A lower bound on the second-order nonlinearity of the class of Maiorana-McFarland bent functions

Boolean functions used in stream ciphers and block ciphers should have high second-order nonlinearity to resist several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also play an important role in coding theory, since its maximal value equals the covering radius of the second-order Reed-Muller code. But it is a extremely hard task to calculate and even to bound the second-order nonlinearity of Boolean functions. In this paper, we present a lower bound on the second-order nonlinearity of the class of Maiorana-McFarland (M-M) bent functions. As applications of our bound, we provide simpler and direct proofs for two known lower bounds on the second-order nonlinearity of functions in the class of M-M bent functions. We also derive a lower bound on the second-order nonlinearity of the functions which were conjectured bent by Canteaut and whose bentness was proved by Leander, by further employing our bound.