On Isotopic Shift Construction for Planar Functions

Abstract

CCZ-equivalence is the most general currently known equivalence relation for functions over finite fields preserving planarity and APN properties. However, for the particular case of quadratic planar functions isotopic equivalence is more general than CCZ-equivalence. A recent construction method for APN functions over fields of even characteristic, so-called isotopic shift construction, was instigated by the notion of isotopic equivalence. In this paper we discuss possible applications of the idea of isotopic shift for the case of planar functions. We show that, surprisingly, some of the known planar functions are actually isotopic shifts of each other. This confirms practically the pertinence of the notion of isotopic shift not only for APN functions but also for planar maps.