We shall first give a survey on the constructions of APN and differentially 4-uniform functions suitable for designing S-boxes for block ciphers. We shall recall why the search for more of such functions is necessary and propose a way of designing functions which can possibly be APN or differentially 4-uniform and be bijective. We shall illustrate it with an example of a differentially 4-uniform (n,n)-permutation for n odd, based on the power function x^3 over the second order Galois extension of GF(2^{n+1}), and related to the Dickson polynomial D_3 over this field. These permutations have optimal algebraic degree and their nonlinearity happens to be rather good.