On many-to-one property of generalized cyclotomic mappings
Yanbin Zheng, Yang Zhang, Zhengbang Zha, Xiangyong Zeng, Qiang Wang
TL;DR
This work advances the understanding of many-to-one properties of generalized cyclotomic mappings over finite fields. It delivers complete $m$-to-$1$ classifications for index $oldsymbol{ l} obreak obreakoldsymbol{ } obreak obreak ext{ with } oldsymbol{ l} obreak obreakoldsymbol{ } obreak ext{ up to }3$, and a full classification of $2$-to-$1$ mappings for any divisor $oldsymbol{ l}$ of $q-1$, by analyzing coset-wise monomial behavior and the interaction of image cosets $f_i(C_i)$. It extends to multiple-branch cases via a unified framework (Lpieces), and leverages polynomial realizations $a_i=h_i(x^s)$ to construct concrete many-to-one mappings, including binomials and trinomials of the form $x^r h(x^{q-1})$ on $oldsymbol{ f^2}^*$, using the subgroup structure of $U_{q+1}$. The paper also synthesizes Hou–Lavorante permutations for $f(x)=x^r h(x^{q-1})$, translating permutation criteria on cosets to $m$-to-$1$ conditions, thereby enabling explicit, spread-ready mappings for coding theory and cryptographic applications. Overall, it provides a rich toolkit for designing finite-field mappings with prescribed fiber sizes via cyclotomic coset decompositions.
Abstract
The generalized cyclotomic mappings over finite fields $\mathbb{F}_{q}$ are those mappings which induce monomial functions on all cosets of an index $\ell$ subgroup $C_0$ of the multiplicative group $\mathbb{F}_{q}^{*}$. Previous research has focused on the one-to-one property, the functional graphs, and their applications in constructing linear codes and bent functions. In this paper, we devote to study the many-to-one property of these mappings. We completely characterize many-to-one generalized cyclotomic mappings for $1 \le \ell \le 3$. Moreover, we completely classify $2$-to-$1$ generalized cyclotomic mappings for any divisor $\ell$ of $q-1$. In addition, we construct several classes of many-to-one binomials and trinomials of the form $x^r h(x^{q-1})$ on $\mathbb{F}_{q^2}$, where $h(x)^{q-1}$ induces monomial functions on the cosets of a subgroup of $U_{q+1}$.