A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel-Zassenhaus

v1 versio

Towards a deeper understanding of APN functions and related longstanding problems

This dissertation is dedicated to the properties, construction and analysis of APN and AB functions. Being cryptographically optimal, these functions lack any general structure or patterns, which makes their study very challenging. Despite intense work since at least the early 90's, many important questions and conjectures in the area remain open. We present several new results, many of which are directly related to important longstanding open problems; we resolve some of these problems, and make significant progress towards the resolution of others. More concretely, our research concerns the following open problems: i) the maximum algebraic degree of an APN function, and the Hamming distance between APN functions (open since 1998); ii) the classification of APN and AB functions up to CCZ-equivalence (an ongoing problem since the introduction of APN functions, and one of the main directions of research in the area); iii) the extension of the APN binomial $x^3 + \beta x^{36}$ over $F_{2^{10}}$ into an infinite family (open since 2006); iv) the Walsh spectrum of the Dobbertin function (open since 2001); v) the existence of monomial APN functions CCZ-inequivalent to ones from the known families (open since 2001); vi) the problem of efficiently and reliably testing EA- and CCZ-equivalence (ongoing, and open since the introduction of APN functions). In the course of investigating these problems, we obtain i.a. the following results: 1) a new infinite family of APN quadrinomials (which includes the binomial $x^3 + \beta x^{36}$ over $F_{2^{10}}$); 2) two new invariants, one under EA-equivalence, and one under CCZ-equivalence; 3) an efficient and easily parallelizable algorithm for computationally testing EA-equivalence; 4) an efficiently computable lower bound on the Hamming distance between a given APN function and any other APN function; 5) a classification of all quadratic APN polynomials with binary coefficients over $F_{2^n}$ for $n \le 9$; 6) a construction allowing the CCZ-equivalence class of one monomial APN function to be obtained from that of another; 7) a conjecture giving the exact form of the Walsh spectrum of the Dobbertin power functions; 8) a generalization of an infinite family of APN functions to a family of functions with a two-valued differential spectrum, and an example showing that this Gold-like behavior does not occur for infinite families of quadratic APN functions in general; 9) a new class of functions (the so-called partially APN functions) defined by relaxing the definition of the APN property, and several constructions and non-existence results related to them.Doktorgradsavhandlin

Order-Theoretic Methods for Space-Time Coding: Symmetric and Asymmetric Designs

Siirretty Doriast

C*-Algebras

[no abstract available

Exploring Neural Networks for computing the Hilbert Class Field of Quadratic Extensions of Q

El Hilbert Class Field también conocido como el cuerpo de clase absoluto de un cuerpo K es la extensión abeliana maximal no ramificada de un cuerpo de números. Métodos para calcularlo explicitamente existen solo para un número reducido de casos. En este trabajo se presenta la teoría algebraíca de números básica para abordar el caso en que K es cuadrático, el cual se encuentra resuelto solo cuando el cuerpo es imaginario. Finalmente se desarrolla marco teorico usando métodos de aprendizaje de máquinas y posiblemente geometría no conmutativa para investigar el caso en que K es cuadrático real.The Hilbert class field, also known as the absolute class field of a field K, is the maximal abelian unramified extension of a number field. Methods to compute it explicitly exist only in a limited number of cases. In this paper we present the basic algebraic number theory needed to study the case where K is quadratic, which is solved only for the imaginary case. Finally, we develop a theoretical framework using machine learning and possibly non-commutative geometry to tackle the case where K is real quadratic.Magíster en MatemáticasMaestrí