## 3 Proposed Chaos-Based Stream Ciphers Based on Secure PRNGs of Chaotic Sequences
### Statistical analysis
#### 3.4.2 Hardware Metrics  
To quantify the performance of the proposed implementation of the chaos-based stream ciphers in regard to the resources utilized (area, DSPs), the speed (Max. Freq., Throughput), the efficiency, and the power consumption, we cipher just four pixels. The proposed PRNGs-CS produce at each clock cycle a 32-bit sample, and we used the least significant 8 bits for XORing with each pixel (8 bits) to generate the corresponding encrypted pixel. Hardware metrics results for the proposed chaos-based stream ciphers are shown in Table 12 and, as expected, are linked to results for the various proposed PRNGs-CS.  
Table 13 provides an overview of the hardware metrics comparison of the chaos-based stream ciphers proposed, the ZUC cipher, and chaotic and non-chaotic systems that are part of the eSTREAM project phase 2-focus hardware profile. It is hard to make sense of this comparison due to the varying characteristics of the used FPGAs. However, considering the clock rate of the FPGA board and the efficiency achieved, this comparison can be made. Therefore, with the exception of the Trivium and ZUC ciphers, the proposed chaos-based stream ciphers have competitive hardware performance when compared to most other chaotic and non-chaotic systems. Nevertheless, since 2007, several kinds of attacks have been launched against the eSTREAM cipher and ZUC stream cipher, revealing several vulnerabilities to cryptanalysis attacks [3; 41; 42; 43; 44].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Image** & **Horizontal** & **Vertical** & **Diagonal** \\ \hline \multirow{4}{*}{Peppers} & \multicolumn{2}{c}{Plain image} & 0.96216 & 0.96712 & 0.95239 \\ \cline{2-5}  & \multirow{2}{*}{Ciphered image} & LSP\(-\)SC & 0.01803 & \(-\)0.01284 & 0.00973 \\ \cline{3-5}  & & LST\(-\)SC & \(-\)0.00927 & \(-\)0.00896 & \(-\)0.00004 \\ \cline{3-5}  & & LSPT\(-\)SC & \(-\)0.00797 & 0.01538 & \(-\)0.00102 \\ \hline \multirow{4}{*}{Baboon} & \multicolumn{2}{c}{Plain image} & 0.95444 & 0.93200 & 0.91957 \\ \cline{2-5}  & \multirow{2}{*}{Ciphered image} & LSP\(-\)SC & \(-\)0.00380 & \(-\)0.00371 & \(-\)0.00347 \\ \cline{3-5}  & & LST\(-\)SC & 0.00374 & \(-\)0.00295 & \(-\)0.01533 \\ \cline{3-5}  & & LSPT\(-\)SC & \(-\)0.01895 & \(-\)0.00289 & \(-\)0.00559 \\ \hline \multirow{4}{*}{Barbara} & \multicolumn{2}{c}{Plain image} & 0.92453 & 0.97042 & 0.90707 \\ \cline{3-5}  & \multirow{2}{*}{Ciphered image} & LSP\(-\)SC & 0.00939 & \(-\)0.02223 & 0.00264 \\ \cline{3-5}  & & LST\(-\)SC & \(-\)0.01218 & \(-\)0.00765 & 0.01441 \\ \cline{3-5}  & & LSPT\(-\)SC & \(-\)0.00748 & 0.00685 & \(-\)0.00556 \\ \hline \multirow{4}{*}{Boats} & \multicolumn{2}{c}{Plain image} & 0.96971 & 0.97280 & 0.94049 \\ \cline{3-5}  & \multirow{2}{*}{Ciphered image} & LSP\(-\)SC & \(-\)0.01362 & 0.00053 & 0.01487 \\ \cline{3-5}  & & LST\(-\)SC & \(-\)0.01202 & \(-\)0.00099 & 0.02195 \\ \cline{3-5}  & & LSPT\(-\)SC & \(-\)0.01370 & \(-\)0.00055 & \(-\)0.01461 \\ \hline \hline \end{tabular}
\end{table}
Table 11: _Cont._Note: Nowadays, promising new fractional chaotic functions have appeared, which seem very robust against cryptographic attacks. We believe they deserve further study to build chaotic fractional systems for data security. The design of such fractional chaotic systems must deal with the trade-offs between safety, performance, and cost.