Seminar

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Next seminars

Tristan Vaccon (XLIM, Université de Limoges)

December 19, 2019 at 10h15, room M7-315 (3rd floor, Monod)

\( p \)-adic precision, examples and applications

\(p\)-adic numbers can usually only be handled with finite precision, which yields the problems of determining the smallest precision needed for a computation or the loss of precision per operation.
With X. Caruso and D. Roe, we have provided a method to handle precision over \(p\)-adics that relies on differentials and first-order approximation. It provides results that are (essentially) optimal and do not depend on the choice of algorithm.
We will present various illustrations of this technique: computation of determinants, characteristic polynomials, \(p\)-adic differential equations,etc…
We will also present a Sagemath implementation to compute automatically the optimal precision on a given computation.

Pierre Lairez (SpecFun, INRIA Saclay – Île-de-France)

January 9, 2020 at 10h15, room M7-315 (3rd floor, Monod)

Robin Larrieu (LMV, Université de Versailles)

January 16, 2020 at 10h15, room M7-315 (3rd floor, Monod)

Laurence Rideau (STAMP, INRIA Sophia Antipolis – Méditerranée)

January 30, 2020 at 10h15, room M7-315 (3rd floor, Monod)

Pascal Koiran (MC2, LIP) and Bruno Salvy (AriC, LIP)

March 12, 2020 at 10h15, room M7-315 (3rd floor, Monod)

2019-2020

Miruna Rosca (AriC, LIP)

December 5, 2019 at 10h15, room M7-315 (3rd floor, Monod)

MPSign: A Signature from Small-Secret Middle-Product Learning with Errors

We describe a digital signature scheme MPSign, whose security relies on the conjectured hardness of the Polynomial Learning With Errors problem (PLWE) for at least one defining polynomial within an exponential-size family (as a function of the security parameter). The proposed signature scheme follows the Fiat-Shamir framework and can be viewed as the Learning With Errors counterpart of the signature scheme described by Lyubashevsky at Asiacrypt 2016, whose security relies on the conjectured hardness of the Polynomial Short Integer Solution (PSIS) problem for at least one defining polynomial within an exponential-size family. As opposed to the latter, MPSign enjoys a security proof from PLWE that is tight in the quantum-access random oracle model. The main ingredient is a reduction from PLWE for an arbitrary defining polynomial among exponentially many, to a variant of the MiddleProduct Learning with Errors problem (MPLWE) that allows for secrets that are small compared to the working modulus. We present concrete parameters for MPSign using such small secrets, and show that they lead to significant savings in signature length over Lyubashevsky’s Asiacrypt 2016 scheme (which uses larger secrets) at typical security levels. As an additional small contribution, and in contrast to MPSign (or MPLWE), we present an efficient key-recovery attack against Lyubashevsky’s scheme (or the inhomogeneous PSIS problem), when it is used with sufficiently small secrets, showing the necessity of a lower bound on secret size for the security of that scheme.
This is joint work with Shi Bai, Dipayan Das, Ryo Hiromasa, Amin Sakzad, Damien Stehlé, Ron Steinfeld and Zhenfei Zhang

Vincent Lefèvre (AriC, LIP)

November 21, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Accurate Complex Multiplication in Floating-Point Arithmetic

We deal with accurate complex multiplication in binary floating-point arithmetic, with an emphasis on the case where one of the operands is a “double-word” number. We provide an algorithm that returns a complex product with normwise relative error bound close to the best possible one, i.e., the rounding unit u. We also discuss variants of this algorithm.
This is a joint work with Jean-Michel Muller.

Fabien Laguillaumie (AriC, LIP)

November 14, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Threshold variant of the digital signature algorithm standard

(EC)DSA is a widely adopted digital signature standard. Unfortunately, efficient distributed variants of this primitive are notoriously hard to achieve and known solutions often require expensive zero knowledge proofs to deal with malicious adversaries. For the two party case, Lindell recently managed to get an efficient solution which, to achieve simulation-based security, relies on an interactive, non standard, assumption on Paillier’s cryptosystem.
In this talk, I will give some recent results to improve Lindell’s solution in terms of security and efficiency, and discuss some possible extension to a full threshold variant.
This is joint works with Guilhem Castagnos, Dario Catalano, Federico Savasta and Ida Tucker.

