*4.2. Cheat Event Detection and Cheater Detection*

The six pairs of image shadows were then applied to test the cheating detection mechanism. In each pair of shadows, shadow 1 was tampered by inserting a small image into a local region while shadow 2 was kept faithful. Four results of the six experiments are provided in Figures 7–10, where (a) is the tampered shadow image 1, (b) is the faithful shadow image 2, and (c) is the result of joint detection. The detected cheat pixel pairs are illustrated by black pixels on the tampered shadow. The joint cheat detection ratio for the six test shadow pairs are listed in Table 3. In each test pair, the detection ratio was calculated by

$$DR\_J = \frac{N(F)}{N},\tag{33}$$

where *N*(*F*) is the number of total detected cheat pixel pairs and *N* is the number of tampered pixels, i.e., the total number of pixels in the inserted small image. As shown in the table, *DRJ* of the joint cheat detection was around 0.42 and independent of the image features.

(**a**) Tampered shadow 1 (**b**) Faithful shadow 2 (**c**) Detection result

(**a**) Tampered shadow 1 (**b**) Faithful shadow 2 (**c**) Detection result **Figure 7.** Joint cheat detection result 1.

**Figure 8.** Joint cheat detection result 2.

(**a**) Tampered shadow 1 (**b**) Faithful shadow 2 (**c**) Detection result

**Figure 9.** Joint cheat detection result 3.

(**a**) Tampered shadow 1 (**b**) Faithful shadow 2 (**c**) Detection result

**Figure 10.** Joint cheat detection result 4. **Table 3.** Joint cheat detection ratio for the six shadow pairs.


The blind cheater detection results for the six tampered shadows are listed in Table 4. The detection ratio for blind cheater detection is defined by

$$DR\_{B1} = \frac{N(F\_1)}{N},\tag{34}$$

where *N*(*F*1) is the number of total detected pixel in shadow 1 by blind cheater detection and *N* is the number of tampered pixels, i.e., the total number of pixels in the inserted small image. As shown in the table, *DRB* of the blind cheater detection is around 0.20 and independent of the image features. Since the image shadow 2 was not tampered, the number of detected pixels *N*(*F*2) and thus *DRB*<sup>2</sup> are both zeros.

**Table 4.** Blind cheater detection ratio for the six tampered shadows.


To investigate the effect of combinatorial tampering, we further designed an experiment in which both image shadows were tampered with dis-aligned regions. Example results are given in Figures 11 and 12, where (a) and (b) are the cover image pair, (c) and (d) are the detection results of joint cheat detection, and (e) illustrates the overview of total detected pixels. The experimental data for all six test shadow image pairs are listed in Table 5, where *DR*1/*DR*<sup>2</sup> is the joint cheating detection ratio (*DRJ*) of the region that shadow 1/shadow 2 is tampered only; *DR*1∩<sup>2</sup> is the *DRJ* of the region that both shadow1 and shadow 2 are tampered; *DR*1∪<sup>2</sup> is the *DRJ* of the union tampered region. The joint cheating detection ratio (*DR*1/*DR*2) was around 43% for single tampered pixel pairs, while it was increased to 72% for combinatorial tampered pixel pairs (*DR*1∩2). Both of the percentage numbers were independent of the image features since the proposed data hiding scheme was a uniform embedding scheme [27]. The detection ratio of the union region (*DR*1∪2) depended on the percentage of overlapped region and was not an intrinsic characteristic of the proposed scheme.

(**c**) Detection result 1 (**d**) Detection result 2 

(**e**) Detection result 3

**Figure 11.** Joint cheating detection for combinatorial tampered shadows: Results for shadow pair 1.

(**a**) Tampered shadow 1 (**b**) Tampered shadow 2

(**c**) Detection result 1 (**d**) Detection result 2

**Figure 12.** Joint cheating detection for combinatorial tampered shadows: Results for shadow pair 3.


**Table 5.** DR values for the six combinatorial tampered shadow pairs.

### *4.3. Comparison with Liu et al.'s Scheme [26]*

The comparison of the proposed maze matrix-based data hiding scheme with the turtle shell matrix-based scheme [26] is provided in Table 6. The new proposed scheme can hide four bits of secret data for each pair of cover pixels, while the turtle shell matrix-based scheme can hide only three bits for each pair. The EC given in the table was measured by bits per pixel pair, one from cover image 1 and the other from cover image 2. Due to different embedding capacity, the PSNR of the proposed scheme was slightly lower than the turtle shell scheme. However, the degradation of visual quality could not be recognized by human eyes.

