On a conjecture on the Kasami APN function: reductions, structure theorems, a proof for $k\bmod n\in\{1,2,n{-}2,n{-}1\}$, and exhaustive verification for $n\le 13$
We study Carlet's cyclic-additive conjecture for 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 $\Delta=\{F(b)+F(b+1)+1: b\in GF(2^n)\}$ and all distinct nonzero $v_1,v_2\in GF(2^n)$, \[ \bigl|\{(x,y,z)\in\Delta^3 : v_1x+v_2y+(v_1+v_2)z=0\}\bigr| \;=\; 2^{2n-3}. \] This exact triple-count condition was first formulated by Carlet in his 2018 cyclic-additive difference-set framework; the Kasami instance was subsequently posed as an open problem at the NSUCRYPTO 2019 cryptographic olympiad, whose individual proposer was not publicly disclosed. We prove the conjecture for $k\bmod n\in\{1,2,n-2,n-1\}$, in particular a complete proof for $k=2$ ($d=13$) via a quadratic-form theory and an exact root-count reduction, and we verify it exhaustively by computer for every admissible $(n,k)$ with $n\le13$.