On the Carlet's Bivariate APN Construction and Representations of Dobbertin Power Functions

In this paper we present necessary and sufficient conditions on some families of bivariate and biprojective APN functions. In the first part, we follow the extended methods given by Calderini et al in 2021, and give sufficient and necessary conditions for Carlet's bivariate APN functions. We also give APN-ness conditions for biprojective functions in some special cases. In the second part, we study the representation of the Dobbertin power function following methods given by Budaghyan et al. in 2022 and give new representations for this kind of functions.