A new class of S-boxes with optimal Feistel boomerang uniformity

The Feistel Boomerang Connectivity Table ($\mathrm{F}\mathrm{B}\mathrm{C}\mathrm{T}$), which is the Feistel version of the Boomerang Connectivity Table ($\mathrm{B}\mathrm{C}\mathrm{T}$), plays a vital role in analyzing block ciphers’ ability to withstand strong attacks, such as boomerang attacks. However, as of now, only four classes of power functions are known to have explicit values for all entries in their $\mathrm{F}\mathrm{B}\mathrm{C}\mathrm{T}$. In this paper, we focus on studying the FBCT of the power function $F(x)=x^{2^{n-2}-1}$ over ${\mathbb{F}}_{2^n}$, where $n$ is a positive integer. Through certain refined manipulations to solve specific equations over ${\mathbb{F}}_{2^n}$ and employing binary Kloosterman sums, we determine explicit values for all entries in the $\mathrm{F}\mathrm{B}\mathrm{C}\mathrm{T}$ of $F(x)$ and further analyze its Feistel boomerang spectrum. Finally, we demonstrate that this power function exhibits the lowest Feistel boomerang uniformity.