Parameterization of Boolean functions by vectorial functions and associated constructions

Too few general constructions of Boolean functions satisfying all cryptographic criteria are known. We investigate the construction in which the support of \begin{document}$ f $\end{document} equals the image set of an injective vectorial function \begin{document}$ F $\end{document} , that we call a parameterization of \begin{document}$ f $\end{document} . Every balanced Boolean function can be obtained this way. We study five illustrations of this general construction. The three first correspond to known classes (Maiorana-McFarland, majority functions and balanced functions in odd numbers of variables with optimal algebraic immunity). The two last correspond to new classes: - the sums of indicators of disjoint graphs of \begin{document}$ (k,n-k $\end{document} )-functions, - functions parameterized through \begin{document}$ (n-1,n) $\end{document} -functions due to Beelen and Leander. We study the cryptographic parameters of balanced Boolean functions, according to those of their parameterizations: the algebraic degree of \begin{document}$ f $\end{document} , that we relate to the algebraic degrees of \begin{document}$ F $\end{document} and of its graph indicator, the nonlinearity of \begin{document}$ f $\end{document} , that we relate by a bound to the nonlinearity of \begin{document}$ F $\end{document} , and the algebraic immunity (AI), whose optimality is related to a natural question in linear algebra, and which may be approached (in two ways) by using the graph indicator of \begin{document}$ F $\end{document} . We revisit each of the five classes for each criterion. The fourth class is very promising, thanks to a lower bound on the nonlinearity by means of the nonlinearity of the chosen \begin{document}$ (k,n-k $\end{document} )-functions. The sub-class of the sums of indicators of affine functions, for which we prove an upper bound and a lower bound on the nonlinearity, seems also interesting. The fifth class includes functions with an optimal algebraic degree, good nonlinearity and good AI.