Table 5 for the three-shadow version, where Gaussian, speckle, Poisson, and salt & pepper noises are tested. The DR values are higher than the tampering attack for the two-shadow version, while they remain 100% for the three-shadow version.
From Table 6, we can see the proposed (2, 3) SIS scheme adds the concept of (k, n) secret sharing. This is a novel framework that has never been used by any SIS schemes before. Our method also provides two very efficient mechanisms for detecting any tampering with the shadow images. Between them, the authentication mechanism with three shadow images can reach a 100% average detection rate when only one shadow image has been tampered with. When the two shadow images are tampered with, we only have a low probability that the detection will fail. However, the average detection rate of the authentication mechanism with the three shadow images provided by Li et al.'s [18] scheme can only reach 98%. In addition, the average detection rate of the authentication mechanism with two shadow images provided in this article has reached 98%, this is the same result as the scheme by Gao et al. [30], while the average detection rate of the schemes of Chang et al. [28] and Chang et al. [29] can only reach 50% and 43% respectively.
The features of the proposed scheme can be compared with the related works listed in Table 6. The modern thin client computing-based SIS schemes all focus on producing meaningful shadows with good visual quality. The irreversible schemes usually adopt multiple cover images to create completely different shadows. While the reversible schemes tend to use a single cover image and produce shadows that look the same. The authentication ability is higher for single cover image schemes, since the duplicated images serve more redundancy pixels for application. Our proposed scheme provides two very notable features, which are the 100% authentication ability and (2, 3) secret sharing with perfect data and image recovery.

VI. CONCLUSION
In this article, we propose a novel (2, 3) -SIS scheme based on the fractal matrix. Our scheme is entirely reversible and has a high level of security. Our scheme has the following four features: (1) Two kinds of fractal matrices can be selected according to the required embedding capacity; (2) using a single cover image, our scheme produces three shadow images with secret data embedded; (3) 100% authentication ability to detect tampered pixels in a shadow image; (4) only two of the three shadow images are required to perfectly extract the secret data and restore the original cover image. The fractal matrix provides excellent (2, 3) sharing scheme within the defined space of fractal groups.
Our future work will focus on improving the shadow image quality and look at the possibility of utilizing multiple cover images to produce different meaningful shadows.