New semifields, PN and APN functions

We describe a method of proving that certain functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f:F\longrightarrow F}$$\end{document} defined on a finite field F are either PN-functions (in odd characteristic) or APN-functions (in characteristic 2). This method is illustrated by giving short proofs of the APN-respectively the PN-property for various families of functions. The main new contribution is the construction of a family of PN-functions and their corresponding commutative semifields of dimension 4s in arbitrary odd characteristic. It is shown that a subfamily of order p4s for odd s > 1 is not isotopic to previously known examples.