Section: Overall Objectives
Overview
The overall objective of AriC is, through computer arithmetic, to improve computing at large, in terms of performance, efficiency, and reliability. Specifically, we focus on the following domains:

Floatingpoint arithmetic: The IEEE 7542008 standard specifies the behavior of floatingpoint 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 multipleprecision 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 seminumerical computation, in finding best or nearly best approximations under special constraints, and in designing efficient algorithms for exact linear algebra. Also, we are working on the development and standardization of interval arithmetic.

Hardware and FPGA arithmetic: The main challenge here is the design of efficient arithmetic hardware. Instead of designing adhoc operators for a given technology and a given target (in terms of speed, accuracy, or power consumption requirements), we aim at building algorithms and programs that automatically design them. This allows one to find better solutions by being able to explore a larger part of the (in general, huge) design space, and to build specific operators for frequent “compound” functions such as, for example, $x/\sqrt{{x}^{2}+{y}^{2}}$.

Cryptography and lattices: Latticebased 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 latticebased 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.