Two-to-one mappings and involutions without fixed points over $\bF_{2^n}$

TL;DR

This paper investigates special functions over finite fields of even characteristic, providing new criteria and constructions for 2-to-1 mappings and fixed-point-free involutions, with potential applications in cryptography.

Contribution

It introduces an AGW-like criterion for 2-to-1 mappings and constructs new classes of such mappings and involutions without fixed points.

Findings

Developed an AGW-like criterion for 2-to-1 mappings

Constructed eight classes of 2-to-1 mappings of specific form

Derived involutions without fixed points from known 2-to-1 mappings

Abstract

In this paper, two-to-one mappings and involutions without any fixed point on finite fields of even characteristic are investigated. First, we characterize a closed relationship between them by implicit functions and develop an AGW-like criterion for 2-to-1 mappings. Using this criterion, some new constructions of 2-to-1 mappings are proposed and eight classes of 2-to-1 mappings of the form $(x^{2^k}+x+\delta)^{s}+cx$ are obtained. Finally, a number of classes of involutions without any fixed point are derived from the known 2-to-1 mappings by the relation between them.