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

TL;DR

This paper introduces a new class of S-boxes based on a power function over finite fields that achieves optimal Feistel boomerang uniformity, enhancing resistance to boomerang attacks in block ciphers.

Contribution

It explicitly computes the Feistel Boomerang Connectivity Table for a specific power function and proves its minimal Feistel boomerang uniformity, a novel result in cryptographic analysis.

Findings

Explicit FBCT values for the power function over finite fields.

The power function has the lowest Feistel boomerang uniformity.

Enhanced resistance to boomerang attacks demonstrated.

Abstract

The Feistel Boomerang Connectivity Table ($\rm{FBCT}$), which is the Feistel version of the Boomerang Connectivity Table ($\rm{BCT}$), 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 $\rm{FBCT}$. 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 $\rm{FBCT}$ of $F(x)$ and further analyze its Feistel boomerang spectrum. Finally, we demonstrate that this power function exhibits the lowest Feistel boomerang uniformity.