  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.
