Consider an elliptic quadric Q-(3 , q) in PG (3 , q). Let E and T denote the set of all lines of PG (3 , q) which meet Q-(3 , q) in 0 and 1 point, respectively. In this paper, we characterize the minimum size (T∪ E) -blocking sets and give a different proof for the characterization of minimum size E-blocking sets in PG (3 , q) which works for all q. We also discuss whether the main results of this paper (Theorems 1.6 and 1.7) can be extended to ovoids in PG (3 , q).