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arXiv · 2605.26263

On planar functions over $\mathbb{F}_{q^3}$

João Paulo Guardieiro·Adler Marques·Luciane Quoos·Guilherme Tizziotti
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Abstract

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.

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João Paulo Guardieiro, Adler Marques, Luciane Quoos, Guilherme Tizziotti. 2026-05-25. On planar functions over $\mathbb{F}_{q^3}$. https://arxiv.org/abs/2605.26263

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