Algebraic Structure of the Iterates of  \(\chi \)

We consider the map \(\chi :\mathbb {F}_2^n\rightarrow \mathbb {F}_2^n\) for n odd given by \(y=\chi (x)\) with \(y_i=x_i+x_{i+2}(1+x_{i+1})\) , where the indices are computed modulo n. We suggest a generalization of the map \(\chi \) which we call generalized \(\chi \) -maps. We show that these maps form an Abelian group which is isomorphic to the group of units in \(\mathbb {F}_2[X]/(X^{(n+1)/2})\) . Using this isomorphism we easily obtain closed-form expressions for iterates of \(\chi \) and explain their properties.