Introduction to finite fields and their applications: Preface

Chaotic dynamics is an important source for generating pseudorandom binary sequences (PRBS) Much efforts have been devoted to obtaining period distribution of the generalized discrete Arnold's Cat map in various domains using all kinds of theoretical methods, including Hensel's lifting approach Diagonalizing the transform matrix of the map, this paper gives the explicit formulation of any iteration of the generalized Cat map Then, its real graph (cycle) structure in any binary arithmetic domain is disclosed The subtle rules on how the cycles (itself and its distribution) change with the arithmetic precision e are elaborately investigated and proved The regular and beautiful patterns of Cat map demonstrated in a computer adopting fixed-point arithmetics are rigorously proved and experimentally verified The results can serve as a benchmark for studying the dynamics of the variants of the Cat map in any domain In addition, the used methodology can be used to evaluate randomness of PRBS generated by iterating any other maps

The Graph Structure of the Generalized Discrete Arnold's Cat Map

Equations Over Finite Fields An Elementary Approach

Chaotic dynamics is an important source for generating pseudorandom binary sequences (PRBS). Much efforts have been devoted to obtaining period distribution of the generalized discrete Arnold&#x0027;s Cat map in various domains using all kinds of theoretical methods, including Hensel&#x0027;s lifting approach. Diagonalizing the transform matrix of the map, this article gives the explicit formulation of any iteration of the generalized Cat map. Then, its real graph (cycle) structure in any binary arithmetic domain is disclosed. The subtle rules on how the cycles (itself and its distribution) change with the arithmetic precision <inline-formula><tex-math notation="LaTeX">$e$</tex-math><alternatives><mml:math><mml:mi>e</mml:mi></mml:math><inline-graphic xlink:href="li-ieq1-3051387.gif"/></alternatives></inline-formula> are elaborately investigated and proved. The regular and beautiful patterns of Cat map demonstrated in a computer adopting fixed-point arithmetics are rigorously proved and experimentally verified. The results can serve as a benchmark for studying the dynamics of the variants of the Cat map in any domain. In addition, the used methodology can be used to evaluate randomness of PRBS generated by iterating any other maps. 

/pdf/the-graph-structure-of-the-generalized-discrete-arnold-s-cat-k6pa965c.pdf

Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function

Elements of high order in Artin-Schreier extensions of finite fields F q

The celebrated Primitive Normal Basis Theorem states that for any $$n\ge 2$$
 and any finite field $${\mathbb {F}}_q$$
 , there exists an element $$\alpha \in {\mathbb {F}}_{q^n}$$
 that is simultaneously primitive and normal over $${\mathbb {F}}_q$$
 . In this paper, we prove some variations of this result, completing the proof of a conjecture proposed by Anderson and Mullen (2014).

Variations of the Primitive Normal Basis Theorem

Let $$\mathbb F_{q}$$
 be the finite field with q elements and $$n\ge 2$$
 be a positive integer. We study the functional graph associated to linear maps over finite fields. In particular, we describe the functional graph $$\mathcal {G}_f(\mathbb F_{q^n})$$
 associated to the map induced by $$L_f$$
 on $$\mathbb F_{q^n}$$
 , where f is any irreducible divisor of $$x^n-1$$
 over $$\mathbb F_q$$
 and $$L_f$$
 is the q-associate of f. This description derives interesting information on the graph $$\mathcal {G}_f(\mathbb F_{q^n})$$
 , such as the number of cycles and the average of the preperiod length. When $$\gcd (f, x^n-1)=1$$
 , $$L_f$$
 is a permutation on $$\mathbb F_{q^n}$$
 and the cycle decomposition of $$\mathcal {G}_f(\mathbb F_{q^n})$$
 is well known. In this case, we present some applications of this result, such as the construction of linear involutions over odd characteristic and permutations with few fixed points.

The functional graph of linear maps over finite fields and applications

Existence of primitive 1-normal elements in finite fields

An element α∈Fqn is normal over Fq if α and its conjugates α,αq,…,αqn−1 form a basis of Fqn over Fq. In 2013, Huczynska, Mullen, Panario and Thomson introduced the concept of k-normal elements, generalizing the normal elements. In the past few years, many questions concerning the existence and number of k-normal elements with specified properties have been proposed. In this paper, we discuss some of these questions and, in particular, we provide many general results on the existence of k-normal elements with additional properties like being primitive or having large multiplicative order. We also discuss the existence and construction of k-normal elements in finite fields, providing a connection between k-normal elements and the factorization of xn−1 over Fq

Lucas Reis

Papers

Elements of high order in Artin-Schreier extensions of finite fields F q

Variations of the Primitive Normal Basis Theorem

The functional graph of linear maps over finite fields and applications

Existence of primitive 1-normal elements in finite fields

Existence results on k-normal elements over finite fields