# Challenging Assumptions of Normality in AES s-Box Configurations under Side-Channel Analysis
## 5 Discussion  
With regard to traditional TVLA testing, it is clear that the application of [PERSON]'s \(t\)-test is intended as a method for establishing differences between two groups of data with some amount of statistical certainty. Cases where the magnitude of a \(\hat{t}\)-value is greater than 4.5, represented by the blue line in Figure 3, have traditionally formed the basis of measuring leakage in power-based side-channel analysis. For all nine s-box configurations, significant power-based leakage is measured, forming the basis of a known, readily verifiable result. This result will serve as a baseline for comparison later in this section. The primary focus of this body of work has been to establish an experimental set-up with sufficient complexity to determine whether the assumption of normality of power trace data associated with AES is well-founded. Due to how Shapiro-Wilk was applied to each dataset, the subsetting could lead to a biased outcome. To combat potential issues from random sampling, the Kolmogorov-Smirnov test was utilized as a secondary test for normality. Examining the results of both the Shapiro-Wilk and the one-sample Kolmogorov-Smirnov tests, there was sufficient evidence to reject the null hypothesis with a 99.99% level of confidence. This indicates that each of the nine datasets are unlikely to be normally distributed. The null hypothesis for a two-sample Kolmogorov-Smirnov states that the two given sets of data come from the same distribution. The results of the two-sample tests indicate that there is no significant evidence to reject the null hypothesis for each of the nine AES datasets. Therefore, the two groups of experimental data for each s-box are sourced from the same distribution with a 99.99% confidence. Under the TVLA specifications, this conclusion is reasonable; it is expected that data samples collected from a single device with an identical firmware share similarities. It is important to examine different statistical tools that do not require assumptions about the distribution of the data, referred to as non-parametric methods. A non-parametric alternative to [PERSON]'s \(t\)-test is the Wilcoxon Signed-Rank test. Examining Figure 4, there are similar trends to those in Figure 3. Despite [PERSON]'s \(t\)-test and the Wilcoxon Signed-Rank test computing a \(\hat{t}\)-statistic and \(W\)-statistic respectively, both give an equivalent, comparable \(p\)-value. Thus, if a \(p\)-value associated with a calculated \(W\)-statistic is less than \(\alpha=0.0001\), the significance between the means is equivalent to having the magnitude of a \(\hat{t}\)-statistic as greater than or equal to 4.5. This evenly balanced range of values is reminiscent of the sinusoidal nature of power trace data, and warrants further study in power-based SCA. Additional exploration was conducted using the Kruskal-Wallis test, a non-parametric method for examining the variance between two populations. Two variants were tested, the first of which focused on examining each group of traces for an s-box configuration as a whole. The results here are neither surprising nor particularly interesting. It is expected there is little variance when two traces are compared as singular entities; there are components of AES that operate independently of the s-box configuration. The overall power trace should reflect such a property. This variant of applying Kruskal-Wallis confirmsthat there is no statistically significant difference between traces from the two data groups for a given s-box configuration when examined as a whole. However, the second test, which focused on comparing pairs of ten-quantiles, had a different result. As seen in Table 1, quantiles 8, 9, and 10, except for the Default and S6 s-box configurations, show statistically significant differences with respect to the variance between the fixed and random data.