The presentation wil be in French (see info here)
May 16th, 2024 at 11:00 am
Modular calculation is used in numerous mathematical applications, such as 
cryptography. Modular reduction in a very general context is costly, requiring mainly division. In practice, however, the modulo is often xed, for example when calculating over a nite eld, and it is therefore possible to implement strategies to reduce the cost of reduction. The rst approaches consist, if you have the choice, in working with particular moduli, such as Mersenne primes or pseudo-Mersenne primes, or even generalizations of these forms. Other approaches involve divisionless modular reductions with moduli that have no special features, such as the Montgomery or Barrett algorithms.
Another type of strategy consists in working on numeration systems to improve computation or reduction. This is the case with so-called Residue Number Systems based on the Chinese remainder theorem, where the gain is made on multiplication and addition, and with which Montgomery’s reduction is adaptable. We can also build modulo-dependent systems where the modulo takes a form similar to pseudo-Mersenne. The reductions here are similar to the search for a nearest vector in a Euclidean lattice. Finally, the modular product can be seen as a change of representation to an Ostrowski basis on integers.
Bio: Jean Claude Bajard is a professor at Sorbonne University. Since 2020, he has been a member of the Ouragan Inria team and the Institut de Mathematiques de Jussieu-Paris Rive Gauche. His eld of research is computer arithmetic. Jean Claude Bajard taught high school mathematics for ten years before completing a doctorate in computer science at ENS Lyon, which he defended in 1993. In 1993 he was recruited as a lecturer in Marseille, then in 1999 he became a professor at the Institut Universitaire de Technologie de Montpellier, a member of LIRMM. In 2009, he joined Universite Pierre et Marie Currie and was a member of LIP6 until 2019.
