- A4.3. Cryptography
- A4.3.1. Public key cryptography
- A4.3.2. Secret key cryptography
- A4.3.3. Cryptographic protocols
- A4.3.4. Quantum Cryptography
- A7.1. Algorithms
- A7.1.4. Quantum algorithms
- A8.1. Discrete mathematics, combinatorics
- A8.3. Geometry, Topology
- A8.4. Computer Algebra
- A8.5. Number theory
- A8.10. Computer arithmetic
- B5.6. Robotic systems
- B9.5.1. Computer science
- B9.5.2. Mathematics
1 Team members, visitors, external collaborators
- Fabrice Rouillier [Team leader, Inria, Senior Researcher, HDR]
- Yacine Bouzidi [Inria, Starting Research Position, until Jun 2020]
- Alban Quadrat [Inria, Senior Researcher, HDR]
- Elias Tsigaridas [Inria, Researcher]
- Jean Bajard [Sorbonne Université, Professor]
- Elisha Falbel [Sorbonne Université, Professor, HDR]
- Antonin Guilloux [Sorbonne Université, Associate Professor, HDR]
- Antoine Joux [Université Pierre et Marie Curie, Associate Professor, HDR]
- Pierre-Vincent Koseleff [Sorbonne Université, Associate Professor, HDR]
- Pascal Molin [Université de Paris, Associate Professor]
- Josue Tonelli Cueto [Inria, from Feb 2020]
- Raphael Alexandre [Sorbonne Université]
- Thibauld Feneuil [Crypto Experts, CIFRE, from Oct 2020]
- Christina Katsamaki [Inria]
- Mahya Mehrabdollahei [Inria]
- Grace Younes [Inria]
Interns and Apprentices
- Valerian Hatey [Inria, from Apr 2020 until Jul 2020]
- Laurence Bourcier [Inria]
- Julien Guieu [Inria]
2 Overall objectives
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
- 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.
2.1 Scientific ground
2.1.1 Basic computable objects and algorithms
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 75, subresultants 96, critical point methods 52, 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 11514, rational parameterizations of solutions of zero-dimensional systems with rational coefficients 6013 or discriminant varieties for solving systems depending on parameters 11, but we are also power users of existing software (mainly Sage 1, Maple 2, Pari-GP 3,Snappea 4) and libraries (mainly gmp 5, mpfr 6, flint 7, arb 8, etc.) to which we contribute when it makes sense.
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 110 (for example to make progress on the computation of class groups in number fields 97), network reductions by means of LLL variants 86 and, obviously, various computations in linear algebra, for example dedicated to almost sparse matrices 111.
Finally, for the algorithmic approach to algebraic analysis of functional equations 56113114, we developed the effective study of both module theory and homological algebra 147 over certain noncommutative polynomial rings of functional operators 4, of Stafford's famous theorems on the Weyl algebras 137, of the equidimensional decomposition of functional systems 134, etc.
2.1.2 Computational Number Theory
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 -functions which are deeply used in many problems in number theory (for example the Riemann Zeta function). We participate, for example to the L-functions and Modular Forms Database9 a world wide collaborative project.
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 82, where they proposed a first instantiation of their paradigm based on the discrete logarithm problem in prime fields. The invention of RSA 145, based on the hardness of factoring came as a second example. The introduction of discrete logarithms on elliptic curves 116149 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 elliptic curve method (ECM) 127
- the quadratic field for factoring 131 and its variant for discrete log called the Gaussian integers method 124
- the number field sieve (NFS) 126
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 computation10. 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 109.
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 110, 98.
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 9.
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 97, 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 55.
2.1.3 Topology in small dimension
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 10 implements this strategy 151 and is regularly used as a starting point in our work. Along the same philosophy of implementation, we can also cite Regina 11. A specific trait of SnapPy is that it focuses on hyperbolic structures on the 3-dimensional manifolds. This setting is the object of a huge amount of theoretical work that were used to speed up computations. For example, some Newton methods were implemented without certification for solving a system of equations, but the theoretical knowledge of the uniqueness of the solution made this implementation efficient enough for the target applications. In recent years, in part under the influence of our team 12, more attention has been given to certified computations (at least with an error control) and now this is implemented in 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 8 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 module13 for SnapPy.
