Further Investigation on Cyclotomic Mapping Permutation Polynomials over Finite Fields

TL;DR

The paper investigates permutation polynomials over finite fields in the cyclotomic-mapping framework by linking polynomials of the form $P(x)=x^r f(x^s)$ to $r$-th order cyclotomic mappings of index $l$, with $q-1=ls$. It introduces a new necessary condition in terms of $Ind_\gamma$ on $A_i=f(\xi^i)$ and then derives a new sufficient condition, establishing a complete set of necessary-and-sufficient criteria for $P(x)=x^r f(x^s)$ to be a permutation polynomial and highlighting the role of index structure and $l$-th roots of unity. The results are demonstrated on explicit examples over $\mathbb{F}_{13}$, showing permutation trinomial examples for certain $r$, and on permutation binomials $x^r(x^{es}+1)$, including nonexistence results for several primes and a general index-based framework. This work provides concrete, verifiable criteria for permutation behavior of cyclotomic-mapping polynomials with potential cryptographic and coding applications.

Abstract

We explore the connection between cyclotomic mapping permutation polynomials and permutation polynomials of the form $x^rf(x^{\frac{q-1}{l}})$ over finite fields. We present a new necessary and a new sufficient condition to verify permutation behavior of such polynomials over finite field. As its application, for particular values of $r$, we point out some permutation trinomials of the form $P(x)=2x^{r+8}+x^{r+4}+2x^r \in \mathbb{F}_{13}[x]$, and work on few classes of permutation binomials.

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Lemma 2.2
• Definition 2.1
• Lemma 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• Theorem 3.1
• Theorem 3.2