  The use of permutation polynomials has appeared, along to their compositional
inverses, as a good choice in the implementation of cryptographic systems.
Hence, there has been a demand for constructions of these polynomials which
coefficients belong to a finite field. As a particular case of permutation
polynomial, involution is highly desired since its compositional inverse is
itself. In this work, we present an effective way of how to construct several
linear permutation polynomials over $\mathbb{F}_{q^n}$ as well as their
compositional inverses using a decomposition of
$\displaystyle{\frac{\mathbb{F}_q[x]}{\left\langle x^n -1 \right\rangle}}$
based on its primitive idempotents. As a consequence, an immediate construction
of involutions is presented.
