Inversion of circulant matrices over $\mathbf{Z}_m$

In this paper we consider the problem of inverting an $n\times n$ circulant matrix with entries over $\mathbf{Z}_m$. We show that the algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over $\mathbf{Z}_m$. We present three different algorithms which do not use this approach. Our algorithms require different degrees of knowledge of $m$ and $n$, and their costs range, roughly, from $n\log n\log\log n$ to $n \log^2n\log\log n \log m$ operations over $\mathbf{Z}_m$. Moreover, for each algorithm we give the cost in terms of bit operations. We also present an algorithm for the inversion of finitely generated bi-infinite Toeplitz matrices. The problems considered in this paper have applications to the theory of linear cellular automata.