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A general framework for secondary constructions of bent and plateaued functions

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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.

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Funding

Funding were provided by the Slovenian Research Agency (Grant Nos. research program P3-0384 and Young Researchers Grant, research program P3-0384 and research project J1-9108), the European Commission for funding the InnoRenew CoE project (Grant No. Grant Agreement no. 739574), the National Key R&D Program of China (Grant No. 2017YFB0802000), the Natural Science Foundation of China (Grant Nos. 61572148, 61872103) and Guangxi Science and Technology Foundation (Grant Nos. Guike AB18281019, 2019GXNSFGA245004).

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Authors and Affiliations

  1. University of Primorska, FAMNIT, Koper, Slovenia

    S. Hodžić

  2. University of Primorska, FAMNIT & IAM, Koper, Slovenia

    E. Pasalic

  3. Guilin University of Electronic Technology, Guilin, People’s Republic of China

    Y. Wei

Authors
  1. S. Hodžić
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  2. E. Pasalic
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  3. Y. Wei
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Correspondence to Y. Wei.

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Communicated by A. Pott.

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Hodžić, S., Pasalic, E. & Wei, Y. A general framework for secondary constructions of bent and plateaued functions. Des. Codes Cryptogr. 88, 2007–2035 (2020). https://doi.org/10.1007/s10623-020-00760-9

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