EXPLICIT INFRASTRUCTURE FOR REAL QUADRATICFUNCTION FIELDS AND REAL HYPERELLIPTIC CURVES

摘要

In 1989, Koblitz first proposed the Jacobian of a animaginary hyperelliptic curve for use in public-key cryptographic proto-cols. This concept is a generalization of elliptic curve cryptography. It canbe used with the same assumed key-per-bit strength for small genus. Morerecently, real hyperelliptic curves of small genus have been introduced asanother source for cryptographic protocols. The arithmetic is more in-volved than its imaginary counterparts and it is based on the so-calledinfrastructure of the set of reduced principal ideals in the ring of regularfunctions of the curve. This infrastructure is an interesting phenomenon.The main purpose of this article is to explain the infrastructure in explicitterms and thus extend Shanks' infrastructure ideas in real quadratic num-ber fields to the case of real quadratic congruence function fields and theircurves. Hereby, we first present an elementary introduction to the contin-ued fraction expansion of real quadratic irrationalities and then generalizeimportant results for reduced ideals.