Unicyclic Strong Permutations

For positive integers $n$ and $k$ such that $0\leq k\leq n-1$, we study some properties of a certain kind of permutations $\sigma$ over $\mathbb{F}_{2}^{n}$. The permutation $\sigma$ is given as the composition of intermediate permutations $\sigma_{k}$ of a specific form. The properties that hold simultaneously are: (1) the algebraic degree of $\sigma_{k}$ is $n-1$; (2) the permutations $\sigma_{k}$ are unicyclic; (3) the number of terms of the algebraic normal form of each $\sigma_{k}$ is at least $2^{n-1}$. We call these unicyclic strong permutations. In this paper, we provide a construction for unicyclic strong permutations. We also notice a dichotomy about the cycle structure of $\sigma$ between odd and even values of $n\leq 30$. For the composition $\sigma$, we also study empirically the differential uniformity for all values of $n\leq 16$ and notice that in almost all cases it never exceeds $6$. For the specific cases of $n=17$ and $n=19$, we report counts of the number of equal entries of their difference table and linear approximation table.