Geodesic
A note on the properties of associated Boolean functions of quadratic APN functions
Prikladnaâ diskretnaâ matematika, no. 1 (2020), pp. 16-21

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - A. A. Gorodilova
TI  - A note on the properties of associated Boolean functions of quadratic APN functions
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2020
SP  - 16
EP  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/
LA  - en
ID  - PDM_2020_1_a1
ER  - 
%0 Journal Article
%A A. A. Gorodilova
%T A note on the properties of associated Boolean functions of quadratic APN functions
%J Prikladnaâ diskretnaâ matematika
%D 2020
%P 16-21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2020_1_a1/
%G en
%F 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/