On the Carlitz Rank of Permutation Polynomials Over Finite Fields: Recent Developments

Permutation polynomials: iteration of shift and inversion maps over finite fields

“…Moreover, if we find 0 = wCrk(ax + b) ≤ (p − 1)/2, then we have Crk(ax + b) = 0 from a result in [2]. Hence, we conclude that wCrk(ax + b) > (p − 1)/2.…”

“…An obvious relation between these two ranks is Crk(Σ) ≤ wCrk(Σ) for any Σ ∈ S p . The key lemma of this paper implies Crk((1 2)(3 4)) ≤ wCrk((1 2)(3 4)) ≤ 6 for all p. These papers (see [2] and [20]) give a survey about the recent development on the Carlitz rank. In Section 4, we will provide an algebraic geometry point of view on low Carlitz rank, and use it to say certain permutations cannot appear when the Carlitz rank is low.…”

“…where 1 ≤ c i,j < p are integers. The Crk(Σ) = 6 follows directly from Lemma 2 in [2], which says that if P n represents Σ with n < (p − 1)/2, then n = Crk(Σ). Lemma 3.3.…”

More constructions of $n$-cycle permutations

“…In this paper, n-cycle permutations are called low-cycle permutations for a small n. When n = 2 or 3, f is also called an involution, or a triple-cycle permutation respectively. Permutation polynomials over finite fields have wide applications in coding theory, cryptography, and combinatorial design theory, and we refer the readers to [2,5,18,28,30,35] and the references therein for more details of the recent advances and contributions to the area. It is a challenging task to find new classes of permutation polynomials.…”