On the iterations of the maps $ax^{2^k}+b$ and $(a x^{2^k} + b)^{-1}$ over finite fields of characteristic two

TL;DR

This paper analyzes the cycle structures of specific quadratic maps over finite fields of characteristic two, relating them to elliptic curve groups and polynomial properties, and extends the study to inverse maps.

Contribution

It characterizes cycle lengths of maps related to supersingular elliptic curves and explores their connections to known polynomial families.

Findings

Cycle lengths are described using elliptic curve group structures.

Relations between inverse maps and specific polynomials are established.

Extensions to inverse maps reveal new structural insights.

Abstract

The maps $x \mapsto ax^{2^k}+b$ defined over finite fields of characteristic two can be related to the duplication map over binary supersingular elliptic curves. Relying upon the structure of the group of rational points of such curves we can describe the possible cycle lengths of the maps. Then we extend our investigation to the maps $x \mapsto (ax^{2^k}+b)^{-1}$. We also notice some relations between these latter maps and the polynomials $x^{2^k+1} + x +a$, which have been extensively studied in literature.