  We provide explicit and unified formulas for the cocycles of all degrees on
the normalized bar resolutions of finite abelian groups. This is achieved by
constructing a chain map from the normalized bar resolution to a Koszul-like
resolution for any given finite abelian group. With a help of the obtained
cocycle formulas, we determine all the braided linear Gr-categories and compute
the Dijkgraaf-Witten Invariants of the $n$-torus for all $n$.
