A Study of Permutation Polynomials over Finite Fields

Author

Neranga Fernando, University of South FloridaFollow

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_q^e. 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