Permütasyon polinomları ve bükülmüş fonksiyonların inşası

This thesis consists of two main parts: In the first part, a study of several classes of permutation 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 trinomials over finite fields. A complete list of permutation polynomials of the form f(x)=x^{(q^n-1)/(q-1)+ 1} + b x in F_{q^n}[x] is obtained for the case n=5, and a criterion on permutation polynomials of the same type is derived for the general case. Furthermore, it is shown that when q is odd, trinomials of the form f(x)= x^5 h(x^{q-1}) in F_{q^2}[x], where h(x)=x^5+x+1 never permutes F_{q^2}. A method of constructing several new classes of bent functions via linear translators and permutation polynomials forms the second part of the thesis. First, a way to lift a permutation over F_{2^t} to a permutation over F_{2^m} is described, where t | m. Then, via this method, 3-tuples of particular permutations that lead to new classes of bent functions are obtained. As a last step, the fact that none of the bent functions obtained here will be contained in Maiorana-McFarland class is proved.