Permutations of finite fields with prescribed properties

For cryptographic applications, the behaviour of the difference map is of importance and since there are no perfect nonlinear (PN) permutations, almost perfect nonlinear (APN) permutations have attracted a lot of attention in the last decades. Costas permutations, which are also extreme with respect to the behaviour of their difference maps, were originally introduced in connection with applications in radar communication. The relation between APN and Costas permutations of the rings Zn has been first investigated in [2]. Enumeration results in [2] show that the sizes of the sets of permutations of Zn, for 1 ≤ n ≤ 18, which are both Costas and APN, vary depending on n without any appearent reason, hence the relation between APN and Costas permutations is described as being ”quite erratic”.