Contributions on plateaued (vectorial) functions for symmetric cryptography and coding theory

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2017
Sınak, Ahmet
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-Hadamard transform values. Indeed, cryptographic algorithms are usually designed by appropriate composition of nonlinear functions, hence plateaued functions have a great effect on the security of these algorithms. Additionally, plateaued functions are closely related to linear codes, the most significant class of codes in coding theory, which have diverse applications in secret sharing schemes, authentication codes, communication, data storage devices and consumer electronics. The main objectives of this thesis are twofold: to study in detail the explicit characterizations for plateaued-ness of functions over finite fields from a cryptographic point of view, and to construct linear codes from weakly regular plateaued functions in coding theory. In this thesis, we first analyse characterizations of plateaued (vectorial) functions over a finite field F_p with p a prime number. More precisely, we obtain a large number of their characterizations in terms of their Walsh power moments, derivatives and autocorrelation functions, with the aim of both clarifying their structure and obtaining information about their construction. In particular, we observe the non-existence of a homogeneous cubic bent function (and in some cases a (homogeneous) cubic plateaued function) over F_p with p an odd prime. Moreover, we show the non-existence of a function whose absolute Walsh transform takes exactly three distinct values (one being zero), and introduce a new class of functions whose absolute Walsh transform takes exactly four distinct values (one being zero). Furthermore, we study partially bent and plateaued functions over a finite field F_q, with q a prime power, and obtain some of their characterizations in order to understand their behaviour over this field. In addition, we introduce the notion of (non)-weakly regular plateaued functions over F_p, with p an odd prime, and provide the secondary constructions of these functions. We then construct three-weight linear p-ary (resp. binary) codes from weakly regular p-ary plateaued (resp. Boolean plateaued) functions and determine their weight distributions. Finally, we show that the constructed linear codes can be used to construct secret sharing schemes with ``nice'' access structures. To the best of our knowledge, the construction of linear codes from plateaued functions over F_p, with p an odd prime, is studied in this thesis for the first time in the literature

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Citation Formats
A. Sınak, “Contributions on plateaued (vectorial) functions for symmetric cryptography and coding theory,” Ph.D. - Doctoral Program, Middle East Technical University, 2017.