Low differentially uniform permutations from Dobbertin APN function over $\mathbb{F}_{2^n}$
This work addresses the need for secure S-boxes in cryptography to resist attacks, but it is incremental as it builds on existing functions.
The paper tackled the problem of constructing S-boxes for block ciphers with low differential uniformity by modifying the Dobbertin APN function over finite fields, resulting in new classes of differentially 4- and 6-uniform permutations.
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.