A note on the properties of associated boolean functions of quadratic APN functions

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Abstract

Let F be a quadratic APN function in n variables. The associated Boolean function γF in 2n variables (γF (a, b) = 1 if a ≠ 0 and equation F(x)+F(x+a) = b has solutions) has the form γF (a, b) = ΦF (a) · b+φF (a)+1 for appropriate functions ΦF : Fn 2 → Fn 2 and φF : Fn 2 → F2. We summarize the known results and prove new ones regarding properties of ΦF and φF. For instance, we prove that degree of ΦF 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 ΦF is n -2. We show that this conjecture is based on two other conjectures of independent interest.

Translated title of the contributionО свойствах ассоциированных булевых функций квадратичных APN-функций
Original languageEnglish
Pages (from-to)16-21
Number of pages6
JournalПрикладная дискретная математика
Issue number47
DOIs
Publication statusPublished - 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

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