Direct Approaches for Generic Constructions of Plateaued Functions and Bent Functions Outside M#

Abstract

The problem of designing explicit bent and plateaued functions has been researched for several decades. However, finding new bent functions outside the well-known completed Maiorana-McFarland class $\mathcal {M}^{\#}$ is still a challenge. Plateaued functions have been characterized in many different ways, but there is no general and rigorous mathematical method to generate them directly, except for the ones in the spirit of the well-known Maiorana-McFarland constructions or those obtained through adaptations of the secondary constructions of bent functions. Jeong and Lee recently made significant advances regarding algorithms for constructing balanced plateaued functions with maximal algebraic degrees in [IEEE Trans. Inf. Theory, 70(2), 1408-1421, 2024]. Due to the gap between our significant interest in the notion of plateaued functions and the knowledge we have on it, our motivation is to bring further results on the constructions of plateaued functions that allow us to understand their structure better. This article creates a framework of new generic constructions of bent and plateaued functions by studying Boolean functions of the form $h(x)=f(x)+F(f_{1}(x),\ldots, f_{r}(x))$ , where $f_{i}(x)=f(x)+f(x+\mu _{i})$ for each $1\leq i\leq r$ . We firstly prove that h and f have the same extended Walsh-Hadamard spectrum if $D_{\mu _{i}}D_{\mu _{j}}f=0$ for any $1\leq i\lt j\leq r$ . This result extends a previous construction of bent functions to any Boolean functions. The strength of such a result is that it allows us to obtain several plateaued functions of high algebraic degrees from known ones with low algebraic degrees, which was a significant and challenging problem raised in the literature. Such a result is a real challenge and breaks a deadlock since no mathematical method allows the general constructions of plateaued functions. We next give an extended affine equivalent form of the function h, which provides us with another compelling perspective to design new bent functions (including those which are outside $\mathcal {M}^{\#}$ from certain known ones inside $\mathcal {M}^{\#}$ ) and plateaued functions. Finally, we present four generic constructions of bent functions outside $\mathcal {M}^{\#}$ from generalized Maiorana-McFarland functions.