Explicit Number Theory

Many classical results and methods of Number Theory are now actively used in important domains of applied mathematics like Cryptography,CodingTheory or Numerical Analysis.To enable these applications one must have explicit versions of these number-theoretical results and tools. This is why Explicit Number Theory emerged and rapidly developed during the last several decades.

In this CIMPA school we plan to introduce the students to this field. The following list is a sample of the topics that will be considered:

• explicit computations in number fields (like finding their class groups, unit groups, etc.);
• explicit solutions of Diophantine equations (like Pell equations, Thue equations, etc.);
• efficient algorithms for prime testing and factorization of integers;
• explicit construction of elliptic curves over finite fields with prescribed number of points.

It is important that development of Explicit Number Theory stimulated progress and posed new problems in the parts of Number Theory which were so far considered "purely theoretical", the Class Field Theory being a notable example. Therefore in our lectures will focus on both theoretical and computational aspects of Explicit Number Theory.