The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications

We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials

Construction of irreducible polynomials through rational transformations

Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over $\mathbb F_q$ of high degree using rational transformations. In particular, given a divisor $D>2$ of $q+1$ and an irreducible polynomial $f\in \mathbb F_{q}[x]$ of degree $n$ such that $n$ is even or $D\not \equiv 2\pmod 4$, we show how to obtain from $f$ a sequence $\{f_i\}_{i\ge 0}$ of irreducible polynomials over $\mathbb F_q$ with $\mathrm{deg}(f_i)=n\cdot D^{i}$.Comment: 21 pages; comments are welcome

A Non-commutative Cryptosystem Based on Quaternion Algebras

We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen's method, the key generation and encryption process is approximately $16/7$ times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions.Comment: Submitted for possible publicatio

Evaluating the generalized Buchshtab function and revisiting the variance of the distribution of the smallest components of combinatorial objects

Let $n\geq 1$ and $X_{n}$ be the random variable representing the size of the smallest component of a combinatorial object generated uniformly and randomly over $n$ elements. A combinatorial object could be a permutation, a monic polynomial over a finite field, a surjective map, a graph, and so on. It is understood that a component of a permutation is a cycle, an irreducible factor for a monic polynomial, a connected component for a graph, etc. Combinatorial objects are categorized into parametric classes. In this article, we focus on the exp-log class with parameter $K=1$ (permutations, derangements, polynomials over finite field, etc.) and $K=1/2$ (surjective maps, $2$-regular graphs, etc.) The generalized Buchshtab function $\Omega_{K}$ plays an important role in evaluating probabilistic and statistical quantities. For $K=1$, Theorem $5$ from \cite{PanRic_2001_small_explog} stipulates that $\mathrm{Var}(X_{n})=C(n+O(n^{-\epsilon}))$ for some $\epsilon>0$ and sufficiently large $n$. We revisit the evaluation of $C=1.3070\ldots$ using different methods: analytic estimation using tools from complex analysis, numerical integration using Taylor expansions, and computation of the exact distributions for $n\leq 4000$ using the recursive nature of the counting problem. In general for any $K$, Theorem $1.1$ from \cite{BenMasPanRic_2003} connects the quantity $1/\Omega_{K}(x)$ for $x\geq 1$ with the asymptotic proportion of $n$-objects with large smallest components. We show how the coefficients of the Taylor expansion of $\Omega_{K}(x)$ for $\lfloor x\rfloor \leq x < \lfloor x\rfloor+1$ depends on those for $\lfloor x\rfloor-1 \leq x-1 < \lfloor x\rfloor$. We use this family of coefficients to evaluate $\Omega_{K}(x)$.Comment: 16 pages, 2 tables, 15 reference

A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics

A ``hybrid method'', dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions--this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.Comment: 31 page

Sistema de Clasificación de Paisajes para Uruguay. Herramienta para la planificación y conservación

253 páginasTesis doctoral (Lectura 29/01/2016). Director: Carlos Montes del Olmo. Tribunal: Estanislao de Luis Calabuig (presidente); Macarena Tejada Tejada (secretaria); Hermelindo Castro Nogueira (vocal). La presente propuesta tiene sus orígenes en un trabajo de relevamiento de suelos a escala 1:40.000 de todo Uruguay que realizáramos en la década de 1970 y su posterior digitalización en SIG. A los efectos de generar un sistema jerárquico con unidades anidadas (de cientos de miles de hectáreas) a partir de la unidad de mayor resolución la que se denominó sitio o complejo de sitios (siguiendo la usanza tradicional); las que se agrupan en Distritos, con el criterio geomorfológico de la relación pendienteprocesos; y finalmente en Eco-zonas siguiendo un criterio de unidades morfo-geoestructurales sobre la base de una revisión de la historia geológica del territorio y los procesos geomorfológicos que actuaron sobre ella desde el Cretácico al Holoceno. Las unidades mayores se clasifican según un criterio climático (Koëpen), aunque todo el Uruguay se encuentre incluido en una única Provincia. Luego a partir de la revisión de formas de vida y la presencia/ausencia de vertebrados y cruzar con los registros de distribución de especies, se generalizan las Eco-zonas en Eco-regiones. Fue dividido el territorio en cuadrículas, a los efectos de determinar la diversidad de especies atribuyendo cada especie a la Eco-zona cuyo hábitat coincidiera con los requerimientos de la especie para luego desarrollar una propuesta de corredores ecológicos. El presente trabajo implicó la clasificación por su similitud de respuesta ecosistémica de 33.608 polígonos que estaban agrupados en 188 unidades de suelo con diferentes resoluciones espacial, en un número menor de Sitios o Complejos de Sitios, los que se agruparon a su vez, en Distritos, Eco-zonas y Eco-regiones en el marco de una Clasificación de Paisaje jerárquica y anidada. El Sitio fue caracterizado por variables intrínsecamente asociadas a procesos genéticos vinculados a las vertientes, como son la edafización y la morfogénesis, tomando en consideración variables ambientales y además la pendiente en la cual se encuentra inserto