Mioara Joldes (CNRS LAAS, Toulouse)

November 7, 2019 at 10h15, room M7-315 (3rd floor, Monod)

An optimization viewpoint for machine-efficient polynomial approximations

Machine implementation of mathematical functions often relies on polynomial approximations. The particularity is that rounding errors occur both when representing the polynomial coefficients on a finite number of bits, and when evaluating it in finite precision. Hence, for finding the best polynomial (for a given fixed degree, norm and interval), one has to take into account both types of errors: approximation and evaluation. By considering a linearized evaluation error model, we formulate a semi-infinite linear optimization problem, whose solution can be obtained by an iterative exchange algorithm. This can be seen as an extension of the Remez algorithm. A discussion of the obtained results and further developments concludes this talk. This is joint work with F. Bréhard and D. Arzelier.

Geoffroy Couteau (CNRS and Paris 7)

October 24, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Efficient Pseudorandom Correlation Generators: Silent OT Extension and More

Secure multiparty computation (MPC) often relies on sources of correlated randomness for better efficiency and simplicity. This is particularly useful for MPC with no honest majority, where input-independent correlated randomness enables a lightweight “non-cryptographic” online phase once the inputs are known. However, since the amount of correlated randomness typically scales with the circuit size of the function being computed, securely generating correlated randomness forms an efficiency bottleneck, involving a large amount of communication and storage. A natural tool for addressing the above limitations is a pseudorandom correlation generator (PCG).
A PCG allows two or more parties to securely generate long sources of useful correlated randomness via a local expansion of correlated short seeds and no interaction. PCGs enable MPC with silent preprocessing, where a small amount of interaction used for securely sampling the seeds is followed by silent local generation of correlated pseudorandomness.
A concretely efficient PCG for Vector-OLE correlations was recently obtained by Boyle et al. (CCS 2018) based on variants of the learning parity with noise (LPN) assumption over large fields. In this work, we initiate a systematic study of PCGs and present concretely efficient constructions for several types of useful MPC correlations. We obtain the following main contributions:
– PCG foundations. We give a general security definition for PCGs. Our definition suffices for any MPC protocol satisfying a stronger security requirement that is met by existing protocols. We prove that a stronger security requirement is indeed necessary, and justify our PCG definition by ruling out a stronger and more natural definition.
– Silent OT extension. We present the first concretely efficient PCG for oblivious transfer correlations. Its security is based on a variant of the binary LPN assumption and any correlation-robust hash function. We expect it to provide a faster alternative to the IKNP OT extension protocol (Crypto ’03) when communication is the bottleneck. We present several applications, including protocols for non-interactive zero-knowledge with bounded-reusable preprocessing from binary LPN, and concretely efficient related-key oblivious pseudorandom functions.
– PCGs for simple 2-party correlations. We obtain PCGs for several other types of useful 2-party correlations, including (authenticated) one-time truth-tables and Beaver triples. While the latter PCGs are slower than our PCG for OT, they are still practically feasible. These PCGs are based on a host of assumptions and techniques, including specialized homomorphic secret sharing schemes and pseudorandom generators tailored to their structure.
– Multiparty correlations. We obtain PCGs for multiparty correlations that can be used to make the circuit-dependent communication of MPC protocols scale linearly (instead of quadratically) with the number of parties.

Benjamin Wesolowski (CWI Amsterdam)

October 10, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Discrete logarithms in quasi-polynomial time in finite fields of small characteristic

We prove that the discrete logarithm problem can be solved in quasi-polynomial expected time in the multiplicative group of finite fields of fixed characteristic. In 1987, Pomerance proved that this problem can be solve in expected subexponential time \(L(1/2)\). The following 30 years saw a number of heuristic improvements, but no provable results. The quasi-polynomial complexity has been conjectured to be reachable since 2013, when a first heuristic algorithm was proposed by Barbulescu, Gaudry, Joux, and Thomé. We prove this conjecture, and more generally that this problem can be solved in the field of cardinality \( p^n \) in expected time \((pn)^{2 \log_2 (n)+O(1)}\).

Théo Mary (University of Manchester, U.K.)

October 3, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Sharper and smaller error bounds for low precision scientific computing

With the rise of large scale, low precision computations, numerical algorithms having a backward error bound of the form \( nu \), for a problem size \( n \) and a machine precision \( u \), are no longer satisfactory. Indeed, with half precision arithmetic, such algorithms cannot guarantee even a single correct digit for problems larger than a few thousands. This has created a need for error bounds that are both sharper and smaller. In this talk, we will discuss recent advances towards this double goal. We will present new error analyses to obtain probabilistic bounds that are sharper on average, and new algorithms that achieve much smaller bounds without sacrificing high performance.

