Geodesic
An estimation of the nonlinearity of balanced Boolean functions generated by generalized Dobbertin's construction
Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 33-35

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A generalization of the Dobbertin’s construction for highly nonlinear balanced Boolean functions is proposed. The Walsh — Hadamard spectrum is studied and estimates of the spectral radius of the proposed functions are obtained. An exact upper bound for the spectral radius (lower bound for nonlinearity) is proved, and a method for constructing a balanced function $\Theta$ in $2n$ variables using a balanced $\theta$ in $n-k$ variables with spectral radius $R_\Theta = 2^n + 2^{k} R_\theta $ is proposed. Here, $ R_\Theta $ and $ R_\theta $ are the spectral radii of $ \Theta $ and $ \theta $ respectively.
Keywords: boolean functions, bent functions, balancedness, nonlinearity, spectral radius. 
@article{PDMA_2020_13_a8,
     author = {I. A. Sutormin},
     title = {An estimation of the nonlinearity of balanced {Boolean} functions generated by generalized {Dobbertin's} construction},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {33--35},
     publisher = {mathdoc},
     number = {13},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2020_13_a8/}
}
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I. A. Sutormin. An estimation of the nonlinearity of balanced Boolean functions generated by generalized Dobbertin's construction. Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 33-35. http://geodesic.mathdoc.fr/item/PDMA_2020_13_a8/