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 Z_p. 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 Z_p, 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 Z_p, where c is a low degree polynomial in log m and log n.