A note on the properties of associated Boolean functions of quadratic APN functions

Let F be a quadratic APN function in n variables. The associated Boolean function gamma(F) in 2n variables (gamma(F) (a, b) = 1 if a not equal 0 and equation F(x) + F(x + a) = b has solutions) has the form gamma(F)(a, b) = phi(F)(a).b + phi(F)(a) + 1 for appropriate functions phi(F) :F-2(n)-> F-2(n) and phi(F) : F-2(n)-> F-2. We summarize the known results and prove new ones regarding properties of phi(F) and phi(F). For instance, we prove that degree of phi(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 phi(F) is n - 2. We show that this conjecture is based on two other conjectures of independent interest.