Littérature scientifique sur le sujet « Structured lattices »

La sécurité des protocoles cryptographiques repose sur la difficulté présumée de problèmes algorithmiques. Parmi ceux qui ont été identifiés jusque-là, les meilleurs problèmes pouvant servir de fondation à une cryptographie résistante aux ordinateurs quantiques sont issus des réseaux euclidiens. Les réseaux euclidiens sont une structure mathématique définie comme un ensemble de vecteurs de l’espace générés par les combinaisons entières d’un nombre fini de vecteurs réels linéairement indépendants (sa base). Un exemple typique de problème de sécurité relié est le Shortest Vector Problem (SVP). Étant donné une base d’un réseau euclidien en dimension n, trouver un vecteur court non nul. Pour des raisons d’efficacité, ces problèmes sont restreints à des réseaux issus de la théorie des nombres, dit structurés. Les hypothèses de sécurité relatives à ces réseaux particuliers étant différentes de celles des réseaux non structurés, il est nécessaire de les étudier spécifiquement, c’est l’objet de cette thèse. Nous avons étudié le cas des modules NTRU et uSVP en rang 2, prouvant que le problème SVP est équivalent sur ces deux familles de réseaux. Nous montrons également une réduction pire-cas vers cas-moyens pour les réseaux uSVP en rang 2. Ensuite nous avons démontré que résoudre SVP sur un idéal premier de petite norme n’était pas plus facile que de résoudre SVP sur n’importe quel idéal<br>The security of cryptographic protocols is based on the presumed difficulty of algorithmic problems. Among those identified so far, some of the best problems to serve as a foundation for quantum-proof cryptography come from lattices. Lattices are a mathematical structure defined as a set of space vectors generated by integer combinations of a finite number of linearly independent real vectors (its basis). A typical example of a related security problem is the Shortest Vector Problem (SVP). Given a base of an n-dimensional lattice, find a non-zero short vector. For efficiency reasons, these problems are restricted to lattices arising from number theory, known as structured lattices. As the security assumptions for these particular lattices are different from those for unstructured lattices, it is necessary to study them specifically, which is the aim of this thesis.We have studied the case of NTRU and uSVP modules in rank 2, proving that the SVP problem is equivalent on these two families of lattices. We also show a worst-case to average-case reduction for rank-2 uSVP lattices. Then we show that solving SVP on a prime ideal of small norm is no easier than solving SVP on any ideal

The fluid-structure interactions of wall-mounted slender structures, such as cilia, filaments, flaps, and flags, play an important role in a broad range of physical processes: from the coherent waving motion of vegetation, to the passive flow control capability of hair-like surface coatings. While these systems are ubiquitous, their coupled nonlinear response exhibits a wide variety of behaviours that is yet to be fully understood, especially when multiple structures are considered. The purpose of this work is to investigate, via numerical simulation, the fluid-structure interactions of arrays of slender structures over a range of input conditions. A direct modelling approach, whereby the individual structures and their dynamics are fully resolved, is realised via a lattice Boltzmann-immersed boundary model, which is coupled to two different structural solvers: an Euler-Bernoulli beam model, and a finite element model. Results are presented for three selected test cases - which build in scale from a single flap in a periodic array, to a small finite array of flaps, and finally to a large finite array - and the key behaviour modes are characterised and quantified. Results show a broad range of behaviours, which depend on the flow conditions and structural properties. In particular, the emergence of coherent waving motions are shown to be closely related to the natural frequency of the array. Furthermore, this behaviour is associated with a lock-in between the natural frequency of the array and the predicted frequency of the fluid instabilities. The original contributions of this work are: the development and application of a numerical tool for direct modelling of large arrays of slender structures; the characterisation of the behaviour of slender structures over a range of input conditions; and the exposition of key behaviour modes of slender structures and their relation to input conditions.

<p> Data structures are vital entities that strongly impact the efficiency of several software applications. Compactness, predictable memory access patterns, and good temporal and spacial locality of the structure's operations are increasingly becoming essential factors in the selection of a data structure for a specific application. In general, the less data we store and move the better for efficiency and power consumption, especially in infrastructure software and applications for hand-held devices like smartphones. In this dissertation, we extensively study a data structure named lattice data structure (LDS) that is as compact and suitable for memory hierarchies as the array, yet with a rich structure that enables devising procedures with better time bounds. </p><p> To achieve performance similar to the performance of the optimal O(log(N)) time complexity of the searching operations of other structures, we provide a hybrid searching algorithm that can be implemented by searching the lattice using the basic searching algorithm when the degree of the sortedness of the lattice is less than or equal to 0.9h, and the jump searching algorithm when the degree of the sortedness of the lattice is greater than 0.9h. A sorting procedure that can be used, during the system idle time, to incrementally increase the degree of sortedness of the lattice is given. We also provide randomized and parallel searching algorithms that can be used instead of the usual jump-and-walk searching algorithms. </p><p> A lattice can be represented by a one-dimensional array, where each cell is represented by one array element. The worst case time complexity of the basic LDS operations and the average time complexity of some of the order-statistic operations are better than the corresponding time complexities of most of other data structures operations. This makes the LDS a good choice for memory-constrained systems, for systems where power consumption is a critical issue, and for real-time systems. A potential application of the LDS is to use it as an index structure for in-memory databases.</p>

The application of lattice structures have become increasingly popular as additive manufacturing (AM) opens up the possibility to manufacture complex configurations. However, modelling such structures can be computationally expensive. The following thesis has been conducted in order for the department of Structural Analysis, at SAAB in Järfälla, to converge with the future use of AM and lattice structures. An approach to model lattice structures using homogenization is presented where three similar methods involving representative volume element (RVE) have been developed and evaluated. The stiffness matrices, of the RVEs, for different sizes of lattice structures, comprising of BCC strut-based units, have been obtained. The stiffness matrices were compared and analysed on a larger solid structure in order to see the deformational predictability of a lattice-based structure of the same size. The results showed that all methods were good approximations with slight differences in terms of boundary conditions (BCs) at the outer edge. The comparative analyses showed that two of the three methods matches the deformational predictability. The BCs in all methods have different influences which makes it pivotal to establish the BCs of the structure before using the approach presented in this thesis.<br>Ökad implementering av gitterstrukturer i komponenter är ett resultat av utvecklingen inom additiv tillverkning. Metoden öppnar upp för tillverkning av komplexa strukturer med färre delmoment. Dock så uppkommer det svårigheter vid simulering av dessa komplexa strukturer då beräkningar snabbt tyngs ner med ökad komplexitet. Följande examensarbete har utförts hos avdelningen Strukturanalys, på SAAB i Järfälla, för att de ska kunna möta upp det framtida behovet av beräkningar på additivt tillverkade gitterstrukturer. I det här arbetet presenteras ett tillvägagångsätt för modellering av gitterstrukturer med hjälp av represantiva volymselement. Styvhetsmatriser har räknats fram, för en vald gitterkonfiguration, som sedan viktats mot tre snarlika representativa volymselement. En jämförelseanalys mellan de olika styvhetsmatriserna har sedan gjorts på en större och solid modell för att se hur väl metoderna förutsett deformationen av en gitterstruktur i samma storlek. Resultaten har visat att samtliga metoder är bra approximationer med tämligen små skillnader från randeffekterna. Vid jämförelseanalysen simulerades gitterstrukturen bäst med två av de tre metoder. En av slutsatserna är att det är viktigt att förstå inverkan av randvillkoren hos gitterstrukturer innan implementering görs med det tillvägagångssätt som presenterats i det här examensarbetet.