International Association
for Cryptologic Research

We first prove the truthfulness of a conjecture on the nonlinearity of monotone Boolean functions in even dimension, proposed in the recent paper ``Cryptographic properties of monotone Boolean functions", by D. Joyner, P. Stanica, D. Tang and the author, to appear in the Journal of Mathematical Cryptology. We prove then an upper bound on such nonlinearity, which is asymptotically much stronger than the conjectured upper bound and than the upper bound proved for odd dimension in this same paper. This bound shows a deep weakness of monotone Boolean functions; they are too closely approximated by affine functions for being usable as nonlinear components in cryptographic applications. We deduce a necessary criterion to be satisfied by a Boolean (resp. vectorial) function for being nonlinear.