Determination of all complete mappings of F_{q^2} of the form aX^{3q}+bX^{2q+1}+cX^{q+2}+dX^3
Zhiguo Ding, Wei Xiong, Michael E. Zieve
TL;DR
The paper tackles the problem of classifying complete mappings over the finite field $\mathbb{F}_{q^2}$ within the family $f(X)=aX^{3q}+bX^{2q+1}+cX^{q+2}+dX^3$, identifying when both $f$ and $f+\text{Id}$ are permutations. It develops a multifaceted approach that combines Hermite’s criterion, Weil bounds, permutation rational function theory, and $\,\mathbb{F}_q$-linear conjugacy to reduce the problem to explicit canonical forms. The main contributions include a complete classification of these complete mappings (with explicit coefficient conditions) and a detailed analysis of several structured bijections on $\mathbb{F}_{q^2}$ or on $\mathbb{F}_q\times\mathbb{F}_q$, expanding the known classes of complete mappings beyond additive and low-degree families. The results have implications for constructing orthogonal Latin squares, cryptographic primitives, and related combinatorial structures, and they provide a framework for classifying complete mappings among high-degree polynomials over arbitrary characteristics.
Abstract
For each prime power q, we determine all polynomials over F_{q^2} of the form f(X) := aX^{3q}+bX^{2q+1}+cX^{q+2}+dX^3 which induce complete mappings of F_{q^2}, in the sense that each of the functions x --> f(x) and x --> f(x)+x permutes F_{q^2}. This is the first result in the literature which classifies the complete mappings among some class of polynomials with arbitrarily large degree over finite fields of arbitrary characteristic. We also determine all permutation polynomials over F_{q^2} of the form X^{q+2}+bX^q+cX, and all permutations of (F_q)^2 induced by maps of the form (x,y) --> (x^3-exy^2-ax-by, y^3-cx-dy) where either e=0 or 3|q. The latter results add to the small number of results in the literature classifying all permutations induced by maps of prescribed forms.