Secondary constructions of bent functions and their enforcement

Abstract

Thirty years ago, Rothaus introduced the notion of bent function andpresented a secondary construction (building new bent functions fromalready defined ones), which is now called the Rothausconstruction. This construction has a strict requirement for itsinitial functions. In this paper, we first concentrate on the designof the initial functions in the Rothaus construction. We show how toconstruct 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 usingRothaus' construction. In the second part of the note, we present anew secondary construction of bent functions which generalizes theRothaus construction. This construction requires initial functionswith stronger conditions; we give examples of functions satisfyingthem. Further, we generalize the new secondary construction of bentfunctions and illustrate it with examples.