Normal bases and factorization of $x^n-1$ in finite fields

Abstract

Let $\mathbb {F}_q$ be the finite field of q elements, and $\mathbb {F}_{q^n}$ its extension of degree n. A normal basis of $\mathbb {F}_{q^n}$ over $\mathbb {F}_q$ is a basis of the form $\{\sigma ^i(\alpha ):i=0,\dots ,n-1\}$ where $\sigma$ denotes the Frobenius automorphism of $\mathbb {F}_{q^n}$ over $\mathbb {F}_q$ and $\alpha \in \mathbb {F}_{q^n}$. Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. Some problems on normal bases are characterized in terms of linearized polynomials. We show that such problems can be finally reduced to the determination of the irreducible factors of $x^n-1$ over $\mathbb {F}_q$. Using this approach, we extend the known results with alternative as well as relatively elementary proofs. And we believe that this approach can be used to deal with other similar problems.