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 547 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 14 funding on the subject, the progress that we did 90 were (much) more significant than expected. In particular, we have introduced new computable objects with an immediate theoretical meaning (let us say rather with a theoretical link established with the usual objects of the domain), namely, the so-called 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 . Every knot admits a polynomial representation and a natural question is to determine explicit parameterizations, minimal degree parameterizations. On the other hand we are interested to determine what is the knot of a given polynomial smooth embedding . These questions involve real algebraic curves. This subject was first considered by Vassiliev in the 90's 150.
A Chebyshev knot 118, is a polynomial knot parameterized by a Chebyshev curve where is the -th Chebyshev polynomial of the first kind. Chebyshev knots are polynomial analogues of Lissajous knots that have been studied by Jones, Hoste, Lamm... It was first established that any knot can be parameterized by Chebyshev polynomials, then we have studied the properties of harmonic nodes 119 which then opened the way to effective computations.
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 ) are two-bridge knots. We were able to give an exhaustive, minimal and certified list of Chebyshev parameterizations of the first rational two-bridge knots, using blind computations 121. On the other hand, we propose the identification of Chebyshev knot diagrams 122 by developing new certified algorithms for computing trigonometric expressions 123. These works share many tools with our action in visualization and computational geometry.
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 ( 120).
We study the lexicographic degree of the two-bridge knots, that is to say the minimal (multi)degree of a polynomial representation of a -crossing two-bridge knot. We show that this degree is with . We have determined the lexicographic degree of the first 362 first two-bridge knots with 12 crossings or fewer 1715. These results make use of the braid theoretical approach developped by Y. Orevkov to study real plane curves and the use of real pseudoholomorphic curves 67, the slide isotopies on trigonal diagrams, namely those that never increase the number of crossings 69.
Visualization and Computational Geometry
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 74, 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 60, 115, 61, 62 which are basic engines for computing the topology 129, 81 and / or plotting.
2.1.4 Algebraic analysis of functional systems
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. 105113114117144132.
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 analysis113, 114, 56, 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 105, 73, 146, differential algebra 5887, Spencer's theory 132, etc.), in 4, 5, 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 73). Based on the effective aspects to algebraic analysis approach, the parametrizability problem 4, the reduction and (Serre) decomposition problems 5, the equidimensional decomposition 134, Stafford's famous theorems for the Weyl algebras 137, etc., have been studied and solutions have been implemented in Maple, Mathematica, and GAP725. 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 -functions.
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.
As mentioned in the section 3.3 the computation of the Mahler measure of an algebraic implicit curve is either a challenging problem in number theory and a new direction in topology. The basic formula requires the study of points of moduli 1 , as for stability problems in Control Theory (stability problems), and certified numerical evaluations of non algebraic functions at algebraic points as for many computations for -Functions.
3 Research program
3.1 Basic computable objects and algorithms
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 81, 125, 61 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.
3.2 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 85 have used them to provide some fast sophisticated factoring algorithms. As a consequence, it is natural to include these objects in our research directions.
Rigorous numerical computations
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 -functions computations (extending the computation range, algorithmic tools for computing algebraic data from the function) will naturally follow.
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.
3.3 Topology in small dimension
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 90, 102; 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 88. 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 91, 93. 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 99 and 102). 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 80, 92, 89.
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 101. Then, theoretical discussions around this conjecture led to a paper on a new approach to the Mahler measure of some 2-variables polynomials 100. 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 , explicit parametrization of trigonal curves corresponding to "dessins d'enfants", etc. The problem seems much more harder when looking for more general knots.
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 16 it is isotopic to.
