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).

Lie algebras of maximal class (or filiform Lie algebras) are the Lie-theoretic analogue of pro-p-groups of maximal class. In particular, they are 2-generated. If one further assumes that the algebras are graded over the positive integers, then over a field of characteristic p it has been shown that a classification is possible provided one generator has degree 1 and the other has either degree 1 or 2. In this thesis I give a classification of graded Lie algebras of maximal class with generators of degree 1 and p, respectively.

Given a linear code, it is important both to identify fast decoding algorithms and to estimate the rst terms of its weight distribution. Ecient decoding algorithms allow the exploitation of the code in practical situations, while the knowledge of the number of small-weight codewords allows to estimate its decoding performance. For ane-variety codes and its subclass formed by Hermitian codes, both problems are as yet unsolved. We investigate both and provide some solutions for special cases of interest. The rst problem is faced with use of the theory of Gröbner bases for zero-dimensional ideals. The second problem deals in particular with small-weight codewords of high-rate Hermitian codes. We determine them by studying some geometrical properties of the Hermitian curve, specically the intersection number of the curve with lines and parabolas.

Channel coding is the branch of Information Theory which studies the noise that can occur in data transmitted through a channel. Algebraic Coding Theory is the part of Channel Coding which studies the possibility to detect and correct errors using algebraic and geometric techniques. Nowadays, the best performing linear codes are known to be mostly algebraic geometry codes, also named Goppa codes, which arise from an algebraic curve over a finite field, by the pioneering construction due to V. D. Goppa. The best choices for curves on which Goppa's construction and its variants may provide codes with good parameters are those with many rational points, especially maximal curves attaining the Hasse-Weil upper bound for the number of rational points compared with the genus of the curve. Unfortunately, maximal curves are difficult to find. The best known examples of maximal curves are the Hermitian curve, the Ree curve, the Suzuki curve, the GK curve and the GGS curve. In the present thesis, we construct and investigate algebraic geometry codes (shortly AG codes), their parameters and automorphism groups.

Public key cryptography allows two or more users to communicate in a secure way on an insecure channel, using two different keys: a public key, which has the function to encrypt the messages, and a private key, employed in the decryption of the ciphertext. Because of the importance of these keys,their generation is a sensible issue and it is often based on an underlying mathematical problem, which is considered hard to be solved. Among these difficult problems, the Integer Factorization Problem (IFP) is one of the most famous: given a composite integer number, recovering its factors is commonly believed to be hard (worst-case complexity). In this thesis, after a brief explaination of the developments on integer factorization and a description of the General Number Field Sieve (GNFS), we will present different strategies to face this well-known problem of Number Theory. First, we will employ elementary remarks on modular arithmetic to produce a formula that describes the remainders of a given integer, starting from just three monotonic remainders and we will link this result to the behaviour of a second-degree interpolating polynomial. Secondly, we will show an attempt to improve GNFS, considering two linearly disjoint quadratic fields and study the relation between first-degree prime ideals. Finally, we will characterize the elements used in GNFS through some systems having integer solutions, that can be found using Groebner Bases.

Historically, the study of modules over finite dimensional algebras has started with the study of the ones with finite dimension. This is sufficient when dealing with a finite dimensional algebra of finite representation type, where there are only finitely many indecomposable modules of finite length. Indecomposable modules of infinite length occur when dealing with algebras of infinite representation type and the study of pure-injective modules over a finite dimensional algebra is crucial for the problem of describing infinite dimensional modules. In this talk, we consider a specific class of finite dimensional algebras of infinite representation type, called "tubular algebras". Pure-injective modules over tubular algebra have been partially classified by Angeleri Hügel and Kussin, in 2016, and we want to give a contribution to the classification of the ones of "irrational slope". In this talk, first, via a derived equivalence, we move to a more geometrical framework, ie. we work in the category of quasi-coherent sheaves over a tubular curve, and we approach our classification problem from the point of view of tilting/cotilting theory. More precisely, we consider specific torsion pairs cogenerated by infinite dimensional cotilting sheaves and we study the Happel-Reiten-Smalø heart of the corresponding t-structure in the derived category. These hearts are locally coherent Grothendieck categories and, in these categories, the pure-injective sheaves over the tubular curve become injective objects. In order to study injective objects in a Grothendieck category is fundamental the classification of the simple objects. In the seminar, we use some techniques coming from continued fractions and universal extensions to provide a method to construct an infinite dimensional sheaf of a prescribed irrational slope that becomes simple in the Grothendieck category given as the heart of a precise t-structure.

The aim of this thesis is to begin the study of graded Lie algebras of maximal class over a field of odd characteristic, which are generated by two elements of different weights.

Gröbner bases are special sets of polynomials, which are useful to solve problems in many fields such as computer vision, geometric modeling, geometric theorem proving, optimization, control theory, statistics, communications, biology, robotics, coding theory, and cryptography. The major disadvantage of algorithms to compute Gröbner bases is that computations can use a lot of computer power. One of the reasons is the amount of useless critical pairs that the algorithm has to compute. Hence, a lot of effort has been put into developing new criteria to detect such pairs in advance. This thesis is devoted to describe efficient algorithms for the computation of Gröbner bases, with particular emphasis to those based on polynomial signatures. The idea of associating each polynomial with a signature on which the criteria and reduction steps depend, has become extremely popular in part due to its good performance. Our main result combines the criteria from Gao-Volny-Wang's algorithm with the knowledge of Hilbert Series. A parallel implementation of the algorithm is also investigated to improve the computational efficiency. Our algorithm is implemented in CoCoALib, a C++ free library for computations in commutative algebra.

We present some upper bounds on the size of nonlinear codes and their restriction to systematic codes and linear codes. These bounds, which are an improvement of a bound by Zinoviev, Litsyn and Laihonen, are independent of other classical known theoretical bounds. Among these, we mention the Griesmer bound for linear codes, of which we provide a partial generalization for the systematic case. Our experiments show that in some cases (the majority of cases for some q) our bounds provide the best value, compared to all other closed-formula upper-bounds. We also present an algebraic method for computing the minimum weight of any nonlinear code. We show that for some particular code, using a non-standard representation of the code, our method is faster than brute force. An application of this method allows to compute the nonlinearity of a Boolean function, improving existing algebraic methods and reaching the same complexity of algorithms based on the fast Fourier transform.

Internet is evolving in both its structure and usage patterns; this work addresses two trends: i) the increasing popularity and the related generated traffic of media streaming applications and ii) the emerging of network portions following different philosophies from the rest of the internet and being characterized by a mesh topology, such as Community Networks. This thesis presents a modeling for decentralized live streaming for mesh networks based on graph theory, considering the different inter-dependent network abstractions involved. It proposes optimization strategies based on popular centrality metrics, such as betweenness and PageRank. Results on real-world datasets validate the theoretical work and the derived optimizing strategies are implemented in open-source streaming platforms.