Functions which are PN on infiitely many extensions of Fp, p odd

Abstract

Let $p$ be an odd prime number. We prove that for $m\equiv 1\mathrm{mod}p$, $x^m$ is perfectly nonlinear over ${\mathbb{F}}_{p^n}$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\in \mathbb{N}$. First, we study singularities of $f(x,y)=\frac{(x+1{)}^m-x^m-(y+1{)}^m+y^m}{x-y}$ and we use Bezout theorem to show that for $m\ne 1+p^l$, $f(x,y)$ has an absolutely irreducible factor. Then by Weil theorem, f(x,y) has rationnal points such that $x\ne y$ which means that $x^m$ is not PN.

References in corpus (1)

• Proof of a Conjecture on the Sequence of Exceptional Numbers, Classifying Cyclic Codes and APN Functions

Cited by in corpus (4)

• Planar functions and perfect nonlinear monomials over finite fields
• Semifields, relative difference sets, and bent functions
• Planar functions over fields of characteristic two
• Exceptional planar polynomials