Shifted plateaued functions and their differential properties

Abstract

A bent₄ function is a Boolean function with a flat spectrum with respect to a certain unitary transform $\mathcal {T}$. It was shown previously that a Boolean function f in an even number of variables is bent₄ if and only if f + σ is bent, where σ is a certain quadratic function depending on $\mathcal {T}$. Hence bent₄ functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent₄ if and only if f + σ is a semibent function satisfying some additional properties. In this article, for the first time, we analyse in detail the effect of the shifts on plateaued functions, on partially bent functions and on the linear structures of Boolean functions. We also discuss constructions of bent and bent₄ functions from partially bent functions and study the differential properties of partially bent₄ functions, unifying the previous work on partially bent functions.