Error correcting codes has become an integral part of the design of reliable data transmissions and storage systems. They are also playing an increasingly important role for other applications such as the analysis of pseudorandom sequences and the design of secure cryptosystems. Cyclic codes form a large class of widely used error correcting codes, including important codes such as the Bose-Chaudhuri-Hocquenghem (BCH) codes, quadratic residue (QR) codes and Golay codes. In this thesis I tackle two problems related to cyclic codes: finding low-weight codewords and decoding. Computing efficiently low-weight codewords of a cyclic code is often a key ingredient of correlation attacks to LFSR-based stream ciphers. The best general purpose algorithm is based on the generalized birthday problem. In the first part of the thesis I show an alternative approach based on discrete logarithms which has - in some cases relevant for applications - much lower memory complexity requirements and a comparable time complexity. The second part of the thesis is devoted to some results concerning efficient bounded-distance decoding for cyclic codes.

In this thesis we consider some problems concerning polynomials over finite fields. The first topic is the action of some groups on irreducible polynomials. We describe orbits and stabilizers. Next, we consider transformations of irreducible polynomials by quadratic and cubic maps and study the irreducibility of the polynomials obtained. Finally, starting from PN functions and monomials, we generalize this concept, introducing k-PN monomials and classifying them for small values of k and for fields of order p, p^2 and p^4.

In this thesis we provide known and new results which explain the relationship between the actual minimum distance of cyclic codes, bounds that use only information on the defining sets of cyclic codes to lower bound the distance (root bounds) and bounds that also need the knowledge of the defining sets of all cyclic subcodes (border bounds). We propose a new bound which is provably better of many known bounds and that can be computed in polynomial time with respect to the length of the code. We sketch how to use the generalized Newton identities to give alternative proofs of known bounds. Finally, we use Groebner bases to prove that the optimal root bound can be computed in finite time.

In this thesis we work on problems related to tensor decomposition from a geometrical perspective. In the first part of the thesis we focus on the identifiability problem, which amounts to understand in how many ways a tensor can be decomposed as a minimal sum of elementary tensors. In particular we completely classify the identifiability of any tensor up to rank 3. In the second part of the thesis we continue to work with specific elementsand we introduce the notion of r-thTerracini locus of a Segre variety. This is the locus containing all points for which the differential of the map between the r-th abstarct secant variety and the r-th secant variety of a Segre variety drops rank. We completely determine the r-th Terracini locus of any Segre variety in the case of r = 2, 3.

I consider the problem of classifying the semisimple orbits of a theta-group. For this purpose, once a preliminary presentation of the theoretical subjects where my problem arises from, I first give an algorithm to compute a Cartan subspace; subsequently I describe how to compute the little Weyl group.

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.