#### Graduation Year

2013

#### Document Type

Dissertation

#### Degree

Ph.D.

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

Xiang-dong Hou

#### Keywords

Dickson polynomial, Finite field, Normal basis, Permutation polynomial, Reversed Dickson polynomial

#### Abstract

Let p be a prime and q = p^{k}. The polynomial g_{n,q} isin F_{p}[x] defined by the functional equation Sigma_{a isin Fq} (x+a)^{n} = g_{n,q}(x^{q}- x) gives rise to many permutation polynomials over finite fields. We are interested in triples (n,e;q) for which g_{n,q} is a permutation polynomial of F_{qe}. In Chapters 2, 3, and 4 of this dissertation, we present many new families of permutation polynomials in the form of g_{n,q}. The permutation behavior of g_{n,q} is becoming increasingly more interesting and challenging. As we further explore the permutation behavior of g_{n,q}, there is a clear indication that g_{n,q} is a plenteous source of permutation polynomials.

We also describe a piecewise construction of permutation polynomials over a finite field F_{q} which uses a subgroup of F_{q}^{*}, a “selection” function, and several “case” functions. Chapter 5 of this dissertation is devoted to this piecewise construction which generalizes several recently discovered families of permutation polynomials.

#### Scholar Commons Citation

Fernando, Neranga, "A Study of Permutation Polynomials over Finite Fields" (2013). *USF Tampa Graduate Theses and Dissertations.*

https://digitalcommons.usf.edu/etd/4484