## Abstract

Let F be a quadratic APN function in n variables. The associated Boolean function γ_{F} in 2n variables (γ_{F} (a, b) = 1 if a ≠ 0 and equation F(x)+F(x+a) = b has solutions) has the form γ_{F} (a, b) = Φ_{F} (a) · b+φ_{F} (a)+1 for appropriate functions Φ_{F} : F^{n} _{2} → F^{n} _{2} and φ_{F} : F^{n} _{2} → F_{2}. We summarize the known results and prove new ones regarding properties of Φ_{F} and φ_{F}. For instance, we prove that degree of Φ_{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 Φ_{F} is n -2. We show that this conjecture is based on two other conjectures of independent interest.

Translated title of the contribution | О свойствах ассоциированных булевых функций квадратичных APN-функций |
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Original language | English |

Pages (from-to) | 16-21 |

Number of pages | 6 |

Journal | Прикладная дискретная математика |

Issue number | 47 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

## Keywords

- A quadratic APN function
- Degree of a function
- The associated Boolean function
- degree of a function
- the associated Boolean function
- a quadratic APN function