Section: Overall Objectives

Overview

The overall objective of AriC (Arithmetic and Computing) is, through computer arithmetic and computational mathematics, to improve computing at large.

A major challenge in modeling and scientific computing is the simultaneous mastery of hardware capabilities, software design, and mathematical algorithms for the efficiency of the computation. Further, performance relates as much to efficiency as to reliability, requiring progress on automatic proofs, certificates and code generation. In this context, computer arithmetic and mathematical algorithms are the keystones of AriC. Our approach conciliates fundamental studies, practical performance and qualitative aspects, with a shared strategy going from high-level problem specifications and normalization actions, to computer arithmetic and the lowest-level details of implementations.

We focus on the following lines of action:

• Design and integration of new methods and tools for mathematical program specification, certification, security, and guarantees on numerical results. Some main ingredients here are: the interleaving of formal proofs, computer arithmetic and computer algebra; error analysis and computation of certified error bounds; the study of the relationship between performance and numerical quality; and on the cryptology aspects, focus on the practicality of existing protocols and design of more powerful lattice-based primitives.

• Generalization of a hybrid symbolic-numeric trend, and interplay between arithmetics for both improving and controlling numerical approaches (symbolic $\to$ numeric), and accelerating exact solutions (symbolic $\leftarrow$ numeric). This trend, especially in the symbolic computation community, has acquired a strategic role for the future of scientific computing. The integration in AriC of computer arithmetic, reliable computing, and algebraic computing is expected to lead to a deeper understanding of the problem and novel solutions.

• Mathematical and algorithmic foundations of computing. We address algorithmic complexity and fundamental aspects of approximation, polynomial and matrix algebra, and lattice-based cryptology. Practical questions concern the design of high performance and reliable computing kernels, thanks to optimized computer arithmetic operators and an improved adequacy between arithmetic bricks and higher level ones.

According to the application domains that we target and our main fields of expertise, these lines of actions are declined in three themes with specific objectives. These themes also correspond to complementary angles for addressing the general computing challenge stated at the beginning of this introduction:

• Efficient approximation methods (§3.1). Here lies the question of interleaving formal proofs, computer arithmetic and computer algebra, for significantly extending the range of functions whose reliable evaluation can be optimized.

• Lattices: algorithms and cryptology (§3.2). Long term goals are to go beyond the current design paradigm in basis reduction, and to demonstrate the superiority of lattice-based cryptography over contemporary public-key cryptographic approaches.

• Algebraic computing and high performance kernels (§3.3). The problem is to keep the algorithm and software designs in line with the scales of computational capabilities and application needs, by simultaneously working on the structural and the computer arithmetic levels.