OURAGAN proposes to focus on the transfer of computational algebraic methods to some related fields (computational geometry, topology, number theory, etc.) and some carefully chosen application domains (robotics, control theory, evaluation of the security of cryptographic systems, etc.), which implies working equally on the use (modeling, know - how) and on the development of new algorithms. The latest breakthrough developments and applications where algebraic methods are currently decisive remain few and very targeted. We wish to contribute to increase the impact of these methods but also the number of domains where the use of computational algebraic methods represent a significant added value. This transfer-oriented positioning does not imply to stop working on the algorithms, it simply sets the priorities.

An original aspect of the OURAGAN proposal is to blend into an environment of fundamental mathematics, at the Institut de Mathématiques de Jussieu – Paris Rive Gauche (IMJ-PRG CNRS 7586), and to be cross-functional to several teams (Algebraic Analysis, Complex Analysis and Geometry, Number Theory to name only the main ones), which will be our first source of transfer of computational know-how. The success of this coupling allows to maintain a strong theoretical basis and to measure objectively our transfer activity in the direction of mathematicians (in geometry, topology, number theory, algebraic analysis, etc.) and to consolidate the presence of Inria in scientific areas among the most theoretical.

We propose three general directions with five particular targets:

Number Theory

Algorithmic Number Theory

Rigorous Numerical Computations

Topology in small dimension

Character varieties

Knot theory

Computational geometry

Algebraic analysis of functional systems

These actions come, of course, in addition to the study and development of a common set of core elements of

Basic theory and algorithms in algebra and geometry [Transverse activity].

This core activity is the invention and study of fundamental algebraic algorithms and objects that can be grouped into 2 categories: algorithms designed to operate on finite fields and algorithms running on fields of characteristic 0; with 2 types of computational strategies: the exactness and the use of approximate arithmetic (but with exact results). This mix also installs joint studies between the various axes and is an originality of the project-team. For example many kinds of arithmetic tools around algebraic numbers have to face to similar theoretical problems such as finding a good representation for a number field; almost all problems related to the resolution of algebraic systems will reduce to the study of varieties in small dimension and in particular, most of the time, to the effective computation of the topology of curves and surfaces, or the certified drawing of non algebraic function over an algebraic variety.

The tools and objects developed for research on algorithmic number theory as well as in computational geometry apply quite directly on some selected connected challenging subjects:

Security of cryptographic systems

Control theory

Robotics

Signal processing

These applications will serve for the evaluation of the general tools we develop when used in a different context, in particular their capability to tackle state of the art problems.

The basic computable objects and algorithms we study, use, optimize or develop are among the most classical ones in computer algebra and are studied by many people around the world: they mainly focus on basic computer arithmetic, linear algebra, lattices, and both polynomial system and differential system solving.

In the context of OURAGAN, it is important to avoid reinventing the wheel and to re-use wherever possible existing objects and algorithms, not necessarily developed in our team so that the main effort is focused on finding good formulations/modelisations for an efficient use. Also, our approach for the development of basic computable objects and algorithms is *application driven* and follows a simple strategy : use the existing tools in priority, develop missing tools when required and then optimize the critical operations.
First, for some selected problems, we do propose and develop general key algorithms (isolation of real roots of univariate polynomials, parametrisations of solutions of zero-dimensional polynomial systems, solutions of parametric equations, equidimensional decompositions, etc.) in order to complement the existing set computable objects developed and studied around the world (Gröbner bases, resultants , subresultants , critical point methods , etc.) which are also deeply used in our developments. Second, for a selection of well-known problems, we propose different computational strategies (for example the use of approximate arithmetic to speed up LLL algorithm or root isolators, still certifying the final result). Last, we propose specialized variants of known algorithms optimized for a given problem (for example, dedicated solvers for degenerated bivariate polynomials to be used in the computation of the topology of plane curves).

In the activity of OURAGAN, many key objects or algorithms around the resolution of algebraic systems are developed or optimized within the team, such as the resolution of polynomials in one variable with real coefficients , rational parameterizations of solutions of zero-dimensional systems with rational coefficients or discriminant varieties for solving systems depending on parameters , but we are also power users of existing software (mainly Sage

For our studies in number theory and applications to the security of cryptographic systems, our team works on three categories of basic algorithms: discrete logarithm computations (for example to make progress on the computation of class groups in number fields ), network reductions by means of LLL variants and, obviously, various computations in linear algebra, for example dedicated to *almost sparse* matrices .

Finally, for the algorithmic approach to algebraic analysis of functional equations , we developed the effective study of both module theory and homological algebra over certain noncommutative polynomial rings of functional operators , of Stafford's famous theorems on the Weyl algebras , of the equidimensional decomposition of functional systems , etc.

Many frontiers between computable objects, algorithms (above section), computational number theory and applications, especially in cryptography are porous. However, one can classify our work in computational number theory into two classes of studies : computational algebraic number theory and (rigorous) numerical computations in number theory.

Our work on rigorous numerical computations is somehow a transverse activity in Ouragan : floating point arithmetic is used in many basic algorithms we develop (root isolation, LLL) and is thus present in almost all our research directions. However there are specific developments that could be labelized *Number Theory*, in particular contributions to numerical evaluations of *L-functions and Modular Forms Database*

Our work in computational algebraic number theory is driven by the algorithmic improvement to solve presumably hard problems relevant to cryptography. The use of number-theoretic hard problems in cryptography dates back to the invention of public-key cryptography by Diffie and Hellman , where they proposed a first instantiation of their paradigm based on the discrete logarithm problem in prime fields. The invention of RSA , based on the hardness of factoring came as a second example. The introduction of discrete logarithms on elliptic curves only confirmed this trend.

