Is Arithmetic Geometry necessary for Public Key Cryptography? Gerhard Frey, University of Duisburg-Essen

One of the most astonishing success stories in recent mathematics is arithmetic geometry, which unifies methods from classical number theory with algebraic geometry („schemes"). In particular, the extremely important role of the Galois groups of base schemes as algebraic equivalent of the topological fundamental group and its representations via étale cohomology groups yielded spectacular results. One of the most important features in this game is the close relation with Galois representations coming from modular forms and the exploitation of well studied attached L-series. At the same time the algorithmic aspect of arithmetical objects like class groups of global fields became more and more important and accessible, stimulated by and stimulating itself the advances in theory. Both aspects are used for the very surprising application of one of the purest parts of mathematics to an applied science: Data security, at least in the area of public key cryptography, depends crucially on results and methods of arithmetical geometry. In this exciting area happens the work of GANITA bringing together the three aspects sketched above, led and inspired by Kumar Murty.

In the lecture we shall discuss how advances concerning the arithmetic in divisor classes of curves over finite fields play a constructive and destructive role in public key systems making, by fast scalar multipli- cation and point counting, the use of such divisor classes possible but, at the same time, constructing algorithms for the computation of discrete logarithms that are in many cases „too fast". As result, we shall see how narrow the range of candidates usable for cryptography is. In addition, we shall discuss recent results and ideas emerging out of the better understanding of isogenies. Again, we have negative implica- tions for security but also new ideas for public key protocols that generalize the Diffie-Hellman key exchange, and there is a vague hope that such protocols could be resistant against quantum computers.

The aim of the lecture will be reached if a vote concerning the question of the title will have a positive answer by a majority of the audience.