| Let p be a prime and q = pk. The polynomial gn,q ∈ Fp [x] defined by the functional equation a∈Fq &parl0;x+a&parr0;n=g n,q&parl0;xq-x&parr0; gives rise to many permutation polynomials over finite fields. We are interested in triples (n, e; q) for which gn,q is a permutation polynomial of Fqe . In Chapters 2, 3, and 4 of this dissertation, we present many new families of permutation polynomials in the form of gn,q. The permutation behavior of gn,q is becoming increasingly more interesting and challenging. As we further explore the permutation behavior of gn,q, there is a clear indication that gn,q is a plenteous source of permutation polynomials.;We also describe a piecewise construction of permutation polynomials over a finite field Fq 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. |
