On bent functions lying outside the completed Maiorana-McFarland class and permutations via translators

One of the most well known classes of bent functions is the Maiorana-McFarland class M. Knowing in which ways it overlaps with other classes of bent functions and determining whether new constructions yield bent functions that lie outside or inside of it are important questions. We will take a look at a special version of the Rothaus'construction and see if it can generate bent functions lying outside the completed Maiorana-McFarland class M#. Sufficient conditions will be presented for functions within C (and D) class to lie outside the M# class and we will demonstrate examples of such functions in C and show two infinite classes of functions in D outside the M# class. Next, translators will be considered – first linear translators and then their generalisation, the Frobenius translator. Constructions of permutations and bent functions relying on translators will be demonstrated. Finally, constructions of vectorial plateaued functions, permutations and complete permutations using the Maiorana-McFarland class.