  In this paper, we highlight that the point group structure of elliptic curves
over finite or infinite fields, may be also observed on singular cubics with a
quadratic component. Starting from this, we are able to introduce in a very
general way a group's structure over any kind of conics. In the case of conics
over finite fields, we see that the point group is cyclic and lies on the
quadric; the straight line component plays a role which may be not explicitly
visible in the algebraic description of point composition, but it is
indispensable in the geometric description. Moreover, some applications to
cryptography are described, considering convenient parametrizations of the
conics. Finally, we perform an evaluation of the complexity of the operations
involved in the parametric groups and consequently in the cryptographic
applications.
