Two Classes of Quadratic APN Binomials Inequivalent to Power Functions

This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are not Carlet-Charpin-Zinoviev (CCZ)-equivalent to power functions (at least for some values of the number of variables). These are two classes of APN binomials from F 2n to F 2n (for n divisible by 3, resp., 4). We prove that these functions are extended affine (EA)-inequivalent to any power function and that they are CCZ-inequivalent to the Gold, Kasami, inverse, and Dobbertin functions when n ges 12. This means that for n even they are CCZ-inequivalent to any known APN function. In particular, for n = 12,20,24, they are therefore CCZ-inequivalent to any power function.