# Hybrid Chaotic-Based PRNG for Secure Cryptography Applications
## 6 Discussion and Conclusions  
This paper introduces a novel 1D chaotic function that exhibits complete chaotic behavior across a broad range of a single control parameter \(c\in[-2,2]\). This new chaotic function was utilized as the core of a pseudorandom number generator (PRNG) in a proposed image encryption scheme, which was subjected to various tests to determine its robustness. The key space analysis indicated that the proposed scheme had a key space size of 256 bits, which is sufficient to withstand brute force attacks. The sensitivity analysis demonstrated that the scheme is sensitive to any small changes in the key, with acceptable values of mean squared error (MSE) and peak signal-to-noise ratio (PSNR). The histogram analysis revealed a uniform distribution of pixel values, indicating good image quality after encryption. The algorithm also passed the entropy analysis test for significant parameter values of \(\alpha=\{0.05,0.01,0.001\}\), indicating high randomness in the generated PRNs. Moreover, the proposed encryption scheme was found to be resistant to differential attacks based on the results of tests using 28 grayscale images of various sizes for NPCR and UACI. Overall, the tests demonstrated that the proposed image encryption scheme, \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{_NPCR Test Result_} & \multicolumn{4}{c}{_UACI Test Result_} \\ _File Name_ & _Score\%_ & _Status_ & **Ref. [26]** & **Ref. [25]** & _Score \%_ & _Status_ & **Ref. [26]** & **Ref. [25]** \\ \hline ruler.512 & 99.6239 & pass & 99.6231 & 99.6021 & 33.4651 & pass & 33.4407 & 33.4163 \\ \hline _Dimentin_ 1024 \(\times\) 1024 & \(\mathcal{N}_{\pm}^{*}\geq\) 99.5952\% & & \(\mathcal{U}_{\pm}^{*-}\),\(\mathcal{U}_{\pm}^{*+}=(33.4040\%,33.5231\%)\) \\ \hline
5.3.01 & 99.6078 & pass & 99.6063 & 99.6078 & 33.4627 & pass & 33.4725 & 33.4706 \\
5.3.02 & 99.6084 & pass & 99.602 & 99.6009 & 33.483 & pass & 33.4983 & 33.4801 \\
7.2.01 & 99.5984 & pass & 99.6073 & 99.6010 & 33.4672 & pass & 33.4723 & 33.4664 \\ \hline _Dimentin_ 1704 \(\times\) 1704 & \(\mathcal{N}_{\pm}^{*}\geq\) 99.602\% & & \(\mathcal{U}_{\pm}^{*-}\),\(\mathcal{U}_{\pm}^{*+}=(33.4326\%,33.4945\%)\) \\ \hline sphinv & 99.6099 & pass & - & - & 33.4517 & pass & - & - \\ \hline _Dimentin_ 2362 \(\times\) 2362 & \(\mathcal{N}_{\pm}^{*}\geq\) 99.604\% & & \(\mathcal{U}_{\pm}^{*-}\),\(\mathcal{U}_{\pm}^{*+}=(33.4408\%,33.4863\%)\) \\ \hline lion & 99.6079 & pass & - & - & 33.4658 & pass & - & - \\ \hline _Dimentin_ 2625 \(\times\) 2625 & \(\mathcal{N}_{\pm}^{*}\geq\) 99.6034\% & & \(\mathcal{U}_{\pm}^{*-}\),\(\mathcal{U}_{\pm}^{*+}=(33.4385\%,33.4886\%)\) \\ \hline bird & 99.6108 & pass & - & - & 33.4701 & pass & - \\ \hline _Dimentin_ 4755 \(\times\) 4755 & \(\mathcal{N}_{\pm}^{*}\geq\) 99.6056\% & & \(\mathcal{U}_{\pm}^{*-}\),\(\mathcal{U}_{\pm}^{*+}=(33.4476\%,33.4794\%)\) \\ \hline pyramids & 99.6081 & pass & - & - & 33.4597 & pass & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**File Name** & **Dimension** & **Encryption Speed (s)** \\ \hline
5.1.09.tiff & 256 \(\times\) 256 & 0.4648439 \\
7.1.01.tiff & 512 \(\times\) 512 & 1.6287242 \\
5.3.01.tiff & 1024 \(\times\) 1024 & 6.9461746 \\ sphinv.tiff & 1704 \(\times\) 2272 & 22.4399294 \\ bird.tiff & 2625 \(\times\) 2250 & 38.3284587 \\ lion.tiff & 2362 \(\times\) 3047 & 46.8042111 \\ pyramids.tiff & 4755 \(\times\) 3090 & 103.5472705 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Encryption speed analysis for various image dimensions. which employs a hybrid chaotic map as its PRNG, exhibits good confusion and diffusion properties. These findings contribute to the field of image encryption and chaos-based cryptography by providing a suitable solution for systems that possess a wide range of chaotic behavior in their control parameters. It also opens the door to further explore the potential of the proposed hybrid chaotic map in other cryptographic applications in future work. Conceptualization, R.A.-E. and E.A.-B.