Vizualisation and Computational Geometry
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 17 under the supervision of the GAMBLE project-team and, on the other hand, we will propose ad-hoc solutions by gluing some of our basic tools (problems of high degrees in robust control theory). The priority is to provide a first software that implements methods that fit as most as possible the very last complexity results we got on several (theoretical) algorithms for the computation of the topology of plane curves.
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).
3.4 Algebraic analysis of functional systems
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 147 which play important roles in algebraic analysis 56, 113, 114 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 133, 136 which play an important role in mathematical systems theory, control theory, and signal processing 133, 142, 138, 139.
The rings of integro-differential operators are highly more complicated than the purely differential case (i.e. Weyl algebras) 12, due to the existence of zero-divisors, or the fact of having a coherent ring instead of a noetherian ring 53. Therefore, we want to develop an algorithmic study of these rings. Following the direction initiated in 136 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 148, algorithmically developed in 12 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 133, 138 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.
4 Application domains
4.1 Security of cryptographic systems
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) 51. 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 104. For example, one can find algebraic proofs for the 40 possible solutions to the direct kinematics problem 128 for steward platforms and companion experiments based on Gröbner basis computations 94. 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 152).
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 71, 70, 106, 108, 107 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.
4.3 Control theory
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 57 on control theory of multidimensional systems. Moreover, the differential algebra approach to nonlinear control theory (see 84, 83 and the references therein) was a major motivation for the algorithmic study of differential algebra 58, 87. Finally, the behaviour approach to linear systems theory 153, 130 advocates for an algorithmic study of algebraic analysis (see Section 2.1.4). 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 57, 130 and infinite-dimensional systems 77 (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 112. While extending those tests to multidimensional systems or differential time-delay systems, one had to tackle multivariate problems recursively with respect to the variables 57. 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 derived from our work in algorithmic geometry and shared with applications in robotics can now handle previously impossible computations and lead to a better understanding of the problems to be solved 63, 64, 66. The previous approaches seem to be blocked on a recursive use of one-variable methods, whereas our approach involves the direct processing of the problem for a larger number of variables.
The structural stability of -D discrete linear systems (with ) is a good source of problems of several kinds ranging from solving univariate polynomials to studying algebraic systems depending on parameters. For instance, we show 65, 64, 66 that the standard characterization of the structural stability of a multivariate rational transfer function (namely, the denominator of the transfer function does not have solutions in the unit polydisc of ) is equivalent to deciding whether or not a certain system of polynomial equations has real solutions. The use state-of-the-art computer algebra algorithms to check this last condition, and thus the structural stability of multidimensional systems has been validated in several situations from toy examples with parameters to state-of-the-art examples involving, e.g., the resolution of bivariate systems 62, 61.
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 59, 135, 140, 143, 141, 142.
4.4 Signal processing
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 76 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 78 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 79. This result advocates for the study of other localization problems based on the computational polynomial techniques developed in OURAGAN.
In collaboration with Safran Tech (Barau, Hubert) and Dagher (Inria Chile), a symbolic-numeric study of the new multi-carrier demodulation method103 has recently been initiated. 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, it is proposed to recover each function from the global signal by means of an optimal reconstruction problem which, based on Fourier analysis, can be rewritten as , where (resp. ) is a given matrix with a special shape (resp. diagonal matrix), is the Frobenius norm, and is the Hermitian transpose of . We have recently obtained closed-form solutions for the exact problem, i.e., , which is a polynomial system with parameters. This first result gives interesting new insides for the study of the non-exact case, i.e. for the above optimization problem.
Our expertise on algebraic parameter estimation problem, developed in the former Non-A project-team (Inria Lille), will be further developed. Following this work 95, the problem consists in estimating a set of parameters of a signal which satisfies a certain dynamics when the signal is observed, where denotes a structured perturbation and a noise. It has been shown that can sometimes be explicitly determined by means of closed-form expressions using iterated integrals of . These integrals are used to filter the noise . Based on a combination of algebraic analysis techniques (rings of differential operators), differential elimination theory (Gröbner basis techniques for Weyl algebras), and operational calculus (Laplace transform, convolution), an algorithmic approach to algebraic parameter estimation problem has been initiated in 138 for a particular type of structured perturbations (i.e. bias) and was implemented in the Maple prototype NonA. The case of a general structured perturbation is still lacking.
