In the first part of this thesis we report results about some “linear” trapdoors that can be embedded in a block cipher. In particular we are interested in any block cipher which has invertible S-boxes and that acts as a permutation on the message space, once the key is chosen. The message space is a vector space and we can endow it with alternative operations (hidden sums) for which the structure of vector space is preserved. Each of this operation is related to a different copy of the affine group. So, our block cipher could be affine with respect to one of these hidden sums. We show conditions on the S-box able to prevent a type of trapdoors based on hidden sums, in particular we introduce the notion of Anti-Crooked function. Moreover we shows some properties of the translation groups related to these hidden sums, characterizing those that are generated by affine permutations. In that case we prove that hidden sum trapdoors are practical and we can perform a global reconstruction attack. We also analyze the role of the mixing layer obtaining results suggesting the possibility to have undetectable hidden sum trapdoors using MDS mixing layers. In the second part we take into account the index coding with side information (ICSI) problem. Firstly we investigate the optimal length of a linear index code, that is equal to the min-rank of the hypergraph related to the instance of the ICSI problem. In particular we extend the the so-called Sandwich Property from graphs to hypergraphs and also we give an upper bound on the min-rank of an hypergraph taking advantage of incidence structures such as 2-designs and projective planes. Then we consider the more general case when the side information are coded, the index coding with coded side information (ICCSI) problem. We extend some results on the error correction index codes to the ICCSI problem case and a syndrome decoding algorithm is also given.

This thesis deals with the classical problem of prime numbers represented by polynomials. It consists of three parts. In the first part I collected many results about the problem. Some of them are quite recent and this part can be considered as a survey of the state of the art of the subject. In the second part I present two results due to P. Pleasants about the cubic polynomials with integer coefficients in several variables. The aim of this part is to simplify the works of Pleasants and modernize the notation employed. In such a way these important theorems are now in a more readable form. In the third part I present some original results related with some algebraic invariants which are the key-tools in the works of Pleasants. The hidden diophantine nature of these invariants makes them very difficult to study. Anyway some results are proved. These results make the results of Pleasants somewhat more effective.

In this thesis, I show that the representation of prime integers by reduced binary quadratic forms of given discriminant can be obtained in polynomial complexity using Schoof's algorithm for counting the number of points of elliptic curves over finite fields. It is a remarkable fact that, although an algorithm of Gauss' solved the representation problem long time ago, a solution in polynomial complexity is very recent and almost unnoticed in the literature. Further, I present a viable alternative to Gauss' algorithm, which constitutes the main original contribution of my thesis. This alternative way of computing in polynomial time an explicit solution of the representation problem is particularly suitable whenever the number of not equivalent reduced forms is small. Lastly, I report that, in the efforts of improving Schoof's algorithm, a marginal incompleteness in its original formulation was identified. This weakness was eliminated by a slight modification of the algorithm suggested by Schoof himself.

In this thesis, we study the degree of the canonical map of surfaces of general type. In particular, we give the first examples known in the literature of surfaces having degree d=10,11, 13, 14, 15, and 18 of the canonical map. They are presented in a self-contained and independent way from the rest of the thesis. We show also how we have discovered them. These surfaces are product-quotient surfaces. In this thesis, we study the theory of product-quotient surfaces giving also some new results and improvements. As a consequence of this, we have written and run a MAGMA script to produce a list of families of product-quotient surfaces having geometric genus three and a self-intersection of the canonical divisor large. After that, we study the canonical map of product-quotient surfaces and we apply the obtained results to the list of product-quotient surfaces just mentioned. In this way, we have discovered the examples of surfaces having degree d=10,11,14, and 18 of the canonical map. The remaining ones with degrees 13 and 15 do not satisfy the assumptions to compute the degree of the canonical map directly. Hence we have had to compute the canonical degree of these two families of product-quotient surfaces in a very explicit way through the equations of the pair of curves defining them. Another work of this thesis is the classification of all smooth surfaces of general type with geometric genus three which admits an action of a group G isomorphic to \mathbb Z_2^k and such that the quotient is a projective plane. This classification is attained through the theory of abelian covers. We obtained in total eleven families of surfaces. We compute the canonical map of all of them, finding in particular a family of surfaces with a canonical map of degree 16 not in the literature. We discuss the quotients by all subgroups of G finding several K3 surfaces with symplectic involutions. In particular, we show that six families are families of triple K3 burgers in the sense of Laterveer. Finally, in another work we study also the possible accumulation points for the slopes K^2/ \chi of unbounded sequences of minimal surfaces of general type having a degree d of the canonical map. As a new result, we construct unbounded families of minimal (product-quotient) surfaces of general type whose degree of the canonical map is 4 and such that the limits of the slopes K^2/ \chi assume countably many different values in the closed interval [6+2/3, 8].

