Infinite classes of vectorial plateaued functions, permutations and complete permutations

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• 作者：E. Pasalic^a ; ^b ; ^{enes.pasalic6@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; N. Cepak^a ; ^{nastja.cepak@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Y. Wei^c ; ^d ; ^{walker_wei@msn.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Maiorana&ndash ; McFarland class ; Boolean functions ; Plateaued functions ; Permutations ; Complete mappings
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 December 2016
• 年：2016
• 卷：215
• 期：Complete
• 页码：177-184
• 全文大小：396 K

文摘

We use the well-known Maiorana–McFarland class to construct several important combinatorial structures. In the first place, we easily identify infinite classes of vectorial plateaued functions $\left\{F\right\}:{\mathbb{F}}_2^n\to {\mathbb{F}}_2^n$ such that all non-zero linear combinations of its component functions are also plateaued. More importantly, by setting certain restrictions on the component functions, the same approach also yields many infinite classes of permutations for any n≥6$n\ge 6$. Finally, we deduce some infinite classes of complete permutations, as a subclass of these permutations. Most notably, all these classes are of variable and controllable degree, the property being intrinsic to the construction method. The construction method is highly tweakable giving rise to many variations that again provide us with infinite classes of these structures.