Parameterization of Boolean functions by vectorial functions and associated constructions

Abstract

<p style='text-indent:20px;'>Too few general constructions of Boolean functions satisfying all cryptographic criteria are known. We investigate the construction in which the support of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> equals the image set of an injective vectorial function <inline-formula><tex-math id="M2">\begin{document}$ F $\end{document}</tex-math></inline-formula>, that we call a parameterization of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Every balanced Boolean function can be obtained this way. We study five illustrations of this general construction. The three first correspond to known classes (Maiorana-McFarland, majority functions and balanced functions in odd numbers of variables with optimal algebraic immunity). The two last correspond to new classes:</p><p style='text-indent:20px;'>- the sums of indicators of disjoint graphs of <inline-formula><tex-math id="M4">\begin{document}$ (k,n-k $\end{document}</tex-math></inline-formula>)-functions,</p><p style='text-indent:20px;'>- functions parameterized through <inline-formula><tex-math id="M5">\begin{document}$ (n-1,n) $\end{document}</tex-math></inline-formula>-functions due to Beelen and Leander.</p><p style='text-indent:20px;'>We study the cryptographic parameters of balanced Boolean functions, according to those of their parameterizations: the algebraic degree of <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula>, that we relate to the algebraic degrees of <inline-formula><tex-math id="M7">\begin{document}$ F $\end{document}</tex-math></inline-formula> and of its graph indicator, the nonlinearity of <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula>, that we relate by a bound to the nonlinearity of <inline-formula><tex-math id="M9">\begin{document}$ F $\end{document}</tex-math></inline-formula>, and the algebraic immunity (AI), whose optimality is related to a natural question in linear algebra, and which may be approached (in two ways) by using the graph indicator of <inline-formula><tex-math id="M10">\begin{document}$ F $\end{document}</tex-math></inline-formula>. We revisit each of the five classes for each criterion. The fourth class is very promising, thanks to a lower bound on the nonlinearity by means of the nonlinearity of the chosen <inline-formula><tex-math id="M11">\begin{document}$ (k,n-k $\end{document}</tex-math></inline-formula>)-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.</p>