A consequence of the dualty theorems in Arithmetic Geometry is that all systems of discrete logarithms used today can be "embedded" into Brauer groups of local and global fields.
A first challenge is to compute this embedding. Till now this is only possible if the fields under considerations have enough roots of unity.
This is mainly due to the fact that we cannot compute effectively inside of Brauer groups. So the challenge behind is to compute the invariant of cyclic algebras over local fields.
A possible approach is to use the Hasse-Brauer-Noether sequence which gives a relation between local invariants of global central simple algebras. As a result we get an index-calculus algorithm to compute discrete logarithms in finite fields and to compute the Euler totient function. It has the same complexity as the "usual" algorithms but till now it is slower in practice. Can this be improved?
Finally the above sequence implies that discrete logarithms obtained by reduction of globally defined group varieties modulo various primes are not independent. The open problem is to determine (all) locally given varieties for which one can find a global one making this dependency effectively computable. The hope is that only tori can be used.
Audio (MP3 File, Podcast Ready)
Copyright. All Rights Reserved.