We give a survey on the constructions of APN and differentially 4-uniform functions suitable for designing S-boxes for block ciphers. We recall why the search for more of such functions is necessary. We propose a way of designing functions which can possibly be APN or differentially 4-uniform and be bijective. We illustrate it with an example of a differentially 4-uniform (
odd, based on the power function
over the second order Galois extension of
, and related to the Dickson polynomial
over this field. These permutations have optimal algebraic degree and their nonlinearity happens to be rather good (but worse than that of the multiplicative inverse functions).
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