The functional graph of $f(X)=(cX^q+aX)(X^{q}-X)^{n-1}$ over quadratic extensions of finite fields
Fabio E. Brochero Martínez, Hugo R. Teixeira
TL;DR
This work studies the dynamical graphs of the polynomial-like map $f(X)=(cX^q+aX)(X^q-X)^{n-1}$ over the quadratic extension $\mathbb{F}_{q^2}$ of a finite field, with $a,c\in\mathbb{F}_q$ and odd prime $p$; it provides a complete description of the functional graph, including the cycle structure, fixed points, and the preimage trees. The analysis treats separately the cases of even and odd $n$, employing a vector-space decomposition $X=x+y\beta$ with $\{1,\beta\}$ as a basis and $\beta^q=-\beta$ to reduce $f$ to a form $f(x,y)=\langle \delta_1 y^n,\delta_2 x y^{n-1}\rangle$ (even $n$) or $f(x,y)= (4\beta^2)^{(n-1)/2}y^{n-1}\langle \delta_1 x,\delta_2 y\rangle$ (odd $n$). The paper derives explicit cycle-count formulas using quadratic characters, Möbius inversion, and order calculations in $\mathbb{F}_q^*$, and it characterizes the hanging trees attached to cyclic elements via levels corresponding to $n^i$-th roots of unity in $\mathbb{F}_q$, giving canonical tree shapes $\mathscr{T}_n(i)$ (and $\mathscr{T}_n^0(i)$ in degenerate cases). A key finding is that all hanging trees attached to nonzero cycles are isomorphic, with the exception of the tree anchored at zero, yielding a complete classification of the functional graph of $f$. The results generalize known dynamics for simpler maps and provide exact structural insight with potential applications in pseudorandomness and cryptography over finite fields.
Abstract
Let $\mathbb{F}_q$ be the finite field with $q=p^s$ elements, where $p$ is an odd prime and $s$ a positive integer. In this paper, we define the function $f(X)=(cX^q+aX)(X^{q}-X)^{n-1}$, for $a,c\in\mathbb{F}_q$ and $n\geq 1$. We study the dynamics of the function $f(X)$ over the finite field $\mathbb{F}_{q^2}$, determining cycle lengths and number of cycles. We also show that all trees attached to cyclic elements are isomorphic, with the exception of the tree hanging from zero. We also present the general shape of such hanging trees, which concludes the complete description of the functional graph of $f(X)$.