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In 1994, the well-known Diffie-Hellman key exchange protocol was for the first 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 field. 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 fields, 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). 1991 Mathematics Subject Classification. 11A55, 11R58, 11T71, 11Y16, 68Q25, 94A60..

Equivalences between elliptic curves and real quadratic congruence function fields