New lattice-based protocols for proving correctness of a shuffle

In an electronic voting procedure, mixing networks are used to ensure anonymity of the casted votes. Each node of the network re-encrypts the input and randomly permutes it in a process named shuffle, and must prove that the process was applied honestly. State-of-the-art classical proofs achieve logarithmic communication complexity on N (the number of votes to be shuffled) but they are based on assumptions which are weak against quantum computers. To maintain security in a post-quantum scenario, new proofs are based on different mathematical assumptions, such as lattice-based problems. Nonetheless, the best lattice-based protocols to ensure verifiable shuffling have linear communication complexity on N. In this thesis we propose the first sub-linear post-quantum proof for the correctness of a shuffe, for which we have mainly used two ideas: arithmetic circuit satisfiability and Benes networks to model a permutation of N elements

Direct sums decompositions: applications

In this thesis we study the behavior of direct sum decomposition\u27s in the category of modules. We present some of the most important classical results involving direct sum decomposition\u27s for modules (e.g. Krull-Schmidt theorem, decomposition theorems for finitely generated modules over PID etc.). In the last part of the thesis we obtain new results, namely isomorphic refinement theorems for direct sum decomposition\u27s of regular modules. We also obtain a link between regular modules and the exchange property

N=4\mathcal{N}=4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries

The structure of half-BPS representations of psu$(2,2|4)$ leads to the definition of a super-polynomial ring $\mathcal{R}(8|8)$ which admits a realisation of psu$(2,2|4)$ in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of $\mathcal{R}(8|8)$. The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu$(2,2|4)$ in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrate the construction using simple examples of representations of su(3) and su(4). These results lead to the definition of a notion of quantum mechanical emergence for oscillator realisations of symmetries, which is based on ideals in the ring of polynomials in the creation operators.Comment: 60 pages, no figure

Classical Algebraic Geometry

[no abstract available

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree $1$ for a complete intersection standard Artinian Gorenstein algebra of codimension $6$ presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them, we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree $1$ and the "lifting" of a weak Lefschetz property to an algebra from one of its quotients.Comment: 21 page

Partial generalized crossed products and a seven term exact sequence (expanded version)

Given a non-necessarily commutative unital ring $R$ and a unital partial representation $\Theta$ of a group $G$ into the Picard semigroup $\mathbf{PicS} (R)$ of the isomorphism classes of partially invertible $R$-bimodules, we construct an abelian group $\mathcal{C}(\Theta /R)$ formed by the isomorphism classes of partial generalized crossed products related to $\Theta$ and identify an appropriate second partial cohomology group of $G$ with a naturally defined subgroup $\mathcal{C}_0(\Theta /R)$ of $\mathcal{C}(\Theta /R).$ Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings $R\subseteq S$ with the same unity and a unital partial representation $G \to \mathcal{S}_R(S)$ of an arbitrary group $G$ into the monoid $\mathcal{S}_R(S)$ of the $R$-subbimodules of $S.$ This generalizes the works by Kanzaki and Miyashita

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients

Elliptic curves with complex multiplication and applications to class field theory

One of the aims of algebraic number theory is to describe the field of algebraic numbers and the extensions of number fields. This problem appears as the 12° of the 23 Hilbert's problems, and is essentially an extension of the Kronecker-Weber theorem, from the field of rational numbers to a generic number field. Although the problem is still open, the particular case of quadratic imaginary fields is completely understood, thanks to the theory of elliptic curves with complex multiplication. The purpose of this dissertation is to introduce some definitions and properties of elliptic curves (in Chapter 1), of the complex multiplication on them (in Chapter 2), of the class field theory (in Chapter 3) and then to give a characterization of the maximal abelian extension and then of any abelian extension of quadratic imaginary fields, with some other interesting properties about elliptic curves with complex multiplication (in Chapter 4)