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Bent Functions in \(\mathcal C\) and \(\mathcal D\) Outside the Completed Maiorana-McFarland Class

  • F. Zhang
  • E. PasalicEmail author
  • N. Cepak
  • Y. Wei
Conference paper
  • 656 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10194)

Abstract

Two new classes of bent functions derived from the Maiorana-McFarland (\(\mathcal {M}\)) class, so-called \({\mathcal {C}}\) and \( {\mathcal {D}}\), were introduced by Carlet [2] two decades ago. However, apart from the subclass \({\mathcal {D}}_0\), some explicit construction methods for these functions were not provided in [2]. Assuming the possibility of specifying a bent function f that belongs to one of these two classes (apart from \({\mathcal {D}}_0\)), the most important issue is then to determine whether f is still contained in the known primary classes or lies outside their completed versions. In this article we partially solve this question by providing sufficient conditions on the permutation and related characteristic function (used to define f in these classes) so that f is provably outside the completed \(\mathcal {M}\) class. To give some existence results, we employ recent results in [12] where some instances of bent functions in \({\mathcal {C}}\) were identified by providing specific permutations and related characteristic functions. More precisely, using our sufficient conditions that apply to both \({\mathcal {C}}\) and \({\mathcal {D}}\), it is shown that these identified classes of \({\mathcal {C}}\) functions described in [12] do not belong to the completed \(\mathcal {M}\) class, whereas the question (which is more difficult) whether these functions are also outside the completed partial spread class remains open. We also propose some generic methods for specifying bent functions in \({\mathcal {D}}\) outside the completed Maiorana-McFarland class.

Keywords

Bent functions \({\mathcal {C}}\) and \({\mathcal {D}}\) class Maiorana-McFarland class 

Notes

Acknowledgements

Fengrong Zhang is supported in part by National Science Foundation of China (Grant No. 61303263), and in part by the Fundamental Research Funds for the Central Universities (Grant No. 2015XKMS086), and in part by the China Postdoctoral Science Foundation funded project (Grant No. 2015T80600). Enes Pasalic is partly supported by the Slovenian Research Agency (research program P3- 0384 and research project J1-6720). Yongzhuang Wei is supported in part by the Natural Science Foundation of China (61572148), in part by the Guangxi Natural Science Found (2015GXNSFGA139007), in part by the project of Outstanding Young Teachers Training in Higher Education Institutions of Guangxi. Nastja Cepak is supported in part by the Slovenian Research Agency (research 25 program P3-0384 and Young Researchers Grant).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.FAMNIT and IAMUniversity of PrimorskaKoperSlovenia
  3. 3.FAMNITUniversity of PrimorskaKoperSlovenia
  4. 4.Guilin University of Electronic TechnologyGuilinPeople’s Republic of China

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