Further study on the maximum number of bent components of vectorial functions

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In 2018, Pott, at al. have studied in [IEEE Transactions on Information Theory. Volume: 64, Issue: 1, 2018] the maximum number of bent components of vectorial function. They have presented serval nice results and suggested several open problems in this context. This paper is in the continuation of their study in which we solve two open problems raised by Pott et al. and partially solve an open problem raised by the same authors. Firstly, we prove that for a vectorial function, the property of having the maximum number of bent components is invariant under the so-called CCZ equivalence. Secondly, we prove the non-existence of APN plateaued having the maximum number of bent components. In particular, quadratic APN functions cannot have the maximum number of bent components. Finally, we present some sufficient conditions that the vectorial function defined from F₂²ᵏ to F₂²ᵏ by its univariate representation: αx²ⁱ(x+x²ᵏ+∑ⱼ₌₁ργ⁽ʲ⁾x²ᵗʲ +∑ⱼ₌₁ργ⁽ʲ⁾x²ᵗʲ⁺ᵏ) has the maximum number of components bent functions, where ρ≤ k. Further, we show that the differential spectrum of the function x²ⁱ(x+x²ᵏ+x²ᵗ¹+x²ᵗ¹⁺ᵏ+x²ᵗ²+x²ᵗ²⁺ᵏ) (where i,t₁,t₂ satisfy some conditions) is different from the binomial function Fⁱ(x)= x²ⁱ(x+x²ᵏ) presented in the article of Pott et al. Finally, we provide sufficient and necessary conditions so that the functions Tr₁²ᵏ(αx²ⁱ(Tr²ᵏₑ(x)+∑ⱼ₌₁ργ⁽ʲ⁾(Tr²ᵏₑ(x))²ʲ)) are bent.