These crypto-systems attracted a lot of interest on the problems of factoring and discrete log. Their study led to the invention of fascinating new algorithms that can solve the problems much faster than initially expected :

the quadratic field for factoring and its variant for discrete log called the Gaussian integers method

Since the invention of NFS in the 90’s, many optimizations of this algorithm have been performed. However, an algorithm with better complexity hasn’t been found for factoring and discrete logarithms in large characteristic.

While factorization and discrete logarithm problems have a long history in cryptography, the recent post-quantum cryptosystems introduce a new variety of presumably hard problems/objects/algorithms with cryptographic relevance: the shortest vector problem (SVP), the closest vector problem (CVP) or the computation of isogenies between elliptic curves, especially in the supersingular case.

Members of OURAGAN started working on the topic of discrete logarithms around 1998, with several computation records that were announced on the *NMBRTHRY* mailing list.
In large characteristic, especially for the case of prime fields, the best current method is the number field sieve (NFS) algorithm. In particular, they published the first NFS based record computation.
Despite huge practical improvements, the prime field case algorithm hasn't really changed since that first record.
Around the same time, we also presented small characteristic computation record based on simplifications of the Function Field Sieve (FFS) algorithm .

In 2006, important changes occurred concerning the FFS and NFS algorithms, indeed, while the algorithms only covered the extreme case of constant characteristic and constant extension degree, two papers extended their ranges of applicability to all finite fields. At the same time, this permitted a big simplification of the FFS, removing the need for function fields.

Starting from 2012, new results appeared in small characteristic. Initially based on a simplification of the 2006 result, they quickly blossomed into the Frobenial representation methods, with quasi-polynomial time complexity , .

An interesting side-effect of this research was the need to revisit the key sizes of pairing-based cryptography. This type of cryptography is also a topic of interest for OURAGAN. In particular, it was introduced in 2000 .

The computations of *class groups in number fields* has strong links with the computations of discrete logarithms or factorizations using the NFS (number field sieve) strategy which as the name suggests is based on the use of number fields. Roughly speaking, the NFS algorithm uses two number fields and the strategy consists in choosing number fields with small sized coefficients in their definition polynomials. On the contrary, in class group computations, there is a single number field, which is clearly a simplification, but this field is given as input by some fixed definition polynomial. Obviously, the degree of this polynomial as well as the size of its coefficients are both influencing the complexity of the computations so that finding other polynomials representing the same class group but with a better characterization (degree or coefficient's sizes) is a mathematical problem with direct practical consequences. We proposed a method to address the problem , but many issues remain open.

Computing generators of principal ideals of cyclotomic fields is also strongly related to the computation of class groups in number fields. Ideals in cyclotomic fields are used in a number of recent public-key cryptosystems. Among the difficult problems that ensure the safety of these systems, there is one that consists in finding a small generator, if it exists, of an ideal. The case of cyclotomic fields is considered .

There is a tradition of using computations and software to study and understand the topology of small dimensional manifolds, going back at least to Thurston's works (and before him, Riley's pioneering work). The underlying philosophy of these tools is to build combinatorial models of manifolds (for example, the torus is often described as a square with an identification of the sides). For dimensions 2, 3 and 4, this approach is relevant and effective. In the team OURAGAN, we focus on the dimension 3, where the manifolds are modelized by a finite number of tetrahedra with identification of the faces. The software SnapPy

This philosophy (modelization of manifolds by quite simple combinatoric models to compute such complicated objects as representations of the fundamental group) was applied in a pioneering work of Falbel when he begins to look for another type of geometry on 3-dimensional manifolds (called CR-spherical geometry). From a computational point of view, this change of objectives was a jump in the unknown: the theoretical justification for the computations were missing, and the number of variables of the systems were multiplied by four. So instead of a relatively small system that could be tackled by Newton methods and numerical approximations, we had to deal with/study (were in front of) relatively big systems (the smallest example being 8 variables of degree 6) with no a priori description of the solutions.

Still, the computable objects that appear from the theoretical study are very often outside the reach of automated computations and are to be handled case by case. A few experts around the world have been tackling this kind of computations (Dunfield, Goerner, Heusener, Porti, Tillman, Zickert) and the main current achievement is the *Ptolemy module*

From these early computational needs, topology in small dimension has historically been the source of collaboration with the IMJ-PRG laboratory.
At the beginning, the goal was essentially to provide computational tools for finding geometric structures in triangulated 3-dimensional varieties.
Triangulated varieties can be topologically encoded by a collection of tetrahedra with gluing constraints (this can be called a triangulation or mesh, but it is not an approximation of the variety by simple structures, rather a combinatorial model).
Imposing a geometric structure on this combinatorial object defines a number of constraints
that we can translate into an algebraic system that we then have to solve to study
geometric structures of the initial variety, for example in
relying on solutions to study representations of the fundamental group of the variety.
For these studies, a large part of the computable objects or algorithms we develop are required, from the algorithms
for univariate polynomials to systems depending on parameters. It should be noted that most of the
computational work lies in the modeling of problems that have strictly no
chance to be solved by blindly running the most powerful black boxes: we usually deal here
with systems that have 24 to 64 variables, depend on 4 to 8 parameters and with degrees
exceeding 10 in each variable. With an ANR *deformation variety*.

