Determinant modulo m What are the known efficient algorithms for computing a determinant of an integer matrix with coefficients in $\mathbb{Z}_m$, the ring of residues modulo $m$. The number $m$ may not be prime but composite (so computations are performed in ring, not a field).

As far as I know (read below), most algorithms are modifications of Gaussian elimination. The question is about the computational efficiency of these procedures.

If it happened that there is some different approach, I also curious about it.

Thanks in advance.

Update:

Let me explain the source of this question. Assume, $m$ is a prime number. So $\mathbb{Z}_m$ is a field. And in this case we can perform all computations using numbers less than $m$, so we have some nice upper bound on all operations on numbers: addition, multiplication and inversion --- all needed operations to run Gaussian elimination.

On the other hand we can not perform inversion for some numbers in case $m$ not a prime. So we need some tricks to compute determinant.

And now I am curious what are the known tricks to do the job and whether such tricks could be found in and papers of books.
There is a combinatorial algorithm by Mahajan and Vinay that works over commutative rings: http://cjtcs.cs.uchicago.edu/articles/1997/5/contents.html