On the Carlitz rank of permutation polynomials

Authors:
Esen Aksoy;AyçA ÇEşMelioğLu;Wilfried Meidl;Alev TopuzoğLu
Affiliations:
Sabancı University, Orhanlı, 34956 Tuzla, İstanbul, Turkey;Sabancı University, Orhanlı, 34956 Tuzla, İstanbul, Turkey;Sabancı University, Orhanlı, 34956 Tuzla, İstanbul, Turkey;Sabancı University, Orhanlı, 34956 Tuzla, İstanbul, Turkey
Venue:
Finite Fields and Their Applications
Year:
2009

Citing 3
Cited 2

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Abstract

A well-known result of Carlitz, that any permutation polynomial @?(x) of a finite field F"q is a composition of linear polynomials and the monomial x^q^-^2, implies that @?(x) can be represented by a polynomial P"n(x)=(...((a"0x+a"1)^q^-^2+a"2)^q^-^2...+a"n)^q^-^2+a"n"+"1, for some n=0. The smallest integer n, such that P"n(x) represents @?(x) is of interest since it is the least number of ''inversions''x^q^-^2, needed to obtain @?(x). We define the Carlitz rank of @?(x) as n, and focus here on the problem of evaluating it. We also obtain results on the enumeration of permutations of F"q with a fixed Carlitz rank.