Knot theory is a wide area of mathematics. We are interested in polynomial representations of long knots, that is to say polynomial embeddings

A Chebyshev knot , is a polynomial knot parameterized by a Chebyshev curve

Our activity in Knot theory is a bridge between our work in computational geometry (topology and drawing of real space curves) and our work on topology in small dimensions (varieties defined as a knot complement).

Two-bridge knots (or rational knots) are particularly studied because they are much easier to study. The first 26 knots (except

We made use of Chebyshev polynomials so as Fibonacci polynomials which are families of orthogonal polynomials. Considering the Alexander-Conway polynomials as continuant polynomials in the Fibonacci basis, we were able to give a partial answer to Hoste's conjecture on the roots of Alexander polynomials of alternating knots ( ).

We study the lexicographic degree of the two-bridge knots, that is to say the minimal (multi)degree of a polynomial representation of a

The drawing of algebraic curves and surfaces is a critical action in OURAGAN since it is a key ingredient in numerous developments. For example, a certified plot of a discriminant variety could be the only admissible answer that can be proposed for engineering problems that need the resolution of parametric algebraic systems: this variety (and the connected components of its counter part) defines a partition of the parameter’s space in regions above which the solutions are numerically stable and topologically simple. Several directions have been explored since the last century, ranging from pure numerical computations to infallible exact ones, depending on the needs (global topology, local topology, simple drawing, etc.). For plane real algebraic curves, one can mention the cylindrical algebraic decomposition , grids methods (for ex. the marching square algorithm), subdivision methods, etc.

As mentioned above, we focus on curves and surfaces coming from the study of parametric systems. They mostly come from some elimination process, they highly (numerically) unstable (a small deformation of the coefficients might change a lot the topology of the curve) and we are mostly interested in getting qualitative information about their counter part in the parameter's space.

For this work, we are associated with the GAMBLE EPI (Inria Nancy Grand Est) with the aim of developing computational techniques for the study, plotting and topology. In this collaboration, Ouragan focuses on CAD-Like methods while Gamble develops numerical strategies (that could also apply on non algebraic curves). Ouragan's work involves the development of effective methods for the resolution of algebraic systems with 2 or 3 variables , , , which are basic engines for computing the topology , and / or plotting.

Systems of functional equations or simply functional systems are systems whose unknowns are functions, such as systems of ordinary or partial differential equations, of differential time-delay equations, of difference equations, of integro-differential equations, etc.

Numerical aspects of functional systems, especially differential systems, have been widely studied in applied mathematics due to the importance of numerical simulation issues.

Complementary approaches, based on algebraic methods, are usually upstream or help the numerical simulation of systems of functional systems. These methods also tackle a different range of questions and problems such as algebraic preconditioning, elimination and simplification, completion to formal integrability or involution, computation of integrability conditions and compatibility conditions, index reduction, reduction of variables, choice of adapted coordinate systems based on symmetries, computation of first integrals of motion, conservation laws and Lax pairs, Liouville integrability, study of the (asymptotic) behavior of solutions at a singularity, etc. Although not yet very popular in applied mathematics, these theories have lengthy been studied in fundamental mathematics and were developed by Lie, Cartan, Janet, Ritt, Kolchin, Spencer, etc. .

Over the past years, certain of these algebraic approaches to functional systems have been investigated within an algorithmic viewpoint, mostly driven by applications to engineering sciences such as mathematical systems theory and control theory. We have played a role towards these effective developments, especially in the direction of an algorithmic approach to the so-called *algebraic analysis* , , , a mathematical theory developed by the Japanese school of Sato, which studies linear differential systems by means of both algebraic and analytic methods. To develop an effective approach to algebraic analysis, we first have to make algorithmic standard results on rings of functional operators, module theory, homological algebra, algebraic geometry, sheaf theory, category theory, etc., and to implement them in computer algebra systems. Based on elimination theory (Gröbner or Janet bases , , , differential algebra , Spencer's theory , etc.), in , , we have initiated such a computational algebraic analysis approach for general classes of functional systems (and not only for holonomic systems as done in the literature of computer algebra ). Based on the effective aspects to algebraic analysis approach, the parametrizability problem , the reduction and (Serre) decomposition problems , the
equidimensional decomposition , Stafford's famous theorems for the Weyl algebras , etc., have been studied and solutions have been implemented in `Maple`, `Mathematica`, and `GAP` . But these results are only the first steps towards computational algebraic analysis, its implementation in computer algebra systems, and its applications to mathematical systems, control theory, signal processing, mathematical physics, etc.

Outside applications which can clearly be seen as transversal acitivies, our development directions are linked at several levels : shared computable objects, computational strategies and transversal research directions.

**Sharing basic algebraic objects**
As seen above, is the well-known fact that the elimination theory for functional systems is deeply intertwined with the one for polynomial systems so that, topology in small dimension, applications in control theory, signal theory and robotics share naturally a large set of computable objects developped in our project team.

Performing efficient basic arithmetic operations in number fields is also a key ingredient to most of our algorithms, in Number theory as well as in topology in small dimension or , more generally in the use of roots of polynomials systems. In particular, finding good representations of number fields, lead to the same computational problems as working with roots of polynomial systems by means of triangular systems (towers of number fields) or rational parameterizations (unique number field). Making any progress in one direction will probably have direct consequences for almost all the problems we want to tackle.

