On Finding the Eigenvalues of the Matrix of Rotation Symmetric Boolean Functions

We consider the action on ${\mathbb{F}}_2^n$ by cyclic permutations ($\mathbb{Z}\mathrm{/}n\mathbb{Z}$). Two elements $x,y\in {\mathbb{F}}_2^n$ are in the same orbit if they are cyclic shifts of each other. Cryptographic properties of rotation symmetric Boolean functions can be efficiently computed using the square matrix ${}_n\mathcal{A}$, the construction of which uses orbit representatives of the cyclic shifting action. In 2018, Ciungu and Iovanov proved that ${}_n{\mathcal{A}}^2=2^n\cdot I$, the identity matrix of dimension $g_n\times g_n$ where $g_n$ is the number of orbits. In this paper, we answer the open question of the precise number of positive and negative eigenvalues of ${}_n\mathcal{A}$.