A larger mathematical structure always can be built on top of a smaller mathematical structure. For example, a pair of sets, together with a function relating those two sets, becomes a larger mathematical package when the ensemble is viewed collectively. Thus, a large elliptic curve can be mapped into a large finite field by mapping each point of the elliptic curve into one point of the finite field. But we want to go beyond this: we want to map a pair of r-torsion points of an elliptic curve into one point of a finite field. More precisely, we want to map a pair of subgroups, each of the same prime order r of an elliptic curve, into a subgroup, also of prime order r, of the finite field. This is the structure that comprises this chapter's subject. A pair of points – one point from each of the two additive subgroups of order r, denoted G1 and G2, of a large elliptic curve under the operation of point addition – is mapped into one point of a subgroup, denoted GT or Gx, of the multiplicative group of a finite field. The mapping with the pair of groups as the domain and the single group as the range, taken as a package, becomes the new mathematical structure that we will want to explore.

We will study a special class of such mappings, called bilinear pairings, and the application of pairings in cryptography.