On the ranks of bent functions

On symplectic matrices of cubic Boolean forms and connections with second order nonlinearity

“…, a im ), and the fact c i jk = · · · = c k ij = c ijk for any ordering of i, j, k. From the above we see that rank(f ), as defined by means of C , is invariant under the action of GL(n, 2). Note that the notion of the rank of a cubic Boolean function is also defined in [7] and [15]; the notion of the rank introduced here is the same (as a result of Theorem 1) with the one provided in [7], although they are defined differently. Note that rank(f ) serves the same purpose with 2h of quadratic functions; however, it will be shown (in Section V) that it is not the only factor that controls their 2nd order nonlinearity.…”

“…, x n ) for the rest of the proof. From (17) and Remark 2, we yield that f a (z, w) = z(a 1 Q 1 + · · · + a s Q s )w T where Q i are r × (n − s − r) Hankel matrices as provided in (15). If we further apply Shannon expansion to f a with respect to the variables in z, we get f a = f a,0 · · · f a,2 r −1 , where f ab (w) = b (a 1 Q 1 + · · ·+ a s Q s )w T for all a ∈ F s 2 and b ∈ F r 2 .…”

The ranks of Maiorana-McFarland bent functions

“…The rank of a bent function is defined to be the 2-rank of the symmetric 2-design, which is deduced by the difference set f −1 (1). In [7], the authors proved that it is an invariant under the equivalence relation among bent functions. Furthermore, some upper and lower bounds of ranks of Maiorana-McFarland bent functions and Desarguesian partial spread bent functions are given.…”

“…Furthermore, some upper and lower bounds of ranks of Maiorana-McFarland bent functions and Desarguesian partial spread bent functions are given. As a consequence, the authors showed that almost every Desarguesian partial spread bent function is not equivalent to a Maiorana-McFarland one in [7].…”

“…The authors have proved that the lower bound of the ranks of Desarguesian partial spread bent functions is 2 t+1 − 2 in [7]. So this special kind of Maiorana-McFarland bent functions cannot be equivalent to any Desarguesian partial spread one when t 3.…”