Geodesic
An affine property of Boolean functions on subspaces and their shifts
Prikladnaya Diskretnaya Matematika. Supplement, no. 6 (2013), pp. 15-16

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Let a Boolean function in $n$ variables be affine on an affine subspace of dimension $\lceil n/2 \rceil$ if and only if $f$ is affine on any its shift. It is proved that algebraic degree of $f$ can be more than $2$ only if there is no affine subspace of dimension $\lceil n/2 \rceil$ that $f$ is affine on it.
Keywords: Boolean functions, bent functions, quadratic functions. 
@article{PDMA_2013_6_a6,
     author = {N. A. Kolomeec},
     title = {An affine property of {Boolean} functions on subspaces and their shifts},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {15--16},
     publisher = {mathdoc},
     number = {6},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2013_6_a6/}
}
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N. A. Kolomeec. An affine property of Boolean functions on subspaces and their shifts. Prikladnaya Diskretnaya Matematika. Supplement, no. 6 (2013), pp. 15-16. http://geodesic.mathdoc.fr/item/PDMA_2013_6_a6/