**Symbolic-numeric strategies**.
Several general low-level tools are also shared such as the use of approximate arithmetic to speed up certified computations. Sometimes these can also lead to improvement for a different purpose (for example computations over the rationals, deeply used in geometry can often be performed in parallel combining computations in finite fields together with fast Chinese remaindering and modular evaluations).

As simple example of this sharing of tools and strategies, the use of approximate arithmetic is common to the work on LLL (used in the evaluation of the security of cryptographic systems), resolutions of real-world algebraic systems (used in our applications in robotics, control theory, and signal theory), computations of signs of trigonometric expressions used in knot theory or to certified evaluations of dilogarithm functions on an algebraic variety for the computation of volumes of representations in our work in topology, numerical integration and computations of

**Transversal research directions**. The study of the topology of complex algebraic curves is central in the computation of periods of algebraic curves (number theory) but also in the study of character varieties (topology in small dimension) as well as in control theory (stability criteria). Very few computational tools exists for that purpose and they mostly convert the problem to the one of variety over the reals (we can then recycle our work in computational geometry).

As for real algebraic curves, finding a way to describe the topology (an equivalent to the graph obtained in the real case) or computing certified drawings (in the case of a complex plane curve, a useful drawing is the so called associated amoeba) are central subjects for Ouragan.

The development of basic computable objects is somehow *on demand* and depends on all the other directions. However, some critical computations are already known to be bottlenecks and are sources of constant efforts.

Computations with algebraic numbers appear in almost all our activities: when working with number fields in our work in algorithmic number theory as well as in all the computations that involve the use of solutions of zero-dimensional systems of polynomial equations. Among the identified problems: finding good representations for single number fields (optimizing the size and degree of the defining polynomials), finding good representations for towers or products of number fields (typically working with a tower or finding a unique good extension), efficiently computing in practice with number fields (using certified approximation vs working with the formal description based on polynomial arithmetics). Strong efforts are currently done in the understanding of the various strategies by means of tight theoretical complexity studies , , and many other efforts will be required to find the right representation for the right problem in practice. For example, for isolating critical points of plane algebraic curves, it is still unclear (at least the theoretical complexity cannot help) that an intermediate formal parameterization is more efficient than a triangular decomposition of the system and it is still unclear that these intermediate computations could be dominated in time by the certified final approximation of the roots.

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 have used them to provide some fast sophisticated factoring algorithms. As a consequence, it is natural to include these objects in our research directions.

Some studies in this area will be driven by some other directions, for example, the rigorous evaluation of non algebraic functions on algebraic varieties might become central for some of our work on topology in small dimension (volumes of varieties, drawing of amoeba) or control theory (approximations of discriminant varieties) are our two main current sources of interesting problems. In the same spirit, the work on

On the other hand, another objective is to extend existing results on periods of algebraic curves to general curves and higher dimensional varieties is a general promising direction. This project aims at providing tools for integration on higher homology groups of algebraic curves, ie computing Gauss-Manin connections. It requires good understanding of their topology, and more algorithmic tools on differential equations.

The brute force approach to computable objects from topology of small dimension will not allow any significant progress. As explained above, the systems that arise from these problems are simply outside the range of doable computations. We still continue the work in this direction by a four-fold approach, with all three directions deeply inter-related. First, we focus on a couple of especially meaningful (for the applications) cases, in particular the 3-dimensional manifold called Whitehead link complement. At this point, we are able to make steps in the computation and describe part of the solutions , ; we hope to be able to complete the computation using every piece of information to simplify the system. Second, we continue the theoretical work to understand more properties of these systems . These properties may prove how useful for the mathematical understanding is the resolution of such systems - or at least the extraction of meaningful information. This approach is for example carried on by Falbel and his work on configuration of flags , . Third, we position ourselves as experts in the know-how of this kind of computations and natural interlocutors for colleagues coming up with a question on such a computable object (see and ). This also allows us to push forward the kind of computation we actually do and make progress in the direction of the second point. We are credible interlocutors because our team has the blend of theoretical knowledge and computational capabilities that grants effective resolutions of the problems we are presented. And last, we use the knowledge already acquired to pursue our theoretical study of the CR-spherical geometry , , .

Another direction of work is the help to the community in experimental mathematics on new objects. It involves downsizing the system we are looking at (for example by going back to systems coming from hyperbolic geometry and not CR-spherical geometry) and get the most out of what we can compute, by studying new objects. An example of this research direction is the work of Guilloux around the volume function on deformation varieties. This is a real-analytic function defined on the varieties we specialized in computing. Being able to do effective computations with this function led first to a conjecture . Then, theoretical discussions around this conjecture led to a paper on a new approach to the Mahler measure of some 2-variables polynomials . In turn, this last paper gave a formula for the Mahler measure in terms of a function akin to the volume function applied at points in an algebraic variety whose moduli of coordinates are 1. The OURAGAN team has the expertise to compute all the objects appearing in this formula, opening the way to another area of application. This area is deeply linked with number theory as well as topology of small dimension. It requires all the tools at disposition within OURAGAN.

We will carry on the exhaustive search for the lexicographic degrees for the rational knots. They correspond to trigonal space curves: computations in the braid group

On the other hand, a natural direction would be: given an explicit polynomial space curve, determine the under/over nature of the crossings when projecting, draw it and determine the known knot

As mentioned above, the drawing of algebraic curves and surfaces is a critical action in OURAGAN since it is a key ingredient in numerous developments. In some cases, one will need a fully certified study of the variety for deciding existence of solutions (for example a region in a robot's parameter's space with solutions to the DKP above or deciding if some variety crosses the unit polydisk for some stability problems in control-theory), in some other cases just a partial but certified approximation of a surface (path planning in robotics, evaluation of non algebraic functions over an algebraic variety for volumes of knot complements in the study of character varieties).

