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Secondary constructions of bent functions and their enforcement

• Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus construction. This construction has a strict requirement for its initial functions. In this paper, we first concentrate on the design of the initial functions in the Rothaus construction. We show how to construct Maiorana-McFarland's (M-M) bent functions, which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We deduce that at least $(2^n!\times 2^{2^n})(2^{2^n}\times2^{2^{n-1}})^2$ bent functions in $2n+2$ variables can be constructed by using Rothaus' construction. In the second part of the note, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.
Mathematics Subject Classification: 06E30, 94A60.

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