Large class of many-to-one mappings over quadratic extension of finite fields

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 $f(x)=h(a{x}^q+bx+c)+u{x}^q+vx$, 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.