Calculating the sign of a NxN determinant

Question

Is there a more efficient way to determine the sign (negative or positive or zero) of a determinant than calculating the full value of the determinant and comparing it with zero?

Answer 1

There are methods, that can approximate the determinat of an integer matrix faster, than computing the exact value. These methods are usually used to compute the sign, since there is a great probability of a correct result. See this paper for much more details.

However AFAIK there is no exact method of computing sign of determinant faster than computing the value itself.