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

TL;DR

This paper provides a detailed analysis of the differential and Feistel boomerang properties of specific power functions over binary finite fields, aiding in the design of more secure block ciphers.

Contribution

It explicitly computes the DDT, FBCT, FBDT, and FBET for a class of power functions, advancing understanding of their cryptographic resistance.

Findings

Explicit values of all entries of DDT, FBCT, FBDT, FBET for the studied power functions.

Enhanced understanding of differential and Feistel boomerang uniformities for these functions.

Guidance for selecting functions resistant to various cryptographic attacks.

Abstract

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.