On obtaining regular, weakly regular and non-weakly regular bent functions over finite fields and ring of integers modulo Pm

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2014

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Çelik, Dilek

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Bent functions over the finite fields of odd characteristics received a lot of attention of late years. Over the finite fields with characteristic 2, a method is given to construct bent functions using near bent functions. This method is then generalized to finite fields with p elements for an odd prime p by Cesmelioglu et al. The idea is constructing a bent function F by glueing the near-bent functions in such a way that Walsh spectrum of F do not include zero value. This can be achieved by combining the near-bent functions having no common element in supports of their Walsh transforms and the union of their support of Walsh transforms should be equal to domain of near-bent functions. In this thesis, we aim to construct regular, weakly regular and non-weakly regular bent functions. For this purpose, we first give an adaptation of the method given in, to the finite fields with p^m elements and ring of integers modulo p^m, where m is a positive integer greater than 1. Then, we generalize the method by using s plateaued functions, for an integer s> 1, instead of using near bent functions over ring of integers modulo p^m. It is notable to emphasize that, we obtain completely different results in every adaptation process.To apply the method of construction, we compute the Walsh spectrum of quadratic functions over finite fields with p^m elements and ring of integers modulo p^m. We evaulate the quadratic Gauss sum over Z_q to achieve the computation over the ring of integers. Also, we give a technique to classify the constructed bent functions as regular, weakly regular and non weakly regular.

Subject Keywords

Functions., Rings of integers., Fourier transformations.

URI

http://etd.lib.metu.edu.tr/upload/12617794/index.pdf
https://hdl.handle.net/11511/23796

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Graduate School of Applied Mathematics, Thesis

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Citation Formats

D. Çelik, “On obtaining regular, weakly regular and non-weakly regular bent functions over finite fields and ring of integers modulo Pm,” Ph.D. - Doctoral Program, Middle East Technical University, 2014.