A note on the properties of associated Boolean functions of quadratic APN functions | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2020. № 47. DOI: 10.17223/20710410/47/2

Let F be a quadratic APN function in n variables. The associated Boolean function γF in 2n variables (γF (a, b) = 1 if a = 0 and equation F(x) +F(x +a) = b has solutions) has the form y_f(a, b) = Φ_f(a) ∙ b + φ_f(a) + 1 for appropriate functions Φ_f : Fn → Fn and ψF : Fn → F₂. We summarize the known results and prove new ones regarding properties of Φ_f and ψ_f. For instance, we prove that degree of Φ_f is either n or less or equal to n - 2. Based on computation experiments, we formulate a conjecture that degree of any component function of ΦF is n - 2. We show that this conjecture is based on two other conjectures of independent interest.