TLS/SSL (currently version TLS 1.2) is one of the 3 essential cryptographic protocols used today (together with SSH and IPSec). Despite its central role in securing e-commerce, Internet browsing, email, VoIP, etc., despite the fact that almost every search and connection query in every browser in the world requires its use, this protocol still presents security flaws in its conception. To overcome recent attacks, such as FREAK, LogJam, 3Shake, SLOTH, or DROWN, a new version i.e. TLS 1.3 has recently been drafted. Our project, SafeTLS, addresses the security both of TLS 1.3 and of TLS 1.2 as they are (expected to be) used, in three important ways: (1) by providing a better understanding -- from the point of view of provable security – of the TLS 1.3 and 1.2 handshakes as they are used in real life. One important, and new aspect of our work concerns formalizing and proving the privacy properties attained by the newly-designed TLS 1.3 draft; another concerns the security of secure-channel establishment protocols against mass surveillance threats, in which a powerful adversary called Big Brother can learn "confidential" data exchanged between users. Another important, innovative goal of our work concerns understanding the degradation of security of the TLS handshake when it is used with middleware infrastructures – which is predominantly the case nowadays. Finally, we will assess and provide new primitives for use in TLS and D-TLS, by looking at the candidates of the CAESAR competition and by studying new elliptic curves fashioned specifically for use in this protocol. (2) by providing clients with a tool that detects the quality of each TLS connection at runtime, and instructs the client what type of data can be safely exchanged across such a channel. In particular, the explanations given to the client must be understandable and as short as possible, making this tool an aid to a safer use of Internet browsing. Another aspect of our work concerns informing clients whether middleware is, in fact, being used in their TLS-secured connections. We note that, while middleware decreases latency for the client and storage and bandwidth needs for servers, it may represent an additional risk to clients, of which they are actually not informed. Indeed, most middleware is designed to pose as the original server that the client wanted to reach. (3) by addressing the problem of secure TLS implementations. We first propose to analyze the security offered by a number of available TLS 1.2 (and earlier) implementations, such as s2n, BoringSSL, and mbedTLS. By furthermore using the automatic verification tool EasyCrypt to formulate and prove the security of the TLS 1.3 handshake (with all its modes of operation), we can also use tools that transform EasyCrypt proofs to certified code, giving explicit guidelines for a secure future TLS 1.3 implementation. Our results will be manifested in the following types of results: (1) security proofs (using formal methods and provable security methodologies), indicating lower bounds on security; (2) impossibility results and upper bounds on security, in particular for security against mass-surveillance and for downgrading due to middleware; (3) the tool (application) designed to assess the quality of each TLS connection at runtime, which will be open-source and made available to any user; (4) the certified code corresponding to a secure implementation of TLS 1.3.

The p-adic Langlands correspondence has become nowadays one of the deepest and the most stimulating research programs in number theory. It was initiated in France in the early 2000's by Breuil and aims at understanding the relationships between the p-adic representations of p-adic absolute Galois groups on the one hand and the p-adic representations of p-adic reductive groups on the other hand. Beyond the case of GL2(Qp) which is now well established, the p-adic Langlands correspondence remains quite obscure and mysterious new phenomena enter the scene; for instance, on the $GLn(F)$-side one encounters a vast zoology of representations which seems extremely difficult to organize. The CLap--CLap ANR project aims at accelerating the expansion of the p-adic Langlands program beyond the well-established case of GL2(Qp). Its main originality consists in its very constructive approach mostly based on algorithmics and calculations with computers at all stages of the research process. We shall pursue three different objectives closely related to our general aim: (1) draw a conjectural picture of the (still hypothetical) p-adic Langlands correspondence in the case of GLn,, (2) compute many deformation spaces of Galois representations and make the bridge with deformation spaces of representations of reductive groups, (3) design new algorithms for computations with Hilbert and Siegel modular forms and their associated Galois representations (in which the p-adic Langlands correspondence is supposed to be realized). This project will also be the opportunity to contribute to the development of the mathematical software SageMath and to the expansion of computational methodologies.

Our project is to treat multiscale models which are both infinite-dimensional and stochastic with a theoretic and computational approach. Multiscale analysis and multiscale numerical approximation for infinite-dimensional problems (partial differential equations) is an extensive part of contemporary mathematics, with such wide topics as hydrodynamic limits, homogenization, design of asymptotic-preserving schemes. Multiscale models in a random or stochastic context have been analysed and computed essentially in finite dimension (ordinary/stochastic differential equations), or in very specific domains, mainly the propagation of waves, of partial differential equations. The technical difficulties of our project are due to the stochastic aspect of the problems (this brings singular terms in the equations, which are difficult to understand with a pure PDE's analysis approach) and to their infinite-dimensional character. These two aspects, combined, typically raise compactness and computational issues. Our aim is to create the new tools, analytical, probabilistic and numerical ones, which are required to understand a large class of stochastic multiscale partial differential equations, that includes some kinetic and dispersive equations. Our aim is to derive reduced equations. In the different regimes we are interested in, and, particularly, in the diffusive regime (diffusion-approximation), that leads to limit equations with white noise. We will investigate the derivation of reduced equations in different context and for various models: - collisional kinetic equations with a Vlasov forcing term induced by (resp. a collisional kernel perturbed by) an external, or coupled, Markov process (kinetic equations for plasmas or fluids, resp. modelling of motion by run-and-tumble), - limit Boltzmann to Navier-Stokes under stochastic forces, - collisional or non-collisional kinetic equations with a stochastic drag force term also induced by an external, or coupled, Markov process (models of sprays in turbulent flows, stochastic Cucker-Smale models, stochastic Landau damping), - dispersive models for the propagation of waves (e.g. Zakharov system, Klein-Gordon-Zakharov system, stochastic NLS equations). The numerical approximation of these models raises the following issues, which we will investigate. First, the construction and analysis of asymptotic-preserving schemes for equations in which the small parameter affects equally the deterministic ans stochastic terms. This concerns the design of schemes unconstrained by the small parameters for the original equations. Numerical schemes for the approximation of the reduced equations furnished by the theoretical analysis are another approach that we will develop. This requires the computation of the coefficients of the reduced equations. We will build accurate Heterogeneous Multiscale Methods (HMM) to that purpose. HMM in our context (kinetic, dispersive stochastic equations) have never been developed. Several questions of numerical analysis are related. Regarding these questions, we will analyse the efficiency of the the numerical schemes in the approximation of invariant measures or auto-correlations, their orders of convergence, and develop strategies to reduce the overall cost, like high-order integrators for invariant distributions, variance reduction strategies in Monte-Carlo methods.

