2016

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

Dymtro Savchuk, Ph.D.

#### Keywords

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

#### Abstract

Let $p$ be a prime, $q$ a power of $p$ and $\Bbb F_q$ the finite field with $q$ elements. Any function $\phi:\f_q\rightarrow \f_q$ can be unqiuely represented by a polynomial, $f_\phi$ of degree $In the second chapter we are concerned the permutation behavior of the polynomial$g_{n,q}$, a$q$-ary version of the reversed Dickson polynomial, when the integer$n$is of the form$n=q^a-q^b-1$. This leads to the third chapter where we consider binomials and trinomials taking special forms. In this case we are able to give explicit conditions that guarantee the given binomial or trinomial is a permutation polynomial. In the fourth chapter we are concerned with permutation polynomials of$\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.

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