A general framework for secondary constructions of bent and plateaued functions
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A general framework for secondary constructions of bent and plateaued functions S. Hodži´c1 · E. Pasalic2 · Y. Wei3 Received: 13 February 2019 / Revised: 23 March 2020 / Accepted: 9 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work, we employ the concept of composite representation of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying new secondary constructions of bent/plateaued functions. This representation gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects. This framework essentially gives rise to an infinite number of possibilities to specify such secondary construction methods (with some induced sufficient conditions imposed on initial functions) and in particular we solve several open problems in this context. We provide several explicit methods for specifying new classes of bent/plateaued functions and demonstrate through examples that the imposed initial conditions can be easily satisfied. Our approach is especially efficient when defining new bent/plateaued functions on larger variable spaces than initial functions. For instance, it is shown that the indirect sum methods and Rothaus’ construction are just special cases of this general framework and some explicit extensions of these methods are given. In particular, similarly to the basic indirect sum method of Carlet, we show that it is possible to derive (many) secondary constructions of bent functions without any additional condition on initial functions apart from the requirement that these are bent functions. In another direction, a few construction methods that generalize the secondary constructions which do not extend the variable space of the employed initial functions are also proposed.
Communicated by A. Pott.
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Y. Wei [email protected] S. Hodži´c [email protected] E. Pasalic [email protected]
1
University of Primorska, FAMNIT, Koper, Slovenia
2
University of Primorska, FAMNIT & IAM, Koper, Slovenia
3
Guilin University of Electronic Technology, Guilin, People’s Republic of China
123
S. Hodži´c et al.
Keywords Secondary constructions · Indirect sums · Rothaus construction · Bent functions · Plateaued functions Mathematics Subject Classification 94A60 · 06E30
1 Introduction Introduced by O. S. Rothaus in 1976, bent (or maximally non-linear) functions became one of the most interesting and important combinatorial objects, due to their wide range of applications (for instance coding theory, difference set theory, cryptography). During the last four decades bent functions have been intensively studied, which resulted in several general (primary) classes: Maiorana–McFarland class (MM) [27], Partial spread class (PS ) of Dillon [19], and Dobbertin’s H class [21]. A somewhat related class of functions, characterized by the property that their Walsh spectrum
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