2018-2019

Miruna Rosca (AriC)

July 4, 2019 at 10h15, room M7-315 (3rd floor, Monod)

On the Middle-Product Learning With Errors Problem and its applications in cryptography

In this talk, we introduce a new variant MP-LWE of the Learning With Errors problem (LWE) making use of the middle product between polynomials modulo an integer q, we exhibit a reduction from the Polynomial-LWE problem (PLWE) parametrized by a polynomial f, to MP-LWE, which works for a large family of polynomials, and we analyze the applications of MP-LWE in cryptography.
This is joint work with A. Sakzad, D. Stehlé and R. Steinfeld.

François Morain (LIX, École polytechnique)

June 13, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Fractions continues avec des algorithmes rapides

Les fractions continues sont un outil très important pour l’approximation des nombres réels, avec de nombreuses applications en théorie algorithmique des nombres ainsi qu’en cryptanalyse. Une des applications importantes est l’algorithme de Cornacchia qui résout élégamment le problème de la représentation des nombres premiers sous la forme p=x^2+d y^2$ avec d > 0. Cet exposé présentera l’utilisation des algorithmes rapides de pgcd d’entiers pour fournir une version rapide de l’algorithme de Cornacchia.

Vincent Neiger (XLIM, Université de Limoges)

May 2, 2019 at 10h15, room M7-315 (3rd floor, Monod)

On the complexity of modular composition of generic polynomials

This talk is about algorithms for modular composition of univariate polynomials, and for computing minimal polynomials. For two univariate polynomials \(a\) and \(g\) over a commutative field, modular composition asks to compute \(h(a) \bmod g\) for some given \(h\), while the minimal polynomial problem is to compute \(h\) of minimal degree such that \( h(a) = 0 \bmod g\). For generic \(g\) and \(a\), we propose algorithms whose complexity bound improves upon previous algorithms and in particular upon Brent and Kung’s approach (1978); the new complexity bound is subquadratic in the degree of \(g\) and \(a\) even when using cubic-time matrix multiplication. Our improvement comes from the fast computation of specific bases of bivariate ideals, and from efficient operations with these bases thanks to fast univariate polynomial matrix algorithms. We will also report on software development and comment on implementation results for the main building blocks in our composition algorithm.
Contains joint work with Seung Gyu Hyun, Bruno Salvy, Eric Schost, Gilles Villard.

Bruno Grenet (ECO, LIRMM)

April 11, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Multiplications polynomiales sans mémoire

Le problème de la multiplication de polynômes a été très étudié depuis les années 1960, et différents algorithmes ont été proposés pour atteindre les meilleures complexités en temps.
Plus récemment, certains de ces algorithmes ont été étudiés du point de vue de leur complexité en espace, et modifiés pour n’utiliser aucun espace supplémentaire autre que les entrées et sorties, tout en gardant la même complexité en temps asymptotiquement.
Dans ce travail, nous étendons ces résultats de deux façons. D’une part, nous nous demandons si tout algorithme de multiplication polynomiale admet une variante « en place », c’est-à-dire n’utilisant aucun espace supplémentaire, de manière générique. D’autre part, nous considérons deux variantes importantes de ce problème qui ne produisent qu’une partie du résultat, les produits dits court et médian, et nous nous demandons si ces opérations peuvent également être effectuées en place.
Pour répondre de manière (essentiellement) affirmative à ces deux questions, nous proposons une série de réductions ayant comme point de départ n’importe quel algorithme de multiplication de complexité en espace linéaire. Pour le produit complet et le produit court, ces réductions fournissent des variantes en place des algorithmes avec la même complexité en temps asymptotiquement. Pour le produit médian, la réduction implique un facteur logarithmique supplémentaire dans la complexité en temps, quand celle-ci est quasi-linéaire.
Travail en commun avec Pascal Giorgi et Daniel Roche

Damien Stehlé (AriC)

April 4, 2019 at 10h15, room M7-315 (3rd floor, Monod)

A survey on security foundations of fast lattice-based cryptography

The Learning With Errors problem (LWE) captures the asymptotic hardness of some standard lattice problems, and enables the design of cryptographic schemes. However, these LWE-based schemes are relatively inefficient.
To address this issue, algebraic variants of LWE have been introduced, such as Polynomial-LWE, Ring-LWE, Module-LWE and MiddleProduct-LWE, whose definitions involve polynomial rings and number fields.
In this talk, I will survey the state of the art on these problems.

