Characterizations and constructions of plateaued functions on finite abelian groups

Plateaued functions have been studied in many papers. They can be candidates for designing cryptographic functions and have been used to construct linear codes. They also have close connections to combinatorics and design theory. Plateaued functions on finite abelian groups were studied in Xu (J Comb Des 27:756-783, 2019; J Comb Des 2021, https://doi.org/10.1002/jcd.21281). In this paper, we continue the research in Xu (2019, 2021). We will first study the characterizations of plateaued functions in terms of the derivatives and autocorrelation functions. We will also characterize the plateaued-ness of a function by its distance to affine functions. Then we investigate constructions of plateaued functions. In particular, we will give two general methods to construct plateaued functions and prove the existence of plateaued functions on finite abelian groups whose orders are not prime numbers. We will also show how to construct plateaued functions from a finite abelian group to a group of prime order. As applications of plateaued functions to combinatorics, we will show the existence of two new infinite families of directed strongly regular graphs.