Properties of subfunctions of self-dual bent functions

Boolean functions in an even number of variables with flat Walsh — Hadamard spectrum are called bent functions. For every bent function, say $f$, its dual bent function, denoted by $\widetilde{f}$, is uniquely defined. If ${\widetilde{f}=f}$, then $f$ is called self-dual bent, and in the case ${\widetilde{f}=f\oplus 1}$ it is called an anti-self-dual bent. In this paper, we study subfunctions of self-dual bent functions obtained by a fixation of the first and the first two coordinates. We characterize subfunctions in $n-1$ variables considering their Rayleigh quotients. A sufficient condition for all subfunctions in $n-2$ variables to be bent is obtained. We propose new iterative constructions of self-dual bent functions in $n$ variables comprising the usage of bent functions in ${n-4}$ variables. Based on them, a new iterative lower bound on the cardinality of the set of self-dual bent functions is obtained.