Jean-Michel Muller (AriC)

March 7, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Error analysis of some operations involved in the Fast Fourier Transform

We are interested in obtaining error bounds for the classical FFT algorithm in floating-point arithmetic, for the 2-norm as well as for the infinity norm. For that purpose we also give some results on the relative error of the complex multiplication by a root of unity, and on the largest value that can take the real or imaginary part of one term of the FFT of a vector x, assuming that all terms of x have real and imaginary parts less than some value b.
This is a joint work with N. Brisebarre, M. Joldes, A.-M. Nanes and J. Picot.

Assia Mahboubi (Gallinette, INRIA Rennes – Bretagne Atlantique, LS2N)

February 14, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Formally Verified Approximations of Definite Integrals

In this talk, we discuss the problem of computing values of one-dimensional definite integrals, with the highest possible guarantee of correctness. Finding an elementary form for an antiderivative is often not an option, and numerical integration is a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. These rigorous algorithms are implemented in software like INTLAB, VNODE-LP, Arb, etc. But the highest possible guarantee of correctness on such approximations, even those obtained by rigorous means, would in fact be provided by a formal proofs, machine-checked using a proof assistant. Proof assistants are pieces of software for representing mathematical definitions, statements, algorithms and proofs in a digital form, suitable for computer processing. In particular, they can be used to devise formal-proof-producing implementations of programs. But numerical integration is still missing from the toolbox when performing formal proofs. We thus describe and discuss a method for automatically computing and proving bounds on some definite integrals, implemented inside the Coq proof assistant.
This is a joint work with Guillaume Melquiond and Thomas Sibut-Pinote.

Ida Tucker (AriC)

February 7, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Practical fully secure unrestricted inner product functional encryption modulo a prime p

Functional encryption (FE) is an advanced cryptographic primitive which allows, for a single encrypted message, to finely control how much information on the encrypted data each receiver can recover. To this end many functional secret keys are derived from a master secret key. Each functional secret key allows, for a ciphertext encrypted under the associated public key, to recover a specific function of the underlying plaintext.
However constructions for general FE are far from practical, or rely on non-standard and ill-understood cryptographic assumptions.
In this talk I will focus on the construction of efficient FE schemes for linear functions (i.e. the inner product functionality), and the framework in which our constructions hold. Such schemes yield many practical applications, and our constructions are the first FE schemes for inner products modulo a prime that are both efficient and recover the result whatever its size. I will also describe an instantiation of the framework in using class groups of imaginary quadratic fields.
This is a joint work with Guilhem Castagnos and Fabien Laguillaumie.

Éric Goubault (LIX, École Polytechnique)

January 24, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Finding Positive Invariants of Polynomial Dynamical Systems – some experiments

Synthetising positive invariants of non-linear ODEs, switched systems or even hybrid systems is a hard problem that has many applications, from control to verification. In this talk, I will present two « exercices de style » for dealing with it, revisiting the classical Lyapunov function approach. The first one is based on algebraic properties of polynomial differential systems (Darboux polynomials, when they exist), for finding polynomial, rational or even some log extensions to rational functions whose level sets or sub-level sets describe positive invariants of these systems, or provide interesting « change of bases » for describing their solutions. The second one is based on topological properties (Wazewski property, mostly) which ensure the existence, in some region of the state space, of a non-empty maximal invariant set. The interest is that there is then in general no need to find complicated functions for precisely describing the invariant set itself, instead we rather use simple template shapes in which a possibly very complicated invariant set lies. The topological criterion can be ensured by suitable SoS relaxations, for polynomial differential systems, that can be implemented using LMI solvers.

Alain Passelègue (AriC)

January 17, 2019 at 10h15, room M7-315 (3rd floor, Monod)

New candidate pseudorandom functions and their applications

In this talk, I will present new and simple candidate pseudorandom functions (PRFs) introduced in a recent work. In this work, we depart from the traditional approaches for building PRFs used in provable security or in applied cryptography by exploring a new space of plausible PRF candidates. Our guiding principle is to maximize simplicity while optimizing complexity measures that are relevant to advanced cryptographic applications. Our primary focus is on weak PRFs computable by very simple circuits (depth-2 ACC circuits).
The advantage of our approach is twofold. On the theoretical side, the simplicity of our candidates enables us to draw many natural connections between their hardness and questions in complexity theory or learning theory. On the applied side, the piecewise-linear structure of our candidates lends itself nicely to applications in secure multiparty computation (MPC). In particular, we construct protocols for distributed PRF evaluation that achieve better round complexity and/or communication complexity compared to protocols obtained by combining standard MPC protocols with practical PRFs (included MPC-friendly ones).
Finally, we introduce a new primitive we call an encoded-input PRF, which can be viewed as an interpolation between weak PRFs and standard (strong) PRFs. As we demonstrate, an encoded-input PRF can often be used as a drop-in replacement for a strong PRF, combining the efficiency benefits of weak PRFs and the security benefits of strong PRFs. We give a candidate EI-PRF based on our main weak PRF candidate. 

