  Substitution boxes (S-boxes) play a significant role in ensuring the
resistance of block ciphers against various attacks. The Upper Boomerang
Connectivity Table (UBCT), the Lower Boomerang Connectivity Table (LBCT) and
the Double Boomerang Connectivity Table (DBCT) of a given S-box are crucial
tools to analyze its security concerning specific attacks. However, there are
currently no related results for this research. The inverse function is crucial
for constructing S-boxes of block ciphers with good cryptographic properties in
symmetric cryptography. Therefore, extensive research has been conducted on the
inverse function, exploring various properties related to standard attacks.
Thanks to the recent advancements in boomerang cryptanalysis, particularly the
introduction of concepts such as UBCT, LBCT, and DBCT, this paper aims to
further investigate the properties of the inverse function $F(x)=x^{2^n-2}$
over $\gf_{2^n}$ for arbitrary $n$. As a consequence, by carrying out certain
finer manipulations of solving specific equations over $\gf_{2^n}$, we give all
entries of the UBCT, LBCT of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$.
Besides, based on the results of the UBCT and LBCT for the inverse function, we
determine that $F(x)$ is hard when $n$ is odd. Furthermore, we completely
compute all entries of the DBCT of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$.
Additionally, we provide the precise number of elements with a given entry by
means of the values of some Kloosterman sums. Further, we determine the double
boomerang uniformity of $F(x)$ over $\gf_{2^n}$ for arbitrary $n$. Our in-depth
analysis of the DBCT of $F(x)$ contributes to a better evaluation of the
S-box's resistance against boomerang attacks.
