Bilinear pairings on elliptic curves

Abstract

We give an elementary and self-contained introduction to pairings on elliptic curves over finite fields. The three different definitions of the Weil pairing that can be found in the literature are stated 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.