Lattice-point enumerators of ellipsoids

Minkowski's second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded above by a quantity involving all the successive minima of K with respect to \Lambda. We will prove here that the number of lattice points inside K can also accept an upper bound of roughly the same size, in the special case where K is an ellipsoid. Whether this is also true for all K unconditionally is an open problem, but there is reasonable hope that the inductive approach used for ellipsoids could be extended to all cases.Comment: 9 page

Attacking (EC)DSA scheme with ephemeral keys sharing specific bits

In this paper, we present a deterministic attack on (EC)DSA signature scheme, providing that several signatures are known such that the corresponding ephemeral keys share a certain amount of bits without knowing their value. By eliminating the shared blocks of bits between the ephemeral keys, we get a lattice of dimension equal to the number of signatures having a vector containing the private key. We compute an upper bound for the distance of this vector from a target vector, and next, using Kannan's enumeration algorithm, we determine it and hence the secret key. The attack can be made highly efficient by appropriately selecting the number of shared bits and the number of signatures

Minkowski’s successive minima in convex and discrete geometry

In this short survey we want to present some of the impact of Minkowski’s successive minima within Convex and Discrete Geometry. Originally related to the volume of an o-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel’s lemma

Lattice point inequalities and face numbers of polytopes in view of central symmetry

Magdeburg, Univ., Fak. für Mathematik, Diss., 2012von Matthias Henz

Adelic convex geometry of numbers

Magdeburg, Univ., Fak. für Mathematik, Diss., 2014von Carsten Thie

Analyse statique des systèmes de contrôle-commande : invariants entiers et flottants

A critical software is a software whose malfunction may result in death or serious injury to people, loss or severe damage to equipment or environmental harm.Software engineering for critical systems is particularly difficult, and combines different methods to ensure the quality of produced software.Among them, formal methods can be used to prove that a software obeys its specifications.This thesis falls within the context of the validation of safety properties for critical software, and more specifically, of numerical properties for embedded software in control-command systems.The first part of this thesis deals with Lyapunov stability proofs.These proofs rely on computations with real numbers, and do not accurately describe the behavior of a program run on a platform with machine arithmetic.We introduce a generic, theoretical framework to adapt the arguments of Lyapunov stability proofs to machine arithmetic.A tool automatically translates the proof on real numbers to a proof with floating-point numbers.The second part of the thesis focuses on linear relation analysis, using an abstract interpretation based on the approximation by convex polyhedrons of valuations associated with each control point in a program.We present ALICe, a framework to compare different invariant generation techniques.It comes with a collection of test cases taken from the program analysis literature, and interfaces with three tools, that rely on different algorithms to compute invariants: Aspic, iscc and PIPS.To refine PIPS results, two code restructuring techniques are introduced, and several improvements are made to the invariant generation algorithms and evaluated using ALICe.Un logiciel critique est un logiciel dont le mauvais fonctionnement peut avoir un impact important sur la sécurité ou la vie des personnes, des entreprises ou des biens.L'ingénierie logicielle pour les systèmes critiques est particulièrement difficile et combine différentes méthodes pour garantir la qualité des logiciels produits.Parmi celles-ci, les méthodes formelles peuvent être utilisées pour prouver qu'un logiciel respecte ses spécifications.Le travail décrit dans cette thèse s'inscrit dans le contexte de la validation de propriétés de sûreté de programmes critiques, et plus particulièrement des propriétés numériques de logiciels embarqués dans des systèmes de contrôle-commande.La première partie de cette thèse est consacrée aux preuves de stabilité au sens de Lyapunov.Ces preuves s'appuient sur des calculs en nombres réels, et ne sont pas valables pour décrire le comportement d'un programme exécuté sur une plateforme à arithmétique machine.Nous présentons un cadre théorique générique pour adapter les arguments des preuves de stabilité de Lyapunov aux arithmétiques machine.Un outil effectue automatiquement la traduction de la preuve en nombres réels vers une preuve en nombres a virgule flottante.La seconde partie de la thèse porte sur l'analyse des relations affines, en utilisant une interprétation abstraite basée sur l'approximation des valuations associées aux points de contrôle d'un programme par des polyèdres convexes.Nous présentons ALICe, un framework permettant de comparer différentes techniques de génération d'invariants.Il s'accompagne d'une collection de cas de tests tirés de publications sur l'analyse de programmes, et s'interface avec trois outils utilisant différents algorithmes de calcul d'invariants: Aspic, iscc et PIPS.Afin d'affiner les résultats de PIPS, deux techniques de restructuration de code sont introduites, et plusieurs améliorations sont apportées aux algorithmes de génération d'invariants et évaluées à l'aide d'ALICe

Atomistic Studies of Structure and Dynamics of Aggregation in Solution: From Traditional Surfactant Micelles to Surfactant-Free Microemulsions

