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On Symmetry and Differential Properties of Generalized Boolean Functions

  • Thor Martinsen
  • Wilfried Meidl
  • Alexander Pott
  • Pantelimon StănicăEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11321)

Abstract

In this paper we investigate various differential properties of generalized Boolean functions defined on \({\mathbb F}_2^n\) with values in \({\mathbb Z}_{2^k}\), \(k\ge 2\). We characterize linear structures for the generalized Boolean functions in terms of their binary expansion components, and find all symmetric generalized bent functions. Next, we show that there are no symmetric balanced functions defined on \({\mathbb F}_2^n\) with values in a group of order \(2^k, k\ge 2\), a contrast to the classical case for \(k=1\), commonly known as the bisection of binomial coefficients. Further, we characterize the avalanche features of a generalized Boolean function in terms of differentials. Lastly, we show that a partially gbent function is plateaued.

Keywords

Generalized Boolean functions Linear structures Generalized bent Semibent Partially bent Avalanche features Plateaued 

Notes

Acknowledgement

This paper was started while the second and fourth named authors visited the third named author at the Institute of Algebra and Geometry, of Otto von Guericke University Magdeburg. They thank the host and the institute for hospitality and excellent working conditions.

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Copyright information

© This is a U.S. government work and not under copyright protection in the United States; foreign copyright protection may apply 2018

Authors and Affiliations

  • Thor Martinsen
    • 1
  • Wilfried Meidl
    • 2
  • Alexander Pott
    • 3
  • Pantelimon Stănică
    • 1
    Email author
  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  3. 3.Institute of Algebra and Geometry, Faculty of MathematicsOtto von Guericke University MagdeburgMagdeburgGermany

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