On many-to-one mappings over finite fields

2024/08/08 by Yanbin Zheng, Zheng, Yanbin, Ding, Yanjin +6 · 1 citation
Computer Science · Mathematics · #Coding theory and cryptography #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #Information Theory (cs.IT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2408.04218

openalex publication_date 2024/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

The definition of many-to-one mapping, or m-to-1 mapping for short, between two finite sets is introduced in this paper, which unifies and generalizes the definitions of 2-to-1 mappings and n-to-1 mappings. A generalized local criterion is given, which is an abstract criterion for a mapping to be m-to-1. By employing the generalized local criterion, three constructions of m-to-1 mapping are proposed, which unify and generalize all the previous constructions of 2-to-1 mappings and n-to-1 mappings. Then the m-to-1 property of polynomials f(x) = x^r h(x^s) on \mathbbF_q^* is studied by using these three constructions. A series of explicit conditions for~f to be an m-to-1 mapping on \mathbbF_q^* are found through the detailed discussion of the parameters m, s, q and the polynomial h. These results extend many conclusions in the literature.

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