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# Next seminars

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

March 12, 2020 at 10h15, Amphi B (3rd floor, Monod)

Bruno Salvy: Absolute root separation

This is joint work with Yann Bugeaud, Andrej Dujella, Wenjie Fang and Tomislav Pejkovic.

Pascal Koiran: Root separation for trinomials

As an algorithmic application, we show that the number of real roots of a trinomial \(f\) can be computed in time polynomial in the size of the sparse encoding of \(f\). The same problem is open for 4-nomials.

Abinand Gopal (Oxford University)

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

An accelerated, high-order accurate direct solver for two-dimensional acoustic scattering

Gleb Pogudin (MAX, LIX, École Polytechnique)

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

Amaury Pouly (IRIF, Paris)

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

Tristan Vaccon (XLIM, Université de Limoges)

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

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

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)

May 7, 2020 at 10h15, room M7-315 (3rd floor, Monod)

Calcul des périodes et des cycles algébriques de surfaces complexes

# 2019-2020

Robin Larrieu (LMV, Université de Versailles)

February 13, 2020 at 10h15, room M7-315 (3rd floor, Monod)

Fast polynomial reduction for generic bivariate ideals

We will consider two situations where these new ideas apply, leading to different algorithms:

- First, there is a class called “vanilla Gröbner bases” for which there is a so-called terse representation that, once precomputed, allows to reduce any polynomial \( P \) in time \( O(n^2)\). In this setting, assuming suitable precomputation, multiplication and change of basis can therefore be done in time \( O(n^2)\) in the quotient algebra \(K[X,Y] / ⟨A,B⟩\).
- Then, we assume that \( A \) and \( B \) are given in total degree and we consider the usual degree lexicographic order. Although the bases are not vanilla in this case, they admit a so-called concise representation with similar properties. Actually, the precomputation can also be done efficiently in this particular setting: from the input \(A, B\), one can compute a Gröbner basis in concise representation in time \( O(n^2)\). As a consequence, multiplication in \( K[X,Y] / ⟨A,B⟩\) can be done in time \( O(n^2) \) including the cost of precomputation.

Laurence Rideau (STAMP, INRIA Sophia Antipolis – Méditerranée)

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

Formalisation in Coq of the correctness of double-word arithmetic algorithms and their errors bounds

We show how this formalisation made it possible to highlight some errors and some inaccuracies in the proofs of the paper.

I will focus in particular on the dangers of the “wlog”, which is used extensively in this type of proofs.

We will also discuss the advantages and disadvantages of such formalization, and how this work has improved confidence in the results of the article, despite the errors detected, and has also improved the Flocq library (intensively used for it).

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

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

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

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

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

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

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

# 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

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

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

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

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

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

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

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

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

Alain Passelègue (AriC)

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

New candidate pseudorandom functions and their applications

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

(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

Nicolas Brunie (Kalray)

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

Overview of arithmetic at Kalray: metalibm and the rest

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

Radu Titiu (AriC and BitDefender)

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

Adaptively secure PRFs from LWE

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

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

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 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.