Construction of orthomorphic $\mathrm{MDS}$ matrices with primitive characteristic polynomial

Matrices having the Maximum Distance Separable property ($\mathrm{MDS}$ matrices) are a vital component for the design of symmetric-key algorithms to achieve the diffusion property. In a number of papers the construction and characterization of $\mathrm{MDS}$ matrices with a low implementation cost in the context of the so-called lightweight schemes were considered. However, small attention was paid to the influence of reducibility of the proposed $\mathrm{MDS}$ matrices; this property may be used by an adversary to exploit the nontrivial invariant subspaces associated to corresponding mappings. We propose some methods for constructing $\mathrm{MDS}$ matrices with primitive characteristic polynomial that provide better resistance against the so-called invariant subspaces attacks.