In this work we deal with both Coding Theory and Entropy Extraction for Random Number Generators to be used for cryptographic purposes. We start from a thorough analysis of known bounds on code parameters and a study of the properties of Hadamard codes. We find of particular interest the Griesmer bound, which is a strong result known to be true only for linear codes. We try to extend it to all codes, and we can determine many parameters for which the Griesmer bound is true also for nonlinear codes. In case of systematic codes, a class of codes including linear codes, we can derive stronger results on the relationship between the Griesmer bound and optimal codes. We also construct a family of optimal binary systematic codes contradicting the Griesmer bound. Finally, we obtain new bounds on the size of optimal codes. Regarding the study of random number generation, we analyse linear extractors and their connection with linear codes. The main result on this topic is a link between code parameters and the entropy rate obtained by a processed random number generator. More precisely, to any linear extractor we can associate the generator matrix of a linear code. Then, we link the total variation distance between the uniform distribution and the probability mass function of a random number generator with the weight distribution of the linear code associated to the linear extractor. Finally, we present a collection of results derived while pursuing a way to classify optimal codes, such as a probabilistic algorithm to compute the weight distribution of linear codes and a new bound on the size of codes.

In the first part of the thesis I produce and implement an algorithm for obtaining generators of the unit group of the integral group ring ZG of finite abelian group G. We use our implementation in MAGMA of this algorithm to compute the unit group of ZG for G of order up to 110. In the second part of the thesis I show how to construct multiplication tables of the semisimple real Lie algebras. Next I give an algorithm, based on the work of Sugiura, to find all Cartan subalgebra of such a Lie algebra. Finally I show algorithms for finding semisimple subalgebras of a given semisimple real Lie algebra.

Cryptography is used to protect data and communications. The basic tools are cryptographic primitives, whose security and efficiency are widely studied. But in real-life applications these primitives are not used individually, but combined inside complex protocols. The aim of this thesis is to analyse various cryptographic protocols and assess their security in a formal way. In chapter 1 the concept of formal proofs of security is introduced and the main categorisation of attack scenarios and types of adversary are presented, and the protocols analysed in the thesis are briefly introduced with some motivation. In chapter 2 are presented the security assumptions used in the proofs of the following chapters, distinguishing between the hardness of algebraic problems and the strength of cryptographic primitives. Once that the bases are given, the first protocols are analysed in chapter 3, where two Attribute Based Encryption schemes are proven secure. First context and motivation are introduced, presenting settings of cloud encryption, alongside the tools used to build ABE schemes. Then the first scheme, that introduces multiple authorities in order to improve privacy, is explained in detail and proven secure. Finally the second scheme is presented as a variation of the first one, with the aim of improving the efficiency performing a round of collaboration between the authorities. The next protocol analysed is a tokenization algorithm for the protection of credit cards. In chapter 4 the advantages of tokenization and the regulations required by the banking industry are presented, and a practical algorithm is proposed, and proven secure and compliant with the standard. In chapter 5 the focus is on the BIX Protocol, that builds a chain of certificates in order to decentralize the role of certificate authorities. First the protocol and the structure of the certificates are introduced, then two attack scenarios are presented and the protocol is proven secure in these settings. Finally a viable attack vector is analysed, and a mitigation approach is discussed. In chapter 6 is presented an original approach on building a public ledger with end-to-end encryption and a one-time-access property, that make it suitable to store sensitive data. Its security is studied in a variety of attack scenarios, giving proofs based on standard algebraic assumptions. The last protocol presented in chapter 7 uses a proof-of-stake system to maintain the consistency of subchains built on top of the Bitcoin blockchain, using only standard Bitcoin transactions. Particular emphasis is given to the analysis of the refund policies employed, proving that the naive approach is always ineffective whereas the chosen policy discourages attackers whose stake falls below a threshold, that may be adjusted varying the protocol parameters.

We provide an explicit procedure to glue (not necessarily compact) silting objects along recollements of triangulated categories with coproducts having a ‘nice’ set of generators, namely, well generated triangulated categories. This procedure is compatible with gluing co-t-structures and it generalizes a result by Liu, Vitória and Yang. We provide conditions for our procedure to restrict to tilting objects and to silting and tilting modules. As applications, we retrieve the classification of silting modules over the Kronecker algebra and the classification of non-compact tilting sheaves over a weighted noncommutative regular projective curve of genus 0.

Block ciphers and their security are the main subjects of this work. In the first part it is described the impact of differential cryptanalysis, a powerful statistical attack against block ciphers, when operations different from the one used to perform the key addition are considered on the message space. It is proven that when an alternative difference operation is carefully designed, a cipher that is proved secure against classical differential cryptanalysis can instead be attacked using this alternative difference. In the second part it is presented a new design approach of round functions for block ciphers. The proposed round functions can give to the cipher a potentially better level of resistance against statistical attacks. It is also shown that the corresponding ciphers can be proven secure against a well-known algebraic attack, based on the action of the permutation group generated by the round functions of the cipher.

Let C be a compact Riemann surface. Let us consider a finite group acting on CxC, having some elements that exchange the factors, and assume that the subgroup of those elements that do not exchange the factors acts freely. We call the quotient a Semi-isogenous Mixed Surface. In this work we investigate these surfaces and we explain how their geometry is encoded in the group. Based on this, we present an algorithm to classify the Semi-isogenous Mixed Surfaces with given geometric genus, irregularity and self-intersection of the canonical class. In particular we give the classification of Semi-isogenous Mixed Surfaces with K^2>0 and holomorphic Euler-Poincaré characteristic equal to 1, where new examples of minimal surfaces of general type appear. Minimality of Semi-isogenous Mixed Surfaces is discussed using two different approaches. The first one involves the study of the bicanonical system of such surfaces: we prove that we can relate the dimension of its first cohomology group to the rank of a linear map that involves only curves. The second approach exploits Hodge index theorem to bound the number of exceptional curves that live on a Semi-isogenous Mixed Surface.

