Vectorial Boolean functions with the maximum number of bent components beyond the Nyberg’s bound

Abstract

Recently, several interesting constructions of vectorial Boolean functions with the maximum number of bent components (MNBC functions, for short) were proposed. However, many of them have component functions from the completed Maiorana-McFarland class M#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}^{\\#}$$\\end{document}. Moreover, no examples of MNBC functions containing component functions provably outside M#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}^{\\#}$$\\end{document} are known. In this paper, we classify all MNBC functions in six variables. Based on the analysis of the obtained equivalence classes, we propose several infinite families of MNBC functions with component functions outside the M#\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}^{\\#}$$\\end{document} class. In particular, two of our new constructions are solutions to the open problem [Bapić et al (eds) Proceedings of the twelfth international workshop on coding and cryptography, 2022, Item 1., p. 9].