A note on the properties of associated Boolean functions of quadratic APN functions
Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 16-21
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Let $F$ be a quadratic APN function in $n$ variables. The associated Boolean function $\gamma_F$ in $2n$ variables ($\gamma_F(a,b)=1$ if $a\neq\mathbf{0}$ and equation $F(x)+F(x+a)=b$ has solutions) has the form $\gamma_F(a,b) = \Phi_F(a) \cdot b + \varphi_F(a) + 1$ for appropriate functions $\Phi_F:\mathbb{F}_2^n\to \mathbb{F}_2^n$ and $\varphi_F:\mathbb{F}_2^n\to \mathbb{F}_2$. We summarize the known results and prove new ones regarding properties of $\Phi_F$ and $\varphi_F$. For instance, we prove that degree of $\Phi_F$ is either $n$ or less or equal to $n-2$. Based on computation experiments, we formulate a conjecture that degree of any component function of $\Phi_F$ is $n-2$. We show that this conjecture is based on two other conjectures of independent interest.
Keywords:
a quadratic APN function, the associated Boolean function, degree of a function.
@article{PDM_2020_1_a1,
author = {A. A. Gorodilova},
title = {A note on the properties of associated {Boolean} functions of quadratic {APN} functions},
journal = {Prikladna\^a diskretna\^a matematika},
pages = {16--21},
publisher = {mathdoc},
number = {1},
year = {2020},
language = {en},
url = {http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/}
}
A. A. Gorodilova. A note on the properties of associated Boolean functions of quadratic APN functions. Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 16-21. http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/