Bent functions embedded into the recursive framework of $${\mathbb{Z}}$$ -bent functions

Suppose that n is even. Let $${\mathbb{F}_2}$$ denote the two-element field and $${\mathbb{Z}}$$ the set of integers. Bent functions can be defined as ± 1-valued functions on $${\mathbb{F}_2^n}$$ with ± 1-valued Fourier transform. More generally we call a mapping f on $${\mathbb{F}_2^n}$$ a $${\mathbb{Z}}$$ -bent function if both f and its Fourier transform $${\widehat{f}}$$ are integer-valued. $${\mathbb{Z}}$$ -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and $${\widehat{f}}$$ . It is shown how $${\mathbb{Z}}$$ -bent functions of lower level can be built up recursively by gluing together $${\mathbb{Z}}$$ -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of $${\mathbb{Z}}$$ -bent functions and give some guidelines for further research.