**Table 6.** Comparison of the proposed maze matrix-based scheme with the turtle shell-based scheme.


The joint cheat detection ratio of the turtle shell scheme was 50% in both single tampered or combinatorial tampered cases. Although only the single tampered data was provided by the authors, the combinatorial tampered detection ratio can be analyzed easily. Since legal hiding locations are the back elements of turtle shells and such elements occupy 50% of the entire matrix, the theoretic cheating detection ratio was 50%. Our cheating detection mechanism outperformed the turtle shell scheme in combinatorial tampering, while the detection ratio was lower in single tampering.

The most creative part of the proposed scheme is the function of blind cheater detection. Without information of the other shadow, we detected 20% of tampered pixels in the shadow shared by a cheater. Meanwhile, the turtle shell scheme can only identify a cheater by a faithful participant.

#### *4.4. Time E*ffi*ciency Evaluation*

To assess the time efficiency of the proposed secret image sharing scheme, we listed the execution time required for the share construction program in Table 7 and the execution time for secret data extraction program in Table 8. The conventional reference matrix-based data hiding scheme and share construction scheme usually embed secret data by searching the nearest element that matches the intended secret digit and modify the pixel values accordingly. This type of searching procedures is often time-consuming. In this paper, a pair of Lagrange polynomials was derived to compute the coordinates of the matched element. Thus, the running time for share construction was drastically reduced. Referring to Table 7, up to 39% of execution time can be saved by leveraging the proposed approach. The execution time required for data extraction grogram is relatively short in comparison with the share construction program as shown in Table 8.


**Table 7.** Efficiency comparison of the proposed embedding scheme with conventional scheme.



#### **5. Conclusions**

In this paper, we proposed a secret image sharing scheme based on a novel maze matrix. A pair of distinct cover images was used to carry secret data and a pair of shadow images was constructed under the guidance of the maze matrix. The secret data is extracted only if both authentic shadows are presented.

A two-layered cheat detection mechanism was devised to examine cheating behaviors as well as to ascertain the inauthentic shadow. In the outer cheat detection layer, the corresponding pair of pixels retrieved from the two shares was jointly used for detecting cheat events. The detection ratio was 43% for the cases in which single shadow was tampered and was 72% for the cases in which both shadows were tampered. In the inner blind cheater identification layer, the cheater's image share could be spotted without the information from the other share. The detection ratio of tampered pixels was 20% for the blind cheater identification.

An additional merit of the proposed scheme is time efficiency. By computing the pixel values of the image shadows with Lagrange polynomials instead of conventional searching algorithms, the proposed approach can save up to 39% of program execution time. In view of the effectiveness and low power consumption of the proposed scheme, the outlook for integrating it with massive IoT systems as a data security module shall be positive.

In the future world where massive IoT environment is fully established, secret image sharing will no longer be restricted to share secrets among human participants. The image shadows produced by the dealer can be transmitted via different routes to devices located at different sites. The shadow production and secret extraction will be executed via APPs installed on smartphones of the dealer and receiver. Uploading and downloading image shadows through IoT links will permit secret data to be communicated securely without the use of a preshared key or password system.

**Author Contributions:** Conceptualization, C.-C.C. (Ching-Chun Chang) and J.-H.H.; Data curation, C.-S.S.; Formal analysis, J.-H.H.; Funding acquisition, J.-H.H.; Investigation, C.-S.S.; Methodology, C.-C.C. (Ching-Chun Chang) and J.-H.H.; Project administration, C.-C.C. (Chin-Chen Chang); Resources, C.-S.S.; Software, C.-S.S.; Supervision, C.-C.C. (Chin-Chen Chang); Validation, J.-H.H.; Visualization, J.-H.H., C.-C.C. (Ching-Chun Chang) and C.-C.C. (Chin-Chen Chang); Writing: original draft, J.-H.H.; Writing: review & editing, J.-H.H., C.-C.C. (Ching-Chun Chang) and C.-C.C. (Chin-Chen Chang). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no competing interests.
