(2026-06) A family of invertible shift-invariant maps with strong arithmetic properties
2026-06-12
Shift-invariant maps have been employed to design nonlinear layers in many symmetric cryptographic schemes, such as the -map used in Keccak. In this paper, we study the shift-invariant maps on , whose defining functions come from a family of -variable Boolean functions induced by a bifix-free sequence with , which we denote by . It is shown that forms a commutative monoid with respect to the composition. Moreover, if , then is isomorphic to the unit group of ; if , then the unit group of is isomorphic to that of . The isomorphic relation transforms the composition of functions on into the multiplication of polynomials on the quotient ring of , where the algebraic properties of the latter are well-understood. As a straightforward application, we focus on the algebraic properties of , which includes the -map as well as several other known maps studied in earlier literature. It is shown that is invertible if and only if . Also the inverse and the cycle structure of (if invertible) can be fully characterized. The construction of generalizes previous works given by Kriepke et al. and Lyu et al. It is hoped that could provide a deeper insight into the study of invertible shift-invariant maps.