## Section: Application Domains

### Cryptology

Public key cryptology has become a major application domain for algorithmic number theory. This is already true for the ubiquitous RSA system, but even more so for cryptosystems relying on the discrete logarithm problem in algebraic curves over finite fields [7] . For the same level of security, the latter require smaller key lengths than RSA, which results in a gain of bandwidth and (depending on the precise application) processing time. Especially in environments that are constrained with respect to space and computing power such as smart cards and embedded devices, algebraic curve cryptography has become the technology of choice. Most of the research topics of the Lfant team concern directly problems relevant for curve-based cryptology: The difficulty of the discrete logarithm problem in algebraic curves determines the security of the corresponding cryptosystems. Complex multiplication, point counting and isogenies provide, on one hand, the tools needed to create secure instances of curves. On the other hand, isogenies have been found to have direct cryptographic applications to hash functions [32] and encryption [37] . Pairings in algebraic curves have proved to be a rich source for novel cryptographic primitives. Class groups of number fields also enter the game as candidates for algebraic groups in which cryptosystems can be implemented. However, breaking these systems by computing discrete logarithms has proved to be easier than in algebraic curves; we intend to pursue this cryptanalytic strand of research.

Apart from solving specific problems related to cryptology, number theoretic expertise is vital to provide cryptologic advice to industrial partners in joint projects. It is to be expected that continuing pervasiveness and ubiquity of very low power computing devices will render the need for algebraic curve cryptography more pressing in coming years.