Permutations of finite fields with prescribed properties

Classes of permutations of finite fields with various specific properties are often needed for applications. We use a recent classification of permutation polynomials using their Carlitz rank with advantage, to produce examples of classes of permutations of Fp, for odd p, which for instance are “random”, have low differential uniformity, prescribed cycle structure, high polynomial degree, large weight and large dispersion. They are also easy to implement. We indicate applications in coding and cryptography.