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Codes Cryptogr."],"published-print":{"date-parts":[[2023,7]]},"abstract":"Abstract<\/jats:title>During the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a \u201cnew\u201d family belongs to the completed Maiorana\u2013McFarland ($${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion in $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and eventually we obtain many infinite families of bent functions that are provably outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The fact that a bent function f<\/jats:italic> is in\/outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if and only if its dual is in\/outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is employed in the so-called 4-decomposition of a bent function on $${\\mathbb {F}}_2^n$$<\/jats:tex-math>\n \n F<\/mml:mi>\n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004\u20132019, 2003) in terms of the second-order derivatives and later reformulated in (Hod\u017ei\u0107 et al. in IEEE Trans Inf Theory 65(11):7554\u20137565, 2019) in terms of the duals of its restrictions to the cosets of an $$(n-2)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n n<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-dimensional subspace V<\/jats:italic>. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For instance, for the elementary case of defining a bent function $$h(\\textbf{x},y_1,y_2)=f(\\textbf{x}) \\oplus y_1y_2$$<\/jats:tex-math>\n \n h<\/mml:mi>\n \n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n \n y<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n y<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n f<\/mml:mi>\n \n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2295<\/mml:mo>\n \n y<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n y<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on $${\\mathbb {F}}_2^{n+2}$$<\/jats:tex-math>\n \n F<\/mml:mi>\n 2<\/mml:mn>\n \n n<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using a bent function f<\/jats:italic> on $${\\mathbb {F}}_2^n$$<\/jats:tex-math>\n \n F<\/mml:mi>\n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we show that h<\/jats:italic> is outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if and only if f<\/jats:italic> is outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $$f_1||f_1||f_2||(1\\oplus f_2)$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n |<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n f<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n |<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n f<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n |<\/mml:mo>\n |<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n \u2295<\/mml:mo>\n \n f<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> also gives bent functions outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if $$f_1$$<\/jats:tex-math>\n \n f<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or $$f_2$$<\/jats:tex-math>\n \n f<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outside $${{{\\mathcal {M}}}{{\\mathcal {M}}}}^\\#$$<\/jats:tex-math>\n \n \n M<\/mml:mi>\n M<\/mml:mi>\n <\/mml:mrow>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are provided.\n<\/jats:p>","DOI":"10.1007\/s10623-023-01204-w","type":"journal-article","created":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T14:06:14Z","timestamp":1679493974000},"page":"2365-2393","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Explicit infinite families of bent functions outside the completed Maiorana\u2013McFarland class"],"prefix":"10.1007","volume":"91","author":[{"given":"Enes","family":"Pasalic","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3568-4321","authenticated-orcid":false,"given":"Amar","family":"Bapi\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fengrong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongzhuang","family":"Wei","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,3,22]]},"reference":[{"key":"1204_CR1","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1016\/j.dam.2022.02.010","volume":"314","author":"A Bapi\u0107","year":"2022","unstructured":"Bapi\u0107 A., Pasalic E.: Constructions of (vectorial) bent functions outside the completed Maiorana\u2013McFarland class. 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