Properties of associated Boolean functions of quadratic APN functions

For a function F : Fn Fn, it is defined the associated Boolean function yF in 2n variables as follows: yF(a, b) = 1 if a = 0 and equation F(x) + F(x + a) = b has solutions. A vectorial Boolean function F from F2n to F2n is called almost perfect nonlinear (APN) if equation F(x) + F(x + a) = b has at most 2 solutions for all vectors a, b 6 F2n, where a is nonzero. In case when F is a quadratic APN function its associated function has the form yF(a, b) = Фр(a) • b + ^F(a) + 1 for appropriate functions Фр : Fn Fn and : Fn F2. We study properties of functions Фр and ^F, in particular their degrees