Motivated by recent results of Tao–Ziegler [Discrete Anal. 2016] and Greenfeld–Tao (2022 preprint) on concatenating affine-linear functions along subgroups an abelian group, we show three recovering affine linearity $$f: V \rightarrow W$$ from their restrictions to lines, where V, W are $${\mathbb {F}}$$ -vector spaces $$\dim \geqslant 2$$ . First, if < |{\mathbb {F}}|$$ {\mathbb is when restri...

Boolean function in even number of variables is called bent if it is at the maximal possible Hamming distance from the class of all affine Boolean functions. We have proven that every isometric mapping of the set of all Boolean functions into itself that transforms bent functions into bent functions is a combination of an affine transform of coordinates and an affine shift.