In this thesis we study the relation between algebraic torus actions on complex projective varieties and the birational geometry of their geometric quotients. Given a C*-action on a normal projective variety X, there exist two unique connected components of the fixed point locus, called the sink Y− and the source Y+, containing the limit at ∞ and 0 of the general orbit. Let GX− (resp. GX+) be the variety parametrizing the orbits converging to the sink (resp. the source). Since there exists an open subset of points converging to Y±, we obtain a birational map ψ: GX->GX+. By choosing different linearizations of ample line bundles on X, we obtain a factorization of the birational map ψ among inner geometric quotient, parametrizing different open subsets of stable points. In this setting, we investigate the local analytic geometry of the birational map ψ. On one hand we link certain birational transformations, called rooftop flips, with varieties with two projective bundles structures. On the other we study when the birational map ψ can be locally described by a toric flip of Atiyah type. If on one side a C*-action naturally induces a birational map among geometric quotients, it is meaningful to study the opposite direction: more precisely, given a birational map φ: Z+->Z− among normal projective varieties, how can we construct a normal projective variety X, endowed with a C*-action, such that Z− is the sink, Z+ is the source, and the natural birational map ψ constructed above coincide with φ? Such an X is called a geometric realization of the birational map φ. We propose a construction of a geometric realization of φ, whose geometry reflects the factorization of the map as a composition of flips, blow-ups and blow-downs. We describe in particular the case in which φ is a small modification of dream type, namely a birational map which is an isomorphism in codimension 1 associated to a finitely generated multisection ring. Moreover, we show that the cone of divisors associated to such multisection rings admits a chamber decomposition where the models are the geometric quotients of the C*-action. If in addition Z± are assumed to be toric varieties, we construct a function in SageMath to compute the polytope of the associated toric geometric realization.

The influence of Chow groups on singular cohomology is motivated by classical results by Mumford and Roitman and has been investigated extensively. On the other hand, the converse influence is rather conjectural and it takes place in the framework of the ``philosophy of mixed motives'', which is mainly due to Grothendieck, Bloch and Beilinson. In the spirit of exploring this influence, Voisin formulated in 1996 a conjecture on 0--cycles on the self--product of surfaces of geometric genus one. There are few examples in which Voisin's conjecture has been verified, but it is still open for a general $K3$ surface. Our aim is to present a new example in which Voisin's conjecture is true, a family of Todorov surfaces. We give an explicit description of the family as quotient of complete intersection of four quadrics in $mathbb{P}^{6}$. We verify Voisin's conjecture for the family of Todorov surfaces of type $(2,12)$. Our main tool is Voisin's ``spreading of cycles'', we use it to establish a relation between 0--cycles on the Todorov surface and on the associated K3 surface. We give a motivic version of this result and some interesting motivic applications.

In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.

The work is situated within the theory of slice analysis, a generalization of complex analysis for hypercomplex numbers, considering function of both quaternionic and Clifford variables, in both one and several variables. %We first characterize some partial slice sets of The primary focus of the thesis is on the harmonic and polyharmonic properties of slice regular functions. We derive explicit formulas for the iteration of the Laplacian on slice regular functions, proving that their degree of harmonicity increases with the dimension of the algebra. Consequently, we present Almansi-type decompositions for slice functions in several variables. Additionally, using the harmonic properties of the partial spherical derivatives and their connection with the Dirac operator in Clifford analysis, we achieve a generalization of the Fueter and Fueter-Sce theorems in the several variables context. Finally, we establish that regular polynomials of sufficiently low degree are the unique slice regular functions in the kernel of the iteration of the Laplacian, whose power is less than Sce index.

