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arXiv · 1809.06271

Low-degree planar polynomials over finite fields of characteristic two

Daniele Bartoli·Kai-Uwe Schmidt
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Abstract

Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of $q$ and we consider the case that $q$ is even. We classify polynomials of degree at most $q^{1/4}$ that induce planar functions on $\mathbb{F}_q$, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over $\mathbb{F}_q$ that induce planar functions on infinitely many extensions of~$\mathbb{F}_q$. The proof strategy is to study the number of $\mathbb{F}_q$-rational points of an algebraic curve attached to a putative planar function.~Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear~functions.

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Daniele Bartoli, Kai-Uwe Schmidt. 2018-09-17. Low-degree planar polynomials over finite fields of characteristic two. https://arxiv.org/abs/1809.06271

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