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Дискретные функции
Properties of associated Boolean functions of quadratic APN functions
A. A. Gorodilovaab a Novosibirsk State University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
Аннотация:
For a function $F:\mathbb{F}_2^n\to \mathbb{F}_2^n$, it is defined the associated Boolean function $\gamma_F$ in $2n$ variables as follows: $\gamma_F(a,b)=1$ if $a\neq\mathbf{0}$ and equation $F(x)+F(x+a)=b$ has solutions. A vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$ is called almost perfect nonlinear (APN) if equation $F(x) + F(x + a)=b$ has at most $2$ solutions for all vectors $a,b\in\mathbb{F}_2^n$, where $a$ is nonzero. In case when $F$ is a quadratic APN function its associated function 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 study properties of functions $\Phi_F$ and $\varphi_F$, in particular their degrees.
Ключевые слова:
APN functions, associated Boolean functions, differential equivalence.
Образец цитирования:
A. A. Gorodilova, “Properties of associated Boolean functions of quadratic APN functions”, ПДМ. Приложение, 2019, no. 12, 77–79
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/pdma439 https://www.mathnet.ru/rus/pdma/y2019/i12/p77
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