There are siblings of $χ$ which are permutations for n even

Let $1$ be the all-one vector and denote the component-wise multiplication of two vectors in F₂ⁿ. We study the vector space Γₙ over F₂ generated by the functions γ₂ₖ: F₂ⁿ → F₂ⁿ, k≥ 0, where γ₂ₖ = S²ᵏ(1+S²ᵏ⁻¹)(1+S²ᵏ⁻³)…(1+S) and S: F₂ⁿ→ F₂ⁿ is the cyclic left shift function. The functions in Γₙ are shift-invariant and the well known $χ$ function used in several cryptographic primitives is contained in Γₙ. For even n, we show that the permutations from Γₙ with respect to composition form an Abelian group, which is isomorphic to the unit group of the residue ring F₂[X]/(Xⁿ +Xn/2). This isomorphism yields an efficient theoretic and algorithmic method for constructing and studying a rich family of shift-invariant permutations on F₂ⁿ which are natural generalizations of $χ$. To demonstrate it, we apply the obtained results to investigate the function γ₀ +γ₂+γ₄ on F₂ⁿ.