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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 $\F_q$. 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 $Crk(f)>q- \max\{3 Ind(f),(3q)^{1/2}\}$. Moreover we show that the permutation polynomial which represents the discrete logarithm guarantees both a large index and a large Carlitz rank.