Inversion of two level circulant matrices over Zp
Carlo J. Accettella, Gianna M. Del Corso, Giovanni Manzini · Linear Algebra and its Applications · 2003
We consider the problem of inverting block circulant with circulant blocks BCCB matrices with entries over the field $Zp$. This problem arises in the study of of two-dimensional linear cellular automata. Since the standard reduction to diagonal form by means of FFT has some drawbacks when working over $Zp$, we solve this problem by transforming it into the equivalent problem of inverting a circulant matrix with entries over a suitable ring~$R$. We show that a BCCB matrix of size $mn$ can be inverted in $O{m n, c(m,n)}$ operations in $Zp$, where $c$ is a low degree polynomial in $log m$ and $log n$.