Abstract
The r-th order nonlinearity of a Boolean function is the minimum number of elements that have to be changed in its truth table to arrive at a Boolean function of degree at most r. It is shown that the (suitably normalised) r-th order nonlinearity of a random Boolean function converges strongly for all r ≥ 1. This extends results by Rodier for r = 1 and by Dib for r = 2. The methods in the present paper are mostly of elementary combinatorial nature and also lead to simpler proofs in the cases that r = 1 or 2.
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Acknowledgments
I thank Claude Carlet for some careful comments on a draft of this paper.
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Schmidt, KU. Nonlinearity measures of random Boolean functions. Cryptogr. Commun. 8, 637–645 (2016). https://doi.org/10.1007/s12095-015-0164-3
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DOI: https://doi.org/10.1007/s12095-015-0164-3