On many-to-one property of generalized cyclotomic mappings

2025-03-09

AI Summary

This paper investigates the many-to-one property of generalized cyclotomic mappings over finite fields. Departing from the classical focus on injectivity, it establishes, for the first time, a comprehensive characterization and classification framework for their many-to-one behavior. Methodologically, the work integrates structural analysis of finite fields, cyclotomic coset decomposition, power-function-induced mechanisms, and algebraic curve techniques. Key contributions include: (1) necessary and sufficient conditions for many-to-one behavior when ℓ ≤ 3; (2) a complete classification of 2-to-1 mappings for arbitrary ℓ dividing q−1; and (3) explicit constructions of several families of many-to-one polynomials—including binomials and trinomials—over 𝔽_{q²}. These results fill a fundamental gap in the systematic classification of such mappings and provide a new paradigm for studying non-injective polynomial mappings over finite fields.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $U_{q+1}$.

Problem

• Characterize many-to-one generalized cyclotomic mappings for small indices
• Classify 2-to-1 generalized cyclotomic mappings for any divisor
• Construct many-to-one binomials and trinomials over finite fields

Innovation

• Characterize many-to-one generalized cyclotomic mappings
• Classify 2-to-1 generalized cyclotomic mappings
• Construct many-to-one binomials and trinomials