In this thesis I've explored the theory of quaternionic slice regular functions. More precisely I've studied some rigidity properties, some differential issues and an application in complex differential geometry. This application concerns the constructions and classifications of orthogonal complex structures on open domains of the four dimensional euclidean space.

The Advanced Encryption Standard (AES) is nowadays the most widespread block cipher in commercial applications. It represents the state-of-art in block cipher design and provides an unparalleled level of assurance against all known cryptanalytic techniques, except for its reduced versions. Moreover, there is no known efficient way to distinguish it from a set of random permutations. The AES (and other modern block ciphers) presents a highly algebraic structure, which led researchers to exploit it for novel algebraic attacks. These tries have been unsuccessful, except for academic reduced versions. Starting from an intuition by I. Toli, we have developed a mixed algebraic-statistical attack. Using the internal algebraic structure of any AES-like cipher, we build an algebraic setting where a related-key (statistical) distinguishing attack can be mounted. Our data reveals a significant deviation of the full AES-128 from a set of random permutations. Although there are recent successful related-key attacks on the full AES-192 and the full AES-256 (with non-practical complexity), our attack would be the first-ever practical distinguishing attack on the full AES-128 (to the best of our knowledge).

In this thesis we define and study the mixed quasi-étale surfaces. In particularwe classify all the mixed quasi-étale surfaces whose minimal resolution of the singularities is a regular surface with p_g=0 and K^2>0. It is a well known fact that each Riemann surface with p_g=0 is isomorphic to P^1. At the end of XIX century M. Noether conjectured that an analogous statement holds for the surfaces: in modern words, he conjectured that every smooth projective surface with p_g=q=0 be rational. The first counterexample to this conjecture is due to F. Enriques (1869). He constructed the so called Enriques surfaces. The Enriques-Kodaira classification divides compact complex surfaces in four main classes according to their Kodaira dimension k: -oo, 0, 1, 2. A surface is said to be of general type if k=2. Nowadays this class is much less understood than the other three. The Enriques surfaces have k=0. The first examples of surfaces of general type with p_g=0 have been constructed in the 30's by L. Campedelli e L. Godeaux. The idea of Godeaux to construct surfaces was to consider the quotient of simpler surfaces by the free action of a finite group. In this spirit, Beauville proposed a simple construction of surfaces of general type, considering the quotient of a product of two curves C_1 and C_2 by the free action of a finite group G. Moreover he gave an explicit example with p_g=q=0 considering the quotient of two Fermat curves of degree 5 in P^2. There is no hope at the moment to achieve a classification of the whole class of the surfaces of general type. Since for a surface in this class the Euler characteristic of the structure sheaf \chi is strictly positive, one could hope that a classification of the boundary case \chi=1 is more affordable. Some progresses in this direction have been done in the last years through the work of many authors, but this (a priori small) case has proved to be very challenging, and we are still very far from a classification of it. At the same time, this class of surfaces, and in particular the subclass of the surfaces with p_g=0 contains some of the most interesting surfaces of general type. If S is a surface of general type with \chi=1, which means p_g=q, then p_g = q < 5, and if p_g=q=4, then S is birational to the product of curves of genus 2. The surfaces with p_g = q = 3 are completely classified. The cases p_g = q < 3 are still far from being classified. Generalizing the Beauville example, we can consider the quotient (C_1 x C_2)/G, where the C_i are Riemann surfaces of genus at least two, and G is a finite group. There are two cases: the mixed case where the action of G exchanges the two factors (and then C_1 = C_2); and the unmixed case where G acts diagonally. Many authors studied the surfaces birational to a quotient of a product of two curves, mainly in the case of surfaces of general type with \chi=1. In all these works the authors work either in the unmixed case or in the mixed case under the assumption that the group acts freely. The main purpose of this thesis is to extend the results and the strategies of the above mentioned papers in the non free mixed case. Let C be a Riemann surface of genus at least 2, let G be a finite group that acts on C x C with a mixed action, i.e. there exists an element in G that exchanges the two factors. Let G^0 be the index two subgroup of the elements that do not exchange the factors. We say that X=(C x C)/G is a mixed quasi-étale surface if the quotient map C x C -> (C x C)/G has finite branch locus. We present an algorithm to construct regular surfaces as the minimal resolution of the singularities of mixed quasi-étale surfaces. We give a complete classification of the regular surfaces with p_g=0 and K^2>0 that arise in this way. Moreover we show a way to compute the fundamental group of these surfaces and we apply it to the surfaces we construct. Some of our construction are more interesting than others. We have constructed two numerical Campedelli surfaces (K^2 = 2) with topological fundamental group Z/4Z. Two of our constructions realize surfaces whose topological type was not present in the literature before. We also have three examples of Q-homology projective planes, and two of them realize new examples of Q-homology projective planes.

ABSTRACT: Roughly speaking, the moduli space of higher spin curves parametrizes equivalence classes of pairs (C, L) where C is a smooth genus g algebraic curve and L is a line bundle on it whose r-th tensor power is isomorphic to the canonical bundle of the curve. The aim of the talk is to discuss important geometrical properties of these spaces under different points of view: one possible compactification together with the description of the rational Picard group, their birational geometry in some low genus cases and their relation with some special locus inside the classical moduli spaces of curves.