Shifted plateaued functions and their differential properties

A bent(4) function is a Boolean function with a flat spectrum with respect to a certain unitary transform T. It was shown previously that a Boolean function f in an even number of variables is bent(4) if and only if f + sigma is bent, where sigma is a certain quadratic function depending on T. Hence bent(4) functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent(4) if and only if f + sigma 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 bent(4) functions from partially bent functions and study the differential properties of partially bent(4) functions, unifying the previous work on partially bent functions.