## 5 Conclusions  
In this paper, we propose a new and innovative method for encrypting images that combines a variety of techniques to provide robust security. The proposed mechanism includes image scrambling, Hill cipher encryption, and the use of a sigmoid logistic map and Kronecker xor product techniques. The modified sigmoid logistic map, in particular, generates complex and secure pseudo-random numbers that are ideal for use in encryption. This enhances our understanding of digital encryption and helps to ensure the security of the proposed system.  
The use of the Kronecker xor product in encryption can significantly increase the size of the data being stored. While the increased storage space required by the Kronecker xor product may be a consideration, it is important to balance this against the value of the data being protected. In today's digital age, data breaches and cyber attacks are becoming increasingly common, and the consequences of such events can be severe. By integrating techniques such as the Kronecker xor product to encrypt data, we can ensure that the proposed encryption algorithm is secure and can protect information and minimize the risk of data breaches and cyber attacks.  
To verify the effectiveness and reliability of the proposed mechanism, we conducted a series of experiments and analyzed the results using various methods such as statistical attack analysis, differential attack analysis, brute force attack analysis, and information entropy analysis. The results of these tests demonstrate that the proposed system is highly secure and efficient and meets all the requirements necessary for protecting information stored using images. Overall, our proposed image encryption mechanism represents a significant advancement in the field and offers a reliable and effective way to secure sensitive information.  
Conceptualization, D.E.M. and Y.X.; data curation, D.E.M., X.F., and X.W.; formal analysis, D.E.M. and X.F.; funding acquisition, X.F.; investigation, D.E.M. and X.W.; methodology, D.E.M.; project administration, X.W.; resources, D.E.M.; software, D.E.M.; supervision, X.F. and X.W.; validation, D.E.M. and X.W.; visualization, D.E.M. and X.F.; writing--original draft, D.E.M.; writing--review and editing, D.E.M. and Y.X. All authors have read and agreed to the published version of the manuscript.  
This study was supported by the National Natural Science Foundation of China (No: 61672124), Password Theory Project of the 13 th Five-Year Plan National Cryptography Development Fund (No: MMJJ20170203), Liaoning Province Science and Technology Innovation Leading Talents Program Project (No: XLYC1802013), Key R&D Projects of Liaoning Province (No: 2019020105-JH2/103), and Jinan City 20 Universities Funding Projects Introducing Innovation Team Program (No: 2019 CXRC031).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Cameraman** & **Lena** & **Pepper** & **Rice** & **House** \\ \hline MSE & 107.856 & 89.2446 & 94.24519 & 101.1521 & 130.143 \\ PSNR & 8.3741 & 8.5655 & 8.4310 & 9.3546 & 9.2214 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Mean squared error (MSE) and peak signal-to-noise ratio (PSNR).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Available on request.  
**Conflicts of Interest:** The authors declare that they have no known competing financial interest on personal relationships that could have appeared to influence the work reported in this paper.