Geodesic
On $1$-stable perfectly balanced Boolean functions
Diskretnaya Matematika, Tome 28 (2016) no. 2, pp. 117-126

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The paper is concerned with relations between the correlation-immunity (stability) and the perfectly balancedness of Boolean functions. It is shown that an arbitrary perfectly balanced Boolean function fails to satisfy a certain property that is weaker than the $1$-stability. This result refutes some assertions by Markus Dichtl. On the other hand, we present new results on barriers of perfectly balanced Boolean functions which show that any perfectly balanced function such that the sum of the lengths of barriers is smaller than the length of variables, is $1$-stable.
Keywords: perfectly balanced functions, barriers of Boolean functions, correlation-immunity, cryptography. 
@article{DM_2016_28_2_a10,
     author = {S. V. Smyshlyaev},
     title = {On $1$-stable perfectly balanced {Boolean} functions},
     journal = {Diskretnaya Matematika},
     pages = {117--126},
     publisher = {mathdoc},
     volume = {28},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2016_28_2_a10/}
}
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S. V. Smyshlyaev. On $1$-stable perfectly balanced Boolean functions. Diskretnaya Matematika, Tome 28 (2016) no. 2, pp. 117-126. http://geodesic.mathdoc.fr/item/DM_2016_28_2_a10/