On constructions and enumeration of bent and semi-bent functions

Koçak, Neşe
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.


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Sınak, Ahmet; Özbudak, Ferruh; Mesnager, Sihem; Department of Cryptography (2017)
Plateaued functions, used to construct nonlinear functions and linear codes, play a significant role in cryptography and coding theory. They can possess various desirable cryptographic properties such as high nonlinearity, low autocorrelation, resiliency, propagation criteria, balanced-ness and correlation immunity. In fact, they provide the best possible compromise between resiliency order and nonlinearity. Besides they resist against linear cryptanalysis and fast correlation attacks due to their low Walsh...
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Plateaued and bent functions play a significant role in cryptography, sequence theory, coding theory and combinatorics. In 1997, Coulter and Matthews redefined bent functions over any finite field F-q where q is a prime power, and established their properties. The objective of this work is to redefine the notion of plateaued functions over F-q, and to present several explicit characterizations of those functions. We first give, over F-q, the notion of q-ary plateaued functions, which relies on the concept o...
Analysis of boolean functions with respect to Walsh spectrum
Uyan, Erdener; Doğanaksoy, Ali; Department of Cryptography (2013)
Boolean functions appear in various scientific disciplines including coding theory, combinatorics, complexity theory, cryptography, graph theory, etc. In cryptography, the design and analysis of Boolean functions possessing a range of cryptographic characteristics has often been the focus of attention. A productive ground of research for most of these cryptographic characteristics is Walsh spectrum, one of the most common representations of a Boolean function. This thesis presents an analysis of Boolean fun...
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Partially bent and plateaued functions over finite fields have significant applications in cryptography, sequence theory, coding theory, design theory and combinatorics. They have been extensively studied due to their various desirable cryptographic properties. In this paper, we study on characterizations of partially bent and plateaued functions over finite fields, with the aim of clarifying their structure. We first redefine the notion of partially bent functions over any finite field Fq , with q a prim...
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Boolean functions are amongst the vital ingredients of any symmetric cryptosystem in order to implement principles of confusion and di usion. These are utilized as non-linear filtering functions or combiner functions in LFSR-based stream ciphers and as s-box component functions or non-linear encryption functions in Fiestel structure based block ciphers. Consequently, the cryptographic properties of Boolean functions are amongst the main contributors to the strength of these ciphers against cryptanalysis. Th...
Citation Formats
N. Koçak, “On constructions and enumeration of bent and semi-bent functions,” Ph.D. - Doctoral Program, Middle East Technical University, 2015.