Low differentially uniform permutations from Dobbertin APN function over $\mathbb{F}_{2^n}$

TL;DR

This paper constructs new low differential uniformity permutations for cryptographic S-boxes by modifying Dobbertin APN functions over finite fields, enhancing resistance to various cryptanalytic attacks.

Contribution

It introduces novel classes of differentially 4- and 6-uniform permutations derived from Dobbertin APN functions, with analysis of their algebraic degree and nonlinearity.

Findings

Constructed permutations with differential uniformity 4 and 6.

Analyzed algebraic degree and nonlinearity bounds.

Enhanced cryptographic resistance of S-boxes.

Abstract

Block ciphers use S-boxes to create confusion in the cryptosystems. Such S-boxes are functions over $\mathbb{F}_{2^{n}}$. These functions should have low differential uniformity, high nonlinearity, and high algebraic degree in order to resist differential attacks, linear attacks, and higher order differential attacks, respectively. In this paper, we construct new classes of differentially $4$ and $6$-uniform permutations by modifying the image of the Dobbertin APN function $x^{d}$ with $d=2^{4k}+2^{3k}+2^{2k}+2^{k}-1$ over a subfield of $\mathbb{F}_{2^{n}}$. Furthermore, the algebraic degree and the lower bound of the nonlinearity of the constructed functions are given.