Additive index and Carlitz rank

We compare several complexity measures for self-mappings of finite fields. In particular, we show that Carlitz rank and additive index cannot be small simultaneously up to trivial exceptions. That is, these two measures detect cryptographic weaknesses of different classes of functions. We also study the relationship between additive index and degree or weight, respectively, complementing earlier results of Aksoy et al. and G\'omez-P\'erez et al. on the relationship between Carlitz rank and degree or weight, respectively. Finally, we show that a function closely related to the discrete logarithm provides an example in which all four complexity measures, degree, weight, additive index and Carlitz rank, are large.