On Bilinear Structures on Divisor Class Groups

摘要

It is well known that duality theorems are of utmost importance for the arithmetic of local and global fields and that Brauer groups appear in this context unavoidably. The key word here is class field theory.In this paper we want to make evident that these topics play an important role in public key cryptopgraphy, too. Here the key words are Discrete Logarithm systems with bilinear structures.Almost all public key crypto systems used today based on discrete logarithms use the ideal class groups of rings of holomorphic functions of affine curves over finite fields $mathbf{F}_q$ to generate the underlying groups. We explain in full generality how these groups can be mapped to Brauer groups of local fields via the Lichtenbaum-Tate pairing, and we give an explicit description.Next we discuss under which conditions this pairing can be computed efficiently.If so, the discrete logarithm is transferred to the discrete logarithm in local Brauer groups and hence to computing invariants of cyclic algebras. We shall explain how this leads us in a natural way to the computation of discrete logarithms in finite fields.To end we give an outlook to a globalisation using the Hasse-Brauer-Noether sequence and the duality theorem ot Tate-Poitou which allows to apply index-calculus methods resulting in subexponential algorithms for the computation of discrete logarithms in finite fields as well as for the computation of the Euler totient function (so we have an immediate application to the RSA-problem), and, as application to number theory, a computational method to “describe” cyclic extensions of number fields with restricted ramification.