A note on the algebraic immunity of the Maiorana–McFarland class of bent functions

Abstract

In Gupta et al. (2011) [5], the authors proved that the algebraic immunity of a subclass of Maiorana–McFarland functions is at most ⌈n4⌉+2 and claimed that this bound is tight. The main theorem of the upper bound is correct. However, their proof is incomplete and the bound is not tight. We will prove a more general theorem of a much larger subclass of Maiorana–McFarland functions and find that its algebraic immunity cannot achieve the optimum value. However, we find an 8-variable Maiorana–McFarland function which is not in that larger subclass of Maiorana–McFarland functions achieving the optimum algebraic immunity (this is the first time that a nontrivial Maiorana–McFarland function with the optimum algebraic immunity is given). Hence, this shows that there exist the Maiorana–McFarland functions achieving the optimum algebraic immunity.