A direct proof of APN-ness of the Kasami functions

Abstract

Using recent results on solving the equation $X^{2^k+1}+X+a=0$ over a finite field ${\mathbb{F}}_{2^n}$, we address an open question raised by the first author in WAIFI 2014 concerning the APN-ness of the Kasami functions $x\mapsto x^{2^{2k}-2^k+1}$ with $gcd(k,n)=1$, $x\in {\mathbb{F}}_{2^n}$.