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Permutation Rational Functions over Quadratic Extensions of Finite Fields

Ruikai Chen, Sihem Mesnager

TL;DR

This paper introduces a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic polynomials, whose numerical values are determined by calculating character sums related to quadratic forms of F2.

Abstract

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over $\mathbb F_{q^2}$, whose numerators are so-called $q$-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special $q$-quadratic polynomial in $\mathbb F_{q^2}$, by calculating character sums related to quadratic forms of $\mathbb F_{q^2}/\mathbb F_q$. Then given some rational function, we can demonstrate whether it induces a permutation of $\mathbb F_{q^2}$.

Permutation Rational Functions over Quadratic Extensions of Finite Fields

TL;DR

This paper introduces a class of permutation rational functions over , whose numerators are so-called -quadratic polynomials, whose numerical values are determined by calculating character sums related to quadratic forms of F2.

Abstract

Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over , whose numerators are so-called -quadratic polynomials. To this end, we will first determine the exact number of zeros of a special -quadratic polynomial in , by calculating character sums related to quadratic forms of . Then given some rational function, we can demonstrate whether it induces a permutation of .
Paper Structure (5 sections, 9 theorems, 147 equations)

This paper contains 5 sections, 9 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. On the Number of Zeros of $q$-quadratic Polynomials in $\mathbb F_{q^2}$
  3. The First Kind of Polynomials
  4. The Second Kind of Polynomials
  5. Permutation Rational Functions over $\mathbb F_{q^2}$

Key Result

Lemma 1

Let $A,B,C\in\mathbb F_{q^2}$ with $B\ne0$ and $D=\mathop{\mathrm{Tr}}\nolimits(A)^2-4\mathop{\mathrm{N}}\nolimits(B)$. If there does not exist $\theta\in\mathbb F_{q^2}$ such that $\mathop{\mathrm{Tr}}\nolimits(A)\theta^q+2B\theta+C=0$, then $D=0$ and If such $\theta$ exists, then where in the case $D\ne0$.

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Proposition 2
  • Proposition 3
  • Lemma 4: williams1975note
  • Theorem 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • ...and 5 more