5 Highlights of the year
We participated the INRIA COVID-19 Mission with the project Parlons Maths;
Our contract with Maple has been renewed for two years;
One contract and one NDA have been signed with Safran Defense.
Sudarshan Shinde defended his PhD 39 (directed by Pierre-Vincent Koseleff and Razvan Barbulescu);
A new CIFRE grant with Crypto experts started in October. Thibauld Feneuil has joined the team.
The the book 38 has been published.
6 New software and platforms
6.1 New software
- Name: 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.
isotop. gamble. loria. fr/
- Publications: hal-00809430, hal-00809425, inria-00329754, inria-00580431, hal-00992634, hal-01342211, inria-00425383, inria-00517175, hal-01468796, hal-00977671
- Authors: Luis Penaranda, Marc Pouget, Sylvain Lazard
- Contacts: Sylvain Lazard, Marc Pouget
- Participants: Luis Penaranda, Marc Pouget, Sylvain Lazard
- Functional Description: Real Roots isolation for algebraic systems with rational coefficients with a finite number of Complex Roots
team. inria. fr/ ouragan/ software/
- Author: Fabrice Rouillier
- Contact: Fabrice Rouillier
- Participant: Fabrice Rouillier
6.1.3 A NewDsc
- Name: A New Descartes
- Keyword: Scientific computing
- Functional Description: Computations of the real roots of univariate polynomials with rational coefficients.
anewdsc. mpi-inf. mpg. de
- Authors: Fabrice Rouillier, Alexander Kobel, Michael Sagraloff
- Contact: Fabrice Rouillier
- Partner: Max Planck Institute for Software Systems
- Keywords: Robotics, Kinematics
- Functional Description: Library of functions for certified computations of the properties of articulated mechanisms, particularly the study of their singularities
siropa. gforge. inria. fr/
- Authors: Damien Chablat, Fabrice Rouillier, Guillaume Moroz, Philippe Wenger
- Contacts: Fabrice Rouillier, Guillaume Moroz
- Partner: LS2N
- Keyword: Arithmetic
- Functional Description: MPFI is a C library based on MPFR and GMP for multi precision floating point arithmetic.
mpfi. gforge. inria. fr
- Contacts: Fabrice Rouillier, Nathalie Revol
7 New results
7.1 Number Theory
7.1.1 Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's p-rationality conjecture
In 16, we make a series of numerical experiments to support Greenberg's p-rationality conjecture, we present a family of p-rational biquadratic fields and we find new examples of p-rational multiquadratic fields. In the case of multiquadratic and multicubic fields we show that the conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the conjecture of Hofmann and Zhang on the p-adic regulator, and we bring new numerical data to support the extensions of these conjectures. We compare the known algorithmic tools and propose some improvements.
7.1.2 An asymptotically faster version of FV supported on HPR
State-of-the-art implementations of homomorphic encryption exploit the Fan and Vercauteren (FV) scheme and the Residue Number System (RNS). While the RNS breaks down large integer arithmetic into smaller independent channels, its non-positional nature makes operations such as division and rounding hard to implement, and makes the representation of small values inefficient. In 25, we propose the application of the Hybrid Position-Residues Number System representation to the FV scheme. This is a positional representation of large radix where the digits are represented in RNS. It inherits the benefits from RNS and allows to accelerate the critical division and rounding operations while also making the representation of smaller values more compact. This directly benefits the decryp-tion and the homomorphic multiplication procedures, reducing their asymptotic complexity, in dimension n, from to and from to , respectively and has resulted in noticeable speedups when experimentally compared to related art RNS implementations.
