My thesis was on the fast computation of theta functions, in time which is quasi-linear in the number of digits of precision wanted. The method generalizes a link between theta constants and the arithmetico-geometric mean, using Newton’s method to achieve the desired complexity; this method appears to generalize nicely to theta functions of genus 2. You can take a look at my manuscript, which was written in English.
My research works was published in the following papers:
- Computing Jacobi’s Theta in quasi-linear time (2015) ; to appear in Mathematics of Computation.
- An implementation of this algorithm using the MPC library is available on the Github repository.
- Computing theta functions in genus 2 and above in quasi-linear time (2016), with Emmanuel Thomé; accepted at ANTS-XII.
- An implementation of this algorithm in MAGMA is available on the Github repository.
- Plane Quartics over Q with Complex Multiplication (2017), with Pınar Kılıçer, Reynald Lercier, Christophe Ritzenthaler, Jeroen Sijsling and Marco Streng; submitted.
- A MAGMA implementation of the fast computation of genus 3 theta constants is available on the Github repository; however there seems to be a few problems with some period matrices.
I also published a research article during my first year of master’s, on genus 2 point multiplication:
- Improved scalar multiplication on hyperelliptic Koblitz curves (2011), with Michael J. Jacobson, Jr.; published at SAC 2011 ; Lecture Notes in Computer Science, 2012, Volume 7118, Selected Areas in Cryptography 2011, Pages 399-411.
Finally, during my Ph.D., I also wrote an article, which was accepted as a short talk at SSTIC 2015; this article deals with building quickly a dictionary of famous sentences and expressions, using Wikipedia, to be used for password cracking. Here is the full paper, an appendix with more informations, and the presentation slides.