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 yf in 2n variables (yF(a, b) = 1 if a = 0 and equation F(x) + F(x + a) = b has solutions) has the form yF(a, b) = Ф,р(a) • b + ^F(a) +1 for appropriate functions Ф,р : Fn Fn and ^f : Fn F2. We summarize the known results and prove new ones regarding properties of Ф,р and ^F. For instance, we prove that degree of Ф,р 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 Ф,р is n — 2. We show that this conjecture is based on two other conjectures of independent interest