7.2 Computer Algebra
7.2.1 Multilinear Polynomial Systems: Root Isolation and Bit Complexity
In 21, we exploit structure in polynomial system solving by considering polyno-mials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded) resultant of such systems as a determinant of Sylvester-like matrices, aka generalized Sylvester matrices. We construct these matrices by means of Weyman homological complexes, which generalize the Cayley-Koszul complex. The computation of the determinant of the resultant matrix is the bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector multiplication, which corresponds to multivariate polynomial multiplication, by extending the seminal work on Macaulay matrices of Canny, Kaltofen, and Yagati to the multi-homogeneous case. We compute a rational univariate representation of the roots, based on the primitive element method. In the case of 0-dimensional systems we present a Monte Carlo algorithm with probability of success , for a given , and bit complexity for any , where is the number of variables, equals the multilinear Bézout bound, is the number of variable subsets, and is the maximum coefficient bitsize. We present an algorithmic variant to compute the isolated roots of overdetermined and positive-dimensional systems. Thus our algorithms and complexity analysis apply in general with no assumptions on the input.
7.2.2 Separation bounds for polynomial systems
In 22, we rely on aggregate separation bounds for univariate polynomials to introduce novel worst-case separation bounds for the isolated roots of zero-dimensional, positive-dimensional, and overde- termined polynomial systems. We exploit the structure of the given system, as well as bounds on the height of the sparse (or toric) resultant, by means of mixed volume, thus establishing adaptive bounds. Our bounds improve upon Canny’s Gap theorem . Moreover, they exploit sparseness and they apply without any assumptions on the input polynomial system. To evaluate the quality of the bounds, we present polynomial systems whose root separation is asymptotically not far from our bounds. We apply our bounds to three problems. First, we use them to estimate the bitsize of the eigenvalues and eigenvectors of an integer matrix; thus we provide a new proof that the problem has polynomial bit complexity. Second, we bound the value of a positive polynomial over the simplex: we improve by at least one order of magnitude upon all existing bounds. Finally, we asymptotically bound the number of steps of any purely subdivision-based algorithm that isolates all real roots of a polynomial system.
7.2.3 Matrix formulae for Resultants and Discriminants of Bivariate Tensor-product Polynomials
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 19, 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.
7.2.4 Certified lattice reduction
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 23, 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.
7.3 Algebraic Analysis
7.3.1 Computing polynomial solutions and annihilators of integro-differential operators with polynomial coefficients
In 37, we study algorithmic aspects of the algebra of linear ordinary integro-differential operators with polynomial coefficients. Even though this algebra is not Noetherian and has zero divisors, Bavula recently proved that it is coherent, which allows one to develop an algebraic systems theory over this algebra. For an algorithmic approach to linear systems of integro-differential equations with boundary conditions, computing the kernel of matrices with entries in this algebra is a fundamental task. As a first step, we have to find annihilators of integro-differential operators, which, in turn, is related to the computation of polynomial solutions of such operators. For a class of linear operators including integro-differential operators, we present an algorithmic approach for computing polynomial solutions and the index. A generating set for right annihilators can be constructed in terms of such polynomial solutions. For initial value problems, an involution of the algebra of integro-differential operators then allows us to compute left annihilators, which can be interpreted as compatibility conditions of integro-differential equations with boundary conditions. We illustrate our approach using an implementation in the computer algebra system Maple.
7.3.2 Equivalences of linear functional systems
In 36, within the algebraic analysis approach to linear systems theory, we investigate the equivalence problem of linear functional systems, i.e., the problem of characterizing when all the solutions of two linear functional systems are in a one-to- one correspondence. To do that, we first provide a new characterization of isomorphic finitely presented modules in terms of inflation of their presentation matrices. We then prove several isomorphisms which are consequences of the unimodular completion problem. We then use these isomorphisms to complete and refine existing results concerning Serre’s reduction problem. Finally, different consequences of these results are given. All the results obtained here are algorithmic for rings for which Gröbner basis techniques exist and the computations can be performed by the Maple packages OreModules and OreMorphisms or the Mathematica package OreAlgebraicAnalysis.
