I am looking for the best upper bounds on the bit complexity for testing the singularity of an integer $n\times n$ matrix, where each integer is represented with $k$ bits. 


I know the fast method for computation of the determinant in 
[Storjohann,
The shifted number system for fast linear algebra on integer matrices](https://cs.uwaterloo.ca/~astorjoh/shifted.pdf),
but it assumes that the determinant is nonzero. 


What about lower bounds on the complexity?



testing singularity of integer matrices
This is addressed in [Storjohann's paper on computing Smith Normal Form.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.8839&rep=rep1&type=pdf)


