A note on the properties of associated Boolean functions of quadratic APN functions A. A. Gorodilova
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ArticleSubject(s): APN-функции | булевы функцииGenre/Form: статьи в журналах Online resources: Click here to access online
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Прикладная дискретная математика № 47. С. 16-21Abstract: Let F be a quadratic APN function in n variables. The associated Boolean function yf in 2n variables (yF(a, b) = 1 if a = 0 and equation F(x) + F(x + a) = b has solutions) has the form yF(a, b) = Ф,р(a) • b + ^F(a) +1 for appropriate functions Ф,р : Fn Fn and ^f : Fn F2. We summarize the known results and prove new ones regarding properties of Ф,р and ^F. For instance, we prove that degree of Ф,р 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 Ф,р is n — 2. We show that this conjecture is based on two other conjectures of independent interest.
Библиогр.: 10 назв.
Let F be a quadratic APN function in n variables. The associated Boolean function yf in 2n variables (yF(a, b) = 1 if a = 0 and equation F(x) + F(x + a) = b has solutions) has the form yF(a, b) = Ф,р(a) • b + ^F(a) +1 for appropriate functions Ф,р : Fn Fn and ^f : Fn F2. We summarize the known results and prove new ones regarding properties of Ф,р and ^F. For instance, we prove that degree of Ф,р 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 Ф,р is n — 2. We show that this conjecture is based on two other conjectures of independent interest.
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