Joint work with Dan Boneh, Yuval Ishai, Amit Sahai, and David J. Wu, published at TCC 2018

Chee Yap (New-York University)

January 9, 2019 at 10h15, room M7-315 (3rd floor, Monod)

Subdivision Path Planning in Robotics: Theory and Practice

Motion planning is a fundamental problem in robotics. We propose to design path planners based on three foundations:
(1) The notion of “resolution-exact” planners. Conceptually, it avoids the zero problem of exact computation.
(2) The use of “soft predicates” for achieving such algorithms in the subdivision approach.
(3) The “feature-based technique” for constructing such soft predicates.
We formulate an algorithmic framework called “Soft Subdivision Search” (SSS) that incorporates these ideas. There are many parallels between our framework and the well-known Sampling or Probabilistic Roadmap framework. Both frameworks lead to algorithms that are
* practical
* easy to implement
* flexible and extensible
* with adaptive and local complexity
In contrast to sampling and previous resolution approaches, SSS confers strong theoretical guarantees, including halting. 

In a series of papers we demonstrated the power of these ideas, by producing planners for planar robots with 2, 3 and 4 degrees of freedom (DOF) that outperform or matches state-of-art sampling-based planners. Most recently, we produced a planner for two spatial robots (rod and ring) with 5 DOFs. Non-heuristic planners for such robots has been considered a challenge for the subdivision approach. We outline a general axiomatic theory underlying these results, including subdivision in non-Euclidean configuration spaces,

Joint work with Y.J. Chiang, C.H. Hsu, C. Wang, Z. Luo, B. Zhou, J.P. Ryan.

Elena Kirshanova (AriC)

December 13, 2018 at 10h15, room M7-315 (3rd floor, Monod)

Practical sieving algorithms for the Shortest Vector Problem

In this talk I present recent results on sieving algorithms for the Shortest Vector Problem. First, I explain why this problem is important and how sieving algorithms work. Then, I present recent advances in memory efficient versions of sieving algorithms. I explain locality-sensitive techniques for these types of algorithms. The part of the talk is based on joint works with Gottfried Herold and Thijs Laarhoven. Finally, I present recent advances in practical aspects of sieving algorithm for SVP. I describe technical challenges that arise when one tries to make sieving algorithms practical, and how one can overcome some of them. This part of the talk is on-going work with Martin R. Albrecht, Leo Ducas, Gottfried Herold, Eamonn W. Postlethwaite, Marc Stevens.

Nicolas Brunie (Kalray)

December 6, 2018 at 10h15, room M7-315 (3rd floor, Monod)

Overview of arithmetic at Kalray: metalibm and the rest

Kalray’s version of Metalibm “lugdunum” has recently been open sourced. It is an interesting tool to developp elementary functions. In this presentation we will present the tool and show how it can be used to explore the design space of a few elementary functions. Then we will present in more details how Metalibm is used at Kalray to developp both the next generation Hardware and the mathematical libraries through the example of CRC reduction and OpenCL-C (kernel code) elementary functions. Finally we will survey the arithmetic at Kalray outside Metalibm through a description of the next generation processor and what is envisioned for the future.

Sylvie Putot (LIX, École Polytechnique)

November 29, 2018 at 10h15, room M7-315 (3rd floor, Monod)

Forward Inner-Approximated Reachability of Non-Linear Continuous Systems

We propose an approach for computing inner-approximations of reachable sets of dynamical systems defined by non-linear, uncertain, ordinary differential equations. This is a notoriously difficult problem, much more intricate than outer-approximations, for which there exist well known solutions, mostly based on Taylor models.  Our solution builds on rather inexpensive set-based methods, namely a generalized mean-value theorem combined with Taylor models outer-approximations of the flow and its Jacobian with respect to the uncertain inputs and parameters. The combination of such forward inner and outer Taylor-model based approximations can be used as a basis for the verification and falsification of properties of cyber-physical systems.

