Paper 2022/1126
Explicit infinite families of bent functions outside $\mathcal{MM}^\#$
Abstract
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 ``new'' family belongs to the completed MaioranaMcFarland ($\mathcal{MM}^\#$) 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 $\mathcal{MM}^\#$ and eventually we obtain many infinite families of bent functions that are provably outside $\mathcal{MM}^\#$. The fact that a bent function $f$ is in/outside $\mathcal{MM}^\#$ if and only if its dual is in/outside $\mathcal{MM}^\#$ is employed in the socalled 4decomposition of a bent function on $\mathbb{F}_2^n$, which was originally considered by Canteaut and Charpin \cite{Decom} in terms of the secondorder derivatives and later reformulated in \cite{HPZ2019} in terms of the duals of its restrictions to the cosets of an $(n2)$dimensional subspace $V$. For each of the three possible cases of this 4decomposition of a bent function (all four restrictions being bent, semibent, or 5valued spectra functions), we provide generic methods for designing bent functions provably outside $\mathcal{MM}^\#$. For instance, for the elementary case of defining a bent function $h(\mathbf{x},y_1,y_2)=f(\mathbf{x}) \oplus y_1y_2$ on $\mathbb{F}_2^{n+2}$ using a bent function $f$ on $\mathbb{F}_2^n$, we show that $h$ is outside $\mathcal{MM}^\#$ if and only if $f$ is outside $\mathcal{MM}^\#$. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $f_1f_1f_2(1\oplus f_2)$ also gives bent functions outside $\mathcal{MM}^\#$ if either $f_1$ or $f_2$ is outside $\mathcal{MM}^\#$. The cases when the four restrictions of a bent function are semibent or 5valued spectra functions are also considered and several design methods of designing infinite families of bent functions outside $\mathcal{MM}^\#$, using the spectral domain design are proposed.
Metadata
 Available format(s)
 Category
 Foundations
 Publication info
 Preprint.
 Keywords
 4decomposition Class inclusion 5valued functions Bent functions Dual functions Plateaued functions Walsh support
 Contact author(s)
 amar bapic @ famnit upr si
 History
 20220831: approved
 20220830: received
 See all versions
 Short URL
 https://ia.cr/2022/1126
 License

CC BY
BibTeX
@misc{cryptoeprint:2022/1126, author = {Enes Pasalic and Amar Bapić and Fengrong Zhang and Yongzhuang Wei}, title = {Explicit infinite families of bent functions outside $\mathcal{{MM}}^\#$}, howpublished = {Cryptology ePrint Archive, Paper 2022/1126}, year = {2022}, note = {\url{https://eprint.iacr.org/2022/1126}}, url = {https://eprint.iacr.org/2022/1126} }