7.3.3 Effective algebraic analysis approach to linear systems over Ore algebras
The purpose of 35 is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra – based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras – and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OreAlgebraicAnalysis, based on the Mathematica package HolonomicFunctions, is demonstrated.
7.4.1 Computing the Homology of Semialgebraic Sets. II: General formulas.
In 18, we describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in Part I to arbitrary semialgebraic sets. All previous algorithms proposed for this problem have doubly exponential complexity.
7.4.2 Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces
The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. In 30, initiate a condition-based complexity framework based on the norm of the cube, that is a step in this direction. We present this framework for real hypersurfaces. We demonstrate its capabilities by providing a new probabilistic complexity analysis for the Plantinga-Vegter algorithm, which covers both random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. We present explicit results with structured random polynomials for problems with two or more dimensions. Additionally, we provide some estimates of the separation bound of a univariate polynomial in our current framework.
7.4.3 Computing the topology of a plane or space hyperelliptic curve
In 15,We present algorithms to compute the topology of 2D and 3D hyperelliptic curves. The algorithms are based on the fact that 2D and 3D hyperelliptic curves can be seen as the image of a planar curve (the Weierstrass form of the curve), whose topology is easy to compute, under a birational mapping of the plane or the space. We report on a Maple implementation of these algorithms, and present several examples. Complexity and certification issues are also discussed.
7.4.4 PTOPO: A Maple package for the topology of parametric curves
In 24, we present PTOPO, a MAPLE package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second for most examples in the literature.
7.4.5 On the Geometry and the Topology of Parametric Curves
In 31, we consider the problem of computing the topology and describing the geometry of a parametric curve in . We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most and maximum bitsize of coefficients , then the worst case bit complexity of PTOPO is . This bound matches the current record bound for the problem of computing the topology of a planar algebraic curve given in implicit form. For planar and space curves, if , the complexity of PTOPO becomes , which improves the state-of-the-art result, due to Alcázar and Díaz-Toca, by a factor of . However, visualizing the curve on top of the abstract graph construction, increases the bound to . We have implemented PTOPO in maple for the case of planar curves. Our experiments illustrate its practical nature.
7.4.6 Sampling the feasible sets of SDPs and volume approximation
In 20, we present algorithmic, complexity, and implementation results on the problem of sampling points in the interior and the boundary of a spectrahedron, that is the feasible region of a semidefinite program. Our main tool is random walks. We define and analyze a set of primitive geometric operations that exploits the algebraic properties of spectrahedra and the polynomial eigenvalue problem, and leads to the realization of a broad collection of efficient random walks. We demonstrate random walks that experimentally show faster mixing time than the ones used previously for sampling from spectrahedra in theory or applications, for example Hit and Run. Consecutively, the variety of random walks allows us to sample from general probability distributions, for example the family of log-concave distributions which arise frequently in numerous applications. We apply our tools to compute (i) the volume of a spectrahedron and (ii) the expectation of functions coming from robust optimal control. We provide a C++ open source implementation of our methods that scales efficiently up to to dimension 200. We illustrate its efficiency on various data sets.
7.4.7 The lexicographic degree of the first two-bridge knots
In 17 study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane curves, and in particular to use the braid theoretical method developed by Orevkov.
7.5 Signal Processing
7.5.1 Algebraic aspects of a rank factorization problem arising in vibration analysis
The article 28 continues the study of a factorization problem arising in gear fault surveillance. The structure of a class of solutions – interesting in practice – of this factorization problem is studied. We show that these solutions can be parametrized. The parameter space P is proved to be the complementary of an algebraic set that is explicitly characterized based on module theory and computer algebra. A finite open cover of P is obtained and for each basic open subset of the cover, a closed-form solution is computed using computer algebra. Hence, the local structure of the solution space can be finely studied. Finally, we show that the existence of a single closed-form solution defined on the whole parameter space P is related to difficult problems in module theory.
