On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for $k\mathrm{mod}n\in \{1,2,n-2,n-1\}$, and exhaustive verification for $n\le 13$

Gábor P. Nagy, Attila Vajda

We study a conjecture on the Kasami almost perfect nonlinear (APN) function $F(x)=x^{4^k-2^k+1}$ on $GF(2^n)$, $\gcd (k,n)=1$: for the $2^{n-1}$-element set $Δ=\{F(b)+F(b+1)+1: b\in…