Special Monomials on Finite Fields

Numerous objects in coding theory, combinatorics or cryptology can be described as and/or constructed from special types of mappings on finite fields. The first step in understanding mappings with particular properties is the study of such monomials, which surprisingly often yield even optimal solutions. In this talk we describe monomials which were used to construct Kakeya sets in finite vector spaces. Further we survey progress on classification and constructions of monomials satisfying certain non-linearity criteria like being APN, crooked, planar or bent.