Radu Titiu (AriC and BitDefender)

November 22, 2018 at 10h15, room M7-315 (3rd floor, Monod)

Adaptively secure PRFs from LWE

In distributed pseudorandom functions (DPRFs), a PRF secret key SK is secret shared among N servers so that each server can locally compute a partial evaluation of the PRF on some input X. A combiner that collects t partial evaluations can then reconstruct the evaluation F (SK, X) of the PRF under the initial secret key. So far, all non-interactive constructions in the standard model are based on lattice assumptions. One caveat is that they are only known to be secure in the static corruption setting, where the adversary chooses the servers to corrupt at the very beginning of the game, before any evaluation query. In this work, we construct the first fully non-interactive adaptively secure DPRF in the standard model. Our construction is proved secure under the LWE assumption against adversaries that may adaptively decide which servers they want to corrupt. We also extend our construction in order to achieve robustness against malicious adversaries. 

This is joint work with Benoit Libert and Damien Stehlé.

Martin Kumm (Uni. Kassel, Germany)

November 8, 2018 at 10h15, room M7-315 (3rd floor, Monod)

Exact Computation of Monotonic Functions with Large Input Word Sizes using Look-Up Tables

The exact evaluation of arbitrary functions by using look-up tables (LUTs) is typically limited to small input word sizes. This is due to the fact that the storage requirements grow exponentially with the input word size \(N\) and linear with the output word size \(M\), i.e., \(O(2^N M)\). However, many applications require the computation of elementary functions with a large precision of the input argument but a lower precision of the result. One example is direct digital frequency synthesis (DDFS) with typically \(N=32..48\) bit and \(M=8..12\) bit. Another example are tone mapping methods for high-dynamic range (HDR) imaging with typ. \(N=16..19\) bit and \(M=8\) bit. In this talk, alternative architectures for evaluation of monotonic functions using LUTs are discussed which memory requirements scale linear with the input word size and exponentially with the output word size, i.e., \(O(2^M N)\). This is achieved by using \(M\) additional comparators or adders. First experimental results from FPGA synthesis show that this also translates to resource reductions for those applications where \(M\) is just larger than \(N\).

Silviu Filip (CAIRN, Inria Rennes Bretagne Atlantique)

October 25, 2018 at 10h15, room M7-315 (3rd floor, Monod)

A High Throughput Polynomial and Rational Function Approximations Evaluator

We present an automatic method for the evaluation of functions via polynomial or rational approximations and its hardware implementation, on FPGAs. These approximations are evaluated using Ercegovac’s iterative E-method adapted for FPGA implementation. The polynomial and rational function coefficients are optimized so that they satisfy the constraints of the E-method. We present several examples of practical interest; in each case a resource-efficient approximation is proposed and comparisons are made with alternative approaches.

Florent Bréhard (AriC)

October 18, 2018 at 10h15, room M7-315 (3rd floor, Monod)

Rigorous Numerics for Function Space Problems and Applications to Aerospace

A wide range of numerical routines exist for solving function space problems (like ODEs, PDEs, optimization, etc.). But in most cases, one lacks information about the reliability of the results, e.g., how many of the returned digits are correct. While most applications focus on efficiency, some safety-critical tasks, as well as computer assisted mathematics, need rigorous mathematical statements about the computed result such as automatic tight error bounds. 

A relevant practical example occurs in the spacecraft rendezvous problem, which consists in determining the optimal command law for a spacecraft equipped with thrusters to be transferred from its original orbit to a target orbit within a given time interval. Computing rigorous trajectories is of high interest to guarantee a posteriori the correctness of the numerical command law returned by the optimization algorithm used to solve this problem.

In this talk we discuss a rigorous framework called Chebyshev models to provide validated enclosures of real-valued functions defined over a compact interval. After having presented the basic arithmetic operations on them, we focus on an algorithm that computes validated solutions of linear ordinary differential equations, specifically, approximate truncated Chebyshev series together with a rigorous uniform error bound. The method relies on an a posteriori validation based on a Newton-like fixed-point operator, which also exploits the almost-banded structure of the problem in an efficient way. We provide an open-source C implementation (https://gforge.inria.fr/projects/tchebyapprox/).

Finally, certified enclosures of spacecraft trajectories arising in the rendezvous problem will be computed using the tools introduced during the talk.

Archives of the seminar

For seminars in previous years, see former AriC seminars.

AriC project – Arithmetic and Computing