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

05/21/2020

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by   Anastasiya Gorodilova, et al.

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Let F be a quadratic APN function of 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_2^n→𝔽_2^n and φ_F:𝔽_2^n→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.