A note on multiplicative-inverse chaining in finite fields

TL;DR

This paper investigates the structure of multiplicative-inverse chains in finite fields using multiple polynomial bases, revealing disjoint cycles and permutation interpretations, with extensions to more than two bases.

Contribution

It introduces a novel analysis of inverse chaining in finite fields with multiple bases, highlighting cycle structures and permutation properties.

Findings

Disjoint cycles of even length in inverse chains with two bases

Permutation cycle interpretation of the chains

Extension of chaining analysis to multiple bases

Abstract

We consider chaining multiplicative-inverse operations in finite fields under alternating polynomial bases. When using two distinct polynomial bases to alternate the inverse operation we obtain a partition of $\mathbb F_{p^n}\setminus \mathbb F_p$ into disjoint cycles of even length. This allows a natural interpretation of the cycles as permutation cycles. Finally, we explore chaining under more than two polynomial bases.