## 5 Conclusions  
This paper presents a novel image encryption scheme that utilizes a hybrid chaotic map and fuzzy mathematics concept. The proposed approach employs a fuzzy triangular membership method to provide values that induce slight changes in all the initial condition parameters of the logistic sine map as well as the fuzzy Henon map.  
The utilization of the triangular membership function in generating the secret key allowed for the generation of additional secret keys by combining the triangular membership function with the Henon map and the logistic sine map. The incorporation of the hybrid chaotic map design with triangular membership methods expands the potential secret key space, demonstrating a robust capability against sensitivity attacks after its use in image encryption.  
While the proposed scheme in the paper does not directly fulfill the security requirements of non-repudiation and authentication, it effectively addresses the vital aspects of confidentiality, integrity, and availability. The system is intelligently designed to ensure continuous availability, even in the presence of faults such as noise data or data loss. Our techniques guarantee the encrypted image remains consistently accessible to legitimate entities. To uphold integrity, any modifications made to the encrypted text through this proposed scheme may produce different outcomes, primarily due to the sensitivity of the encryption keys. Maintaining confidentiality is a paramount concern as the information is meticulously safeguarded and accessible only to authorized parties.  
The proposed scheme underwent various experiments, and the results were extensively analyzed and evaluated to verify its security capability. The testing and verification processes included statistical attack analysis, NIST tests, differential attack analysis, mean signal-to-noise ratio, signal-to-noise distortion ratio and mean error square, brute force attack analysis, and information entropy analysis. The mechanism demonstrated a high level of efficiency and met all the requirements necessary for the secure transmission of information by means of images.  
Although the proposed scheme offers several advantages, there is still room for potential improvements in the future. The current design is limited to gray-scale images of size \(256\times 256\) and the use of triangular membership functions. It may be beneficial to explore other membership functions such as trapezoidal, Gaussian, quadratic, exponential, or even combinations of multiple methods to enhance the scheme's robustness. Another limitation of this study is that it may not be suitable for real-time applications due to its computational complexity. In upcoming research, we aim to address these limitations and further improve the encryption scheme.  
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: 616/2124), 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), and Jinan City 20 University Funding Projects Introducing Innovation Team Program (No: 2019 GXRC03).