*4.1. EC and PSNR Comparison*

Table 3 compares the PSNR of different reference matrices under the same EC. As shown in Table 3, Xie et al.'s scheme has the lowest average PSNR at 41.87 dB. The average PSNR of the proposed scheme is 46.37 dB, which is nearly 4 dB higher than the average PSNR of the Xie et al.'s scheme. Moreover, compared with the other two schemes, our scheme also achieves the highest PSNR with the same EC. It can be seen that the proposed scheme outperforms the related works.


**Table 3.** Comparison with related works.

In order to further understand the performance of our scheme, it is compared with three other schemes [19,22,23] based on the SuDoKu reference matrix. As shown in Table 4, the EC of the Chang et al.'s scheme is 393,216 bits, and its average PSNR is 44.83 dB. Regardless of the EC or image quality, the proposed scheme outperforms the Chang et al.'s scheme. Comparing with the other two schemes, although their image quality are better than the proposed scheme, their embedding capacity are far lower than our scheme. Therefore, it can be concluded that the overall performance of the proposed scheme is better than the SuDoKu-based data-hiding schemes.

**Table 4.** Comparison among SuDoKu-based data-hiding schemes.


In addition, we also compare the proposed CMSM-based scheme with the 3D SuDoKu-based scheme. As shown in Table 5, the average PSNR of the scheme proposed by Xia et al. is 41.31 dB, while the average PSNR of our scheme is 5 dB higher than their scheme. Based on the same frame structure of using 3D reference matrix, our scheme has a relatively small modification of pixel values under the same EC.

**Table 5.** Comparison with the 3D SuDoKu matrix-based scheme.


Our scheme is inspired by the He et al.'s scheme. Except for expanding the 2D mini-SuDoKu to a 3D CMSM, our scheme effectively improved the complexity of reference matrix and efficiently reduced the computation time. As shown in Table 6, although our scheme has no advantage over the He et al.'s scheme in terms of EC and PSNR. However, the improvement in time consumption (TC) is obvious. As discussed in Section 2.2, to achieve a translation invariant property for minimizing the distortion of pixel value modification, the mini-SuDoKu has to repeat a same basic 4 × 4 submatrix. This severely damages the complexity of a reference matrix. In our scheme, in spite of CMSM or NMSM, each basic structure is a completely random permutation of 2*<sup>n</sup>* distinct numbers. The possible combinations of a CMSM is an enormous figure. In the construction of NMSM, referring to Section 3.2.1, we even truncate the size of a randomly generated array and make the whole NMSM by repetition to reduce computational load. The key difference with the He et al.'s scheme is the design of range locator. It releases the constraint on the diversity of reference matrix. An additional benefit of the range locator is that the basic structure for embedding can be efficiently located without applying a time-consuming searching process. The cooperation of range locater in the outer embedding and the matrix operation in the inner embedding frees the hiding scheme from intensive loops of searching. According to the experimental data in Table 6, the required embedding time is less than half of the compared scheme.


**Table 6.** Comparison with the mini-SuDoKu matrix-based scheme.

#### *4.2. Time E*ffi*ciency Comparison*

To investigate the time efficiency of the proposed algorithm, we try to compare the time consumption of the proposed algorithm with the traditional approach. Although we present the three-dimensional (3D) CMSM and the n-dimensional (n−D) NMSM, our approach can also be de-generalized back to two-dimensional (2D) and one-dimensional (1D) version.

This experiment uses 8 typical grayscale images for testing and compares the proposed algorithm with a traditional one. A computer with a Dual i7-920 CPU and 8 GB memory is adopted for the experiment. The tic and toc commands in MATLAB are used to record the time cost in seconds. As shown in Table 7, due to the large number of loops used in the search step of the traditional algorithm, it takes significantly longer time to embed secret data. As the dimension rises, the time consumption increases rapidly. On the other hand, our algorithm has consistent performance as the dimension rises. Note that, as the dimension rises, the complexity of the reference matrix increases and thus the security level raises.


**Table 7.** Comparison of time efficiency between the proposed scheme and traditional algorithm.

### **5. Conclusions**

This study introduces an efficient multidimensional secret data-embedding scheme based on the mini SuDoKu matrix. In the proposed scheme, a CMSM RM with high complexity is first constructed to guarantee the security, and then a range locator function and the matrix operation are adopted to enhance the embedding efficiency. The reference matrix is further expanded to multidimension in order to obtain even higher embedding capacity and, meanwhile, still preserve good security and efficiency. The proposed scheme is compared with state-of-the-art RM-based data-hiding schemes and the experimental results show that the proposed scheme achieved higher than 46 dB in terms of the image quality and two bits per pixel in terms of the embedding capacity. In addition, the time consumption of the proposed algorithm is less than half of the traditional approach and keeps consistency as the dimension and security level raises. It is shown that the proposed scheme is advantageous in both embedding efficiency and security compared to the original mini-SuDoKu matrix.

We also provide a set of true color test images to demonstrate that the proposed scheme performs equal well to multi-channel images. By leveraging CMSM, each pixel of three-color channels—i.e., R, G, and B—can exactly match with the requirement of embedding a secret segment of data. The flexibility in dimension of RM meets the diverse data structure of cover media in the future word of massive IoT.

**Author Contributions:** Conceptualization, J.-H.H. and S.X.; Data curation, S.X.; Formal analysis, S.X.; Funding acquisition, J.-H.H.; Investigation, S.X.; Methodology, J.-H.H. and S.X.; Project administration, C.-C.C. (Chin-Chen Chang); Resources, S.X.; Software, S.X.; Supervision, C.-C.C. (Chin-Chen Chang); Validation, S.X.; Visualization, J.-H.H., C.-C.C. (Ching-Chun Chang) and C.-C.C. (Chin-Chen Chang); Writing: original draft, S.X.; 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.

#### **Appendix A**

The pseudo code for constructing the CMSM is provided as follows:


for *x* = 0 : 2 : 255, for *y* = 0 : 2 : 255, for *z* = 0 : 2 : 255,

$$A = [V(0)\ V(1); V(2)\ V(3)];\\B = [V(4)\ V(5); V(6)\ V(7)];\\\mathcal{M}(\mathbf{x}:\mathbf{x}+1, y:y+1, z:z+1) = A;$$

end

end end

The MATLAB functions rng() and randperm(8) are used to initialize random number generator and produce random permutations of 1 to 8, respectively. Although we apply the coding format and the functions of MATLAB programming language, the index of an array in the pseudo code follows the convention of zero leading value.
