Section: New Results

Elliptic curve cryptology

Participants : Jean-Marc Couveignes, Andreas Enge, Damien Robert.

With J.-G. Kammerer, J.-M. Couveignes has given in [14] an appropriate geometric method for studying and classifying encodings into elliptic curves in a cryptographic context. Such encodings were first proposed by Icart in 2009, and later on by Farashahi, Kammerer, Lercier, and Renault. But it was a little bit disappointing to see that it was no more than an application of Tartaglia’s result without any geometrical explanations for the existence of such “parameterisations” of elliptic curves. Couveignes and Kammerer have filled this gap by giving exactly what can be expected from geometry: a clear explanation. Moreover, they unify all the recent “parameterisations” of elliptic curves under the same geometric point of view. The approach described in this article uses dual curves with some results coming from intersection theory. The main originality of this work is that these geometrical tools are employed to explain symbolic computations used in cryptography, that is, encoding on elliptic curves.

The survey [20] , to be published in the Handbook of Finite Fields, presents the state of the art of the use of elliptic curves in cryptography.