In 2020 Parusiński and Rond proved that every algebraic set X ⊂ R^n is homeomorphic to an algebraic set X’ ⊂ R^n which is described globally (and also locally) by polynomial equations whose coefficients are real algebraic numbers. In general, the following problem was widely open: Open Problem. Is every real algebraic set homeomorphic to a real algebraic set defined by polynomial equations with rational coefficients? The aim of my PhD thesis is to provide classes of real algebraic sets that positively answer to above Open Problem. In Chapter 1 I introduce a new theory of real and complex algebraic geometry over subfields recently developed by Fernando and Ghiloni. In particular, the main notion to outline is the so called R|Q-regularity of points of a Q-algebraic set X ⊂ R^n. This definition suggests a natural notion of a Q-nonsingular Q-algebraic set X ⊂ R^n. The study of Q-nonsingular Q-algebraic sets is the main topic of Chapter 2. Then, in Chapter 3 I introduce Q-algebraic approximation techniques a là Akbulut-King developed in collaboration with Ghiloni and the main consequences we proved, that are, versions ‘over Q’ of the classical and the relative Nash-Tognoli theorems. Last results can be found in in Chapters 3 & 4, respectively. In particular, we obtained a positive answer to above Open Problem in the case of compact nonsingular algebraic sets. Then, after extending ‘over Q’ the Akbulut-King blowing down lemma, we are in position to give a complete positive answer to above Open Problem also in the case of compact algebraic sets with isolated singularities in Chapter 4. After algebraic Alexandroff compactification, we obtained a positive answer also in the case of non-compact algebraic sets with isolated singularities. Other related topics are investigated in Chapter 4 such as the existence of Q-nonsingular Q-algebraic models of Nash manifolds over every real closed field and an answer to the Q-algebrization problem for germs of an isolated algebraic singularity. Appendices A & B contain results on Nash approximation and an evenness criterion for the degree of global smoothings of subanalytic sets, respectively.

Very few standards exist for tting products to people. Footwear fit is a noteworthy example for consumer consideration when purchasing shoes. As a result, footwear manufacturing industry for achieving commercial success encountered the problem of developing right footwear which is fulfills consumer's requirement better than it's competeries. Mass customization starts with understanding individual customer's requirement and it finishes with fulllment process of satisfying the target customer with near mass production efficiency. Unlike any other consumer product, personalized footwear or the matching of footwear to feet is not easy if delivery of discomfort is predominantly caused by pressure induced by a shoe that has a design unsuitable for that particular shape of foot. Footwear fitter have been using manual measurement for a long time, but the combination of 3D scanning systems with mathematical technique makes possible the development of systems, which can help in the selection of good footwear for a given customer. This thesis, provides new approach for addressing the computerize footwear fit customization in industry problem. The design of new shoes starts with the design of the new shoe last. A shoe last is a wooden or metal model of human foot on which shoes are shaped. Despite the steady increase in accuracy, most available scanning techniques cause some deficiencies in the point cloud and a set of holes in the triangle meshes. Moreover, data resulting from 3D scanning are given in an arbitrary position and orientation in a 3D space. To apply sophisticated modeling operations on these data sets, substantial post-processing is usually required. We described a robust algorithm for filling holes in triangle mesh. First, the advance front mesh technique is used to generate a new triangular mesh to cover the hole. Next, the triangles in initial patch mesh is modified by estimating desirable normals instead of relocating them directly. Finally, the Poisson equation is applied to optimize the new mesh. After obtaining complete 3D model, the result data must be generated and aligned before taking this models for shape analysis such as measuring similarity between foot and shoe last data base for evaluating footwear it. Principle Component Analysis (PCA), aligns a model by considering its center of mass as the coordinate system origin, and its principle axes as the coordinate axes. The purpose of the PCA applied to a 3D model is to make the resulting shape independent to translation and rotation asmuch as possible. In analysis, we applied "weighted" PCA instead of applying the PCA in a classical way (sets of 3D point-clouds) for alignment of 3D models. This approach is based on establishing weights associated to center of gravity of triangles. When all of the models are aligned, an efficient algorithm to cut the model to several sections toward the heel and toe for extracting counters is used. Then the area of each contour is calculated and compared with equal sections in shoe last data base for finding best footwear fit within the shoe last data base.