On some quadratic APN functions

A construction of APN functions using the bent function \(B(x,y)=xy\) is proposed in Carlet (Des Codes Cryptogr 59:89–109, 2011). At this time, two families of APN functions using this construction are known, that is, the family of Carlet (2011) and the family of Zhou and Pott (Adv Math 234:43–60, 2013). In this note, we propose another family of APN functions with this construction, which are not CCZ equivalent to the former two families on \({{\mathbb {F}}}_{2^8}\). We also propose a family of presemifields and determined the middle, left, right nuclei and the center of the associated semifields.