General Kasami APN triple-correlation conjecture

Prove the Kasami APN triple-correlation conjecture for all remaining residue classes k modulo n with gcd(k,n)=1, namely that for every pair of distinct nonzero v₁,v₂ in the binary field F of order 2ⁿ, the set of triples (x,y,z) in Δ³ satisfying v₁x+v₂y+(v₁+v₂)z=0 has cardinality 2²ⁿ⁻³; equivalently, for odd k and every ρ∈F\{0,1}, establish the vanishing of the character sum ∑_{t,a∈F}(-1)^{Tr(ψ(t)+ψ(t+a)+ψ(t+ρa))}, where ψ is the inverse of the Müller–Cohen–Matthews permutation MCM.

Background

The paper studies the Kasami APN function F(x)=x^{4^k−2^k+1} over the binary field F with gcd(k,n)=1, defines Δ={F(b)+F(b+1)+1:b∈F}, and conjectures a uniform triple count for all distinct nonzero coefficients v₁ and v₂. The conjecture is proved in the paper for k modulo n belonging to {1,2,n−2,n−1}, and it is verified exhaustively for all admissible parameters with n≤13.

For odd k, the conjecture is reduced to the vanishing of a two-variable character sum involving ψ=MCM^{-1}. The paper identifies this equivalent formulation as the sharpest known open form and notes that the general proof must exploit cancellation, because the proposed termwise support-disjointness mechanism fails in the remaining cases.