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][1],
but it assumes that the determinant is nonzero. 

What about lower bounds on the complexity?
    

  [1]: https://cs.uwaterloo.ca/~astorjoh/shifted.pdf