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Substitution boxes (S-boxes) play a significant role in ensuring theresistance of block ciphers against various attacks. The DifferenceDistribution Table (DDT), the Feistel Boomerang Connectivity Table (FBCT), theFeistel Boomerang Difference Table (FBDT) and the Feistel Boomerang ExtendedTable (FBET) of a given S-box are crucial tools to analyze its securityconcerning specific attacks. However, the results on them are rare. In thispaper, we investigate the properties of the power function$F(x):=x^{2^{m+1}-1}$ over the finite field $\gf_{2^n}$ of order $2^n$ where$n=2m$ or $n=2m+1$ ($m$ stands for a positive integer). As a consequence, bycarrying out certain finer manipulations of solving specific equations over$\gf_{2^n}$, we give explicit values of all entries of the DDT, the FBCT, theFBDT and the FBET of the investigated power functions. From the theoreticalpoint of view, our study pushes further former investigations on differentialand Feistel boomerang differential uniformities for a novel power function $F$.From a cryptographic point of view, when considering Feistel block cipherinvolving $F$, our in-depth analysis helps select $F$ resistant to differentialattacks, Feistel differential attacks and Feistel boomerang attacks,respectively.

In-depth analysis of S-boxes over binary finite fields concerning their differential and Feistel boomerang differential uniformities