Carlitz rank and index of permutation polynomials

Carlitz rank and index are two important measures for the complexity of a permutation polynomial f(x) over the finite field Fq. In particular, for cryptographic applications we need both, a high Carlitz rank and a high index. In this article we study the relationship between Carlitz rank Crk(f) and index Ind(f). More precisely, if the permutation polynomial is neither close to a polynomial of the form ax nor a rational function of the form ax−1, then we show that Crk(f)>q−max⁡{3Ind(f),(3q)1/2}. Moreover we show that the permutation polynomial which represents the discrete logarithm guarantees both a large index and a large Carlitz rank.