be an elliptic curve defined over F
, the finite field of
elements. It is known that the set of
-rational points of
has a structure of an abelian group. This fact, since the works of Koblitz  and Miller , underlies all known applications of elliptic curves to cryptography, see [3, 15, 16, 50, 73] and references therein. We give a survey of recent results about the structure of this group as well as techniques used.