A family of invertible shift-invariant maps with strong arithmetic properties

Shift-invariant maps have been employed to design nonlinear layers in many symmetric cryptographic schemes, such as the $\chi$-map used in Keccak. In this paper, we study the shift-invariant maps on $\mathbb{F}_2^n$, whose defining functions come from a family of $n$-variable Boolean functions induced by a bifix-free sequence $\underline{a}=(a_1,a_2,\ldots,a_m)\in \mathbb{F}_2^m$ with $2\leq m