In this thesis we study products of linear error correcting codes. Error correcting codes are used to correct the errors introduced by some noisy communication channel and are essential in all... Show moreIn this thesis we study products of linear error correcting codes. Error correcting codes are used to correct the errors introduced by some noisy communication channel and are essential in all communications that, due to economic or practical constraints, do not allow data retransmission: for instance deep space communications, broadcasting and mass storage. Their products, throughout the last forty years, have appeared in many different fields, such as cryptography, complexity theory, additive combinatorics and cryptanalysis. We study such products and discuss applications to cryptography. First, we prove that typically the product operation generates trivial codes; then, we investigate and characterize some class of codes whose products are non trivial and satisfy interesting properties. Our methods are algebraic-combinatorial in nature, though sometimes probabilistic techniques will be involved. Show less
This dissertation presents new cryptographic protocols, which can be divided into two families. Protocols in the first family achieve unilateral security: this means that they protect legitimate... Show moreThis dissertation presents new cryptographic protocols, which can be divided into two families. Protocols in the first family achieve unilateral security: this means that they protect legitimate users against an external attacker. Concretely, we assume that two users wish to communicate securely over a given communication system, where an external attacker eavesdrops and tampers with some of the wires of the system. We contribute to the topic by presenting protocols with improved efficiency and a simpler definition compared to previous work, and we design interactive protocols that achieve security against a stronger attacker.Protocols of the second type achieve multilateral security, meaning that they protect users against each other. This is the case for multi-party computation or MPC, where several users wish to compute a function on private inputs while keeping inputs private and without appealing to a trusted third party; we contribute to this topic by adding a cheater-detection functionality to a well-established MPC protocol.A key component that underlies these scenarios is secret sharing; we investigate this topic by casting in particular a new light on its connections with coding theory. This allows us to better harness the features of recent code constructions to obtain improved secret-sharing schemes. Show less
This thesis deals with properties of Jacobians of genus two curves that cover elliptic curves. If E is an elliptic curve and C is a curve of genus two that covers it n-to-1 so that the covering... Show moreThis thesis deals with properties of Jacobians of genus two curves that cover elliptic curves. If E is an elliptic curve and C is a curve of genus two that covers it n-to-1 so that the covering does not factor through an isogeny, then C also covers another elliptic curve n-to-1 in such a way and the Jacobian of C is isogenous to the product of the two elliptic curves. The Jacobian is said to be (n,n)-split and the elliptic curves are said to be glued along their n-torsion. The first chapter deals with the geometric aspects of this setup. We describe two approaches to constructing (n,n)-split Jacobians and we investigate which curves can appear in the setup. The second chapter deals with the arithmetic aspects, focusing on height functions and the Lang-Silverman conjecture in particular. We show that this conjecture holds for families of (n,n)-split Jacobians if and only if it holds for the corresponding families of elliptic curves that can be glued along their n-torsion. Show less
This thesis concerns the relation bettween the good reduction of Shimura varieties and the associated loop groups. To be precise, by studying the Breuil-Kisin modules of p-divisible groups, we... Show moreThis thesis concerns the relation bettween the good reduction of Shimura varieties and the associated loop groups. To be precise, by studying the Breuil-Kisin modules of p-divisible groups, we construct a direct morphism from the special fibre of a Shimura varieties of Hodge type to an fpqc subquotient of the associated loop group, and show that the geometric fibres of the morphism gives back the Ekedahl-Oort strata of the Shimura varieties in question. This can be seen as an alternative definition of the Ekedahl-Oort stratification of Shimura varieties, which gives a conceptual explanation of E. Viehmann's new invariants of "truncation of level 1" of loop groups. Show less