7.5.2 On a rank factorisation problem arising in gearbox vibration analysis
Given a field , r matrices , a matrix of rank at most , in 29, we study the problem of factoring as follows , where and , for . This problem arises in modulation-based mechanical models studied in gearbox vibration analysis (e.g., amplitude and phase modulation). We show how linear algebra methods combined with linear system theory ideas can be used to characterize when this polynomial problem is solvable and if so, how to explicitly compute the solutions.
7.6 Control Theory
7.6.1 Computation of the L-Infinity norm of finite-dimensional linear systems
In 26, we study the computation of the -norm for finite-dimensional linear systems. This problem is first reduced to the computation of the maximal -projection of the real solutions of a bivariate polynomial system . We then apply computer algebra methods to solve the problem. We alternatively study a method based on rational univariate representations, a method based on root separation and finally a method based on the sign variation of the leading coefficients of the signed subresultant sequence and on the identification of an isolating interval for the maximal -projection of the real solutions of the system.
7.6.2 On the effective computation of stabilizing controllers of 2D systems
In 32, we show how stabilizing controllers for 2D systems can effectively be computed based on computer algebra methods dedicated to polynomial systems, module theory and homological algebra. The complete chain of algorithms for the computation of stabilizing controllers, implemented in Maple, is illustrated with an explicit example.
7.6.3 Symbolic Methods for Solving Algebraic Systems of Equations and Applications for Testing the Structural Stability
In 33, we provide an overview of the classical symbolic techniques for solving algebraic systems of equations and show the interest of such techniques in the study of some problems in dynamical system theory, namely testing the structural stability of multidimensional systems.
7.7.1 TRPLP – Trifocal Relative Pose From Lines at Points
In 27, we present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Grobner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver, which dramatically speeds up previous HC solving by specializing HC methods to generic cases of our problems. We show in simulated experiments that our solvers are numerically robust and stable under image noise. We show in real experiment that (i) SIFT features provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches where the state of the art structure from motion initialization fails.
7.7.2 Using Maple to analyse parallel robots
in 34, we present the SIROPA Maple Library which has been designed to study serial and parallel manipulators at the conception level. We show how modern algorithms in Computer Algebra can be used to study the workspace, the joint space but also the existence of some physical capabilities w.r.t. to some design parameters left as degree of freedom for the designer of the robot.
8 Bilateral contracts and grants with industry
8.1 Bilateral contracts with industry
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.
- A research contract was signed with the company Safran Electronics & Defense on the study of a geolocalization problem.
- A NDA was signed with the company Safran Electronics & Defense on the study of parallel mecanisms.
9 Partnerships and cooperations
9.1 International initiatives
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
9.1.1 Inria associate team not involved in an IIL
- Title: Mathematics and Algorithms for Cryptographic Advanced Objects
- Duration: 2019
- Coordinator: Antoine Joux
- isgDPRMN, University of Wollongong (Australia)
- Inria contact: Antoine Joux
- Summary: 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.
9.1.2 Inria international partners
Declared Inria international partners
University of Wollongong (Australia)
Informal international partners
- 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)
- Gamble INRIA project-team
- Datashape INRIA project-team
9.2 European initiatives
9.2.1 FP7 & H2020 Projects
Program: H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
Project acronym: Almacrypt
Project title: Algorithmic and Mathematical Cryptology
Duration: 01/2016 - 06/2021
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.
9.3 National initiatives
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.
10.1 Promoting scientific activities
Chair of conference program committees
Jean-Claude Bajard was co-chair of the 8th International Workshop, WAIFI 2020, Rennes, France, July 6-8, 2020
Member of the conference program committees
Alban Quadrat was member of the Program Committee of the 2020 Maple Conference, 02-06/11/2020
Elias Tsigaridas was a member of the program committee of the 21th International Workshop on Computer Algebra in Scientific Computing (CASC) 2020.
