A plane curve X over the field F is the set of points (x, y) in the plane F2 that are the zeros of some fixed irreducible bivariate polynomial p(x, y) over F. If one can define a pairwise operation (x, y) + (x′, y′) taking any two points (x, y) and (x′, y′) of the curve into a third point of the curve so as to form an abelian group, then one can use this group operation to deine a public-key cryptography system in various ways. Of course, one then requires assurance that such a cryptosystem is secure. These topics comprise the subjects of elliptic-curve cryptography and elliptic-curve cryptanalysis. Together they form the subject of elliptic-curve cryptology.

Elliptic curves on finite ields are a very attractive class of plane curves that allow one to define a well-behaved operation on any two points of the curve. This operation, called point addition, forms a finite abelian group whose cyclic subgroups are used to form public-key cryptosystems, called elliptic-curve cryptosystems. Elliptic-curve cryptography is attractive because, in part, index calculus methods of attack have not been found for elliptic curves and are not expected because the notion of a smooth integer does not have a parallel for the points of an elliptic curve. In fact, no satisfactory subexponential algorithm is known for solving the discrete-log problem on an elliptic curve.