#### Graduation Year

2016

#### Document Type

Thesis

#### Degree

Ph.D.

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Xiang-dong Hou, Ph.D.

#### Committee Member

Brian Curtin, Ph.D.

#### Committee Member

Mohamed Elhamdadi, Ph.D.

#### Committee Member

Dymtro Savchuk, Ph.D.

#### Keywords

Binomial, Finite Fields, Monomial Graph, Permutation Polynomial, Trinomial

#### Abstract

Let *p* be a prime, *p* a power of *p* and 𝔽_{q} the finite field with *q* elements. Any function φ: 𝔽_{q} → 𝔽_{q} can be unqiuely represented by a polynomial, 𝔽_{φ} of degree < *q*. If the map *x* ↦ *F _{φ}*(

*x*) induces a permutation on the underlying field we say

*F*is a

_{φ}*permutation polynomial*. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial

*f*, when does the map

*x*↦

*F*(

*x*) induce a permutation on 𝔽

_{q}.

In the second chapter we are concerned the permutation behavior of the polynomial *g _{n,q}*, a

*q*-ary version of the reversed Dickson polynomial, when the integer

*n*is of the form

*n*=

*q*-

^{a}*q*- 1. This leads to the third chapter where we consider binomials and trinomials taking special forms. In this case we are able to give explicit conditions that guarantee the given binomial or trinomial is a permutation polynomial.

^{b}In the fourth chapter we are concerned with permutation polynomials of 𝔽_{q}, where *q* is even, that can be represented as the sum of a power function and a linearized polynomial. These types of permutation polynomials have applications in cryptography. Lastly, chapter five is concerned with a conjecture on monomial graphs that can be formulated in terms of polynomials over finite fields.

#### Scholar Commons Citation

Lappano, Stephen, "Some Results Concerning Permutation Polynomials over Finite Fields" (2016). *Graduate Theses and Dissertations.*

https://scholarcommons.usf.edu/etd/6293