  We give an elementary and self-contained introduction to pairings on elliptic
curves over finite fields. For the first time in the literature, the three
different definitions of the Weil pairing are stated correctly and proved to be
equivalent using Weil reciprocity. Pairings with shorter loops, such as the
ate, ate$_i$, R-ate and optimal pairings, together with their twisted variants,
are presented with proofs of their bilinearity and non-degeneracy. Finally, we
review different types of pairings in a cryptographic context. This article can
be seen as an update chapter to A. Enge, Elliptic Curves and Their Applications
to Cryptography - An Introduction, Kluwer Academic Publishers 1999.