The properties of the micellar structures formed by surfactant molecules in aqueous solution has been a widely discussed topic for the last century. Numerous hypotheses evolved in the effort to characterize the shape, internal structure, and driving force in the emergence of these aggregates. Many of the disputes can be ascribed to the fact that precise examination would require observations on a molecular level, which is difficult to achieve with experimental techniques in such dynamic colloidal systems. Therefore, theories often necessarily rely on the interpretation of circumstantial evidence rather than direct proof. In this study, the subject is approached with atomistic molecular dynamics simulations, that provide atomistic insight into the assembly of micelles formed by the ionic surfactants n-octyltrimethylammonium bromide, n-decyltrimethylammonium bromide, n-dodecyltrimethylammonium bromide, sodium octanoate, and sodium decanoate in water. In the course of these simulations, the spontaneous aggregation of surfactant molecules from initially homogeneous distribution is sampled and data acquisition is progressed far beyond the point where dynamic equilibrium is reached. This converged stage is in each case characterized by a noticeable degree of polydispersity that describes a bell-shaped peak in a histogram of aggregation numbers. Targeted analysis of micelles close to the most probable aggregation number reveals objects of globular shape with a diffuse interfacial region. In this assembly, the molecular subunits preferably occupy certain radial positions with the apolar alkyl tails in the micelle center and the polar groups oriented towards the solvent. Compared to idealized representations, however, this ordering is significantly defused and even the terminal methyl groups have a finite probability to be located at the micelle/water interface. The internal configuration that leads to this occurrence is a combination of a few surfactant molecules constituting a radially aligned molecular scaffold and a number of tangentially attached monomers. In case of the Cₓ-trimethylammonium bromide micelles, a consequence of this mixed array is a rough aggregate surface with deep crevices that allow for significant water penetration up to a buried hydrophobic core. The anionic species form a notably smoother boundary by tighter packing and comprise a larger water-free region in the micelle center, that, however, still only makes up 40% of the micelle volume. The immediate solvent shell of the aggregates is characterized by a substantial decrease of water-water hydrogen bonds that is more pronounced in case of the anionic compounds, where the carboxylate group represents a dedicated hydrogen-bonding site. The monomer exchange rate is found to be strongly decelerated with increasing aliphatic tail length and ranges between 10⁷ s⁻¹ and 10⁹ s⁻¹ for the cationic and 10⁷ s⁻¹ and 10⁸ s⁻¹ for the anionic micelles. The acquired insights on traditional surfactant micelles provide a useful background for the characterization of aggregation in solutions without surfactants. Speculation on such surfactant-free microemulsions dates back to the 1970s but the idea was dismissed for a long time, not least due to the absence of experimental evidence. Recent studies, however, confirmed this conjecture with direct proof of micelle-like oil-in-water aggregates in the ternary system water/ethanol/octan-1-ol. The weakly associated structures occur for a certain range of compositions that constitute the so-called “pre-Ouzo region” in the monophasic part of the ternary phase diagram. The hydrotrope ethanol enables the stability of these octanol droplets on the nanometer scale by accumulating to a film at the interface similar to that formed by surfactant molecules in conventional microemulsions. A variety of aggregation numbers is observed, that encompasses a continuous spectrum of nanoordering. The smaller clusters are of globular shape and feature a roughly radial symmetric assembly with a core region composed of mostly terminal carbon units and an outer shell enriched with hydroxyl groups. This resemblance to traditional micelles fades towards larger aggregation numbers, where increasingly amorphous structures are detected. The system is governed by quick rearrangements and the mechanisms of micelle growth due to inter-particle collisions and shrinkage by decomposition into smaller aggregates play a major role. Not all of the cluster sizes that occur during these fluctuations are equally stable and certain aggregation numbers can be characterized as transitory stages during processes of swelling or depletion. The surfactant-free microemulsion exhibits excellent solubilizing properties for the apolar substances propane and squalene. For the first, the solvation free energy in the micelle interior is close to the favorable value predicted for a similarly composed binary octanol/ethanol solution, whereas the aqueous pseudo-phase poses a poor solvent. In the presence of the C₃₀ compound, octanol domains expand to enclose the large organic molecules in comparatively stable structures. This leads to significantly enhanced solubilization over a binary water/ethanol mixture of equal proportions. The primary consequence of the addition of the electrolytes NaI and LiCl is a stabilization of the aggregates via the salting-out of ethanol. Preferential adsorption of I⁻ or Li⁺ at the interface is only weakly observed due to the local accumulation of ethanol and octanol hydroxyl groups that leave little apolar surface exposed. In further analogy to traditional microemulsion systems, oil-in-water aggregation reverses to water-in-oil aggregation in excess of octanol. These two regimes are linked by a bicontinuous region, where the octanol and water pseudo-phases entwine to form a sponge-like network

Marine Ecosystem Challenges & Opportunities (MECOS 3)

The Marine Biological Association of India (MBAI), established in 1958, is proud to gear up for MECOS3, the third symposium on Marine Ecosystems- Challenges and Opportunities during 7-10 January, 2020. The MBAI besides organising MECOS1 (2009) and MECOS2 (2014) has inculcated active interest and participation among its members by handling several national and international symposia/seminars, since its formation. The MBAI has 794 life members and 20 institutional members. The mandate of the MBAI is promotion of scientific research in the field of marine biology and allied sciences