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On many-to-one mappings over finite fields

Yanbin Zheng, Yanjin Ding, Meiying Zhang, Pingzhi Yuan, Qiang Wang

TL;DR

This work introduces a unified framework for many-to-one mappings over finite fields by defining a generalized local criterion that encompasses and extends prior 2-to-1 and $n$-to-$1$ notions. It then develops three constructions that reduce the problem to simpler subproblems, enabling explicit, verifiable conditions for when a map $f$ is $m$-to-$1$ on $ F_q^*$, and specializes these results to maps of the form $f(x)=x^r h(x^s)$ with reductions to the subgroup $U_ell$. The paper provides concrete necessary and sufficient criteria for $m$-to-$1$ behavior via parameters $(m,m_1,ell)$ and the auxiliary map on $U_ell$, thereby extending and unifying a broad set of results on polynomial and rational maps, including new $3$-to-$1$ and $5$-to-$1$ rational functions on $U_{ obreak q+1}$ and extensions to $ F_{q^2}$ and $ F_{q^n}$. Overall, the results offer a comprehensive, technique-driven approach with many new classes of $m$-to-$1$ mappings and potential cryptographic and coding applications.

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 $\mathbb{F}_{q}^{*}$ is studied by using these three constructions. A series of explicit conditions for~$f$ to be an $m$-to-$1$ mapping on $\mathbb{F}_{q}^{*}$ are found through the detailed discussion of the parameters $m$, $s$, $q$ and the polynomial $h$. These results extend many conclusions in the literature.

On many-to-one mappings over finite fields

TL;DR

This work introduces a unified framework for many-to-one mappings over finite fields by defining a generalized local criterion that encompasses and extends prior 2-to-1 and -to- notions. It then develops three constructions that reduce the problem to simpler subproblems, enabling explicit, verifiable conditions for when a map is -to- on , and specializes these results to maps of the form with reductions to the subgroup . The paper provides concrete necessary and sufficient criteria for -to- behavior via parameters and the auxiliary map on , thereby extending and unifying a broad set of results on polynomial and rational maps, including new -to- and -to- rational functions on and extensions to and . Overall, the results offer a comprehensive, technique-driven approach with many new classes of -to- mappings and potential cryptographic and coding applications.

Abstract

The definition of many-to-one mapping, or -to- mapping for short, between two finite sets is introduced in this paper, which unifies and generalizes the definitions of -to- mappings and -to- mappings. A generalized local criterion is given, which is an abstract criterion for a mapping to be -to-. By employing the generalized local criterion, three constructions of -to- mapping are proposed, which unify and generalize all the previous constructions of -to- mappings and -to- mappings. Then the -to- property of polynomials on is studied by using these three constructions. A series of explicit conditions for~ to be an -to- mapping on are found through the detailed discussion of the parameters , , and the polynomial . These results extend many conclusions in the literature.
Paper Structure (23 sections, 60 theorems, 119 equations, 8 figures)

This paper contains 23 sections, 60 theorems, 119 equations, 8 figures.

Table of Contents

  1. Introdution
  2. The progress of many-to-one mapping
  3. The constructions of many-to-one mappings
  4. The organization of the paper
  5. Notations
  6. Some properties of m-to-1 mappings
  7. Three constructions for m-to-1 mappings
  8. The first construction
  9. The second construction
  10. The third construction
  11. m-to-1 mappings of the form x^ rh(x^ s)
  12. The case m = 2, 3
  13. The case ell = 2, 3
  14. Monomials
  15. m-to-1 mappings on Fq2
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $A$, $S$, and $\bar{S}$ be finite sets with $\#S=\#\bar{S}$, and let $f: A\rightarrow A$, $\bar{f}: S\rightarrow\bar{S}$, $\lambda: A\rightarrow S$, and $\bar{\lambda}: A\rightarrow \bar{S}$ be mappings such that $\bar{\lambda}\circ f= \bar{f}\circ\lambda$. If both $\lambda$ and $\bar{\lambda}$

Figures (8)

  • Figure 1: Schematic diagrams of many-to-one mappings
  • Figure 2: Commutative diagram of the AGW criterion
  • Figure 3: $2$-to-$1$ in 2to1-MesQ19
  • Figure 4: $n$-to-$1$ in GAO211612
  • Figure 5: Our \ref{['constr1']}
  • ...and 3 more figures

Theorems & Definitions (125)

  • Definition 1.1
  • Definition 1.2
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.1: The AGW criterion
  • Theorem 1.2: Local criterion Yuan243070
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 115 more