Title: Self-dual bent functions

Authors: Claude Carlet, Lars Eirik Danielsen, Matthew G. Parker, Patrick Sole

Addresses: LAGA, Universities of Paris 8 and Paris 13 and CNRS, Department of Mathematics, University of Paris 8, 2, rue de la liberte, Saint-Denis Cedex 93526, France. ' Department of Informatics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway. ' Department of Informatics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway. ' Telecom ParisTech, Department of Comelec, 46 rue Barrault, Paris 75013, France

Abstract: A bent function is called self-dual if it is equal to its dual. It is called anti-self-dual if it is equal to the complement of its dual. A spectral characterisation in terms of the Rayleigh quotient of the Sylvester Hadamard matrix is derived. Bounds on the Rayleigh quotient are given for Boolean functions in an odd number of variables. An efficient search algorithm based on the spectrum of the Sylvester matrix is derived. Primary and secondary constructions are given. All self-dual bent Boolean functions in ≤ 6 variables and all quadratic such functions in eight variables are given, up to a restricted form of affine equivalence.

Keywords: Boolean functions; bent functions; Walsh-Hadamard transform; self-dual codes.

DOI: 10.1504/IJICOT.2010.032864

International Journal of Information and Coding Theory, 2010 Vol.1 No.4, pp.384 - 399

Published online: 25 Apr 2010 *

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