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Characterizations of a Class of Planar Functions over Finite Fields

Ruikai Chen, Sihem Mesnager

TL;DR

This work investigates planar functions of the form $f(x)=Tr(a x^{q+1})+ell(x^2)$ over odd-characteristic extension fields. It derives a general planarity criterion, then specializes to quadratic and cubic extensions, achieving explicit constructions and complete characterizations in these cases, while establishing nonexistence results for higher degrees under natural conditions. The analysis combines trace-linearized polynomial structure with discriminant-type tests and, for larger degrees, analytic tools, to delineate precisely when such planar functions can exist. The findings advance the understanding of planar function existence and construction in finite field settings, with potential implications for cryptography and coding theory.

Abstract

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ on an extension of the finite field $\mathbb F_{q^n}/\mathbb F_q$. Specifically, we investigate those functions on $\mathbb F_{q^2}/\mathbb F_q$ and construct several typical kinds of planar functions. We also completely characterize them on $\mathbb F_{q^3}/\mathbb F_q$. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.

Characterizations of a Class of Planar Functions over Finite Fields

TL;DR

This work investigates planar functions of the form over odd-characteristic extension fields. It derives a general planarity criterion, then specializes to quadratic and cubic extensions, achieving explicit constructions and complete characterizations in these cases, while establishing nonexistence results for higher degrees under natural conditions. The analysis combines trace-linearized polynomial structure with discriminant-type tests and, for larger degrees, analytic tools, to delineate precisely when such planar functions can exist. The findings advance the understanding of planar function existence and construction in finite field settings, with potential implications for cryptography and coding theory.

Abstract

Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form on an extension of the finite field . Specifically, we investigate those functions on and construct several typical kinds of planar functions. We also completely characterize them on . When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.
Paper Structure (5 sections, 10 theorems, 116 equations)

This paper contains 5 sections, 10 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. The New Class of Planar Functions
  3. Planar Functions on Quadratic Extensions
  4. Planar Functions on Cubic Extensions
  5. Non-existence Results of Higher Degree

Key Result

Proposition 1

The function $x^{q+1}+\ell(x^2)$ is planar on $\mathbb F_{q^2}$ if and only if $\ell(u)^2-\mathop{\mathrm{N}}\nolimits(u)$ is a nonzero square in $\mathbb F_q$ for all $u\in\mathbb F_{q^2}^*$ such that $\ell(u)\in\mathbb F_q$.

Theorems & Definitions (21)

  • Proposition 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Example 5
  • Theorem 6
  • proof
  • ...and 11 more