In this thesis we classify the pairs (p,G), where p is a prime number and G is a finite p-group possessing an intense automorphism, i.e. an automorphism that sends each subgroup of G to a G... Show moreIn this thesis we classify the pairs (p,G), where p is a prime number and G is a finite p-group possessing an intense automorphism, i.e. an automorphism that sends each subgroup of G to a G-conjugate, that is non-trivial and whose order is coprime to p. Show less
In this thesis we develop a theory of relative displays and give hence a description of stratifications of Hilbert modular varieties with parahoric level structure in the case of bad... Show moreIn this thesis we develop a theory of relative displays and give hence a description of stratifications of Hilbert modular varieties with parahoric level structure in the case of bad reduction. Show less
Computed Tomography (CT) is an imaging technique that is used to calculate the interior of an object using X-rays under multiple projection angles. A well-known application is medical imaging with... Show moreComputed Tomography (CT) is an imaging technique that is used to calculate the interior of an object using X-rays under multiple projection angles. A well-known application is medical imaging with a CT-scanner. The reconstruction methods can roughly be divided into two categories: analytical reconstruction methods and algebraic reconstruction methods (ARMs). An example of an algorithm from the first category is Filtered Backprojection (FBP). This method has a high computational efficiency and it performs well in cases with many equiangularly distributed projection angles and high signal-to-noise ratio. ARMs require in general more computation time. They are more robust with respect to noise and can handle few projection angles or a limited angular range better. In this dissertation, the new algorithm Algebraic filter – Filtered Backprojection (AF-FBP) is introduced, which uses an ARM to create filters that can be used in FBP. The reconstruction quality of AF-FBP approximates that of the corresponding (locally) linear ARM, while the reconstructions are obtained with the computational efficiency of FBP. In cases with a small number of different scanning geometries, using AF-FBP enables the reconstruction of images of relatively high quality for few projection angles, limited angular range, or low signal-to-noise ratio. Show less
This thesis is composed of papers on four topics: Bayesian theory for the sparse normal means problem, specifically for the horseshoe prior (Chapters 1-3), Bayesian theory for community detection ... Show moreThis thesis is composed of papers on four topics: Bayesian theory for the sparse normal means problem, specifically for the horseshoe prior (Chapters 1-3), Bayesian theory for community detection (Chapter 4), nested model selection (Chapter 5), and the application of competing risk methods in the presence of time-dependent clustering (Chapter 6). Show less
In this dissertation, a primitive recursive algorithm is given for the computation of the étale Euler-Poincaré characteristic (which is the alternating sum of the étale cohomology groups in... Show moreIn this dissertation, a primitive recursive algorithm is given for the computation of the étale Euler-Poincaré characteristic (which is the alternating sum of the étale cohomology groups in the Grothendieck group of Galois modules) with finite coefficients, and on arbitrary varieties over a field. For smooth curves, a primitive recursive algorithm is given for the computation of the étale cohomology groups themselves, using a geometric interpretation of the elements of the first etale cohomology. The general case is then reduced to the case of smooth curves by making the standard dévissage techniques explicit. Show less
This thesis covers different problems concerning the evaluation of DNA evidence. It is mainly divided into two parts: the first regards the DIP-STR genotyping techniques. It addresses the... Show moreThis thesis covers different problems concerning the evaluation of DNA evidence. It is mainly divided into two parts: the first regards the DIP-STR genotyping techniques. It addresses the imperative need of developing a model to assign the likelihood ratio for DIP-STR results, and compares, from a statistical and forensic perspective, the advantages of this novel marker system compared to traditional marker systems, such as STR and Y-STR. The second part deals with several more general statistical aspects involved in the evaluation of DNA evidence. It aims at defining the differences between full Bayesian methods and ad hoc plug-in approximations, and at solving the rare type match problem for Y-STR data. The issues of the different reductions of data and of the levels of uncertainty involved in frequentist solutions are also discussed. These two parts are connected in the final project, by developing a Bayesian solution for the rare type match problem for DIP-STR marker system. Moreover, the initial model for DIP-STR data is improved in the light of the statistical discussion of the second part: any ad hoc solution is avoided to obtain a full Bayesian approach. Show less
