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The arithmetic of certain cubic function fields. (English) Zbl 1053.11087
Summary: We discuss the properties of curves of the form \(y^3=f(x)\) over a given field \(K\) of characteristic different from 3. If \(f(x)\) satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the Jacobian of such curves. From a purely mathematical perspective, this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory. At the same time, it provides another source of groups which are suitable for Diffie-Hellman type protocols in cryptographic applications.

MSC:
11R58 Arithmetic theory of algebraic function fields
94A60 Cryptography
14H05 Algebraic functions and function fields in algebraic geometry
11Y40 Algebraic number theory computations
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