On the existence of distortion maps on ordinary elliptic curves

An important problem in cryptography is the so called Decision Diffie-Hellman problem (henceforth abbreviated DDH). The problem is to distinguish triples of the form (g, g, g) from arbitrary triples from a cyclic group G = 〈g〉. It turns out that for (cyclic subgroups of) the group of m-torsion points on an elliptic curve over a finite field, the DDH problem admits an efficient solution if there exists a suitable endomorphism called a distortion map (which can be efficiently computed) on the elliptic curve.