Equivalences Between Elliptic Curves And Real Quadratic Congruence Function Fields

In [11] Scheidler, Stein and Williams proposed a key exchange protocol which makes use of the set of reduced principal ideals of a real quadratic congruence function field as its underlying structure. The security of the protocol depends on a corresponding discrete logarithm problem (DLP). In this article we show that the DLP for real quadratic congruence function fields of genus one is equivalent to the DLP for elliptic curves over finite fields. We explicitly give the one-to-one correspondence between the set of reduced principal ideals of a real quadratic congruence function field of genus one and the group generated by a point on the corresponding elliptic curve (without the point itself) such that this one-to-one correspondence can be easily computed. 1. Introduction In [11], it is shown how the theory of real quadratic congruence function fields can be used to produce a secure key distribution protocol. The method is an extension of the recent ideas of Scheidler, Buchmann and Wi..