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, which can then be checked for being zero. Is there any faster method?

What about lower bounds on the complexity?

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, but it assumes that the determinant is nonzero.

What about lower bounds on the complexity?