Equivalences between elliptic curves and real quadratic congruence function fields
The Diffie-Hellman key exchange protocol has previously been considered in the principal ideal class of a real quadratic number field as well as real quadratic congruence function fields. This set does not possess a group structure but rather an infrastructure. The security of the scheme depends on that of the discrete logarithm problem in this setting. This article shows that the discrete logarithm problem in real quadratic congruence function fields of genus 1, called real elliptic congruence function fields, is equivalent to that for elliptic curves. The properties and arithmetic of the set of reduced principal ideals in elliptic congruence function fields are discussed. The connection between elliptic curves and real quadratic congruence function fields is made. The one-to-one correspondence between the set of reduced principal ideals of an elliptic congruence function field and the group \(\langle{\mathcal P}\rangle/\{ {\mathcal P} \}\) is made, where \({\mathcal P}\) denotes an \({\mathbb{F}}_q\)-rational point on the corresponding elliptic curve. Several questions of importance that arise in establishing this correspondence are also discussed.
real quadratic congruence function fields
scientific article; zbMATH DE number 1132889
scientific article