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

TL;DR

This paper characterizes when the monomial function x^m over finite fields is perfectly nonlinear infinitely often, showing it occurs precisely when m is of the form p^l+1 for odd prime p.

Contribution

It proves a complete characterization of exponents m for which x^m is PN over infinitely many extensions of F_p, using algebraic geometry and irreducibility arguments.

Findings

x^m is PN over infinitely many extensions iff m=p^l+1 for some l

For other m, the associated polynomial has an absolutely irreducible factor

Application of Weil's theorem links irreducibility to the non-PN property

Abstract

Let $p$ be an odd prime number. We prove that for $m\equiv1\mod p$, $x^m$ 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 $m\neq 1+p^l$, $f(x,y)$ has an absolutely irreducible factor. Then by Weil theorem, f(x,y) has rationnal points such that $x\neq y$ which means that $x^m$ is not PN.