Graphs of vectorial plateaued functions as difference sets

Abstract

A function F : F-pn -> F-pm , is a vectorial s-plateaued function if for each component function F-b(mu) = Tr-n(bF(x)), b is an element of F-pm* and mu is an element of F-pn , the Walsh transform value vertical bar(F-b) over cap(mu)vertical bar is either 0 or p(n+s/2) . In this paper, we explore the relation between (vectorial) s-plateaued functions and partial geometric difference sets. Moreover, we establish the link between three-valued cross-correlation of p-ary sequences and vectorial s-plateaued functions. Using this link, we provide a partition of F-3n into partial geometric difference sets. Conversely, using a partition of F-3n into partial geometric difference sets, we construct ternary plateaued functions f : F-3n -> F-3. We also give a characterization of p-ary plateaued functions in terms of special matrices which enables us to give the link between such functions and second-order derivatives using a different approach. (C) 2020 Elsevier Inc. All rights reserved.