Related papers: Connected components of the graph generated by pow…

We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…

Interesting properties of the partitions of a finite field $\mathbb F_q$ induced by the combination of involutions and trace maps are studied. The special features of involutions of the form $\frac{u}{z}$, $u$ being a fixed element of…

Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are…

Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional graph of $f$ as a directed graph on $q$ nodes labelled by the elements of ${\mathbb F}_q$ where there is an edge from $u$ to $v$ if and only if…

In recent work, Pomerance and Shparlinski have obtained results on the number of cycles in the functional graph of the map $x \mapsto x^a$ in $\mathbb{F}_p^*$. We prove similar results for other families of finite groups. In particular, we…

Given a prime $p$, we consider the dynamical system generated by repeated exponentiations modulo $p$, that is, by the map $u \mapsto f_g(u)$, where $f_g(u) \equiv g^u \pmod p$ and $0 \le f_g(u) \le p-1$. This map is in particular used in a…

Let $\mathbb{F}_q$ be the finite field with $q=p^s$ elements, where $p$ is an odd prime and $s$ a positive integer. In this paper, we define the function $f(X)=(cX^q+aX)(X^{q}-X)^{n-1}$, for $a,c\in\mathbb{F}_q$ and $n\geq 1$. We study the…

There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called…

We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power…

Generating random and pseudorandom numbers with a deterministic system is a long-standing challenge in theoretical research and engineering applications. Several pseudorandom number generators based on the inversive congruential method have…

Let $F=\mathbb{F}_q(T)$ be the field of rational functions with $\mathbb{F}_q$-coefficients, and $A=\mathbb{F}_q[T]$ be the subring of polynomials. Let $D$ be a division quaternion algebra over $F$ which is split at $1/T$. Given an…

We derive several existence results concerning cycle types and, more generally, the "mapping behavior" of complete mappings. Our focus is on so-called first-order cyclotomic mappings, which are functions on a finite field $\mathbb{F}_q$…

For polynomials and rational maps of fixed degree over a finite field, we bound both the average number of connected components of their functional graphs as well as the average number of periodic points of their associated dynamical…

For $E \subset {\Bbb F}_q^d$, $d \ge 2$, where ${\Bbb F}_q$ is the finite field with $q$ elements, we consider the distance graph ${\mathcal G}^{dist}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$…

In this paper, we consider various graphs, namely: power graph, cyclic graph, enhanced power graph and commuting graph, on a finite semigroup $S$. For an arbitrary pair of these four graphs, we classify finite semigroups such that the…

In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences,…

We investigate additive cyclic codes over the alphabet $\mathbb{F}_{q}\mathbb{F}_{q^2}$, where $q$ is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of $\mathbb{F}_{q^2}$-additive…

For an integer $r$, a prime power $q$, and a polynomial $f$ over a finite field ${\mathbb F}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of $f$ which fall in a proper…

Let $\mathbb{F}_q$ be the finite field with $q$ elements and $char(\mathbb{F}_q)$ odd. In this article we will describe completely the dynamics of the map $f(X)=c(X^{q+1}+aX^2)$, for $a=\{\pm1\}$ and $c\in\mathbb{F}_q^*$, over the finite…

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power…