Since 2016 and the introduction of the exTNFS (extended Tower Number Field Sieve) algorithm, the security of cryptosystems based on non-prime finite fields, mainly the paring and torus-based one, is being reassessed. The feasibility of the relation collection, a crucial step of the NFS variants, is especially investigated. It usually involves polynomials of degree one, i.e., a search space of dimension two. However, exTNFS uses bivariate polynomials of at least four coefficients. If sieving in dimension two is well described in the literature, sieving in higher dimension received significantly less attention. We describe and analyze three different generic algorithms to sieve in any dimension for the NFS algorithms. Our implementation shows the practicability of dimension four sieving, but the hardness of dimension six sieving.