Section: Research Program

Algorithmic Number Theory

Participants : Luca de Feo, François Morain, Benjamin Smith, Mathilde de La Morinerie, Antonin Leroux, Guénaël Renault.

Algorithmic Number Theory is concerned with replacing special cases with general algorithms to solve problems in number theory. In the Grace project, it appears in three main threads:

• fundamental algorithms for integers and polynomials (including primality and factorization);

• algorithms for finite fields (including discrete logarithms);

• algorithms for algebraic curves.

Clearly, we use computer algebra in many ways. Research in cryptology has motivated a renewed interest in Algorithmic Number Theory in recent decades—but the fundamental problems still exist per se. Indeed, while algorithmic number theory application in cryptanalysis is epitomized by applying factorization to breaking RSA public key, many other problems, are relevant to various area of computer science. Roughly speaking, the problems of the cryptological world are of bounded size, whereas Algorithmic Number Theory is also concerned with asymptotic results.