A Generalized χ n -Function

The mappingχnfrom Fn2to itself defined byy=χn(x) withyi=xi+xi+2(1 +xi+1), where the indices are computed modulo n, has been widely studied for its applications in lightweight cryptography. However,χnis bijective on Fn2only whennis odd, restricting its use to odd-dimensional vector spaces over F2. To address this limitation, we introduce and analyze the generalized mappingχn,mdefined byy=χn,m(x) withyi=xi+xi+m(xi+m−1+ 1)(xi+m−2+ 1) · · · (xi+1+ 1), wheremis a fixed integer withm∤n. To investigate such mappings, we further generalizeχn,mto θm,k, where θm,kis given byyi=xi+mkПmk−1j=1, m∤j(xi+j+ 1) , fori∈ {0, 1, . . . ,n− 1}. We prove that these mappings generate an abelian group isomorphic to the group of units in F2[z]/(z⌊n/m⌋+1). This structural insight enables us to construct a broad class of permutations over Fn2for any positive integern, along with their inverses. We rigorously analyze algebraic properties of these mappings, including their iterations, fixed points, and cycle structures. Additionally, we provide a comprehensive database of the cryptographic properties for iterates ofχn,mfor small values ofnandm. Finally, we conduct a comparative security and implementation cost analysis amongχn,m,χn,χχn(EUROCRYPT 2025 [3]) and their variants, and prove Conjecture 1 proposed in [3] as a by-product of our study. Our results lead to generalizations ofχn, providing alternatives toχnandχχn.