This project is set in the blooming field of singular stochastic partial differential equations (singular SPDEs) that has undergone a revolution two-three years ago, with the joint introduction by Hairer and Gubinelli-Imkeller-Perkowski of entirely new methods. These works have opened a whole field and offer now the possibility to investigate a number of problems that were out of reach so far. We aim in this project to push further the study of their theoretical foundations and to investigate a number of challenging open questions about some special singular equations. 1. Developing the theory. The theory of regularity structures offers a very clean setting for the study of a class of parabolic singular SPDEs, called sub-critical. So far, the probabilistic structure has mainly been used as a tool to set the study of such an equation into the framework of regularity structures, by enriching the noise into a model. It is very likely that the stochastic cancellations inherent to the probabilistic objects will be instrumental in analyzing a number of problems beyond the local well-posedness problem, such as the global well-posedness problem, or the use of Malliavin calculus tools to investigate the existence and regularity of densities for solutions of some classes of singular sub-critical SPDEs. On the paracontrolled side, the theory was originally written as a 'first order Taylor expansion' theory. Despite its successes in recovering a number of results obtained via the theory of regularity structures, its present form prevents a priori its use in a number of problems and hides one of Hairer's other breakthrough, which is the introduction in his theory of a renormalization group. We intend to develop a higher order paracontrolled calculus and introduce in this setting an analogue of the renormalization group. 2. Qualitative properties of particular singular SPDEs. The powerful tools of regularity structures and paracontrolled calculus have mainly been used so far to derive existence, uniqueness and regularity results for singular PDEs. We feel this is the right time to explore further the full power of these techniques, and other techniques, to get a number of qualitative properties of solutions of important examples of singular equations, such as intermittency for the parabolic Anderson model equation, localization for the Anderson Hamiltonian, the study of the KPZ equation in space dimension greater than 1, or the stochastic Yang-Mills equation that comes from Parisi-Wu Euclidean quantization scheme. At the same time, we shall also investigate a number of PDEs for which totally different tools need to be used, such as fully nonlinear parbolic equations.

The consortium Thales R&T (TRT-Fr), LAAS, Institute Fresnel and IRMAR aims to go beyond the actual technological limits in the domain of phase-noise performances of microwave reference signals generated by optoelectronic oscillators (OEO) devices, in order to answer to the actual needs in telecommunication and defense. Indeed their actual evolution requires equipments that are even more embedded, compact and agile with growing bandwidth and optimized noise performances. This consortium brings together the necessary expertise for the sizing, modeling, fabrication and optimization of fibered Fabry-Perot mini-resonators with a centimeter length, delimited with innovative thin-film technology mirrors. Hence, we aim to reach with these resonators a quality factor (Q) higher than 10^8 for passive resonators and beyond by using erbium doped fiber, which will allow us to reach transparency and selective amplification regimes which are highly interesting regimes for the Q factor exaltation. Moreover, these resonators will not only answer to a compromise of finesse, but also a good reproducibility of fabrication and long term robustness of the coupling function. On top of that, the optical fibers show an excellent transparency (ranked at the second position behind crystalline material CaF2) and a large diversity of dispersive non-linear properties. Adding to this, there exists a wide scale of erbium doped amplification fibers. Hence optical fibers offer an unrivalled variety of resonant waveguides that can’t be proposed by alternative technologies developed nowadays. It is through the association of fiber technology with the worldwide excellence and know-how of Fresnel Institute in the realization of thin-film mirrors that we intend to develop an alternative technology of innovative mini-resonators showing a state of the art Q factor and a high coupling ability. We will demonstrate at TRT-Ft that the displayed qualities of such type of resonator can benefit to oscillators devices in order to synthesize 10 GHz microwave signals with state of the art phase noise performances that guarantee a simplification of oscillators systems (no need for selective RF filter, limitation of optical and microwave amplification systems). Two adaptations of the OEO device will be studied. The first one will be based on passive mini-resonator if Q factor is high enough. The other one will be based on erbium doped active resonator with an external pumping scheme that will guarantee the necessary inversion of population that is required for selective amplification. Last but not least, the state of the art know how of LAAS in the stabilization of laser sources on high Q (Q > 10^9) resonators will contribute to the qualification of the resonators made throughout the project. Then a new topology of coherent microwave frequency generation at multiples of 10 GHz will be experimentally demonstrated by the generation of Kerr combs in non-linear mini-resonators. A dedicated design of chirped thin-film mirrors adapted to the non-linear fiber properties will be studied in order to lower the Kerr comb power threshold to milliwatt range.