Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Permutation Trinomials and Complete Permutation Polynomials via Fiber Criteria over Finite Fields

Chahrazade Bouyacoub, Asmae El-Baz, Omar Kihel

Abstract

We give new, short proofs of recent permutation polynomial results of Bousalmi, Bayad, and Derbal by reducing the verification to explicit computations on a three-element multiplicative subgroup via Zieve's fiber criterion. Building on this approach, we develop a general framework -- combining Zieve's theorem with the AGW criterion -- for constructing complete permutation polynomials over finite fields through a fiber decomposition over the cube roots of unity. A scalar specialization of the criterion yields families that are easy to produce and verify. We illustrate the construction with concrete examples and show through counterexamples that the underlying arithmetic conditions are sharp.

On Permutation Trinomials and Complete Permutation Polynomials via Fiber Criteria over Finite Fields

Abstract

We give new, short proofs of recent permutation polynomial results of Bousalmi, Bayad, and Derbal by reducing the verification to explicit computations on a three-element multiplicative subgroup via Zieve's fiber criterion. Building on this approach, we develop a general framework -- combining Zieve's theorem with the AGW criterion -- for constructing complete permutation polynomials over finite fields through a fiber decomposition over the cube roots of unity. A scalar specialization of the criterion yields families that are easy to produce and verify. We illustrate the construction with concrete examples and show through counterexamples that the underlying arithmetic conditions are sharp.
Paper Structure (9 sections, 7 theorems, 32 equations)

This paper contains 9 sections, 7 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and standing assumptions
  4. Short proofs of the Bousalmi--Bayad--Derbal theorems
  5. A general criterion for index-3 complete permutation polynomials
  6. The scalar fiber specialization
  7. Examples and counterexamples
  8. CPP examples
  9. Counterexamples outside $q\equiv 1\pmod{9}$

Key Result

Theorem 2.1

Let $d$ and $r$ be positive integers with $d\mid q-1$, and let $h\in\mathbb{F}_q[X]$. Then is a permutation polynomial of $\mathbb{F}_q$ if and only if

Theorems & Definitions (15)

  • Theorem 2.1: Zieve Zieve2009
  • Theorem 2.2: AGW criterion AGW2011
  • Theorem 3.1: Theorem 3 of BBD2021
  • proof
  • Theorem 3.2: Theorem 4 of BBD2021
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3
  • ...and 5 more