The Weil Pairing, first introduced by André Weil in 1940, plays an
important role in the theoretical study of the arithmetic of
elliptic curves and Abelian varieties. It has also recently become
extremely useful in cryptologic constructions related to those
objects. This paper gives the definition of the Weil Pairing,
describes efficient algorithms to calculate it, gives two
applications, and describes the motivation to considering it.