On constructions and enumeration of bent and semi-bent functions

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2015

Öğrenci: NEŞE KOÇAK

Danışman: ALİ DOĞANAKSOY

Özet:

Bent and semi-bent functions play an important role in cryptography and coding theory. They are widely studied as parts of building blocks in symmetric key cryptosystems because they provide resistance to fast correlation attacks and linear cryptanalysis due to their high nonlinearity. Besides, they can possess other desirable cryptographic properties such as low autocorrelation, propagation criteria, resiliency and high algebraic degree. Therefore, parallel to the advances in cryptanalysis techniques, the need for finding and constructing such functions increases day by day. However, as the number of inputs gets higher, it becomes impossible to search exhaustively all bent/semibent functions on the entire space. This limitation prompts researchers to deduce new methods to obtain bent/semi-bent functions with reasonable amount of computation power. A lot of research has been devoted to construction, characterization or enumeration of bent/semi-bent functions. For these reasons, we aim to contribute to the knowledge of bent and semi-bent functions by presenting new results on constructions, characterization and enumeration of these functions. In this thesis, characterization of a class of quadratic Boolean functions for semi-bentness is given and it is proved that semi-bent functions exist only when the input number is a multiple of 6. Furthermore, a generic method for enumeration of semi-bent and bent functions in certain classes is presented. Using this method, exact number of characterized semi-bent functions is found. Moreover, with this method some previous partial and incomplete enumeration results for three other classes of semi-bent/bent functions in the literature are completed. Explicit constructions of bent and semi-bent functions of Maiorana-McFarland class via linear structures and linear translators are proposed. Also, by using these explicit constructions as well as other algebraic structures like derivatives, certain quadratic and cubic functions, new secondary constructions of bent and semi-bent functions are obtained.