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Homogeneous bent functions, invariants, and designs. (English) Zbl 1026.06015
Bent functions are special polynomial functions over the 2-element Boolean ring. They have been studied extensively for the last 30 years and play an important role in coding theory and cryptography. In this paper, the authors present new methods of constructing homogeneous bent functions. These methods are based on invariant theory and combinatorics (Nagy graphs and their cliques) and provide a great computational advantage over an unstructured search.
In particular, the authors show how the bent functions of degree three in six variables, presented by C. Qu, J. Seberry and J. Pieprzyk [Lect. Notes Comput. Sci. 1587, 26-35 (1999; Zbl 0919.94019)] (which – besides the quadratic bent functions – are the only previously known homogeneous bent functions) fit into the new framework, and they apply their machinery in order to construct new homogeneous bent functions of degree three in eight, ten, and twelve variables. Furthermore, they consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.

MSC:
06E30 Boolean functions
94A60 Cryptography
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
Software:
Magma
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