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Finding nonnormal bent functions. (English) Zbl 1091.94021
Summary: The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in \(n\) variables is called normal if there exists an affine subspace of dimension \(n\)/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in J.F. Dillon and H. Dobbertin [Finite Fields Appl. 10, No. 3, 342–389 (2004; Zbl 1043.05024)], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions.

MSC:
94A60 Cryptography
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
03B70 Logic in computer science
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