## 5 Conclusions  
In this paper, we analyzed the cryptographic properties of binomial power functions. Through our observation of the redundancy table, we discover that all binomial power functions generated over a finite field \(\mathbb{F}_{2\text{s}}\) are not bijective. We modify and enhance the method called the _Redundancy Removal Algorithm_ to obtain cryptographically strong S-boxes. Our proposed S-box exhibits cryptographic properties of (108, 4, 7, 20) for (NL, DU, AD, LA), respectively; thus all our prerequisite requirements are fulfilled and the performance of the original algorithm proposed by [PERSON] et al. and [PERSON] et al. is surpassed. From our comparative analysis, the mathematical function remains the best method to construct a cryptographically strong S-box. The production of a new mathematical function that can challenge multiplicative inversion is an interest of many researchers. However, recent research developments on combination methods of mathematical functions and the heuristic approach show an encouraging trend in the construction of a cryptographically \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Rank** & **S-Box** & **NL** & **DU** & **AD** & **LA** & **\#Term** & **Design Method** \\ \hline \multirow{9}{*}{1} & AES [8] & 112 & 4 & 7 & 16 & 9 & \\  & Hierarchy [40] & 112 & 4 & 7 & 16 & 255 & \\  & Camelia [21] & 112 & 4 & 7 & 16 & 254 & \\  & ARIA [23] & 112 & 4 & 7 & 16 & 9 & \\  & CLEFIA [41] & 112 & 4 & 7 & 16 & 254 & \\  & HyRAL [26] & 112 & 4 & 7 & 16 & 253 & \\  & [PERSON] et al. [80] & 112 & 4 & 7 & 16 & 1 & Mathematical \\  & [PERSON] et al. [82] & 112 & 4 & 7 & 16 & 253 & \\  & [PERSON] et al. [57] & 112 & 4 & 7 & 16 & 254 & \\  & [PERSON] and [PERSON] [90] & 112 & 4 & 7 & 16 & 254 & \\  & [PERSON] et al. [54] & 112 & 4 & 7 & 16 & 251 & \\  & [PERSON] and [PERSON] [79] & 112 & 4 & 7 & 16 & 252 & \\  & [PERSON] et al. [89] & 112 & 4 & 7 & 16 & 254 & \\  & [PERSON] et al. [84] & 112 & 4 & 7 & 16 & 255 & \\ \hline
2 & [PERSON] et al. [91] & 112 & 4 & 5 & 16 & 217 & Mathematical + Heuristic \\ \hline \multirow{2}{*}{3} & [PERSON] et al. [80] & 112 & 6 & 7 & 16 & 250 & Mathematical \\  & [PERSON] et al. [70] & 112 & 6 & 7 & 16 & 253 & Mathematical \\ \hline
5 & [PERSON] et al. [80] & 110 & 6 & 7 & 18 & 253 & Mathematical \\  & [PERSON] et al. [70] & 110 & 6 & 7 & 18 & 254 & Mathematical + Heuristic \\ \hline \multirow{2}{*}{6} & This Paper & 108 & 4 & 7 & 20 & 255 & Mathematical + Heuristic \\  & [PERSON] et al. [93] & 108 & 4 & 7 & 20 & 3 & Mathematical \\ \hline \multirow{2}{*}{7} & [PERSON] et al. [70] & 108 & 6 & 7 & 20 & 253 & Mathematical + Heuristic \\  & [PERSON] et al. [77] & 108 & 6 & 7 & 20 & 253 & Mathematical \\ \hline \multirow{2}{*}{8} & [PERSON] et al. [93] & 108 & 6 & 4 & 20 & 3 & Mathematical \\ \hline \multirow{3}{*}{9} & [PERSON] et al. [70] & 106 & 6 & 7 & 22 & 252 & \\  & [PERSON] et al. [28,92] & 106 & 6 & 7 & 22 & 254 & Mathematical + Heuristic \\  & Hierarchy [40] & 106 & 6 & 7 & 22 & 253 & \\  & [PERSON] et al. [93] & 106 & 6 & 7 & 22 & 3 & Mathematical \\ \hline \multirow{2}{*}{10} & [PERSON] et al. [12] & 104 & 6 & 7 & 24 & 255 & \\  & [PERSON] et al. [70] & 104 & 6 & 7 & 24 & 255 & Mathematical + Heuristic \\ \hline \multirow{2}{*}{11} & [PERSON] [42] & 104 & 8 & 7 & 24 & 254 & Heuristic \\  & [PERSON] et al. [43] & 104 & 8 & 7 & 24 & 254 & Random \\ \hline \multirow{2}{*}{12} & [PERSON] et al. [11] & 102 & 8 & 7 & 26 & 254 & Mathematical + Heuristic \\ \hline \multirow{2}{*}{13} & [PERSON] et al. [43] & 100 & 8 & 7 & 28 & 254 & Random \\ \hline \multirow{2}{*}{14} & [PERSON] et al. [69] & 100 & 10 & 7 & 28 & 254 & Heuristic \\ \hline \multirow{2}{*}{15} & CLEFIA [41] & 100 & 10 & 6 & 28 & 246 & Mathematical \\ \hline \multirow{2}{*}{16} & [PERSON] et al. [69] & 100 & 12 & 7 & 28 & 253 & \\  & SKIPACK [32] & 100 & 12 & 7 & 28 & 255 & Heuristic \\ \hline \hline \end{tabular}
\end{table} 
Table 8: S-Boxes Ranks. strong S-box. As for random generation, this method is unfavorable since there are no recent developments driven by this approach.