Normal Bases on Galois Ring Extensions

Abstract

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension $$\mathbf{R}/{\mathrm{Z}}_{p^r}$$, where $$\mathbf{R}=\mathrm{GR}(p^r,n).$$ We present a criterion on the normal basis for $$\mathbf{R}/{\mathrm{Z}}_{p^r}$$ and reduce this problem to one of finite field extension $$\overline{\mathbf{R}}/{\overline{\mathrm{Z}}}_{p^r}={\mathbb{F}}_q/{\mathbb{F}}_p(q=p^n)$$ by Theorem 1. We determine all optimal normal bases for Galois ring extension.