On the one hand, we will contribute to general tools like ISOTOP

A particular effort will be devoted to the resolution of overconstraint bivariate systems which are useful for the studies of singular points and to polynomials systems in 3 variables in the same spirit : avoid the use of Gröbner basis and propose a new algorithm with a state-of-the-art complexity and with a good practical behavior.

In parallel, one will have to carefully study the drawing of graphs of non algebraic functions over algebraic complex surfaces for providing several tools which are useful for mathematicians working on topology in small dimension (a well known example is the drawing of amoebia, a way of representing a complex curve on a sheet of paper).

We want to further develop our expertise in the computational aspects of algebraic analysis by continuing to develop effective versions of results of module theory, homological algebra, category theory and sheaf theory which play important roles in algebraic analysis , , and in the algorithmic study of linear functional systems. In particular, we shall focus on linear systems of integro-differential-constant/varying/distributed delay equations , which play an important role in mathematical systems theory, control theory, and signal processing , , , .

The rings of integro-differential operators are highly more complicated than the purely differential case (i.e. Weyl algebras) , due to the existence of zero-divisors, or the fact of having a coherent ring instead of a noetherian ring . Therefore, we want to develop an algorithmic study of these rings. Following the direction initiated in for the computation of zero divisors (based on the polynomial null spaces of certain operators), we first want to develop algorithms for the computation of left/right kernels and left/right/generalized inverses of matrices with entries in such rings, and to use these results in module theory (e.g. computation of syzygy modules, (shorter/shortest) free resolutions, split short/long exact sequences). Moreover, Stafford's results , algorithmically developed in for rings of partial differential operators (i.e. the Weyl algebras), are known to still hold for rings of integro-differential operators. We shall study their algorithmic extensions. Our corresponding implementation will be extended accordingly.

Finally, within a computer algebra viewpoint, we shall continue to algorithmically study issues on rings of integro-differential-delay operators , and their applications to the study of equivalences of differential constant/varying/distributed delay systems (e.g. Artstein's reduction, Fiagbedzi-Pearson's transformation) which play an important role in control theory.

The study of the security of asymmetric cryptographic systems comes as an application of the work carried out in algorithmic number theory and revolves around the development and the use of a small number of general purpose algorithms (lattice reduction, class groups in number fields, discrete logarithms in finite fields, ...). For example, the computation of generators of principal ideals of cyclotomic fields can be seen as one of these applications since these are used in a number of recent public key cryptosystems.

The cryptographic community is currently very actively assessing the threat coming for the development of quantum computers. Indeed, such computers would permit tremendous progress on many number theoretic problems such as factoring or discrete logarithm computations and would put the security of current cryptosystem under a major risk. For this reason, there is a large global research effort dedicated to finding alternative methods of securing data. For example, the US standardization agency called NIST has recently launched a standardization process around this issue. In this context, OURAGAN is part of the competition and has submitted a candidate (which has not been selected) . This method is based on number-theoretic ideas involving a new presumably difficult problem concerning the Hamming distance of integers modulo large numbers of Mersenne.

Algebraic computations have tremendously been used in Robotics, especially in kinematics, since the last quarter of the 20th century . For example, one can find algebraic proofs for the 40 possible solutions to the direct kinematics problem for steward platforms and companion experiments based on Gröbner basis computations . On the one hand, hard general kinematics problems involve too many variables for pure algebraic methods to be used in place of existing numerical or semi-numerical methods everywhere and everytime, and on the other hand, global algebraic studies allow to propose exhaustive classifications that cannot be reached by other methods,for some quite large classes.

Robotics is a long-standing collaborative work with LS2N (Laboratory of Numerical Sciences of Nantes). Work has recently focused on the offline study of mechanisms, mostly parallel, their singularities or at least some types of singularities (cuspidals robots ).

For most parallel or serial manipulators, pose variables and joints variables are linked by algebraic
equations and thus lie an algebraic variety. The two-kinematics problems (the direct kinematics problem - DKP- and the inverse kinematics problem - IKP) consist in studying the preimage of the projection of this algebraic variety onto a subset of unknowns. Solving the DKP remains to computing the possible positions for a given set of joint variables values while solving the IKP remains to computing the possible joints variables values for a given position. Algebraic methods have been deeply used in several situations for studying parallel and serial mechanisms, but finally their use stays quite confidential in the design process. Cylindrical Algebraic Decomposition coupled with variable's eliminations by means of Gröbner based computations can be used to model the workspace, the joint space and the computation of singularities. On the one hand, such methods suffer immediately when increasing the number of parameters or when working with imprecise data. On the other hand, when the problem can be handled, they might provide full and exhaustive classifications.
The tools we use in that context , , , , depend mainly on the resolution of parameter-based systems and therefore of study-dependent curves or flat algebraic surfaces (2 or 3 parameters), thus joining our thematic *Computational Geometry*.

