Permutation polynomials and construction of bent functions

Ongan, Pınar
This thesis consists of two main parts: In the first part, a study of several classes ofpermutation and complete permutation polynomials is given, while in the second part,a method of construction of several new classes of bent functions is described.The first part consists of the study of several classes of binomials and trinomialsover finite fields. A complete list of permutation polynomials of the formf(x) =xqn−1q−1+1+bx∈Fqn[x]is obtained for the casen= 5, and a criterion on permutationpolynomials of the same type is derived for the general case. Furthermore, it is shownthat whenqis odd, trinomials of the formf(x) =x5h(xq−1)∈Fq2[x], whereh(x) =x5+x+ 1never permutesFq2.A method of constructing several new classes of bent functions via linear translatorsand permutation polynomials forms the second part of the thesis. First, a way to lifta permutation overF2tto a permutation overF2mis described, wheret|m. Then,via this method,3-tuples of particular permutations that lead to new classes of bentfunctions are obtained. As a last step, the fact that none of the bent functions obtainedhere will be contained in Maiorana-McFarland class is proved.
Citation Formats
P. Ongan, “Permutation polynomials and construction of bent functions,” Ph.D. - Doctoral Program, Middle East Technical University, 2021.