Group law on affine conics and applications to cryptography

Abstract

In this paper, we highlight that the point group structure of elliptic curves, over finite or infinite fields, may be also observed on reducible cubics with an irreducible quadratic component. Starting from this, we introduce in a very general way a group’s structure over any kind of conic. In the case of conics over finite fields, we see that the point group is cyclic and lies on the quadratic component. Thanks to this, some applications to cryptography are described, considering convenient parametrizations of the conics. We perform an evaluation of the complexity of the operations involved in the parametric groups and consequently in the cryptographic applications. In the case of the hyperbolas, the Rédei rational functions can be used for performing the operations of encryption and decryption, and the More’s algorithm can be exploited for improving the time costs of computation. Finally, we provide also an improvement of the More’s algorithm.