A construction of weakly and non-weakly regular bent functions over the ring of integers modulo \(p^m\)
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Bent functions over the finite fields of an odd characteristic received a lot of attention of late years. In Çeşmelioğlu et al. (J Comb Theory Ser A 119:420–429, 2012), Çeşmelioğlu and Meidl (Des Codes Cryptogr 66:231–242, 2013), an efficient method of construction of weakly regular and non-weakly regular bent functions defined over a finite field with odd characteristic is presented. In this paper, we give an adaptation of this method to the ring of integers modulo \(p^m\), where p is an odd prime and m is a positive integer. We emphasize that different results than the results of the finite field case are obtained in every application process. First, we give a method that constructs bent functions using plateaued functions by increasing the dimension. Then, in order to give concrete examples, we compute Walsh spectrum of some specific quadratic functions defined over the ring of integers modulo \(p^m\) and apply the construction method on these functions. There are notable differences between the cases when m is odd and even. Also, we explain how to determine weakly regular and non-weakly regular bent functions among the bent functions that are constructed by the method.
KeywordsBent Near-bent Plateaued functions Weakly regular Non-weakly regular Finite rings Walsh spectrum
We would like to express our gratitude to the anonymous reviewers for their valuable comments which improved the final version of the paper.
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