## Section: Research Program

### Low level implementation and Dedicated Algebraic Computation and Linear Algebra.

Participants : Jean-Charles Faugère, Elias Tsigaridas, Olive Chakraborty, Jocelyn Ryckeghem.

Here, the primary objective is to focus on *dedicated*
algorithms and software for the linear algebra steps in Gröbner
bases computations and for problems arising in Number Theory. As
explained above, linear algebra is a key step in the process of
computing efficiently Gröbner bases. It is then natural to develop
specific linear algebra algorithms and implementations to further
strengthen the existing software. Conversely, Gröbner bases
computation is often a key ingredient in higher level algorithms
from Algebraic Number Theory. In these cases, the algebraic problems
are very particular and specific. Hence dedicated Gröbner bases
algorithms and implementations would provide a better efficiency.

**Dedicated linear algebra tools.**
The FGb library is
an efficient one for Gröbner bases computations which can be
used, for instance, via Maple . However, the library is
sequential. A goal of the project is to extend its efficiency to new
trend parallel architectures such as clusters of multi-processor
systems in order to tackle a broader class of problems for several
applications. Consequently, our first aim is to provide a durable,
long term software solution, which will be the successor of the
existing FGb library. To achieve this goal, we will first
develop a high performance linear algebra package (under the LGPL
license). This could be organized in the form of a collaborative
project between the members of the team. The objective is not to
develop a general library similar to the Linbox (http://www.linalg.org/) project but to
propose a dedicated linear algebra package taking into account the
specific properties of the matrices generated by the Gröbner bases
algorithms. Indeed these matrices are sparse (the actual sparsity
depends strongly on the application), almost block triangular and
not necessarily of full rank. Moreover, most of the pivots are known
at the beginning of the computation. In practice, such matrices are
huge (more than ${10}^{6}$
columns) but taking into account their shape may allow us to speed
up the computations by one or several orders of magnitude. A variant
of a Gaussian elimination algorithm together with a corresponding C
implementation has been presented. The main peculiarity is the
order in which the operations are performed. This will be the kernel
of the new linear algebra library that will be developed.

Fast linear algebra packages would also benefit to the transformation of a Gröbner basis of a zero–dimensional ideal with respect to a given monomial ordering into a Gröbner basis with respect to another ordering. In the generic case at least, the change of ordering is equivalent to the computation of the minimal polynomial of a so-called multiplication matrix. By taking into account the sparsity of this matrix, the computation of the Gröbner basis can be done more efficiently using a variant of the Wiedemann algorithm. Hence, our goal is also to obtain a dedicated high performance library for transforming (i.e. change ordering) Gröbner bases.

**Dedicated algebraic tools for Algebraic
Number Theory.** Recent results in Algebraic Number Theory tend to
show that the computation of Gröbner basis is a key step toward
the resolution of difficult problems in this domain
( P. Gaudry, *Index calculus for abelian
varieties of small dimension and the elliptic curve discrete
logarithm problem*, Journal of Symbolic Computation 44,12 (2009)
pp. 1690-1702). Using existing resolution methods is simply not
enough to solve relevant problems. The main algorithmic bottleneck
to overcome is to adapt the Gröbner basis computation step to the
specific problems. Typically, problems coming from Algebraic Number
Theory usually have a lot of symmetries or the input systems are
very structured. This is the case, in particular, for problems coming
from the algorithmic theory of Abelian varieties over finite fields
( e.g. point counting, discrete logarithm, isogeny.)
where the objects are represented by polynomial system and are
endowed with intrinsic group actions. The main goal here is to
provide dedicated algebraic resolution algorithms and
implementations for solving such problems. We do not restrict our
focus on problems in positive characteristic. For instance, tower of
algebraic fields can be viewed as triangular sets; more generally,
related problems (e.g. effective Galois theory) which can be
represented by polynomial systems will receive our attention. This
is motivated by the fact that, for example, computing small integer
solutions of Diophantine polynomial systems in connection with
Coppersmith's method would also gain in efficiency by using a
dedicated Gröbner bases computations step.