Homogeneous Bent Functions, Invariants, and Designs

  • Authors:

  • Chris Charnes;Martin Rötteler;Thomas Beth

  • Affiliations:

  • Department of Computer Science and Software Engineering, University of Melbourne, Parkville, Vic, 3052, Australia charnes@cs.mu.oz.au;Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, D-76128 Karlsruhe, Germany roettele@ira.uka.de;Institut für Algorithmen und Kognitive Systeme, Universität Karlsruhe, Am Fasanengarten 5, D-76128 Karlsruhe, Germany eiss_office@ira.uka.de

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2002

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Abstract

We establish a new connection between invariant theory and the theory of bent functions. This enables us to construct Boolean functions with a prescribed symmetry group action. Besides the quadratic bent functions the only other previously known homogeneous bent functions are the six variable degree three functions constructed in [16]. We show that these bent functions arise as invariants under an action of the symmetric group on four letters and determine the stabilizer which turns out to be a matrix group of order 10752. We apply the machinery of invariant theory in order to construct homogeneous bent functions of degree three in 8, 10, and 12 variables. This approach gives a great computational advantage over the unstructured search problem and yields Boolean functions which have a concise description in terms of certain designs and graphs. We consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.