# A Novel Fractional Sine Chaotic Map and Its Application to Image Encryption and Watermarking
## 6 Conclusions  
A novel discrete fractional sine chaotic map was developed and compared with existing chaotic maps, such as the Tent map, Tinkerbell map, and other fractional chaotic maps proposed in the literature. The comparison of dynamic characteristics confirms that the proposed map exhibits high non-linearity and complex behaviour, making it suitable for image encryption applications. The bifurcation diagrams demonstrate chaotic behaviour over a wide range of parameters, while the positive Lyapunov exponents further support the chaotic nature of the proposed algorithm. To assess the randomness of the chaotic time series generated by the map, approximate entropy results were calculated and tabulated for different tolerance values. These results highlight the randomness and unpredictability of the chaotic sequences. An image encryption algorithm based on the principles of blockchain logic was proposed to enhance the security of information transmission. This scheme ensures a higher level of security for authorized communication between senders and receivers. The incorporation of the transform methods for embedding secret images as watermarks at various levels enhances the transmission of secret messages.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Encrypted Image**} & \multicolumn{6}{c}{**Correlation Coefficient**} \\ \cline{2-7}  & **Entropy** & _NPCR (\%)_ & _UACI (\%)_ & _Horizontal_ & **Vertical** & **Diagonal** \\ \hline _Proposed_ & 7.9977 & 99.6902 & 33.5929 & \(-\)0.0043 & \(-\)0.0103 & \(-\)0.0004 \\
[49] & 7.9972 & 99.6050 & 33.5062 & 0.0025 & 0.0040 & \(-\)0.0015 \\
[48] & 7.9970 & 99.6048 & 33.4539 & \(-\)0.0011 & 0.0014 & \(-\)0.004 \\
[50] & 7.9969 & 99.6196 & 33.4150 & \(-\)0.0103 & \(-\)0.0127 & 0.0084 \\
[51] & 7.9976 & 99.6024 & 33.4975 & 0.0045 & 0.0028 & 0.0010 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Comparisonsults based on the peppers image.  
Figure 15: Key sensitivity for peppers image. (**a**) Input image, (**b**) decrypted images with original key \(x_{0},y_{0}\), (**c**) decrypted images with key \(x_{0}+10^{-16}\), (**d**) decrypted images with key \(y_{0}+10^{-16}\).  
For objective images of size \(256\times 256\), the proposed encryption scheme significantly increases the randomness of pixel distribution, as evidenced by the entropy results presented in Tables 6 and 7. The impact of the chaotic sequences generated by the fractional chaotic map is evident from the high entropy values and the _NPCR_ exceeding 99.69% for a single bit change in the input image. This demonstrates the effectiveness of the proposed encryption scheme for secure information communication through images. Additionally, the incorporation of watermarking techniques extends its applicability to areas such as copyright protection, e-voting, and securing medical and military data. In future research, the concept of genesis block-based encryption can be extended to RGB images, and the inclusion of multi-block image storage can be explored to further enhance the encryption scheme's capabilities.  
Conceptualization, D.V., N.A.A.F. and S.B.; methodology, D.V., N.A.A.F. and S.B.; software, D.V.; validation, D.V., N.A.A.F. and S.B.; investigation, D.V. and N.A.A.F.; writing--original draft preparation, D.V., N.A.A.F. and S.B.; writing--review and editing, N.A.A.F. and S.B.; supervision, N.A.A.F. and S.B. All authors have read and agreed to the published version of the manuscript.  
We would like to thank UPNM for supporting this research via Postdoctoral and postgraduate research grant UPNM/2022/GPPP/SG/13.  
Not applicable.  
Not datasets are generated or analysed during the current study.  
The authors declare no conflicts of interests.