On the nonlinearity of monotone Boolean functions

Claude Carlet

Paper 2016/133

On the nonlinearity of monotone Boolean functions

Claude Carlet

Abstract

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.

Metadata

Available format(s): PDF
Category: Secret-key cryptography
Publication info: Preprint. MINOR revision.
Contact author(s): claude carlet @ gmail com
History: 2016-02-15: received
Short URL: https://ia.cr/2016/133
License: CC BY

BibTeX

@misc{cryptoeprint:2016/133,
      author = {Claude Carlet},
      title = {On the nonlinearity of monotone Boolean functions},
      howpublished = {Cryptology {ePrint} Archive, Paper 2016/133},
      year = {2016},
      url = {https://eprint.iacr.org/2016/133}
}