Large class of many-to-one mappings over quadratic extension of finite fields
Yanbin Zheng, Meiying Zhang, Yanjin Ding, Zhengbang Zha, Qiang Wang
TL;DR
The paper addresses the problem of characterizing many-to-one mappings on the quadratic extension $\mathbb{F}_{q^{2}}$ of a finite field, focusing on the class $f(x)=h(a x^{q}+b x+c)+u x^{q}+v x$ with $a^{q+1}=b^{q+1}$ and $a v\neq b u$. It develops a unifying reduction via a commutative diagram built from trace maps to relate $f$ on $\mathbb{F}_{q^{2}}$ to an associated polynomial $g$ on $\mathbb{F}_{q}$, such that $f$ is $m$-to-$1$ on $\mathbb{F}_{q^{2}}$ iff $m|q$ and $g$ is $m$-to-$1$ on $\mathbb{F}_{q}$. By taking $h(x)=x^{r}$ with $r$ in various families, $g$ reduces to low-degree polynomials or linearized forms, enabling explicit classifications of $m$-to-$1$ mappings for $f$ across degrees 2, 3, 4, and linearized cases; the paper also derives inverses for all $1$-to-$1$ instances and constructs involutions from certain $2$-to-$1$ mappings, thereby unifying and extending prior results. Practical implications lie in systematic construction and inversion of such mappings for cryptographic and coding-theoretic applications. The work provides a comprehensive framework for translating high-degree, finite-field mappings into tractable small-degree or linearized subproblems on $\mathbb{F}_{q}$.
Abstract
Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as $f(x) = h(a x^q + b x + c) + u x^q + v x$, where $h(x) \in \mathbb{F}_{q^2}[x]$ and $a$, $b$, $c$, $u$, $v \in \mathbb{F}_{q^2}$. Using a commutative diagram satisfied by $f(x)$ and trace functions over finite fields, we reduce the problem whether $f(x)$ is a many-to-one mapping on $\mathbb{F}_{q^2}$ to another problem whether an associated polynomial $g(x)$ is a many-to-one mapping on the subfield $\mathbb{F}_{q}$. In particular, when $h(x) = x^{r}$ and $r$ satisfies certain conditions, we reduce $g(x)$ to polynomials of small degree or linearized polynomials. Then by employing the many-to-one properties of these low degree or linearized polynomials on $\mathbb{F}_{q}$, we derive a series of explicit characterization for $f(x)$ to be many-to-one on $\mathbb{F}_{q^2}$. On the other hand, for all $1$-to-$1$ mappings obtained in this paper, we determine the inverses of these permutation polynomials. Moreover, we also explicitly construct involutions from $2$-to-$1$ mappings of this form. Our findings generalize and unify many results in the literature.