# Challenging Assumptions of Normality in AES s-Box Configurations under Side-Channel Analysis
## 4 Results  
When applying traditional TVLA, significant data leakage was detected in all nine s-box configurations. From [PERSON]'s \(t\)-test applied in R, the computed values are summarized in Figure 3. A \(t\)-value having a magnitude greater than 4.5 is equivalent to a \(p\)-value of less than 0.0001. To test for normality, the Shapiro-Wilk test was run on each of the nine data groups. Across all nine configurations, the largest \(p\)-value reported across all traces was \(1.443\times 10^{-31}\). The one-sample Kolmogorov-Smirnov test produced a maximal \(p\)-value of \(2\times 10^{-16}\) for each point in each power trace, indicating that the null hypothesis should be rejected, meaning that the data is not normally distributed. The two-sample variant of the test reported high \(p\)-values of at least 0.9, so that it cannot be said that the two data sources originate from different distributions. The approximation of \(p\)-values is due to a limitation in the implementation of these tests in base R.  
In order to evaluate the dataset for variance, the Wilcoxon Signed-Rank test was utilized, with the results summarized in Figure 4. The results of the Wilcoxon Signed-Rank test results in a \(W\)-statistic. The \(W\)-statistic behaves similarly to the \(t\)-statistic in that it is inversely related to a \(p\)-value calculation based on the statistic. That is, a higher \(W\)-statistic is associated with a lower \(p\)-value, inviting the conclusion that there are significant differences when our \(p\)-value is less than the chosen \(\alpha=0.0001\) significance level. Finally, the results of applying the Kruskal-Wallis test, focusing on the variance within the data, is provided in Table 1.  
Figure 4: Boxplot describing the \(W\)-values calculated by the Wilcoxon Signed-Rank test. Figure 3: An example \(t\)-test output showing the \(\hat{t}\)-values based on the magnitudes of their values. \begin{table}
\begin{tabular}{c c} \hline \hline
**Data** & **K-W \(p\)-Value** \\ \hline \hline  & (a) Default s-box \\ \hline Full Default & 0.9178411 \\ \hline Quantile 1 & 0.7973211 \\ Quantile 2 & 0.9492303 \\ Quantile 3 & 0.4865965 \\ Quantile 4 & 0.003551987 \\ Quantile 5 & 0.4060664 \\ Quantile 6 & 0.2482653 \\ Quantile 7 & 0.03989401 \\ Quantile 8 & \(\mathbf{4.704231\times 10^{-5}}\) \\ Quantile 9 & \(\mathbf{1.787921\times 10^{-28}}\) \\ Quantile 10 & 0.0002557376 \\ \hline  & (b) S1 s-box \\ \hline Full S1 & 0.8889263 \\ \hline Quantile 1 & 0.9047302 \\ Quantile 2 & 0.9721271 \\ Quantile 3 & 0.5588611 \\ Quantile 4 & 0.01427206 \\ Quantile 5 & 0.5762426 \\ Quantile 6 & 0.1448012 \\ Quantile 7 & 0.1547279 \\ Quantile 8 & \(\mathbf{1.198919\times 10^{-7}}\) \\ Quantile 9 & \(\mathbf{6.674018\times 10^{-24}}\) \\ Quantile 10 & \(\mathbf{6.324433\times 10^{-6}}\) \\ \hline  & (c) S2 s-box \\ \hline Full S2 & 0.915456 \\ \hline Quantile 1 & 0.8438535 \\ Quantile 2 & 0.9775268 \\ Quantile 3 & 0.5287456 \\ Quantile 4 & 0.01392253 \\ Quantile 5 & 0.5347051 \\ Quantile 6 & 0.1436789 \\ Quantile 7 & 0.1304291 \\ Quantile 8 & \(\mathbf{1.721812\times 10^{-7}}\) \\ Quantile 9 & \(\mathbf{3.478914\times 10^{-25}}\) \\ Quantile 10 & \(\mathbf{5.773031\times 10^{-6}}\) \\ \hline  & (d) S3 s-box \\ \hline Full S3 & 0.9500724 \\ \hline Quantile 1 & 0.9117693 \\ Quantile 2 & 0.9705669 \\ Quantile 3 & 0.7504782 \\ Quantile 4 & 0.04621217 \\ Quantile 5 & 0.9294591 \\ Quantile 6 & 0.03665044 \\ Quantile 7 & 0.8746878 \\ Quantile 8 & \(\mathbf{1.247497\times 10^{-10}}\) \\ Quantile 9 & \(\mathbf{3.693989\times 10^{-11}}\) \\ Quantile 10 & \(\mathbf{2.67089\times 10^{-11}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The \(p\)-values calculated using the Kruskal-Wallis test, associated with each of the nine s-boxes. Values that are in bold are statistically significant for \(\alpha=0.0001\).