# A Novel Method for Constructing the S-Box Based on Spatiotemporal Chaotic Dynamics
## 5 Performance Analysis of the Constructed S-Box
### The Equiprobable Input/Output XOR Distribution  
The work in [7] proposed differential cryptanalysis for an S-box using the imbalances in the input/output XOR distribution table. Output variations can be obtained from input variations. In the differential approximation table, the probability of all the input XOR values was the same as that of all the output XOR values. An S-box should have differential uniformity. To determine the differential uniformity of the obtained S-box, we calculated its differential probability (DP) by:  
\[DP_{g}=max_{\Delta w\
eq 0,\Delta z}\left(\frac{\sharp\{w\in W\mid g(w)\oplus g (w\oplus\Delta w)=\Delta z\}}{2^{n}}\right), \tag{10}\]  
where \(2^{n}\) is a count of all the elements in the S-box and \(W\) contains all the input values. The smaller the value of \(DP_{g}\), the stronger the ability to resist differential attacks.  
Table 9 is the differential approximation table. We can find that the maximum value was 12. This indicates that our S-box has a strong ability to resist differential attacks. Moreover, the maximum DP for different S-boxes is analyzed in Table 10. For the maximum DP, the obtained S-box was consistent with other S-boxes. Hence, the obtained S-box had a good ability to resist differential attacks.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline - & 106 & 106 & 106 & 104 & 102 & 104 \\
106 & 106 & 108 & 100 & - & 104 & 108 & 106 \\
104 & 102 & 108 & 104 & 106 & 102 & 106 & - \\ \hline \end{tabular}
\end{table}
Table 6: The Bit Independence Criterion (BIC)-nonlinearity analysis of our S-box.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline - & 0.5332 & 0.4785 & 0.5117 & 0.5215 & 0.5059 & 0.5137 & 0.5078 \\
0.5332 & - & 0.5117 & 0.5078 & 0.5137 & 0.5332 & 0.5137 & 0.5039 \\
0.4785 & 0.5117 & - & 0.5195 & 0.5000 & 0.5098 & 0.5039 & 0.5098 \\
0.5117 & 0.5078 & 0.5195 & - & 0.4961 & 0.5137 & 0.4922 & 0.5059 \\
0.5215 & 0.5137 & 0.5000 & 0.4961 & - & 0.4766 & 0.5020 & 0.5020 \\
0.5059 & 0.5332 & 0.5098 & 0.5137 & 0.4766 & - & 0.4902 & 0.5117 \\
0.5137 & 0.5137 & 0.5039 & 0.4922 & 0.5020 & 0.4902 & - & 0.5195 \\
0.5078 & 0.5039 & 0.5098 & 0.5059 & 0.5020 & 0.5117 & 0.5195 & - \\ \hline \end{tabular}
\end{table}
Table 7: The BIC-SAC analysis of our S-box.  
\begin{table}
\begin{tabular}{c c c} \hline \multicolumn{1}{c}{**S-Boxes**} & **BIC-SAC** & **BIC-Nonlinearity** \\ \hline The obtained S-box & 0.5075 & 104.64 \\ S-box based on CML & 0.5039 & 103.57 \\ S-box proposed in [11] & 0.5031 & 104.29 \\ S-box proposed in [13] & 0.5044 & 102.96 \\ S-box proposed in [14] & 0.5024 & 103.14 \\ S-box proposed in [16] & 0.5010 & 103.71 \\ S-box proposed in [17] & 0.5022 & 104.07 \\ S-box proposed in [18] & 0.4989 & 104.07 \\ S-box proposed in [19] & - & 108.21 \\ S-box proposed in [20] & 0.5032 & 104.5 \\ S-box proposed in [21] & 0.4970 & 103.78 \\ S-box proposed in [22] & 0.4996 & 103.8 \\ S-box 2 proposed in [23] & 0.5050 & 103 \\ S-box proposed in [24] & 0.5010 & 100.36 \\ S-box proposed in [25] & 0.5006 & 104 \\ \hline \end{tabular}
\end{table}
Table 8: The BIC analysis of different S-boxes.