On the second-order nonlinearities of some bent functions

The rth-order nonlinearity of Boolean functions plays a central role against several known attacks on stream and block ciphers. It plays also an important role in coding theory, since its maximum equals the covering radius of the rth-order Reed–Muller code. But it is difficult to calculate and even to bound. In this paper, we show lower bounds on the second-order nonlinearity of two subclasses of well-known bent functions. We first improve a known lower bound on the second-order nonlinearity of the simplest partial spread bent functions, whose nonlinearity profile has been bounded by the second author. This improvement allows obtaining a better bound for the whole profile. We subsequently give a lower bound on the second-order nonlinearity of some infinite class of Maiorana–McFarland (M–M) bent functions, which generalizes a result by Gangopadhyay et al.