Carlitz Rank and Index of Permutation Polynomials

TL;DR

This paper investigates the relationship between Carlitz rank and index of permutation polynomials over finite fields, establishing bounds and analyzing the properties of the discrete logarithm permutation.

Contribution

It provides new bounds linking Carlitz rank and index for permutation polynomials, and shows the discrete logarithm permutation has both high rank and index.

Findings

Permutation polynomials not close to linear or rational functions have high Carlitz rank relative to their index.

The discrete logarithm permutation guarantees both large index and Carlitz rank.

Established bounds improve understanding of permutation polynomial complexity in cryptography.

Abstract

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.