Abstract
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.
Highlights
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 Fq to generate the underlying groups
One can suspect that DL-systems based on divisor class groups of rings of holomorphic functions on curves over finite fields are endowed with a bilinear structure. (Note that in this case Pic(O) is finite and so
Has singularities, or if K is a finite field which is the interesting case for cryptography since in this case H 2 (GK, (Fq )s ) = 0
Summary
Letbe a prime number and A a group of ordersuch that i): the elements in A are presented in a compact way, for instance by O(log(`)) bits, ii): it is easy to implement the group composition ◦ such that it is very fast, for instance has complexity O(log(`)), but iii): to compute, for randomly chosen elements g1 , g2 ∈ A, a number k ∈ Z such that g2k = g1 (the discrete logarithm problem (DLproblem)) is hard. In the ideal case this complexity were exp(O(log(`))). This is obtained in black box groups, and, as we hope, in certain groups related to elliptic curves and abelian varieties. Typical examples are systems related to the multiplicative group of quotients of rings of integers (“classical” DL). A group (A, ◦) satisfying conditions i),ii) and iii) is called a DL-system
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