Explicit factorization of $x^{2^nd}-1$ over a finite field

Let $\mathbb{F}_q$ be a finite field of odd characteristic containing $q$ elements and integer $n\ge 1$. In this paper, the explicit factorization of $x^{2^nd}-1$ over $\mathbb{F}_q$ is obtained when $d$ is an odd divisor of $q+1$.