A note on multiplicative-inverse chaining in finite fields

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.