On irreducible binary polynomials

In the article [1], Michon and Ravache define a group action of S3 on the set of irreducible polynomials of degree ≥ 2 over F2, and seeing that the orbits can have 1, 2, 3, or 6 elements, they give answers to the following two questions: Which polynomials have i ∈ {1, 2, 3, 6} elements in their orbits? Within the orbits of the irreducible polynomials of degree n ≥ 2, how many of them consist of i ∈ {1, 2, 3, 6 } elements? After their article, the next step seems to generalize their results to the Fq-case, however, their de nition of the group action is not so suitable for such an extension. Therefore it is defined in a slightly different approach in this master thesis so that it can be easily generalized to the Fq-case later. Furthermore, the results of the article [1] are reacquired using the new definition. Additionally, in the light of the articles [2] by Meyn and [3] by Michon and Ravache, the construction of irreducible polynomials of a higher degree which remain invariant under the group action of a given element forms a part of this thesis

Permütasyon polinomları ve bükülmüş fonksiyonlarının inşası

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.Bu tez iki ana bölümden olu ̧smaktadır: ̇Ilk kısımda, çe ̧sitli permütasyon ve tam per-mütasyon polinom sınıflarının incelenmesi verilirken; ikinci kısımda, birkaç yeni bü-külmü ̧s fonksiyon sınıfı in ̧sası için bir yöntem tarif edilmektedir. ̇Ilk kısım; sonlu cisimler üzerinde tanımlı çe ̧sitli iki terimli ve üç terimli polinom sınıf-larının incelenmesini içermektedir.Fqnüzerinde tanımlıf(x) =x(qn−1)/(q−1)+1+bxformundaki permütasyon polinomlarınınn= 5iken tamamı bir liste hâlinde sunu-lurken, herniçin geçerli bir kriter de elde edilmektedir. Dahası;qtek iken,h(x) =x5+x+ 1durumundaf(x) =x5h(xq−1)∈Fq2[x]formundaki üç terimlilerin aslaFq2cismini permüte etmeyece ̆gi gösterilmektedir.Do ̆grusal öteleyiciler ve permütasyon polinomları aracılı ̆gıyla birkaç yeni bükülmü ̧sfonksiyon sınıfı in ̧sa etme yöntemi ise tezin ikinci bölümünü olu ̧sturmaktadır. ̇Ilk ola-rak;m’yi bölentler için,F2t[x]’teki bir permütasyonuF2m[x]’teki bir permütasyonayükseltmenin bir yolu tarif ediliyor. Daha sonra, bu yöntemle yeni bükülmü ̧s fonksi-yon sınıflarının in ̧sasında kullanılacak olan çe ̧sitli permütasyon3’lüleri elde edilmek-tedir. Son olarak, burada elde edilen bükülmü ̧s fonksiyonların hiçbirinin Maiorana-McFarland sınıfında yer almadı ̆gı kanıtlanmaktadır.Ph.D. - Doctoral Progra

New bent functions from permutations and linear translators

Starting from the secondary construction originally introduced by Carlet ["On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities", Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 2006], that we shall call "Carlet` ssecondary construction", Mesnager has showed how one can construct several new primary constructions of bent functions. In particular, she has showed that three tuples of permutations over the finite field F2m such that the inverse of their sum equals the sum of their inverses give rise to a construction of a bent function given with its dual. It is not quite easy to find permutations satisfying such a strong condition (Am). Nevertheless, Mesnager has derived several candidates of such permutations in 2015, and showed in 2016 that in the case of involutions, the problem of construction of bent functions amounts to solve arithmetical and algebraic problems over finite fields. This paper is in the line of those previous works. We present new families of permutations satisfying (Am) as well as new infinite families of permutations constructed from permutations in both lower and higher dimensions. Our results involve linear translators and give rise to new primary constructions of bent functions given with their dual. And also, we show that our new families are not in the class of Maiorana-McFarland in general