Certain problems studied in mathematical systems theory and control theory can be better understood and finely studied by means of algebraic structures and methods. Hence, the rich interplay between algebra, computer algebra, and control theory has a long history. For instance, the first main paper on Gröbner bases written by their creators, Buchberger, was published in Bose's book on control theory of multidimensional systems. Moreover, the differential algebra approach to nonlinear control theory (see , and the references therein) was a major motivation for the algorithmic study of differential algebra , . Finally, the behaviour approach to linear systems theory , advocates for an algorithmic study of algebraic analysis (see Section ). More generally, control theory is porous to computer algebra since one finds algebraic criteria of all kinds in the literature even if the control theory community has a very few knowledge in computer algebra.

OURAGAN has a strong interest in the computer algebra aspects of mathematical systems theory and control theory related to both functional and polynomial systems, particularly in the direction of robust stability analysis and robust stabilization problems for multidimensional systems , and infinite-dimensional systems (such as, e.g., differential time-delay systems).

Let us shortly state a few points of our recent interests in this direction.

In control theory, stability analysis of linear time-invariant control systems is based on the famous Routh-Hurwitz criterion (late 19th century) and its relation with Sturm sequences and Cauchy index. Thus, stability tests were only involving tools for univariate polynomials . While extending those tests to multidimensional systems or differential time-delay systems, one had to tackle multivariate problems recursively with respect to the variables . Recent works use a mix of symbolic/numeric strategies, Linear Matrix Inequalities (LMI), sums of squares, etc. But still very few practical experiments are currently involving certified algebraic computations based on general solvers for polynomial equations. We have recently started to study certified stability tests for multidimensional systems or differential time-delay systems with an important observation: with a correct modelization, some recent algebraic methods

The structural stability of

The rich interplay between control theory, algebra, and computer algebra is also well illustrated with our recent work on robust stabilization problems for multidimensional and finite/infinite-dimensional systems , , , , , .

Due to numerous applications (e.g. sensor network, mobile robots), sources and sensors localization has intensively been studied in the literature of signal processing. The *anchor position self calibration problem* is a well-known problem
which consists in estimating the positions of both the moving sources and a set of fixed sensors (anchors) when only the distance information between the points from the different sets is available. The position self-calibration problem is a particular case of the *Multidimensional Unfolding* (MDU) problem for the Euclidean space of dimension 3. In the signal processing literature, this problem is attacked by means of optimization problems (see and the references therein). Based on computer algebra methods for polynomial systems, we have recently developed a new approach for the MDU problem which yields closed-form solutions and a very efficient algorithm for the estimation of the positions based only on linear algebra techniques. This first result, done in collaboration with Dagher (Inria Chile) and Zheng (DEFROST, Inria Lille), yielded a recent patent . This result advocates for the study of other localization problems based on the computational polynomial techniques developed in OURAGAN.

Our expertise on *algebraic parameter estimation problem*, developed in the former Non-A project-team (Inria Lille), will be further developed. Following this work , the problem consists in estimating a set `Maple` prototype `NonA`. The case of a general structured perturbation is still lacking.

Two new projects have started this year

the MACAO associated team in collaboration with the University of Wollongong (Australia) - see

*Topology and geometry of planar algebraic curves*

Keywords: Topology - Curve plotting - Geometric computing

Functional Description: Isotop is a Maple software for computing the topology of an algebraic plane curve, that is, for computing an arrangement of polylines isotopic to the input curve. This problem is a necessary key step for computing arrangements of algebraic curves and has also applications for curve plotting. This software has been developed since 2007 in collaboration with F. Rouillier from Inria Paris - Rocquencourt.

Participants: Luis Penaranda, Marc Pouget and Sylvain Lazard

Contact: Marc Pouget

Publications: Rational Univariate Representations of Bivariate Systems and Applications - Separating Linear Forms for Bivariate Systems - On The Topology of Planar Algebraic Curves - New bivariate system solver and topology of algebraic curves - Improved algorithm for computing separating linear forms for bivariate systems - Solving bivariate systems using Rational Univariate Representations - On the topology of planar algebraic curves - On the topology of real algebraic plane curves - Bivariate triangular decompositions in the presence of asymptotes - Separating linear forms and Rational Univariate Representations of bivariate systems

Functional Description: Real Roots isolation for algebraic systems with rational coefficients with a finite number of Complex Roots

Participant: Fabrice Rouillier

Contact: Fabrice Rouillier

*A New Descartes*

Keyword: Scientific computing

Functional Description: Computations of the real roots of univariate polynomials with rational coefficients.

Authors: Fabrice Rouillier, Alexander Kobel and Michael Sagraloff

Partner: Max Planck Institute for Software Systems

Contact: Fabrice Rouillier

Keywords: Robotics - Kinematics

Functional Description: Library of functions for certified computations of the properties of articulated mechanisms, particularly the study of their singularities

Authors: Damien Chablat, Fabrice Rouillier, Guillaume Moroz and Philippe Wenger

Partner: LS2N

Contact: Guillaume Moroz

Keyword: Arithmetic

Functional Description: MPFI is a C library based on MPFR and GMP for multi precision floating point arithmetic.

Contact: Fabrice Rouillier

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovász reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants—where the rational arithmetic required by Gram–Schmidt orthogonalization is replaced by floating-point arithmetic—are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In , we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields.

Recent progress on NFS imposed a new estimation of the security of pairings. In , we study the best attacks against some of the most popular pairings. It allows us to propose new pairing-friendly curves of 128 bits and 192 bits of security.

The construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since mixed systems of equations arise frequently in many problems. Nevertheless, resultant formulae for mixed polynomial systems is a very challenging problem. In , we present a square, Koszul-type, matrix, the determinant of which is the resultant of an arbitrary (mixed) bivariate tensor-product polynomial system. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the elements of the corresponding matrix are up to sign the coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. In addition we prove an impossibility result which states that for tensor-product systems with more than two (affine) variables there are no universal degree-one formulae, unless the system is unmixed. Last but not least, we present applications of the new construction in the efficient computation of discriminants and mixed discriminants.

Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space

The objective of our Agrement with WATERLOO MAPLE INC. is to promote software developments to which we actively contribute.

On the one hand, WMI provides man power, software licenses, technical support (development, documentation and testing) for an inclusion of our developments in their commercial products. On the other hand, OURAGAN offers perpetual licenses for the use of the concerned source code.

As past results of this agreement one can cite our C-Library *RS* for the computations of the real solutions zero-dimensional systems or also our collaborative development around the Maple package *DV* for solving parametric systems of equations.

For this term, the agreement covers algorithms developed in areas including but not limited to: 1) solving of systems of polynomial equations, 2) validated numerical polynomial root finding, 3) computational geometry, 4) curves and surfaces topology, 5) parametric algebraic systems, 6) cylindrical algebraic decompositions, 7) robotics applications.

In particular, it covers our collaborative work with some of our partners, especially the Gamble Project-Team - Inria Nancy Grand Est.

In 2019, a contract was signed with the company *Safran Tech*. Its goal is to bring our scientific expertise on mathematical and algorithmic aspects on certain problems studied in gearbox vibration analysis. Gear fault diagnosis is
an important issue in aeronautics industry since a damage in a
gearbox, which is not detected in time, can have dramatic effects on
the safety of a plane. Since the vibrations of a spur gear can be modeled as a product of two periodic functions related to the gearbox kinematic,
has proposed to recover each function from the
global signal by means of an optimal reconstruction problem which, by
means of Fourier analysis, yields a Frobenius norm
minimization problem for structured matrices. The goal of the collaboration is to use symbolic-numeric to study this problem.

FMJH Program, PGMO grant

ALMA (Algebraic methods in games and optimization).

Duration: 2018 – 2020. (2 years project)

Coordinator: Elias Tsigaridas, with Stéphane Gaubert and Xavier Allamigeon (CMAP, École Polytechnique)

ANR JCJC GALOP (Games through the lens of ALgebra and OPptimization)

Coordinator: Elias Tsigaridas

Duration: 2018 – 2022

GALOP is a Young Researchers (JCJC) project with the purpose of extending the limits of the state- of-the-art algebraic tools in computer science, especially in stochastic games. It brings original and innovative algebraic tools, based on symbolic-numeric computing, that exploit the geometry and the structure and complement the state-of-the-art. We support our theoretical tools with a highly efficient open-source software for solving polynomials. Using our algebraic tools we study the geometry of the central curve of (semi-definite) optimization problems. The algebraic tools and our results from the geometry of optimization pave the way to introduce algorithms and precise bounds for stochastic games.

Program: H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)

Project acronym: Almacrypt

Project title: Algorithmic and Mathematical Cryptology

Duration: 01/2016 - 12/2010

Coordinator: Antoine Joux

Abstract: Cryptology is a foundation of information security in the digital world. Today's internet is protected by a form of cryptography based on complexity theoretic hardness assumptions. Ideally, they should be strong to ensure security and versatile to offer a wide range of functionalities and allow efficient implementations. However, these assumptions are largely untested and internet security could be built on sand. The main ambition of Almacrypt is to remedy this issue by challenging the assumptions through an advanced algorithmic analysis. In particular, this proposal questions the two pillars of public-key encryption: factoring and discrete logarithms. Recently, the PI contributed to show that in some cases, the discrete logarithm problem is considerably weaker than previously assumed. A main objective is to ponder the security of other cases of the discrete logarithm problem, including elliptic curves, and of factoring. We will study the generalization of the recent techniques and search for new algorithmic options with comparable or better efficiency. We will also study hardness assumptions based on codes and subset-sum, two candidates for post-quantum cryptography. We will consider the applicability of recent algorithmic and mathematical techniques to the resolution of the corresponding putative hard problems, refine the analysis of the algorithms and design new algorithm tools. Cryptology is not limited to the above assumptions: other hard problems have been proposed to aim at post-quantum security and/or to offer extra functionalities. Should the security of these other assumptions become critical, they would be added to Almacrypt's scope. They could also serve to demonstrate other applications of our algorithmic progress. In addition to its scientific goal, Almacrypt also aims at seeding a strengthened research community dedicated to algorithmic and mathematical cryptology.

Partenariat Hubert Curien franco-turc (PHC Bosphore) with Gebze Technical University, Turkey.

Title: "Gröbner bases, ResultAnts and Polyhedral gEometry” (GRAPE)

Duration: 2019 – 2020 (2 years project)

Coordinator: Elias Tsigaridas

Title: Mathematics and Algorithms for Cryptographic Advanced Objects

International Partner (Institution - Laboratory - Researcher):

University of Wollongong (Australia) - Thomas Plantard

Start year: 2019

Since quantum computers have the ability to break the two main problems on which current public cryptography relies, i.e., the factoring and discrete logarithm problem, every step towards the practical realization of these computers raises fears about potential attacks on cryptographic systems. By scrutinizing the techniques proposed to build post-quantum cryptography, we can identify a few candidate hard problems which underly the proposals. One objective of this international project is to precisely assess the security of these cryptographic algorithms. First, by analyzing in a systematic manner the existing resolution algorithms and by assessing their complexity as a function of security parameters. Then, we will consider new algorithmic techniques to solve these candidate hard Post-Quantum problems, both on classical computers and quantum machines aiming at the discovery of new and better algorithms to solve them.

