Our overall objective is, through computer arithmetic, to improve computing at large, in terms of performance, efficiency, and reliability. We work on arithmetic algorithms (integer and floating-point arithmetic, complex arithmetic, multiple-precision arithmetic, finite-field arithmetic) and their implementation, approximation methods, Euclidean lattices and cryptology, certified computing and computer algebra.
Specifically, we focus on the following domains:
- Floating-point arithmetic.
The IEEE 754-2008 standard specifies the behavior of floating-point arithmetic. We are interested in preparing future evolutions of the standard, in implementing it efficiently on embedded processors, in exploring its “low level” properties for better numerical analysis (for instance by finding certified and tight error bounds of numerical algorithms), and in building correctly rounded mathematical function programs. We are also interested in designing efficient algorithms and software for multiple-precision arithmetic and complex arithmetic.
- Certified computing and computer algebra.
We are interested in computing certified approximations using computer algebra and formal proof systems, in analyzing the fundamental algorithms of semi-numerical computation, in finding best or nearly best approximations under special con- straints, and in designing efficient algorithms for exact linear algebra. Also, we are working on the development and standardization of interval arithmetic.
- Cryptography and lattices.
Lattice-based cryptography (LBC) is a fast developing field, raising fascinating questions both on cryptography and lattices. Lattice algorithmics is an established research area that is being revived by the amazing application that is LBC and by the new tools and concepts that it introduced. We aim at contributing to a major technological switch, from conventional to lattice-based cryptography. This will help suppress the main limitation to the expansion of the cloud economy that are the privacy concerns. Further, thanks to the ubiquity of lattices, our work may significantly impact several other fields, including coding, computer algebra, and computer arithmetic.