Low differentially uniform permutations from the Dobbertin APN function over F2n

Substitution boxes (S-boxes) play a central role in block ciphers. In substitution-permutation networks, the S-boxes should be permutation functions over ${\mathbb{F}}_{2^n}$ to realize the invertibility of the encryption. More importantly, the S-boxes 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 ${\mathbb{F}}_{2^n}$. In addition, the algebraic degree and the lower bound of the nonlinearity of the constructed functions are given.