Equivalences between Elliptic Curves and Real Quadratic Congruence Function Fields

In 1994, the well-known Diie-Hellman key exchange protocol was for the rst time implemented in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number eld. This set does not possess a group structure, but instead exhibits a so-called infrastructure. More recently, the scheme was extended to real quadratic congruence function elds, whose set of reduced principal ideals has a similar infrastructure. As always, the security of the protocol depends on a certain discrete logarithm problem (DLP). In this paper, we show that for real quadratic congruence function elds of genus one, i.e. elliptic congruence function elds, this DLP is equivalent to the DLP for elliptic curves over nite elds. We present the explicit corresponce between the two DLPs and prove some properties which have no analogues for real quadratic number elds. Furthermore, we show that for elliptic congruence function elds, the set of reduced principal ideals is even \closer" to a group than in the general case, but still fails to be a group.