Overall Objectives
Research Program
Application Domains
Highlights of the Year
New Software and Platforms
New Results
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
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Section: Overall Objectives

Overall Objectives

A general keyword that could encompass most of our research objectives is arithmetic. Indeed, in the Caramel team, the goal is to push forward the possibilities to compute efficiently with objects having an arithmetic nature. This includes integers, real and complex numbers, polynomials, finite fields, and, last but not least, algebraic curves.

Our main application domains are public-key cryptography and computer algebra systems. Concerning cryptography, we concentrate on the study of the primitives based on the factorization problem or on the discrete-logarithm problem in finite fields or (Jacobians of) algebraic curves. Both the constructive and destructive sides are of interest to Caramel . For applications in computer algebra systems, we are mostly interested in arithmetic building blocks for integers, floating-point numbers, polynomials, and finite fields. Also some higher level functionalities like factoring and discrete-logarithm computation are usually desired in computer algebra systems.

Since we develop our expertise at various levels, from most low-level software or hardware implementation of basic building blocks to complicated high-level algorithms like integer factorization or point counting, we have remarked that it is often too simple-minded to separate them: we believe that the interactions between low-level and high-level algorithms are of utmost importance for arithmetic applications, yielding important improvements that would not be possible with a vision restricted to low- or high-level algorithms.

We emphasize three main directions in the Caramel team:

Polynomial system solving is a transverse theme to these research directions. It is rather natural with algebraic curves, and occurs also in NFS-related contexts, that many important challenges can be represented via polynomial systems, which have structural specificities. We also intend to develop algorithms and tools that, when possible, take advantage of these specificities.