Characterizations and constructions of n-to-1 mappings over finite fields

TL;DR

This paper characterizes and constructs various n-to-1 mappings over finite fields, providing new criteria, explicit examples, and applications in cryptography and finite geometry.

Contribution

It introduces a Walsh transform-based characterization, an AGW-like criterion, and explicit constructions of n-to-1 mappings over finite fields.

Findings

Characterization of n-to-1 mappings via Walsh transform.

Complete determination of 3-to-1 polynomials up to degree 4.

New explicit constructions of n-to-1 mappings using cyclotomic and polynomial forms.

Abstract

$n$-to-$1$ mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of $n$-to-$1$ mappings over finite fields are studied. First, we provide a characterization of general $n$-to-$1$ mappings over $\mathbb{F}_{p^m}$ by means of the Walsh transform. Then, we completely determine $3$-to-$1$ polynomials with degree no more than $4$ over $\mathbb{F}_{p^{m}}$. Furthermore, we obtain an AGW-like criterion for characterizing an equivalent relationship between the $n$-to-$1$ property of a mapping over finite set $A$ and that of another mapping over a subset of $A$. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit $n$-to-$1$ mappings. Especially, three explicit constructions of the form $x^rh\left( x^s \right) $ from the cyclotomic perspective, and several classes of $n$-to-$1$ mappings of the form $ g\left( x^{q^k} -x +\delta \right) +cx$ are provided.