Inversion of two level circulant matrices over Z p

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(mn c(m,n)) operations in Zp, where c is a low degree polynomial in log m and log n.