Section: Research Program

Algorithmic Number Theory

Concerning algorithmic number theory, the main problems we will be considering in the coming years are the following:

  • Number fields. We will continue working on the problems of class groups and generators. In particular, the existence and accessibility of good defining polynomials for a fixed number field remain very largely open. The impact of better polynomials on the algorithmic performance is a very important parameter, which makes this problem essential.

  • Lattice reduction. Despite a great amount of work in the past 35 years on the LLL algorithm and its successors, many open problems remain. We will continue the study of the use of interval arithmetic in this field and the analysis of variants of LLL along the lines of the Potential-LLL which provides improved reduction comparable to BKZ with a small block size but has better performance.

  • Elliptic curves and Drinfeld modules. The study of elliptic curves is a very fruitful area of number theory with many applications in crypto and algorithms. Drinfeld modules are “cousins” of elliptic curves which have been less explored in the algorithm context. However, some recent advances  [44] have used them to provide some fast sophisticated factoring algorithms. As a consequence, it is natural to include these objects in our research directions.