## Section: Application Domains

### Cryptology

The main application domain of our project-team is cryptology. Algebraic curves have taken an increasing importance in cryptology over the last ten years. Various works have shown the usability and the usefulness of elliptic curves in cryptology, standards (for instance, IEEE P1363 [24] ) and real-world applications (like the electronic passport).

We study the suitability of higher genus curves to cryptography (mainly hyperelliptic curves of genus two, three). In particular, we work on improving the arithmetic of those curves, on the point counting problem, and on the discrete logarithm problem.

We also have connections to cryptology through the study and development of the integer LLL algorithm, which is one of the favourite tools to cryptanalyze public-key cryptosystems. Examples are the cryptanalysis of knapsack-based cryptosystems, the cryptanalyses of some fast variants of RSA, the cryptanalyses of fast variants of signature schemes such as DSA or Elgamal, or the attacks against lattice based cryptosystems like NTRU. The use of floating-point arithmetic dramatically speeds up this algorithm, which renders the aforementioned cryptanalyses more feasible.

Finally, we are studying integer factoring algorithms which are of utmost importance for the evaluation of the security of the still widely used RSA cryptosystem. In the context of our ANR CADO grant, we are investigating the Number Field Sieve algorithm, which is the best known algorithm for factoring numbers of the kind used in practical RSA instances.