Member of the editorial boards
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 associated editor of Journal of Symbolic Computations, Elsevier
10.1.2 Leadership within the scientific community
Alban Quadrat co-organized the invited session Algebraic and Symbolic Methods in Mathematical Systems Theory at the 21st IFAC World Congress, 13-17 July 2020
10.1.3 Scientific expertise
Alban Quadrat is member of the Scientific committee of the Journées Nationales de Calcul Formel (JNCF)
10.1.4 Research administration
Alban Quadrat is 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.
Elias Tsigaridas is an elected member of the Commission d’évaluation d’Inria (CE).
10.2 Teaching - Supervision - Juries
Alban Quadrat, Project Démodulation (2 groups), 10, L3, Sorbonne University.
Fabrice Rouillier: Course in Algebraic Computations, M1, 24h, Sorbonne Université.
Antonin Guilloux: Courses L1 at Sorbonne Université; Courses and organization of the "Préparation à l'Option C de mathématiques", at Sorbonne Université
Pascal Molin collaborates to the funded (20K euros) project AATEM for implementing a Jupyter+Cocalc server for teaching Maths and coputer science à Paris University.
Pierre-Vincent Koseleff did propose anew course in Master 1 at Sorbonne Université : Algèbre Algorithmique
Alban Quadrat supervised the Master thesis of Valérian Hatey, Etude effective de l'algèbre des opérateurs intégro-différentiels ordinaires à coefficients polynomiaux, M2 Algèbre Appliquée, University of Paris Saclay
Pierre-Vincent Koseleff supervised the Master thesis of Andrea Negro, some properties of the Alexander polynomials of two-bridge knots, M2 Mathématiques fondamentales, Sorbonne Université
- 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: Raphael Alexandre, 09/2019, directed by Elisha Falbel.
- PhD in progress : Thibauld Feneuil, 10/2020, directed by Jean-Claude Bajard.
- PhD in progress : Carles Checa, 10/2020, directed by Elias Tsigaridas (co-supervision with Ioannis Emiris)
- Sudarshan Shinde, defense in July 2020 was directed by Razvan Barbulescu and Pierre-Vincent Koseleff
J.-C. Bajard was reviewer for the PhD of Mohamad Ali Mehrabi on hardware implementation of Elliptic Curve Cryptography based on Residue Number Systems , Macquarie University, Australia
J.-C. Bajard was president of the Jury for the PhD of Timo ZIJLSTRA on accélérateurs matériels sécurisés pour la cryptographie post- quantique, Université de Bretagne Sud.
J.-C. Bajard was was member of the jury for the PhD of Natalia Kharchenko on lattice algorithms and lattice-based cryptography , Sorbonne University.
Antonin Guilloux: Interview for a short movie presented to "Je filme le métier qui me plaît" concourse on the work of a teacher/researcher in mathematics.
10.3.1 Internal or external Inria responsibilities
Alban Quadrat is member of the technical committee Linear Systems of the International Federation of Automatic Control (IFAC)
Fabrice Rouillier is the President of the association Animath.
Fabrice Rouillier is Chargé de mission Médiation for the Inria Paris research center.
Fabrice Rouillier is in the editoria board of Interstices.
Fabrice Rouillier is a member of the comité de pilotage de l'année des matématiques
Fabrice Rouillier is a member of the comité de pilotage de la semaine des mathématiques
Fabrice Rouillier est membre du Jury des Olympiades Nationales de Mathématiques
Fabrice Rouillier participated to the semaine des mathématiques for the Paris Inria Paris research center.
11 Scientific production
11.1 Publications of the year
International peer-reviewed conferences
National peer-reviewed Conferences
Conferences without proceedings
Scientific book chapters
Edition (books, proceedings, special issue of a journal)
Doctoral dissertations and habilitation theses
Reports & preprints
Other scientific publications