  Substitution boxes (S-boxes) play a significant role in ensuring the
resistance of block ciphers against various attacks. The Difference
Distribution Table (DDT), the Feistel Boomerang Connectivity Table (FBCT), the
Feistel Boomerang Difference Table (FBDT) and the Feistel Boomerang Extended
Table (FBET) of a given S-box are crucial tools to analyze its security
concerning specific attacks. However, the results on them are rare. In this
paper, 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, by
carrying out certain finer manipulations of solving specific equations over
$\gf_{2^n}$, we give explicit values of all entries of the DDT, the FBCT, the
FBDT and the FBET of the investigated power functions. From the theoretical
point of view, our study pushes further former investigations on differential
and Feistel boomerang differential uniformities for a novel power function $F$.
From a cryptographic point of view, when considering Feistel block cipher
involving $F$, our in-depth analysis helps select $F$ resistant to differential
attacks, Feistel differential attacks and Feistel boomerang attacks,
respectively.
