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.

弯曲₄函数是一个布尔函数，相对于某个单一变换\（\ mathcal {T} \）具有平坦频谱。先前已证明，当且仅当f + σ弯曲时，偶数个变量的布尔函数f才弯曲₄，其中σ是取决于\（\ mathcal {T} \）的某个二次函数。因此，弯曲₄函数也称为移位弯曲函数。类似地，当且仅当f + σ时，奇数个变量的布尔函数f弯曲_4。是满足某些其他属性的半弯曲函数。在本文中，我们首次详细分析了移位对平稳函数，部分弯曲函数和布尔函数的线性结构的影响。我们还将讨论部分折弯函数的折弯和折弯₄函数的构造，并研究部分折弯₄函数的微分性质，以统一先前有关部分折弯函数的工作。