University of Wollongong (Australia)

CQT Singapour (UMI CNRS Majulab)

UFPA - Para -Brésil (José Miguel Veloso)

Institut Joseph Fourier - Université Grenoble Alpes (Martin Deraux, V. Vitse et Pierre Will)

Max-Planck-Institut für Informatik - Saarbrücken - Germany (Alex. Kobel)

Holon Institute of Technology, Israel (Jeremy Kaminsky)

Department of Informatics, National Kapodistrian University of Athens, Greece (Ioannis Emiris)

Elias Tsigaridas was the program chair of Mathematical Aspects of Computer and Information Sciences (MACIS 2019), which was held at Istanbul, 13-15 November 2019.

A. Quadrat was a member of the organization committee of the
*Journées Nationales de Calcul Formel* (JNCF), Luminy,
France, 04-08/02/2019. He is now a member of the scientific committee of JNCF.

Elias Tsigaridas was a program committee member of Computer Algebra in Scientific Computing (CASC 2019).

Elisha Falbel is a member of the editorial board of *São Paulo Journal of Mathematical Sciences - Springer*

Antoine Joux is a member of the editorial board of *Designs, Codes and Cryptography*

Alban Quadrat is associate editor of *Multidimensional Systems and
Signal Processing*, Springer

Fabrice Rouillier is a member of the editorial board of *Journal of Symbolic Computation*

Alban Quadrat co-organized the invited sessions
*Algebraic and symbolic methods for mathematical systems theory* and *New trends in computational methods for time-delay systems* at the Join
*Joint IFAC Conference 7th Symposium on Systems Structure and Control and 15th Workshop on Time Delay Systems*, 9–11 September 2019, Sinaia, Romania

Elias Tsigaridas organized
a session on *Algebraic and geometric tools for optimisation and statistics* during the PGMO days 2019,
at the EDF’Lab Palaiseau, December 2019.

Alban Quadrat is a member of the Conseil d'Administration of the Société Mathématique de France (SMF).

Fabrice Rouillier is a member of the scientific commitee of the Indo French Centre for Applied Mathematics.

Elisha Falbel is director of the "École Doctorale Sciences Mathématiques de Paris Centre - ED 386".

Fabrice Rouillier: Course in Algebraic Computations, M1, 24h, Sorbonne Université

Fabrice Rouillier: Course in "Agrégation Option - C", M2, 31 heures, Sorbonne Université

Antonin Guilloux: Courses and administration of the general math courses in L1 at Sorbonne Université

Elisha Falbel: First year 'mathématiques approfondie' and master class on Riemann surfaces.

Pierre-Vincent Koseleff : head of the Master 2 “pr éparation à l'agrégation de mathématiques” at Sorbonne Université.

Pierre-Vincent Koseleff : Course in M2 "Agrégation de Mathématiques", Computer Algebra, Option C, 100h. Sorbonne Université.

Pierre-Vincent Koseleff : Course in L3, "algèbre appliquée".

PhD in progress: Thomas Espitau, 09/2016, directed by Antoine Joux

PhD in progress: Natalia Kharchenko, 09/2016, directed by Antoine Joux

PhD in progress: Mahya Mehrabdollahei, 09/2018, directed by Antonin Guilloux and Fabrice Rouillier

PhD in progress: Grace Younes, 09/2018, directed by Alban Quadrat and Fabrice Rouillier

PhD in progress: Christina Katsamaki, 09/2019, directed by Elias Tsigaridas and Fabrice Rouillier

PhD in progress: Sudarshan Shinde, 09/2016, directed by Razvan Barbulescu and Pierre-Vincent Koseleff

PhD in progress: Raphael Alexandre, 09/2019, directed by Elisha Falbel.

Alban Quadrat was a member of the PhD defense committee of Elisa Hubert, *Surveillance vibratoire d'une transmission de puissance aéronautique*, University of Lyon & Safran Tech, 28/06/2019.

Elisha Falbel was a member of the Habilitation defense committee of Junyan Cao, *Positivité des images directees, extension d'Ohsawa-Takegoshi et applications*, Sorbonne Université , 25/10/2019.

Alban Quadrat co-edited the book
*Le Jeu de rôle sur table : un laboratoire de l'imaginaire*, Carrefour des lettres modernes, n. 7, Classiques Garnier, 2019.

Fabrice Rouillier is a member of the editorial board of *Interstices*

Fabrice Rouillier is *chargé de mission médiation* at Inria Paris

Fabrice Rouillier is a member of the *comité de pilotage de l'année des mathématiques*

Fabrice Rouillier is a member of the *comité de pilotage de la semaine des mathématiques*

Fabrice Rouillier is the president of the association *Animath*

Fabrice Rouillier is member of the scientific board of the SMF *Mathématiques étonnantes*

Alban Quadrat is a member of the technical committee *Linear Systems* of the *International Federation of Automatic Control* (IFAC)

Elias Tsigaridas is a member of the Evaluation committee (Commission d'Évaluation) of Inria.

Fabrice Rouillier participated to the *semaine des mathématiques* for the Paris Inria Paris research center

Fabrice Rouillier participated to the welcoming of schoolchildren for their *semaine d'observation* at Inria Paris research center.