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Thanks to a new upper bound, we study more precisely the nonlinearities of Maiorana-McFarland's resilient functions. We characterize those functions with optimum nonlinearities and we give examples of functions with high nonlinearities. But these functions have a peculiarity which makes them potentially cryptographically weak. We study a natural super-class of Maiorana-McFarland's class whose elements do not have the same drawback and we give examples of such functions achieving high nonlinearities.

A Larger Class of Cryptographic Boolean Functions via a Study of the Maiorana-McFarland Construction